supply chain planning ||

Supply Chain Planning

EditorsHans-Otto Günther Herbert Meyr

Supply Chain Planning

•

and Advanced Planning SolutionsQuantitative Decision Support

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© 2009 Springer-Verlag Berlin Heidelberg

Prof. Dr. Hans-Otto GüntherTU Berlin H 95Department of Production ManagementStraße des 17. Juni 13510623 [email protected]

Prof. Dr. Herbert MeyrTU Darmstadt

Hochschulstr. 164289 [email protected]

ISBN 978-3-540-93774-6 e-ISBN 978-3-540-93775-3

DOI 10.1007/978-3-540-93775-3

Library of Congress Control Number: 2008943995

and Supply Chain ManagementDepartment of Production

Preface

In recent years, supply chain planning has emerged as one of the most challengingproblems in the industry. As a consequence, the planning focus is shifting from themanagement of plant-specific operations to a holistic view of the various logisticsand production stages, that is an approach in which suppliers, production plants andcustomers are considered as constituents of an integrated network. A major driv-ing force behind this development lies in the globalization of the world economy,which has facilitated the co-operation between different partners working togetherin world-wide logistics networks. Hence, considerable cost savings can be gainedfrom optimizing the structure and the operations of complex supply networks link-ing plants, suppliers, distribution centres and customers. Consequently, to improvethe performance of the entire logistic chain, more sophisticated planning systemsand more effective decision support are needed.

Clearly, successful applications of supply chain management have driven thedevelopment of advanced planning systems (APS), which are concerned with sup-porting decision-making activities at the strategic, tactical and operational decisionlevel. These software packages basically rely on the application of quantitativemethods, which are used to model the underlying complex decision problems con-sidering the limited availability of resources and the need to react on time tocustomer orders. The core module at the mid-term level of APS comprises oper-ational supply chain planning. In many industries, production stages are assigned todifferent plants and distribution centres have been established at geographically dis-persed locations. Supply chain planning aims at coordinating production activitieswithin such multi-site logistics networks to avoid excessive inventories, inefficientcapacity utilization and poor customer service. In APS, advanced optimization tech-niques are provided to determine the quantities to be produced, stored, transportedand procured in the supply network.

This book is divided into five parts. The first one is concerned with decision-making in supply chains. The paper by Hartmut Stadtler addresses issues ofcollaboration in inter-organizational supply chains. It analyzes and clusters the ap-proaches that have been developed since the advent of supply chain managementabout two decades ago and gives an overview on the state-of-the-art. Moreover,the paper provides a framework for collaborative planning in supply chains, with

v

vi Preface

special emphasis on model-based decision support at the operational level of inter-organizational supply chains.

In their paper, Christian Almeder, Margaretha Preusser and Richard F. Hartlpresent a general framework for combining an optimization model and discreteevent simulation to support supply chain planning. Both models are applied in aniterative fashion until convergence is reached. It is shown that this approach deliv-ers competitive results much faster compared to mixed-integer linear programmingmodels in a stochastic environment.

Part 2 comprises five papers on demand management and order fulfillment insupply chains. Revenue management is a topic that has received great attentionin service and recently also in manufacturing industries. In their paper, RainerQuante, Herbert Meyr and Moritz Fleischmann analyze the underlying supply chainprocesses of revenue management and demand fulfilment in different business envi-ronments. In particular, they discuss and clarify the relationships between availablesoftware solutions and applications as well as scientific models in this field andhighlight directions of future research.

A systematic mathematical programming approach for active demand manage-ment in process industries is presented by Aaron A. Levis and Lazaros Papageorgiou.They develop an iterative algorithm for supporting decision-making on pricingstrategies as well as on output levels for substitute products. Several case studiesare used to obtain game-theoretical insights for a duopolistic market situation.

The paper by Herbert Meyr focuses on the segmentation of customers and theallocation of available quantities. It is shown that the current practice of rule-basedorder promising can be improved by exploiting information on customer hetero-geneity and customer demand. As a practical tool, deterministic linear programmingmodels to support available-to-promise decisions in a make-to-stock environmentare presented.

In their paper, Thomas R. Ervolina, Markus Ettl, Young M. Lee and Daniel J.Peters propose a novel management process for determining marketable productalternatives in a supply chain. Their approach aims at better integrating the supplychain horizontally by connecting the activities of customers, business partners andsales teams to procurement and manufacturing capabilities of a firm. The proposedmathematical optimization based approach has contributed to substantial businessimprovements at IBM.

The paper by Richard Pibernik and Prashant Yadav focuses on real-time orderpromising in a make-to-stock environment. An integrated approach is developedthat exploits the structure of order arrivals and material receipts to determine inven-tory reservations for high-priority orders in real time. In a comprehensive numericalstudy, the impact of inventory reservation and order promising is investigated undervarying system parameters.

The three papers included in Part 3 of this book focus on inventory management.In supply chains, safety stocks are needed to be more responsive to customer ordersand to meet the target service levels. Youssef Boulaksil, Jan C. Fransoo and ErnicoN. G. van Halm propose a combination of an optimization and a simulation model.They apply this hybrid modeling approach to determine safety stock levels in a

Preface vii

multi-item multi-stage inventory system. As a case study, the supply chain of abiopharmaceutical company is considered.

The paper by Pieter L.M. van Nyen, J. Will M. Bertrand, Henny P.G. van Ooijenand Nico J. Vandaele investigates the impact of different variants of supplier-managed inventory on costs in a supply chain consisting of a parts supplier and anoriginal equipment manufacturer’s assembly plant. Numerical experiments revealedthat substantial system-wide cost savings could be gained compared to a situationin which the assembly plant manages the inventories.

In the subsequent paper on vendor-managed inventory, Bogdan C. Bichescu andMichael J. Fry analyze decentralized supply chains under different degrees of chan-nel power, that is the agents’ ability to control the decision-making process in asupply chain. Game-theoretical models are used to compare the effectiveness ofvendor-managed inventory and to analyze different channel power relationships un-der a variety of environmental conditions.

Applications of supply chain planning in the chemical industry are presentedin Part 4. The concept of value chain management can be seen as another chal-lenging extension of classical supply chain management. Considering a case studyfrom the chemical industry, Matthias Kannegiesser, Hans-Otto Gunther, Paul vanBeek, Martin Grunow and Christoph Habla present an operative planning modelfor coordinating sales, distribution and production activities throughout a globalvalue chain. Specifically, the optimization model addresses spot demand for chem-ical commodities with volatile and uncertain sales prices and evaluates the impactof the respective price-quantity elasticity.

An optimization model based on mixed-integer linear programming to schedulecampaigns in a specialty chemicals plant is presented by Marcus Brandenburg andFranz-Josef Tolle. They focus on a real-world problem, which is characterized bya variety of chemical processes with sequence-dependent setup conditions, com-plex material flows, flexible use of resources and facility-dependent batch sizes. Tosolve this complex scheduling problem, a two-stage solution procedure is applied,which determines near-optimal schedules even for large-sized real-world probleminstances within reasonable CPU time.

Finally, two papers with applications in the automotive industry are presentedin Part 5. The paper by Herbert Meyr, originally published in 2004, first gives anoverview of short- and mid-term approaches for supply network planning in theautomotive industry and specifically discusses the application of OR methodologyto support the various planning tasks involved. Afterwards, the author discusses theimpact of the ongoing change in strategy, namely the change from a built-to-stockoriented to a customized built-to-order production, on the future application of ORmethods.

The final paper by Ralf Bihlmaier, Achim Koberstein and Rene Obst considersstrategic flexibility and capacity planning under uncertain demand in productionnetworks of automobile manufacturers. Their solution approach is integrated into adecision support system, which determines minimum-cost product allocations anddevelops tactical workforce plans. The practicality of their approach is demonstratedby the use of an industrial case study.

viii Preface

The primary objective of this book is to reflect the recent developments of sup-ply chain planning and to examine new research issues. It presents recent researchresults on collaborative planning in supply chains, demand and inventory manage-ment in logistics networks as well as industrial applications. The specific focus ofthis book is on the application of quantitative methods, which also form the ba-sis of commercial advanced planning software systems. Fourteen papers previouslypublished in “OR Spectrum – Quantitative Approaches in Management” have beenselected for publication in this volume. All papers have been peer-reviewed accord-ing to the standards of the journal.

This book has greatly benefited from the cooperation among the authors, review-ers and editors. We express our sincere thanks to the reviewers for their excellentand timely refereeing. Last, but not least, we thank all authors for their contribu-tions, which made this book possible.

Hans-Otto GuntherHerbert Meyr

Berlin and Darmstadt,February 2009

Contents

Part I: Decision Making in Supply Chains

A framework for collaborative planning and state-of-the-art ........................... 3

H. Stadtler

Simulation and optimization of supply chains: alternative or complementary approaches?.......................................................................... 29

C. Almeder, M. Preusser, R.F. Hartl

Part II: Demand Management

Revenue management and demand fulfillment: matching applications, models and software............................................................................................. 57

R. Quante, H. Meyr, M. Fleischmann

Active demand management for substitute products through price optimisation........................................................................................................... 89

A.A. Levis, L.G. Papageorgiou

Customer segmentation, allocation planning and order promising in make-to-stock production ............................................................................. 117

H. Meyr

Managing product availability in an assemble-to-order supply chain with multiple customer segments ........................................................... 145

T.R. Ervolina, M. Ettl, Y.M. Lee, D.J. Peters

ix

Part III: Inventory Management

Setting safety stocks in multi-stage inventory systems under rolling horizon mathematical programming models................................................................. 199

Y. Boulaksil, J.C. Fransoo, E.N.G. van Halm

Supplier managed inventory in the OEM supply chain: the impact of relationship types on total costs and cost distribution................................ 219

P.L.M. van Nyen, J.W.M. Bertrand, H.P.G. van Ooijen, N.J. Vandaele

Vendor-managed inventory and the effect of channel power ........................ 247

B.C. Bichescu, M.J. Fry

Part IV: Applications in the Chemical Industry

Value chain management for commodities: a case study from the chemical industry................................................................................ 283

M. Kannegiesser, H.-O. Günther, P. van Beek, M. Grunow, C. Habla

MILP-based campaign scheduling in a specialty chemicals plant: a case study.......................................................................................................... 315

M. Brandenburg, F.-J. Tölle

Part V: Applications in the Automotive Industry

Supply chain planning in the German automotive industry.......................... 343

H. Meyr

Modeling and optimizing of strategic and tactical production planning in the automotive industry under uncertainty................................................. 367

R. Bihlmaier, A. Koberstein, R. Obst

Inventory reservation and real-time order promising in a make-to-stock system.................................................................................. 169

R. Pibernik, P. Yadav

x Contents

Part IDecision Making in Supply Chains

A framework for collaborative planningand state-of-the-art

Hartmut Stadtler

Abstract Inter-organizational supply chain management incurs the challenge toalign the activities of all members which contribute to the value creation of a productor service offered to customers. In general, a supply chain faces the “problem” ofinformation asymmetry, members having their own objectives and constraints whichmay be in conflict with those of the other members. Still, activities have to be alignedin such a way that the supply chain as a whole stays or becomes competitive whileeach member wins by cooperating. A number of collaborative planning schemes havebeen put forward in the last two decades with different assumptions and differentareas of application. This paper intends to provide a framework and an overview onthe state-of-the-art of collaborative planning. The criteria of the framework will allowus to position existing concepts and to identify areas where more research is needed.The focus of the literature reviewed here will be on model-based decision support atthe operational planning level.

Keywords Collaborative planning · Supply chain management · State-of-the-art

1 Introduction and definitions

Supply chain management (SCM) is concerned with the coordination of material,information and financial flows within and across legally separated organizationalunits (Christopher 1998). One important way to achieve coordination in an inter-orga-nizational supply chain (SC) is the alignment of future activities of SC members,hence the coordination of plans. The aim of this paper is to present a new framework

H. Stadtler (B)Institute for Logistics and Transportation, University of Hamburg,Von-Melle-Park 5, 20146 Hamburg, Germanye-mail: [email protected]

H.O. Gunther, H. Meyr, Supply Chain Planningc

3

DOI 10.1007/s00291-007-0104-5OR Spectrum (2009) 31:5–30Originally published in:

© Springer-Verlag Berlin Heidelberg 2009

H. Stadtler

of collaborative planning (CP) together with a state-of-the-art overview of conceptsfor CP from the literature with special emphasis on a model-based decision supportat the operational planning level of an inter-organizational SC.

Coordination of flows requires adapting plans of supply chain members at variouslevels of a planning hierarchy. Planning is regarded as an activity to choose, sequenceand evaluate future activities for a specific decision making unit (e.g. a company). Aprocedure for aligning plans of two or more decision-making units is called a coor-dination scheme . The terms procedure and process are regarded synonyms here fordescribing the interaction of activities of at least two SC members (e.g. a negotiationprocess). In order to become a CP scheme we further require that individual plans areadapted in an effort of joint decision making, i.e. a willingness to cooperate and tocontribute to the generation of a plan which will be accepted by these SC members(which may well be a subset of the overall SC). In other words, we exclude pure cen-tral planning by a single (focal) SC member without the active contribution of otherSC members. This is usually required to overcome information asymmetry, where noSC member possesses all the information and preferences of the other SC members(see Schneeweiss 2003, p. 29 for a similar definition). This private information may berevealed to the other SC members in the course of joint decision making provided ade-quate incentives for true information providing exist (e.g. Feldmann and Müller 2003).Note that for central planning at least one SC member must possess all the informationrelevant for generating an overall SC plan that is accepted by all members.

In summary, we define collaborative planning as a joint decision making pro-cess for aligning plans of individual SC members with the aim of achievingcoordination in light of information asymmetry.

But when does CP result in a state of SC coordination? The most stringent answerrefers to the contract literature: a contract coordinates the SC if (and only if) “the set ofsupply chain optimal actions is a Nash equilibrium, i.e., no firm has a profitable unilat-eral deviation from the set of supply chain optimal actions” (Cachon 2003, p. 230 seeMyerson 1991 for the definition of a Nash equilibrium). Here, coordination requiresa solution which represents both a (central) supply chain optimum as well as a Nashequilibrium.

Omitting the game theoretic perspective the overall SC perspective remains: now,a (central) SC optimum solution suffices for coordination. A third and even “softer”definition of coordination results if the implemented actions lead to an improved planfor the SC as a whole compared to a default (or initial) solution. Such a definition isimplicitly supported by Corbett and de Groote who compare their (suboptimal) coordi-nation mechanism with the default solution (no coordination) (Corbett and de Groote2000, p.449). Finally, a fourth alternative even calls the default situation coordinated,which seems to be favoured by Schneeweiss (“worst-case” coordination, Schneeweiss2003, p.278).

The first two definitions of SC coordination imply that a large number of CP schemeswill be left in a state of non-coordination although solutions generated may be near-optimal. On the other hand, the fourth proposal will call all solutions coordinatedwithout looking at feasibility or solution quality. Hence, we regard the third proposala good compromise.

4

A framework for collaborative planning and state-of-the-art

The alignment of plans will take place at a certain planning level (e.g. master plan-ning) with a given degree of detail (level of aggregation) and planning horizon. Hence,CP is further specified, like collaborative master planning.

CP not only applies to a SC partnership—as described by five criteria by Landerosand Monczka (1989) based on a survey in the automotive industry—but also to a morecompetitive environment. Hence, the term SC member seems more neutral. Still, weregard an arms-length type of interaction (see Dyer et al. 1998) as well as actionsof moral hazard to be counterproductive for an effective CP. Each SC member is incharge of a specific planning domain. It comprises a part of the SC and the relatedplanning processes that are under the control and in the responsibility of a distinct SCmember (Kilger and Reuter 2005).

While intended to be applied to inter-organizational SCs, CP schemes might wellbe applied in an intra-organizational setting where SC members belong to the samecompany. The main requirement is that CP takes place in the absence of a centralplanning instance which may ultimately enforce coordination. If the coordination ofactivities (like transport or production activities) is achieved across different SCs this iscalled horizontal collaboration (for an example in the distribution of consumer goodssee Fleischmann 1999). However, if the activities considered belong to one single SCthis requires vertical coordination—which will be the focus of this paper.

CP software modules are already offered by some software vendors which basi-cally support the exchange of (demand and procurement) data plus some additionalfeasibility checks. According to Schneeweiss (2003, p. 5) this is only the starting pointof CP.

In the following we will present an overview of research areas where CP schemeshave originated from (Sect. 2). Sections 3, 4 and 5 describe our framework for CP.Section 6 discusses the issue of fairness which seems to be an important condition forengaging in and accepting a CP scheme. Finally, Sect. 7 summarizes our findings andprovides some ideas for future research.

2 Related research areas

There are a number of research areas which are closely linked to CP in that theseprovide mechanisms to coordinate decentralized decision units. These areas will bementioned briefly below with their main focus and limitations. These areas will be dis-cussed sequentially starting with the area with the largest number of papers consideredhere (see Table 1): mathematical decomposition. Within this broad area we discrim-inate exact mathematical decomposition, heuristic mathematical decomposition andmeta-heuristics.

In principle (exact) mathematical decomposition techniques are applicable for CP(like Dantzig–Wolfe decomposition (Dantzig and Wolfe 1960) or Benders decompo-sition (Benders 1962) to name only a few). These techniques usually require a specificstructure of the underlying decision model (coefficient matrix) and aim at findingan optimal solution to the overall (central) model with less computational efforts.Mathematical decomposition can be interpreted as a model of a divisionalized orga-nization with individual and hidden constraints at the divisions as well as common

5

H. Stadtler

Tabl

e1

Pape

rsap

plie

dto

the

fram

ewor

kfo

rC

Pan

das

soci

ated

rese

arch

area

s

Rel

ated

rese

arch

area

sL

u(1

995)

Cor

bett

and

deG

root

e(2

000)

Bar

bar-

osog

lu(2

000)

Ert

ogra

lan

dW

u(2

000)

Fran

soo

etal

.(2

001)

Gje

rdru

met

al.(

2002

)K

arab

ukan

dW

u(2

002)

Fink

(200

3,20

04)

Schn

eew

eiss

and

Zim

mer

(200

4)

Suck

y(2

004b

)D

udek

and

Stad

tler

(200

5,20

07)

Jung

etal

.(2

005)

Shir

od-

kar

and

Kem

pf(2

006)

Exa

ctm

athe

mat

ical

X

deco

mpo

sitio

n

Heu

rist

icX

XX

mat

hem

atic

al

deco

mpo

sitio

n

Met

a-he

uris

tics

X

EO

Qm

odel

s/X

XX

sim

ple

cont

ract

s

Inve

ntor

ysy

stem

sX

Hie

rarc

hica

lX

Plan

ning

Pape

rfu

lfils

CP

defi

nitio

n�

�N

o(c

entr

alm

odel

)

��

No

(cen

tral

mod

el)

��

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��

No

(cen

tral

mod

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6


constraints at the organizational level. The organizational level serves as a central deci-sion unit, which knows the interdependencies among divisions and guides the searchfor the organization’s optimal solution by a price-budget directive planning procedure(Meijboom 1986). Limitations of this area have been lifted in recent years (e.g.Vanderbeck 2000). A paper originating from this area is Karabuk and Wu (2002)making use of Augmented Lagrangean techniques (Table 1).

Similar to the Dantzig–Wolfe decomposition heuristic mathematical decomposi-tion techniques are based on individual mathematical programming models for eachSC member comprising the constraints and objectives of their planning domain. Thesemodels may be used to generate and evaluate purchase and supply proposals (for anexample see Dudek 2004, Dudek and Stadtler 2005, 2007). The models’ decisions arealigned by heuristic search techniques with the aim of finding a feasible, near optimalsolution for the overall SC (e.g. Jung et al. 2005). CP schemes differ in the type ofinformation to be exchanged. Here, resource- or price-related information can be madeuse of (e.g. Ertogral and Wu 2000). Sometimes mathematical programming modelsare solved by a standard MIP solver or by a meta-heuristic (like in Fink 2003, 2004)(all included in Table 1).

Coordination of static lot-sizing models [also known as economic order quantity(EOQ) models] in a two party setting (usually termed buyer and vendor) have a longhistory (starting with Goyal 1976 and Monahan 1984) and provide a number of valu-able insights (e.g. how to devise a quantity discount pricing schedule that results inthe SC optimal solution). Recently, research has been extended to three-level supplychains. Reviews can be found in Thomas and Griffin (1996), Sarmah et al. (2006)and Li and Wang (2007). They state that research has been limited to “. . .one productand one machine only and thereby fails to capture the essence of real supply chain”(Sarmah et al. 2006, p. 13). Papers that originate from this research area and assumeasymmetric information and thus fit into this paper (see Table 1) are from Lu (1995),Corbett and de Groote (2000) as well as Sucky (2004a,b).

The origin of research into the coordination of decentralized decision units datesback to the paper of Clark and Scarf (1960). They study a serial multi-echelon inven-tory system and devise a recursive decomposition approach for determining the optimalparameters of an echelon stock policy. In the meantime both convergent and divergentinventory systems have been studied with remarkable results. For a review of thisresearch area we refer to Thomas et al. (1996) and Minner (2000). A paper whichexplicitly deals with decentralized inventory decisions and asymmetric information isFransoo et al. (2001) (see Table 1).

Decomposition and coordination is also the topic of Hierarchical Planning. Firstintroduced by Anthony (1965) and specialized for applications in the area of pro-duction planning by Hax and Meal (1975) this area has gained great acceptance forintra-organizational SCs. The planning tasks facing a company are split into severallevels. The closer a planning level is to the actual object of planning (e.g. the shopfloor) the greater the degree of detail and the shorter the planning horizon will be.Coordination will be achieved by anticipation of lower level decisions, directives andfeedback. In any case there is a central planning unit at the top coordinating the overallSC (for more details see Schneeweiss 2003). Considering elements from hierarchical

7

H. Stadtler

planning Schneeweiss and Zimmer (2004), Zimmer (2001) devised a CP scheme foran inter-organizational SC with two SC members (see Table 1).

Papers mentioned above and included in Table 1 fulfill our definition of CP andadvocate model-based CP. However, Table 1 contains three further papers, namely byBarbarosoglu (2000), Gjerdrum et al. (2002) as well as Shirodkar and Kempf (2006)which ultimately apply a central model and impose the resultant decisions on the SC(see Table 1). Since the cases described in these papers contain a number of interest-ing ideas to overcome information asymmetry, we hesitated to totally exclude thesepapers from our presentation. Hence, we show the underlying structure of the SC, therelationships among SC members and the decision situation facing each SC memberin the Appendix (Table 5).

Finally, a few related research areas will be mentioned which are not included heredue to the following reasons:

A field that has attracted many researchers in recent years is SC coordination withsimple contracts. A contract “. . . makes the terms of the relationship explicit” (Tsayet al. 1998, p. 306). Often idealized decision problems are considered to gain structuralinsights, like the news vendor model. Tsay et al. (1998) distinguish eight different con-tract clauses (like the specification of decision rights, pricing or buyback and returnpolicies).

Only a few papers deal with several of these clauses simultaneously (Tsay et al.1998, p. 306). Although there is evidence of a successful application of a clausein practice (Cachon and Lariviere (2001) mention a revenue-sharing contract in thevideo-rental industry) most results are still theoretic and based on stylized models.Hence, “they fail to address a variety of issues that become relevant to actual imple-mentation” (Tsay et al. 1998, p. 330). Although models have been extended in themeantime (see the paper by Wang and Gerchak 2003 and the overview by Cachon2003) this conjecture still holds. Hence, contract theory will not be considered hereany further. This does not preclude that CP schemes analyzed here incorporate resultsfrom this research area ( Lu 1995; Corbett and de Groote 2000; Sucky 2004a, b).

Also, we will not deal with pure auctions (for an introduction see Milgrom 2004).In our view auctions are most applicable in pure market interactions at the boundariesof a SC but not within a SC. Note that auction mechanisms may be utilized to coordi-nate decentralized (often detailed) schedules (like Fox et al. 2000). This then can beregarded as a substitute for a central optimization approach.

Likewise, we will not consider Collaborative Planning, Forecasting and Replen-ishment (CPFR) here since its emphasis (so far) is on the exchange of information andless on model-based decision support. A short outline of some of the developmentsof CPFR should suffice: CPFR is a formalized process which has been worked out bythe standardization committee VICS (Voluntary Interindustry Commerce Standards)and implemented over 300 companies (VICS 2004, p.5). The CPFR process modelconsists of eight planning tasks, which can be subsumed under four main activities:strategy and planning, demand and supply management, execution and analysis (VICS2004, S.9). Here “planning” does not refer to the alignment of operational plans butto the identification and communication of events which may affect demand, such aspromotional activities or product introductions.

8


While in the original model collaboration is restricted to mere informationexchange, some authors extend the scope of CPFR to joint decision-making of the SCmembers involved. Danese (2005, p.458) mentions a concept called “limited CPFRcollaboration”, where plans are synchronized jointly by the SC members (e.g. replen-ishment plans between a central company and a distribution centre) and exceptionsare managed.

Multi-agent systems may be one way to deal with CP. Here “an agent is an auton-omous, goal oriented software process that operates asynchronously, communicatingand coordinating with other agents as needed” (Fox et al. 2000, p. 16). While agentsmay be in charge of only a single element of a (production) system [like a productionagent representing a single machine which is able to produce a certain number ofproducts (Grolik et al. 2001)] there may also be very versatile (software) agents repre-senting a whole decision unit (like Master Planning) of a specific SC member (Stadtler2004). While we are not interested in the software architecture of these multi-agentsystems the basic logic and models guiding the search for a SC solution may well beof interest here (e.g. see the proposal of Jung and Jeong 2005, Jung et al. 2005). Sinceany CP scheme considered here could be implemented along the guidelines of agenttechnology, we will not make use of it as a discriminating criterion.

A first taxonomy of coordination has been advocated by Whang (1995). He first dis-criminates organizational units according to coordination “within operations”, “cross-functional” and “inter-organizational”. The second criterion relates to the behaviour ofthe people in the organization, namely “single-person perspective”, “team perspective”and “nexus-of-contract perspective”. A single-person perspective incurs the advantagethat there is a single (central) decision-making unit who has access to all relevant infor-mation. In the team perspective each party has limited information and action sets.This requires to communicate and coordinate activities to achieve the global (team)objective, i.e. in the team perspective the organizational units act separately but sharethe same objective. This is in contrast to the “nexus-of-contract perspective” wherethere are separate decision making units with private information and individual goals(opportunism). For each of the resulting nine subcategories Whang presents examplesfrom literature. One of his conclusions is that in 1995 the research in Operations andInformation Management has heavily leaned towards the single-person perspective oforganizations.

In this paper we will further elaborate the two subcategories defined by “inter-organizational coordination” and “team-perspective” as well as “nexus of-contractperspective”. Since the latter involves multiple organizations and has no relevance to anorganizational perspective Whang (1995, p. 420) renamed it into “inter-organizationalinteraction”.

Although the taxonomy of Whang (1995) is a first attempt to categorize differenttypes of coordination it does not span the large variety of characteristics necessary todescribe CP problems. Subsequently, we will present a framework (synonym “typol-ogy”) of CP problems and schemes intended to describe their main characteristics asobserved in the literature in greater detail. Still, we do not claim to have extractedall possible characteristics in its totality as would be required for a classification (seeDyckhoff and Finke 1991, pp. 1 for a discussion of these terms).

9

H. Stadtler

Ideally, a CP scheme should contain a set of activities and rules applicable to a widerange of decision problems. However, CP schemes presented in the literature often areclosely linked to a specific decision problem. Hence, our framework of CP consistsof three broad categories:

(1) the structure of the SC and the relationships among SC members (Sect. 3),(2) the decision situation facing each SC member (Sect. 4) and(3) the characteristics of the CP scheme itself (Sect. 5).

The first two categories specify a CP problem facing a SC. This allows us to clusterthose application areas where CP schemes are already available. Ideally, the third cat-egory should not depend on the two former categories. Not surprisingly it turned outthat CP schemes actually are designed for specific CP problems.

3 SC structure and relationships among SC members

The structure of the SC constitutes a major factor for CP and the complexity of align-ing plans. Furthermore, the relationships among SC members will have an influenceon the type and reliability of the information exchanged. Thirdly, requirements for aSC solution may comprise an objective to be followed by the supply chain as a wholeincluding the notion of fairness. These issues will be described in the following (seeTable 2). Please note that the information presented in tables often can only provide afirst idea and not a complete description of a criterion specification:

3.1 Criteria (1.1): Structural elements of the SC

For defining the structure of a SC we consider

(1.1.1) the number of SC tiers(1.1.2) the number of SC members on each tier and(1.1.3) the business functions SC members fulfil.

The easiest situation, where collaboration can be applied is a two party situation, usu-ally termed a supplier and a buyer (depicted by “1–1”). In general one has to mentionthe number of tiers considered and the number of SC members ni (individual planningdomains) on each tier i (where i = 0 depicts the SC member most downstream theSC). Additionally one can discriminate the structure of flows between SC members asserial, divergent, convergent, mixed and cyclic similar to definitions well known fromthe bill of materials. Note, that we do not consider the structure of flows within theplanning domain of a SC member. The difficulty in the context of CP is not only thecomplexity resulting from the number of links to handle but also the additional deci-sions to make, e.g. consider the case of a two-level divergent SC (1−n0) with a scarceresource on the side of the supplier. Now we may have to decide on the allocation ofthe scarce material in light of the (unfilled) demand from the buyers.

According to the business functions to coordinate Bhatnagar and Chandra (1993)distinguish “multi-plant coordination” and “general coordination” within the contextof intra-organizational collaboration. If decisions have to be linked within the same

10


Tabl

e2

Cri

teri

afo

rdi

scri

min

atin

gSC

stru

ctur

esan

dre

latio

nshi

ps

Cri

teri

afo

rSC

stru

ctur

ean

dre

latio

nshi

ps

Lu

(199

5)C

orbe

ttan

dde

Gro

ote

(200

0)E

rtog

ral

and

Wu

(200

0)

Fran

soo

etal

.(2

001)

Kar

abuk

and

Wu

(200

2)

Fink

(200

3,20

04)

Schn

eew

eiss

and

Zim

mer

(200

4)Su

cky

(200

4b)

Dud

ekan

dSt

ad-

tler

(200

5,20

07)

Jung

etal

.(2

005)

1.St

ruct

ural

elem

ents

1.1.

1N

o.of

tiers

22

32

22

22

22

1.1.

2N

o.of

mem

bers

ontie

rs1–

1,1–

n 01–

1n 2

–n1–n

01–

n 01–

1n 1

–11–

11–

11–

n 01–

1

1.1.

3B

usin

ess

func

tions

prod

.pr

od.

prod

.pr

od.

and

reta

iler

prod

.an

dm

ar-

ketin

g

prod

.pr

od.

prod

.pr

od.

prod

.an

ddi

str.

1.2.

Rel

atio

nshi

ps

1.2.

1Po

wer

S?

–?

?–

B?

??

1.2.

2B

ehav

iour

team

opp.

team

team

team

team

team

opp.

team

opp.

1.2.

3L

earn

ing

eff.

–+∞

––

––

–+∞

–ye

s

1.2.

4R

oll.

sche

d.–

––

––

––

–ye

s–

1.3.

Req

uire

dop

t.S

fair

SCSC

SCSC

SSC

SC

SCso

lutio

nfo

rS,

1–1

near

opt.

near

opt.

(nea

r)op

t.ne

arop

t.ne

arop

t.ne

arop

t.ne

arop

t.ne

arop

t.

Bbu

yer,

Ssu

pplie

r,di

str.

Dis

trib

utio

n,–

noor

not

appl

icab

le,i

nv.

inve

ntor

y,+∞

solu

tion

last

sfo

reve

r,n i

num

ber

ofSC

mem

bers

ontie

ri,

?op

enqu

estio

nsi

nce

noSC

mem

ber

poss

esse

sal

lthe

mec

hani

sms

toex

erci

sepo

wer

,opt

.opt

imal

,opp

.opp

ortu

nism

,pro

d.pr

oduc

tion,

roll

.sch

ed.r

ollin

gsc

hedu

les

11

H. Stadtler

business function at different echelons in an organization this is termed multi-plantcoordination. Although not limited, the term focuses on the production function. Ifseveral functions, such as production and marketing have to be aligned then the termgeneral coordination applies.

This idea can be transferred to inter-organizational collaboration, also. Hence, wedistinguish two specifications of the business function within a SC: the planningdomains of all SC members are

– of the same type or– of a different type.

Consequently, the name of the business function(s) involved have to be indicated.For instance Barbarosoglu (2000) and Fink (2003, 2004) consider the same businessfunction—production—for all SC members. Fransoo et al. (2001) deal with coordi-nating a manufacturer and several retailers while Jung and Jeong (2005) show how tocoordinate production with the distribution of final products.

3.2 Criteria (1.2): Relationships among SC members

The relationships among SC members comprise the following issues:

(1.2.1) power of each SC member,(1.2.2) extent of self-interest governing a SC member’s behaviour,(1.2.3) learning effects and(1.2.4) rolling schedules.

The power of a SC member and its relative position in a SC may result from differentsources, like

– product and (production) process know-how,– number of competitors,– portion of the value creation with respect to the value of the final product,– access to the customer base (market) and– financial resources.

Listing and describing these attributes is rather simple, however, measuring power ismuch more difficult.

In the literature on CP the notion of power (or leadership as it is called sometimes)is usually not discussed explicitly. However, the premises and CP scheme advocatedmay reveal in which way power is used by SC members. Still, generalizations oftenare difficult: For instance a SC member making the first offer may on the one handfix his minimum profit while on the other hand reveal new information which maybe exploited by another SC member when making a counter proposal. (If informa-tion about the most powerful SC member is provided by the authors explicitly this isindicated in Table 2, otherwise a “?” is placed.)

Furthermore, the distribution of power may and usually will change over time—itmay even change in the course of a single instance of collaboration. Also, power maybe exercised in different phases of the collaboration (see criterion (2.2)), i.e.

12


• when designing a SC and fixing the conditions of a collaboration strategy (note,this may also concern the distribution of the efficiency gain),

• in the planning phase the most powerful member may choose to make the firstoffer and thereby fixing his minimum profit (like in Dudek and Stadtler 2005) orenforce a certain stopping rule,

• in the execution phase the way exceptions are handled and resultant costs are splitmay be to the disadvantage of the weaker SC members and finally

• in the evaluation phase the most powerful SC member may impose penalties forthose SC members not conforming to expected performance targets.

The extent of self-interest governing a SC member’s behaviour has already been men-tioned in Sect. 2 when referring to the taxonomy of Whang (1995). Here, a teambehaviour exists if all actions (or decisions) being favourable for the SC as a whole areaccepted, irrespective of the SC members’ individual interests. Opportunism prevails,if SC members only implement actions which are in their own interest and may even becheating when passing information to others. Note that providing information (eithertruthfully or not) can be regarded an action, too (see Hirshleifer and Riley 1992).

According to a survey by Landeros and Monczka (1989) a supplier–buyer part-nership rests (among other things) on a trustworthy commitment of future conduct—hence, a team perspective may not be unrealistic.

Looking at the papers considered here, it is not easy to distinguish between teamperspective and opportunism. Dudek and Stadtler (2005, 2007) assume that all partiesprovide requested information truthfully while searching for an overall supply chainoptimum (hence, a team perspective applies). The paper of Fransoo et al. separatesthe retailers in cooperative (team) and non-cooperative (opportunistic) retailers. Thelatter do not share (demand) information and impose a β-service level constraint themanufacturer has to fulfil (thereby exercising their power over the manufacturer).Considering this β-service level as a constraint, the remaining SC members exerciseteam behaviour. According to Table 2 CP schemes for opportunism are the minority(Corbett and de Groote 2000; Sucky 2004b; Jung et al. 2005).

Rolling schedules play a role in the planning phase. It is most popular in industry inorder to cope with uncertainty (e.g. of demand). In a SC setting this not only involvesupdating and extending an existing plan by one SC member but also renegotiating allchanges with all other SC affected members. One question here is, who will bear thecosts resulting from these changes if there is already a previously approved plan? Themajority of papers assume that a plan once agreed upon will be executed unaltered upto the planning horizon (an exception is Dudek 2004, pp. 115).

Closely related to rolling schedules is the notion of learning effects. If the nego-tiation procedure is repeated, then a party may make use of information gained inprevious negotiations. This is especially true if there is an overlap of decisions intwo successive plans (like in rolling schedules). In the extreme this may lead to atotally different situation, like in Corbett and de Groote (2000), p. 447: once thebuyer has decided to choose a specific purchasing contract all data are revealed to thesupplier. Thus there is no renegotiation and the conditions of the contracts must last“forever”.

13

H. Stadtler

3.3 Criteria (1.3): Requirements for a SC solution

The objective(s) governing the generation of plans in the CP scheme should not bemixed with the members individual objectives. Here, we are interested in the waythe—often conflicting—objectives are handled. We discriminate three broad types ofobjectives:

– the alignment of flows,– the search for the SC optimum and– the search for a fair solution.

If “only” the alignment of (material) flows is looked for, a feasible supply and pro-curement plan has to be established such that customer orders are fulfilled in duetime. No monetary objectives are considered here (e.g. Friedrich et al. 2002; Stäbleinand Baumgärtel 2006). However, aiming at the SC optimum often means maximizingprofits or minimizing costs for the SC as a whole. Such a solution may incur a lossfor some members and high profits for some others. In these cases side-payments ordiscounts may become an issue to yield a win–win situation for each member.

Searching for a fair solution requires a definition of the term fair. Here, a numberof proposals exist which will be discussed in detail in Sect. 6.

Having described the SC structure and the relationships between SC members wecan continue in characterizing the decision situation each member is facing.

4 Characteristics of the decision situation of each SC member

Characteristics for discriminating decision situations fall into four broad categorieswhich stem from answering the following question: which decisions take place, when,with which objectives and which information? These four Ws of CP will be describedin greater detail in the following (also see Table 3).

4.1 Criteria (2.1): Decision models—which decisions take place?

Here the real world decision situation (planning tasks) of each SC member should bedescribed. However, if standard decision models are deployed, their names provide aclue for the underlying decision situation.

Often it is assumed that each member in the SC faces the same type of deci-sion model, e.g. both the supplier and the buyer deploy an EOQ model for a singleproduct. The buyer calculates the order quantity while the supplier calculates opti-mal production orders. But there are also examples where SC members face different(basic) decision situations: a resource-constrained-project scheduling problem(RCPSP) for the buyer and a capacitated lot-sizing problem (CLSP) for the supplier(e.g. Schneeweiss and Zimmer 2004).

We would like to add that in the course of collaboration both the goal(s) as well asthe action set of a SC member may change. This may require additional constraintswhich often destroy the “typical” structure of a standard decision model. As a resulta solution technique applicable for the standard decision model may no longer be

14


Tabl

e3

Cha

ract

eris

tics

for

disc

rim

inat

ing

the

deci

sion

situ

atio

n

Cri

teri

afo

rde

cisi

onsi

tuat

ion

Lu

(199

5)C

orbe

ttan

dde

Gro

ote

(200

0)

Ert

ogra

lan

dW

u(2

000)

Fran

soo

etal

.(2

001)

Fink

(200

3,20

04)

Kar

abuk

and

Wu

(200

2)

Schn

eew

eiss

and

Zim

mer

(200

4)Su

cky

(200

4b)

Dud

ekan

dSt

adtle

r(2

005,

2007

)

Jung

etal

.(2

005)

2.1.

Dec

isio

nm

odel

sE

OQ

EO

QM

LC

LSP

(R,S

)-in

ven-

tory

MIP

2-st

age

sto-

chas

ticpr

ogra

m

CL

SP,R

CPS

PE

OQ

ML

CL

SPL

P

2.2

Phas

esof

colla

bora

tion

cond

.co

nd.

oper

.op

er.

oper

.op

er.

oper

.co

nd.

oper

.op

er.

2.3.

Info

rmat

ion

stat

us

2.3.

1In

form

atio

nhi

dden

(fro

mB

orS)

s&h

cost

Bh

cost

Bal

lco

sts

all

all

cap.

Ss&

hco

stB

all

all

2.3.

2In

form

atio

nex

chan

ged

orde

rsor

der

men

uor

ders

serv

ice

leve

lta

rget

s

rej./

ac.p

lans

pric

es&

supp

lyqu

anti-

ties

orde

rsor

der

men

uor

ders

orde

rs

2.3.

3D

egre

eof

unce

rtai

nty

det.

det.

det.

stoc

h-as

tic(m

arke

tde

man

d)

det.

stoc

h-as

tic(y

ield

,de

man

d)

det.

stoc

h-as

ticde

t.de

t.

2.4.

Obj

ectiv

e(s)

min

.co

sts

min

.cos

tsm

in.

cost

sm

in.

cost

sar

bitr

ary

max

.pr

ofit

min

.cos

tsm

in.

cost

sm

in.c

osts

min

.co

sts

Bbu

yer,

oper

.op

erat

iona

lpl

anni

ngph

ase,

cap.

capa

citie

s,re

j./ac

.pl

ans

gene

rate

dby

am

edia

tor

are

tran

sfer

red

toSC

mem

bers

and

eith

erre

ject

edor

acce

pted

,co

nd.

colla

bora

tion

cond

ition

s,de

t.de

term

inis

tic,S

supp

lier,

inv.

Inve

ntor

y,s&

hse

tup

and

hold

ing,

–no

15

H. Stadtler

skeletoncontract

negotiationsabout opera-tional details

operational execution phase evaluation phase

adaptationsduringexecution

renegotiate orrenew contract

collaborationconditions planning phase

time

Fig. 1 Phases of collaboration

applicable. As an example consider the EOQ model where the non-linear objectivefunction can be minimized by taking the first derivative. In a situation where a supplierwould like to generate a menu of supply proposals to be presented to the buyer addi-tional constraints result. Now the constrained non-linear optimization model requiresthe application of the Karush–Kuhn–Tucker conditions (Sucky 2004b).

The complexity of describing CP schemes becomes clear when dealing with thedecision problem at hand. For each decision problem there may exist a distinct clas-sification or typology [e.g. for the RCPSP see Brucker et al. (1999) and for lot-sizingsee Drexl and Kimms (1997)]. Further typologies for decision problems in the area ofproduction have been put forward which so far have not been addressed in conjunctionwith CP schemes [e.g. cutting and packing (Dyckhoff and Finke 1991) and assemblyline balancing (Boysen et al. 2007)].

4.2 Criteria (2.2): Phases of collaboration—when does collaboration take place?

Decisions are made at different points in time in the course of collaboration (see Fig. 1).We discriminate four phases, starting with specifying the

(2.2.1) conditions of collaboration followed by(2.2.2) planning,(2.2.3) execution and(2.2.4) evaluation phase.

Determining the conditions of collaboration often incurs negotiating the terms of askeleton or detailed contract valid over a certain period of time. The wholesale price(Barbarosoglu 2000) or a service level (Fransoo et al. 2001) as well as the extent ofinformation exchange can be set. Furthermore, the auditing and evaluation proceduremay be agreed upon by the parties. In the (operational) planning phase the procure-ment, production and distribution plans at different levels of the planning hierarchyhave to be aligned.

During the execution phase (of the contract) it might be advantageous to reconsidercertain obligations (like the fulfillment of an order) due to unexpected incoming orders.Finally, once the duration of a contract comes to an end, an ex post evaluation maytake place. This may give rise to renegotiating the terms of the contract and possiblyits prolongation. An example for the evaluation phase is presented in the paper byJammernegg and Kischka (2005) where the contract’s attributes may be renegotiatedin the course of time in order to improve the performance of a SC.

16


CP schemes proposed in the literature often address decisions relating to just oneplanning phase. CP schemes presented in Table 3 are restricted to the first two phases.Obviously, it would be ideal to take a holistic view and to devise a concept coveringall four phases of collaboration.

4.3 Criteria (2.3): Information status—what is the information status of each SCmember?

The information status in an inter-organizational collaboration is assumed to be asym-metric. The reasons for asymmetric information can be manifold. On the one hand,there are practical reasons such as administering a decentralized database may be moreeconomical and faster than a central database. Also, information gathered decentrally(like the productivity of specific worker/job assignments) may remain the expertiseof local decision makers. On the other hand, some information might be hidden fromthe other SC member(s) in order not to weaken the (future) bargaining power (e.g. asupplier disclosing large slack capacities may run the risk that the buyer will ask forprice reductions). In an asymmetric information situation the question remains

(2.3.1) which information is hidden to the other party?

Another aspect regards the

(2.3.2) type of information to be exchanged in the course of collaboration.

Here, one can discriminate three subcategories

– quantities (like purchase orders or supplies of a product),– monetary values,– additional key performance indicators (KPIs).

As monetary data we may have data from cost accounting as well as data directlyapplicable for decision making (like a product’s holding cost coefficient or penaltycosts for late delivery). These data are usually regarded sensitive data, i.e. data thatcan do harm to the owner of the data if exploited by a third party. Another type of costis the total cost of a “plan” (which may also be regarded a KPI) or a side-payment orcompensation required for accepting a SC member’s proposal.

A special case has been reported by Shirodkar and Kempf (2006). Here, suppliershave collaborated with the buyer in specifying a model of their planning domain. Themodel has been transferred to the buyer in order to combine all these submodels into acentral model. For the operational planning phase the data have to be maintained andupdated by each supplier. In this way information asymmetry is lifted.

The third sub-category, the additional KPIs, is often agreed upon at the start of aSC relationship. KPIs, like service levels, are calculated continuously or in certainintervals of time and serve to measure whether the SC is operating as expected.

A final discrimination of the data is the

(2.3.3) degree of uncertaintyA decision model used by a SC member may contain uncertain data due to the

environment (like market demand) or due to the behaviour of SC members. The latter

17

H. Stadtler

has been addressed in the SC literature very often and various ways to overcome thissource of uncertainty have been proposed (e.g. the uncertainty of demand of a sup-plier can be reduced by transferring the buyer’s production plan in a vendor-managedinventory (VMI) situation (see Holweg et al. 2005)). As a result either

– deterministic or– stochastic models

will be constructed.

4.4 Criteria (2.4): Objectives—what are the objectives of the decision problem?

The last category concerns the objective(s) of the decision problem a SC memberstrives for. Basically, either profit maximization or cost minimization is considered.The majority of papers analysed here advocate minimizing costs. If a standard deci-sion problem is considered then the objective is often also standard (e.g. minimizingof the sum of setup and stock holding costs per unit time for the EOQ). Models usedin industrial practice often make use of soft constraints. Violating these constraints ispenalized in the objective function (Shirodkar and Kempf 2006).

A time-oriented objective function is used in Fink (2003, 2004) where the objectiveof one member is to minimize throughput times.

5 Characteristics of collaborative planning schemes

CP schemes will be described here by only a few structural elements that can be“observed” and agreed upon by SC members and thus can also be regarded as char-acteristics of the CP problem (Table 4). Also, we will not go into algorithmic detailssince there may be several internal options to generate solutions (like for an LP model).The interaction between the parties involved in a CP scheme can be documented by aprotocol (see Fink 2004 or Stadtler 2004 for examples). The structural elements to bepresented here define the parties involved, the starting point as well as their “interface”.Hence, the following four characteristics will be analysed:

(3.1) the incorporation of a mediator,(3.2) the initial solution,(3.3) the number of rounds and the number of offers to be exchanged (stopping

criteria) and(3.4) final results SC members can expect.

A mediator is a third party controlling the rules of the game, e.g. by controllingthe (timing of) interactions among members. A mediator may have the capability ofgenerating plans and presenting these to all SC members for evaluation and even maybe entitled to propose the distribution of efficiency gains among SC members. Animportant issue is the proliferation of data to a mediator required for generating plansfor the SC as a whole, i.e. a mediator must be a trusted entity. A mediator differs froma central planning function (executed by one SC member) in that a mediator shouldnot be biased and has to take into account preference or objectives of each SC member

18


Tabl

e4

Cha

ract

eris

tics

for

disc

rim

inat

ing

CP

sche

mes

Cri

teri

afo

rC

Psc

hem

esL

u(1

995)

Cor

bett

and

deG

root

e(2

000)

Ert

ogra

lan

dW

u(2

000)

Fran

soo

etal

.(2

001)

Kar

abuk

and

Wu

(200

2)

Fink

(200

3,20

04)

Schn

eew

eiss

and

Zim

mer

(200

4)

Suck

y(2

004b

)D

udek

and

Stad

tler

(200

5,20

07)

Jung

etal

.(2

005)

3.1.

Med

iato

r–

–ye

sye

s–

yes

––

––

3.2

Initi

also

lutio

n–

–up

––

rand

om–

–up

dow

n

3.3.

No.

ofpl

ans

exch

ange

d

3.3.

1N

o.of

roun

ds1

1n b

ig–

n big

n big

11

n sm

all

n sm

all

3.3.

2Pa

ralle

loff

ers

1n ?

1–

11

1n ?

11

→to

taln

o.of

fers

1n ?

n big

–n b

ign b

ig1

n ?n s

mal

ln s

mal

l

3.4.

Fina

lres

ults

3.4.

1Q

ualit

yof

sol.

anal

.–

com

p.–

proo

fco

mp.

com

p.–

com

p.co

mp.

3.4.

2Si

de–p

aym

ents

–ye

s–

–ye

s–

yes

yes

yes

–

anal

.ana

lytic

alpr

oof,

com

p.co

mpu

tatio

nalt

ests

,dow

ndo

wns

trea

mpl

anni

ng,n

big

big

num

ber

(e.g

>>

10),

snsm

all

smal

lnum

ber

(e.g

.≤10

),n ?

num

ber

unkn

own,

proo

fpr

oof

ofco

nver

genc

eto

the

loca

lopt

imum

ofea

chSC

mem

ber,

up.u

pstr

eam

plan

ning

,–no

,?no

tmen

tione

din

the

pape

r

19

H. Stadtler

adequately. In industrial practice such a mediator in the area of planning has becomeknown as an application service providing company (Knolmayer et al. 2002, p. 10). Sofar, a mediator has rarely been considered in the literature on CP [with the exceptionof Ertogral and Wu (2000), Fransoo et al. (2001) and Fink (2003, 2004)].

When starting a negotiation process the solution to start from may be very impor-tant. In rolling schedules one can expect that the plan agreed to previously (e.g. theweek before) will form the basis for updating, renegotiation and extending a plan inlight of newly available information (e.g. customer orders).

If no such plan exists a plan has to be generated from scratch. A general and easyway is to generate an initial plan for the SC members randomly (Fink 2003, 2004).The drawback is that these initial plans probably are infeasible and the CP schememust take care to finally obtain feasibility.

A procedure often used in practice is upstream planning (Simpson and Erenguc2001). Here, the SC member closest to the final customer starts to generate his (opti-mal) plan with demand forecasts as an input. The supply of input materials is assumedto be unconstrained. From this (individually optimal) plan the required material sup-plies are derived and transferred to the suppliers on the next upstream tier. The mate-rial supplies requested now become the demand of the first tier suppliers. These firsttier suppliers then apply the same logic for generating their (optimal) plans. In thisway supply plans are propagated upstream the whole SC. This procedure assumes thatsuppliers will always be able to supply the materials required (Dudek and Stadtler2005).

In case the sequence of generating plans is reversed, i.e. starting with the mostupstream supplier, downstream planning results. The problem here is to indicate thedemand a supplier has to fulfill: given the supplier gets access to the demand fore-casts of final products a fixed lead time offset can be applied to estimate when thecorresponding parts demands will take place. This logic is applied by Jung and Jeong(2005) to a producer and a distributor. We would like to add that downstream planningis only favourable if the most expensive bottlenecks exist upstream and fixed leadtimes can be estimated with sufficient accuracy.

The expected number of rounds to run and the number of offers to be exchangedin the course of negotiations largely differ among CP schemes analysed. A round iscompleted if one party generates an offer or several parallel offers to choose from andthe subsequent evaluation by the other SC member(s). We say there is “no” round ifthere is only one instruction (and no “offer”) by one SC member the other membershave to follow, which is the case in classical central planning. Limited coordinationexists (one round) if there is only a single offer or a menu of offers where the otherparty has the right to choose from or to “take-it-or-leave-it” (see Corbett and de Groote2000; Sucky 2004b). The “burden” of collaboration largely depends on the numberof offers to generate and evaluate. Hence, an indicator is the expected total number ofoffers resulting from the expected number of rounds times the number of parallel offers(see Table 4). The number of offers is regarded “small”, if each offer (or plan) can beevaluated by a human decision maker before it is presented to the other members inthe SC. If this is the case then an interactive CP scheme can be designed, otherwise theCP scheme must be automated (like in the case of Fink 2003, 2004). An interactive

20


CP scheme with a small number of offers (e.g. at most ten) increases the chances ofacceptance by the decision maker(s).

The final results of a CP scheme are described by the

(3.4.1) quality of solutions and(3.4.2) the use of side-payments.

Ideally a CP scheme will result in the SC optimum and a win–win situation for eachSC member while the gains of collaboration are distributed among the members insuch a way that no SC member has a desire to deviate from the SC solution generated(e.g. a Nash equilibrium). A Nash equilibrium is often looked for in game-theoreticapproaches (like in Corbett and de Groote 2000; Sucky 2004b). If the decision situa-tion becomes complex an optimality proof rarely exists [Lu (1995) proved optimalityof the solution for the one supplier – one buyer case analytically]. Karabuk and Wu(2002) present a proof of convergence to the individual optimum of each SC memberbased on Augmented Lagrangean theory. However, in the majority of cases computa-tional tests come into play showing the performance within a certain decision context(e.g. multi-level capacitated lot-sizing). Given the tests are based on a suitable set ofparameter combinations, results of these tests can provide a good indication for thesuitability of a CP scheme in this context. However, the transfer of results to othertypes of decision problems may be misleading.

In case neither optimality nor an equilibrium can be proved certain properties of thefinal solution might be stated, like performance bounds. However, no such structuralresults have been reported in the literature analysed here.

Obviously the distribution of gains resulting from collaboration plays a major rolefor the active involvement and the acceptance of the results of a CP scheme. Side-payments (sometimes called compensations) may be one instrument to achieve a win–win situation for each SC member. This leads us to the notion of fairness which willbe discussed in the next section.

6 The notion of fairness

In experimental economics the ultimatum game has gained much attention (Güth1995). It provides insights into human behaviour and the evaluation of a fair solu-tion. The ultimatum game assumes that there are two players A and B which play thegame only once. Player A receives an amount of money X which he has to share withplayer B. The share s offered to B remains the decision of A, it may range from 0%< s ≤ 100%. In case player B accepts the share the amount X is shared accordinglyamong the two players. However, if player B rejects, the amount X is withdrawn andneither A nor B get any money.

In the ultimatum game information asymmetry exists, since player A does not knowthe threshold value of player B separating the acceptance of share s and its refusal.From the point of rationality player B should accept any positive share, since the alter-native is to get nothing. However, experiments have shown that if the share offeredto B is “not substantial” player B prefers “no payment” because B does not grant thelarger share to A – or in our terminology here regards the distribution of paymentsunfair. Assuming that A knows the attitude of B, he should offer a fair share in order

21

H. Stadtler

not to lose all the money. In Western culture this results in player A offering a shareof 66.7% on average (Güth 1995, p. 331). Interestingly, experiments have shown thatthere are some cultures which regard a significantly smaller share for B to be fair(because A is regarded the leader of the game).

Our conjecture from these experiments for CP is that fairness is depending on theculture, role and decision situation a SC member is in.

A proposal considering fair shares explicitly in dyadic channel coordination by awholesale price contract (news vendor model) is presented in Cui et al. (2007). Here,the retailer requires an a priori given fraction (share) of the manufacturer’s payoff inorder to reduce his disutility from inequity. The authors provide formulas—dependingon the fraction required by the retailer—for setting a wholesale price that coordinatesthe SC.

Ertogral and Wu (2000) define several objective functions for different measuresof fairness. They analyse a vertical SC where each SC member aims at minimizinghis costs. Costs comprise setup and holding costs up to a given planning horizon. Asa benchmark the minimum cost solution is calculated for each member provided theprimary and secondary customer demands are fulfilled in time (assuming that nei-ther a lead-time offset nor lot-sizing takes place at downstream production facilities).Now, any solution generated for the SC as a whole can be evaluated for each memberwith respect to the absolute deviation of costs (an increase or decrease) compared tothe benchmark. This also allows us to calculate the average cost deviation for all themembers in the SC. Based on these measures, unfairness results if one member facesan absolute cost deviation larger than the average cost deviation.

The authors consider three objectives:

– minimizing (the sum of) unfairness across members,– minimizing the relative cost increase above the member’s-best solution (bench-

mark) and– minimizing the distance between the member with the least unfair and the most

unfair solution.

Ertogral and Wu (2000) favour the first objective. However, even for the first objec-tive their computational tests showed that fair solutions sacrifice 37.15% on averagein solution quality [compared with the minimum cost solution of the SC as a whole(central optimization)].

Our conjecture from this study is that a CP scheme which aims at achieving “fair-ness” solely by considering the members’ individual cost functions may be counter-productive with respect to SC competitiveness. It seems much more favourable tolook for a CP scheme which aims at finding the SC optimal solution first and then toallow side-payments (or discounts) such that a fair solution is reached (which shouldbe a win–win situation). In order to calculate fair side-payments the above-mentionedobjectives (criteria) may be applied a posteriori.

Gjerdum et al. (2002) propose to introduce fairness into their central model for coor-dinating two inter-organizational SC members e = 1, 2. At the start each SC memberindicates the minimum profit πmin

e to the central planner that must be reached. Thena non-linear objective function is constructed

22


∏

e

(πimprovede − πmin

e ) (1)

which allocates profits πimprovede in such a way that, ideally, each SC members will

get the same absolute additional amount.Fleischmann (1999) analyses the collaboration of two logistic service providers

e = 1, 2 which may collaborate in the distribution of goods to the same customersby, e.g. sending just one truck (horizontal collaboration). Fleischmann favours thefollowing rule to allocate the resultant cost savings ex post: a solution is neutral tocompetition if the savings S are split such that costs κe are cut by the same percentage α

α = S

κ1 + κ2(2)

Note that this allocation rule requires calculating (initial) costs κe of each memberassuming no collaboration.

Transferring the proposal of Gjerdum et al. (2002) to a cost minimization objectiveand contrasting it with the one of Fleischmann (1999) reveals that granting the sameabsolute cost savings to each SC member will not be neutral to competition providedinitial local costs are different.

While formal rules or axioms for the allocation of gains of a collaboration may beshown to be fair one should keep in mind that the perception of fairness by the partiesinvolved will often incur some subjective elements and will be situation dependent(e.g. on the distribution of power among parties).

7 Summary and outlook

We have presented a framework for collaborative planning (CP) which allows to con-trast and cluster various contributions in this relatively new research area. A selectednumber of papers dealing with operational, model-based CP has been analysed accord-ingly.

These ten papers primarily address the following characteristics of collaborativeplanning (see the mind-map type Fig. 2. Here, the underlined characteristics indicatethat at least five of the ten CP approaches analysed possess this characteristic): a twotier SC with one SC member on each tier is addressed. Production decisions haveto be aligned with the aim of obtaining a near optimal SC solution. Mainly lot siz-ing decisions are addressed—both static and dynamic—for the operational planningphase. Orders—including order menus and sequences of orders—are exchanged in adeterministic environment. The CP schemes do without a mediator and do not need a(specific) initial solution. The number of rounds ranges from 1 to a very large numberwith only one plan exchanged per party per round. Computational tests provide someinsights into the quality of solutions.

Unfortunately, a widely applicable or even generic CP scheme for more complexSCs is still missing (with Fink 2004 coming very close to this challenge). Also, a holis-tic view or concept covering several phases of collaboration has still to be elaborated.

23

H. Stadtler

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The performance of a CP scheme can be evaluated in two different ways. One wayis a comparison with an optimal central solution. Some might argue that decentralplanning in practice may even result in better solutions than a central one because acentral unit will not dispose of the level of detail and timeliness of information of adecentral decision unit. Also, the experience how to handle soft data (like preferencesfor job assignments) may be larger in decentral decision units. We will not contradictthese arguments. However, in a laboratory experiment or in theory we can assume thatthere is an ideal central decision unit which possesses all the decentralized knowledge.If there is evidence that a CP scheme performs well in these laboratory experimentsthen we expect favourable results also in real-world applications. In the absence of acentral solution a CP scheme will be evaluated according to the improvements overthe initial solution versus the additional efforts incurred.

Research contributions in CP are limited to assumptions typical in the operationsmanagement literature, like people are predictable in their actions, emotionless andobservable (Boudreau et al. 2003). In other words, organizational behaviour, communi-cation or cultural aspects are mostly omitted in designing CP schemes. Incorporatingresults from game theory (like in Corbett and de Groote 2000) is one step in thisdirection but should be extended in future [the reader is referred to Akkermans et al.(2004), Bendeloy et al. (2006) and Marble and Lu (2007) for valuable insights andrecommendations in this area].

Once adopted by industry CP schemes could be the object of empirical research[like empirical research on Advanced Planning Systems, see Buxmann et al. (2004)

24


and Roussel et al. (2002)]. But so far CP is more a research area than a ready-to-useproduct. Future research in CP should come up with CP schemes that

• work well for more than two tiers,• work well in a great number of decision situations (problems) even in a mix across

a SC,• address, support or even secure a fair distribution of the gains among SC members

and• allow renegotiations of already accepted plans in rolling schedules.

Still, there is a further important obstacle to overcome: while companies have realizedthat a SC perspective is necessary for improving competitiveness and that isolatedplanning domains lead to local optima they are reluctant to share information and toconsider compensations to SC members required to reach a win–win situation as aresult of CP.

Appendix

Table 5

Table 5 Criteria for discriminating SC structures, relationships and decision situations

Criteria for SC structures and Barbarosoglu (2000) Gjerdrum et al. (2002) Shirodkar and Kempf (2006)relationships anddecision situation of eachSC member

1.1 Structural elements

1.1.1 No. of tiers 2 2 2

1.1.2 No. of members on tiers 1− n0 1–1 n1 − 1

1.1.3 Business functions production production production

1.2. Relationships

1.2.1 Power S ? B

1.2.2 Behaviour S Opportunistic team B Opportunistic

1.2.3 Learning effects, – – –

1.2.4 Rolling schedules – – –

1.3. Required solution S opt. (fair) SC fair B opt.

2.1. Decision models CLSP/ LP MIP MIP

2.2 Phases of collaboration conditions oper. oper.

2.3. Information status

2.3.1 Information hidden all, except: none none

2.3.2 Information exchanged orders expected from B prices model of S

2.3.3 Degree of uncertainty stochastic deterministic deterministic

2.4. Objective(s) min. costs min. costs min. costs+ viol.

B buyer, ? neither B nor S dominate, MIP individual mixed integer programming model, – no, ni numberof SC members on tier i , oper. operational planning phase, opt. optimal solution, S supplier, viol. violationof soft constraints

25

H. Stadtler

Acknowledgements The author is indebted to Martin Albrecht and Carolin Püttmann for their valuablecontributions to this paper.

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28

Simulation and optimization of supply chains:alternative or complementary approaches?

Christian Almeder · Margaretha Preusser ·Richard F. Hartl

Abstract Discrete-event simulation and (mixed-integer) linear programming arewidely used for supply chain planning. We present a general framework to supportthe operational decisions for supply chain networks using a combination of an optimi-zation model and discrete-event simulation. The simulation model includes nonlinearand stochastic elements, whereas the optimization model represents a simplified ver-sion. Based on initial simulation runs cost parameters, production, and transportationtimes are estimated for the optimization model. The solution of the optimization modelis translated into decision rules for the discrete-event simulation. This procedure isapplied iteratively until the difference between subsequent solutions is small enough.This method is applied successfully to several test examples and is shown to delivercompetitive results much faster compared to conventional mixed-integer models ina stochastic environment. It provides the possibility to model and solve more real-istic problems (incorporating dynamism and uncertainty) in an acceptable way. Thelimitations of this approach are given as well.

Keywords Supply chain management · Optimization · Discrete-event simulation ·Hybrid method

C. Almeder (B) ·M. Preusser · R. F. HartlUniversity of Vienna, Brünnerstr 72, 1210 Vienna, Austriae-mail: [email protected]

M. Preussere-mail: [email protected]

R. F. Hartle-mail: [email protected]

Originally published in:


29

DOI 10.1007/s00291-007-0118-zOR Spectrum (2009) 31:95–119


C. Almeder et al.

1 Introduction

In recent years intra-company supply chains have been growing significantly spanningproduction and distribution sites all over the world. At the same time global competi-tion has increased, such that there is a strong demand for new decision support toolson strategic, tactical and operational levels. Biswas and Narahari (2004) classified therelevant research on such decision support systems into three categories:

(a) Optimization models mainly for multi-echelon inventory control. In most casesthese models are deterministic and used for strategic or tactical decisions.

(b) Analytical performance models, which consider a dynamic and stochastic envi-ronment. They are used to investigate design or principal management decisions.Such systems are represented as Markov chains, Petri nets or queuing models.

(c) Simulation and information models, which are used to analyze complex dynamicand stochastic situations and to understand issues of supply chain decision mak-ing.

For the first and the second categories, it is often necessary to make several simpli-fications from the real-world case in order to develop solvable models. Neverthelessthe problem size is usually very limited. Although there are promising developmentsof combinations of these two categories (cf. Sect. 2), many of them remain on a stra-tegic level and the stochastic property is considered by a small number of differentscenarios.

In this paper we develop a new solution approach by applying a LP/MIP formu-lation in the context of a discrete-event simulation. So we are able to combine theadvantages of models from all categories mentioned above by considering a detailedrepresentation of a dynamic and stochastic environment and allow the application ofoptimization methods in this context. Our investigations are based on a general supplychain network model with different facilities (suppliers, manufacturers, distributors)and different transportation modes connecting these facilities. We assume that there isa central planner with perfect information such as for intra-company supply chains orsupply chains with a dominant member. This problem setting is motivated by a casestudy about a global supply chain network in the paper industry (Gronalt et al. 2007).The goal is to reduce costs by simultaneously optimizing the production/transportationschedule and reducing inventory levels. We are aiming for a robust solution, in thesense that a stochastic environment is considered. Comparing our problem to the tasksin the supply chain matrix (cf. Stadtler 2005), the problem is a combination of severaloperational tasks: production planning, distribution planning and transport planning.In addition to other approaches, we assume a stochastic simulation model for thesetasks and combine it with classical optimization approaches.

Our goal is to achieve an optimal operation plan for supply chain networks by com-bining optimization models and simulation models. We do not use the optimization ontop of the simulation, where an optimization algorithm uses the simulation model as ablack-box evaluation function (cf. Glover et al. 1999). Instead we include simulationand optimization in an iterative process in order to gain the advantages of optimiza-tion (exact solution) and simulation (nonlinearities, complex structure, stochasticity).In the previous research (Almeder and Preusser 2004; Preusser et al. 2005a,b) we

30

Simulation and optimization of supply chains

Solutions of simulation

experiments

Optimization

model

Decision rules in

D-E model

aggregate

Linear solutionoptimize

simulate

derive

Fig. 1 Interaction between simulation and optimization

developed a rough idea of this concept. In the current paper we extended this con-cept, such that it is possible to apply it to a wide range of supply chain problems.Furthermore we analyze in detail the advantages and disadvantages of this approachand present results for different test cases.

The supply chain is represented as a discrete-event model (D-E model) and a sim-plified version is modeled as an optimization model. We start by performing severalsimulation runs in order to get average values of the parameters (e.g., unit transportationcosts) which are then fed into the optimization model. After solving the optimizationmodel the result is transformed into decision rules that are used in the discrete-eventmodel. Then we start again with further simulation experiments (see Fig. 1), and soon.

Our contribution is twofold:

• Development and analysis of a general framework (Fig. 1) and a toolbox for thecombination of discrete-event simulation and optimization of supply chains.

• For stochastic supply chains, an iterative combination of simulation and linearprogramming is empirically shown to be competitive compared to deterministicMIP-models.

The paper is organized as follows: We start with a literature review in Sect. 2, fol-lowed by a description of the general model framework in Sect. 3. In Sect. 4 we explainthe linkage between the simulation model and its linear version. Finally, we report ondifferent test results in Sect. 5 and give conclusions and an outlook for possible futureresearch in Sect. 6.

2 Literature review

2.1 Supply chains

Aspects of the integration of transport and production planning within supply chainshave been investigated in several papers (cf. Erengüc et al. 1999). Combined plan-ning approaches for different decision levels (e.g., tactical and operational decisions)can be found in Meyr (2002) and Schneeweiss (2003). There are numerous papersdealing with linear or mixed-integer programs for supply chain networks and networkflows. Yaged (1971) discussed in his paper a static network model which includes

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C. Almeder et al.

nonlinearities. He tries to optimize the product flow by solving a linearized version ofthe network and to improve the flow in the network. Paraschis (1989) discussed severaldifferent possibilities to linearize such networks and Fleischmann (1993) presentedseveral applications of network flow models, which are solved through linearization.Pankaj and Fisher (1994) showed that based on an MIP model the coordination ofproduction and distribution can reduce the operating cost substantially. Dogan andGoetschalckx (1999) showed that larger supply chain design problems can be solvedusing decomposition. A recent case study about a supply chain of the pulp indus-try modeled as a MIP is given by Gunnarsson et al. (2007). In general the problemssolved with LPs and MIPs usually include several simplifications in order to keepthem solvable.

Recent publications also included stochastic elements in the optimization models.Santoso et al. (2005) considered a stochastic programming approach for the sup-ply chain network design. They used a sample average approximation and Bendersdecomposition to solve design problems for a supply chain while considering futureoperational costs. For that purpose they developed a linear model with uncertain costfactors and demand. Although they used a fast algorithm, realistic problems withsample sizes of up to 60 scenarios need several hours to be solved. Alonso-Ayuso etal. (2003) considered a similar combined design and operation problem. Their sto-chastic programming approach was able to solve medium sized problems with about100 binary decisions within 1 h. Leung et al. (2007) presented a robust optimizationmodel for a simultaneous production planning for several sites in a supply chain underuncertainty. But still they are restricted to rather small models and consider only fourdifferent scenarios.

In the field of supply chain simulation Kleijnen (2005) gave a short overview ofsimulation tools and techniques used for supply chains. He distinguished betweenfour different approaches: spreadsheet simulation, system dynamics, discrete-eventdynamic systems simulation, and business games. Clearly, discrete-event simulationis the most powerful tool to consider complex stochastic systems. Numerous softwarepackages for discrete-event simulation are available, both very specialized ones for aspecific part of the supply chain and general ones with a high functionality in modelingand visualization of supply chains (cf. Kelton et al. 2002; Kuhn and Rabe 1998). Oneexample is the Supply Net Simulator presented by Stäblein et al. (2007). It allowssimulating the behavior of individual members in a supply chain network. They usedan agent-based approach, where each member optimizes its own operations in thesense of an advanced planning system. But there is no interaction between simulationand optimization.

2.2 Optimization and simulation

Most of today’s simulators include possibilities to do a black-box parameter optimiza-tion of a simulation model. Glover et al. (1999) presented the successful developmentof OptQuest (© OptTek Systems, Inc.1), an optimization toolbox containing different

1 http://www.opttek.com.

32


algorithms (mainly metaheuristics) designed to optimize configuration decisions insimulation models. The simulation model is used only for the evaluation of the objec-tive value, no further structural information is used. Swisher et al. (2000) and Fu (2002)stated in their papers that there is still a big gap between optimization methods forsimulation-based optimization used in commercial software and methods available inresearch literature.

Truong and Azadivar (2003) developed an environment for solving supply chaindesign problems, where they combine simulation with genetic algorithms and mixed-integer programs. Strategic decisions regarding facility location and partner selectionare considered.

The work by Lee and Kim (2002), possibly the most related work in this context,shows a combination of simulation and optimization for the case of a production-distribution system. They use simulation to check the capacity assumptions used for asimpler linear model in a more realistic environment with stochastic machine break-downs and to update these capacity parameters for the optimization. After severaliterations they end up with a solution of the optimization model which is also withinthe constraints of the stochastic simulation model. Their method is quite similar toour approach, but they aim for more realistic capacity estimation for the optimizationmodel. In contrast, we try to find a robust plan for production, stocking, and transpor-tation considering stochastic and nonlinear operations and costs by estimating delaysand cost factors based on simulation experiments.

3 The supply chain network model

The general description of the supply chain originates from a case study about a supplychain in the paper industry (cf. Gronalt et al. 2007). Several production sites are usedto manufacture different paper products, which are delivered either directly or viahubs to customers all over the world. The main task in this case study was to developa 1-year plan for production quantities and transportation links. In this case study astatic model was developed, which was used to get rough estimates quickly.

Inspired by this case study we formulate the following problem setting. The basisfor our supply chain model is a predefined network, i.e., the locations of all actors andthe connections between them are given. Within the network we differentiate betweenthree types of participants connected by transportation links:

• suppliers providing raw materials;• customers who demand certain products at a specific time;• production/warehouse sites where production, stocking, and transshipment takes

place.

The whole supply chain is order-driven, that means products are manufactured ortransported only if a subsequent member of the supply chain requests it. So the originfor all activities is the predefined deterministic demand of the customers. All activitiesare based on time periods, which might be days or shorter time periods.

The suppliers are used as source for raw materials, which are sent to production sitesif requested. Production/warehouse sites can store incoming products. These products

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C. Almeder et al.

can be used to manufacture new products, or they are simply transferred to the outputinventory. From there they are sent to subsequent members of the supply chain.

The simulation model is implemented using AnyLogic (© XJ Technologies), aJava-based simulation tool. The model is constructed as a library including severaldifferent modules. These modules represent the four types of participants in the supplychain network plus a general control module necessary for controlling the simulationexperiments as well as the communication with the optimization model which wasdeveloped using Xpress-MP (© Dash Optimization). This model is a simplified deter-ministic version of the library modules of the simulation model. In this section we willexplain the different modules of our supply chain network.

3.1 Module supplier

Simulation model. This module is used to generate certain products, store them, anddeliver them if demanded. It has one input port to receive orders for products andone output port to deliver products. If this module receives an order through the inputport, then it sends the requested amount of products via the output port. If the amountexceeds the current inventory level, only the available amount is sent. As soon as newproducts arrive in the inventory they are delivered until the whole order has been ful-filled. The costs arising in this module are only inventory costs for storing productsprior to delivery. These costs may have any user-defined functional form. Accordingto the given parameters in each period, new products are generated and added to thestock.

Optimization model. We also developed a simplified representation as an optimi-zation model. We denote by JS the set of suppliers within a network, by P the setof products, and by T the number of periods. The representation of the supplier’sbehavior in the optimization model can be formulated as follows (If p and t are freeindices, i.e. not used as a summation index, then the set of equations is meant to bevalid ∀t = 1, . . . , T, p ∈ P .):

TCSi =

∑

p∈P

T∑

t=1

out H pi

(outl pi (t)

) ∀i ∈ JS, (1)

outl pi (t) = outl p

i (t − 1) − out f pi (t)+ S p

i (t) ∀i ∈ JS, (2)outl p

i (t) ≥ 0 ∀i ∈ JS . (3)

For a complete list of parameters and variables, see Appendix C. The overall costof supplier i is denoted by TCS

i , consisting only of the holding cost out H pi (·) of the

output inventory, expressed by the right-hand side of (1). Equations (2) are the inven-tory balance equations for the output inventories outl p

i (t). The stock is diminished bythe outflow of materials, out f p

i (t), and increased by the given supply S pi (t). The last

set of constraints (3) guarantees that the inventory level cannot be negative.The simulation and the optimization model are connected via the holding costs in

(1) which represent the user-defined cost function in the simulation.

34


3.2 Module production

Simulation model. This module is the core of the whole model. It represents aproduction site as well as a transshipment point. It consists of an input and an out-put storage. Items are either transformed into new items or simply transferred fromthe input to the output storage. This module has one input port and one output portfor orders, as well as one input and one output port for products. The input storageis replenished by ordering products via the output port for orders from a supplieror another production module. The ordering policy may be either autonomous (e.g.,an (s,S)-policy or any user-defined policy) or it is determined by the result of theoptimization model. Products are received through the product input port and storedin the input inventory. The production of new products or the transfer of productsis initiated by an internal order placed by the output inventory (either autonomousor based on the solution of the optimization model). The delay for production andtransfer is a user-defined function. It may contain stochastic elements and dependon other parameters (e.g., the current load). Production and transfer have limitedcapacities and furthermore production is restricted to the availability of raw materials(other products). If these capacities do not allow producing (or transferring) a lot as awhole, it is split into several batches. Through the input order port the modulereceives orders from other production or customer modules. Products are sent throughthe output product port according to these orders and based on availability. Costs arisein this model for inventory holding (input and output), for production, and for transfer.

Optimization model. The optimization model for the production node is as follows(we denote by JI the set of production nodes in the supply chain network):

TCIi =

∑

p∈P

T∑

t=1

W pi

(m p

i (t))+

∑

p∈P

T∑

t=1

Z pi

(u p

i (t))

+∑

p∈P

T∑

t=1

in H pi

(inl p

i (t))+

∑

p∈P

T∑

t=1

out H pi

(outl pi (t)

) ∀i ∈ JI , (4)

m pi (t) ≤ prodCapp

i (t),∑

p∈P

a pi · m p

i (t) ≤ prodCi (t) ∀i ∈ JI , (5)

u pi (t) ≤ taCapp

i (t),∑

p∈P

d pi · u p

i (t) ≤ taCi (t) ∀i ∈ JI , (6)

inl pi (t) = inl p

i (t − 1) + in f pi (t)

−∑

p′∈P

αpi (p′) · m p′

i (t)− u pi (t)+ r p

i (t) ∀i ∈ JI , (7)

outl pi (t) = outl p

i (t − 1) − out f pi (t)+ χt≥δ

pi· m p

i (t − δpi )

+χt≥σp

i· u p

i (t − σp

i )+ s pi (t) ∀i ∈ JI , (8)

inl pi (t) ≤ invinCapp

i (t),∑

p∈P

q pi · inl p

i (t) ≤ in Li (t) ∀i ∈ JI , (9)

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C. Almeder et al.

outl pi (t) ≤ invoutCapp

i (t),∑

p∈P

q pi · outl p

i (t) ≤ out Li (t) ∀i ∈ JI , (10)

m pi (t) ≥ 0, u p

i (t) ≥ 0, inl pi (t) ≥ 0, outl p

i (t) ≥ 0 ∀i ∈ JI . (11)

The overall cost of a production node i is represented by TCIi . These costs

consist of production costs (the production amounts are denoted by m pi (t)) trans-

fer costs (the transfer amounts are denoted by u pi (t)) and the holding costs of the

input and the output inventory. Constraints (5) and (6) restrict the production and thetransfer for each product individually, as well as for the total production and transfer.In the latter case the amounts are multiplied by the resource requirements. The dis-tinction between individual capacity constraints for each product and global capacityconstraints are necessary to cover general situations where different resources as wellas common resources are necessary for production. Equations (7) are the inventorybalance equations for the input inventories. The current inventory level is determinedby the inventory level of the previous period, the inflow from other nodes, the requiredraw materials for production, the transfer amount and some external inflow (from out-side of the system); α

pi (p′) represents the units of raw material p which is necessary

to produce one unit of product p′. The inventory balance equations for the outputinventories (8) are similar, but the production and transfer delays (δ p

i , σp

i ) have to beconsidered before a new product arrives in the output inventory. Function χt≥ε is anindicator function, used in order to avoid the use of production and transfer amountsfor negative periods. Equations (9) and (10) are used to restrict the stock of the inputand the output inventory (for each product separately and accumulated using the spacerequirements q p

i ). These two types of restrictions allow modeling a dedicated-storageas well as a random-storage policy.

The simulation model and the optimization model are connected through the costfactors in Eqs. (4) and production and transfer delays (δ p

i , σp

i ), which are user-definedfunctions in the simulation model possibly containing stochastic and nonlinearelements. Furthermore the production and transfer amounts (m p

i (t), u pi (t)) of the

optimization model are used to determine production plans in the simulation model.For example, if m p

i (t) > 0 for a specific product and period, then in the simulationmodel the amount given by m p

i (t) is ordered in period t .

3.3 Module customer

Simulation model. According to a given demand table, the customer places orders atthe production sites. Due to stochastic features within the simulation, it is not possibleto time deliveries exactly. Therefore the customer has an input inventory, which isused to satisfy the demand. The inventory level can be negative (shortages), as well aspositive (oversupply). In both cases penalty costs occur, which are higher for shortages.The module has one output port for sending requests and one input port for receiv-ing products. The orders are sent either according to the demand table (including astandard delay time for transportation) or according to the solution of the optimizationmodel.

36


Optimization model. The optimization model for the customers’ behavior can bewritten as follows (we denote by JC the set of customer nodes in the supply chainnetwork):

TCCi =

∑

p∈P

T∑

t=1

R pi (inbp

i (t)) ∀i ∈ JC , (12)

inl pi (t)− inbp

i (t) = inl pi (t − 1)− inbp

i (t − 1)+ in f pi (t)

−D pi (t)+ r p

i (t) ∀i ∈ JC , (13)inl p

i (t) = 0 ∀i ∈ JC . (14)

In (12) we calculate the cost at the supplier which consists only of penalty costfor back orders inbp

i (t). Equations (13) are the inventory balance equations where thecustomers’ demands D p

i (t) are considered. It is assumed that all customers are just-in-time customers. Therefore, constraints (14) ensure that no oversupply (positivestock level) is possible, i.e. it is not allowed to send more products than demandedby the customers. This JIT assumption may be dropped and holding costs for positivestock may be included. In the simulation model the JIT assumption is weakened,because stochastic transportation times may cause an unwanted early delivery. Theseearly deliveries are penalized.

The differences between the simulation model and its representation as an optimi-zation model are the penalty cost factors in (12) and the JIT assumption expressed in(14).

3.4 Module transport

Simulation model. This module is used to transport products between different mod-ules. It receives products through its single input port and sends it (according to sometime delay) through the output port to the next module (Production or Customer). Ithas a limited capacity and organizes the transports according to a FIFO rule. It is alsopossible to split shipments if the available capacity does not allow single shipment.The time delay may be stochastic and may depend on other parameters. User-definedcosts arise for transportation and may include transportation time, amounts, and fixedcharge parts.

Optimization model. The representation of the transport modules as an optimizationmodel can be formulated as follows. Each transport module is identified by the indi-ces of the nodes, which it connects. Furthermore, we need an additional index v ∈ Vdenoting the different transport modes (if v is not used as a summation index, theequations are valid for all v ∈ V ):

TCTi j =

T∑

t=1

∑

v∈V

∑

p∈P

vC pi j

(vx p

i j (t))∀i ∈ JS ∪ JI , j ∈ JI ∪ JC , (15)

37

C. Almeder et al.

vx pi j (t) ≤ vCapp

i j (t),∑

p∈P

vg p · vx pi j (t) ≤ vCi j (t) ∀i ∈ JS ∪ JI , j ∈ JI ∪ JC ,

(16)in f p

j (t) =∑

i∈Js∪JIvτi j <t

∑

v∈V

vx pi j (t − vτi j ) ∀ j ∈ JI ∪ JC , (17)

out f pi (t) =

∑

j∈JI∪JC

∑

v∈V

vx pi j (t) ∀i ∈ JS ∪ JI , (18)

vx pi j (t) ≥ 0 ∀i ∈ JS ∪ JI, j ∈ JI ∪ JC . (19)

The total transportation cost for transports from member i to member j of thesupply chain network is denoted by TCT

i j and vx pi j (t) gives the transportation amount

for each period, each product, and each transportation mode. Constraints (16) limitthe transportation to a product-specific and an overall capacity limit. Equations (17)and (18) represent the inflow of products to member j and the outflow of productsfrom member i .

The connection between the simulation and the optimization model is establishedby the transportation cost functions in (15) and the transportation delays vτi j in (17) onthe one hand, and through the transportation amounts vx p

i j (t) on the other hand. Thesetransportation amounts are used to define ordering schemes for the Production andCustomer modules in the simulation model. That means if vx p

i j (t) > 0, then member

j of the supply chain sends a request for vx pi j (t) units of product p to member i of the

supply chain at time t .

3.5 Supply chain optimization model

The optimization model of the whole supply chain network is defined by minimizingthe total cost

min∑

i∈JS

TCSi +

∑

i∈JI

TCIi +

∑

i∈JC

TCCi +

∑

i∈JS∪JI

∑

j∈JI∪JC

TCTi j (20)

subject to the constraints (1)–(19). If we assume that all cost functions are linear, i.e.that the objective (20) is a linear function, we can write it as follows:

min∑

i j∈J

∑

p∈P

∑

t=1,..T

∑

v∈V

vcpi j · vx p

i j (t)+∑

i∈JI

∑

p∈P

∑

t=1,..T

wpi · m p

i (t)

+∑

i∈JI

∑

p∈P

∑

t=1,..T

z pi · u p

i (t)+∑

i∈JI

∑

p∈P

∑

t=1,..T

inh pi · inl p

i (t)

+∑

i∈JS∪JI

∑

p∈P

∑

t=1,..T

outh pi · outl p

i (t)+∑

i∈JC

∑

p∈P

∑

t=1,..T

ρpi · inbp

i (t). (21)

38


supplier customer production

transport_1 transport_2

param (LP) param (LP) param (LP) piw , ,

, out ,

,DV (Sim)

, u

piz

i

(t)

pi

inhp

i

m pi

pih

p

(t)pi

pi

outh pi

in

param (LP) param (LP) pij

vc ,

DV (Sim) ij

v

(t)x pij

v

pij

vc ,

DV (Sim) ij

v

(t)x

sdtt

r

pij

v

Fig. 2 An example of the module configuration for a simple supply chain consisting of one supplier, oneproduction site, and one customer (dashed lines indicate information flow and solid lines indicate materialflow). For the case of a linear program the information below the modules represents the parameters whichare calculated during the simulation runs and transferred to the LP, param (LP), and the decision variablesof the LP used as decision rules in the simulation model, DV (Sim.)

Hence, we get a linear programming model which we will use in connection withthe simulation model as depicted in Figure 1. A detailed description of the linear modelcan be found in Preusser et al. (2005a,b).

This model is a pure linear program which can be solved easily with any standardLP-solver within very short time. If necessary, it is possible to extend the model formu-lation to consider more features, e.g., fixed-cost transportation, binary decisions, stepfunctions, etc. These extensions lead to a mixed-integer formulation, thus increasingcomputational time (cf. Appendix A).

3.6 Supply chain simulation model

Implementing a simulation model in AnyLogic means to arrange the according mod-ules and connect them. In Fig. 2 an illustrative example shows, how these modulescan be connected in order to maintain information flow (direct connections bet-ween Supplier, Production, and Customer) and material flow (via Transport modules).Furthermore the cost and delay functions for each module must be specified.

4 Connecting the optimization with the simulation

In order to couple the optimization model and the simulation model, we first have todefine the required data and the way they should be exchanged. We decided to use anMS Access database to store all necessary information which is:

• General network structure: This includes the number of actors in the supply chainand the according links between them.

• General parameters used in the simulation and optimization models: These sets ofparameters include all capacity limitations, resource parameters, bill-of-materials,predefined supply at the suppliers, and predefined demand at the customers.

39

C. Almeder et al.

decision rules (ordering plans, production schedules,…)

generalparameters

generalparameters MS Access

database

(ODBC)

simulation

model

(AnyLogic)

optimization

model

(Xpress)

aggregated results (transportation delays, production delays,…)

Fig. 3 This scheme shows the data exchange between the simulation and the optimization model via theMS Access database in the middle

• Results of the optimization model (= parameters for the simulation): The resultsof the optimization used as decision rules in the simulation are production andtransfer quantities, as well as transportation amounts (m p

i (t), u pi (t), vx p

i j (t)).• Results of the simulation model (= parameters for the optimization model): Themain results of the simulation experiments used in the optimization model are thecost parameters and the delays for production, transfer and transport.

The simulation model is designed as the master process, which controls the datacommunication and the LP/MIP-solver. The simulation model and the optimizationmodel retrieve and store values from and to the database using the Open DatabaseConnectivity (ODBC) interface (see Fig. 3).

To initiate the optimization process in our system, a few simulation runs are per-formed using the data from the database. Missing decision rules which, in later iter-ations, are generated using the results of the optimization model, are substituted byautonomous decision rules (like an (s,S)-policy for the replenishment). These firstsimulation runs are only necessary to generate initial parameter values for the opti-mization model, but their results will be ignored in further iterations in order to avoidbiasing effects caused by the autonomous decision rules. The results of the initial runs(delays, per unit costs, etc.) are aggregated and according means and variances arestored in the database (see Sect. 4.1). Afterwards Xpress-MP is executed. It loads thegeneral data and the simulation results from the database, computes the solution ofthe optimization model and stores the results (ordering and delivery plans, productionand transfer schedules, etc.) in the database. Then we start again with five simulationexperiments using now the newly computed decision rules (see Sect. 4.2), based onthe solution of the optimization model. Further on we will denote this algorithm bySimOpt (or SimLP, if a pure LP model is used for the optimization part, and SimMIP,if a mixed-integer formulation is used). In Table 1 a pseudo code of this SimOptalgorithm is given.

4.1 Aggregating simulation results

Since the simulation model may contain stochastic and nonlinear elements, it isnecessary to perform several simulation runs and combine the results. For the costparameters, necessary for a linear optimization model, we calculate average per unitcost. That means, e.g., for the production costs we accumulate the total cost for the

40


Table 1 Pseudo code for the combined simulation optimization approach SimOpt

SimOpt:

Load necessary simulation parameters from the database

Perform a few simulation runs using autonomous decision rules

Aggregate results and store them in the database

while stopping criteria are not met

Load aggregated parameters into LP/MIP-Solver

Solve the optimization model

Write new decision rules to the database

Load new decision rules into simulation model

Perform simulation runs using these decision rules

Aggregate results and store them in the database

end-while

whole planning horizon for a certain product and divide these costs by the number ofproducts produced. Other parameters, called critical parameters, have a direct influ-ence on the material flow (e.g., transportation delay). The use of average values forthose parameters would most probably lead to bad results. In about half of the cases thedelay would be longer than assumed and would cause additional delays in subsequentoperations. Therefore, it seems reasonable to use, e.g., a 90%-quantile (based on anormal distribution with estimated mean and variance calculated from the differentsimulation runs) for such delay parameters. This results in an overestimation of thedelays for the optimization model, because the time is determined such that 90% ofthe observed delays will be shorter, but it ensures that a smooth material flow throughthe network is possible.

For the critical parameters it is useful to combine results from the previous itera-tions with current ones, in order to enlarge the sample size and to get better estimatesof the mean and the variance.

4.2 Decision rules based on the solution of the optimization model

There are several possible ways to use the solution of the LP-model within the simula-tion model. One method, which we apply here, is to use the transportation, productionand transfer results (vx p

i j (t), m pi (t), u p

i (t)) as a given ordering plan. According to the

transportation values vx pi j (t) module j sends a request to module i at time t for the

given amount vx pi j (t) of products of type p using the transportation mode v. Simi-

larly, production and transfer results can be used. In some cases, due to the stochasticfeatures of the simulation model, it may happen that some of the modules are out-of-stock for a specific product. Since unfulfilled orders are backlogged, these requestsare fulfilled as soon as the products are available.

More complex procedures would be, e.g., to use results of the sensitivity analysis(dual variables, reduced costs) to determine the critical parameters, to observe theseparameters during the simulation runs, and to adapt decision rules (use a different

41

C. Almeder et al.

10000

60000

110000

160000

210000

260000

310000

miS-

1

tp

O-1

miS-

2

tp

O-2

miS-

3

tp

O-3

miS-

4

tp

O-4

miS-

5

tp

O-5

miS-

6

tp

O-6

iteration

tota

l co

st

simulation

optimization

Fig. 4 Objective values of the optimization model and the simulation model for each iteration for a deter-ministic model considering fixed costs for production, transfer and transport

solution of the optimization model) if the observed parameters reach a certain thresh-old.

For our examples we use the first approach for translating the solution of the opti-mization model into decision rules for the simulation model (cf. Sect. 3). In this paperwe wish to investigate the direct interactions between the solution of the optimizationmodel and the simulation results. The analysis of more complex decision rules goesbeyond the scope of this paper and might by a subject for further research.

5 Tests and results

We wish to investigate the following research questions with empirical tests using aset of test instances:

• Does this method converge in practice for realistic test cases?• If we can observe convergence, is the result optimal or at least a good approxima-

tion?• Is this method advantageous compared with traditional planning methods?

Although it is not possible to prove general convergence for all our test instances,we observe fast convergence of the objective values of the simulation and optimizationmodel. Figure 4 shows a typical situation using the results of the deterministic testinstance D1-L described in Sect. 5.1.

We start with the simulation model using an autonomous rule for replenishing theinventories. Since we start with all inventories empty, it takes a long time, until theorders are fulfilled. This causes high costs and an overestimation of transportationand production delays. Therefore, the first solution of the linear model leads also to ahigh objective value mainly consisting of penalty costs for late (or even no) deliveries.Consequently, the simulation model leads to a similar objective function in iteration2, because it uses the delivery plans of the solution of the linear model. Due to the factthat the solution of the linear model causes a somehow synchronized material flow,

42


the measured delays are much lower now. Therefore, the cost of the solution of thelinear model in the second iteration decreases. After three iterations the simulationand the linear model have converged to the same solution.

5.1 Deterministic problems with fixed costs

In order to verify the quality of the solutions, we create a set of 12 examples. For thesetest instances we consider a simple supply chain consisting of three actors (a supplier,a producer and a customer) and a time horizon of 30 periods. For transportation ofproducts two transport modules are used, which connect the supplier and the produceras well as the producer and the customer. Two types of products are demanded by thecustomer: product 1 which is provided by the supplier and sent via the producer to thecustomer and product 2 which is manufactured by the producer using product 1 as araw material. The cost structure is as follows:2

• The transportation costs consist of fixed costs per delivery, which are subject to astep function. A transport costs 100, 200 or 300 monetary units, depending on theamount delivered.

• The costs of production and transfer are separated into variable costs and fixedcosts. The variable production costs are set to 30 and the fixed part is 50 monetaryunits. Transferring products costs 15 per product unit plus a fixed part of 10.

• Delayed deliveries are penalized by 100 monetary units per product unit and period.

Concerning the demand at the customer we distinguish between instances with highdemand and others with low demand. The difference lies in the frequency of orderssent off by the customer. In high demand cases the occurring orders in each periodare around the maximum possible quantities which could be delivered considering thecapacities of the supplier and the producer. In low demand models the ordered amountscover approximately 70% of the possible deliveries in each period. Instances D1-Lto D5-L (see Table 2) represent five different realizations for low demand models.Accordingly, D6-H to D10-H correspond to five different realizations of high demandmodels. The last two instances, D1a-L and D6a-H, are modifications of instances D1-Land D6-H, respectively. The former ones consider exactly the same ordering amountsas D1-L and D6-H but the fixed costs for production and transfer at the intermediatenode are increased to 1000 and 500, respectively.

For examples of this size it is possible to formulate an exact mixed-integer modeland determine the optimal solution. The corresponding MIP formulation consists of1,342 constraints, 1,080 continuous and 300 binary decision variables. For the simu-lation approach the nonlinear parts are only considered in the simulation model itself,the connected linear (non-integer) model does not include any of them. See Table 2for the resulting total costs of the simulation and the optimal solution (MIP).

The gap between our SimLP approach the optimal solutions gained by solving theexact MIP formulation varies between 0.44 and 3.46% and averages in 1.87%. Aswe would expect, for the two test instances with high fixed cost the gap increases.

2 All datasets are available at http://www.univie.ac.at/bwl/prod/download/SCM-Data.

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Table 2 Comparison of totalcosts between oursimulation-based optimizationapproach SimLP and the exactMIP-model for deterministictest cases classified by theoccurrence of customer demand(H—high demand, L—lowdemand)

MIP solutions marked with (*)are best solutions found afterone hour calculation time

Instance SimLP Exact MIP Difference (%)

D1-L 53640 52947 1.31

D2-L 55032 53860 2.18

D3-L 52626 52394 0.44

D4-L 54442 53600 1.57

D5-L 55198 54057 2.11

D6-H 59885 58830 1.79

D7-H 61257 60129 1.88

D8-H 59028 58347 1.17

D9-H 60403 59501 1.52

D10-H 61436 60365 1.77

D1a-L 63720 61587 3.46

D6a-H 76165 73761* 3.26

Average 77165 73760 1.87

For test instances with low demand the variation of the gap seems higher than for thetest instances with high demand. But on average there seems no significant differencebetween the results of those two groups of instances.

Based on the above results we may conclude that the error caused by neglectingfixed cost is low as long as the fixed costs are low compared with other costs. If thefixed costs increase (relative to the other costs), the nonlinear properties should beconsidered in the optimization model used for the SimOpt approach, i.e. a SimMIPshould be used instead of the SimLP (see also the following subsection).

5.2 Test instances with stochastic transport delays

In order to measure the quality of our solutions in a stochastic environment we preparea set of test examples including stochastic transportation times. We compare our Sim-LP approach using a simplified linear model without binary variables with an exactMIP-model. This MIP-model does not cover stochastic features and we have to provideestimated values of the transportation times. Within the simulation we consider uni-formly distributed transportation delays between 1 and 9 for transportations from thesupplier to the producer and between 1 and 5 for transportations from the producer tothe customer. For estimating the delay parameters we perform runs using 90%-, 70%-,and 50%-quantiles. The corresponding transportation delays for the MIP-model areset according to the used quantile. For the small test cases there would probably beno noticeable difference between the results of a 99%- and a 90%- quantile. There-fore, we test a high quantile (90%, risk averse), the average value (50%), and someintermediate value (70%). The maximum runtime is set to 30 minutes for the MIP-model, i.e. we report the best solution found after this time limit, while the simulationapproach converges after a few seconds. For the simulation approach we again process8 iterations, each consisting of 5 simulation runs and one LP computation. Finally thesolution of the MIP-model and the solution of the SimLP are evaluated by performing

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Table 3 Difference of the mean total costs of 20 runs between the SimLP and SimMIP method and thesolution found by a deterministic MIP model classified by the occurrence of customer demand (H—highdemand, L—low demand)

Instance MIP SimLP SimMIP

Cost Quant. (%) Cost Quant. (%) Diff. (%) Cost Quant. (%) Diff. (%)

S1-L 66400 90 62601 90 −5.72 61637 90 −7.17

S2-L 61338 90 60635 90 −1.15 60282 90 −1.72

S3-L 63323 90 63566 70 0.38 63618 70 0.47

S4-L 63122 90 64067 90 1.50 64060 90 1.49

S5-L 60954 90 62399 90 2.37 62229 90 2.09

S6-H 72485 90 72342 90 −0.20 70871 90 −2.23

S7-H 70928 90 70751 90 −0.25 71040 90 0.16

S8-H 73257 90 77537 70 5.84 74999 70 2.38

S9-H 73501 90 74637 70 1.55 72845 90 −0.89

S10-H 71606 90 70230 90 −1.92 73934 90 3.25

S1a-L 71511 90 71686 90 0.25 70350 90 −1.62

S6a-H 88442 70 90582 90 2.42 88442 70 0.00

Av.-L 64441 64159 −0.39 63696 −1.08

Av.-H 75037 76013 1.24 75355 0.44

Average 69739 70086 0.42 69526 −0.32

The total costs are reported in the columns Cost. The quantile which lead to the best results is reportedin the column Quant. The difference with respect to the solution of the deterministic MIP is denoted incolumn Diff.

20 independent simulation runs. Furthermore, we replace the simplified linear modelwith the MIP model (SimMIP approach) and perform the same tests. The results forall three methods are displayed in Table 3, where negative percentage values implythat the simulation achieves a better result than the exact MIP-model.

The results of the SimLP approach, where we combine the pure LP model withthe simulation model, are on average slightly worse compared with the deterministicMIP approach. For the low demand cases alone we can observe a small improvement.Considering the variation of the results these differences are not significant.

Furthermore, we test a second approach, the SimMIP approach, where all nonlinearfeatures of the model are also considered in the optimization part which is representedby a MIP model. So the difference between the SimMIP and the exact approach is only,that in the first case the parameters are estimated based on simulation experiments andin the latter case the parameters are determined using the known distribution functions.Here we see that the solution quality can be raised for 9 out of 12 test instances andon average the result is slightly better than the deterministic MIP approach.

Even if for some instances the 70%-quantile yields the best results, the 90%-quantile leads to only slightly higher costs. Using the 50%-quantile, i.e. the expectedvalue, always caused much higher costs.

Additionally we analyze the variation of the 20 final simulation runs for eachmethod. There is no significant difference of the variation for all methods. The coef-ficient of variation for the total costs is always around 0.07.

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For larger test instances it would not be possible to solve the MIP or to apply the Sim-MIP method. In a preliminary study (cf. Mitrovic 2006) we focused on that issue andtried to find the approximate limits of solving MIP formulations of supply chain prob-lems by the means of three state-of-the-art LP/MIP-solvers. The tests were conductedon a PC (Intel P4 2.4 GHz, 1 GB RAM) using Windows 2000. We used a set of differentsized supply chain network problems considering fixed costs in transportation, pro-duction and transfer. The best performing solver succeeded in solving problems with20 supply chain actors, 8 products, 5 periods and 2 transportation modes, considering3,360 binary variables before and 250 binary variables after presolving. The next insize, which included 10 products instead of 8, could not be solved within a time limitof one hour. Thus, there is a trade-off between a good approximation resulting fromMIP-models or fast computational times. Definitely, important decisions involvinghigh fixed costs should be considered within the optimization model of our SimOptmethod.

5.3 Quantile tests on larger instances

In addition to the small instances used in the previous subsections we generate a set of12 instances representing larger supply chain networks. Using these test instances weanalyze the influence of the quantile on the results if the uncertainty is concentrated ina specific part of the supply chain. The size and structure of these test cases is shownin Fig. 5.

This fictitious supply chain network consists of 10 actors: 3 suppliers, 4 productionnodes, and 3 customers. The intermediate nodes are separated into two layers and allof them are authorized to produce and also transfer products. All actors are connected

sup 1

prod 1

cust 1

sup 2

sup 3

prod 2 prod 4

prod 3

cust 2

cust 3

Fig. 5 Exemplary supply chain network. For simplicity the transport modules have been omitted

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Table 4 Total costs of SimLPfor test example with ten supplychain actors and stochastictransportation delays at thebeginning of the supply chain

The results for the 90%-quantileare taken as basic values. For theremaining quantiles thedifference to the correspondingbasic value is given. The Sindicates that there is morestochasticity close to the supplier

Instances Quantile

90% 70% 50%

L1-L-S 274239 −0.19% 11.20%

L2-L-S 274241 9.98% 6.00%

L3-L-S 274995 5.76% 10.85%

L4-L-S 275366 5.83% 11.51%

L5-L-S 273214 1.33% 9.80%

L6-H-S 270286 2.72% 10.10%

L7-H-S 270491 2.23% 4.51%

L8-H-S 270438 −0.59% 10.92%

L9-H-S 267766 2.71% 8.65%

L10-H-S 270155 −0.57% 10.12%

L1a-L-S 333772 2.77% 4.04%

L6a-H-S 346953 1.42% 4.52%

Total Avg. 283493 2.78% 8.52%

by one transportation mode. The customers request 4 different products. Products 1and 2 are on the one hand final products, which have to be delivered to the customersand on the other hand raw materials used to produce products 3 and 4.

First we consider the case with the stochasticity concentrated at the beginning ofthe supply chain. Hence, for the connection between the suppliers and the first layerof production sites we assume stochastic transportation times, which are uniformlydistributed between 1 and 5. The transportation times between the two layers of pro-duction nodes are uniformly distributed between 1 and 3. For the remaining links weassume deterministic transportation times of 1. The costs functions for production andtransport consist of fixed costs and variable costs. If all binary decisions would be con-sidered in a MIP-model, this would lead to more than 3,800 binary variables, whichis beyond the size of problems we could solve with the best MIP/LP-solvers withinseveral hours. In comparison, our SimLP algorithm takes about 12 min for one testinstance to converge to a solution. We evaluate three different values for the quantileused for the estimation of the delay parameters: 90, 70, and 50%. See Table 4 for theresults.

In this case it seems that the 90%-quantile is most robust choice, although the 70%-quantile delivers only slightly worse results on average and in some cases even betterones, whereas the 50%-quantile leads to the worst results for all instances. Due to thefact that there is less stochastic near to the customer, it is possible to reduce the safetyfactors for the delays to some extent without increasing the costs too much, becauselost time at the beginning of the supply chain can be made up at the end.

We also conduct experiments where the transportation delays at the end of the sup-ply chain are stochastic, i.e. the connection between production nodes and customersare uniformly distributed between 1 and 5. Transportation times between the two lay-ers of production nodes are again uniformly distributed between 1 and 3. Remainingtransportation times are set to 1. The corresponding results are summarized in Table 5.

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Table 5 Total costs of SimLPfor test example with ten supplychain actors and stochasticityconcentrated near the end of thesupply chain

The results for the 90%-quantileare taken as basic values. For theremaining quantiles thepercentage difference to thecorresponding basic value isgiven. The C indicates that thereis more stochastic at thecustomer

Instance Quantile

90% 70% 50%

L1-L-C 263957 11.70% 43.96%

L2-L-C 263252 16.81% 45.02%

L3-L-C 263948 13.79% 48.81%

L4-L-C 263114 12.65% 50.60%

L5-L-C 263707 13.98% 51.41%

L6-H-C 258440 18.11% 52.09%

L7-H-C 257963 14.60% 59.29%

L8-H-C 260606 12.67% 41.68%

L9-H-C 258762 24.74% 52.44%

L10-H-C 258936 17.52% 43.16%

L1a-L-C 335837 7.15% 32.56%

L6a-H-C 335973 9.61% 32.95%

Total Avg. 273708 14.45% 46.16%

For these instances the best choice would be to use the highest safety factor (90%),because if there are delays at the end of the supply chain, there is no chance to catchup.

For the test instances L1a-H-C and L6a-H-C with high fixed cost, we also apply aSimMIP method where we include only the high fixed-charge production costs. Thelow fixed-charge transportation costs are still neglected. For all quantiles the Sim-MIP method delivers slightly lower costs (L1a-L-C: −4.48%/−10.86%/−1.19% for90%/70%/50%-quantiles; L6a-H-C: −3.47%/−2.96%/−0.37% for 90%/70%/50%-quantiles) but the calculation times are more than five times longer.

If we assume stochastic transportation times for the whole supply chain (all trans-portation delays are uniformly distributed between 1 and 5), the results are similar tothose in Table 5, i.e. the 90%-quantile is always the best choice.

6 Conclusions

In this paper we have presented a new approach that combines the advantages of com-plex simulation models and abstract optimization models. We have shown that ourmethod is able to generate competitive solutions quickly, even compared with tradi-tional planning approaches that are much more time consuming. Our investigationscan be summarized as follows:

• In many cases the SimLP method seems to be a good trade-off between solu-tion quality and computational time. If the nonlinear elements in the model aredominating it is better to apply the SimMIP approach and consider these nonlin-earities in the optimization model as along as the computational time for solvingthe optimization model is acceptable.

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• Furthermore, we investigated the impact of safety times for delays on the solutionquality. If we use the 90%-quantile, we can generate robust plans, but for specificsituations we might get better results with less safety time. Only for the case ifstochastic is near the customer, then the 90%-quantile is clearly the best. Never-theless, the choice of the quantile depends on the structure of the supply chain andhas to be fine-tuned in each case.

• Using the 50%-quantile, i.e. the expected values for the delays, always leads topure results. If the uncertainty is concentrated far away from the customer, thecost increase by using the expectation value is about 10% whereas the increase isalmost 50% if the uncertainty occurs close to the customer.

Further research for different aspects of this method is still possible and neces-sary. The aggregation step and the generation of new decision rules is an open field.One possibility is to interpret the solution of the optimization model only as a tar-get strategy and use adaptive decision rules to approximate this target strategy in anuncertain environment. The use of sensitivity results of the optimization model mightlead to improved decision rules. Further investigations are possible for the bound-aries between the simulation and the optimization model. The question, which aspectsshould be included in the optimization model, is not completely answered yet. If morecomplex models are used, other fast solution methods (e.g., heuristics, metaheuristics,etc.) should be taken into consideration.

We conclude by answering the question posed in the title of this paper: simulationand optimization are complementary approaches and it is worthwhile combining them.

Acknowledgment We wish to thank Martin Grunow and two anonymous referees for their valuablecomments on this manuscript.

Appendix A: MIP formulation for fixed-charge transportation cost

The objective (20) of the optimization model can be transformed into a mixed-integerprogram considering for example fixed transportation costs. In case, the transportation

cost functions vC pi j

(vx p

i j (t))

can be written as follows:

vC pi j

(vx p

i j (t))=

{vcp

i j if vx pi j (t) > 0

0 otherwise∀i, p, t, v. (22)

In order to capture this situation, it is necessary to introduce binary decision vari-ables v�

pi j (t) which indicate if there is positive transportation. So by adding the fol-

lowing constraints:

vx pi j (t) ≤ G · v� p

i j (t) ∀i, p, t, v (23)

it is possible to formulate the transportation costs as the linear functions

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vC pi j (t) = vcp

i jv�

pi j (t) ∀i, p, t, v. (24)

The resulting mixed-integer linear program includes now |JI | × |P| × T × |V |binary decision variables.

A similar approach can be used for modeling step functions like

vC pi j

(vx p

i j (t))=

⎧⎨

⎩

vcpi j if vx p

i j (t) ∈ (v X pi j ,

vY pi j ]

vd pi j if vx p

i j (t) ∈ (0, v X pi j ]

0 otherwise∀i, p, t, v. (25)

Here we need 2 different binary decision variables v�pi j (t) and v

pi j (t) to represent

this situation. If we add two additional constraints

vx pi j (t) ≤ G · v� p

i j (t) ∀i, p, t, v, (26)vx p

i j (t) ≤ G · vpi j (t)+ v X p

i j (t) ∀i, p, t, v, (27)

the cost functions can be written as

vC pi j (t) = vd p

i jv�

pi j (t)+

(vcp

i j − vd pi j

)v

pi j (t) ∀i, p, t, v. (28)

The resulting mixed-integer linear program includes now 2×|JI |× |P|× T ×|V |binary decision variables.

Appendix B: Notation

Notation used for the optimization modelJ set of locations J = JS ∪ JI ∪ JC

j ∈ JS raw-material supplier (starting nodes)j ∈ JC customer (end nodes)j ∈ JI nodes between supplier and customerP set of productsV set of transportation modesT number of periods

Decision variablesm p

i (t) amount of product p (product p is the end product of the produc-tion process at location i) that starts to be produced at location iin period t

u pi (t) amount of product p that starts to be transacted in location i in

time period tvx p

i j (t) flow of product p from location i to location j with transportationmode v (sent away in period t)

50


Costs, delays, and general parametersa p

i factor indicating the amount of capacity units required to produceone unit of product p at location i

αpi (p′) amount of product p′ required to produce one unit of product p at

location iinbp

i (t) amount of backorders of product p at customer i in period tvC p

i j (·) transportation cost function of product p transported from locationi to location j with transportation mode v

vCi j (t) maximum transportation capacity of transportation mode v on theway from location i to location j

prodCi (t) maximum production capacity at location i in period ttaCi (t) maximum transaction capacity at location i in period tinvinCap p

i (t) maximum amount of product p that can be held in the inboundinventory of intermediate i in period t

invoutCap pi (t) maximum amount of product p that can be held in the outbound

inventory of intermediate i in period tvCap p

i j (t) amount of product p that transportation mode v can transport fromlocation i to location j in period t

prodCap pi (t) amount of product p that can be produced at location i in period t

taCap pi (t) amount of product p that can be transacted at location i in period t

vcpi j cost factor used in case of linear transportation costs for deliveries

of product p between location i and location j with transportationmode v

D pi (t) demand for product p at location i in period t

d pi factor indicating the amount of capacity units required to transact

one unit of product p at location iδ

pi amount of periods required to produce product p at location i

in f pj (t) amount of product p arriving at location j in period t

out f pj (t) amount of product p sent away at location j in period t

vg p factor indicating the amount of capacity units required to transportone unit of product p with transportation mode v

in H pi (·) inbound inventory cost function for product p at location i

out H pi (·) outbound inventory cost function for product p at location i

inh pi cost factor used in case of linear inventory costs for the inbound

inventory of actor i and for product pouth p

i cost factor used in case of linear inventory costs for the outboundinventory of actor i and for product p

in Li (t) maximum capacity of inbound inventory at location i in period tout Li (t) maximum capacity of outbound inventory at location i in period tinl p

i (t) inbound inventory level of product p at location i in period toutl p

i (t) outbound inventory level of product p at location i in period tq p

i factor indicating the amount of capacity units required to hold oneunit of product p at the inventory of location i

R pi (·) penalty cost function at location i for product p

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r pi (t) amount of product p which is already transported at period 0 and will arrive

at location i in period t (or external increase of inventory)ρ

pi cost factor used in case of linear penalty costs at customer c and for product

pS p

j (t) supply of product p at location j in period ts p

i (t) amount of product p which is already in production process in period 0 andwill be finished in period t (or external increase of inventory)

σp

i amount of periods required to transact product p at location ivτi j amount of periods transportation mode v requires to go from location i to

location jW p

i (·) production cost function of product p at location iw

pi cost factor used in case of linear production costs at site i and for product

pZ p

i (·) transaction cost function of product p at location iz p

i cost factor used in case of linear transaction costs at site i and for productp

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53

Part IIDemand Management

Revenue management and demand fulfillment:matching applications, models, and software

Rainer Quante · Herbert Meyr ·Moritz Fleischmann

Abstract Recent years have seen great revenue management successes, notably

industries, including manufacturers and retailers, are exploring ways to adopt simi-lar concepts. Software companies are taking an active role in promoting the broa-dening range of applications. Additionally technological advances, including smartshelves and radio frequency identification (RFID), are removing many of the barriersto extended revenue management. The rapid developments in supply chain planningand revenue management software solutions, scientific models, and industry appli-cations have created a complex picture, which is not yet well understood. It is notevident which scientific models fit which industry applications and which aspectsare still missing. The relation between available software solutions and applicationsas well as scientific models appears equally unclear. The goal of this paper is tohelp overcome this confusion. To this end, we structure and review three dimensions,namely applications, models, and software. Subsequently, we relate these dimensionsto each other and highlight commonalities and discrepancies. This comparison alsoprovides a basis for identifying future research needs.

R. Quante (B)Institute for Production Management, Vienna University of Economics and Business Administration,Nordbergstraße 15, 1090 Vienna, Austriae-mail: [email protected]

H. MeyrChair of Production and Supply Chain Management, Technical University of Darmstadt,Hochschulstr. 1, 64289 Darmstadt, Germanye-mail: [email protected]

M. FleischmannRSM Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlandse-mail: [email protected]



57

in the airline, hotel, and car rental businesses. Currently, an increasing number of

DOI 10.1007/s00291-008-0125-8OR Spectrum (2009) 31:31–62


R. Quante et al.

Keywords Revenue management · Demand fulfillment · Manufacturing · Software ·Advanced planning systems

1 Introduction

Recent years have seen great revenue management successes, notably in the airline,hotel, and car rental businesses. These successes essentially rely on identifying andexploiting differences in the customers’ willingness to pay. Some approaches exploitthese differences by offering multiple product variants, tailored to different customersegments, such as different fare classes in the airline industry. Other approaches stick toa single product variant but adjust its price dynamically over time. This is practiced, forexample, by many budget airlines and fashion retailers during end-of-season clearancesales. Currently, an increasing number of industries, including manufacturers andretailers, are exploring ways to adopt similar concepts. Software companies are takingan active role in promoting the transfer of revenue management concepts to a broaderrange of applications. Technological advances, such as smart retail shelves and RFIDtags for real-time inventory visibility further support this development by decreasingmany potential barriers. Finally, increasing customer service and revenues throughintelligent demand fulfillment provides a way for companies to respond to the everincreasing pressure of global competition.

The growing interest in revenue management applications is also reflected in intensi-fied scientific research, as documented by a rapidly increasing number of publications(see e.g. Fleischmann et al. 2004). Supply chain planning and advanced planningsoftware is also gradually incorporating ideas of revenue management. For example,advanced planning systems (APS) extend the traditional available-to-promise andcapable-to-promise-logic of demand fulfillment modules to a profitable-to-promiselogic (SAP 2003). This development coincides with an ongoing consolidation in theAPS market, from a multitude of small vendors, like Red Pepper or Numetrix, to a fewbig business application and business intelligence software companies, like Oracle orSAP. Nevertheless, many niche players remain successful, due to their “greater abilityto manage the complexities of the supply chain, superior calculation power, greateragility, and improved integration capabilities achieved through open standards andservice-oriented architectures” (ARC Advisory Group 2006). Due to acquisitions, bigsoftware companies offer a wide range of supply chain management modules (severaldozens to more than a hundred), instead of just a few supply chain planning softwarepackages. Consequently, these companies need to re-arrange and re-structure theirsupply chain management (SCM) software portfolios, as illustrated by the example ofOracle, that needs to position its own advanced planning solutions together with thesupply chain planning, pricing, and demand management modules obtained throughthe acquisitions of Peoplesoft (including the former Red Pepper and Numetrix soft-ware), Retek, ProfitLogic, and Demantra (Oracle 2007).

Given these rapid developments, it is not surprising that not only software com-panies, their customers, but also the scientific community is struggling to maintain aclear picture of the resulting situation: which software modules serve which planningpurposes in which business applications? Which scientific models that have proven

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RM and DF: matching applications, models, and software

successful in a certain type of industry are transferable or adaptable to which othertypes of business with similar characteristics? The goal of this paper is to help over-come these confusions. To this end, we structure and review three dimensions, namelyapplications, models, and software. Subsequently, we relate these dimensions to eachother highlighting commonalities and discrepancies. This comparison also provides abasis for identifying future research needs.

We do not pretend to cover every potential application, nor every modeling detail.Instead, we focus on a few prototypical cases (“types”) on each of the three aforemen-tioned dimensions. Moreover, within each dimension we build on existing reviews andclassifications whenever possible.

The scope of our analysis includes short- and mid-term demand fulfillmentdecisions in a supply chain. For this delineation we follow Fleischmann and Meyr(2004) who define the planning tasks of demand fulfillment relative to the position ofthe decoupling point, which divides the supply chain into forecast-driven and order-driven processes (Sharman 1984; Hoekstra and Romme 1992). Demand fulfillment, asunderstood in this article, comprises the decisions at and downstream of the decou-pling point. These decisions are based on customer orders and primarily deal withmanaging the due dates of these orders.

We complement demand fulfillment with the concept of revenue management.According to Talluri and van Ryzin (2004), revenue management concerns demandmanagement decisions aimed at increasing a firm’s revenues. The authors distinguishquantity-based and price-based revenue management approaches. The first approachrelies on exploiting customer heterogeneity. It segments customers into multiple classesand prioritizes them when allocating scarce capacity. The key idea is that giving prio-rity to high-margin segments yields higher revenues than selling scarce capacity on afirst-come-first-served basis. The second revenue-management approach uses pricingdecisions as a lever for demand management. This includes adjusting prices dynami-cally over time in response to non-stationary demand or a finite selling season or viaauctions as a price-discovery mechanism.

Traditionally, the demand fulfillment and APS perspective is common in manu-facturing, whereas revenue management applications are mainly found in the serviceindustries. In this paper we argue that the planning tasks of both concepts are actuallyvery similar, and we systematically compare them. By highlighting analogies and dis-tinctions, we aim to provide a basis for expanding the traditional domains of applicationfor both concepts.

To summarize, our contribution is threefold. First, we unite the currently distinctconcepts of demand fulfillment and revenue management and compare them to eachother. Second, we link the three dimensions of applications, models, and softwarediscussing similarities and differences between them. We note that Elmaghraby andKeskinocak (2003) address similar aspects for dynamic pricing in a retail environment.Our paper differs from their’s by a broader scope in terms of planning tasks and appli-cations. At the same time, we consider aggregated model types in the literature, ratherthan reviewing individual modeling contributions. Third, we provide a supply chainframework for revenue management and demand fulfillment. We believe that manyother applications beyond the examples illustrated in this paper are worth exploring.Our presented framework provides a means for doing so in a systematic way.

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The remainder of the paper is structured as follows. Section 2 introduces ourframework, which we use for structuring the three dimensions of applications,models, and software. Sections 3–5 analyze each of these dimensions separately.Section 6 then links the three dimensions and identifies alignments and discrepanciesbetween them. Section 7 summarizes our main insights and discusses opportunitiesfor future research.

2 A supply chain framework for revenue management and demand fulfillment

In this section, we present a framework, which we use in the remainder of the paper for astructured analysis of revenue management and demand fulfillment (RM&DF) aspectsin each of the three dimensions of applications, models, and software. The frameworkis based on the elements of the supply chain depicted in Fig. 1. It is motivated by theaforementioned definition of demand fulfillment by Fleischmann and Meyr (2004).Recall that demand fulfillment encompasses supply chain decisions downstream of thedecoupling point (DP). At the same time, we can also represent revenue managementdecisions in this framework. We link pricing decisions to the item “final product” andcapacity allocation decisions to the item “demand”. We explain these elements in moredetail below.

Our framework includes supply chain elements that are directly related to revenuemanagement and demand fulfillment decisions. Specifically, we consider the followingelements, from upstream to downstream: Replenishment represents either an externalsupplier or internal production. According to the definition of demand fulfillment, thereplenishment policy at the DP of the receiving party is based on demand forecasts.This is a key point in our subsequent analysis. The decoupling point itself is thenext element of our supply chain framework. It holds the inventory that is needed tohedge against forecast errors and replenishment uncertainty. The DP plays a pivotalrole in our analysis since many RM&DF decisions are dependent on the availableinventory and on future replenishment orders. For a more detailed discussion of theDP concept and its impact in different production environments [including make-to-order (MTO), assemble-to-order (ATO) and make-to-stock (MTS)] we refer toFleischmann and Meyr (2004). The supply chain may contain additional productionprocesses downstream of the DP. For MTS production this is not the case, sincethe DP holds the final product. In contrast, limited downstream capacity, possibly ofmultiple production stages, plays a critical role in an MTO supply chain. By definition,production downstream of the DP is order driven. Therefore, it has to be considered

Final

__Pro

duct

Replenishment Demand

Decou

pling

Point

(Pro

ducti

on)

Capac

ity

Forecast-driven ( Demand Planning) Order-driven ( Demand Fulfillment)

Fig. 1 Supply chain framework

60


in the analysis to capture the effects of production lead times or order fulfillment.The elements furthest downstream in our framework pertain to the final product andcorresponding customer demand.

RM&DF are closely tied to decisions in the depicted supply chain and thereforedepend on its specific characteristics. For example, the current inventory at the DP orthe remaining production capacity may influence pricing decisions or promised duedates. By capturing these characteristics, the framework provides a systematic basis foridentifying RM&DF requirements in different applications. Moreover, by structuringmodels and software tools in a similar way, we can compare the three dimensions toeach other.

In order to characterize different types of supply chains we describe each elementof the framework by a number of attributes, which are relevant to RM&DF. Eachsupply chain then corresponds with a specific value for each attribute. For example,the product life cycle is an attribute of the final product, which can take the values shortor long. We do not seek to describe every potential attribute of any given supply chainelement but rather to focus on those attributes that are the most relevant to RM&DF.

Using these attributes allows us to reduce the complexity of the intended compa-rison. Specifically, we group instances together by types that have the same or simi-lar values in many attributes. In this way, we identify applications that have similarRM&DF requirements. In addition, we can compare application, model, and softwaretypes to each other to see which tools are available for supporting RM&DF decisionsin a given context.

We proceed as follows. Sections 3–5 address applications, models, and software,respectively. In each case, we first introduce corresponding supply chain attributesand briefly discuss their potential values and their relevance to RM&DF. Then wecharacterize a set of instances in terms of these attributes and cluster them into types.Section 6 finally compares the types across the three dimensions.

3 Industry applications

In this section we analyze and compare RM&DF decisions in different industries.As explained in Sect. 1, we do not seek completeness, but rather explore a set ofexamples covering a broad range of different applications. To this end, we includeexamples from the service industries, retail, and manufacturing. Specifically, withinthe service industries, we look at the airline industry as a classical user of revenuemanagement. Given the diverse airline demand management strategies, we make adistinction between premium and budget airlines. The retail sector is a driving forcein many recent pricing strategies. We include examples of fashion retail and consumerpackaged goods in our analysis. Finally, we include three manufacturing examplesthat reflect different DPs, namely MTS, ATO, and MTO.

3.1 Application-oriented supply chain attributes

In order to analyze and compare RM&DF decisions in these different environmentswe characterize them in terms of our supply chain framework. To this end, we consider

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PerishableDurable

ShortLong

At order timeShort termMid termNone

Profitheterogeneity

πtk =rtk – ctk + sk

π: Profitr: Revenuesc: Costss: Strategic

component

t: Time indexk: Customer class

index

HighLow

MTOATOMTS

Buyer-drivenrefillingVendor-drivenrefilling


Decou

pling

Point

(Pro

ducti

on)

Capac

ity

Refilling ofinventory

Decouplingpoint

Flexibility ofcapacity

Perishability

Life cycle

Pricingflexibility

Final

__Pro

duct

Fig. 2 Supply chain framework: application dimension

the attributes depicted in Fig. 2, which we explain from upstream to downstream inwhat follows. Again, we do not claim completeness of our selection but rather focuson arguing why the selected attributes are relevant to RM&DF. Table 1 summarizesthe evaluation of the attributes for the aforementioned industry applications.

3.1.1 Replenishment-related attributes

At the upstream end of the framework, we distinguish different roles of the supplychain members with respect to the forecast-based replenishment of inventory located atthe DP. We capture these differences in the attribute refilling of inventory. The famous“rationing game” describes a powerful supplier who distributes scarce supply amonghis customers proportional to their ordered quantities. This mechanism encouragescustomers to inflate orders in order to obtain the quantity they truly desired. We denotesituations in which suppliers have the power to decide on production and deliveriesby the attribute value vendor-driven refilling. Rationing only occurs in the case ofscarce supply capacity. If supply exceeds demand, orders will, in general, be fulfilledas requested. We describe this situation as buyer-driven refilling.

If the supplier has the power to decide on deliveries this results in unreliable deli-very times and limited replenishment capacity from the buyer’s perspective. A morepowerful buyer implies more reliable deliveries and a perception of unlimited supplycapacity. In conclusion, the attribute affects the buyer’s supply flexibility and therebyhis fulfillment decisions.

To illustrate the application of this attribute, consider the automotive industry,fashion retail, and airlines. In automotive production, suppliers are usually in a weakposition due to the extreme competition in the market—hence refilling is buyer dri-ven. A similar situation occurs in the case of fashion retail, although this may differbetween cheap and luxury fashion. For airlines, the concept of DP refilling does not

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Tabl

e1

App

licat

ion

exam

ples

Attr

ibut

eA

pplic

atio

n

Man

ufac

turi

ng:

Man

ufac

turi

ng:

Man

ufac

turi

ng:

Serv

ice:

Ret

ail:

cons

umer

good

sco

nfigu

rabl

em

achi

nes

(bud

get/p

rem

ium

)(f

ashi

on/c

onsu

mer

com

pute

rsai

rlin

espa

ckag

edgo

ods)

Refi

lling

ofin

vent

ory

Ven

dor-

driv

enPa

rtly

vend

or-d

rive

nB

uyer

-dri

ven

Dec

oupl

ing

poin

tM

TS

AT

OM

TO

MT

OM

TS

Flex

ibili

tyof

capa

city

Hig

hH

igh

Low

Peri

shab

ility

Dur

able

Dur

able

Dur

able

Peri

shab

leD

urab

le

Lif

ecy

cle

Lon

gSh

ort

Lon

gL

ong

Shor

t/Lon

g

Pric

ing

flexi

bilit

yM

id-t

erm

Ato

rder

time

Ato

rder

time

Shor

t-te

rm/M

id-t

erm

Shor

t-te

rm/M

id-t

erm

Profi

thet

erog

enei

tyπ

tk=

r t−

c tk

+s k

πtk

=r t

k−

c k+

s kπ

tk=

r tk

−c k

+s k

πt=

r t−

c/π

k=

r k−

cπ

t=

r t−

c

63

R. Quante et al.

really apply since seats are “sold” only temporarily (i.e., for a given flight) and noorders are placed to get new seats.

3.1.2 Decoupling-point-related attributes

As discussed in Sect. 2, the decoupling point represents a bundle of characteristics thatseparate forecast-driven from order-driven processes. Another common term is orderpenetration point. See Sharman (1984); Hoekstra and Romme (1992); Fleischmannand Meyr (2004) and Meyr (2003) for a detailed analysis of the DP concept. Wedistinguish three attribute values.

In MTO production systems the DP lies upstream in the supply chain. Productionis triggered by incoming customer orders. The inventory at the DP consists of rawmaterials, which are used in downstream production processes. The bottleneck inMTO is usually the production capacity downstream of the DP. MTO production isappropriate primarily for high-value and customer-specific products.

In ATO systems, production downstream of the DP includes the final product assem-bly. Inventory at the DP essentially consists of components that are usually deliveredby external suppliers. Bottlenecks may occur at either the DP inventory or the downs-tream assembly process.

MTS production is based on forecasts. Final products are produced on stock. Thisis a common strategy for standard products without customer-specific requirements.

The location of the DP has a strong impact on RM&DF decisions by influencingcustomer service times, order fulfillment, and control of replenishment orders. Forexample, deciding on a (promised) delivery date becomes more important as the DPmoves upstream in the supply chain. At the same time, customer service times increase.Furthermore, the different production environments require different types of forecastswith different levels of aggregation.

The literature provides several real-life examples of RM&DF in supply chains withdifferent DPs. For example, Spengler et al. (2007) show an MTO application in the ironand steel industry. Harris and Pinder (1995) mention custom-made textile and customequipment manufacturing as ATO examples. Meyr (2008) shows an application ofMTS in the lighting industry. Given the scope of our paper, it is worth pointing outhow the DP concept can be applied not only to manufacturing industries, but alsoto retail and service industries. In retail, inputs are the final products, while outputsare the sold products. The “production” process involves bundling and offering theright products at the right time, corresponding with MTS “production”. In the caseof a service business, “production” inputs include working time, material, etc., andthe output coincides with the transformation process. Since the “production” step isdependent on the customer’s presence it can be characterized as MTO.

3.1.3 Capacity-related attributes

Similar to DP inventory replenishment, the degree of flexibility of downstreamcapacity also impacts a firm’s fulfillment decisions by providing an alternative leverfor matching supply and demand. Possibilities to adapt capacity levels to short-termdemand fluctuations include, e.g., shutting down unnecessary machines or production

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lines in the case of excess capacity and hiring temporary workers or extending regularworking hours in the case of shortages. General strategies for balancing capacities anddemand can be found in, e.g., Chase and Aquilano (1995) or Bertrand (2003). Theseoptions may be further restricted by technical or regulatory requirements.

Evaluating this attribute for our application examples, we observe that airlinecapacity tends to be very inflexible in the short term, due to long planning lead timesfor flight and crew schedules. In manufacturing, flexibility depends strongly on thetechnology used and on working-time agreements with labor unions. In the processindustry, machines cannot easily be switched off due to the long and costly running-upphase. In contrast, computer manufacturing is highly flexible due to a high number ofmanual tasks. In retail, capacity flexibility concerns changing opening hours and thusworking times.

3.1.4 Product-related attributes

We observe several characteristics of the final product that impact RM&DF. The firstone is the degree of perishability, i.e., the maximum storage time. The values of thisattribute range from perishable to durable. Perishability describes the flexibility regar-ding the time horizon for selling the product. An inherent characteristic of perishableproducts is their low salvage value after expiration.

Product perishability has important consequences for RM&DF. For perishable pro-ducts, sales in the current period usually do not influence future sales. This is intuitivefor necessity items that are frequently consumed. Moreover, since necessity items areconsumed repeatedly, there is hardly a possibility for the customer to wait until theprice decreases. Perishability also increases the importance of optimal replenishment.Since overstocking is expensive for perishable products, proper demand forecasts andcorresponding optimal order quantities are crucial. In the case of durable goods, to-day’s purchases may affect future sales. For example, a customer typically buys acomputer only once within a few years. Therefore, price discounts early in the productlife cycle may hurt future sales (Elmaghraby and Keskinocak 2003).

Another relevant product-related attribute concerns the length of the product lifecycle, i.e., the duration of the selling season. It can range from a few months, as in thecase of fashion goods, to multiple years, as for basic food items.

This attribute also has a strong impact on RM&DF. Short life cycles limit theavailability of historical demand data and thereby complicate reliable forecasting.Moreover, due to the high frequency of new product releases, the customer will learnto anticipate the shape of the future price path. For example, video games usually havea short selling season and new ones are released frequently. Therefore, prices of videogames are rapidly decreasing since they out-date quickly. Anticipating future prices,customers behave strategically and buy at their individual best price. In the case oflong life cycles, anticipating future prices is more difficult for the consumer, due to alack of experience. We refer to Elmaghraby and Keskinocak (2003) for a more detaileddiscussion of this attribute.

A third product-related attribute worth considering is the degree of pricing flexi-bility, i.e., a company’s possibilities to change prices. A customized product usuallyalso has a customized price, as opposed to a standard product that sells for the same

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R. Quante et al.

price during a longer time horizon. The attribute values range from changing at ordertime, as in the case of highly customized products, over short-term, up to mid-termprice changes.

An online store may serve as an example of almost costless short-term price changes.Mid-term price changes are found in industries where the physical or organizationalinfrastructure for frequent price changes do not exist; for example, retail prices publi-shed in catalogs. Additionally, not only the frequency of price changes but also theirpotential magnitude characterizes the degree of pricing flexibility.

The impact of this attribute on RM&DF decisions is large since, for example, price-based demand management relies crucially on pricing flexibility. If this flexibility isnot given, quantity-based demand management by reserving products for specificcustomer segments becomes more important.

3.1.5 Demand-related attributes

On the downstream end of our supply chain framework, we distinguish multiple waysin which profitability may differ between orders. We denote this attribute as profitheterogeneity. The formula in Fig. 2 displays three factors for differentiating customerorder profitability, namely revenues r , costs c, and strategic importance s. Each of thesefactors may vary over time (denoted by the index t) and between customers (denotedby the index k).

For example, airlines often charge different prices according to the remaining boo-king time and other factors like remaining capacity, which implies different revenuesrt at different points in time. Different fare classes, such as business and economy,represent an example of different revenues rk from different customer segments k atthe same point in time. In retail, different prices rt are charged at different points intime t according to the remaining selling season however they typically apply to allcustomers.

Similarly, the costs of serving a customer order may also be a differentiator. Notethat only those costs which can still be influenced when accepting the order are relevanthere. This includes, for example, transportation costs, taxes, and any variable costsof downstream production. Again, these costs can be invariant (c), differ betweencustomer classes (ck) and/or differ between different points in time (ctk /ct ).

Finally, customers may differ in their strategic importance, which may go beyondimmediate costs and revenues. We capture this in our framework through the parameters. For example, loyal customers may be extremely important and should be treatedbetter than occasional customers.

As discussed in Sect. 1, heterogeneity lies at the very heart of RM. Price-basedapproaches are driven by the time dependence of the above profit elements. Quantity-based approaches seek to exploit customer segmentation. It is worth noting thatsegmentation may be attractive even in the case of constant unit revenues if theoverall profit π is heterogeneous due to varying costs (e.g. πtk = r − ctk) and/ordifferent strategic importance (e.g. πk = r + sk). This contradicts the commonclaim that sunk costs of unused capacity are a major prerequisite for applying RM(Weatherford and Bodily 1992). Meyr (2008) and Fischer (2001) describe applicationsof quantity-based approaches with strategically important customers.

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Retail:Consumer

Pack. Goods

Manufacturing:Machines

Manufacturing:Consumer

Goods

Manufacturing:ConfigurableComputers

Retail: Fashion

Service:BudgetAirlines

Service:PremiumAirlines

Supply Flexibility

Pri

cin

gF

lexi

bili

ty

FlexibleInflexible FlexibleInflexible

Capacity-based (MTO) Inventory-based (MTS + ATO)

Fle

xib

leIn

flex

ible

Fig. 3 Different application types

3.2 Application types

Table 1 summarizes the various attribute values for the selected applications brieflyintroduced at the beginning of this section. Missing attribute values indicate that anattribute is not applicable to the specific case. For example, the capacity flexibilityattribute is not applicable to make-to-stock production since there is no productionprocess downstream of the DP in this case. Capacity may be crucial upstream of theDP (e.g., in consumer goods manufacturing where scarce production capacities oftenlead to push-based, “vendor-driven” refilling of finished goods inventories) but theproduction process can no longer be influenced upon order arrival and therefore doesnot make part of RM&DF, as defined in Sect. 1. Similarly, refilling of inventory is notapplicable to the MTO cases. These examples illustrate that the different attributes inTable 1 are not mutually independent. We can therefore further simplify the charac-terization of the selected applications. Figure 3 highlights that the applications differfrom each other with respect to the criteria supply flexibility and pricing flexibility.The first criterion combines the first three attributes of Table 1, whereas the secondone corresponds directly with the original attribute pricing flexibility. The supply isconsidered as inflexible if there is either a partly vendor-driven refilling of inventoryor low flexibility of capacity. Note also that these criteria correspond with the mainplanning tasks among the list of attributes whereas the remaining attributes furtherdescribe the context.

We exploit the classification displayed in Fig. 3 for matching the application withavailable models and software in Sect. 6. In particular, we discuss the implications ofthe different degrees of flexibility on RM&DF requirements. Before doing so however,we first describe the model and the software dimensions in Sects. 4 and 5, respectively.

4 Models and methods

4.1 Model-oriented supply chain attributes

This section addresses the model dimension of RM&DF. Analogous with the previoussection, we first discuss a number of model-oriented attributes of our underlying supply

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NoneDataDecision variable*

Perishable / Durable

*Discrete set, continuous

NoneDecision variable

DeterministicStochastic

NoneDataDecisionvariable



DeterministicStochastic


Decou

pling

Point

(Pro

ducti

on)

Capac

ity

Final

__Pro

duct

Replenishmentquantity yt

Level ofinventory It

Capacity at Price ptk Customer orderacceptance xtk

Demand reliabilityPerishability

Replenishmentlead time

Fig. 4 Supply chain framework: Model dimension

chain framework and then derive and review a set of distinct model types. Figure 4 listsa number of attributes that we believe are relevant for characterizing RM&DF models,as we discuss below. Again, we do not claim the list of attributes to be exhaustive.

We derive our model attributes from the basic elements of mathematical optimiza-tion models, namely decision variables, objective function, constraints, and input data.We do not address the objective function separately since the decision variables includesufficient information for our analysis. In addition, we jointly consider constraints anddata as given model inputs. Figure 4 displays potential decisions in straight bold cha-racters and pure data attributes in italic bold characters.

A basic set of decision variables is derived from the product flow through the supplychain. These pertain to the replenishment quantity at the DP yt , the resulting inventorylevel It , and the number of customer orders accepted xtk . As in the previous section,the index t denotes the time dependence and k indicates a distinction by customerclass. These quantities are interrelated through the basic inventory balance equationIt = It−1 + yt − ∑

k xtk , stating that the inventory at the end of period t is equalto the inventory at the end of the preceding period plus replenishment minus totalaccepted demand from all customer classes. Each of the factors in this equation maybe modeled either as decision variables, as exogenous data, or it may not be consideredat all (none).

We make the same distinction for the sales price ptk and also consider whether itsdomain is discrete or continuous. With respect to the production capacity downstreamof the DP at we distinguish between capacitated models (data), models with capacityas a decision (e.g. overtime), and uncapacitated models, which do not consider scarcecapacity downstream of the DP (none).

In addition, we consider a few pure data attributes whose different values appearto be relevant to RM&DF (see Fig. 4). These include the replenishment lead time,which may be completely known (deterministic) or uncertain (stochastic), productperishability, and demand reliability. Many other attributes will be valuable for refining

68


StochasticInventoryControl

OrderPromising

-

InventoryRationing

aATPTraditional RM

IntegratedPricing

TradePromotions

Markdown /Pricing /Auctions

Replenishment consideration

Dem

and

/Pri

ceco

nsi

der

atio

n

Quantity-based•Price pk: Data•Customer order acc. xtk:Decision var.

Price-based•Price pt: Decision var.•Customer order acc. xt:Decision var.

Data•Price p: Data•Customer order acc. xt:Decision var.

None•Replen. yt: None

Data•Replen. yt: Data

Decision variable•Replen. yt: Decisionvariable

Fig. 5 Different model types

the analysis, such as salvage values, number of different products, backordering or lostsales, or the number of sales channels. However, in order not to overload our modelwe do not include all of these details in our analysis below.

4.2 Model types

Table 2 characterizes different streams of literature in terms of the attributes listedin Fig 4. As for the different application types in Sect. 2, the different attributes inTable 2 are also not independent. We therefore again condense the representation byconcentrating on the most distinctive attributes. It turns out that pricing and orderacceptance, on the one hand, and control over the replenishment quantity, on the otherhand, suffice to reasonably characterize most model types. Re-arranging the modeltypes along these two variables results in the compressed view shown in Fig. 5.

Models in the first row of Fig. 5 take demand and price as entirely exogenous. Theysatisfy demand first-come-first-served (FCFS) at a given price. In particular, they donot involve any customer segmentation. The next two rows of Fig. 5 correspond with amore active demand management. Models in the second row still consider only a singlecustomer class, but allow price changes which provide a lever for influencing demand.The last row shows models that explicitly recognize heterogeneous customers. Inresponding to a customer request they have to make a trade-off between accepting acurrent, low-priority order now versus reserving the resources for high-priority ordersexpected in the future.

The demand side of Fig. 5 corresponds with the classification of RM-models ofTalluri and van Ryzin (2004). We follow their terminology and label models as price-based or quantity-based. The columns reflect the way the different models handleinventory replenishment at the decoupling point. Note that both dimensions of thefigure correspond with key decision variables regarding demand and supply. Also notethat these dimensions are closely related to those of the application types discussed inthe previous section. We analyze this analogy in detail in Sect. 6.

In the remainder of this section, we briefly discuss each of the above model types.Given the scope of our analysis, we primarily build on available review papers in

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R. Quante et al.

Tabl

e2

Mod

elex

ampl

es

Attr

ibut

eM

odel

type

Ord

erSt

och.

inve

ntor

yPr

icin

gM

arkd

own

Tra

deIn

tegr

ated

aAT

PT

radi

tiona

lRM

Inve

ntor

yA

uctio

nspr

omis

ing

cont

rol

man

agem

ent

prom

otio

nspr

icin

gra

tioni

ng

Rep

leni

shm

entq

uant

ityD

ata

Dec

isio

nN

one

Non

eD

ata

Dec

isio

nD

ata

Non

eD

ecis

ion

Non

e

Rep

leni

shm

entl

ead

time

Det

/Sto

chD

et.

Det

.D

et.

Det

.D

et/S

toch

Lev

elof

inve

ntor

yD

ecis

iona

Dec

isio

naN

one

Dat

aD

ata

Dec

isio

naD

ecis

iona

Non

eD

ecis

iona

Non

e

(Pro

duct

ion)

Cap

acity

Dat

aN

one

Non

eN

one

Non

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the literature, rather than review the individual models. Again, we see a more refinedanalysis as a valuable issue for future research.

4.3 Review of model types

4.3.1 Single-class exogenous demand models

We have not found any models fitting in the upper left cell of Fig. 5. This is notsurprising since models with a given price and no consideration of replenishment orinventory, respectively, have nothing to decide on, neither on the demand nor on thereplenishment side of the supply chain.

In the next cell to the right, the so-called order promising models consider price(or demand), current inventory, and future replenishment quantities as given. Thisresults in information about product availability and delivery times. For each incomingcustomer order the model decides real-time on the due date. The decision is made ina greedy fashion, purely based on availability. An introduction and overview of thisso-called “real-time mode” or “single-order-processing” models is given in Ball et al.(2004), Chen et al. (2001), and Fleischmann and Meyr (2004). Additionally, a broadoverview of due date management models with an emphasis on stochastic models isincluded in the work of Keskinocak and Tayur (2004).

The upper right cell of Fig. 5 holds the large class of stochastic inventory control(SIC) models, which focus on optimal inventory replenishment. Some of these modelsprimarily address the structure of optimal replenishment policies, as for example thefamous (s, S)-policy proven by Scarf (1960) (if inventory position is below s, order upto S). Other models seek to determine optimal control parameters of such policies, suchas the optimal ordering time, order quantity, and inventory review intervals. Many SICmodels build on the classical newsvendor model, which seeks to determine the optimalorder quantity for a perishable product under stochastic demand. An overview ofsingle-period newsvendor problems is given by Khouja (1999). Silver (1981) providesan overview and typology of many standard inventory problems, such as the onesmentioned above. General up-to-date overviews of inventory models can be found inthe textbooks by Silver et al. (1998), Porteus (2002) and Tempelmeier (2006).

4.3.2 Price-based models

The model types in the middle row of Fig. 5 treat price as a decision, which influencesthe demand. Pure pricing models aim to determine an optimal selling price, withoutconsidering replenishments. For example, given a price–demand relation, the goal isto find the price which maximizes total revenues. Mild et al. (2006) review factorsinfluencing demand and show how to find optimal prices.

Markdown models determine the right price path for inventory clearance for agiven amount of inventory, which cannot be replenished during the planning horizon.Elmaghraby and Keskinocak (2003) classify several dynamic pricing models with andwithout replenishment decisions, the latter ones including markdown models.

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Auctions, as discussed for example by Talluri and van Ryzin (2004, Sect. 6), take afundamentally different approach to pricing. They provide a price-discovery mecha-nism and thereby an alternative to posting fixed prices. This approach is particularlyvaluable if little demand information is available. The aforementioned authors discussthe close connection between auctions and dynamic pricing.

Trade promotion models represent a type of pricing model that considers reple-nishments as an exogenous input. These models therefore fit in the second column ofFig. 5. Neslin (2002) provides an overview and discusses the reasons for promotions.

Research in integrated pricing (IP) models dates back to Whitin (1955) who extendsthe EOQ-formula (economic order quantity) as well as the classical newsvendor modelwith price decisions. This field has seen extensive research in the last decades, whichis summarized, for example, by Petruzzi and Dada (1999). Recent research focuses onmultiple period models, which are discussed in the well-known literature reviews ofChan et al. (2004), Elmaghraby and Keskinocak (2003) and Yano and Gilbert (2003).Few models exist for environments in which replenishment, prices, and due dates areset simultaneously. Some models of this type and other models dealing with settingdue dates can be found, for example, in the previously mentioned review paper byKeskinocak and Tayur (2004).

From an application-oriented perspective it is worthwhile comparing IP and asuccessive application of pricing and SIC models. While IP models recognize theinterdependence between pricing and replenishment and therefore determine decisionssimultaneously, they do so at the cost of a more simplified demand and supply repre-sentation. Pure pricing models may include sophisticated demand functions, includingreference price effects, promotion effects, and competition (Mild et al. 2006). Simi-larly, SIC models consider factors such as multiple suppliers and quantity-discounts.IP models typically cannot deal with these factors due to tractability (Elmaghraby andKeskinocak 2003, Sect. 4).

4.3.3 Quantity-based models

Models in the bottom row of Fig. 5 take prices as exogenous but manage demand bymeans of rationing strategies. In contrast with the models of the top row, the modelsdistinguish multiple customer classes and prioritize them rather than fulfilling ordersin an FCFS manner.

The type traditional revenue management (TRM) in the first cell of the third rowrefers to models that are common in airline applications. In these models, a givennumber of units of a perishable product (e.g., seats on a flight on a specific day) areallocated to customers with different priorities or different willingness to pay. Thebasic question is whether to accept a given order or to reserve capacity in anticipationof more profitable future orders. McGill and van Ryzin (1999) and Pak and Piersma(2002) provide an overview and a short history of research in traditional revenuemanagement with a focus on airline applications. Boyd and Bilegan (2003) discussmodels focusing on e-commerce applications. The recent review by Chiang et al.(2007) includes an overview of RM practices in different industries.

Allocated available-to-promise models (aATP) are similar to the order promisingtype of the top row except for differentiating between multiple customer classes. Scarce

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resources (inventory on hand, planned stock at the DP or capacity downstream of theDP) are allocated to these classes according to customer profitability or other prioritymeasures. Within each class, customer requests are usually handled FCFS, just as intraditional order promising. Guerrero and Kern (1988) introduce the general problemof accepting and refusing orders and discuss the requirements and implications oforder promising mechanisms. For reviews of the mostly deterministic models of thistype we refer to Kilger and Meyr (2008), Pibernik (2005) and Meyr (2008).

If customer requests do not have to be answered instantaneously several customerorders can be collected and jointly promised in a batch, thereby creating higher degreesof freedom for selecting the most important or profitable orders within a simultaneousoptimization process. Overviews of these so-called “batch order promising models”can again be found in the work of Ball et al. (2004), Chen et al. (2001) or Fleischmannand Meyr (2004).

A review of integrated due-date management and job-scheduling models withdeterministic orders is provided by Gordon et al. (2002). Their article considers batchmodels in which due dates are determined according to current capacity and the desi-red delivery date. Keskinocak and Tayur (2004) give a general overview of due-datesetting models.

aATP and TRM models are similar in that they decide about demand fulfillmentwith respect to different customer classes. The most significant difference between thetwo models is the perishability of resources. TRM considers “perishable” products,e.g., empty seats on a specific flight, which are lost after the departure date, whereas theATP quantities managed in aATP models are storable, in general. Another differenceconcerns the time horizon. TRM models typically consider a fixed day of capacityavailability, e.g., the departure date of a flight. In contrast, aATP models considermultiple periods linked through the storability of excess inventory. Furthermore, aATPmodels usually assume deterministic demand whereas demand in TRM models isstochastic.

The last model type within our framework concerns inventory rationing (IR) models.Similar to the relationship between aATP and traditional order promising, IR modelsextend traditional SIC models by distinguishing and prioritizing multiple customerclasses. For an early review we refer to Kleijn and Dekker (1998). Like traditionalSIC models, IR models may consider deterministic or stochastic replenishment leadtimes. A further distinction within this class of models is the number of demand classesconsidered, which may be general or limited to two classes.

For lack of recent reviews, we refer to a few individual articles that reflect two broadresearch streams within the type of IR models. Ha (1997) and De Véricourt et al. (2002)propose models with multiple demand classes and stochastic replenishment times,thus assuming limited production (= replenishment) capacities. In contrast, Melchiorset al. (2000) and Arslan et al. (2005) model deterministic replenishment lead timesand unbounded replenishment quantities. All these models take decisions on orderingand rationing levels, which are typically expressed in policies like (s, S, R) where s isthe reorder point, S the order-up-to level, and R the protection level between customerclasses.

IR and aATP models differ in terms of exogenous versus endogenous replenishment.Specifically, IR models consider replenishment decisions with stationary deterministic

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or stochastic lead times. In contrast, aATP models typically focus on capacitated,dynamic and deterministic arrivals of push-based production (=replenishment) quan-tities. To this end, aATP usually assumes deterministic and dynamic demand forecastswhereas IR models assume stochastic demand.

In addition to the model types captured in Fig. 5, a few recent research streamscombine several types by simultaneously considering multiple attributes. For example,Pibernik and Yadav (2008) present a model uniting the distinct features of the typesaATP (deterministic production quantities) and IR (stochastic demand). Kocabıyıkogluand Popescu (2005) jointly analyze price and allocation decisions with two customerclasses. Since most quantity-based models assume exogenous prices, this seems tobe a promising direction for future research. Bitran and Caldentey (2003) formulatea general model of this problem and review the current state of research. Anotherapproach is pursued in Ding et al. (2006) in which trade promotion models are com-bined with inventory rationing models. The authors denote the resulting new pro-blem type by ADP, referring to the allocation of available stock, discounting andprioritization of customers.

5 Software applications

5.1 Software types

The software market for demand and supply chain solutions has changed in the recentyears. For many years the focus was on the supply side. The interest is now, however,turning to end-to-end solutions including the demand side. Big supply chain solutionproviders like Oracle and SAP are investing large amounts into the acquisition ofdemand-related know-how. For example, in 2005 SAP took over Khimetrics, a leadingvendor of markdown, price, and promotion-optimization solutions. As noted in Sect. 1,Oracle—after taking over one of its largest competitors in supply chain solutions,Peoplesoft, in 2005—simultaneously invested in the demand solutions of Demantra(2006), ProfitLogic (2005) and Retek (2005), all of them leading vendors of retailrevenue management software. Another big consolidation occurred in 2006 whenJDA Software—a provider of specialized retail solutions—took over Manugistics, asupply chain solution provider focusing on profit optimization in the consumer goodsindustry.

The scope of our current analysis is restricted to software supporting short-termdecision making in RM&DF. These solutions draw data from other software systems,such as customer relationship management systems on the demand side (Buttle 2004)and enterprise resource planning systems (Stadtler and Kilger 2008) on the supplyside. Since these systems themselves do not focus on decision making we do notinclude them in our analysis.

As discussed in the previous section, scientific optimization models are fairly welldescribed in the literature. One can easily identify data, decision variables, restrictions,and solution strategies. Moreover, the solution quality is often analyzed in detailednumerical studies. This is different for commercial software solutions. Usually, avai-lable information is scarce and reveals little of the underlying technology. Software

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Our analysis of software modules reflects this limited availability of objective infor-mation. We build our characterization of software types and functionalities primarilyon available software reviews and whitepapers. As a starting point we use essentiallythe same attributes and values as for the scientific models (see Fig. 4). Model data anddecisions roughly correspond with software input and output, respectively. Due to thiscorrespondence, we omit a detailed discussion of attributes in this section.

Figure 6 structures software types along the same axes as the model types in Sect. 4.We choose names according to the functionality of commercial software modules onthe market. The remainder of this section briefly reviews each of these software types.

5.2 Review of software types

5.2.1 Single-class exogenous demand solutions

The mid-upper cell of Fig. 6 denoted by traditional order promising contains traditionalsoftware modules for short-term order promising under known inventory availability.When a customer order arrives, the software simply determines whether the order canbe satisfied out of available inventory. If not, the order is backlogged according to astandard lead time without considering future capacity or additional incoming supply.It is easy to see that this approach can lead to an order peak after the standard lead timeand thus to severe capacity problems in the future. Kilger and Meyr (2008) illustratethis situation in a simple example.

Refilling of inventory is usually left to purchasing and materials requirementsplanning modules, which are part of enterprise resource planning (ERP) systems.Essentially, these systems support refilling of non-bottleneck material and compo-nents from a single vendor. An overview of these classical systems can be found,for example, in the textbook of Vollmann et al. (2005). Since these classical systemsprovide sufficient solution quality only for very simple settings, specialized inventory

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modules consider extensions such as capacitated replenishment, stochastic demand,and multiple suppliers (Stadtler 2008). Such modules usually are part of largeradvanced planning and supply chain planning software suites. Additionally, thereare specialized vendors of supply-chain wide inventory optimization tools, such asOptiant (2007) with its inventory suite Powerchain and Smartops Smartops (2007).

5.2.2 Price-based solutions

Markdown management systems are mainly used in retail, for example for end-of-season stock clearance. An example of markdown management systems is B_Line,described by Mantrala and Rao (2001) under the name MARK. The system takespossible prices and corresponding demand probability distributions for each period asinputs and can find both markdown and markup price paths. The output consists ofa specific price in each period. Furthermore, MARK is capable of finding a suitableamount of initial inventory by iterating through a discrete set of possible inventorylevels. Elmaghraby and Keskinocak (2003, Sect. 3.2) describe the capability of mark-down solutions.

Software systems of the type pricing management are relatively new. This is dueto improvements in computing power and increased availability of past sales data.The rise of data warehouses and cheap computing power has recently allowed the useof automated pricing systems for many applications. Pricing management systemsare based on complex price–demand functions for which suitable parameters haveto be estimated, a process requiring vast amounts of past sales data. For example,to estimate price elasticity, the sales data must include a certain degree of diversity,corresponding with at least a few past price changes. Capacity or inventory restrictionsare usually not considered in these types of software (see for example Mild et al.2006).

The quick expansion of e-commerce applications has boosted the use of auctionsystems. The large number of different systems merits a review in its own right andexceeds the scope of our analysis. We refer to Kambil and van Heck (2002) for asystematic introduction to this field. Vakali et al. (2001) discuss the characteristics ofinternet-based auction systems and present a short survey of popular applications.

Similar to the previously described markdown systems, promotion optimization isalso used in retail environments, as described by Elmaghraby and Keskinocak (2003,Sect. 3.2). Very detailed information about the capability of such systems can be foundonline, for example from the vendors mentioned at the beginning of this section.

The term enterprise profit optimization (EPO) was coined by the software companyManugistics, who claims to be the first vendor offering an integrated pricing and supplysolution (Manugistics 2002). Furthermore, Manugistics software is meant to be ableto allocate scarce resources to the most profitable customers, thus simultaneouslyapplying ideas of quantity-based RM&DF. Demand and supply planning is realized inmany solutions, but not in an integrated way and not including price decisions. MostAPS forecast demand for different price levels and then successively analyze—withinthe context of mid-term planning—several what–if scenarios and their effects on thetotal supply chain.

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5.2.3 Quantity-based solutions

APS software modules that support mid-term, aggregated supply and demand decisionsare known as master planning modules (Meyr et al. 2008b). They receive determi-nistic demand forecasts and prices as inputs from the demand planning module ofAPS (Kilger and Wagner 2008) and then determine the best combinations of sales,production and replenishment quantities and the corresponding inventories undergiven capacity constraints. Quantities can be allocated to different customer classes.In terms of our supply chain framework in Fig. 1, master planning modules dealwith forecast-driven demand planning (i.e. push-based, vendor-driven refilling of theDP) and therefore fall outside the scope of our definition of RM&DF. However,we feel that they deserve mention since their resulting allocations serve as the pri-mary input for the short-term, capacity-checked order promising, executed by theDemand Fulfillment and ATP modules of APS. A detailed list of options conside-red in master planning modules can be found in the work of Rohde and Wagner(2008).

By taking capacity and inventory replenishments into account, demand fulfillmentand ATP modules of APS extend the aforementioned traditional order promising. Theydetermine due dates for incoming customer orders, which promise to be more reliablethan simple standard lead times. In addition, if ATP quantities are allocated to custo-mer priority classes—in the usually implemented aggregated way—order promisingdifferentiates with respect to customer importance, based on customer profitability orstrategic impact.

To find a reliable due date for a customer order, the software searches for demandfulfillment alternatives according to pre-defined “search dimensions”. These includethe time dimension, i.e., checking for ATP back- or forwards in time, the productdimension, i.e., substitute products, the location dimension, and the customer dimen-sion, i.e., checking for availability in other priority classes (Kilger and Meyr 2008).Usually, the software systems do not take the profitability of different fulfillmentalternatives into account during this search. However, recent systems not only consideravailable-to-promise quantities (available inventory) or capable-to-promise quantities(available capacity), but also follow a profitable-to-promise (PTP) logic that enablesthem to compare customer orders and fulfillment alternatives according to their prio-rity. Usually, simple rules are defined as search strategies for the different dimensions(Meyr et al. 2008a Sect. 18.3.1).

Revenue management software is widely used by airlines, hotel chains, and carrental agencies. RM software systems basically take the given capacity and offe-red tariffs as input and decide on acceptance or rejection of customer orders. Oneof the main differences with demand fulfillment and ATP is that RM softwarefocuses on revenues rather than costs. Furthermore, RM systems usually forecastdemand in much more detail than demand-fulfillment modules, e.g., for each flight,on each day, and for each customer class. These forecasts require a large amountof historical sales data in order to be reliable. Modern revenue management sys-tems can handle many additional industry-specific issues, such as overbooking andconnecting flights in the airline context (Talluri and van Ryzin 2004 Sects. 10.1.3,11.2).

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6 Matching applications, models, and software

In this section, we match the three previously discussed dimensions applications,models, and software against each other in order to identify alignments and misali-gnments. In particular, our goal is to identify the most appropriate models and soft-ware types for RM&DF decisions in the different applications introduced in Sect. 3,namely service, retail, and manufacturing. We also seek to highlight remainingresearch needs.

We build our discussion around the structure of Figs. 3, 5, and 6. Specifically,we compare the supply flexibility observed in a given application with the way thatreplenishment decisions are supported by different model and software types. Simi-larly, we tie the observed demand flexibility to the supported demand management. Inthis way, we identify for each application the most appropriate cell(s) in Figs. 5 and 6.Subsequently, we discuss the match/mismatch with models and software within thatcell in more detail by including the additional attributes highlighted in the precedingsections. This allows us to recognize empty spots and future research needs.

Figure 7 summarizes the match between the different types of applications and themain model and software attributes. In the remainder of this section we explain thismatch by application type.

6.1 Service industries

Service industries can be characterized as MTO since the “production” step requiresthe presence of the customer. Hence, replenishment decisions do not play a role andone primarily needs to consider available capacity downstream of the decoupling point(currently available seats in an airplane). This suggests that the first column in bothFigs. 5 and 6 is the most relevant for service industries.

Moreover, since capacity is inflexible and cannot be substituted nor stored, themeans for matching supply to demand are limited, which makes demand managementthe main lever in the short term. Models and methods in the top row of our figurehardly support this step and therefore appear insufficient for these applications. A moreintelligent way of demand management is required, which recognizes the customers’

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willingness to pay. This can be achieved either through price-based (second row)or quantity-based (third row) RM&DF, which renders models and software of thetype markup/-down, pricing, and (traditional) revenue management the most intuitivecandidates (see Figs. 5, 6 and 7).

Which of these types are best suited essentially depends on the pricing flexibilityand possibility of customer segmentation. This corresponds with the two examplesof service applications sketched in Sect. 3, namely budget and premium airlines.Because of their online pricing flexibility, budget airlines can use markup modelsand software to determine the (increasing) mid-term price path of a certain flight,as well as short-term temporary discounts. Premium airlines benefit from tailoringfares to specific customer segments, which leads to the quantity-based approaches oftraditional revenue management.

Of course, these matches are hardly surprising since they confirm well-knowncommon practice. However, the examples show that our approach is able to identifythese matches correctly. This supports the application of the same approach to lessobvious examples as in the following cases.

6.2 Retail

Inventory replenishment in retail environments is primarily buyer-driven (see Table 1)and thus endogenous. This suggests that the third column of Figs. 5 and 6 is the mostappropriate in this context. There are, however, exceptions. For example, when reple-nishment decisions are made on the mid-term and cannot be revised on the short-term(e.g. in the fashion industry due to long lead times and short life cycles) replenishmenthas to be taken for granted at the time of order fulfillment or it may be non-existentaltogether. Similarly, replenishment may be of little concern in the case of amplecapacity and short lead times. Thus, the first two columns may also be relevant toretail.

Under the general assumption of buyer-driven inventory replenishment and the MTSdecoupling point, supply, and production can be considered as relatively unproblematicin retail (see Sect. 3). On the other hand, retail is close to the final consumer andtherefore has a traditional focus on price setting, even if prices can only be changed onthe mid-term. Therefore, the second row of our scheme is of primary interest. Becauseof the relatively simple supply and production requirements, all of the price settingmodels in this row appear relevant under certain circumstances. This includes pricingmodels (e.g., if inventory cannot or needs not be considered), markdown models (forshort life cycles), trade promotion models (for long life cycles with given replenishmentcontracts), and integrated pricing models (with simultaneous replenishment decisions).

Similarly, the corresponding software types, namely pricing, markdown manage-ment, promotion optimization, and EPO software can be applicable, with the appro-priate choice depending on the detailed application attributes in Table 1. All thesemodels and software can be applied on the mid-term, or on the short term if pricingflexibility is high enough.

Opportunities for customer segmentation (third row) are generally more limitedin a retail environment. It is hard to collect sufficient data for identifying natural

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consumer segments or to define and fence off “artificial” consumer segments analogousto the service industries. Fencing usually requires additional services (e.g., businesscustomers receiving bonus miles, preferred transfer, etc.) which may not be obvious inretail—essentially requiring a shift from a purely product-oriented retailer to a moregeneral service provider. Other examples of segmentation in retail include the sellingof a product as part of a bundled package, or the use of multiple distribution channels(Agatz et al. 2007). Inventory rationing models can then be applied for refilling andallocating stock to different channels.

Online retailers face particularly promising opportunities for gathering data for cus-tomer segmentation as well as for changing prices almost instantaneously. Therefore,they can integrate price- and quantity-based RM with flexible replenishment strate-gies. Models (and software) for this type of application are rare and offer promisingresearch opportunities (see also Sect. 4).

6.3 Manufacturing

Our third main application type is the manufacturing industry. Unlike in the precedingapplication types, production processes are the most important, and usually mostcostly, process steps. The decoupling point as the interface between forecast-drivendemand planning and customer-oriented demand fulfillment (see Fig. 1 and Sect. 3)describes whether a certain production process is operated under demand (un)certainty,what type of stocks (raw material, components, final products) must be held, wherethe main bottlenecks (stocks, production capacity) can be expected, and the length ofcustomer service times. Because of these significant differences we consider MTS,ATO, and MTO manufacturing applications separately.

6.3.1 Make-to-stock

In MTS environments, all production processes are executed based on forecasts. Dueto upstream capacity limitations, production planning decides on short-term reple-nishment of DP inventories in a push-based, “vendor-driven” manner. Thus, modelsincluding replenishment decisions (third column of Fig. 5) can only support mid-term,forecast-based demand planning, but not short-term demand fulfillment. In order tomake use of the (uncertain) information on future DP inventory replenishments, asimplied by the production plans, demand fulfillment models of the second columnappear the most appropriate.

Because of the MTS market conditions and contracting practice, pricing deci-sions typically have to be taken on a mid-term basis (see Table 1). For example,the demand planning module of an APS forecasts several price-demand scenariosincluding, e.g., different alternatives for price discounts or promotions. These scena-rios are passed to the master planning module, which checks each of them with respectto supply chain constraints, selects the most profitable one, and generates directives forthe (forecast-based) short-term production planning. Thus, short-term pricing flexibi-lity is rather limited, which rules out the models and software in the second row ofFig. 5 for demand fulfillment in MTS manufacturing. Price-based approaches appear

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mainly applicable on the mid-term planning level, e.g., to determine demand forecastsin conjunction with optimal prices.

Thus order promising and aATP models remain as the most applicable modelsfor RM&DF in MTS manufacturing. Both models consider the current level of DPinventory. Order promising in a MTS environment searches through the ATP quantitiesin an FCFS manner to be able to fulfill a customer order. Newer approaches processseveral customer orders in a batch and allocate ATP to the most profitable customerorders. Due date setting is not relevant in this environment. aATP models overcome thedisadvantage of batch order promising, namely not providing a real-time order promiseand forcing the customer to wait. These models are, however, dependent on forecast-based information on DP inventories as provided by the master and production plans,and on the possibility of customer segmentation. The ATP search rules that are used toconsume the allocated ATP quantities of the different customer classes follow similarideas as traditional RM methods. However, since products in MTS manufacturing aredurable and can be stored (see Table 1), they also have to be able to “search over time”,i.e., to take future inventory replenishments into account.

6.3.2 Assemble-to-order

In ATO production, the most important supply chain elements are the inventory kept atthe DP (components to be assembled) and the scarce assembly capacity downstreamof it. Since customer order service times are longer than in MTS environments, orderpromising becomes more difficult and more important. In addition, new customer-oriented planning tasks arise, such as supply and demand matching and the short-termproduction planning of the assembly process (see Fleischmann and Meyr 2004).

To determine the suitable columns in Figs. 5 and 6, one should consider whether ornot replenishments can be controlled in the short term. In the aforementioned exampleof configurable computer manufacturing, long replenishment lead times essentiallyrule out this option. If availability of the computer components, e.g., high-end CPUs,is limited and they cannot be replenished as required (i.e., vendor-driven), an integrateddemand fulfillment and replenishment is impossible, which rules out the third column.On the other hand, DP inventory is as important in ATO as in MTS and thus informationabout future replenishments should not be neglected. Otherwise promised due dateswill not be reliable. Therefore, again, models of the first column (especially traditionalRM methods) are not appropriate or at least have to be adapted to these needs.

In general, the capacity downstream of the DP involves some degree of flexibility.Therefore, demand management is less important for matching supply and demandthan in entirely inflexible environments (see service applications above). However,ATO usually concerns customized products, and therefore yields opportunities forcustomer segmentation and exploiting differences in willingness to pay. Therefore,the focus of ATO RM&DF should lie on selling the available resources to the differentcustomer segments in the most profitable way. aATP models and software (third rowof Figs. 5 and 6) outperform the simpler FCFS order promising (first row) in thatrespect.

For real-time order promising, the advantages of aATP over an FCFS logic aresimilar to the ones in the MTS case, except that multi-stage bills-of-material have to be

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taken into account. Batch order (re-)promising models, however, are more important inthe ATO case because they can also be applied for supply-demand-matching (which ispart of the short-term production planning), i.e., to select the most important customerorders to release next to the shop floor for assembly. Standard APS software can beused for this purpose (see Kilger 2008), but usually does not provide as sophisticatedmethods as the corresponding optimization models (Ball et al. 2004).

For customized products, pricing can also be part of order promising because pricesoften are only communicated upon customer request, due to the high number of poten-tial product configurations. However, traditional trade promotion models and softwareas listed in the central cell of Figs. 5 and 6 are not appropriate for this planning task.On the other hand, traditional pricing or EPO models/software usually do not take intoaccount future inventory availability at the DP, as determined by the mid-term pro-duction planning. Therefore, we see room for future research on price-based RM&DFwith given replenishments.

Even though their business model has suffered recent criticism, Dell has shownthat direct selling can build a bridge between traditional manufacturing and traditionalretail and that pricing flexibility—at least for non-standard components such as addi-tional memory—can be increased through direct access to customers (Kraemer et al.2000). In this case, both pricing flexibility and customer segmentation can be exploi-ted simultaneously. Therefore, as noted for the case of online retailing in the previoussubsection, integrated price- and quantity-based approaches offer further researchopportunities, also for ATO manufacturing.

6.3.3 Make-to-order.

In MTO production all—usually multi-stage and very complex—production processesare executed to order and production capacity is critical. In contrast, DP inventory onlyconsists of basic materials that can be easily replenished. Therefore, replenishmentdecisions are of lesser importance and we can concentrate on the models and softwareof the first and second column of Figs. 5 and 6. Although short-term productionplanning in MTO is also customer-oriented and thus part of demand fulfillment, wemainly focus on the order promising aspects of MTO in what follows. The key task isto quote reliable due dates and to decide which orders to accept in order to maximizeprofit.

Pricing flexibility in an MTO setting is high. However, automatic price-basedapproaches are not generally applicable, because prices for the usually complex andexpensive MTO products must be negotiated. Standard pricing software (second row)cannot really be used for determining minimum acceptable prices since sufficient pastsales data are hardly available for the highly customized products in MTO. A moreappropriate way of finding minimum acceptable prices may be using the productionplanning software to assess marginal costs, e.g., by simulating production plans withand without the given order. Due to the complexity of the production system this canoften only be done in an aggregate, approximate manner.

Similar to ATO, the high customization of MTO products offers opportunities forsegmentation strategies, provided that the customer base is sufficiently heterogeneousto identify and separate different segments. Customers of MTO products are naturally

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segmented by the differences in production costs between different orders. Therefore,fencing may not be needed.

We further narrow the choice of appropriate cells of Figs. 5 and 6 by noting that theunderlying models and software should include available production capacity. Thisleaves us with the model types of order promising, aATP, and traditional revenuemanagement. Traditional RM assumes fixed production costs, non-flexible capacityand perishability. These assumptions apply to only a few, exceptional MTO industries(e.g., perishable products in some chemical industries). In the majority of cases, tra-ditional RM must at a minimum be modified to be able to support due-date setting inMTO production.

Traditional order promising models for MTO focus on due date setting under limitedproduction capacities. They either check capable-to-promise (CTP) quantities in anFCFS manner (Dickersbach 2004; Kilger and Meyr 2008), estimate due dates by meansof stochastic queuing theory (Keskinocak and Tayur 2004), or simulate the short-termproduction planning deterministically, as discussed above. Whereas the first and thelast types of models are usually implemented in standard APS, the second type is onlyfound in specialized software.

Differences in the profitability of customer orders are not exploited by traditionalorder promising models. Some APS allow a profitability assessment of different ful-fillment alternatives by evaluating their revenues and costs during the rule-based (ATPand) CTP check. The aATP logic can be extended similarly for additional CTP checks.This again allows prioritizing different customer segments, thereby transferring RMideas to MTO manufacturing. However, since CTP quantities represent productioncapacities only in a very aggregate manner, the resulting due dates and thus also theestimated profits are not very reliable for complex production systems. In this casea more detailed simulation of short-term, customer-oriented production planning, asdescribed above, appears necessary. This, however, often yields prohibitively longcomputation times, which implies long reaction times to customer requests.

Of course, short-term production planning models and software can also be appliedto batch promising of multiple customer orders, which offers additional degrees offreedom. However, the approach suffers from the same inherent problems as in thesingle-order case. We conclude that the biggest challenge in MTO order promising isnot so much the availability of models and software but how to implement them inpractice to yield a good balance between solution quality/reliability and short responsetimes.

7 Conclusion

In this paper, we have analyzed and structured revenue management (RM) and demandfulfillment (DF) decisions. We have presented a framework that covers the underlyingsupply chain processes and allows for a systematic comparison of different businessenvironments.

The conceptual integration of revenue management and demand fulfillment is amajor contribution of this paper. Both concepts have emerged in different industries.Demand fulfillment is a standard component of advanced planning systems that are

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mainly applied in manufacturing industries. Revenue management, on the other hand,is tightly linked to service industries, notably to airline ticket sales. These differentbackgrounds and associated terminology and connotations complicate a systematiccomparison and thus a mutual exchange of ideas between both concepts. In this paper,we argue that revenue management and demand fulfillment actually concern essen-tially the same supply chain processes.

The main conceptual distinction is the differing degree of demand management.Traditional DF essentially treats demand as exogenous. It assumes given prices andtreats customer orders in a first-come-first-served manner. For each incoming customerorder it searches for the fastest fulfillment possibility and promises a correspondingdue date to the customer. In contrast, RM seeks to more actively manage demand. Itconsiders short-term price adjustments and prioritizes different customer segments,thereby exploiting differences in the customers’ willingness to pay.

The framework introduced in Fig. 1 in Sect. 2 allows for a uniform treatment ofRM and DF decisions. In this paper, we have applied it to characterize and comparea few exemplary application environments. The selected examples are far from com-prehensive. Yet they serve to demonstrate the applicability of our framework, whichwe hope will be useful for analyzing many other examples. Moreover, the frameworkalso supports a classification of RM and DF models and software systems. We haveidentified two key distinctive factors, namely the aforementioned demand manage-ment and the replenishment strategy at the supply chain decoupling point. Similar tothe former, the latter can be endogenous or exogenous. We have used both of thesefactors (see Figs. 5, 6, 7) to structure reviews of optimization models and softwaresystems, which are relevant for RM&DF, and to compare them to the requirements ofselected applications in the service, retail, and manufacturing industries in Sect. 6.

This led to the following main insights (cp. also Fig. 7):

– Replenishment is not an issue in service industries. Scarce capacities (e.g., ofseats in an airplane, hotel rooms, etc.), which cannot be stored and thus also notbe replenished, are the center of attention. Therefore, fewer means are availablefor balancing supply (=capacities) and demand than in other industries, whichmotivated the early use of RM techniques. On the other hand, adapting thesetraditional RM techniques for manufacturing or retail applications requires theintegration of replenishment and storability.

– Retail is closest to the end consumer. Production and limited production capacitiesdo not play a noteworthy role, but replenishment is important and can take ondifferent shapes. Various types of price-based demand management are applicablehere. Possibilities for differentiated stock allocation are still limited. Exceptionsinclude multi-channel retailing, where some channels may have priority overothers, or online retailing, where a lot of information about customer behavioris available and the customers do not see the physical inventories.

– Manufacturing industries are heterogeneous. Inventory holding, replenishment,and limited production capacities can occur separately or in any combination,and the means for balancing supply, capacity, and demand are manifold. Thus, anactive demand management in the sense defined above was not as important in thepast as in service industries and is still in its infancy today. However, it arouses

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more and more interest. The decoupling point concept helps structure the varietyof manufacturing applications. The major potential for a more active demandmanagement seems to lie in differentiated capacity allocation, i.e., quantity-basedRM&DF. A short-term pricing potential is mainly given for ATO manufacturing,due to the high degree of customization in combination with a pressure for shortcustomer order lead times.

Unlike in service environments, order promising in manufacturing industries is a multi-period problem, i.e., production earlier or deliveries later than the customer’s requesteddate are possible. Therefore, traditional RM techniques typically cannot be applied asis, except possibly in a few MTO manufacturing industries. Otherwise, they have tobe adapted to deal with the holding and (future) replenishment of decoupling pointinventory (MTS, ATO), costs (MTS, ATO, MTO) and throughput-time estimates fordownstream processes (ATO, MTO). The allocation and ATP consumption rules cur-rently used in APS serve this purpose, but are very basic. We see a need for futureresearch here. Incorporating further ideas of traditional (quantity-based) RM mightlead to more sophisticated methods and software.

Additionally, we have identified further promising research opportunities: In onlineretail the integration of price- and quantity-based demand management ideas withflexible replenishment strategies seems to offer interesting potentials. The same istrue for direct sellers in ATO manufacturing who additionally have to take care of theproduct assembly.

For due-date re-promising and demand-supply-matching, which are further plan-ning tasks of ATO manufacturing, batch optimization models appear helpful and havealready been proposed in the scientific literature. However, to develop optimizationmethods that are scalable to practical needs and can find their way into commercialplanning software, further research seems necessary.

Last but not least, short-term production planning and scheduling modules of APSalready offer the basic functionality to estimate the due dates and costs of a cer-tain customer request in complex MTO environments. However, for most practicalapplications the effort in terms of modeling complexity and computation time is stilltoo high. Thus we see an interesting trade-off between detailed but complex cost andthroughput time projection models as a basis for demand management and simplerbut possibly less accurate projections. Even if this problem can eventually be handledthe question will remain how to use these projections in price negotiations with thecustomer and whether another customer is about to call who is willing to pay an evenhigher price.

Acknowledgements The authors are grateful to the Vienna Science and Technology Fund (WWTF) forfunding this research.

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88

Active demand management for substitute productsthrough price optimisation

Aaron A. Levis · Lazaros G. Papageorgiou

Abstract This paper presents a systematic mathematical programmingapproach for active demand management in process industries. The proposedmethodology aims to determine optimal pricing policies as well as output levelsfor substitute products, while taking into consideration manufacturing costs,resource availability, customer demand elasticity, outsourcing and market com-petition. First, profit maximisation analytical formulae are derived for deter-mining Nash equilibrium in prices for a duopolistic market environment whereeach company produces only one product. An iterative algorithm is then pro-posed so as to determine the decision-making process by solving a series of non-linear mathematical programming (NLP) models before determining the Nashequilibrium in prices for the competing companies. The proposed algorithm isextended in order to accommodate the case of multi-product companies, eachone selling a set of substitute products at different prices. The applicability of theproposed methodology is demonstrated by a number of illustrative examples.

Keywords Active demand management · Substitute products · Priceoptimisation · Nash equilibrium · Non-linear mathematical programming

1 Introduction

In today’s global marketplace, process industries no longer compete as inde-pendent entities but rather as integral part of supply chain links. The ultimatesuccess of a firm depends on its managerial ability to integrate and co-ordinate

A. A. Levis · L. G. Papageorgiou (B)Centre for Process Systems Engineering, Department of Chemical Engineering,UCL (University College London), London WC1E 7JE, UKe-mail: [email protected]



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DOI 10.1007/s00291-006-0064-1OR Spectrum (2007) 29:551–577


A. A. Levis, L. G. Papageorgiou

the complex network of business relationships among supply chain members(Min and Zhou 2002). The recent wave of mergers and acquisitions (M & A) hasled a number of smaller companies to consolidate into a few giant supply chainfirms (e.g. Unilever, Procter & Gamble) that provide close substitute products(e.g. fast moving consumer goods FMCG) to a wide range of customers. Theintense competition among different companies is evident and occurs in almostevery market sector nowadays.

In this competitive environment, the customer demand is usually satisfied bya small number of companies, each one manufacturing and selling its individualsubset of products. The goal of every company is to obtain the highest possibleprofit by determining optimal price levels for its portfolio products. In that casean oligopolistic price competitive market environment is established that needsto account for both competitors’ activities and customers’ willingness to buy.

Duopolistic market competition is the natural starting point for investigat-ing the behaviour of oligopolies. Consider two companies, company A andcompany B, each one manufacturing its own subset of substitute products. Bysubstitute products we mean slightly differentiated product brands that belongto the same product-class (e.g. lubricants, detergents, cosmetics, carbonated softdrinks, etc.). In that case, increased sales of one product result in reduced salesof another, thus forming a market environment where products brands competewith each other over a common customer base.

The manufacturing of products usually takes place in production sites ownedby the company (in-house manufacturing). Every site has a limited amountof available resources used for production. Alternatively, each company mayhave the option to allocate manufacturing of a certain amount of products toa third-party company (outsourcing). As shown in Fig. 1, final products fromeach company (in-house manufactured and outsourced) are then transportedto the marketplace in order to satisfy the anticipating customer demand at givenproduct prices.

A crucial precondition of effective price competition is that customers areinclined to search for lower-priced substitute products. Low prices howevercan kill profit margins and jeopardise the overall company profitability. On theother hand, high prices will drive away potential customers and inevitably putcompany’s market share at risk. Pricing decisions are of crucial importance andunless taken seriously, they can pose a major threat to the sustainability of thecompany.

Traditional approaches for customer demand management assume fixedproduct prices and usually rely on forecasting tools, trying to predict customerdemand based on historical sales patterns (Markidakis and Wheelwright 1982).Passive demand management (PDM) approaches ignore the importance offlexible product pricing and usually lead to poor customer demand satisfaction.

Modern industrial enterprises are typically multi-product, multi-purpose andmulti-site facilities operating in different countries and dealing with a global-wide international clientele. In such enterprise networks, the issue of optimalproduct pricing policy plays a key role in business performance and necessitatesthe appropriate attention. A new trend towards active demand management

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Active demand management for substitute products

COMPANY A COMPANY B

SITE 1

SITE 2

OUTSOURCE

MARKET

SITE 4

SITE 3

OUTSOURCE

Fig. 1 Duopolistic market environment

(ADM) has recently emerged focusing on how to actually drive customerdemand away from traditional baseline forecasts so as to maximise both over-all business performance and customer satisfaction. Many companies rang-ing from the automotive industry (Ford Motor Co.) to the Internet dotcoms(Amazon.com) have recently realised the potential benefits of adopting such amarketing-based concept of smart pricing (Coy 2000). Some firms (e.g. thoseoperating petrol forecourts) do not hesitate to go even a step further ahead andemploy such clever “dynamic pricing” strategies almost on a daily basis. Accord-ing to Manugistics (Manugistics Website 2003), a leading company in pricingand revenue optimisation, pricing is the next battleground for competitiveness.

However, product pricing is not a trivial task. Successful pricing strategiesshould consider simultaneously rapidly changing customer expectations, fast-reacting competitors, complex product interactions and fluctuating manufactur-ing capacity constraints. Accelerating product lifecycles and increasing productmix diversity further magnify the complexity of capturing an accurate understat-ing of the pricing environment and managing a comprehensive strategy aroundit (Rapt Website 2003). Lanning et al. (2000) allow demand to be determinedby prices via a constant-elasticity demand function. Prices are then optimisedjointly with capacity investment decisions. Optimal capacity levels and prices forsubstitutable products are considered by Birge et al. (1998) in a single-periodmodel while joint co-ordination of production and marketing decisions areinvestigated by Eliashberg and Steinberg (1987). In market-oriented program-ming, Kaihara (2001) proposes negotiation mechanisms that can lead to paretooptimal resources allocation in supply chain management. In a subsequentresearch paper, Kaihara (2003) formulates the supply chain model as a discreteresource allocation problem under dynamic environment and demonstratesthe applicability of the virtual market concept and the analysis of the systembehaviour in economic terms.

Although the problem of product pricing is not new in the applied economicsand operational research literature, previous studies adopt a somehow simplis-tic approach to the problem. They focus their attention on single-product firmsand therefore cannot accommodate the nature of multi-product firms whichare predominant nowadays. Another common drawback is that many studiesconsider product pricing in isolation of the market competition, thus ignoring

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the reaction effect of rival companies offering substitute products in themarketplace. Even in the case where competition between single-product firmsis addressed, the number of firms is restricted to two (duopolistic competi-tion) while joint production and pricing decision-making is based on unreal-istic assumptions such as unlimited manufacturing resources that render theproposed solution of impractical value and inapplicable to real-life businessproblems.

There exists a clearly identified need to address product pricing issues in amore realistic context that is able not only to consider simultaneously multi-product firms competing in an oligopolistic market environment but also pro-pose alternative pricing policies and production modes such as outsourcingoptions. Our proposed methodology of active demand management throughprice optimisation is able to capture the dominant trade-off between productprice and product market share so as to deliver value to the customer whileensuring high profitability for the company.

The rest of the paper is structured as follows. In the next section, the role ofprice as a marketing tool is briefly described, while the main characteristics of anefficient pricing strategy are also discussed. Section 3 presents the case of single-product price competition between two firms. Analytical formulae are derivedfor determining Nash equilibrium in prices while we propose an iterative algo-rithm validated by a motivating example. In Sect. 4, we extend the proposedalgorithm in order to accommodate the case of multi-product firms operatingin an oligopolistic market environment and address customer demand forecastswhile also considering outsourcing options. A number of illustrative examplesare then studied to demonstrate the applicability of the proposed approach.Finally, some concluding remarks are drawn in Sect. 5.

2 Pricing strategy for active demand management

The marketing mix is defined as the set of controllable tactical marketing toolsthat the firm blends to produce the response it wants in the market place. Themarketing mix consists of everything a firm can do to influence the demand forits product. The many possibilities gather into four groups of variables knownas the “four P’s”: product, price, place and promotion (Kotler et al. 1996).

In the narrowest sense, price is the amount of money charged for a product orservice. More broadly, price is the sum of all the values that consumers exchangefor the benefits of having or using a specific product or service. Price is the onlyelement in the marketing mix that produces revenue, while all other elementsrepresent costs (Kotler et al. 1996). Product, promotion and place are value-cre-ating activities while pricing can be viewed as the firm’s attempt to capture someof the created value in the profits earned (Nagle and Holden 1995). Therefore,pricing is identified as the most flexible element of the marketing mix, since itis the fastest and most cost-effective way to enhance company profits.

Every company nowadays is operating with a different set of business objec-tives. Many companies for example set profit maximisation as their ultimate

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goal. Other companies however, seek to increase their market share or eventry to augment their customer satisfaction levels. Different business objectivescan be achieved through the employment of alternative pricing strategies suchas skim pricing, penetration pricing, neutral pricing, etc.

Irrespective of the business objectives, an effective pricing strategy shouldconsider simultaneously the following three main aspects: costs, customers andcompetition. Integrating cost management, customer behaviour and marketcompetition into a unified framework is the key in developing a successfulpricing strategy for active demand management.

2.1 Costs

Costs play a significant role in formulating an efficient pricing strategy. Therecan be variable and/or fixed costs. Manufacturing costs are usually variablecosts depending on the sales volume. Traditional pricing strategies are based ona cost-driven approach as shown in Fig. 2. According to the cost-based pricingstrategy, every product is priced so as to cover its own costs plus make a fair mar-ginal profit. Although such a strategy seems as a simple guide to profitability, inpractice it does not deliver the desired results. The fundamental problem withcost-driven pricing is that unit costs cannot be calculated before determiningthe product price. The reason for that is that pricing affects sales volumes andsales volumes in turn affect unit costs (Nagle and Holden 1995).

2.2 Customers

In order to capture the trade-off between price and sales volume, a value-basedpricing strategy can be employed as shown in Fig. 2.

The main difference in this case is the inverse order of decision-makingallowing for a value-based pricing strategy that is more customer-oriented.Unlike, cost-based pricing, customer’s perceived value is now the driving forcefor product pricing. Conjoint-analysis is a market research tool concernedwith understanding how customers perceive product value and how they make

Customer Value Price Cost Product

Product Cost Price Value Customer

Cost-based Pricing

Value-based Pricing

Fig. 2 Cost-based vs. value-based pricing strategies

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choices between products based on their individual attributes. BPTO (brand-price trade-off) is a variation of the conjoint analysis used for testing pricesensitivity in the context of brands available on the market so as to assess brandpreference at any given price scenario.

Price sensitivity can be measured by using the concepts of demand elas-ticity and cross-elasticity. Price elasticity measures the percentage change inthe quantity demanded relative to the percentage change in price (Pashigian1998). When there exists a certain degree of substitution between differenti-ated products, cross-elasticity can be used to measure the percentage change inthe quantity demanded relative to the percentage change in price of anotherproduct, so as to quantify the competition effect between close substitutesbrands (Pindyck and Rubinfield 1992). Estimating elasticity and cross-elasticityparameters is an active research area while many market research companiesare developing their own methodologies. A paper by Stavins (1997) adoptedfrom the differentiated-product literature is an illustrative example of demandelasticity estimation in the personal computer (PC) market while Acutt andDodgson (1996) present a method for calculating cross-elasticities betweendifferent public transport modes. Shankar and Krishnamurthi (1996) relateprice sensitivity and price policy from a retailer point of view. Alves and Bueno(2003) estimate the cross-elasticity between gasoline and alcohol while Tellis(1988) confirms the negative sign of elasticity parameters. Besanko et al. (1998)provide an example of price elasticity parameter estimation for two productcategories (yogurt and catsup). Their approach is based on weekly sales dataanalysis with main focus on prices and market shares for a 102-week period.Their framework provides explicit estimates of customers’ willingness to payfor a brand while taking into account product price responses.

2.3 Competition

Oligopolistic competition has received a great deal of attention in the researchliterature (Varian 1992). However, the “oligopoly problem” has proved to beone of the most resilient problems in the history of economic thought (Vives1999).

A very early paper written by Hotelling (1929) describes competition amonga small number of firms. His work focuses on spatial competition where thelocations of the products differ. He also makes reference of two earlier devel-oped models that proceed from different assumptions, namely the Cournot(1838) and the Bertrand (1883). According to those models, competing firmsonly act once and also act simultaneously to determine the outcome of compe-tition among them. The Cournot model treats output (quantity) as the strategicdecision variable of each firm while the Bertrand model focuses on price as thestrategic decision variable to be determined by each firm.

Smithies (1941) generalised the theory of spatial competition by assuming anelastic linear demand function at every point of the market and compared differ-ent cases ranging from monopoly to full competition. Smithies also considers

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the effect of the magnitude of the freight rates and changes in marginal costsfor one or both producers.

Kreps and Scheinkman (1983) considered a two-stage duopolistic game. Inthe first stage, the two firms determine their capacities while in the second stagethey engage into a Bertrand-type of price competition subject to the capacitiesconstrains determined previously. Their study emphasises not only the impor-tance of strategic variables selection (quantity vs price) but also the context ofthe (game form) in which those variables are employed.

Apart from the aforementioned Cournot and Bertrand models, oligopolisticcompetition formulations include the repeated and the sequential games. Therepeated games can be viewed as series of Cournot-type or Bertrand-type mod-els not related to each other and solved independently. The sequential gameson the other hand, involve a sequence of decision-making between the firmswhere the outcome of competition derives from the interaction of logic-basedfirm policies. The Stackelberg model (1934), also known as the leader-followermodel, constitutes an extension of the Cournot model that can be classified asa sequential game of oligopolistic competition. Output decisions are taken inturns with the leader-firm making the first move and the follower-firm actingupon observation of the previous move, resulting in a two-stage game.

A well-respected solution concept for non-cooperative games in oligopolisticcompetition is the Nash equilibrium point (Nash 1951) which is defined as thepoint where all players in the game do their best given the choice of all otherplayers. Sherali et al. (1983) study the supply side of an oligopolistic marketsupplying an homogeneous product noncooperatively. They characterise thenature of Stackelberg–Nash–Cournot equilibria and they prescribe methodsfor their computation. Sherali and Leleno (1988) present a mathematical pro-gramming approach for Nash–Cournot equilibrium analysis of oligopolies andderive equilibrium solutions in various market structures.

The coordination of pricing and production decisions in the face of pricecompetition is studied by Zhao and Wang (2002). They examine a supply chainthat consists of a manufacturer and a retailer in a leader-follower setting whereboth firms try to maximise their respective profits. According to their analysis,the Stackelberg solution itself will not lead in general to channel optimality andthey provide managerial insight on how to achieve a channel-optimal pricingpolicy where both competing parties can benefit from. More recently, Parlar andWeng (2006) study the effect of coordinating pricing and production decisionson the improvement of a firm’s position in a price-competitive environment.They formulate game-theoretical models in order to analyse duopolistic com-petition between firms facing price-sensitive demand and manage to quantifythe effects of coordinating price and production decisions.

Choi et al. (1990) present a product pricing and positioning methodology inthe face of price competition. They propose both an analytical and a numericalapproach in order to provide qualitative and quantitative solutions respectively.Despite the difficulty to derive closed form solutions for multi-firms competi-tion, they suggest a numerical solution approach for single-product firms thatresults in an oligopolistic Stackelberg–Nash equilibrium in prices.

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A comprehensive review of theories of oligopolistic behaviour (Shapiro1989) suggests that for similar firms with constant marginal costs and homog-enous products, the only Nash equilibrium in prices (Bertrand equilibrium)exists where each firm prices its product at marginal costs. In our case how-ever, firms are not similar and furthermore their products are not perfect sub-stitutes and therefore the Bertrand equilibria involve prices above marginalcosts.

Our proposed methodology can be classified as a sequential Bertrand-typeprice optimisation approach that aims to determine optimal price levels andproduction plans. Our analysis examines the duopolistic case where two firmsproduce only one product each. Apart from the derived analytical formulae, acomprehensive algorithm is also developed. Furthermore, our proposed meth-odology is extended from single-product to multi-products firms. In additionto market competition, our proposed mathematical is taking into account cus-tomer demand forecasts and also considers outsourcing options. Outsourcingoptions in conjunction with market competition provide an interesting game-theoretical insight into the price competition problem as described in Sect. 4.2.5of this paper.

3 Single-product price competition

In this section, we focus our attention on the specific case of single-productfirms operating in a duopolistic marketplace. Analytical formulae are derivedfor that special oligopolistic case, while we propose an iterative algorithmfor determining optimal product prices. A motivating example is then solvedin order to validate the applicability of both formulae and the proposedalgorithm.

3.1 Analytical formulae

Consider two firms 1 and 2, each one offering a single product to the market.Suppose their products are close substitutes and compete with each other overthe same customer base. However, there is at least some degree of differentia-tion between the two products and therefore each firm faces different demandcurves (Q1, Q2) and different variable (VC1, VC2) and fixed (FC1, FC2) man-ufacturing costs while the products are sold for different prices (P1, P2). Thesales volume for every firm is defined as a linear function of its own price (P1)

and the competitor’s price (P2):

Firm 1: Q1 = a1 − b1 · P1 + c12 · P2 (1a)

and

Firm 2: Q2 = a2 − b2 · P2 + c21 · P1 (1b)

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where a1, a2 are demand coefficients, b1, b2 are demand elasticity parametersand c12, c21 are demand cross-elasticity parameters. All parameters in our for-mulation take positive values.

Note that the quantity each firm can sell decreases when the firm raises itsown price, but increases when its competitor charges a higher price. If bothfirms set their prices at the same time, we can use a Bertrand-type model todetermine the resulting equilibrium. Each firm will choose its own price, takingthe competitor’s price as fixed. The profit of firm 1 equals its revenue minus thevariable and fixed manufacturing costs:

�1 = (P1 −VC1) ·min(Q1, Cap1)− FC1 (2)

where Cap1 is the available capacity. Depending on the values of Q1 and Cap1,the profit of Firm 1 equals to

�1 ={

(P1 −VC1) ·Q1 − FC1 when Q1 < Cap1,(P1 −VC1) · Cap1 − FC1 otherwise

(3)

If Q1 equals Cap1, then

a1 − b1 · P1 + c12 · P2 = Cap1 (4)

and the critical value for the price of Firm 1 equals to

Pc1 =

a1 − Cap1 + c12 · P2

b1(5a)

Similarly for Firm 2:

Pc2 =

a2 − Cap2 + c21 · P1

b2(5b)

3.1.1 Case 1: Unconstrained–unconstrained

In that case both firms have unlimited resource capacity, meaning that Q1 <

Cap1, Q2 < Cap2, P1 > Pc1 and P2 > Pc

2. Using Eq. (2) and substituting Q1from Eq. (1a), the profit for Firm 1 is calculated as follows:

�1 = a1 · P1 − b1 · P21 + c12 · P2 · P1

− a1 ·VC1 + b1 · P1 ·VC1 − c12 · P2 ·VC1 − FC1 (6)

Firm’s 1 profit is maximised when the incremental profit from a very smallincrease in its own price is zero. Taking P2 as fixed, Firm 1’s profit is concave inP1 and therefore the optimal price is given by

∂�1/∂P1 = a1 − 2 · b1 · P1 + b1 ·VC1 + c12 · P2 = 0 (7)

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This can be rewritten to give the following pricing rule or reaction curve forFirm1:

P1 = a1 + b1 ·VC1 + c12 · P2

2 · b1(8a)

This equation dictates the price Firm 1 should set, given the price P2 that Firm 2is setting. Similarly, we can derive the pricing rule (reaction curve) for Firm 2:

P2 = a2 + b2 ·VC2 + c21 · P1

2 · b2(8b)

The point where the two reactions curves cross determines the Nash equilib-rium in prices. At that point, each firm is doing the best it can, given the priceits competitor has set and therefore, neither firm has the incentive to change itsprice.

By substituting Eq. (8b) in (8a), the Nash equilibrium in prices is determinedat point (P∗1, P∗2):

P∗1 =2 · b2 · (a1 +VC1 · b1)+ c12 · (a2 +VC2 · b2)

4 · b1 · b2 − c12 · c21(9a)

and

P∗2 =a2 + b2 ·VC2 + c21 · P∗1

2 · b2(9b)

3.1.2 Case 2: Constrained–constrained

In that case both firms have limited capacity resources meaning that Q1 ≥ Cap1,Q2 ≥ Cap2, P1 ≤ Pc

1 and P2 ≤ Pc2. The resulting profit for Firm 1 is calculated

as follows:

�1 = (P1 −VC1) · Cap1 − FC1 (10)

In that case, the firm’s profit is a monotonically increasing function of price P1.Therefore, reaction curve for Firm 1 is given by the critical price Pc

1.

P1 = a1 − Cap1 + c12 · P2

b1(11a)

and similarly for Firm 2 we have

P2 = a2 − Cap2 + c21 · P1

b2. (11b)

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Again, Nash equilibrium in prices is determined at the point where the tworeactions curves cross each other.

By substituting Eq. (11b) in (11a) in Firm 1’s reaction curve, we get

P∗1 =a1 · b2 − Cap1 · b2 + c12 · a2 − c12 · Cap2

b1 · b2 − c12 · c21(12a)

and

P∗2 =a2 − Cap2 + c21 · P∗1

b2(12b)

Nash equilibrium in prices is determined at point (P∗1, P∗2).

3.1.3 Case 3: Unconstrained–constrained

In that case it is assumed that Firm 1 has unlimited capacity resources (uncon-strained) while Firm 2 has a limited amount of capacity resources (constrained),meaning that Q1 < Cap1, Q2 ≥ Cap2, P1 > Pc

1 and P2 ≤ Pc2. The reaction curve

for Firm 1 is given by Eq. (8a) as in Case 1.While the reaction curve of Firm 2 is calculated is given by Eq. (11b) as in

Case 2.By substituting Eq. (11b) in (8a) the Nash equilibrium in prices is determined

at point (P∗1, P∗2):

P∗1 =a1 · b2 + b2 ·VC1 · b1 + c12 · a2 − c12 · Cap2

2 · b1 · b2 − c12 · c21(13a)

and

P∗2 =a2 − Cap2 + c21 · P∗1

b2(13b)

3.2 Algorithm A1

In the previous section, the analytical form of the Nash equilibrium was derivedfor the case of price competition between two firms that manufacture and selltwo substitute products. Based on the capacity resource levels of each company,three different cases were studied, namely the unconstrained-unconstrained,the constrained-constrained and the unconstrained–constrained case, respec-tively. For each case, the closed form of the resulting Nash equilibrium in priceswas calculated.

In this section, we propose an iterative algorithm [Algorithm A1] able toaccommodate all the aforementioned cases and derive the Nash Equilibriumpoint by employing mathematical programming techniques. In any iteration ofthe algorithm, each company f decides on its individual pricing policy while

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taking into account the price its competitor is currently charging (P0f ). The

pricing decision-making process for each firm is formulated as a non-linearprogramming (NLP) mathematical model that tries to maximise the companyprofit given the competitor’s price. The algorithm terminates when the pricechanges become infinitesimal so that a convergence criterion is satisfied for theprices of both firms. The optimisation problem for company f and rival companyf ′ in iteration m is mathematically formulated as follows:

[Model M1]

max �mf = (Pm

f − VCf ) ·Qmf − FCf

Subject to:

Demand constraints

Qmf = af − bf · Pm

f + cff ′ · Pm−1f ′

Capacity constraints

Qmf ≤ Capf

The proposed algorithm [Algorithm A1] comprises the following steps:

[Algorithm A1]

Step 1. Set price levels to current market prices and initialiseiterations counter m:=0.

Step 2. Set iterations counter to m:= m+1. If m> mmax then STOP.

Step 3. For every company f compute the prices for all products of thecompany using Model M1.

Step 4. If�m

f −�m−1f

�mf

≤ ε for all companies then STOP. Otherwise, go to

Step 2.

The proposed algorithm determines the pricing decision-making process be-tween two competing firms. It should be noted that the equilibrium productprices are computed by a central decision-making by applying algorithm A1.It is also assumed that the central decision-maker has knowledge of the costsand demand functions of all companies involved. Each firm decides on itsoptimal pricing policy while taking into account the observable current pricecharged by its competitor firm. Therefore, the algorithm is able to capture thegame-theoretical nature of the pricing problem and successfully determines thesequential decision-making process between the two firms. The algorithm ter-minates at a point where neither company wants to change its pricing policygiven the price of its competitor. At that point both companies are doing theirbest, therefore neither company wants to deviate from that point and that isby definition, the Nash equilibrium point in prices. The applicability of the pro-posed algorithm is demonstrated by solving a motivating example as describedin the following section.

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Table 1 Additional inputdata for the motivatingexample

Parameter Firm 1 Firm 2

Variable cost (VC) 0.5 0.4Fixed cost (FC) 20 25Capacity (Cap) in Case 1 Unlimited UnlimitedCapacity (Cap) in Case 2 60 50Capacity (Cap) in Case 3 Unlimited 50

3.3 Example for single-product firms

Consider two firms that offer two differentiated products that are close substi-tutes to each other. Suppose that the two companies are facing the followingdemand curves:

Firm 1: Q1 = 160− 30 · P1 + 4 · P2 andFirm 2: Q2 = 180− 40 · P2 + 3 · P1

The additional input data concerning the two companies is shown in Table 1.Three different cases, namely Case 1, Case 2 and Case 3 are examined based onthe capacity resource availability. In Case 1, both firms have unlimited capacityresources. According to the analytical form equations derived in the previ-ous section, the Nash Equilibrium in prices is determined at point (P∗1, P∗2) =(3.09, 2.57). Also, Algorithm A1 successfully predicts the same Equilibriumpoint within less than three iterations depending on the starting point (initialprice vector1) as illustrated in Fig. 3. Most importantly, product prices convergeto the same equilibrium point irrespective of the starting point, thus illustratingthe robustness of the proposed methodology.

In Case 2, both firms have limited amounts of capacity resources, thereforetheir output levels are restricted by the resource availability of every firm.Consequently, the Nash equilibrium in prices is also influenced by the lack ofunlimited resources. According to the theoretically derived equations, Nashequilibrium in prices is now determined at point (P∗1, P∗2) = (3.80, 3.54). Theproposed algorithm derives the exact same equilibrium point irrespective ofthe initial price vector employed as shown in Fig. 4. It is very interesting tonotice that the equilibrium prices in this case are slightly higher than the equi-librium prices in Case 1. This is mainly attributed to the fact that the outputsin Case 2 are restricted to the available resource levels. At the equilibriumpoint, both companies make full utilisation of their resources, producing 50and 60 units of product respectively which are less compared to the equilib-rium outputs in Case 1 (77.6 and 86.6, respectively). In order to compensatefor the decreased output levels, both firms are now forced to raise their pricesso as to maximise their profits. Finally in Case 3, Firm 1 has unlimited amountof capacity resource while Firm 2 has a finite level of capacity resource. Theclosed form equations predict that the Nash equilibrium in prices lies at point

1 For all cases examined, four different initial price vectors (P01, P0

2) were used as follows (1, 1), (2, 2),(3, 3) and (4, 4).

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3.18 3.09 3.09 3.09 3.09 3.09 3.09 3.09

4

2.57 2.57 2.57 2.57 2.57 2.57 2.57 2.57

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 3 5 72 4 6 8

Iteration

Pri

ce le

vel

P1_1 P2_1

P1_2 P2_2

P1_3 P2_3

P1_4 P2_4

Fig. 3 Nash equilibrium in prices (Case 1)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 3 5 72 4 6 8

Iteration

Pri

ce le

vel

43.87 3.81 3.8 3.8 3.8 3.8 3.8 3.8

4

3.54 3.54 3.54 3.54 3.54 3.54 3.543.54

P1_1 P2_1

P1_2 P2_2

P1_3 P2_3

P1_4 P2_4


(P∗1, P∗2) = (3.15, 3.49). The proposed algorithm converges at the exact sameequilibrium point as shown in Fig. 5.

Unlike Cases 1 and 2, the equilibrium price for Firm 1 is now slightly lowerthan the price charged by Firm 2. Firm 2 has a limited capacity resource andtherefore its equilibrium output is restricted to 50 product units. The lack ofresources for Firm 2 is inevitably reflected on the resulting high price. On theother hand, Firm 1 is able to produce a larger output and charge a lower price

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4

3.18 3.15 3.15 3.15 3.15 3.15 3.15 3.15

3.49 3.49 3.49 3.49 3.49 3.49 3.49 3.49

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 3 5 72 4 6 8

Iteration

Pri

ce le

vel

P1_1 P2_1

P1_2 P2_2

P1_3 P2_3

P1_4 P2_4


for its product so as to benefit from the economy of scale. The results from Case3 clearly illustrate what takes place in a real life marketplace, where it is verycommon that big companies supply large quantities in relatively low prices andconsequently outrun the small companies who struggle to cover their costs bycharging high prices.

The Nash equilibrium points for Cases 1, 2 and 3 are summarised in Table 2.In all three cases the proposed algorithm A1 successfully determines the sameNash equilibrium point as the one predicted from the closed form equations.Algorithm A1 is further extended so as to accommodate the case of multi-prod-uct competing firms trying to satisfy the anticipated customer demand forecastwhile considering outsourcing options as described in the next section.

4 Multi-product price competition

In the previous sections, we investigated the case of price competition in aduopolistic market environment where each company is producing only oneproduct. However, process industries nowadays usually operate multi-product

Table 2 Nash equilibrium points for the motivating example

Case 1 Case 2 Case 3

Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2

Price 3.09 2.57 3.80 3.54 3.15 3.49Output 77.6 86.6 60 50 79.47 50Profit 180.89 162.63 178.28 131.77 190.53 129.31

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plants producing a set of differentiated products (e.g. different paints, deter-gents, carbonated drinks, etc.). These products belong to the same family ofproducts (product class) and they share a number of common characteristics(e.g. water-based paints). On the other hand, products are differentiated fromeach other in such a way so as to cover a broad range of customer preferences(e.g. different paint colours/quality).

Market segmentation is a widely used marketing strategy that recognisesthe different ways customers perceive product value and make their purchasechoices accordingly. In order to deliver value to the different existing customersegments, most companies decide to market launch a wide variety of slightlydifferentiated products so as to attract customers via a tailor-based market-ing approach. Each one of the company products is a unique brand name withunique features that clearly differentiates itself from the rest of the family prod-ucts. The unique attributes of each product appeal to a very distinct customerbase that is choosing to buy that specific product over the entire range of prod-ucts present in the marketplace. Customers are willing to buy their preferredproduct as long as the product price charged by the company reflects theirperceived value of the product. Product brand loyalty is expressed by repeatpurchases of the installed customer base. Alternatively, the customer may wellswitch to a lower-priced substitute product offered by the same company or arival company.

Pricing in a multi-product competitive market environment is not an easytask. The analytical formulae presented in Sect. 3 for the two products pric-ing problem cannot be applied to the multi-product pricing problem so as toderive a meaningful Nash equilibrium in prices. In the multi-product case, pricecompetition exists not only between company and competitor products but alsobetween differentiated brands belonging to the very same company. Moreover,company products are manufactured by utilising a common pool of availableresources. Family products are therefore competing with each other for scarceand shared manufacturing resources. Therefore, product cannibalisation effectshave to be seriously taken into account when determining an optimal pricingpolicy. On the same time though, the company has to account for the marketcompetition by considering the pricing policy adopted by the rival company forits products also present in the marketplace. On top of that modern processindustries have recently realised the benefits of adopting outsource manufac-turing policies in an attempt to drive manufacturing costs further down andavoid any unnecessary capacity expansion overheads. Such outsource optionsshould be addressed in a proper manner before deciding on a comprehensivepricing strategy.

Reaction curve analysis cannot be applied in a straightforward way as in theprevious two-products case. However, in order to capture the trade-off betweenproduct price and market share in a multi-product environment, an extension ofthe previously developed non-linear programming (NLP) mathematical modelis proposed. Based on that mathematical model, Algorithm A1 is extended inorder to determine optimal pricing polices for multi-product competing com-panies.

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4.1 Mathematical model

The following nomenclature is used in our mathematical model formulation:

Indicesf companiesi, j productss production sitesr resources

SetsSPf set of products i for company fSf set of production sites s for company fRf set of resources r for company fZf set of products i using resource r at site s in company f

Parametersai demand coefficient for product ibi demand elasticity coefficient for product icij demand cross-elasticity coefficient between products i and jrfcs relative fixed cost coefficient for site srtcs relative transportation cost coefficient for site srvcs relative variable cost coefficient for site sVCi variable manufacturing cost for product iFCi fixed manufacturing cost for product iTCi unit transportation cost for product iOCi unit outsource cost for product iρir unit consumption coefficient for product i using resource rArs availability level of resource r at site sDF total market demand forecast

VariablesPi price for product iVi sales volume for product iQis amount of product i manufactured at site sOi amount of product i outsourced�f total profit for company f

The derivation of the general mathematical model for company f [ModelM2] is described next. The sales volume for every product i is a monotonicallydecreasing function of its price and a monotonically increasing function of theprice of all other competing products, including substitute products belongingto company f as well as competitor products. The sales volume for every producti is given by the following linear function:

Vi = ai − bi · Pi +∑

j �=i

cij · Pj ∀i (14)

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The total sales volume of every product i equals to the amount manufacturedin-house at all production sites belonging to company f plus the amount manu-facturing from outsourcing:

Vi =∑

s∈Sf

Qis +Oi ∀i ∈ SPf (15)

The amount of company products manufactured in-house at every productionsite s is limited by the availability of the shared company resources. The follow-ing constraints safeguard that the resource availability levels are not exceeded:

∑

i∈Zf

ρir ·Qis ≤ Ars ∀r ∈ Rf , s ∈ Sf (16)

Market research surveys are conducted periodically so as to assess the currenttrends and predict future customer demand of a specific product class. The totalsales volume of all products present in the marketplace should be greater orequal to the forecasted customer demand:

∑

i

Vi ≥ DF (17)

The objective function employed in our mathematical model corresponds tothe net profit generated by the subset of the products belong to company f.The net profit is calculated as sales revenue minus the different costs, namelyvariable and fixed manufacturing costs, transportation and outsourcing costs.Mathematically we have

max

�f =∑

i∈SPf

Pi · Vi −∑

i∈SPf

∑

s∈Sf

rvcs ·VCi ·Qis −∑

i∈SPf

∑

s∈Sf

rfcs · FCi

−∑

i∈SPf

∑

s∈Sf

rtcs · TCi ·Qis −∑

i∈SPf

OCi ·Oi (18)

4.1.1 Summary of the mathematical model

In the general case, the optimisation problem for company f is mathematicallyformulated as follows:

[Model M2]max

�f =∑

i∈SPf

Pi · Vi − ∑i∈SPf

∑s∈Sf

rvcs ·VCi ·Qis − ∑i∈SPf

∑s∈Sf

rfcs · FCi

− ∑i∈SPf

∑s∈Sf

rtcs · TCi ·Qis − ∑i∈SPf

OCi ·Oi

106


Subject to:

Vi = ai − bi · Pi +∑

j �=i

cij · Pj ∀i

Vi =∑

s∈Sf

Qis +Oi ∀i ∈ SPf

∑

i∈Zf

ρir ·Qis ≤ Ars ∀r ∈ Rf , s ∈ Sf

∑

i

Vi ≥ DF

Clearly, the restrictions imposed by the analytical formulae are now alleviatedand Model M2 is able to accommodate the case of oligopolistic market compe-tition where more than two firms are competing. Furthermore, every companypresent in the marketplace is allowed to manufacture in-house and/or outsourcemore than one product. The Nash equilibrium in prices is obtained by employingthe same steps as the previously proposed algorithm [Algorithm A1] with thedifference being the optimisation model [Model M2] used for each company.The proposed methodology is able to determine optimal production policiesand prices for all products, as it is demonstrated by the illustrative exampledescribed in the next section.

4.2 Illustrative examples for multi-product firms

Consider two firms, namely company A and company B that manufacture andsell products P1, P2, P3, P4 and P5, P6, P7, respectively, as shown in Fig. 1.Products P1–P7 are close substitutes to each other and therefore each producthas a unique demand function curve associated with it, as described by Eq. (14).Demand coefficients include parameter αi, elasticity bi and cross-elasticity cijparameters as shown in Tables 3 and 4. Every company has two available man-ufacturing sites (sites 1 and 2 for company A and sites 3 and 4 for company B).The products can be manufactured in-house by using shared manufacturingresources available at each site (in-house manufacturing). Resource utilisationcoefficients for every product are given at Table 5 while resource availabilitylevels for every resource at each site are given at Table 6. Manufacturing sitesare geographically distributed facilities, therefore relative manufacturing costand transportation cost coefficient are used so as to capture the effect of differ-ent manufacturing locations (see Table 7). Final products are transported fromthe manufacturing sites to the end-customers at a given transportation cost TCi.Alternatively, a certain amount of production can be outsourced to a third-partycompany at a given outsource cost OCi. Note that since we are dealing withproducts belonging to the same product class, fixed costs are assumed to bethe same for all products, therefore they are not considered explicitly in theillustrative example.

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Table 3 Product input datafor the illustrative example Product ai bi VCi TCi OCi P0

i

P1 160 25 4 1 5.3 8P2 200 30 3 1 5.3 10P3 150 25 4 1 5.3 9P4 120 20 5 1 5.3 11P5 170 30 3 1 4.2 9P6 110 25 4 1 4.2 10P7 180 30 3 1 4.2 8

Table 4 Cross-elasticityparameters (cij) for theillustrative example

Product P1 P2 P3 P4 P5 P6 P7

P1 – 4 3 2 6 7 4P2 2 – 5 3 5 4 2P3 3 3 – 2 3 2 5P4 4 2 4 – 2 6 3P5 2 4 3 5 – 3 2P6 5 3 4 3 2 – 3P7 3 3 2 2 4 3 –

Table 5 Resource utilisationcoefficients

Product res1 res2 res3 res4 res5 res6

P1 1 1.1 0.8 – – –P2 0.7 1.2 0.7 – – –P3 1.2 1.4 0.9 – – –P4 1.1 1.3 0.4 – – –P5 – – – 0.9 1.2 0.7P6 – – – 0.8 1.4 0.6P7 – – – 1 1.6 0.8

Table 6 Initial resourceavailability levels acrossmanufacturing sites

Resources Site 1 Site 2 Site 3 Site 4

Res1 200 340 – –Res2 300 370 – –Res3 150 270 – –Res4 – – 140 150Res5 – – 210 220Res6 – – 110 130

Table 7 Manufacturing siterelated data

Manufacturing. Site rvcs rtcs

Site 1 1 1Site 2 0.8 1.2Site 3 0.7 1.4Site 4 0.9 1.1

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Table 8 Initial state ofcompany A

Products Site 1 Site 2 Outsource

P1 – 205 –P2 95 – –P3 68 20 –P4 – 90 –

Total 163 315 0

Table 9 Initial state ofcompany B

Products Site 3 Site 4 Outsource

P5 84 – –P6 41 – –P7 32 68 –

Total 157 68 0

Given an initial price vector (current market prices P0i ) for all products, the

problem is to determine optimal product prices, output levels and outsourceamounts so as to derive a comprehensive Nash equilibrium point for companiesA and B that neither company would wish to deviate from.

Initially, both companies A and B manufacture their products in-house byonly relying on the manufacturing capabilities of their production sites while nooutsourcing is considered. In particular, the allocation of production betweenthe different sites is shown in Tables 8 and 9. The total amount of sales forthe specific product class equals the combined manufacturing volume of bothcompanies (703 units). Given the initial price vector and output levels for allproducts, the initial profit is 2,194 rmu2 and 1,129 rmu for company A and B,respectively.

A recent market research survey has estimated future customer demand forthe product family under investigation to be equal to 712 units and thereforethe two companies are competing over the anticipated customer demand. Everycompany has the strategic choice to consider outsourcing options or rely entirelyon its own in-house manufacturing capabilities, thus resulting in four distinctcases as explained in the following sections. All four cases were implementedin GAMS (Brooke et al. 1998) using the CONOPT NLP solver (Drud 1985)while all runs were performed on an IBM RS/6000 workstation.

4.2.1 Case 1: In-house/in-house

In this case both companies A and B manufacture their products in-house whileno outsourcing is allowed to take place. Model M2 is solved with the outsourcevariable fixed to zero for both companies. As shown in Fig. 6, Nash equilibriumis reached after five iterations resulting in profits 2,259 rmu and 1,262 rmu for

2 rmu = relative monetary units.

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21942405

2250 2259 2259 2259

11291211 1261 1262 1262 1262

0

500

1000

1500

2000

2500

3000

543210

Iteration

Pro

fit

Lev

el

Company ACompany B

Fig. 6 Nash equilibrium for Case 1

0

2

4

6

8

10

12

Pri

ce L

evel

P1 P2 P3 P4 P5 P6 P7

Product

Initial PriceEquilibrium Price

Fig. 7 Product prices for Case 1

company A and B, respectively. Optimal product price levels are determinedas illustrated in Fig. 7. More specifically, equilibrium prices for P1 and P3 lieabove their original levels while a price decline is suggested for products P2, P5and P6. Finally, the optimal prices of products P4 and P7 are very close to theiroriginal values.

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Table 10 Nash equilibrium results across all cases

Company A Company B

Profit P1 P2 P3 P4 Profit P5 P6 P7

Initial 2194 8 10 9 11 1129 9 10 8Case 1 2259 10.46 8.52 9.50 11.06 1262 8.09 8.74 7.78Case 2 2241 10.45 8.54 9.51 11.05 1284 8.07 8.57 7.78Case 3 2288 10.49 8.55 9.53 10.92 1259 8.08 8.74 7.77Case 4 2269 10.48 8.57 9.54 10.90 1281 8.06 8.56 7.77

4.2.2 Case 2: In-house/outsource

In this case company A manufactures its products entirely in-house while com-pany B has the option to outsource a certain amount of production. Model M2is solved with the outsource variable fixed to zero only for company A. Start-ing from the same initial state as in case 1, Nash equilibrium results in profits2,241 rmu and 1,284 rmu for company A and B, respectively. In this case, com-pany B outsources 84 units of product P6. Optimal product prices are shown inTable 10.

4.2.3 Case 3: Outsource/in-house

This case is the exact inverse of case 2. Company B manufactures all of its prod-ucts in-house while company A has the option to outsource a certain amountof production. According to the results, profits of 2,288 rmu and 1,259 rmu forcompany A and B, respectively, are achieved at the Nash equilibrium point.Company A outsources 91 units of product P4 while optimal product prices aregiven in Table 10.

4.2.4 Case 4: Outsource/outsource

In this case both companies A and B have the option to manufacture productsin-house and/or outsource a certain amount. Nash equilibrium results in profits2,269 rmu and 1,281 rmu for company A and B, respectively. In this case, com-pany A outsources 90 units of product P4 and company B outsources 84 unitsof product P6. Optimal product prices can be found in Table 10. The alloca-tion of production between manufacturing sites and outsourcing in Case 4 aregiven as pie charts in Fig. 8 for both companies. Notice that the largest share ofproduction is allocated to site 2 and site 3 since they both offer low variable man-ufacturing cost compared with sites 1 and site 4, respectively. According to theobtained results, outsourcing activity constitutes over 20% of total productionfor both companies.

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Company A

Site 15%

Site 273%

Outsource22%

Company B

Site 352%

Site 420%

Outsource28%

Fig. 8 Allocation of production in Case 4

4.2.5 Game-theoretical insight

In the previous subsections we examined four different cases of duopolisticcompetition. By comparing the derived Nash equilibria, useful insight can begained from a game-theoretical point of view. The duopolistic game under inves-tigation is defined as follows. The competing companies are regarded as twoplayers. Each player in the game has a number of possible strategies, courses ofaction that he may choose to follow. In our particular case, companies have thechoice to either produce their products entirely in their own manufacturing sites(in-house strategy) or produce a certain amount in-house while outsourcing acertain percentage of production (outsource strategy). The strategies chosen byeach player determine the so-called outcomes of the game. In our example, weend up with four different outcomes, namely in-house/in-house, in-house/out-source, outsource/in-house and outsource/outsource, each one representing acase examined in the previous sections. In every formally stated game, there isa collection of numerical payoffs, one to each player, associated with every pos-sible outcome of the game. Those payoffs represent the value of the outcome tothe different players. In our example, Nash equilibrium profits can play the roleof companies payoffs for every particular case examined. Overall, we are deal-ing with a two-person game with two strategies per player and a game payoffmatrix as shown in Table 11. The values in parentheses are the Nash equilibriumprofits determined previously for all four cases, with the first number being theprofit for company A and the second one the profit of company B.

Game theory is the study of how players should rationally play games. Eachplayer would like the game to end in an outcome which offers him the largestpossible payoff. He has some control over the outcome, since his choice ofstrategy will influence it. However, the outcome is not determined by his choicealone, but also depends upon the choices of all other players. In general, there

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Table 11 Game payoff matrix

Company A

In-house Outsource

Company B In-house Case 1 (2,259;1,262) Case 3 (2,288;1,259)Outsource Case 2 (2,241;1,284) Case 4 (2,269;1,281)

might be conflict because different players value outcomes differently (Straffin1993).

In our example companies are faced with the question of which strategyto adopt in order to reach the Nash Equilibrium associated with the highestprofit for the company under investigation. First, let us consider company A.Company A does not have any indication of which policy rival company B willadopt. If company B adopts a strictly in-house manufacturing policy, then com-pany A has a choice between Case 1 and 3. Since the profit for company A inCase 3 is higher than the one in Case 1 (2,288 vs 2,259), company A decides toadopt an outsourcing strategy. In case company B adopts an outsourcing policy,company A has a choice between Case 2 and 4. Case 4 offers company A with aprofit of 2,269 which is higher that the Case 2 profit (2,241). So, in both scenar-ios, company A is better off by choosing to outsource a certain amount of itsproduction, irrespective of the production policy adopted by rival company B.Similarly, we can prove that the exact same rule applies for company B as well.Without any prior knowledge of the production policy adopted by company A,company B always earns a higher profit by adopting an outsourcing strategy.

It is very interesting to notice that Case 3 provides the highest profit forcompany A while Case 2 provides the highest profit for company B. However,Case 4 is considered to be the most likely outcome of the game since the out-source/outsource policy guarantees higher profits for both companies no matterwhat policy the rival company decides to adopt, thus providing a robust Nashequilibrium for both companies.

5 Concluding remarks

A systematic mathematical programming approach for active demand manage-ment through price optimisation was presented in this paper. First, we derivedanalytical formulae for calculating Nash equilibrium points in a duopolisticmarket environment where each company produces and sells only one product.An iterative algorithm was then proposed that derived the exactly same equi-librium points as predicted by the closed-form formulae. Following that, theproposed algorithm was further extended in order to accommodate the caseof multi-product firms and also consider additional features such as customerdemand forecast and mixed in-house and outsourcing production policies. Anillustrative example was solved in order to demonstrate the applicability of the

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proposed methodology across four different case studies. Finally, a comparisonamong the different cases provided us with valuable game-theoretical insightconcerning the problem of duopolistic competition coupled with outsourcingoptions.

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Customer segmentation, allocation planning and orderpromising in make-to-stock production

Herbert Meyr

Abstract Modern advanced planning systems offer the technical prerequisites foran allocation of “available-to-promise” (ATP) quantities—i.e. not yet reserved stockand planned production quantities—to different customer segments and for a realtime promising of incoming customer orders (ATP consumption) respecting allocatedquota. The basic idea of ATP allocation is to increase revenues by means of customersegmentation, as it has successfully been practiced in the airline industry. However,as far as manufacturing industries and make-to-stock production are concerned, it isunclear, whether, when, why and how much benefits actually arise. Using practical dataof the lighting industry as an example, this paper reveals such potential benefits. Fur-thermore, it shows how the current practice of rule-based allocation and consumptioncan be improved by means of up-to-date demand information and changed customersegmentation. Deterministic linear programming models for ATP allocation and ATPconsumption are proposed. Their application is tested in simulation runs using thelighting data. The results are compared with conventional real time order promisingwith(out) customer segmentation and with batch assignment of customer orders. Thisresearch shows that—also in make-to-stock manufacturing industries—customer seg-mentation can indeed improve profits substantially if customer heterogeneity is highenough and reliable information about ATP supply and customer demand is available.Surprisingly, the choice of an appropriate number of priority classes appears moreimportant than the selection of the ATP consumption policy or the clustering methodto be applied.

H. Meyr (B)Chair of Production and Supply Chain Management,Technical University of Darmstadt,Hochschulstr. 1, 64289 Darmstadt, Germanye-mail: [email protected]



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OR Spectrum (2009) 31:229–256DOI 10.1007/s00291-008-0123-x


H. Meyr

Keywords Available-to-Promise (ATP) · Advanced planning systems · Clustering ·Integer and linear programming · Order promising

1 Introduction

One of the biggest challenges in airline industry is to avoid that a hasty, high marginbusiness class customer cannot get a seat because a low price economy customerhas booked the last one a few minutes ago. Revenue management has developedtechniques to treat such problems adequately, e.g. to establish and fence off customersegments in form of booking classes and to determine booking limits. The situation isdifferent in make-to-stock (MTS) supply chains of consumer goods industries wherefinal item stocks are built up on basis of forecasts and customer requests are served fromthis stock. But not too different. Here, too, exist more important and less importantcustomers yielding higher and lower profit margins. Here, too, occur shortages. Anda service level of 98 percent also implies that two percent of the customers have notbeen served as desired. This may concern several dozens of orders per day, for a singleitem only. “Not as desired” not necessarily means that the customers are not suppliedat all. However, late deliveries lead to customer annoyance and customer migration inthe long term. Thus here, too, it is important to consider carefully who gets its goodson time and—even more crucially—who does not.

Actually “order promising”, i.e. communicating the customer a reliable andhopefully soon delivery date, is the planning task to be considered. However, inMTS situations order promising also means deciding about— and for short-termorders simultaneously releasing—delivery (see Fleischmann and Meyr 2003a). Thus,these decisions about actual deployment can hardly be re-thought. In order to promisereliable delivery dates, modern enterprise resources planning (ERP) systems or advan-ced planning systems (APS) build on up-to-date information about stock on hand andplanned supply of the distribution centers that both not yet have been assigned tocustomers. Such unreserved quantities are called “available-to-promise”(ATP). Sinceproduction has to be planned on basis of forecasts (push concept implied by MTS),unused production capacity, sometimes called “capable–to–promise”, and stock re–filling are no more concern at this point in time. The information about the plannedsupply of the distribution centers either stems from the short–term master productionschedule of a single, corresponding production plant or—for a longer preview—evenfrom a mid–term production and delivery plan (“master plan”) of the overall supplychain (see e.g. Kilger and Meyr 2008).

Usually two different modes of promising ATP to incoming customer orders aredistinguished, “batch order processing” and real time “single order processing” (seee.g. Ball et al. 2004; Fleischmann and Meyr 2003a; Pibernik 2005). In batch mode,an order is not promised immediately upon request, but held back. It is then assignedto ATP inventories together with several other orders in a “batch”. Thus, there mustbe enough time to gather these orders and a customer must be willing to wait for ananswer. Often, this “batching horizon” comprises several hours or a whole day.

Sometimes customers expect an immediate answer for their order query. In thiscase batching of orders is not possible. Thus, each single order has to be processed inreal time and ATP is consumed in a first-come-first-served (FCFS) manner.

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Customer segmentation, allocation planning

As addressed in the airline example above, in shortage situations, where demandis higher than capacities—i.e. in this case than ATP inventories—single order proces-sing entails the danger of promising scarce inventory to the wrong customers, e.g. toless important customers or to customers showing smaller profit margins. Allocationplanning, as propagated by APS vendors like i2 and SAP (see Kilger and Meyr 2008),promises to be a way to improve real time single order processing by reserving sharesof the ATP, the so-called “quotas” or “allocated ATP”, for important customers in themedium term and afterward promising orders with respect to these allocated quotasin the short term. That means ATP is held back in anticipation of later arriving, moreprofitable orders even if a less profitable order already requests this stock. Such anallocation of quotas shall take advantage of a customer segmentation into low andhigh priority customers as it has shown to be successful in airline industries. Thisleads to a two step ATP allocation and ATP consumption process, in the followingcalled “allocation planning and ATP consumption” (AP&C).

It is important to note that such a segmentation already appears useful if the sameproduct is sold for different profits or with different priorities. For example, varioussales channels might generate different profit margins because sales prices vary due tocountry-specific tax levels or due to differing transport costs. Or in-house customersmight show other strategic importance for a company than external customers. All inall, the AP&C approach promises to be useful for companies which produce storablestandard products in high volume on an MTS basis and whose multitude of customersare heterogeneous in the above sense. Then, there is the hope that the same or evenbetter profits as in batch mode can be achieved, even though a customer gets his answerimmediately.

The intention of this paper is to structure the AP&C process and to reveal thepotential benefits of allocation planning as compared to the common practices of FCFSsingle order processing or batch order processing. However, before the contributionof the paper can be specified in some more detail, a brief review of current practicesand existing literature is necessary.

1.1 Literature review

For a literature review we will concentrate on ATP support for commercial (ERP and)advanced planning systems and especially discuss papers which tackle ATP allocationor consumption in more detail. Note that we focus on MTS situations, i.e. the ATPsupply of finished items is assumed to be fixed because it bases on the stock on handand on the production quantities that have already been planned in the short-termproduction scheduling module of the APS and/or a mid–term master planning module(see e.g. Meyr et al. 2008). This rules out literature on make–to–order (MTO) andassemble–to–order (ATO) supply chains, which most of the due date setting (see e.g.Keskinocak and Tayur 2004) and batch order promising models (see e.g. Chen et al.2001, 2002) have been developed for. In these situations customers are usually willingto wait longer for an order promise than in MTS supply chains. This also rules outinventory rationing (see e.g. de Vericourt et al. 2002), which explicitly allocates stockson hand to several customer classes, but assumes that the refilling of the stock can stillbe influenced by means of orders. Finally, it also excludes revenue management (see

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e.g. Talluri and Van Ryzin 2004), where “capacities” are assumed to be perishableand thus stocks cannot be held at all. A deeper discussion of the relationship betweenthese various, but similar types of models and their applications in industry wouldgo beyond the scope of this paper. Instead, the reader who is interested is referred toQuante et al. (2008).

Demand fulfillment and order promising on the basis of ATP information is oneof the most popular planning tasks (see Kilger and Wetterauer 2008, Table 16.1)covered by commercial APS. A general overview regarding APS and the role of ATPtherein is given by Fleischmann and Meyr (2003b) and Stadtler and Kilger (2008).Fleischmann and Meyr (2003a) classify different situations of demand fulfillment withrespect to the three order penetration points MTO, ATO and MTS. They also pointout that—as opposite to MTO and ATO — in MTS situations it often is sufficientto consider each product separately. Pibernik (2005) also characterizes different ATPapplications and models. He implicitly uses a similar categorization by distinguishingthe operating mode (real time/batch), the availability level of goods and the interactionwith manufacturing planning, where the two latter ones are usually used to characterizethe different order penetration points. ATP software modules of several APS vendorsare presented by Meyr et al. (2008). Dickersbach (2004, Sect. 11) and Knolmayeret al. (2002, Sect. 3.1.5), however, put a special emphasis on the Global ATP moduleof SAP’s advanced planner and optimizer (APO).

The paper of Kilger and Meyr (2008) is basic for the following sections because itpresents the implementation of demand fulfillment in APS in a sufficiently high detail.Kilger and Meyr (2008) especially describe the simple rules that are usually appliedin APS for both allocation planning (Sect. 9.4) and ATP consumption (Sect. 9.5).Whereas their argumentation mainly bases on experiences with software of the APSvendor i2, Dickersbach (2004, Sects. 11.2 and 11.3) shows that a similar approachhas also been favored by SAP/APO. Allocation planning rules, for example, quotean overall ATP quantity to different customer classes on basis of priority rankings,with respect to some pre-defined fixed shares or proportional to the original forecastsof different customers or markets. ATP consumption rules, for instance, allow accessto allocated ATP of an order’s corresponding class or to ATP of classes showinglower priority. If customers have not been segmented—and thus the above allocationplanning is useless—ATP that has been assigned to other time buckets, to substituteproducts or to other locations (e.g. distribution centers or regional warehouses) issearched for in an user–defined sequence.

Fischer (2001) compares such ATP consumption rules for single order processingwith a linear programming (LP) based batch order processing for a practical caseof the lighting industry and shows advantages of the batch mode. It is interesting tonote that this lighting company originally distinguished eight classes of customersshowing different importance, which have—for sake of simplicity — been reduced tothree by Fischer. In a similar MTS environment Pibernik (2006) compares differentATP consumption rules for managing the stock outs of a pharmaceutical company.He suggests to change from a single order to a batch order processing mode only ifshortage is foreseeable. Even though this company also segments their customers intofive priority groups, allocation planning is tested only rudimentarily by Pibernik, usinga “naive” allocation scheme reserving stock for the two most important groups only.

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As mentioned above, the APS allocation rules either make no assumptions aboutdemand (for example priority rankings) or use short–term demand forecasts in a ratherdoubtful manner, e.g. by allocating production quantities and ATP proportionally tothe demand forecasts, which has been shown to increase the bullwhip effect withinsupply chains (see Lee et al. 1997). Instead, Ball et al. (2004, Chap. 15.4.2) propose anLP based deterministic allocation model. Basically, it summarizes linear and mixedinteger programming models of hierarchical production planning that are used toallocate aggregate inventory of product families and/or limited production capacity tovarious items within a family. Obviously, this general idea can be transferred to allocateATP to different customer classes. Although the model proposed by Ball et al. oughtto be applied in an MTS environment, it rather fits ATO supply chains because it alsodecides about raw material and capacity usage. A more convenient MTS applicationof this type of models is presented below in Sect. 2.3.

Summing up, modern APS offer the technical prerequisites for ATP allocation andATP consumption, thus hoping to gain similar advantages in manufacturing industriesas have been achieved by revenue management principles in airline or hotel indus-tries. However, they only provide very simple allocation and consumption rules, andfurthermore do not give advices how and when to apply them. Thus, overall benefitsare doubtful. Looking through scientific literature is hardly helpful in this specificsituation because either the model assumptions do not fit (e.g. stochastic inventoryrationing) or the overall performance of both allocation and consumption policieshas not been tested for potential alternatives of customer segmentation (for example,(Fischer 2001; Pibernik 2005) take the segmentation for granted).

1.2 Contribution and organization of the paper

The basic idea of this paper is to improve demand fulfillment in MTS supply chainsby making use of the heterogeneity of different customers through AP&C order pro-mising. The fundamental steps are:

• To segment customers with respect to their importance and profitability into severalpriority classes,

• to allocate ATP to these classes on basis of a deterministic profit maximizationprocess taking advantage of short–term demand information, and

• to promise customer orders, i.e. to consume ATP, in real time with respect to thesecustomer hierarchies.

In order to demonstrate the usefulness, all steps will be executed in a holistic simulationexperiment exploiting practical data of the lighting industry. To our knowledge, sucha comprehensive test, including customer segmentation and allocation, is missing sofar. The aim is to structure the planning tasks concerned with AP&C and to gainideas whether and how a preceding allocation process—making use of the short–terminformation provided by APS—may be advantageous compared to the traditionalfirst-come-first-served single order processing.

The next section introduces appropriate LP models for demand fulfillment in MTSsupply chains. Numerical experiments with data of the lighting industry are run in

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Sect. 3. A summary of the methodology proposed and of the managerial insightsgained concludes the paper.

2 Model formulations

The following section describes the modeling environment that allows to compare thedifferent ways of order promising and ATP assignment. LP models for single and batchorder processing without customer segmentation are proposed in Sect. 2.2, whereasSect. 2.3 introduces the allocation planning model making use of segmentation. Allmodels aim at profit maximization. Their outcome can be compared directly withthe optimal profit that would result from a simultaneous ex–post assignment of allorders arriving within the planning horizon, which is called “global optimization” inthe following.

2.1 Modeling environment

The different order promising alternatives verbally described in the introduction willnow be represented by mathematical models. Figure 1 shows the modeling environment

a-c without customer segmentation:

supply planning(e.g. production

planning)

ATP consumption(“order promising”)

“customer”

ATP

order(s) commit-ment(s)

a) GO: once for all orders of the planning horizon T

b) BOP: several times for all ordersof a batching horizon B<<T

c) SOP: in real-time for each single order

supply planning

ATP allocation(“allocation planning”, AP)

“customer”

ATP

singleorder

singlecommitment

demand planning

forecasts

ATP consumption (SOPA)

allocated ATP(= quotas

for customerclasses k)

d) with customer segmentation:

once

once

real-time

real-time

once forplanninghorizon T

real-time

a-c

Fig. 1 Modeling environment for the models “Global Optimization” (GO), “Batch Order Processing”(BOP) and “Single Order Processing” (SOP) without customer segmentation and “Allocation Planning”(AP) and “SOP after allocation planning” (SOPA) with customer segmentation

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that was chosen to do this. The models (a–c) that do not distinguish customer segmentsshall be compared with the AP&C models (d) which put the revenue management ideaof the introductory example into practice by differentiating different customer classesk, allocating ATP to these customer classes and satisfying customer demand only ifenough allocated ATP of the customer’s corresponding class is available.

For this, the finite, overall planning horizon T is subdivided into discrete timebuckets t = 1, . . . , T . Once, at the beginning of planning (t = 0), on the basis ofsupply information—e.g. from the master production schedule or master plan—it iscalculated how much ATP becomes available in each period t (for this calculation seee.g. Fleischmann and Meyr 2003a,b). Customer orders i arrive one after each otherat different arrival dates ai . For each order i it is known, how much the customerwants to get (“requested delivery quantity” qi ) and when he wants to get this quantity(“requested delivery date” di , i.e. the time bucket t , for which the customer requestshis order i to be delivered). The limited availability of ATP necessitates that not allorders can be served on time. The “order promising” or “ATP consumption” problemis to decide whether, when and to which degree each order will be served from theATP. Not fulfilling an order on time or not filling an order at all will be punished bypenalty costs diminishing the original profit the order would leave. ATP is assumed tobe known deterministically at t = 0, for the whole planning horizon T . Thus it needsonly to be updated when orders are accepted but not because its supply has changedunexpectedly.

The models (a–c) without customer segmentation differ according to the number oforders that are gathered before the orders are processed, i.e. assigned to the differentperiods’ ATP by means of an LP model maximizing the profit of all incoming orders.The “Single Order Processing” model SOP processes each order immediately in realtime and thus is trivial to be solved. The “Batch Order Processing” model BOP gathersall orders arriving within a batching horizon B � T . The “Global optimization” modelGO gathers all orders of the whole planning horizon T . Of course, since T is a quitelong time span (e.g. a month) it is not realistic that customers will wait so long untilgetting a promise. However, because all orders of the whole planning horizon arecovered and optimized simultaneously, this model can serve as a benchmark to judgethe performance of an iterative application of the other models.

Situation (d) is modeled by a sequence of an “Allocation Planning” model AP thatis executed once at t = 0 and several single order processing models—now denoted as“Single Order Processing After allocation planning” (SOPA)—which are executed inreal time when each new customer order arrives. The AP model once allocates ATP tothe different, a priori known customer classes k by means of linear programming. Forthis, up-to-date forecasts of customer demand within each customer class are necessary.Like in (c) each single order is processed in real time, but it is only allocated to thedesired delivery date if enough allocated ATP (aATP) of its respective customer classis available and can be consumed. This corresponds to the revenue management andinventory rationing idea that some portion of scarce stock should be held back formore important orders which might arrive later on.

The motivation for this kind of deterministic, mathematical modeling originatesfrom current practice of APS usage (see e.g. Kilger and Meyr 2008). ATP and demandforecasts are calculated in APS anyway and can be aggregated for different customer

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classes. Also basic allocation and consumption rules are used. Thus, the fundamen-tal technical framework for its application already exists. Furthermore, LP as a moresophisticated allocation method could probably easily be implemented because it isused for mid–term master planning and strategic network design, anyway (Fleisch-mann and Meyr 2003b).

Of course, very simplifying assumptions are made in this modeling environment ascompared to practice. For example partial delivery of orders is assumed to be possible,no bargaining about delivery dates is allowed, customer service can only be expressedin terms of money and uncertainties of demand and supply are excluded. The latterproblem, for instance, could be tackled by introducing a rolling horizon planning onat least two planning levels: a mid-term (e.g. weekly rolling) level for updating supplyinformation and executing allocation planning and a short-term (e.g. daily rolling forBOP or real-time for SOP/SOPA) level for ATP consumption. In this case ATP updateswould be necessary weekly after each update of supply information, but also dailyor for each order (see e.g. Fleischmann and Meyr 2003a, for ATP re-calculation).AP runs would also be necessary weekly, after each supply and subsequent ATPupdate, and would base on the latest forecasts on customer demand. However, a moredetailed discussion of these application issues would go beyond the scope of thispaper because, first, structural insights on the negative impacts of the decompositionof the GO problem into subsequent BOP, SOP or AP/SOPA models should be gained.Thus, the restrictive assumptions are necessary to exclude side effects, e.g. due to badforecasting of supply and demand. Of course, in a next step, these assumptions shouldbe weakened (see Sect. 4).

In the following, the situation (a–c) without customer segmentation is described inmore detail by introducing a single, “basic” order promising model that is applied indifferent ways to gain the models GO, BOP and SOP.

2.2 Models without customer segmentation

The basic order promising model is a simple network flow problem where the requestedquantities qi —in the following also called “demand”—of certain customer ordersi = 1, . . . , I have to be satisfied by ATP inventories AT Pt that become availablein discrete periods t = 1, . . . , T , e.g. days or weeks. In order to ensure feasibilityeven if demand is higher than ATP inventory, a fictitious period T + 1 has beenintroduced being able to serve the surplus demand by setting AT PT+1 :=∑I

i=1 qi −∑Tt=1 AT Pt . The goal is to find the part oit of order i that has to be satisfied by

ATP of period t so that the overall profit is maximized for a given per unit profitpit . This per unit profit can, for example, be computed by subtracting the per unitcosts ci from the per unit revenues ei of the order i and by punishing the use of ATPfrom periods earlier (necessitating storage) or later (backlogging) than the customer’srequested delivery date di . ATP of the fictitious period T +1 models non-delivery andthus cannot generate any profit (pi, T+1 = 0). It may even cause a loss of goodwillbeing punished by negative profits pi, T+1 < 0. Note that costs and revenues ofdifferent customers/orders may vary individually, e.g. due to different transportation

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Table 1 Indices, data and variables of the basic order promising model

Indices

s = 1, . . . , S Iterations

i, j = 1, . . . , I Orders

t = 1, . . . , T Periods

t = T + 1 Dummy period with “infinite” supply

I s Set of orders that are promised in iteration s

Data

ai Arrival date of order i (i.e. when customer requests a promise)

di Date, the customer requests order i to be delivered

qi Quantity, the customer requests to get delivered by order i [SKU]

ei Per unit revenue of order i [$/SKU]

ci Per unit supply costs of order i (e.g. transportation costs) [$/SKU]

AT Pst Not yet assigned supply that becomes available in period t and

can still be promised to customers during iteration s[SKU]

pit Per unit profit of order i if satisfied by ATP of period t [$/SKU]

= ei − ci

- “low holding costs” if t < di , and

- backlogging costs if di < t ≤ T , respectively

= 0 if t = T + 1

(or -penalty costs for loss of goodwill)

Variables

osit ≥ 0 Part of order i which is served by ATP of period t and promised

during iteration s (only defined for i ∈ I s )[SKU]

costs and customized sales prices, which have already been negotiated in the mediumterm.

In practice, orders arrive successively with a continuous arrival date/time ai . Thisdynamic situation will later on be modeled by a simulation run with successive itera-tions s = 1, . . . , S. At a certain point in time, i.e. in a certain iteration s, only a limitedsubset I s of all orders i = 1, . . . , I is usually known and has not yet been promised,e.g. a single order in the SOP case or a batch of all orders of a single day in the BOPcase. Thus, the LP formulation of the basic order promising model shown below isrestricted to this subset I s of orders for a given iteration s. For ease of readability,Table 1 summarizes the indices, data and variables of the LP model. The superscriptss of the data AT Pt indicate that—after consumption in iteration (s − 1)—the ATPremaining for iteration s had to be reduced, accordingly. Whereas, the superscripts sof the variables oit indicate in which iteration s the corresponding order i has beenpromised.Basic order promising model of iteration s:

maximizeT+1∑

i∈I s ,t=1

pit osit (1)

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subject to

T+1∑

t=1

osit = qi ∀ i ∈ I s (2)

∑

i∈I s

osi t ≤ ATPs

t ∀ t = 1, . . . , T (3)

The overall profits of satisfying the orders i ∈ I s from ATP inventory are maximizedby the objective function (1). The requested quantity qi of each order i has to be metexactly, either by “real” supply of a regular period t = 1, . . . , T or by the “fictitioussupply” modeling non–delivery (2). Constraints (3) ensure that the supply capacitycannot be exceeded, i.e. that only the still available ATP of period t can be assignedto yet unpromised orders i ∈ I s .

This basic order promising model will be applied for simulating the three scenarios(a), (b) and (c) of Fig. 1. With i(s) := argmini {ai : i ∈ I s} denoting the order i ∈ I s

having the earliest arrival date during iteration s, the following three situations—justdiffering by the cardinality |I s | of the subsets I s—can be distinguished:

(a) GO: all orders of the planning horizon T are known in advance and are consideredin a single optimization run, i.e. I s := {1, . . . , I } and S := 1.

(b) BOP: only subsets of orders within a “batching horizon” of B periods are consi-

dered, i.e. I s :={

i :⌊

ai(s)

⌋≤ ai <

⌊ai(s)

⌋+ B

}and S := T/B with

⌊ai(s)

⌋

denoting the period t the arrival of order ai(s) is assigned to (assuming that T isan integer multiple of B).

(c) SOP: only a single order is considered during an iteration s, i.e. I s := {i(s)} andS := I . This is the case for real time due date assignment on an FCFS basis.

Since the degree of freedom decreases, it is expected that the overall objective functionvalues of these models decrease, too, i.e. GO� ≥ ∑T/B

s=1 BOP�s ≥

∑Is=1 SOP�

s witha � denoting the optimal solution of a model. As already mentioned, GO� can serveas a benchmark (“first best solution”), showing what profit would be optimal if therewere perfect knowledge of customer demand for the whole planning horizon T . Thevalues SOP� := ∑I

s=1 SOP�s and BOP� := ∑T/B

s=1 BOP�s are directly comparable to

GO�. They show the loss of profit that has to be accepted if, for the sake of customerservice, real time order promising or a short batching horizon B have to be realized.

To compute SOP� and BOP� in a simulation experiment, the remaining ATP has tobe updated according to ATPs+1

t := ATPst −

∑i∈I s os�

i t ∀t = 1, . . . , T in-betweenthe iterations s and s+1. This corresponds to the inventory netting and ATP calculationprocedure, more generally described by Fleischmann and Meyr (2003a), for the specialcase that supply is assumed to be deterministically known in advance. AT P1

t canbe initialized by inventory on hand (t = 0) and the projected supply (accordingto the master production schedule or master plan of the supply chain) of periodst = 1, . . . , T . Without loss of generality, AT Ps

0 = 0 ∀s is assumed in the following.

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2.3 Models with customer segmentation

The above formulas give rise to the suspicion that BOP� can be brought closer to G O�

by simply increasing the batching horizon B. This behavior has already been confirmedby the experiments of Chen et al. (2002, 2001). However, customer expectations ofshort order promising response times set a natural limit to an increase of B. Thusmodern APS follow another approach to close the gap to G O� while simultaneouslyoffering the real time single order response times of SOP. As described by Kilgerand Meyr (2008), they adapt ideas of revenue management for industrial purposes:scarce capacity (in this case ATP) is allocated to certain customer classes with differentpriorities (or profits). Incoming customer orders are allowed to consume capacity oftheir own or a lower priority class only. By doing this, it shall be prevented that alower priority customer order can consume capacity that would later on be needed fora higher priority order gaining higher profits.

Thus, single order promising can still be applied, but it is preceded by an earlierallocation (sometimes called “quoting”) process, reserving ATP for distinct priorityclasses. It is the aim of this paper to model the planning problems arising in such acontext and to demonstrate and quantify the potential benefits of such a procedure.Therefore, the ATP allocation and ATP consumption processes of situation (d) in Fig. 1have been put into the same modeling and simulation environment as GO, BOP andSOP in a–c of Fig. 1 and the LP models AP and SOPA have been designed to representboth partial problems: The allocation planning model AP first assigns ATP to a pre-defined number K of customer (or more generally: priority) classes k = 1, . . . , K . Thesubsequent single order consumption SOPA of the class-specific ATP is also modeledand solved by LP, even if APS usually apply simpler and faster rule-based algorithmsfor the ATP consumption. Section 2.4 finally demonstrates how orders can be assignedto priority classes.

Table 2 shows the indices, data and variables of the AP model. As can be seen,agreements on how much has to be sold at a minimum (lower bound on sales quantity)to a respective priority class k in a certain period t and forecasts on how much can atmost be sold (upper bound on sales quantity) are needed in order to quote ATP withrespect to the expected profits of the respective classes. The lower bounds usuallyrepresent strategic sales targets or mid-term commitments which ensure that certaincustomer groups get a minimum level of service. The upper bounds are estimates ofthe aggregate customer demand of the respective class in a certain period, i.e. forecastson what all customers of this class will buy at a maximum. The degree to which thedemand of a certain class should (in terms of overall profits) actually be satisfied willbe determined by the model. Thus, with respect to the limited ATP capacity, the modelfurther restricts potential sales to certain customer classes by allocating ATP to themost profitable ones.

In detail, the AP problem can be formalized as follows:Allocation planning problem (AP):

maximizeT+1∑

k,t=1

T∑

τ=1

pktτ · zktτ (4)

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Table 2 Indices, data and variables of the allocation planning problem (AP)

Indices

k = 1, . . . , K Priority (or profit) classes of orders/customer groups

(The number of classes K has to be pre–defined in advance).

�k Set of orders i belonging to priority class k

Data

dminkt (≥ 0) Lower bound on sales to priority class k in period t [SKU]

dmaxkt (≥ dmin

kt ) Estimated (maximum) customer demand of class k in period t [SKU]

(= upper bound on sales quantity to priority class k in period t)

pktτ Per unit profit if ATP of period t (= 1, . . . , T + 1) satisfiesdemand of priority class k in period τ (= 1, . . . , T ), e.g.

[$/SKU]

= Per unit revenue ek in priority class k

-supply costs c

-“low holding costs” if t < τ , and

-backlogging costs if τ < t ≤ T , respectively

= 0 if t = T + 1

(or -penalty costs for loss of goodwill)

Variables

zktτ ≥ 0 Part of demand of priority class k in period τ (= 1, . . . , T )

which is satisfied by ATP in period t (= 1, . . . , T + 1)

[SKU]

ft ≥ 0 Still unallocated part of ATP in period t [SKU]

subject to

dminkτ ≤

T+1∑

t=1

zktτ ≤ dmaxkτ ∀ k, τ = 1, . . . , T (5)

T∑

k,τ=1

zktτ + ft = ATP1t ∀ t = 1, . . . , T (6)

ATP is allocated to the priority classes k so that the overall profit is maximized (4).The per unit profits pktτ of a class k can, for example, be computed as the averageprofits pit of the orders i ∈ �k that have been assigned to class k. The totally reservedATP has to be within the upper and lower sales bounds of the respective priorityclass (5). If, due to the upper bounds dmax

kτ , ATP cannot be assigned to one of theclasses, it remains unallocated (6) and thus can be used by any class in the later SOPAconsumption.

As already explained in Sect. 2.1, when facing supply and demand uncertainty, APshould be done on a rolling horizon basis. However, since supply uncertainty shouldnot matter in the simulation experiments of Sect. 3, AP only needs to be executedonce at the beginning of planning in t = 0. Further, to exclude forecast errors (demanduncertainty), the aggregate demand forecast dmax

kτ of class k is initialized with the (lateron) actually requested quantities, i.e. dmax

kτ :=∑

i∈�k :di=τ qi ∀k, τ with �k denoting

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Table 3 Indices, data and variables of the SOPA problem in iteration s

Indices

classi Priority class order i belongs to

�i Set of priority classes which can be consumed by order i

Data (ATP that can be consumed by order i(s) in iteration s)

aATPsktτ ATP that becomes available in period t and has been allocated to orders

in priority class k with a requested delivery date in period τ

[SKU]

uATPst ATP that becomes available in period t but has not yet been allocated to

any priority class or planned delivery date[SKU]

Variables

oskt ≥ 0 Part of allocated ATP of priority class k in period t (= 1, . . . , T + 1)

which is in iteration s assigned to order i(s) showing a requested deliverydate di(s)

[SKU]

xst ≥ 0 Part of unallocated ATP of period t , which is in iteration s assigned to

order i(s) showing a requested delivery date di(s)

[SKU]

the priority class which order i belongs to and di denoting the requested deliveryperiod of order i . For ease of simplicity, the lower bounds on sales are set to zero,i.e. dmin

kτ := 0 ∀k, τ .The optimal solution z�

ktτ of AP allows a very detailed allocation of ATP, not onlyspecifying the period t , the ATP becomes available, but also specifying which priorityclass k it should be reserved for and in which period τ it should be consumed. LetaATPs

ktτ denote allocated ATP that has been defined on the same level of granularityand remains available for consumption in iteration s. Then, the allocated ATP of thefirst period after the allocation procedure AP can be defined according to

aATP1ktτ := z�

ktτ ∀k, t, τ, (7)

thus allowing a very restrictive reservation for important classes. This appears usefulif the forecasts of customer demand are very reliable. Of course, if forecast accuracyis low, also a more aggregate allocation could be applied, e.g. by

aATP1kt :=

T∑

τ=1

z�ktτ ∀k, t. (8)

The quantities uATP1t := f �

t ∀t remain unallocated in case the expected ATP inven-tories are higher than estimated demand. If, on the other hand, estimated demand isexpected to be higher than total ATP inventories, the portion of demand of period t inclass k that has been allocated to z�

kt,T+1 > 0 by (5) cannot be served later on.The LP model (9)–(12) uses these allocated and unallocated ATP quantities as an

input for real time single order processing after allocation planning. The variables ofthis SOPA model are explained in Table 3. Since the SOPA models of the subsequentsimulation iterations consider a single order, each, the only order of iteration s isdenoted by i(s) in the following:

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“SOP after allocation planning” model of iteration s (SOPAs):

maximize∑

k∈�i(s)

T+1∑

t=1

pi(s),t oskt +

T∑

t=1

pi(s),t xst (9)

subject to

∑

k∈�i(s)

T+1∑

t=1

oskt +

T∑

t=1

xst = qi(s) (10)

oskt ≤ aATPs

ktdi(s)∀ k ∈ �i(s), t = 1, . . . , T (11)

xst ≤ uATPs

t ∀ t = 1, . . . , T (12)

In (9) the original profits pit of Table 1 are maximized. Thus, the simulation resultSOPA� := ∑S

s=1 SOPA�s of a preceding AP optimization, followed by S := I ite-

rations of SOPA (with an optimal objective function value SOPA�s of iteration s), is

directly comparable to GO�, BOP� and SOP� as computed in Sect. 2.2. The Eqs. (10)ensure that the requested quantity of order i(s) is either met by (un)allocated ATPor assigned to the fictitious period T + 1 and thus denied, however generating noprofit or even incurring penalty costs. The capacity constraints (11) and (12) limitthe use of allocated and unallocated ATP to their predefined values. An order i(s)can only consume ATP in some dedicated classes �i(s). For example, by setting

�i(s) := {k : classi(s) ≥ k ≥ K } it can be ensured that an order i(s) ∈ �l canonly consume ATP of its own priority class l := classi(s) or other classes k > l sho-wing lower priorities. Thus, also for the AP problem, it is assumed that the classesk = 1, . . . , K have been sorted according to decreasing priorities, e.g. defining k > lif the average profits fulfill

∑t, i∈�k

pit

|�k | ≤∑

t, i∈�lpi t

|�l | . (13)

Such a strategy of allowing access to lower priority ATP has, for example, been appliedby Fischer (2001)—there called “hierarchical cumulated quoting”—or by Kilger andMeyr (2008) using customer hierarchies.

Analogously to the SOP procedure described in Sect. 2.2, in the following simula-tion experiments the (un)allocated ATP remaining after iteration s for use in iterations + 1 can easily be calculated by (14) and (15):

a AT Ps+1ktdi(s)

:= a AT Psktdi(s)

− os�kt ∀ k, t = 1, . . . , T, (14)

u AT Ps+1t := u AT Ps

t − x�t ∀ t = 1, . . . , T . (15)

As already mentioned in Sect. 2.1, this is possible because demand and supply areassumed to be known in advance. Such a data update is more complicated if demand

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and supply are uncertain and if AP is executed on a rolling horizon basis. In this caseinventory netting and ATP calculation as described by Fleischmann and Meyr (2003a)are necessary. Note that in MTS situations late delivery or cancellation of orders is onlypossible for newly arriving orders but not for orders that have already been promised(and thus delivered!). This is opposite to order promising in ATO or MTO situations.

Applying AP/SOPA instead of GO can be seen as a kind of problem decompositionbecause the single problem GO has to be decomposed into the two subproblemsallocation planning and SOPA, which have to be solved subsequently and iteratively.Due to this decomposition, a gap between the GO� and SOPA� may result, even ifall orders were known with certainty. This gap is generated by aggregating individualorders to priority classes. However, if demand was known in advance and each orderi was assigned to its own priority class (�classi = {i}, K = I ), the final objectivefunction values GO� and SOPA� would be identical. Thus, the overall problem is tofind a decomposition that brings the result of AP/SOPA as close as possible to the (inreality only ex post known) result of GO.

Summarizing these structural insights, the following conclusions can be drawn: Inpractice, the result of GO (“first best solution”) cannot be realized because of tworeasons:

• There are demand and supply uncertainties, i.e. orders and supplies cannot beknown in advance. Schneeweiss (2003) denotes a problem decomposition, whichis caused by such a missing information, “time decomposition”.

• For real time order promising, an aggregation of individual orders to priorityclasses is necessary. The impacts of this will further be analyzed in Sect. 3.

However, before, the still open problem of determining priority classes has to bediscussed.

2.4 Identification of customer classes

In the above sequence of AP and SOPA an assignment of orders i to priority classesk was assumed to be predefined, which is expressed by the order sets �k and classindices classi . Usually, such an assignment of orders to classes is not obvious, it mayeven be hard to define a useful number K of classes k. This assignment task is amid-term planning task because the allocation planning AP has also to be done in themedium term. It may sound confusing that an order i can be assigned to a class beforeit actually arrives at date ai . But usually there are quite stable relationships betweenvendors and their customers so that an order can directly be linked to the customersending it and thus the problem reduces to assigning customers to priority classes k inthe medium term. For ease of simplicity, the notation will not further be complicatedby distinguishing between customers and their orders. The reader should just keep this1:n-relationship in mind.

The profits pit as introduced in the above tables usually originate from a time-independent indicator vali of the “value” of order i (or its corresponding customer)and a time-dependent, discrete function p·t that punishes non-delivery or earliness andlateness with respect to di . A piecewise-linear example for such a function, which will

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H. Meyr

be applied in the following experiments, is given by (16):

pit := vali ·[

1− help

(T − 1) · late

](16)

with

help :=

⎧⎪⎨

⎪⎩

(di − t) · early if t < di

(t − di ) · late if di ≤ t ≤ T

(T − 1) · late if t = T + 1

and with penalty costs early for being early and late for being late (usually early <<

late). In the experiments of Sect. 3 early := 1 and late := 10 are used.One should be aware that usually vali is only an artificial measure describing the

overall importance of order i . Besides the per unit profit ei − ci other non-monetaryfactors may contribute to vali as well, for instance, the strategic power of the cus-tomer ordering i . An example for such a procedure is given by Fischer (2001) andin Sect. 3.1. Thus, quantifying the measure vali is a crucial task, depending on thepractical application under consideration.

Knowing the vali , for the assignment of a given set of orders to a predefined numberof classes standard clustering methods can be used. They group all such orders i and jinto the same class which are “similar” according to a certain distance measure disti j ,for example,

disti j := dist ji :=∣∣vali − val j

∣∣ . (17)

Thereby, “similarity” can be expressed by different types of objectives. For exampleMeyr (2007) introduces two alternative clustering models, CS and CM, minimizingthe sum of the distances between any pair of orders within the same class and the sumof the maximum distances of each class, respectively. To solve the CS problem, heproposes three alternative local search heuristics basing on steepest descent (calledSum-DE), threshold accepting (Sum–TA) and tabu search (Sum–TS). For the CM modela simple rule–based heuristic is applied (called MinMax).

Clustering models, including CS and CM, usually assume that the number of classesK is known in advance (see Meyr 2007). This was also the case for the AP and SOPAmodels of the previous section. Obviously, the optimal objective function values ofCS and CM both will decrease to 0 if K is increased to I . This is because, assumingcomplete demand information, in the extreme case K = I the allocation problemAP reserves the necessary ATP for each single order i , separately. Thus it seems tobe useful to choose the number of classes as large as possible. However, one has tobe aware that increasing the number of classes is not only advantageous. First, alsothe complexity of AP and of the clustering problem is increased. Second, and morecrucially, in practice demand information is uncertain. Thus, missing informationabout not yet known orders has to be substituted by demand forecasts. Followingthe law of large numbers, forecast accuracy is the better, the higher the number oforders per class is, i.e. the lower the number of classes is. Altogether, a trade offbetween better allocation/reservation capabilities and lower forecast accuracy has to

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be balanced, which can hardly be formalized. Section 3.5 will give some hints how ahopefully good compromise can be found.

3 Experiments

The described models are tested with a practical example of the lighting industry. Thecase itself and the corresponding data are described in Sect. 3.1. Then, a first overviewof the benefits of allocation planning is given. Different ways of defining the ATPsearch space �i(s) and ATP consumption rules are discussed in Sect. 3.3. The effectsof varying K are tested in Sect. 3.4. The final subsection of Sect. 3 evaluates the overallimpact of clustering on the finally decisive SOPA outcome.

The allocation and ATP assignment problems GO, BOP, SOP, AP and SOPA canall be interpreted as classical transportation problems. Thus, standard LP software orspecialized network flow solvers (see e.g. Ahuja et al. 1993) can be applied withoutany problems. SOP and SOPA show an even simpler structure because of consideringa single order i only. Thus, they can be solved to optimality with fast backward andforward-oriented, rule-based algorithms, which start in period di and class classi andproceed in sequence of descending per unit profits. Similar real time ATP search rulesare usually implemented in APS (as heuristics for more complicated variants of SOPand SOPA). However, for ease of simulation in the following experiments, which havebeen coded with Microsoft Visual C++ 6.0, the standard linear programming solverCPLEX 9.0, the modeling language ILOG OPL Studio 3.7 and its C++ componentlibraries interface (ILOG 2007) have been used for all ATP models, including thesimpler SOP and SOPA problems. The computational tests have been executed on apersonal computer with an Intel Pentium M 1.3 GHz processor and 512 MB RAM,operated by the Microsoft Windows XP Professional system.

3.1 Problem data

The experiments of the following sections use practical data that have been introducedby Fischer (2001) in a case study of lighting production. This business is a classicalMTS-environment where customer orders arrive at the distribution centers and have tobe served from the stock which is already available or at least projected to arrive soon.Six different problems, denoted as P1, …, P6 in the following, have been consideredby Fischer. These problems reflect the demand for six different final items—also calledP1, …, P6 in the following—during one month, i.e. a period of T = 30 days. Note,even if 30 days are simulated by Fischer and in the experiments of Sects. 3.2–3.5,orders usually arrive between day 1 and day 26. The only exceptions are P3, wherethe last order arrives at day 23, and P5, where the first order arrives at day 6.

The characteristics of the problems P1,…, P6 are shown in Table 4. The fourproblems P1, P2, P3 and P5, with less than 40 orders arriving, are rather small. Due tothe infrequent arrival of orders and the resulting low average number of orders per daybetween 0.9 and 2.1, a BOP-horizon of a single day is expected to show only weakimpacts. This might be different for the two larger problems P4 and P6 with 1,305 and509 orders, respectively, and with 72.5 or 28.3 orders per day, on the average.

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H. Meyr

Table 4 Data used by Fischer (2001)

P1 P2 P3 P4 P5 P6

Total no. of orders 37 29 25 1305 17 509

Orders per class 8/14/15 29// 24//1 725/440/140 //17 500/7/2

No. of supplies 19 13 12 19 19 19

Lost sales (percent) 11.3 17.9 17.3 19.9 4.5 22.6

Orders per day 2.1 1.6 1.4 72.5 0.9 28.3

Aver. distance disti j 4.7 0.7 0.0 4.3 0.0 0.9

The number of supplies, i.e. refillings of ATP inventory, within the planning horizonvaries between 12 and 18. The four products P1, P4, P5 and P6 start with positive initialinventory that has been modeled as an additional 19th ATP supply at day t = 0 (seerow “no. of supplies” in Table 4).

The total order quantity within the planning horizon exceeds the total supply signi-ficantly by 4.5–22.6%. The respective shares have been denoted as “lost sales” inTable 4. This indicates that there are indeed shortage situations in this kind of busi-ness. However, usually not all of these sales are really “lost” because some of theorders might be satisfied by supply arriving after the planning horizon of T = 30days. Nevertheless, it seems that the customer service level has been poor for thesesix products.

Table 4 also shows the average distance disti j between all pairs of orders i andj for a certain product. Note that this is not the original distance measure used byFischer. Fischer used up to three priority classes, as indicated in the row “orders perclass” of Table 4, to differentiate customers/orders showing various importance whencomputing order–specific costs. The original data have been normalized in order toallow the application of general clustering models, like CS and CM, also for K �= 3.The distances disti j have been calculated as follows:

Two major attributes contribute to the value indicator vali of a certain customerorder i :

• The normalized per unit profit prof i tnormi of order i has been calculated by means

of

prof i tnormi := (prof i ti − prof i tmin)

prof i tmax − prof i tmin

with prof i ti := ei − ci denoting the per unit profit of order i and prof i tmin :=mini {prof i ti } and prof i tmax := maxi {prof i ti } denoting the minimum andmaximum profit of any order i . The resulting normalized profits are in a range0 ≤ prof i tnorm

i ≤ 1.• According to the varying importance of different customers, Fischer assigned all

customers and their respective orders to the three priority groups mentioned above.Therefore, each customer order has a priority index priori t yi ∈ {1; 2; 3}. Thesepriority indices have also been normalized to a range between 0 and 1 by using

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priori t ynormi := (priori t yi − 1)

3− 1.

Both attributes have been aggregated into the single value indicator vali of order i byweighing them with weights w1 and w2 according to

vali := w1 · prof i tnormi + w2 · priori t ynorm

i . (18)

For the experiments in the following sections identical weights w1 := w2 := 10 havebeen used, resulting in an indicator range 0 ≤ vali ≤ 20.

The profits pit of GO, SOP, BOP, and SOPA and the distance measure disti j of CSand CM (see Meyr 2007) have then finally been calculated by (16) and (17) with penaltycosts early := 1 for being early and late := 10 for being late. Since early ≤ lateand |t − di | ≤ (T − 1), the profits pit also range between 0 and 20. Note that thecustomers’ and orders’ priorities of the problems P2, P3, P5 and P6 seem to be quitesimilar because their average distance is small. In P2 and P5 all orders even have thesame priority values priori t ynorm

i , but the average distance disti j = 0.7 of P2 iscaused by varying per unit profits. However, P3 contains a single order with lowerpriority, but the higher profit prof i tnorm

i of this order causes all value indicators valiof P3 to be equal. Altogether, no real advantage of the allocation process underlyingSOPA can be expected for P3 and P5.

3.2 Benefits of allocation planning

Using the notation of Sects. 2.2 and 2.3, GO�, SOP�, BOP� and SOPA� denote theoverall objective function value of a complete, raw-data driven simulation run overT time periods. The SOPA run is preceded by the allocation planning problem APas described in Sect. 2.3 and uses the original priority classes of Fischer.

Table 5 shows the percentage deterioration of SOP�, BOP� and SOPA� as com-

pared to GO�, e.g. GO�−SOP�

GO� · 100. It can be interpreted as the percentage profit

loss of a short–range order acceptance compared to the ex–post optimal solution. TheBOP� results are varied over a batching horizon of B = 1, . . . , 5 days (and T mod Bfor the last periods, respectively). SOPA� results are shown in two different variants:SOPA�a aggregates allocated ATP according to (8). SOPA�d uses disaggregate aATPas defined by (7), thus also allowing a reservation of ATP becoming available inperiod t for use in another period τ �= t . Therefore, SO P A�a demonstrates the“pure” effect of allocating ATP to the three customer classes pre–defined by Fischer(2001), whereas SO P A�d combines this effect with an additional “temporal” reser-vation of ATP quantities for the periods of their expected use, thus assuming a highforecast quality. The computation times of a single run are negligible, e.g. solvingGO for the biggest problem P4 takes just a few seconds. However, since only thestandard C++ libraries and data conversion routines of the LP software OPL (ILOG2007) are used, a complete SOP- or SOPA-simulation run of P4 may last severalhours.

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Table 5 Percentage profit loss of SO P�, B O P� (B = 1, . . . , 5) and SO P A� (disaggregate, aggregate)as compared to G O�

P1 P2 P3 P4 P5 P6 Average

SOP* 15.0 21.1 0.0 12.4 1.2 6.9 9.4

BOP*1 14.8 21.0 0.0 11.6 1.2 6.8 9.2

BOP*2 14.1 21.0 0.0 11.0 1.2 6.8 9.0

BOP*3 14.1 13.3 0.0 10.7 0.6 6.7 7.6

BOP*4 13.6 20.8 0.0 6.3 1.2 6.7 8.1

BOP*5 12.1 11.2 0.0 10.0 0.3 6.7 6.7

SOPA*a 5.5 21.1 0.0 1.5 1.2 0.3 4.9

SOPA*d 0.5 0.1 0.0 0.2 0.0 0.3 0.2

As expected, batching orders and increasing the batching horizon B is advantageouswhen compared to the FCFS single order processing SOP. But even for a batch horizonof a whole week, the overall improvement is rather disappointing. Astonishingly, thisholds especially true for the problems P4 and P6, which show a high degree of freedombecause of their large number of orders per day. Of course, a simulation horizon of 30days is actually too short for such experiments. However, studying Table 5 it seemslikely that increasing the simulation horizon would stabilize the results, but not reallychange the overall picture.

SOPA�a shows a significant improvement for problems P1, P4 and P6. The diffe-rence to SO P� is only caused by the allocation planning on basis of the three priorityclasses used by Fischer (see Sect. 3.1, Table 4). These results can further be improvedby SO P A�d , which allows a temporal reservation of ATP, too. In this case, near–optimal profits can be gained for all six scenarios. Thus, if companies are able torealize a high forecasting accuracy, defining disaggregate ATP seems reasonable.

SOP solves P3 and P5 almost to optimality because of their corresponding custo-mers’ homogeneity and the distances disti j = 0 between every two orders i and j . Theprofit loss of 1.2% for P5 is caused by inventory or backlogging costs as a consequenceof an unfavorable temporal assignment of ATP and can thus additionally be avoidedby SO P A�d . As opposite to P3 and P5, P2 not only shows a small number of orders,but also a non–zero heterogeneity. This might be the reason for the exorbitant advan-tage of temporal reservation for P2. On the other hand, temporal reservation seems tohave no impact on P6 (0.3 for both SOPA�a and SOPA�d). Summing up both SOPAvariants clearly profit from clustering effects.

3.3 Variation of the ATP search space and consumption rules

Both SOPA variants of the last section assumed that ATP can only be consumed in thepriority class classi(s), the order i(s) belongs to. The subsequent experiments allowa more flexible consumption of ATP by varying the ATP search space �i(s) in thefollowing way:

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cc⇔ ATP can only be consumed in the class classi(s), order i(s) has been assignedto, i.e. �i(s) := {classi(s)} (as done in Sect. 3.2).

cK⇔ ATP can be consumed in the order’s original class or in classes with lowerpriority, i.e. �i(s) := {classi(s), . . . , K } (see Sect. 2.3 regarding the sorting ofclasses).

1K⇔ ATP can be consumed in all classes, i.e. �i(s) := {1, . . . , K }.1c⇔ ATP can be consumed in the order’s original class or in classes with higher

priority, i.e. �i(s) := {1, . . . , classi(s)}.Intuitively, the last variant does not seem to make much sense, but has been imple-mented for ease of validation and comparison.

Note that the constraints above do not specify a sequence for searching this space. Bysolving the SOPA model (9) – (12) using linear programming, ATP can be consumedfreely within the search space �i(s) because—for a given period t—the order’s originalprofit pi(s),t remains the same independently of the class, the ATP quantities actuallycome from. In order to guide the search through various priority classes in an intendedmanner (while still applying LP methods), fictitious gains and losses have been definedthe following way: The objective function (9) is extended by

∑

t,k≥classi(s)

(K − k) · 0.01 · pi(s),t oskt (19)

for the search space cK and by (19) plus

∑

t,k<classi(s)

(k − classi(s)) · 0.01 · pi(s),t oskt (20)

for the search space 1K. Thus, the allowed classes are searched in an order of descen-ding priorities first, starting with the original class classi(s). If no such ATP has beenfound for a search space 1K, higher priority classes are then searched in a sequenceof ascending priorities, starting with classi(s) − 1. Of course, the loss of profit shownin Table 6 has been calculated on basis of the regular profits (9), only. This way, ATPsearch rules, as proposed by Kilger and Meyr (2008) and Fischer (2001) and used inmost APS, can also be simulated within the LP framework of this paper.

Table 6 shows the percentage profit losses for a variation of aATP aggregation(aggregate, disaggregate), of the search space (cc, cK, 1K, 1c) and of the searchsequence (free allocation, search sequence predefined). The two rows marked in boldcorrespond to the respective SOPA� results of Table 5.

When comparing the four a/·/f scenarios among themselves, the best results areachieved for the cc search space, i.e. when staying within an order’s original priorityclass. Access to lower class ATP is only reasonable if search rules are used (a/cK/s).In this case, the a/cc/f results can be equalized but not improved. Free access to higherpriority ATP (a/1K/· and a/1c/f) is indeed proven to be nonsense. A variation of thesearch space or the introduction of search rules (a/·/·) do not show any effects on P2 andP5. For these products a profit increase can only be achieved by temporal reservation(d/·/·). The situation is actually the same for P3. Its anomalies for a/cK/· only occur

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Table 6 Percentage profit loss of SOPA� as compared to GO� for varying temporal reservation (aggregate,disaggregate), ATP search space (cc = original class, cK = lower priority, 1K = all classes, 1c = higherpriority) and ATP search rules (f = free allocation, s = search sequence predefined)

P1 P2 P3 P4 P5 P6 Averagea

a/cc/f 5.5 21.1 0.0 1.5 1.2 0.3 5.9

a/cK/f 5.5 21.1 (24.4) 4.3 1.2 0.3 6.5

a/1K/f 15.0 21.1 0.0 12.5 1.2 6.9 11.3

a/1c/f 18.0 21.1 0.0 18.9 1.2 6.9 13.2

a/cK/s 5.5 21.1 (17.7) 1.5 1.2 0.3 5.9

a/1K/s 15.0 21.1 0.0 12.4 1.2 6.9 11.3

d/cc/f 0.5 0.1 0.0 0.2 0.0 0.3 0.2

d/cK/f 0.5 0.1 0.0 0.2 0.0 0.3 0.2

d/1K/f 0.6 0.1 0.0 6.8 0.0 0.8 1.7

d/1c/f 12.8 0.1 0.0 13.8 0.0 0.8 5.5

d/cK/s 0.5 0.1 0.0 0.2 0.0 0.3 0.2

d/1K/s 0.6 0.1 0.0 6.8 0.0 0.8 1.7

a Without P3

because the penalty holding costs “early = 1” have turned out to be too low for thisproduct. Thus, the average values in the corresponding column of Table 6 have beencalculated without considering P3.

The comparison of the a/·/· with their respective d/·/· scenarios emphasizes theadvantages of a temporal reservation, again. Altogether the picture is similar for thedisaggregate scenarios. The search spaces cc and cK show equal quality, whereas 1Kand 1c compare badly. A positive effect of search rules cannot be recognized, here.

3.4 Variation of the number of classes

All SO P A� results presented so far are based on Fischer’s original assignment ofcustomers to three priority classes (see Sect. 3.1). It will now be investigated whethera variation of the number of priority classes might be advantageous. At the sametime the various ATP consumption alternatives will be compared again. The followingexperiments will be limited to P1. Product P1 has been chosen because

• it comprises only 37 orders and thus can be simulated in short computation times,• Fischer’s assignment of orders to classes showed balanced proportions for P1 (8/14/

15, see Table 4), and because• the SOPA allocation achieved significant and non–identical profit increases for both

variants—those with (0.5% loss) and those without (5.5%) temporal reservation—as compared to the standard SOP (15%) procedure (see Table 5).

Up to 20 priority classes have been generated using the clustering models and heuristicsof Meyr (2007). Table 7 shows the average of the corresponding percentage SOPA�

profit losses.

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Table 7 Percentage profit loss of SOPA� as compared to G O� for P1 with respect to different ATPconsumption rules (see Table 6) and a varying number of priority classes K (missing entry = 0.0)

Reser.: Aggregate Disaggregate

Space: cc cK 1K 1c cK 1K cc cK 1K 1c cK 1K

Search: Free Sequ. Free Sequ.

K = 1 15.0 15.0 15.0 15.0 15.0 15.0 0.9 0.9 0.9 0.9 0.9 0.9

2 10.1 10.1 15.0 15.1 10.1 15.0 0.6 0.6 0.6 1.6 0.6 0.6

3 6.4 6.7 15.0 17.2 6.4 15.0 0.5 0.5 0.5 10.5 0.5 0.5

4 5.7 5.7 15.0 18.0 5.7 15.0 0.4 0.4 0.5 10.6 0.4 0.5

5 4.5 5.2 15.0 20.8 4.5 15.0 0.4 0.4 0.5 10.6 0.4 0.5

6 3.6 4.3 15.0 23.8 3.6 15.0 0.1 12.0 0.1

7 2.7 3.2 15.0 24.7 2.7 15.0 0.2 0.2 0.3 12.1 0.2 0.3

8 2.9 3.5 15.0 26.1 2.9 15.0 0.1 12.0 0.1

9 2.4 2.7 15.0 26.7 2.4 14.9 0.1 12.0 0.1

10 2.4 2.7 15.0 26.2 2.4 14.9 0.1 12.0 0.1

11 2.4 2.7 15.0 26.2 2.4 14.9 0.1 12.0 0.1

12 2.4 2.6 15.0 26.6 2.4 14.9 0.1 12.0 0.1

13 2.4 2.6 15.0 26.6 2.4 14.9 0.1 12.0 0.1

14 2.5 2.6 15.0 26.7 2.5 14.9 0.1 12.0 0.1

15 2.8 3.0 15.0 26.8 2.8 14.9 0.1 12.2 0.1

16 3.0 3.2 15.0 26.7 3.0 14.9 0.1 11.9

17 3.0 3.3 15.0 26.8 3.0 14.8 0.1 12.1

18 2.2 2.7 15.0 26.8 2.2 14.8 0.1 12.1

19 2.2 2.7 15.0 26.9 2.2 14.8 0.1 12.3

20 2.2 2.9 15.0 26.8 2.2 14.7 0.1 12.1

Average 4.05 4.38 15.0 24.0 4.05 14.9 0.16 0.16 0.24 10.7 0.16 0.20

Fischer 5.5 5.5 15.0 18.0 5.5 15.0 0.5 0.5 0.6 12.8 0.5 0.6

The row “average” of Table 7, containing average results of all 20 classes for eachATP search alternative, confirms the findings of the last section. Within the aggregateaATP scenarios (left part of Table 7) the search spaces cc and cK perform best again,also for a varying number of classes K . If access to lower priority classes is allowed(a/cK/·), sequential search rules should be applied instead of a free ATP consumption.The results of the disaggregate aATP (right part of Table 7) show a similar structure.However, the overall solution quality is better. Due to the limited degree of freedomleft after the temporal reservation, the d/1K/· scenarios also behave well. All in all,the a/cK/s–rules for ATP consumption, as proposed by most APS, seem justified bythese experiments. However, simply staying within the original class (a/cc/·) wouldperform equally.

The number of classes K appears more important than the search space and searchrule. This can be seen when studying the profit improvement resulting from increasingK for all ·/cc/· and ·/cK/· scenarios. The absurdity of an 1c search space becomes

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particularly clear in Table 7 where the profit loss even increases for a higher numberof customer classes. The row K = 1 shows the results for a single class only, i.e. theSOP performance without allocation planning. The a/·/· values coincide with the SOP�

value of P1 in Table 5. The d/·/· values for K = 1 illustrate the improvement possibleby solely introducing temporal reservation, without additionally building customerclasses. A profit loss of 0.9% still remains because all orders of the same period areconsidered as being equal. However, for P1 this affects only 2.1 orders on the average(see Table 4).

The two lines marked in italics allow a comparison of the clustering methods (K =3)of Meyr (2007) with the original customer segmentation of Fischer. There seemsto be a small advantage for the automatic methods. Nevertheless, in general bothsegmentations lead to similar results.

Note that the results are based on a single product only and thus can hardly be gene-ralized. Nevertheless, the example shows that profit can be increased by introducingpriority classes. Even if there is no obvious, natural customer segmentation, a cluste-ring into several price classes is valuable, as long as different customer orders showvarious per unit profits. Thus, it seems more important whether a clustering is donethan how it is done. To what extent this assumption is true will be further investigatedin the next section.

3.5 Effects of clustering on SOPA

Table 8 shows the percentage profit loss of single order processing after allocationplanning, as compared to the global optimization result G O�, for each of the clusteringalternatives MinMax, Sum-DE, Sum-TA and Sum-TS of Meyr (2007), individually (seeSect. 2.4). The results are presented for the products P1, P4 and P6 comprising thelargest number of orders (37, 1,305 and 509, respectively) and showing the largestinhomogeneity of distances disti j (see Table 3). For ease of clarity, the simulationhas been restricted to the single d/cK/s scenario, one of the best-performing scenariosof Sect. 3.4. Missing entries in the table indicate a profit loss of 0.00, i.e. that G O�

has been reached. Note that the MinMax results and the results of the CS heuristicsSum-DE, Sum-TA and Sum-TS would not have been directly comparable because theysolve the two different problems CM and CS. However, each product’s profit losses ofTable 8 can immediately be compared with each other, since the clustering heuristicsinfluence SOPA only indirectly by the different ways of cluster building.

Looking at row “aver.”, containing the results averaged over all 20 classes, givesa quick overview of the overall performance of the four heuristics. However, resultsappear nonuniform. While P1 and P6 are dominated by MinMax, the CS heuristics out-perform the CM algorithm clearly for P4. Thus there does not seem to be a significantcorrelation between the clustering objectives, the solution quality of different heuristicsand the profits generated by the respective clusters.

Interestingly, the profit losses of Sum-DE (for P6) and Sum-TA (for P4 and P6)decrease first, but then increase again. A reason for this might be found in a bad overallsolution quality of the CS heuristics, particularly for large problems with many ordersand classes (Meyr 2007). This is, besides forecast accuracy, a second argument forchoosing a not too large class number K .

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Table 8 Percentage profit loss of the SOPA clustering alternatives MinMax (MM), Sum-DE, Sum-TA andSum-TS as compared to GO� for P1, P4 and P6 in the d/cK/s scenario (missing entry = 0.00)

K P1 P4 P6

MM DE TA TS MM DE TA TS MM DE TA TS

1 0.89 0.89 0.89 0.89 11.03 11.03 11.03 11.03 11.39 11.39 11.39 11.39

2 0.49 0.66 0.66 0.66 11.00 10.12 10.12 10.12 0.30 4.13 4.13 4.13

3 0.25 0.52 0.52 0.52 10.10 3.50 3.45 3.50 0.30 2.71 2.71 2.71

4 0.25 0.50 0.50 0.50 0.15 0.15 0.15 0.15 0.29 2.72 2.72 2.72

5 0.20 0.52 0.44 0.50 0.15 0.15 0.15 0.15 0.29 1.77 1.77 1.77

6 0.09 0.03 0.03 0.03 0.15 0.34 0.10 0.34 0.29

7 0.03 0.03 0.03 0.69 0.15 0.10 0.10 0.10 0.29

8 0.03 0.03 0.03 0.03 0.15 0.10 0.07 0.10 0.29

9 0.03 0.03 0.03 0.03 0.15 0.07 0.07 0.07 0.28

10 0.03 0.03 0.03 0.03 0.15 0.07 0.07 0.07 0.03

11 0.03 0.03 0.03 0.03 0.13 0.07 0.04 0.07 0.03

12 0.03 0.03 0.03 0.03 0.12 0.08 0.03 0.08 0.01 0.39

13 0.03 0.03 0.03 0.03 0.12 0.04 0.03 0.04 1.64

14 0.03 0.03 0.03 0.12 0.03 0.03 0.03 1.35

15 0.03 0.14 0.03 0.03 0.03 2.65

16 0.08 0.03 0.03 0.03 0.97 0.67

17 0.08 0.03 0.03 0.03

18 0.08 0.03 0.02 0.03 1.39 0.49

19 0.08 0.03 0.06 0.03 0.91

20 0.06 0.03 0.06 0.03 0.58 1.96

Aver. 0.12 0.17 0.16 0.20 1.71 1.30 1.28 1.30 0.69 1.58 1.34 1.14

The clustering of Fischer often shows better results (0.51 for P1, 0.21 for P4 and0.33 for P6) than the automatic clustering methods for K = 3. However, for K = 4already the MinMax clustering outperforms Fischer’s profits for all three products.Starting with K = 7 the same holds true for all CS heuristics as well. On the whole,all four heuristics show promising results when four or more classes are used.

Summing up this section, SOPA� indeed seems not to be very sensitive with respectto the clustering method used. An increase of the number of classes K leads to higherprofits if orders of the same product are inhomogeneous enough. Considering theexamples of this section at least 4, but better 6–7 classes should be used. However, thenumber of classes should not be chosen too large in order to reduce forecasting errorsand a bad performance of clustering heuristics, especially for CS.

4 Summary, managerial insights and outlook

The exemplary tests of the paper have shown that a first-come–first-served processingof arriving customer orders is hardly the best way of demand fulfillment in shortagesituations if reliable forecasts are available. Gathering data for a certain period of time

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and processing them in a batch can improve the situation. However, often customerservice sets a natural limit to such a procedure because customers increasingly expectshort order confirmation lead times. Another way of improvement can be to precedethe FCFS single order processing by a further allocation planning step. Here, priorityclasses for customer orders are built, available inventory (ATP quantities) is “allocated”to these classes and reserved for later consumption by their respective customers. Sucha customer segmentation has proven its potentials when introducing booking classes inairline yield management. Thus, the basic idea is not new and has also been supportedby advanced planning systems where simple ATP allocation and consumption rules areoffered. However, until now it was largely unclear—in science and practice—whether,when, why and to what extent such a proceeding might be useful in manufacturingindustries, too.

First answers to these questions have been given using an example from the lightingindustry where bulbs, fluorescent lamps etc. are made to stock on the basis of forecasts,first, and then sold from stock as soon as customer orders arrive. In order to demonstrateits potentials the following planning tasks had to be structured, discussed and solvedfirst:

1. Determination of a reasonable number of priority classes,2. clustering, i.e. assignment of customers and customer orders, respectively, to these

classes,3. allocation planning, i.e. allocation of available inventory on hand and planned

production quantities (ATP) to the priority classes, and4. ATP search, i.e. successively consuming this allocated ATP for each incoming

order. In this case both the search space (classes allowed) and the search sequencehave to be specified.

(1) has been tackled by means of simulation by varying the number of classes in areasonable range and (2) by applying standard clustering methods. For (3) and (4)linear programming models have been proposed and solved to optimality. All in all, itwas not intended to discuss each of these planning tasks in all detail and to solve it inthe best possible manner (even though this has not satisfactorily been done in scienceup to now). The primary goal was to bring all four tasks together in a single simulationexperiment to give an impression of the overall potential of allocation planning inmake–to–stock industries of this or similar types.

Since practical data have been used and the test bed was limited one has to be awarethat the results are only exemplary and more general statements would need furtherexperiments. Nevertheless, some interesting insights have been gained by the lightingexample and also common views have been confirmed: Introduction of priority classesand allocation planning can indeed increase revenues and profits, substantially. Themore heterogeneous the customers and their orders, e.g. with respect to the revenuesmade or to the strategic importance of the customers, the higher the advantages are.The number of customer classes plays an important role. Too few classes cause aloss of profits, too many classes make forecasting and clustering difficult. A temporalreservation of stocks, for use in a specific period, would generally be advantageous,but its practical application is only reasonable if customer demand can be forecastreliably enough.

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At least in the lighting case, ATP consumption policies and the clustering methoditself are not as crucial as the choice of an appropriate number of classes. AlthoughLP methods have been applied for ATP consumption, simple ATP search rules wouldperform equally for this example. Such rules should either stay within an order’soriginal class or, as often claimed in Revenue Management and by APS, also allowaccess to lower priority classes. In the latter case, lower classes should better besearched for in order of descending priorities. However, note that LP methods or moresophisticated rules are required in more complex supply chains, e.g. in make-to-stocksupply chains with several stocking points and/or product substitution or in assemble-to-order supply chains with multi–stage bills of materials.

Thus, also in manufacturing industries managers should pay additional attention totheir customers’ varying nature and try to increase their overall customer service byallocating their scarce resources—in make–to–stock environments more specifically:their limited finished item stock—with higher priority to their more important cus-tomers. As the example of the lighting industry has shown, for this, not even activeor for the customer visible measures of customer segmentation (like fencing strate-gies or longer response times for order promises) are necessary. It is sufficient to takeadvantage of the already existing customer heterogeneity by applying standard clus-tering methods for identifying priority classes and by introducing well-coordinatedATP allocation and consumption processes.

Of course, there are still a lot of research challenges. Each of the planning tasksintroduced above should be investigated in more detail for prerequisites of applicationand fitting solution methods. First of all, the sensitivity of the results with respect toless reliable supply information, e.g. concerning the viability of production plans, anddemand information, i.e. to lower forecast accuracy, has to be tested. Furthermore,similar simulation experiments should be executed for more complex types of supplychains with other order penetration points. On the one hand, APS support allocationplanning and ATP consumption in resource- and capacity-constrained manufacturingindustries by offering the deterministic rules mentioned above. On the other hand,there is an obvious affinity to inventory rationing for several customer classes andto quantity-based revenue management, as defined by Talluri and Van Ryzin (2004)and practiced in many service industries like airline, hotel or car rental. Most of theirmethods are of a stochastic nature. Thus the most challenging prospect for futureresearch is to find out whether and how these worlds can learn from each other.

Acknowledgments The author is grateful to Markus E. Fischer and Bernhard Fleischmann for providingthe data, Matthias Mann for supporting the experiments and the Wiener Wissenschafts-, Forschungs- undTechnologiefonds (WWTF) for funding the research.

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Managing product availability in an assemble-to-ordersupply chain with multiple customer segments

Thomas R. Ervolina · Markus Ettl ·Young M. Lee · Daniel J. Peters

Abstract In this article, we propose a novel availability management process calledAvailable-to-Sell (ATS) that incorporates demand shaping and profitable demandresponse to drive better supply chain efficiency. The proposed process aims at findingmarketable product alternatives in a quest to maintain a financially viable and prof-itable product portfolio, and to avoid costly inventory overages and shortages. Theprocess is directly supported by a mathematical optimization model that enables ondemand up-selling, alternative-selling and down-selling to better integrate the sup-ply chain horizontally, connecting the interaction of customers, business partners andsales teams to procurement and manufacturing capabilities of a firm. We outline thebusiness requirements for incorporating such a process into supply chain operations,and highlight the advantages of ATS through simulations with realistic productiondata in a computer manufacturing environment. The models featured in this paperhave contributed to substantial business improvements in industry-size supply chains,including over $100M of inventory reduction in IBM’s server computer supply chain.

T. R. Ervolina ·M. Ettl · Y. M. Lee (B)IBM T.J. Watson Research Center,P.O. Box 218, Yorktown Heights, NY 10598, USAe-mail: [email protected]

T. R. Ervolinae-mail: [email protected]

M. Ettle-mail: [email protected]

D. J. PetersReal Estate Operations, 294 Route 100, Somers, NY 10589, USAe-mail: [email protected]


H.O. Guntc

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OR Spectrum (2009) 31:257–280DOI 10.1007/s00291-007-0113-4

her, H. Meyr, Supply Chain Planning© Springer-Verlag Berlin Heidelberg 2009

T. R. Ervolina et al.

Keywords Availability management · Assemble-to-order · Demand shaping ·Configure-to-order

1 Introduction

In today’s competitive and dynamic business environment, companies need to contin-ually evaluate the effectiveness of their supply chain and look for ways to transformbusiness processes to achieve superior customer service and higher profitability. Imbal-ances between supply and demand are the primary reason for degraded supply chainefficiency, often resulting in delinquent customer orders, missed revenue, and excessinventory. This paper describes a novel availability management process called Avail-able-To-Sell (ATS) that incorporates demand shaping and profitable demand responseto drive better operational efficiency of the supply chain. The proposed methodol-ogy aims at finding marketable product alternatives that replace demand on supply-constrained products while minimizing expected stock-out costs for unfilled productdemand and holding costs for left-over inventory. While most prior related literaturefocuses on the concept of Available-To-Promise (ATP) where a scheduling systemdetermines a particular product’s availability, this paper proposes a new approachwhere product substitutions and up-sell opportunities are considered in the planningphase. The ATS business process is most effective in an assemble-to-order manufactur-ing environment where end products are configured from pluggable components, suchas computer manufacturing where computer systems are assembled from standardizedcomponents such as hard disks, microprocessors, video cards, etc.

Industry best practices for demand shaping and demand response include identify-ing entry level products suitable for up-selling, changing marketed products based onsupply position, providing product alternatives, and methods of continuous up-sellingand cross-selling to meet financial objectives (O’Marah and Souza 2004; Cecere et al.2005). While every industry struggles with instabilities in demand and supply syn-chronization, real-time supply chain software provides continuous visibility to thesupply position of every part. If a part is in short supply due to greater demand, thesales department is informed to check if demand can be moved to alternative products.For example, if a 3.0 GHz processor is in short supply marketing teams could offer anup-sell product with a 3.2 GHz processor to the technologically savvy customer, or adown-sell model with a 2.8 GHz processor to a price-sensitive customer. In a consumersociety driven by having a product in next to real time, improved shipment or arrivallead times can be a compelling factor in a purchase decision.

Consider the following simple example that illustrates the benefits of ATS overa conventional ATP planning process. A firm sells two types of computer products,denoted M1 and M2, to its customers over a given planning period, e.g., a quarter. Eachproduct is assembled from one of two processors, hard drives and memory modules asshown in Table 1. The demand for each product is 1,000 units. The component quan-tities that would be required in order for all customers to receive their choice of prod-uct are 1,000 units of every component. Now, suppose that the firm’s micro supplierhas a short-term manufacturing constraint on 3.0 GHz processors and committed todeliver 500 units less than the required 1,000 units. To compensate for the shortfall, the

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Managing product availability

Table 1 Example of availability management with ATP and ATS

Components Price Profit Component Product portfoliosupply

Group Technology M1 M2 M3

SYSTEM 3.0 GHz/800 MHz Xeon 391 51 500 1 – –

PROCESSORS 3.2 GHz/800 MHz Xeon 504 84 1,500 – 1 1

HARD DRIVES 60 GB 4200 RPM 216 36 1,000 1 – 1

120 GB 7200 RPM 312 72 1,000 – 1 –

MEMORY 512 MB SDRAM 192 32 1,000 1 – 1

1.0 GB SDRAM 260 60 1,000 – 1 –

Demand forecast 1,000 1,000 –

ATP schedule 500 1,000 –

ATS schedule 500 1,000 500

supplier promised additional 500 units of 3.2 GHz processors. Given these componentquantities, a conventional ATP model would match the available component suppliesto the demand and create a so-called ATP schedule as shown at the bottom of theTable 1. As a result of the constrained supply of 3.0 GHz processors, the firm incurs500 unfilled orders of product M1, and 500 units of unallocated inventory of 60 GBhard drives and 512 MB memory modules. The gross profit from sales under the ATPschedule is $275,500.

To effectively deal with supply and demand imbalances, the proposed ATS modelaims at finding marketable product alternatives that replace demand on supply-constrained products while minimizing inventory holding costs from left-over com-ponent supplies. In the above example, the ATS model creates an alternative productM3 shown in the shaded column of the table that can be substituted for the con-strained product M1 (possibly in conjunction with a price discount offered to customersfor accepting the substitute). The ATS schedule eliminates any unfilled orders andexcess component inventories. The gross profit from sales under the ATS schedule is$351,500 which corresponds to a percentage profit gain of 27% over the conventionalATP approach.

The ATS capabilities described in this paper can easily be imbedded into supplychain operations to improve day-to-day flexibility. Direct sales businesses that dealwith customers directly through external websites or telesales systems can use the pro-posed models to highlight featured products on-the-fly based on current componentavailability and steer customers towards product configurations that they can supplyeasily and profitably. Business benefits are increased revenue, profitability, and mar-ket share, and improved client satisfaction. Additional financial benefits that directlyimpact the profit and loss statement are the cost avoidance of brokering, scrappingand inventory obsolescence, reduction of inventory carrying costs, return cash foradditional investments, and improved cash-to-cash cycle times.

The contributions of this paper are twofold. First, we present a linear program-ming model for ATS that determines the optimal planned allocation of products and

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components to market segments based on customer demand. Secondly, we develop asimulation model to evaluate the benefits of the ATS model in an order execution envi-ronment where products and components must be allocated to customer orders overtime. The simulations demonstrate how ATS enables on demand up-selling, alterna-tive-selling and down-selling to better integrate the supply chain horizontally, connect-ing the interaction of customers, business partners and sales teams to the procurementand manufacturing capabilities of a company.

The remainder of this paper is organized as follows. In Sect. 2 we review the relatedliterature. In Sect. 3 we present the underpinning principles of ATP and ATS, and dis-cuss the advantages and disadvantages of each management approach. In Sect. 4 wepropose an ATS planning model that captures customer preferences to effectively mit-igate supply and demand imbalances. In Sect. 5 we present a simulation frameworkfor availability management in assemble-to-order supply chains. Numerical findingsand discussions of results are presented in Sect. 6. These produce several insightsinto how advanced availability management can help proactively coordinating supplyand sales, and quantify several business benefits in the context of assemble-to-ordermanufacturing. Finally in Sect. 7 we present concluding remarks and suggestions forfuture research.

2 Literature review

There are two streams of research that are related to our work: (1) models from the pro-duction planning and operations literature that deal with Available-to-Promise (ATP)systems for order promising and fulfillment, and (2) models from the operations man-agement literature that consider inventory problems with configurable products andproduct substitution. We provide an overview of both research streams.

There is an extensive literature in the production planning area dealing with real-time order promising and ATP (e.g., Kilger and Schneeweiss 2000; Moses et al. 2004;Hoop and Roof 1999). Ball et al. (2004) develop a general modeling framework foravailability promising and present examples of ATP business practices from electron-ics companies including Dell and Toshiba. Chen et al. (2002) present a mixed integerprogramming model that provides an ATP order promising and fulfillment solutionfor batch orders that arrive within a predefined time interval. Ervolina and Dietrich(2001) describe an application of the implosion technology for ATP order promisingin assemble-to-order (ATO) and configure-to-order (CTO) manufacturing environ-ments. The goal is to create a feasible production plan that can be used to scheduleor promise customer orders. Chen-Ritzo (2006) studies a similar availability manage-ment problem in a CTO supply chain with order configuration uncertainty. Akcay andXu (2004) develop a two-stage stochastic integer program with recourse to allocateconstrained components so as to maximize the fraction of orders assembled within aquoted maximum delay. The closest work in this stream is Dietrich et al. (2005) whichdescribes a deterministic implosion model that identifies suitable product configura-tions for an Available-to-Sell process that consume the most surplus inventory andrequire minimal additional component purchasing costs. The focus of this model is onthe perspective of the firm, independent of the customer’s propensity to buy alternative

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products. Market demand, customer preferences, or product substitution policies arenot considered. In contrast, we explicitly model customer expectations in a dynamicsetting, utilizing a customer behavior model that determines how customers evaluateproduct substitutions if their initial product selection is unavailable.

In the operations management literature, there are several papers in which productsubstitutions or flexible customer requirements are important elements. Bassok et al.(1999) study a multi-product inventory problem with full downward substitution whereexcess demand for a product can be filled using a product of higher utility. Hale et al.(2001) extend the analysis of the downward substitution problem to an ATO systemwith two end-products where each product is composed of two components. Substitu-tions are carried out at the component level. Gallego et al. (2006) consider downwardsubstitution to satisfy unmet demand for lower grade products in a semiconductorproduction environment, and propose a heuristic allocation scheme for determiningnear-optimal build plans. Swaminathan and Tayur (1998) determine optimal configu-rations of semi-finished products (vanilla boxes) along with their inventory stockinglevels to enable late customization in an assemble-to-order supply chain for computermanufacturing. Building upon these concepts, Yunes et al. (2007) apply a customermigration model in conjunction with mixed-integer programming to determine theoptimal product portfolio at John Deere and Company. A customer migration listcontains alternative product configuration choices if a customer’s preferred productselection is unavailable. Balakrishnan and Geunes (2000) study a production planningproblem with flexible bills-of-materials and component substitution. A dynamic pro-gramming solution method is developed to find production and substitution quantitiesthat satisfy demands at minimum total cost, comprising setup, production, substitu-tion, and inventory holding cost. Because supply is assumed to be unconstrained, themodel does not address matching of demand and supply. Balakrishnan and Geunes(2003) consider a production planning problem faced by a steel manufacturer whosecustomers allow flexibility in product specification. In a recent paper, Balakrishnanet al. (2005) apply concepts from revenue management to investigate how a firm canmaximize profits by shaping demand through dynamic pricing.

3 Availability management

Availability management is the overarching task of coordinating the planning of prod-uct availability with the real-time promising of customer orders. The most commonapproach to Availability Management is the Available-to-Promise (ATP) process. Thegoal of the ATP process is to generate a single, integrated supply plan that bringstogether the business objectives of finance, sales, and marketing, with the reality ofthe unbiased forecast and the capacity of the supply chain. This integrated supplyplan is called the “ATP schedule” that allocates component supplies to products andcustomer segments. During execution, the ATP process deals with a real-time streamof customer orders. As a customer request arrives, the order scheduling process mustpromise an availability date to the customer. This task involves checking the contentsof the order against the ATP schedule, determining an availability promise date to thecustomer, and decrementing the ATP to accurately reflect the supply committed to new

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customer orders. The ATP process utilizes an analytical technique called implosionto generate an optimized ATP schedule that takes into account supplier commitmentsand limited manufacturing capacities (e.g., Dietrich et al. 2005). The ATP scheduleseeks to satisfy a fixed sales target that is based on a demand forecast. Because ATPprovides no mechanism for dealing with unallocated supply, a separate non-integratedbusiness process is often created to manage inventory excess and overages, e.g. by exer-cising buy-back agreements with component suppliers or other procurement-relatedtechniques.

In today’s environment, customers expect that products are available in a largevariety of configurations, and, with this expanding variety, customers have becomeincreasingly flexible in what they will purchase. When the capacity of the supply chaindoes not directly align with the sales target, imbalances often result in an ATP schedulethat falls short of customer demand. In this paper, we propose a new methodologycalled Available-to-Sell (ATS) that more effectively balances supply and demand bytaking advantage of demand flexibility when determining the allocation of products tocustomer segments. ATS is designed to find alternative product configurations that bestconsume excess supply while minimizing additional procurement investments. ATS ismost effective in an assemble-to-order (ATO) manufacturing environment where endproducts are configured from standard components, and where the simplified productstructure ensures that product substitutions will drive customer interest.

The main output of the ATS process is an “ATS schedule” that comprises opti-mized availabilities of a firm’s core products as well as saleable product alternatives.The ATS schedule takes advantage of up-sell, alternate-sell or down-sell opportunities.An up-sell opportunity is where a customer is sold a more richly configured solutionabove the customer’s initially selected price range; price incentives may be used toentice the customer to agree to an up-sell. An alternative-sell relates to a sale of a sim-ilar product that falls within the selected price range. An alternative-sell is performedwhen an up-sell is not available or the customer opts for a similarly priced product.A down-sell opportunity refers to a sale of a product that falls below the price rangeselected by the customer. ATS can drive further efficiencies if the marketplace can besubdivided into customer segments where customers in the same segment have similarbuying behavior and lifetime values to the company (customer lifetime value is thepresent value of future cash flows attributed to a customer relationship).

4 ATS planning model

In this section, we define the underlying customer behavior model and formulate alinear programming model for ATS. Before we state the problem, we define the rele-vant notation that is used throughout the remainder of the paper.

Customer behavior model

• C : set of customer segments, or customer classes, indexed by c.• αc ≥ 0 : Reservation price parameter for customers in segment c ∈ C .• 0 ≤ βc ≤ 1 : Reservation quality parameter for customers in segment c ∈ C .

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• γc : First-choice probability of customers in segment c ∈ C , denoting the probabil-ity that a customer in segment c will only accept its first-choice product selectionand no product alternatives.

Products and components

• I : set of components, indexed by i .• M : set of core products, indexed by m.• N : set of alternative products, indexed by n.• S : set of products, indexed by s where S := M ∪ N .• uis : usage of component i ∈ I in product s ∈ S (bill-of-material); the usage

values are assumed to be binary.

Supply and demand

• Qi : supply of component i ∈ I .• Dc

m : demand for core product m ∈ M and customer segment c ∈ C ; demand isassumed to be deterministic.

Cost and profit

• hi : inventory holding cost of component i ∈ I per time period, e.g., week.• rs : per-unit retail price of product s ∈ S.• ps : per-unit marginal profit of product s ∈ S.• qc

s : quality of product s ∈ S for customer segment c ∈ C .• bc

m : backorder penalty of a customer order in segment c ∈ C for product m ∈ M ;the backorder penalty is considered a one-time charge that is independent of theduration of the backorder.

• wcmn : per-unit penalty cost for substituting one unit of product n ∈ N for one unit

of product m ∈ M (price discount).

Decision variables

• Xcm : ATS quantity for core product m ∈ M and customer segment c ∈ C .

• Y cmn : ATS quantity of alternative product n ∈ N used as a substitute for core

product m ∈ M in customer segment c ∈ C .

The two sets of decision variables Xcm and Y c

mn represent the optimized ATS schedule.

4.1 Customer behavior model

Similar to the customer migration model described in Yunes et al. (2007), we assumethat a known percentage of customers accept a substitute product if its price and qualityare within a certain range. The first-choice probability γc determines the proportion ofcustomers in segment c ∈ C that will not accept substitutions. The reservation pricedetermines the incremental price that a customer in segment c is willing to pay for analternative product. If the customer’s initial selection is product m and rm denotes theprice of product m, the customer’s reservation price is (1 + αc)rm . Similarly, if qc

mdenotes the quality level of product m for a customer in segment c, the customer’s res-ervation quality is (1−βc)qc

m . A customer is willing to purchase an alternative product

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n ∈ N if its price rn does not exceed the customer’s reservation price (1+ αc)rm , andif its quality qc

n is no less than the customer’s reservation quality (1 − βc)qcm . If no

alternative selections in the desired price and quality range are available, the customerorder is assumed to be backlogged.

4.2 Problem formulation

The ATS planning model is formulated as a linear programming problem. For analyt-ical tractability, we choose a simple single-period model to accomplish the short-termallocation of components and products to customer segments. The model recognizesthe unique customer preferences associated with a segmented market.

Consider a two-level product structure that consists of a set of components I whereeach component i ∈ I has some finite supply Qi , and a product portfolio S = M ∪ Nthat is the union of a set of core products M and a set of alternative products N . The setof core products M contains currently featured products that are offered by the seller.The set N contains alternative products that may be used to fill unsatisfied demandfor certain core products with additional substitution cost incurred. Since some coreproducts could be substituted for other core products, the sets M and N may not bedisjoint. The demand in customer segment c ∈ C for core product m ∈ M equals Dc

m .Given a component i , the number of units of the component in product s ∈ S equalsuis . The output of the model is the ATS schedule, i.e., the allocation of core and alter-native products to customer segments. The model maximizes total supply chain profit,consisting of net profit from sales less component holding costs, backorder costs andsubstitution penalty costs.

Using the above notation, the objective function of the ATS planning problem isformulated as follows:

Max Z(Xcm, Y c

mn) =∑

m∈M

(∑

c∈C

pm Xcm +

∑

n∈N

∑

c∈C

pnY cmn

)

−∑

m∈M

∑

c∈C

bcm

(Dc

m − Xcm −

∑

n∈N

Y cmn

)

−∑

i∈I

hi

(Qi −

∑

m∈M

(∑

c∈C

uim Xcm +

∑

n∈N

∑

c∈C

uinY cmn

))

−∑

m∈M

∑

n∈N

∑

c∈C

wcmnY c

mn (1)

The first term is the net profit derived from sales of core and alternative products. Thesecond term represents backorder costs that are incurred when the allocation of coreand alternative products falls short of the customer demand Dc

m . The third term repre-sents the inventory holding cost incurred for unused components when the componentusage is less than the available supply. The last term represents product substitutioncosts that are incurred as a result of discounts offered to customers for accepting asubstitute.

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Next we formulate the constraints. Given the demand Dcm for core product m ∈ M ,

the total build volume for this product, including the volume substituted by new alter-native products, cannot exceed the demand for core product m:

Xcm +

∑

n∈N

Y cmn ≤ Dc

m for all m ∈ M, c ∈ C (2)

The ATP schedule must be feasible with respect to the component supply, i.e., thenumber of components consumed plus any unallocated inventory must be less than orequal to the available component supply:

∑

m∈M

(∑

c∈C

uim Xcm +

∑

n∈N

∑

c∈C

uinY cmn

)≤ Qi for all i ∈ I (3)

The number of product substitutions for core product m ∈ M cannot exceed thefraction of demand that can be filled with alternative products:

∑

n∈N

Y cmn ≤ (1− γc)Dc

m for all m ∈ M, c ∈ C (4)

Any alternative product n ∈ N that is used to substitute demand for core productm ∈ M in customer segment c ∈ C must meet the reservation price and reservationquality requirements qc

n ≥ (1− βc)qcm and rn ≤ (1+ αc)rm . These requirements are

expressed as linear constraints in the substitution quantities Y cmn as follows (alterna-

tively, for combinations of m, n, and c that do not satisfy the above conditions, theY c

mm variables could be eliminated during pre-processing):

Y cmn

[qc

n − (1− βc)qcm

] ≥ 0 for all m ∈ M, n ∈ N and c ∈ C (5)

Y cmn[(1+ αc)rm − rn] ≥ 0 for all m ∈ M, n ∈ N and c ∈ C (6)

Finally, constraints (7) and (8) impose non-negativity constraints on the decision vari-ables:

Xcm ≥ 0 for all m ∈ M, c ∈ C (7)

Y cmn ≥ 0 for all m ∈ M, n ∈ N , c ∈ C (8)

The optimization problem (1)–(8) is a linear program that can be solved very efficientlyfor large problem sizes.

The single-period model described above has a few limitations which we discussnext. First, the model does not capture situations in which demand and componentsupplies are planned over multiple time periods. In many industrial applications,component suppliers commit to the delivery of component supplies several weeksinto the future. Such multi-period deliveries might create correlations of componentallocations to customer segments that are not captured in a single-period model.Under a time dimension, our model can be used to allocate available components

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to customer segments in a myopic fashion using a rolling horizon. Although this isan approximation, most practical applications of the model focus on maximizing thefirm’s ability to fill orders over a short-term planning horizon (1–2 weeks) in a myopicfashion. The reason is that supply chain managers are often reluctant to reserve supplyfor future weeks because demand forecasts become increasingly uncertain. More-over, because many high-technology component suppliers operate inventory stockingpoints near the OEM’s (original equipment manufacturer) production facilities theyare usually able to respond to supply changes on a weekly basis. If the likelihood of along-term supply constraint is small and supplier lead times are short, we suspect thatthe effect of applying a myopic strategy on the optimal component allocations will besmall.

Second, we assume that all demand in a period is observed before products areallocated to customer segments. In reality, customers orders are placed continuouslyand components would need to be allocated to them on an order by order basis. Thus,our model provides an optimistic estimate of the profitability of a firm under a givendemand, supply plan, and customer behavior model.

Third, the assumption that all customers within a market segment have the samebuying behavior (in terms of their willingness to upgrade to a higher-priced product)may not be true in reality. It would be more realistic to model the customers’ reserva-tion price and quality values as random variables. In our numerical study, we allowedthe customer behavior parameters to randomly deviate from their base values.

Finally, we assume that the component supply quantities are exogenous inputs tothe model that are determined prior to executing the allocation of components to cus-tomer segments. It would be interesting to develop an integrated model that not onlydetermines product allocations but also the ideal component supply mix in terms ofmaximizing the firm’s profit in view of customer’s propensities to upgrade to higher-priced, more profitable products. Such an integrated decision model is beyond thescope of this paper, but is a promising area for research.

5 Simulation framework

The simulation model was built using the Availability Management Simulation Tool(AMST) which has been used at IBM to develop various availability management sim-ulation models (Lee 2006). AMST was developed using the simulation capabilitiesof IBM’s WBM� (WebSphere Business Modeler) as a simulation modeling frame-work for availability management processes. The simulation framework consists ofreusable components and methods which are easily adapted to different availabilitymanagement environments.

The goal of the simulation is to evaluate the potential of the ATS model in a real-istic order execution environment where components must be allocated to productson an order by order basis. The simulation model can be executed in single-period ormulti-period mode. In single-period mode, the ATS planning model is invoked onceto determine the planned allocation of products to customer segments. In multi-periodmode, the ATS planning model is invoked in a myopic fashion at pre-determined timepoints (roll-forward events) after the remaining order backlog and unsold inventory

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Fig. 1 ATS simulation model and data flow

from previous time periods is evaluated and carried forward to the next time period.Subsequently, the simulator generates a randomized arrival sequence of customerorders for a given demand statement. Each customer order is assigned a customersegment, a core product selection, a first-choice probability, and a reservation priceand reservation quality pertaining to the assigned segment. The outputs of the sim-ulation are statistical outcomes of the system performance metrics, including salesprofit, backorders, proportion of orders filled, and inventory holding costs. Figure 1illustrates the data flow of the simulator and its interaction with the ATS model.

The ATS linear programming model assumes that all demand in a period is observedbefore products are allocated to customer segments, whereas in reality customer ordersare placed continuously and components need to be allocated to each order as it arrives.Thus, the results from the ATS model establish a theoretical upper bound of the prof-itability under a given demand, supply, and customer behavior model. To assess thetrue potential of ATS, the optimized allocation must be simulated under an operational“order acceptance” policy. The simulation model therefore embodies different orderacceptance policies for allocating ATS quantities to customer orders. The simplestorder acceptance policy is First-Come-First-Serve (FCFS) shown in Fig. 2.

For each customer order, the FCFS order scheduler checks the ATS quantity of thecustomer’s product selection. If the first-choice product is available, the order is sched-uled and fulfilled. If the first-choice product is not available, the scheduler determineswhether the customer is willing to take a substitution, in which case the schedulerlooks for a substitution product that meets the customer’s price and quality tolerance.In this case, the first available alternative product within the price and quality rangeof a customer is selected. A more sophisticated implementation might consider prof-itability or availability as selection criteria. When a substitution is found, the order isscheduled and fulfilled. If the customer is not willing to take a substitution or a suitablesubstitution is not found, the order is backlogged.

Under a second order acceptance policy called rationing, a substitute product maybe offered even when the preferred product selection is available in order to up-sellan item that is available, acceptable, and may improve the profit from the sale. Theflow diagram is depicted in Fig. 3. If a customer is willing to take a substitution, the

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Fig. 2 First-Come First-Serve (FCFS) order acceptance policy

Fig. 3 Rationing order acceptance policy

scheduler randomly searches for an alternative product that meets the price and qualitytolerance of the customer from the ATS schedule. Again, the first available substituteproduct within the price and quality range of the customer is selected. If no suitableproduct alternative is found and the customer’s first choice product is available, theorder is fulfilled; otherwise the order is backlogged.

In order to incorporate the benefits of risk pooling over multiple customer segmentsinto the simulation, the product allocation quantities Xc

m and Y cm,n that are created by

the ATS planning model are transformed into an “aggregated” ATS schedule. Theallocated product quantities of the ATS schedule are assigned to common inventorybuckets, denoted Xm and Yn , that are derived from the original ATS allocation quan-tities as follows:

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Xm =∑

c∈C

Xcm for all m ∈ M (9)

Yn =∑

c∈C

∑

m∈M

Y cm,n for all n ∈ N (10)

In the simulation experiments presented in Sect. 6, both order acceptance policies usethe aggregated ATS allocation quantities Xm and Yn to determine whether to substitutean alternative product for an item requested in a customer order.

6 Numerical study

The numerical study focuses on prescribing how the firm should adjust its sales strat-egy when faced with different degrees of supply and demand imbalances. To addressthis goal, we implemented the availability management models described in the previ-ous sections in the context of a representative assemble-to-order (ATO) supply chainfor mid-range server computers. In Sect. 6.1 we describe the example scenario for thenumerical study. In Sect. 6.2 we test the conjecture that intelligent demand shapingbased on customer preferences can provide significant financial benefits over a tradi-tional ATP-based approach, particularly in environments where inventory imbalancesexists. We apply the linear programming model to show how a firm can take advan-tage of up-sell opportunities for increased profit, and analyze the effect of customers’price sensitivity on profitability. In Sect. 6.3 we use the linear programming modelin conjunction with simulation to investigate the performance of ATS under differentorder acceptance policies in a realistic order execution environment.

6.1 Supply chain model

The example scenario for the numerical study is derived from an industry-size assem-ble-to-order supply chain of a server computer product line. The product portfolio con-sists of eight mainstream server computer products that represent a whole spectrumof price-performance points. The products and their bills-of-materials are depictedin Table 2. Products M1 and M2 are entry level products, M3 to M6 are mid-rangesystems, and M7 and M8 are high-performance computers. Each product is assembledfrom components of six different commodity groups: system processors, memory,hard drives, optical drives, video adapters and software preloads. For example, prod-uct M1 is assembled from a 2.8 GHz system processor, a 30 GB hard drive, 128 MBmemory, a 48X CD-RW optical drive, an Extreme 3D video card and a system soft-ware preload B. Although in reality each server product is assembled from dozens ofcomponents, the major components represented in this study account for more than80% of the cost of a product. The manufacturing operation is driven by an assemble-to-order process. The table also shows the price, gross profit, and quality score of everycomponent. The quality score of a component depends on its parts worth relative to theother components in the same commodity group. Components with the highest partsworth are assigned the highest quality score, and they carry the highest gross profit.

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Tabl

e2

Bill

-of-

mat

eria

lsst

ruct

ure

used

inth

enu

mer

ical

stud

y

Com

pone

nts

Pric

ePr

ofit

Qua

lity

scor

ePr

oduc

tpor

tfol

io

Gro

upTe

chno

logy

Eco

nom

yV

alue

Perf

orm

ance

M1

M2

M3

M4

M5

M6

M7

M8

SYST

EM

2.8

GH

z/80

0M

Hz

Xeo

n28

626

2025

251

1–

––

––

–

PRO

CE

SSO

RS

3.0

GH

z/80

0M

Hz

Xeo

n39

151

3050

50–

–1

1–

––

–

3.2

GH

z/80

0M

Hz

Xeo

n50

484

4075

75–

––

–1

1–

–

3.4

GH

z/80

0M

Hz

Xeo

n65

015

050

100

100

––

––

––

11

HA

RD

DR

IVE

S30

GB

4200

RPM

110

1010

2020

1–

1–

––

––

40G

B42

00R

PM16

121

2040

40–

1–

–1

––

–

60G

B42

00R

PM21

636

3060

60–

––

1–

1–

–

120

GB

7200

RPM

312

7240

8080

––

––

––

1–

160

GB

7200

RPM

405

105

5010

010

0–

––

––

––

1

ME

MO

RY

128

MB

SDR

AM

888

1010

201

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––

––

–

256

MB

SDR

AM

138

1820

2040

–1

––

1–

––

512

MB

SDR

AM

192

3230

3060

––

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1.0

GB

SDR

AM

260

6040

4080

––

––

–1

1–

2.0

GB

SDR

AM

324

8450

5010

0–

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1

OPT

ICA

LD

RIV

ES

9.5

mm

Slim

DV

D77

730

3030

–1

1–

––

––

48X

CD

-RW

104

1440

4040

1–

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1–

11

48X

CD

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/DV

D12

020

5050

50–

––

1–

1–

–

VID

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138

1330

3030

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Ext

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Exp

ress

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2340

4040

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oad

A11

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C16

828

3030

30–

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D24

040

4040

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Prel

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E32

575

5050

50–

––

––

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1

158


Table 3 Product quality, demand and backorder costs by customer segment

Product Price Profit Economy segment Value segment Performance segmentrm pm

Quality Demand Backlog Quality Demand Backlog Quality Demand Backlogq1

m D1m cost b1

m q2m D2

m cost b2m q3

m D3m cost b3

m

M1 960 92 47 800 44.6 39 200 133.8 37 – 178.4

M2 978 95 43 800 45.5 39 200 136.4 39 – 181.9

M3 1,002 101 43 500 46.8 40 400 140.3 38 100 187.1

M4 1,284 190 70 500 63.0 65 400 188.9 64 100 251.9

M5 1,218 161 57 300 58.8 56 400 176.4 54 300 235.2

M6 1,380 233 70 200 69.1 69 400 207.4 70 400 276.5

M7 1,656 358 83 200 86.9 85 300 260.7 84 500 347.6

M8 1,836 450 93 1000 99.0 95 300 297.0 96 600 396.0

The quality level of a product is computed as the (weighted) average of the qualityscores of the components used in its configuration. Each component in a commoditygroup is assigned a quality score (a value between 0 and 100) based on its quality rela-tive to all other components in the same commodity group. Higher scores are assignedto components with higher parts worth, e.g., a 120 GB hard disk will score higher thana 60 GB hard disk.

We assume that customers belong to one of three market segments, denoted econ-omy, value and performance. Customers in the economy segment tend to purchaseentry-level products and are highly price sensitive (i.e., they have a low reservationprice). Customers in the value and performance segments are moderately price sensi-tive and are more often willing to accept up-sells. The perceived quality of a productdepends on the customer segment. Customers in the economy segment have a balancedvaluation of components in the six commodity groups, with 50 being the highest qualityscore in each group. Customers in the value segment place a higher relative importanceon system processors and hard drives, whereas customers in the performance segmentplace their highest importance on processors, hard drives, and memory as indicatedby a top quality score of 100.

Table 3 summarizes the price and quality scores of the product portfolio for thethree customer segments. The price of product m, rm , is the sum of the prices of thecomponents used in its bill-of-materials. The quality score of product m in customersegment c, qc

m , is the average of the quality scores of all components used in its bill-ofmaterial. The table also shows the backorder penalty, bc

m , and the customer demand,Dc

m , for each product and each customer segment. The backorder penalty is a fractionof the price of a product; backorder penalties are lowest for economy customers andhighest for performance customers.

The reservation price and reservation quality parameters used in the customerbehavior model are assumed to be (α1,α2,α3) = (0.1, 0.2, 0.3) and (β1,β2,β3) =(0.3, 0.2, 0.1). The parameter choice is driven by the fact that customers in the econ-omy segment tend to be highly price sensitive and may compromise on product quality,

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whereas customers in the performance segment are relatively price insensitive butdemand a high quality level of a product. In addition to utilizing a reservation priceand a reservation quality to determine whether a customer will consider an alternativeproduct, we assume that a fraction of customers are committed to their first productchoice and will not accept an alternative configuration. The first-choice probabilityin the baseline scenario is γc = 0.5 for all three customer segments, i.e., 50% ofcustomer orders will not accept product alternatives if their initial product selection isunavailable.

6.2 Comparisons between ATS and ATP

In practice firms often highlight entry-level products to customers to provide an inter-esting price-performance point that will establish a sound brand image and elicit afavorable customer response, i.e., a buy decision. The marketed entry-level productsare usually supplied at a lower rate than actual demand, driving longer product avail-ability lead times (although the seller must have a reasonable supply line for the entry oreconomy level products to meet regulatory and country specific business practices).The goal is to have customers contact the seller which provides the opportunity toup-sell the customer to a more richly configured solution, normally at a higher price-performance point, usually thought of as the market “sweet spot” for the productcategory. In this section, we apply the linear programming model of Sect. 4 to inves-tigate how demand shaping with ATS helps improve the operational performance ofthe supply chain when the component supply deviates from the ideal net componentrequirements. We illustrate the impact of employing different supply schemes on afirm’s profitability. This enables us to identify conditions under which benefits to thefirm are the most significant.

Supply and demand imbalances

To obtain a baseline supply plan for the demand scenario given in Table 3, we firstcalculate the net component requirements by “exploding” the demand through the bill-of-materials in a standard MRP-type calculation (e.g., Hopp and Spearman 2000). Thiscalculation yields supply quantities that would be required in order for all customersto receive their first-choice product. This supply plan is labeled “unbiased componentmix” in Table 4. Component mix denotes the supply of a component relative to thesupply of the other components in the same commodity group. For example, 25%of the first-choice products utilize a 30 GB hard drive, whereas 13% use a 160 GBhard drive. Table 4 shows two biased supply scenarios, denoted “low skew” and “highskew”. In both scenarios, high-value components are procured at a higher rate thanactual customer demand, and low-end components are purchased at a lower rate thanactual demand. For each supply scenario, we generated ten problem instances byapplying small random perturbations to the component mix shown in the table, andsolved each instance using the linear programming model described in Sect. 4. Theresults presented next are averages over these ten problem instances.

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Table 4 Component mix for unbiased, low, and high supply skew

Components Unbiased component Biased component Biased componentmix (no skew) mix (low skew) mix (high skew)

SYSTEM 2.8 GHz/800 MHz Xeon 25% 24% 19%

PROCESSORS 3.0 GHz/800 MHz Xeon 25% 26% 31%

3.2 GHz/800 MHz Xeon 25% 25% 20%

3.4 GHz/800 MHz Xeon 25% 25% 30%

HARD DRIVES 30 GB 4200 RPM 25% 10% 10%

40 GB 4200 RPM 25% 28% 28%

60 GB 4200 RPM 25% 31% 26%

120 GB 7200 RPM 13% 19% 21%

160 GB 7200 RPM 13% 13% 15%

MEMORY 128 MB SDRAM 25% 10% 10%

256 MB SDRAM 25% 28% 28%

512 MB SDRAM 13% 19% 19%

1.0 GB SDRAM 25% 28% 25%

2.0 GB SDRAM 13% 16% 19%

Fig. 4 Attainable net profit achieved by ATS under different supply scenarios

Figure 4 displays the attainable net profit, i.e., sales profit less backorder penaltiesless inventory holding costs, under the ATS strategy where the firm takes advantage ofup-sell opportunities. We can see that the profit gain is monotone increasing with thedegree of supply skew for the two most profitable customer segments (performanceand value). The profit decreases slightly for the economy customer segment which iscaused by supply constraints for low-end components, combined with limited oppor-tunities for up-selling or alternative-selling due to the low reservation prices of entrylevel customers.

Figure 5 shows the percentage profit gain of ATS compared to an ATP-basedapproach. This can be interpreted as the value of demand shaping when the firmhas advance knowledge about the price sensitivity and quality profiles of its customersegments, and under the assumption that demand is known. The figure illustrates that

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Fig. 5 Percentage profit gain of ATS over ATP under different supply scenarios

Fig. 6 Order fill rate and number of substitutions under different supply scenarios

the net profit derived from all customer segments increases as the degree of supplyskew increases. The firm can be significantly better off financially (up to 10%) byrecognizing opportunities for up-selling to the market sweet spots in its customer seg-ments. The profit gain in the firm’s two most profitable customer segments (value andperformance) more than offsets the profit decrease in the entry-level (economy) seg-ment. The modest positive profit gain of ATS under the no skew scenario is a result ofaveraging over multiple random perturbations that were analyzed for each componentmix (i.e., the sampled component mix of each of the ten problem instances deviatesslightly from the unbiased component mix shown in Table 2 which results in a smallorder backlog under ATP). Although not shown here, we note that the profit gainwould eventually decrease as the supply is progressively skewed towards high-valuecomponents because the proportion of customers willing to accept up-sell products atincreasingly higher price points will eventually decline.

For the same experiment, Fig. 6 shows the number of product substitutions in thedifferent customer segments under the ATS regime as well as the proportion of ordersthat are filled with either the customer’s first choice or an alternative product. Theproportion of orders filled is plotted against the secondary vertical axis.

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We can make two key observations. First, the order fill rate decreases from 100% to97%. This decline is driven entirely by backorders in the economy customer segment.Second, the relative proportion of customers that purchase an alternative product ishighest in the value segment (43%), as is the incremental profit gain for this segmentas illustrated in Fig. 5. This observation is consistent with marketing analyses thatsuggest that customers in the mid-range segment are most likely to respond favor-ably to alternative product offerings. Firms should therefore focus their marketingefforts on protecting revenues derived from the mid-range segment and employ effec-tive demand shaping actions to grow the share of the wallet from these customeraccounts.

Effect of customer behavior

To investigate how customer buying preferences can affect a firm’s profitability, weapplied the heavily skewed supply scenario from the previous experiment and analyzedthe firm’s sales performance under different settings of the first-choice probability γc.We examine three scenarios where γc = 0.5, 0.75 and 1.0. Higher values of γc implythat customers are less willing to accept an alternative product if their initial productchoice is unavailable. In the last scenario, customers always prefer their initial productchoice. Figure 7 displays the effect of customers’ first-choice probability on the orderfill rate. We observe that as the first-choice probability increases the order fill ratedecreases. This is expected because a higher first-choice probability translates intofewer customers accepting product substitutions when their first choice is stocked out.The order fill rate in the economy segment decreases much more rapidly than in theother segments because economy customers predominantly buy entry-level productsthat are in short supply.

Table 5 depicts the net profit, backorder cost, and inventory cost for the differ-ent first-choice probability values. In the extreme case where γc = 1.0, the averageprofit penalty over the baseline scenario (γc = 0.5) is 37%. The reason for the largepenalty is a significant sales decline in the economy segment combined with increasedinventory costs that are driven by unsold high-value component supplies.

Fig. 7 Effect of customer first-choice probability on order fill rate

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Table 5 Net profit, backorder cost and inventory costs under different customer behavior models

Customer First choice γ = 0.50 First choice γ = 0.75 First choice γ = 1.00segment

Net Backorder Inventory Net Backorder Inventory Net Backorder Inventoryprofit cost cost profit cost cost profit cost cost

Economy 48,1398 11,242 – 341,867 38,643 – (25,345) 100,707 –

Value 635,144 – – 584,505 3,162 – 551,621 234 –

Performance 705,195 – – 665,350 – – 619,300 – –

All segments 1,821,737 11,242 21,120 1,591,721 41,805 78,540 1,145,576 100,941 189,640

6.3 Impact of order acceptance policy

As discussed earlier, a key assumption of the linear programming model is that alldemand in a period is observed before products are allocated to customer orders. Theanalytical results presented in the previous section should therefore be interpreted asa theoretical upper bound of the firm’s profitability for a given supply and demandscenario. In a realistic supply chain environment, the achievable profitability underATS would depend heavily on the order acceptance policy selected to execute theATS schedule. Since customer orders are placed continuously over time, the orderacceptance policy must match the inflow of customer orders against the optimizedATS schedule and determine (on an order by order basis) which product to allocate toan individual customer order without visibility of future orders.

We developed a discrete-event simulation model described in Sect. 5 to simulatethe performance of ATS under the two different order acceptance policies rationingand FCFS. The simulation results reported here are derived from 120 independentreplications. The simulations are based on the assumption that the realized demandsin each replication are exactly equal to the demand values shown in Table 3, onlythe sequence of order arrivals is varied. All simulation runs were conducted in sin-gle-period mode. In each replication, the simulator executes the following steps: (a)invoke the linear programming model to determine an optimal ATS schedule; (b) com-pute an aggregated ATS schedule where the allocations are derived from Eqs. (9) and(10); (c) generate a randomized arrival sequence of customer orders where the totalnumber of orders is equal to the demand quantities shown in Table 3; and (d) fulfillcustomer orders on an order-by-order basis using the desired order acceptance policy.Recall that throughout this paper demand is deterministic which permits analyzing theeffects of the order acceptance policy on profitability and fill rates without cloudingthe analysis with uncertainty in aggregate demand.

Figure 8 demonstrates how a skewed component supply mix can drive profitability.The results from the linear programming model are included for comparison. Not sur-prisingly, the analytical model dominates the profitability metric and drives the largesttheoretical profit improvements attainable through integrated sales actions. Under therationing and FCFS order acceptance regimes, profitability decreases as the supplymix is skewed to high-value components. The rationing policy seeks suitable up-sellopportunities when customers indicate their receptiveness to other product choices.

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Fig. 8 Expected net profit achieved under rationing and FCFS for different supply scenarios

Table 6 Order fill rate achieved under rationing and FCFS for different supply scenarios

Customer No supply skew Low supply skew High supply skewsegment

Analytical Rationing FCFS Analytical Rationing FCFS Analytical Rationing FCFS

Economy 100.0% 94.7% 93.2% 95.0% 80.5% 73.1% 92.9% 76.7% 70.2%

Value 100.0% 97.5% 96.0% 100.0% 88.9% 80.2% 100.0% 89.0% 79.0%

Performance 100.0% 99.1% 97.4% 100.0% 96.3% 85.9% 100.0% 96.1% 84.6%

All segments 100.0% 96.7% 95.2% 97.9% 87.1% 78.7% 97.0% 85.5% 76.6%

FCFS performs least favorably regardless of supply skew. FCFS may achieve a senseof equality amongst customers, but when supply is skewed or constrained it decreasesthe profitability of the enterprise. Therefore, the order acceptance policy implementedin an enterprise’s application suite can have a profound effect on profitability whenan enterprise seeks to shape demand to market sweet spots.

Table 6 shows the order fill rates achieved under the different order acceptance poli-cies by supply skew and customer segment. The details by customer segment show theATS engine driving sales towards value and performance products as supply is skewedincreasingly to those products. Overall, the order fill rates achieved under rationingand FCFS demonstrate increasing gaps to the analytical fill rates as supply is skewedto value and performance products, thereby leaving more backorders in each salescycle which has an undesirable effect on revenue and profitability. Under a stochasticdemand regime, the order fill rates would deteriorate unless the baseline supply planhad a provision for extra safety stocks to protect against higher than expected demandlevels.

Figure 9 shows the effect on profitability based on the order acceptance regimeand customers’ flexibility on their first choice. Clients with a first-choice fixation (i.e.,probability equals 100%) provide an overall drag on profitability. Clients that are moreflexible in their final product selections increase the profitability of the enterprise, andare more valuable to the business. The linear programming model takes advantage of

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Fig. 9 Expected net profit achieved under rationing and FCFS for different customer behavior models

Table 7 Order fill rate achieved under rationing and FCFS for different customer behavior models

Customer First choice γ = 0.50 First choice γ = 0.75 First choice γ = 1.00segment

Analytical Rationing FCFS Analytical Rationing FCFS Analytical Rationing FCFS

Economy 92.9% 76.7% 70.2% 75.7% 73.9% 67.1% 36.8% 59.3% 59.3%

Value 100.0% 89.0% 79.0% 97.4% 85.7% 75.4% 99.8% 75.9% 75.9%

Performance 100.0% 96.1% 84.6% 100.0% 96.3% 84.8% 100.0% 90.3% 90.3%

All segments 97.0% 85.5% 76.6% 88.8% 83.3% 74.3% 73.1% 72.5% 72.5%

the given supply mix to drive the highest net profit regardless of first-choice probability,followed by the rationing method and FCFS.

Table 7 provides detailed insight into Fig. 9 by evaluating the different order accep-tance methods and customer segments on order fill rates. The analytical model consis-tently outperforms the rationing and FCFS methods across order fill rates regardlessof customer segment.

7 Concluding remarks

In this paper we have developed and simulated an advanced availability managementprocess for assemble-to-order supply chains and have outlined the business require-ments for incorporating such a process into supply chain operations. We have describeda mathematical model that aims at finding marketable product alternatives in a productportfolio that best utilize inventory surplus and replace demand on supply-constrainedproducts, and have highlighted business benefits through simulations with realisticproduction data. Our numerical results point out that more flexible customers aremore profitable customers. Market intelligence and data analytics can identify thesemore flexible customers via market models. The integration of marketing insight withthe number and types of sales recommendations are the key to fully attaining theseresults and are beyond the ability of this simulation construct. For example, a veryprice-sensitive client may only be presented with two sales recommendations—both

166


of which are alternative-sells or one alternative sell and one down sell. A more priceinsensitive client may be presented with five dynamic sales recommendations—threeare up-sells and two are alternative sells (no down sells). This stratification of clientsby price sensitivity and the approach to dynamic sales recommendations will be essen-tial to achieving the business results we have identified. Moreover, the use of the ATSmodel as an intelligent and dynamic engine for sales recommendations on the Internetand for sales professionals, and the integration of the ATS output with sales activitieswill be imperative for attaining a sustainable competitive advantage.

The models featured in this paper have already contributed to substantial businessimprovements in real-world supply chains. In 2002 IBM has implemented a hybridATS/ATP process in its complex-configured server supply chain. In this implementa-tion, a conventional ATP model is executed first, and the ATS model is run subsequentlyagainst the remaining unfilled demand and leftover component supplies. The hybridmodel is focused on managing inventory excesses and overages, i.e., finding saleableproduct offerings that consume aged supplies that might otherwise be salvaged or soldin secondary markets. The realized savings include a $100M reduction of inventorywrite-offs in the first year of implementation, and over $20M reduction annually in thesubsequent years. While this hybrid implementation does not deliver the full potentialof ATS business benefits, it does provide management with a control lever ensuringthat there is sufficient supply for core products.

Future work requires the integration of the ATS engine with demand and supplyprocesses and market intelligence data and applications. This is important when largeproduct portfolios are in place and automation is necessary for speed and accuracyof calculations. The benefit of an integrated process and application architecture isto migrate ATS closer to sales execution and allowing automation to make the com-munication and presentation of credible sales recommendations a push self-servicecapability (instead of a pull) for customers. This will minimize the amount of effortof the customer and supporting sales staffs as automation masks the complexity ofthe product portfolio from the client and business considerations such as profitabil-ity from the sales teams, and presents viable product alternatives that have attractiveprice-performance characteristics.

The major prerequisite to integrate ATS into the process and application architec-ture is a robust market intelligence capability. The rationale for this dependency isto identify the customer’s flexibility concerning their first product selections or ame-nability to another set of sales recommendations. As the customer’s flexibility rangeis identified in the various types of market intelligence models such as propensityto buy and share of the wallets profiles, this business insight into customer buyingbehavior can be exploited by ATS modeling by proposing credible and dynamic salesrecommendations based on the customer’s buying characteristics. The flexibility ofcustomers and their spending patterns are highly relevant inputs into the ATS andmay determine the sequence and number of sales recommendations presented to thecustomer based on enterprise business rules.

Acknowledgements The authors thank the two anonymous referees for their exceptional efforts whichhelped significantly improve the content and presentation of the paper.

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168

Inventory reservation and real-time order promisingin a Make-to-Stock system

Richard Pibernik · Prashant Yadav

Abstract In this paper we consider a Make-to-Stock order fulfillment system facingrandom demand with random due date preferences from two classes of customers. Wedevelop an integrated approach for reserving inventory in anticipation of future orderarrivals from high priority customers and for order promising in real-time. Our researchexhibits three distinct features: (1) we explicitly model uncertain due date preferencesof the customers; (2) we consider multiple receipts in the planning horizon that can beutilized to fulfill customer orders; and (3) we choose to utilize a service level measurefor reserving inventory rather than estimating short- and long-term implications oforder promising with a penalty cost function. We propose an algorithm that exploitsthe time structure in order arrivals and time-phased material receipts to determineinventory reservations for high priority orders. Numerical experiments are conductedto investigate the performance and the benefits of the inventory reservation and orderpromising approach under varying system parameters.

Keywords Make-to-Stock · Inventory reservation · Order fulfillment ·Order promising · Inventory rationing

R. Pibernik (B)European Business School, International University Schloss Reichartshausen,Supply Management Institute, Wiesbaden, Germanye-mail: [email protected]

R. Pibernik · P. YadavMIT-Zaragoza International Logistics Program,Zaragoza Logistics Center, Zaragoza, Spaine-mail: [email protected]



169

OR Spectrum (2009) 31:281–307DOI 10.1007/s00291-007-0121-4



1 Introduction

Today’s Enterprise Resource and Advanced Planning Systems include order fulfill-ment modules that play a critical role in efficiently matching a company’s supply anddemand in the short term. Based on inventory on-hand, planned receipts and prior ordercommitments, these systems calculate the available-to-promise (ATP) quantities andallocate them to incoming customer orders. By processing orders, promising due datesand (pre-) allocating them to available inventory and capacity, they not only impact thecustomer service but also have significant influence on scheduling and execution ofmanufacturing and logistics activities. Because companies are increasingly realizingthe importance of earmarking a portion of the ATP quantities for future (uncertain)demand of important customers, software vendors are enhancing their order fulfill-ment modules by what is termed as “allocation planning” (Meyr 2007). We extend thescope of allocation planning in order fulfillment by (1) developing a novel approachto inventory reservation for future uncertain demand of important high priority cus-tomers, and (2) by integrating this approach with real-time order promising. Considerthe following industry example that motivates our analysis:

An Original Equipment Manufacturer (OEM) of DSL telecommunications equip-ment sources all of its products from Electronic Manufacturing Service (EMS) pro-viders with manufacturing facilities located in low cost production regions. The OEMuses a three month rolling forecast from its customers to determine replenishmentorders. Once the orders are placed, the EMS provides firm delivery dates with a leadtime of eight weeks. In many instances, there is a significant deviation between theforecast provided by the customers to the OEM and the actual orders received. Thisoften results in situations wherein the OEM anticipates that the available inventorywill not be sufficient to fulfill all customer demand on time. Due to the long replen-ishment lead times, however, he has no way to remedy this anticipated shortage byre-ordering. In order to maintain the on-time delivery performance promised to someof the key customers, the OEM resorts to reserving a portion of the available inventoryonly for these high priority customers. Determining these reservations is, however, adifficult task, given that multiple inventory receipts are scheduled and future demandis uncertain with respect to order quantities and the required due dates. Also, the OEMfaces the dilemma that if he reserves too much inventory, he risks not being able tomeet orders of other customers with inventory that was in fact available in the ware-house. Alternatively, if he reserves too little, he may not be able to satisfy the targetedon-time delivery performance for the high priority customers.

The problem described in our example is faced by many companies dealing withlong product replenishment lead times and stringent requirements on due date deliveryperformance from some of their customers. The analysis in this paper attempts to lendinsight into this “inventory allocation/reservation” problem. We develop an approachto inventory reservation with the objective of enhancing a manufacturer’s ability to ful-fill uncertain future demand from high priority customers while keeping the fulfillmentof low priority orders at a reasonable level. We assume that long term consequences ofnot meeting customer due date requirements cannot be accurately estimated throughcost based measures and, therefore, suggest that inventory is reserved with the objec-tive of attaining a specific service level for high priority customers. The reservation

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Inventory reservation and real-time order promising in a Make-to-Stock system

quantities become the input to order promising. We assume that order promising isperformed in real-time, i.e. order acceptance/rejection decisions are made instantly atthe time an order arrives. The order promising engine (that decides upon acceptance orrejection of incoming orders and determines order due dates) can utilize the reservedinventory only for orders placed by high priority customers. The unreserved portioncan either be dedicated exclusively to low priority orders (non-nested reservations) or,more realistically, can also be made available to any high priority demand exceedingthe reserved quantities (nested reservations). Our approach is distinct with respect tothe time structure in the uncertain order stream and the inventory receipts across time.As described in our motivating example, we consider the case when multiple inventoryreceipts are scheduled to occur during the planning horizon. The time interval betweenany two inventory receipts constitutes a cycle for which reservation quantities have tobe determined based on aggregate uncertain demand due in this time interval. Due tothe cumulative nature of inventory, these reservations are not independent across time.We develop a simple algorithm that utilizes aggregate demand forecasts and scheduledinventory receipts to compute interrelated inventory reservation quantities and showhow these can be used for promising orders in real-time.

Through numerical analyses we identify factors determining inventory reservationquantities and provide insights and recommendations on how to set relevant systemparameters of an order fulfillment system. Our analysis provides valuable insight intoa fundamental problem described in our initial example: ensuring a target service levelfor future high priority orders comes at the expense of a decrease in the system’s over-all fulfillment performance. We analyze the non-linear trade-off between the servicelevel for uncertain future high priority demand and the expected loss in the overallsystem’s fulfillment caused by inventory reservation. We also show how this relation-ship is impacted by exogenous parameters such as the number of scheduled receiptsin the planning horizon and the customer due date requirements.

The remainder of this paper is organized as follows. In Sect. 2 we review the litera-ture related to our research. In Sect. 3 we present our inventory reservation and orderpromising approach. Key results and findings of our numerical analysis are presentedin Sect. 4. In Sect. 5 we provide conclusions and managerial insights.

2 Literature review

The research in this paper relates to two different streams of literature: order fulfill-ment and inventory rationing for multiple demand classes. In this section we brieflyreview the previous work in these areas and contrast the contributions of our work.

Many researchers examine the impact of order allocation on profit, cost or servicelevel measures in a Make-to-Order (MTO) or Assemble-to-Order (ATO) environment.Ball et al. (2004) provided an excellent review of work in this area. They present mixed-integer programming models for allocating orders arriving within a pre-determinedtime interval to available components and assembly capacity in an ATO setting. Theobjective of such models is to determine an allocation which maximizes (short-term)profit calculated on the basis of per unit revenue of different demand classes, pro-duction cost, inventory holding cost and penalties for order rejection and capacity

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under-utilization (see also Chen et al. 2001, 2002). They propose a “push–pull-frame-work” for available to promise, which exhibits similarities to the structure of the modelwe develop in this paper. Push-based available to promise allocates available resourcesto forecasted demand of different customer classes. Pull-based available to promiseallocates received orders to available resources. For push-based available to prom-ise, the authors propose a deterministic optimization model for allocating availablecapacity and material to forecasted demand of different customer classes. The authorsalso suggest employing stochastic models for determining booking levels for differentcustomer classes and relate push-based available to promise to revenue management.

Fleischmann and Meyr (2003) developed linear and mixed integer programmingmodels for allocating orders to finished goods inventory as well as available resourcesof an ATO system. Allocation decisions are based on a penalty cost parameter thatsimultaneously captures the costs for backlogging, early allocation and order denial.The proposed approaches are myopic in nature: orders received in a certain time periodare allocated without considering the impact on the fulfillment of orders arriving infuture. The authors acknowledge that resource reservation should be addressed in moredetail: “Further research effort has to be put on the allocation mechanisms that assignthe projected production quantities [. . .] to different order classes [. . .]”. Meyr (2007)provides deterministic model formulations for the allocation of available to promiseinventory to a given number of different customer classes (allocation planning) andalso explores approaches to determine the optimal number of customer classes. Kilgerand Meyr (2008) provide an overview of the order allocation functions provided bycommercial order fulfillment systems. They point out that order allocation is typicallyperformed on the basis of a set of heuristic allocation rules that can be customized toaccommodate company specific requirements. Pibernik (2006) reviewed several basicmechanisms for allocating scarce inventory to customer orders and evaluates theirperformance in a situation where inventory availability is severely constrained. Witha case study from the pharmaceutical industry he illustrates that the logic currentlyemployed in order fulfillment systems does not enable a company to adequately con-sider customer priorities when deciding upon order acceptance/rejection and quotingdue dates.

Whereas order fulfillment research mainly focuses on a deterministic setting inwhich a set of orders has to be allocated to available resources, inventory rationingresearch addresses the allocation of available inventory to uncertain future demand andusually considers the replenishment quantities to be endogenous. Research on inven-tory rationing between customer classes is motivated by the following conception: iffinished goods inventory is scarce (and production capacity is constrained), it maybe reasonable to reject demand from less valuable classes in anticipation of demandfrom higher value classes. In inventory literature, the early work of Topkis (1968)considered how inventory should be allocated between demand classes. Each demandclass is characterized by a different shortage cost and the trade-off involves comparingthe benefit of filling demand for low class items in the current period vs. reservingthe available inventory to fill potential demand for higher class items in subsequentperiods. Nahmias and Demmy (1981) evaluated fill rates for given rationing and reor-der levels in a (Q, r) inventory system with Poisson demand and two demand classes.More recent examples of work in characterizing the optimal inventory rationing policy

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are Ha (1997a,b) and Vericourt et al. (2000). Optimal control policies and rationinglevels are determined on the basis of holding and backorder cost/lost sales cost in thesepapers. Benjaafar et al. (2004) extended these approaches to account for multiple prod-ucts and facilities. For a single product inventory system, Hariharan and Zipkin (1995)considered uncertainty in the due date preferences of the customers. They term thetime from a customer’s order arrival to the desired due date as demand lead time. Fora simple setting, they show that the effect of an increase in the uncertain demand leadtime is precisely the same as an equivalent reduction in the uncertain supply lead time.

The research presented in this paper is closely related to “push-based available topromise”, introduced by Ball et al. (2004). We extend the approach of allocating avail-able to promise quantities of finished goods to future demand. As a major distinction,we consider demand to be stochastic with respect to the time of arrival, the due datepreferences and the customer class. We consider multiple receipts across the planninghorizon and combine elements of inventory rationing and class booking limits knownfrom revenue management. In contrast to more recent inventory rationing research, weassume exogenous replenishment quantities. Also, we consider inventory reservationas a measure for maintaining responsiveness of an order fulfillment system to preventthe negative long-term consequences resulting from not meeting delivery date prefer-ences of high priority customers. As long-term implications of not meeting customerpreferences are extremely difficult to capture, we derive inventory reservations on thebasis of a disaggregated service level for high-priority customer demand. Therefore,our approach is not based on maximizing short term profits or revenues generatedfrom different demand classes but on a service level as measure for the responsivenessof an order promising system in regard to future demand of high priority customers.

3 The model

We consider a facility of a MTS manufacturer that carries inventory of a single prod-uct. We assume a planning horizon spanning over T periods. Typically, the periodst ∈ {1, . . . , T } represent working days during which orders are being received and forwhich delivery dates of individual orders are quoted. In any period t the manufacturerfaces uncertain order streams for a single product from two customer classes.1 Anyorder j , arriving in period t can be characterized by a required due date d j ∈ {1, . . . , T }(specified by the customer) and the customer class (high or low priority).

We assume that the manufacturer employs an automated order fulfillment systemwhich handles orders and promises due dates in real-time, i.e. in the sequence of theirarrival. The system keeps track of the uncommitted, cumulative inventory quantitiesin the planning horizon, the “available to promise” (ATP) quantities. We assume thatthe replenishment schedule is frozen throughout the planning horizon (i.e., inventoryreceipts are fixed) and thus the ATP quantities are only dependent on the number ofcustomer orders promised. When an order j with required due date d j arrives, the

1 We assume that demand occurs one unit at a time and orders for more than one unit can be split intosingle unit orders. Hereafter, we use the term “order” to refer to a single unit of demand.

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&

Determine Inventory Reservations

Forecast Demand Distribution

Order Promising

Update

Roll-over

Determine Available Inventory

Update &

Roll-over

Fig. 1 Inventory reservation and order promising framework

system computes the ATP quantities in the planning horizon. If it finds a positive ATPquantity in period d j , the order is accepted; else the system rejects the order.

To ensure fulfillment of high priority orders in case of inventory scarcity, the manu-facturer wants to protect some portion of the ATP quantities from being consumed bylow priority orders. As described previously, we assume that the manufacturer wants toreserve ATP quantities to ensure that a pre-determined target service level is achievedfor high priority orders. Many variations of service level measures continue to be usedin inventory theory reflecting the different types of costs that arise from not being ableto meet demand in time (for a more involved discussion see for example Ronen 1982,1983, as well as Silver et al. 1998). Although it is difficult to quantify these costs,knowing their nature helps determine an appropriate service level measure. In ourcase we assume the cost associated with not being able to meet high priority customerdemand to be a fixed cost; hence we choose to work with an α-service level. This alsocorresponds to our observations for the case example of the DSL OEM described inSect. 1. Note, however, that our approach is not limited to an α-service level; reser-vation quantities could, for example, also be based on a pre-defined fill rate target.Admittedly, the choice of the target service level is an important decision problem byitself; in this section we assume an exogenously set α. In Sect. 4.2 we provide insightsinto the problem of setting α.

Figure 1 gives an overview of our modeling framework and the individual activitiesin our inventory and order promising approach.

At the beginning of period t = 1, the manufacturer forecasts the demand anddue date distribution for periods t = 1, . . . , T . Based on the demand forecast, calcu-lated available inventory and the pre-defined α-service level, reservation quantities arecomputed for high priority orders throughout the planning horizon. The reservationquantities are used as an input to order promising. We assume that the reservationsfor the whole planning horizon are frozen during a period which is determined by theroll-over interval. During this frozen time period, orders arrive and are either acceptedand quoted their required due date, or rejected. Orders from low priority customerscan only utilize the remaining, unreserved portion of the available inventory. After theroll-over interval has elapsed, the inventory reservations are unfrozen, available inven-tory is updated based on orders committed and any new information about arrivals ofinventory receipts are incorporated into the subsequent calculation of available inven-tory. Also, the demand forecast may be updated to accommodate any new demand

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information received. Finally, the planning horizon is rolled-over and the procedurerepeated. The length of the roll-over interval is determined by the time until the nextinventory receipt occurs. In Sect. 3.3 we show that interrelated reservation quantitieshave to be determined for individual inventory cycles. These cycles are given by thepoints in time for which subsequent inventory receipts are scheduled.

In the following sub-section we first describe how we model the order arrival pro-cess for future demand (forecast of the demand distribution). In Sect. 3.2, we showhow the available inventory is calculated and outline the real-time order promisingmechanism. Thereupon we introduce our approach to reserving inventory for highpriority orders.

3.1 Demand and due date distribution

The manufacturer receives orders for a single product from two classes of customers(high and low) that are uncertain with respect to the time of arrival within period t , andthe due date preference. To model the uncertainty in due date preferences, we assumethat each order has a random demand lead time of �(independent of the arrival time)that follows a discrete probability distribution g(� = τ) with cdf G. We assume thatthe distribution of � has a bounded support (τb, τe) such that for some reasonableτe, g(τe) > 0 and G(τe) = 1. Thus, an order arriving in time period t with a demandlead time of τ has a due date preference d = t+ τ . d = t+ τb is the earliest due date acustomer can request for any order placed in period t . τb is typically governed by thelead time for transportation and order picking. By (Qk

t,d(x) : x ≥ 0) we denotethe count process that represents the demand of class k ∈ {H, L} orders arriving inthe interval (0, x] in time period t with a desired due date of d. We assume that thecount process has finite moments and also has stationary and independent increments(Lévy process). The number of orders that arrive in any period t with a due dated can then be denoted by Qk

t,d with its corresponding distribution function Fkt,d(·).

The knowledge of Fkt,d(·) alone is sufficient for the modeling approach presented in

Sect. 3.3. However, we present here the underlying Lévy process (Qkt,d(x) : x ≥ 0)

for reasons that will become clear in Sect. 3.4.

3.2 Real time order promising

At the beginning of the planning horizon, the inventory on hand, denoted by inv0,as well as planned receipts in periods t ∈ {1, . . . , T }, denoted by st , are known. Weassume that receipts occur at the beginning of the period so that these additional quan-tities can be fully utilized in the period of their arrival. Without loss of generality, adelivery lead time of zero is assumed. By invt we denote the available inventory inperiod t . It can be calculated as

invt = inv0 +t∑

ω=1

sω−t∑

d=1

cd , t = 1, . . . , T, (1)

where cd represents inventory quantities committed to orders accepted previouslywith due date d ∈ {1, . . . , T }. We assume that committed orders are binding; their

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promised due date cannot be changed in favor of new incoming orders. Given invt

we can determine the ATP quantities, denoted by atpd , by setting atpd = invT ford = T and recursively computing

atpd = min{invd , atpd+1

}. for d = T − 1, . . . , 1. (2)

Immediately after order j arrives, atpd jis calculated. If atpd j

> 0, the order isaccepted and due date d j is quoted; cd j is updated and invt is recalculated for theremaining planning horizon. We assume that the order is rejected if atpd j

= 0.

3.3 Inventory reservation without nesting

In this section we show how inventory reservations for future high priority demandcan be determined for the non-nesting case, i.e. when high priority orders can only befulfilled from reserved inventory and any high priority demand exceeding the reservedportion will be rejected.

When determining reservation quantities the time structure of the order arrivals,due date preferences and the time-phased inventory availability have to be taken intoaccount. We begin to outline our reservation approach for the simplest case where noinventory receipts occur during the planning horizon. We further assume that the nextreceipt will occur in period T + 1, i.e. in the first period after our planning horizon.Without loss of generality we assume that at the beginning of the planning horizoncd = 0 for all d ∈ {1, . . . , T }.

Because no receipts are scheduled to occur during the planning horizon, the avail-able inventory can be considered as a single resource for which high and low priorityorders compete. Therefore, we do not need to distinguish different time periods in theplanning horizon and can define inv = inv0 as the inventory available throughoutthe planning horizon. By r we denote the portion of the available inventory the man-ufacturer wants to reserve for high priority demand with a required due date d ≤ T .For the sake of notational ease let Q H =∑T

t=1∑T

d=t Q Ht,d . Given the manufacturer’s

service level objective, r has to be chosen to satisfy constraint (3).

Pr{

Q H ≤ r}≥ α. (3)

Clearly, determining r is equivalent to determining a safety stock level that ensures aprobability α of not stocking out in the lead time. Constraint (3) can be written as

r ≥ F−1Q H (α).2 (4)

Given the service level objective, the manufacturer wants to choose the minimumreservation quantity that satisfies (4). The remaining inventory can then be utilized bylow priority orders. Considering that the available inventory may not suffice to satisfy(4), the minimal (and inventory feasible) reservation quantity

2 For notational simplicity we use F−1X (α) to denote min

xF(X) ≥ α.

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r∗ = min(

F−1Q H (α), inv

).

The order promising mechanism can easily be modified to account for inventoryreservations. The available inventory inv is split into invH = r∗−cH for high priorityorders and invL = inv0 − r∗ − cL for low priority orders. By cH and cL we denoteinventory committed to high priority and low priority orders respectively. The com-mitted quantity is updated after an order has been accepted. Any high (low) priorityorders will be accepted as long as invH > 0 (invL > 0).

Now consider the case where an additional material receipt is scheduled for periodt ′(1 < t ′ ≤ T ). At the beginning of the planning horizon the available inventory isgiven by

invt =

⎧⎪⎪⎨

⎪⎪⎩

inv0 −t∑

d=1cd for t = 1, . . . , t ′ − 1,

inv0 + st ′ −t∑

d=1cd for t = t ′, . . . , T .

(5)

In this case, the manufacturer has to determine two (interrelated) reservation quanti-ties. Although in effect only the inventory reservation for the first cycle is implemented,it is still important to determine the inventory reservation for the second cycle. Thisprevents low priority orders arriving in the first cycle from consuming inventory whichwould still be required to achieve the target service level for high priority orders in thesecond cycle. Also, as some of the inventory reserved in the first cycle may remainunused, it needs to be accounted for in computing the inventory reservation for thesecond cycle. By r1 and r2 we denote the quantity of available inventory reservedfor high priority demand in periods t = 1, . . . , t ′ − 1 and periods t = t ′, . . . , T ,respectively. To simplify our notation we define the demand due before period t ′ asQ H

1 =∑t ′−1

t=1∑t ′−1

d=t Q Ht,d and the demand with a required due date later than t ′ as

Q H2 =

∑Tt=t

∑Td=t ′ Q H

t,d . Following our approach of choosing a reservation quantitybased on the probability of fulfilling all high priority demand in the lead time, r1 hasto be set to satisfy the service level constraint (6):

Pr{

Q H1 ≤ r1

}≥ α. (6)

r2 cannot be determined independent of r1. When determining r2 we have to take intoaccount that some random quantity

(r1 − Q H

1

)+may remain unused by demand with

a due date requirement before t ′ and that both(r1 − Q H

1

)+and r2 can be utilized to

fulfill demand Q H2 (r2 however cannot be utilized by Q H

1 ). Considering this, we canwrite service level constraint (7) for r2:

Pr

{Q H

2 ≤ r2 +(

r1 − Q H1

)+}≥ α (7)

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Ignoring inventory availability constraints, we can determine r1 and r2 by first settingr1 = F−1

Q H1(α) independently and then solving

arg minr2

Pr

{Q H

2−

(r1 − Q H

1

)+ ≤ r2

}≥ α (8)

to determine r2. Because the expression involves a convolution of a random variablewith another truncated random variable, we cannot derive a closed form solution to(8). We employ a two-moment approximation to a normal distribution for the distri-bution of Q H

2 −(r1 − Q H

1

)+which is commonly used in inventory literature (see, for

example, Zipkin 2000, pp. 304–305).From (5) we know that r2 has an upper bound st ′ . If the material receipt st ′ is

less than the required reservation quantity r2, determined from (8), some additionalportion of inventory inv0 has to be reserved to ensure that the probability of fulfillinghigh priority demand Q H

2 is at least α. This implies that r1 has to be increased in caser2 > st ′ . The choice of r1 then has to be based on random demand Q H

1 and the random

shortfall(Q H

2 − st ′)+

. Thus, if r2 > st ′ we first set r2 = st ′ and then solve

arg minr1

Pr

{Q H

1 +(

Q H2 − st ′

)+ ≤ r1

}≥ α (9)

to determine r1.From the previous analysis we can derive the following procedure to determine

reservation quantities r∗1 and r∗2 for the case of one receipt in the planning horizon:

1. r1 = min(inv0; FQ H1(α))

2. r2 = arg minr2

Pr{

Q H2 −

(r1 − Q H

1

)+ ≤ r2

}≥ α

3. If r2 ≤ st ′ then r∗1 = r1 and r∗2 = r2; End

4. Else: r∗2 = st ′ and r∗1 = min

(inv0; arg min

r1

Pr{

Q H1 +

(Q H

2 − st ′)+ ≤ r1

}≥ α

)

Knowing r∗1 and r∗2 it is straightforward to compute the available inventory for highand low priority orders:

invHt =

⎧⎪⎪⎨

⎪⎪⎩

r∗1 −t∑

d=1cH

d for t = 1, . . . , t ′ − 1

r∗1 + r∗2 −t∑

d=1cH

d for t = t ′, . . . , T(10)

invLt =

⎧⎪⎪⎨

⎪⎪⎩

inv0 − r∗1 −t∑

d=1cL

d for t = 1, . . . , t ′ − 1

inv0 + st ′ − r∗1 − r∗2 −t∑

d=1cL

d for t = t ′, . . . , T(11)

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Given invHt and invL

t we can determine separate available to promise quantities atpHd

and atpLd for any due date d as shown in conjunction with Eq. (2) and perform real-time

order promising as described in Sect. 3.2.We now generalize this approach for n > 1 scheduled receipts. Let

n = |{st > 0 |t ∈ {2, . . . , T } }| denote the number of receipts in the planning hori-zon. With n receipts, reservation quantities have to be determined for n + 1 cycles{[

1, t1 − 1],[t1, t2 − 1

], . . . ,

[tn, T

]}, with t i denoting the period in which the i th

receipt occurs. We denote by ri the reservation quantity for cycle i (i = 1, . . . , n+1)

and by Q Hi the high priority demand with a required due date in cycle i .

Following our approach for the one receipt case, we start by calculating the res-ervation quantity for the first cycle and set r1 = min(inv0; FQ H

1(α)). We define by

L1 =(r1 − Q H

1

)+the random reservation quantity of cycle i = 1 that remains

unused, i.e. the quantity that can be utilized by high priority demand with a due daterequirement in subsequent cycles. Given r1 and L1 we can employ a simple recursiveprocedure to calculate the reservation quantities for cycles i = 2, . . . , n + 1:

ri = arg minri

Pr{

Q Hi − Li−1 ≤ ri

}≥ α, (12)

where Li =(

Li−1 + ri − Q Hi

)+. (13)

From the one receipt case we know that r2, . . . , rn+1 are bounded from above byst1, . . . , stn . Therefore, the reservation quantities calculated from (12) are only feasibleiff ri ≤ sti−1 for i = 2, . . . , n + 1.

As before, if ri > sti−1 then ri ← sti−1 , and ri−1 has to be recalculated. From (9)we obtain

ri−1 ← arg minri−1

Pr

{Q H

i−1 +(

Q Hi − sti

)+ ≤ ri−1

}≥ α. (14)

However, the recalculated ri−1 is only feasible iff ri−1 ≤ sti−2 . Otherwise, ri−1 ←sti−2 and

ri−2 ← arg minri−2

Pr

{Q H

i−2 +(

Q Hi−1 − sti−1

)+ +(

Q Hi − sti

)+ ≤ ri−2

}≥ α. (15)

This procedure is continued for ri−2, . . . , r1. Note that through this procedure, any“shortfalls” are transferred into earlier periods. Consequently, inventory feasibilitycan be determined based on the reservation quantity r1 computed for the first cycle.We know that the reservation quantities determined from recursively solving (14) and(15) are only inventory feasible iff inv0 ≥ r1. In cases where r1 ≥ inv0, the reserva-tion quantities are (r∗1 , r∗2 , . . . , r∗n+1) = (inv0, s1, . . . , sn). Based on the developedexpressions we can derive the following algorithm for computing inventory feasiblereservation quantities for the n-receipt case:

1. Set r1 = min(inv0; FQ H1(α)); determine L1

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2. For i = 2, . . . , n + 1,

a. ri = arg minri

Pr{

Q Hi − Li−1 ≤ ri

}≥ α (from (12))

b. Li =(

Li−1 + ri − Q Hi

)+(from (13))

c. For κ = i, . . . , 1 (from (14) and (15))

If rκ > stκ−1

rκ ← stκ−1

rκ−1 ← arg minrκ−1

Pr

{Q H

κ−1 +i∑

ω=κ

(Q H

ω − stω)+ ≤ rκ−1

}≥ α

Else : Next i

Next κ

Next i

3. Set r∗1 = min (inv0, r1); r∗i = ri for i = 2, . . . , n + 1; End

Given r∗1 , . . . , r∗n+1 we can calculate the available inventory quantities for high andlow priority orders for all periods of the planning horizon.

invHt =

i∑

ω=1

r∗ω−t∑

d=1

cHd ∀t ∈

[t i−1, t i

];

i = 1, . . . , n + 1; t0 = 1; tn+1 = T (16)

invLt = inv0 +

t∑

ω=1

sω−i∑

ω=1

r∗ω−t∑

d=1

cLd

∀t ∈[t i−1, t i

]; i = 1, . . . , n + 1; t0 = 1; tn+1 = T (17)

Based on invHt and invL

t we can again determine available to promise quantities atpHd

and atpLd for any due date d based on Eq. (2). atpH

d and atpLd can be used for real-time

order promising as described in Sect. 3.2.

3.4 Inventory reservation with nesting

The analysis in the previous section shows how reservations can be determined byclearly partitioning a portion of the inventory for high priority orders. This helps ourunderstanding of the general approach to reservation in the setting considered here.However, the more practically interesting problem is how to determine inventory reser-vations when the orders from high priority customers, besides the exclusively reservedinventory, can also utilize the unreserved portion. This is similar to nested protectionlevels used in airline and other yield management applications.

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We again begin by first considering the simpler case with no receipts occurringin the planning horizon. As before, we reserve r only for high priority orders andonce (and if) high priority orders have exhausted this quantity, they compete equallywith the low priority orders for the remaining inventory. Computing the expressionfor the probability that all high priority orders are fulfilled is now more involved as itrequires explicitly considering the time structure of the order arrival process from thetwo customer classes.

The probability that the reserved inventory itself is sufficient to fulfill all highpriority orders arriving in the planning horizon is Pr{Q H ≤ r}. When the reservedinventory r is not sufficient to meet all high priority orders, any excess demand canalso be met from the remaining unreserved inventory. By (Qi (x) : x ≥ 0) we de-note a count process to represent the cumulative demand observed in the continuoustime interval [0, x] where x ∈ [0, lT ]. Suppose that the reserved inventory is ex-hausted at time V ∈ [0, lT ]. The inventory available to meet joint demand occurringin the time interval (V, lT ], i.e., after the reserved inventory has been exhausted, is[inv0 − r − QL(V )

]+, where the last term QL(V ) accounts for the inventory con-

sumed by the low priority orders up to time V . Suppose the last high priority customerorder occurs at time W ∈ (V, lT ]. Then all high priority customer orders will befulfilled iff

inv0 − r − QL(V )− Q H (W − V )− QL(W − V ) ≥ 0. (18)

Combining the two cases, the conditional probability of high priority fulfillment isgiven by the following function:

ζ (inv0, r |V, W ) ={

1 i f V ≥ lTPr

{QL(W )+ Q H (W − V ) ≤ inv0 − r

}i f V < lT

(19)

(19) can be simplified and written as

ζ (inv0, r |V, W ) = Pr{

Q H (lT ) ≤ r}+ Pr

{Q H (lT ) > r

}

×Pr{

QL(W )+ Q H (W − V ) ≤ inv0 − r}. (20)

By the independent increment property of the assumed stochastic process for orderarrival we can rewrite Q H (W − V ) as Q H (W ) − Q H (V ). Also, Q H (V ) = r bydefinition of V . Substituting and rearranging terms we can write the unconditionalprobability of fulfilling all high priority orders as3

α=Pr{

Q H ≤ r}+Pr

{Q H > r

} lT∫

w= 0

Pr{

QL(w)+Q H (w) ≤ inv0

}f (W =w)dw

(21)

3 To avoid trivial issues we assume α ≥ Pr{

QL (W )+ Q H (W ) ≤ inv0

}.

181


The first part of the above expression represents the probability of high priority fulfill-ment without nesting. For brevity, we denote this by αN N . The second term representsthe contribution of nesting to the overall service level α. We can express this as

α − αN N = Pr{

Q H > r} lT∫

w=0

Pr{

QL(w)+ Q H (w) ≤ inv0

}f (W = w)dw (22)

Next we analyze the case where an additional material receipt is scheduled for periodt ′(1 < t ′ ≤ T ). As in the previous section, the manufacturer has to determine twocycle specific reservation quantities r1 and r2. For the first cycle, the probability α1 offulfilling all high priority orders can be calculated as for the case of no receipts. Forbrevity we assume that r1 has been determined based on α1. The inventory availablein the second cycle is inv0 + st ′ +

[inv0 − Q H

1 −min(inv0 − r1, QL

1

)]+. All high

priority orders with due date d ∈ {t ′, . . . , T

}will be fulfilled iff

inv0 + st ′ +[inv0 − Q H

1 −min(

inv0 − r1, QL1

)]+ − r2 − QL2 (V2)

−Q H2 (W2 − V2)− QL

2 (W2 − V2) ≥ 0. (23)

where V2 corresponds to the time required to observe a cumulated demand of r2 in thesecond cycle and W2 denotes the time at which the last high priority customer order inthe second cycle arrives. With Q H

2 (W2−V2) being equivalent to Q H2 (W2)− Q H

2 (V2)

and Q H2 (V2) = r2 we can write the conditional probability of high priority fulfillment

in the second cycle as

ζ2(inv0, r1, r2, st ′ |V2, W2)

=⎧⎨

⎩

1 i f V2 ≥ lT

Pr

{Q H

2 (W2)+ QL2 (W2)−[

inv0 − Q H1 −min

(inv0 − r1, QL

1

)]+ ≤ inv0 + st ′

}i f V2 < lT

(24)

From (24) we can derive the unconditional probability α2 of fulfilling all high priorityorders in the second cycle:

α2 = Pr{

Q H2 ≤ r2

}+ Pr

{Q H

2 > r2

}

·lT∫

w2=lt ′Pr

{QL

2 (w2)+ Q H2 (w2)−

[inv0 − Q H

1 −min(

inv0 − r1, QL1

)]+

≤ inv0 + st ′}

f (W2 = w2)dw2 (25)

Clearly, (21) and (25) do not lend themselves for a closed form solution for r1 and r2and the integration over the random variable W makes it difficult to employ simple

182


approximation methods as described in Sect. 3.2. For a larger number of inventoryreceipts n > 2, the expressions for αi become even more complex to approximate.Owing to these difficulties, we choose to conduct numerical experiments to quantifythe effect of nesting on the high priority service level for multiple receipts. The resultsof these experiments are presented in Sect. 4.1.

4 Numerical experiments

In this section we present results and insights from numerical analyses we conductedto investigate the impact and performance of our integrated inventory reservation andorder promising approach. For the purposes of the numerical analysis we assume thatthe order arrival process is a Poisson process and hence in each time period of equallength l, orders from high and low priority customers arrive to the system followingPoisson processes with rates λH and λL respectively.

The tool we employ for numerical analysis is implemented in Microsoft Excel andVisual Basic for Applications (VBA) and includes modules for order generation basedon the Poisson arrival process with the general stochastic demand lead time structuredescribed in Sect. 3.1; inventory reservation as described in Sect. 3.3; and real-timeorder promising (with nesting) based on the orders’ time of arrival (as in Sect. 3.2).

All analyses are conducted for a planning horizon spanning over T = 15 peri-ods. The demand parameters chosen for the simulation experiments are λH = 20and λL = 80. One of the objectives of the numerical analysis is to investigate howthe performance measures of interest behave as we change the preset service level α

and the degree of inventory tightness. For the preset service level (denoted by αN N )we chose four commonly used factor levels {0.75, 0.90, 0.95, 0.99}. For the over-all inventory availability in the planning horizon, we consider eight factor levels:{0.2c, 0.4c, 0.6c, 0.8c, 0.9c, 1.0c, 1.1c, 1.2c}, where c = ∑T

t=1∑T

d=t λtd denotesthe total mean demand with a required due date in the planning horizon. With respectto the number of inventory receipts occurring in the planning horizon we consider fourdifferent cases: (1) one inventory receipt in period t = 1(n = 1), i.e., all inventory isavailable at the beginning of period t = 1;4 (2) n = 3 receipts; (3) n = 5 receipts;and (4) n = 15 receipts occurring in every period. For a given n, we assume equalcycle lengths T/n and equal sizes of material receipts of sti = 0.2c

n , 0.4cn , . . . , 1.2c

n (i =1, . . . , n).

Since our reservation approach is based on an α-service level for high prioritydemand due in individual inventory cycles, we measure the relative frequency of100% high priority fill rate for demand due within an inventory cycle and term it asthe realized service level α. This is used as an estimator of the probability of fulfillingall high priority demand due within an inventory cycle.

To determine the statistical robustness of the sample size for our experiments wecalculated the confidence band for the case of five receipts at a capacity equal to themean demand (1.0c) for 500 sample runs. The standard deviation observed from thetest runs was 0.0049. We inferred that with this standard deviation we can obtain good

4 Note that this is identical to the case of no receipt described in Sect. 3.2.

183


Table 1 Reservation quantities for different levels of αN N

αN N

0.75 0.9 0.95 0.99

i 1 2 3 1 2 3 1 2 3 1 2 3

ri 107 100 99 113 104 102 117 105 105 124 108 107

estimates of our service levels within a bound of ±0.008 with a confidence level of95% and conducted 500 sample runs for all experiments reported in this chapter.

In Sect. 3.3 we were not able to derive closed form expressions for the probabilityof fulfilling all high priority demand within a cycle for the case of nested reserva-tions. In our first set of analyses (presented in Sect. 4.1) we conduct experiments toquantify the effect of nesting and to determine whether αN N (the service level targetwithout nesting) can instead be used as an adequate system parameter for determin-ing reservation quantities. Thereupon, in Sect. 4.2, we perform analyses to quantifythe impact of inventory reservation on the overall performance of the system. Theseexperiments specifically address an important trade-off the manufacturer faces whenchoosing a target service level for high priority orders: high target service levels forhigh priority orders may come at the expense of a significant decrease in the systemperformance. Our results provide insights into the relationship between these two per-formance measures. In Sects. 4.1 and 4.2, we also highlight how the number of receiptsin the planning horizon impacts the effects of nesting and the overall fill rate of thesystem. In Sect. 4.3 we explore the effects of different due date distributions on theperformance of our reservation approach. Finally, in Sect. 4.4 we compare the perfor-mance of our approach to a simple myopic reservation policy in which reservationsare made individually for every cycle without taking carry-over effects into account.

4.1 Effect of nested inventory reservation on the high priority service level

In this section we analyze the impact of reserving inventory for different levels ofinventory availability and explore the effects of nesting. All simulation experimentspresented in this section are conducted for a customer lead time distribution of τe = 0,i.e., all customers require instant delivery.

We first show results for the case of three receipts in the planning horizon. Thereafterwe point out the effects of a varying number of inventory receipts. In Table 1 we pro-vide the reservation quantities which result from the reservation approach developedin Sect. 3.3.

Figure 2 shows the realized service level α vs. preset αN N for different levels ofinventory availability. The realized service level α depends not only on the reservationquantity (determined by αN N ) but also on the extent of inventory availability in thesystem. In a severely constrained scenario, it may constitute an upper bound on thereservation quantities. Also, the effects of nesting are dependent on the availabilityof inventory. At very low inventory availability (0.2c), setting a higher target service

184


0,00,0 0,75 0,9 0,95 0,99

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0.2c 0.4c 0.6c 0.8c 0.9c 1.0c 1.1c 1.2c

α

NNα

Fig. 2 Realized service level α for different αN N dependent on inventory availability

level for high priority orders does not guarantee the desired service level. The requiredreservation quantities exceed the available inventory required in the individual inven-tory cycles. As more inventory becomes available, it is sufficient to meet the requiredreservation quantities and there may also be an unreserved portion of the inventoryavailable for fulfilling low priority and any excess high priority orders.

However, this unreserved quantity may still not lead to a significant increase of thehigh priority service level above αN N ; at the hitting time V , low priority orders havealready consumed most of the unreserved portionof inventory. In such instances thereare little or no benefits from nested reservations (e.g., at inventory levels of 0.6c, 0.8cand 0.9c). Beyond a threshold level of 0.9c, the unreserved portion is large enoughto also fulfill some of the excess high priority demand. Therefore, at levels of inven-tory availability beyond this threshold we realize the benefits of nested reservations.Figure 3 illustrates the effects of nesting dependent on the capacity level for a presetαN N = 0.75. Beyond the threshold level of 0.9c we start realizing benefits of nesting(see highlighted area in Fig. 3); they are increasing in inventory availability. It shouldbe noted that at high levels of inventory availability (1.1c, 1.2c) the differences in α

with reservation and without reservation approach zero; the benefits of nesting becomeirrelevant as we realize (almost) the same service levels with nested reservations aswithout reserving inventory.

The benefits of nested reservations depend not only on the available inventory asdepicted in Fig. 3 but also on the manufacturer’s choice of αN N and the correspondingreservation quantities. In Fig. 4 we illustrate this effect by varying αN N at the inven-tory level 1.0c. We can see that at a value of αN N = 0.99 there are almost no effectsof nesting. Below αN N = 0.99, the realized high priority service levels α are higher

185


0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

With Nesting Without Nesting

α

Inventory Availability

0,1c 0,2c 0,6c0,4c 0,8c 0,9c 1,1c1,0c 1,2c

Fig. 3 High priority service levels for nested and non-nested reservations (αN N = 0.75; n = 3)

than αN N due to the additional benefits of nesting. The benefits increase as αN N isdecreased.

These results can guide the decision maker in choosing an appropriate αN N . Morespecifically, they provide insight into the service level the manufacturer can expectfrom setting an αN N , given a certain inventory availability. The results also identifythe levels of inventory and preset αN N at which the manufacturer will experiencepositive effects of nesting on the realized service level. In our experiments we observethat significant effects of nesting only occur if inventory is not very constrained andif the manufacturer does not aim for very high service levels. From a practical pointof view, however, tight inventory and high target service levels (i.e. α ≥ 0.95)are themore interesting and relevant scenarios for inventory reservation. For such instances,αN N alone is an adequate parameter for determining inventory reservation quantities(even with nesting). Further analyses suggest that this conclusion holds irrespective ofthe number of inventory receipts scheduled for the planning horizon. We do observe,however, that the benefits of nesting increase in the number of receipts in the planninghorizon. In Fig. 5 we give an overview of the benefits of nesting experienced withalternative numbers of receipts at a relevant inventory availability level of 1.0c andtarget service levels of αN N = 0.75 and αN N = 0.9.

In the case of just one receipt, orders are accepted until all remaining inventoryis consumed. From this point on, all orders have to be rejected and there are no fur-ther effects of nesting. In cases with multiple receipts, however, high and low priorityorders will already be rejected if there is not sufficient inventory available within the

186


0,750,75 0,9 0,95 0,99

0,80

0,85

0,90

0,95

1,00

With Nesting Without Nesting

α

NNα

Fig. 4 Effects of nesting dependent on αN N

0,001 3 5 15

0,05

0,10

0,15

0,20

α

Number of Receipts (n)

∆

Fig. 5 Nesting effects dependent on the number of receipts in the planning horizon (�α represents thedifference in the realized service level with and without nesting)

specific cycle in which they are due. Also, any remaining inventory gets carried overto the next cycle. The (Poisson) “race” for the available inventory is re-initiated inevery cycle, providing multiple opportunities to realize positive effects of nesting.

187


0,0%0,2c 0,4c 0,6c 0,8c 0,9c 1,1c 1,21c

0,2%

0,4%

0,6%

0,8%

1,0%

1,2%

1,4%

1,6%

1,8%

2,0%0,75 0,90 0,95 0,99

fr


Fig. 6 Impact of inventory reservations on overall fill rate (� f r represents the difference in expectedoverall fill rate with and without inventory reservation)

4.2 Effect of inventory reservation on the overall system performance

In the previous section we analyzed the impact of inventory reservations on the highpriority service level and characterized the effect of nesting. When setting a ser-vice level target for high priority orders, ex-ante the manufacturer needs to considerthe expected impact on the fulfillment of low priority orders. In general, reservinginventory for high priority orders will decrease the (expected) number of low priorityorders fulfilled. It is reasonable to assume that the manufacturer will always acceptthis decrease as long as it is offset by an equal increase in the expected number of highpriority orders fulfilled, i.e. for every rejected low priority order one additional highpriority is fulfilled. However, as the high priority service level is increased throughhigher reservation quantities, the probability that some portion of the reserved quantityremains unused also increases. This unused portion of the reserved inventory couldhave been utilized to fulfill low priority orders that are currently getting rejected. Themanufacturer therefore has to consider the trade-off between ensuring a certain servicelevel for high priority orders and the expected negative impact on overall order ful-fillment. To measure this negative effect, we utilize the decrease in the overall systemfill rate attributed to reservation. It shows how many additional low priority ordershave to be sacrificed on average to ensure a certain α-service level. Any decrease inthe overall fill rate can be attributed to unused reservation quantities which are notavailable to fulfill low priority orders. Note that the two performance measures (highpriority service level and decrease in overall system fill rate) allow the manufacturerto evaluate the trade-off between the decrease of the “risk” of missing a high priorityorder and the associated decrease in the overall system performance. In Fig. 6 we plotthe difference in the overall system fill rate with and without inventory reservation.

We observe that at very low levels of inventory availability (e.g. 0.2c), reservationcauses only a small decrease in the overall fill rate. At these low levels inventoryis not sufficient to reserve the required quantity. As a consequence, only in very few

188


instances the realized demand is lower than the quantity reserved. Severely constrainedinventory prevents the system from achieving the target high priority service level (seeFig. 2) but also decreases the negative impact on the overall fill rate. We observe asudden increase in � f r as sufficient inventory becomes available to achieve the targetservice level for high priority orders (0.4c). At this level of inventory availability themanufacturer has to sacrifice approximately 1.8% of the overall fill rate to ensure aservice level of 0.99 for high priority orders. Note that at these low levels of inven-tory availability the manufacturer does not benefit from nesting (see Sect. 4.1). Asmore inventory becomes available, the negative impact on the overall fill rate slightlydecreases. This can be explained by very few instances in which benefits of nestingcan be utilized to slightly increase the overall fill rate. Up to an inventory availabilityof up to 0.9c these effects are, however, only marginal. At the nesting threshold of 1.0ctwo effects lead to a significant decrease in � f r . In cases of elevated high prioritydemand, nesting has a positive impact on the overall fill rate. As seen in the previ-ous section (see Figs. 3 and 4), notable nesting effects are only realized beyond thethreshold level of 1.0c. Furthermore, when high priority demand is low and some ofthe reserved inventory remains unused, the unreserved portion may still be sufficientto fulfill all low priority demand. For the same reason the negative effect of inventoryreservation approaches to zero at high levels of inventory availability (e.g. 1.1c, 1.2c).The results of our analysis support the manufacturer in setting an appropriate servicelevel target for high priority orders. Knowing the “cost” of reservation (measured interms of the loss in system fill rate) allows him to evaluate the (non-linear) trade-offbetween different target service levels for high priority fulfillment and overall systemperformance.

Further experiments with a varying number of inventory receipts indicate that thenegative effect of inventory reservation on the overall fill rate increase in the numberof receipts. In Fig. 7 we plot the loss in fill rate caused by inventory reservation (� f r )for n = 1,3,5,15 receipts across different levels of inventory availability and a targetservice level of 0.95.

Given that the benefits of nesting increase in the number of inventory receipts (seeFig. 5), it seems counterintuitive that the loss in overall system fill rate decreases in thenumber of receipts. In the case of one receipt, all inventory is available at the begin-ning of the planning horizon and can be used to fulfill high priority orders and all lowpriority orders up to the protection level. Pooling effects are realized across time up tothe point at which inventory is completely exhausted. In the case of n > 1 receipts theavailable inventory in the planning horizon is allocated to the n + 1 inventory cycles.In every cycle orders are rejected if the cycle-specific inventory is not sufficient tofulfill all orders due in this cycle. Through this “demarcation” of inventory quanti-ties less demand pooling is realized across time. Although the nesting effects havea positive impact on high priority fulfillment, a larger number of low priority ordersis rejected. This effect becomes most evident when considering the case of n = 15receipts. In every single period high priority orders can consume the reserved portionand any remaining unreserved inventory. Low priority orders can only consume theunreserved portion of inventory available in the respective period. In any period, lowpriority demand exceeding the unreserved portion is lost and cannot be compensatedby lower demand realizations in subsequent periods.

189


n=3n=1 n=5 n=15


fr

0,0%

0,2%

0,4%

0,6%

0,8%

1,0%

1,2%

1,4%

1,6%

1,8%

2,0%

0,2c 0,4c 0,6c 0,8c 0,9c 1,1c 1,2c1c

Fig. 7 Impact of the number of receipts on overall fill rate (� f r represents the difference in expectedoverall fill rate with and without inventory reservation)

4.3 Effect of the demand lead time distribution

In our previous analyses we assumed that all orders have to be fulfilled immediatelyin the period of their arrival. Our reservation approach as described in Sect. 3, how-ever, explicitly accounts for uncertain due date requirements of the customers. In thissection we explore the impact of different demand lead time distributions on overallfill rates of the system. We conducted experiments with three alternative demand leadtime distributions: (1) a uniform distribution gu ∼ U (0, 4); (2) a left-skewed distri-bution gl ∼ (0.4, 0.3, 0.1, 0.1, 0.1); and (3) a right-skewed distribution gr ∼ (0.1,0.1, 0.1, 0.3, 0.4). We choose to work with n = 5 receipts so that the length of anyinventory cycle (T/n) < τe, and a target service level of αN N = 0.95. In general, weobserve that demand lead time distribution does not have a very strong impact on theperformance of our reservation approach. This result is rather intuitive since differentdemand lead time distributions only lead to a different allocation of overall demandto the periods of the planning horizon. In the Fig. 8 we exemplarily plot the values of� f r for gu, gl , gr and g0 = g(τe = 0) = 1.

Overall we see an insignificant impact of the due date distributions on the overallfill rate. Only for gr we observe a higher effect on � f r . In this case mean demandis lower in the initial periods of the planning horizon, leading to a lower reservationquantity for the first cycle. As a consequence, the expected remaining quantity carriedover to the subsequent cycle is lower, requiring higher reservation quantities in laterperiods. Only negligible effects on the realized high priority service levels and nestingare caused by different demand lead time distributions.

Although the demand lead time distribution does not have a significant impact onorder fulfillment performance, it plays an important role in the correct calculation

190


grgu glg0

0,0%

0,2%

0,4%

0,6%

0,8%

1,0%

1,2%

1,4%

1,6%

1,8%

2,0%

fr

0,2c 0,4c 0,6c 0,8c 0,9c 1,1c1,0c 1,2c


Fig. 8 Impact of the demand lead time distribution on the overall fill rate (� f r represents the differencein expected overall fill rate with and without inventory reservation)

of the reservation quantities. If the demand lead time distribution is not adequatelyaccounted for (as described in Sect. 3.3), the mean demand quantities due in the indi-vidual inventory cycles are not calculated correctly and significant deviations in thereservation quantities may occur across the inventory cycles. Although this will notcause a negative effect on high priority service level, it may harm the overall fill rateof the system. To exemplify this, we conducted an experiment in which reservationquantities were based on a (wrong) due date distributions g(τe = 0); the true distri-bution which determines the due dates of orders arriving in the planning horizon was,however, assumed to be gr . In Fig. 9 we display the impact on � f r compared to theresults obtained from the previous experiment in which both reservations and actualdue dates were based on gr . The significant additional loss in the overall fill rate iscaused by two effects: (1) Mean demand in the first cycle is severely overestimated,resulting in a very high reservation quantity. A significant amount of reserved inven-tory is not utilized by high priority orders and a large number of low priority orders arerejected although inventory would have been available. (2) The overall mean demandin the planning horizon is overestimated, leading to higher reservation quantities thanrequired for achieving the target service level of 0.95.

4.4 Comparison with a myopic reservation approach

With the final set of analyses we give evidence on how our reservation approach per-forms in comparison to a myopic reservation policy. As explained in Sect. 3, we baseour reservation approach on the concept of a cycle-based service level. A distinctfeature of our approach is, however, that we consider the dynamic carry-over effectsacross a multiple number of inventory cycles, dependent on the number of receiptsscheduled. Reserving inventory in a myopic fashion would imply that these carry over

191


0%

1%

2%

3%

4%

5%

6%

7%

8%and gr g0

and gr gr

0,2c 0,4c 0,6c 0,8c 0,9c 1,1c1,0c 1,2c


fr

Fig. 9 Impact of basing inventory reservations on a “wrong” demand lead time distribution (n = 5 and

αN N = 0.95)

effects are ignored and that reservation quantities are only based on the demand dis-tribution for an individual inventory cycle. Formally, this would imply that for anycycle i , the reservation quantity ri would be set to FQ H

i(α), where Q H

i represents thehigh priority demand due in cycle i . By accounting for the carry-over effect (throughrandom quantity

(ri − Q H

i

)+available in the subsequent inventory cycle i + 1), the

overall reservation quantities are lower as compared to a myopic reservation policy,while the target service levels are still achieved. This clearly leads to a positive impacton the overall fill rate of the system. Lower reservation quantities will inevitably resultin a lower � f r for a given target service level αN N . In Fig. 10 we plot the differ-ences between the � f r incurred with our inventory reservation approach and the � f rincurred with myopic inventory reservations across multiple receipts and αN N = 0.95.

In the case of one receipt, our approach and the myopic approach will lead to thesame results. From Fig. 10 we observe, however, that significant additional negativeconsequences are incurred in the case of multiple receipts under the myopic approach.Also, these negative consequences increase as the number of receipts increases. Thiseffect can be attributed to demand pooling across time. In the case of n = 3 receipts,for example, ri = FQ H

i(α) for i = 1,2,3. The myopic approach does not account for

the fact that(r1 − Q H

1

)+and

(r2 − Q H

2

)+will be available for subsequent cycles.

As the number of receipts increases to n = 5 the reservation quantities increase inrelative terms. Shorter inventory cycles lead to lower mean demand quantities percycle and to higher reservation quantities across the whole planning horizon. Theconsequences of not accounting for carry-over of the unused quantities

(ri − Q H

i

)+

increase as the number of receipts increases. These results show that a “traditional”cycle-based approach leads to an unnecessary negative impact of inventory reservation

192


0%

1%

2%

3%

4%

5%

6%

7%n=3 n=5 n=15

Dif

fere

nces

in

fr

Inventory Availability0,2c 0,4c 0,6c 0,8c 0,9c 1,1c1,0c 1,2c

Fig. 10 Differences in overall fill rate, proposed vs. myopic reservation approach

on the system performance. With the approach presented in this paper we are able toachieve target high-priority service levels with significantly lower negative impact onthe system’s performance.

5 Conclusions

In this paper we extend the scope of traditional order fulfillment systems by integratinginventory reservation for high priority customers and order promising. We develop anintegrated model to assist manufacturers in reserving inventory for future high prioritydemand and determining order acceptance/rejection for incoming orders. Realizingthe practical difficulties associated with assigning each order a penalty cost to capturethe long term effects of not being able to fulfill an order according to the due daterequirements, we utilize a target service level as the basis for determining inventoryreservations. Our study is among the very few studies that utilize target service level asan analogue for capturing the long term effects of not being able to quote the desireddue dates in order promising and inventory rationing. Most previous studies haveassumed that the short term and long term costs associated with due date quoting areexplicitly known.

We develop a model that captures uncertainty in the time of arrival, the desired duedate and the type of customer orders. A first contribution of this paper is the develop-ment of an algorithm to calculate the amount of inventory to reserve in a MTS systemwith multiple inventory receipts within the planning horizon. In addition, expressionsto characterize the probability of high priority fulfillment with nested inventory res-ervations are presented. Unfortunately, these expressions do not yield closed-formsolutions and we performed simulation-based numerical analysis to trace the effectsof nesting.

Our numerical study also shows other interesting findings with strong managerialrelevance. We see that the effect of nested inventory reservations on the high priorityservice level increases as the number of inventory receipts increases. However, the

193


loss in overall fill rate due to inventory reservation also increases in the number ofreceipts. We also demonstrate that not knowing the true due date distribution or usinga myopic method to determine reservations can have strong detrimental effects on theoverall system performance.

In summary, apart from presenting an approach for integrated inventory reserva-tion and order promising, our analysis generates various insights for managers on theimpact of inventory reservation under different system settings.

We have assumed that the time of arrival of scheduled inventory receipts is deter-ministic to avoid an over-parameterized model with limited tractability. Future exten-sions could consider modeling the uncertainty in inventory receipts and its impacton inventory reservation and order fulfillment performance. Also, the demand withinthe planning horizon is assumed to be stationary and identically distributed in eachperiod. This assumption, although not unreasonable in certain settings, may not beapplicable for others. Finally, we assumed that orders are rejected if the customer duedate requirements cannot be met. In many practical settings, customers may acceptsome reasonable delay beyond their required due date. Analyzing the performance ofan inventory reservation approach with different customer behavior may be of theo-retical and practical interest.

Acknowledgment The authors express their gratitude to the Spanish Ministry of Education and Science(Project Reference: PSE-370500-2006-1) for financial support.

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Meyr H (2007) Customer segmentation, allocation planning and order promising in make-to-stock produc-tion. OR Spect (forthcoming)

Nahmias S, Demmy WS (1981) Operating characteristics of an inventory system with rationing. ManageSci 27:1236–1245

Pibernik R (2006) Managing stock-outs effectively with order fulfilment systems. J Manufact TechnolManage 17(6):721–736

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Topkis DM (1968) Optimal ordering and rationing policies in a non-stationary dynamic inventory modelwith n demand classes. Manage Sci 15:160–176

Vericourt FD, Karaesmen F, Dallery Y (2000) Dynamic scheduling in a make-to-stock system: a partialcharacterization of optimal policies. Oper Res 48:811–819

Zipkin P (2000) Foundations of inventory management. McGraw-Hill/Irwin, Boston

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Part IIIInventory Management

Setting safety stocks in multi-stage inventory systemsunder rolling horizon mathematical programmingmodels

Youssef Boulaksil · Jan C. Fransoo ·Ernico N. G. van Halm

Abstract This paper considers the problem of determining safety stocks in multi-itemmulti-stage inventory systems that face demand uncertainties. Safety stocks are nec-essary to make the supply chain, which is driven by forecasts of customer orders,responsive to (demand) uncertainties and to achieve predefined target service levels.Although there exists a large body of literature on determining safety stock levels,this literature does not provide an effective methodology that can address complexmulti-constrained supply chains. In this paper, the problem of determining safetystocks is addressed by a simulation based approach, where the simulation studies arebased on solving the supply chain planning problem (formulated as a mathematicalprogramming model) in a rolling horizon setting. To demonstrate the utility of theproposed approach, an application of the approach at Organon, a worldwide operatingbiopharmaceutical company, will be discussed.

Keywords Safety stocks · Advanced planning and scheduling · Simulation · Supplychain planning · Organon

1 Introduction

Supply chains are exposed to different types of uncertainties that stem from randomyields, processing times or forecast errors. These uncertainties can be covered to a

Y. Boulaksil (B) · J. C. FransooDepartment of Technology Management, Technische Universiteit Eindhoven,Eindhoven, The Netherlandse-mail: [email protected]

J. C. Fransooe-mail: [email protected]

E. N. G. van HalmSupply Chain Management Department, Organon N.V., Oss, The Netherlandse-mail: [email protected]



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large extent by mechanisms like safety time, safety stocks or combinations of these(Whybark and Williams 1976; Wijngaard and Wortmann 1985). This paper focuses onthe determination of safety stocks in multi-item multi-stage inventory systems that facedemand uncertainties. We assume that the inventory system is planned and controlledby a central decision authority that plans the supply chain based on deterministic math-ematical programming models. However, demand uncertainty is an important factorto be considered in supply chain planning. Planning systems based on mathematicalprogramming models are widely implemented in so-called Advanced Planning andScheduling systems (APS) (Stadtler and Kilger 2005).

When a particular supply chain is facing demand uncertainties, stock outs can occurat all stages in the supply chain. A stock out may cause lost sales, emergency ship-ments, or loss of goodwill. Therefore, safety stocks should be kept to increase theservice levels. Traditionally, safety stocks are determined in advance based on modelsfrom inventory theory (Silver et al. 1998). However, it is not obvious how to determinesafety stock levels that cover demand uncertainties in complex supply chains that faceseveral constraints such as batch sizes, capacity constraints, non-stationary demandprocess or forecast errors.

The approach proposed in this paper enables the determination of safety stocksin multi-item multi-stage inventory systems that face demand uncertainties. Thisapproach considers all kinds of constraints that are also considered in supply chainplanning practice such as batch sizes and capacity and materials constraints. Theapproach is based on a simulation of the supply chain planning model in a rollinghorizon setting. Based on target service levels, safety stocks are determined after per-forming simulations, assuming that the demand process and replenishment decisionsare independent of the safety stock levels.

The core of the approach is solving the supply chain planning problem very fre-quently, where the safety stocks are excluded from the supply chain planning modelor by setting them equal to zero. Since we assume that all unsatisfied demand is back-ordered at all stages in the supply chain, backorder quantities are recorded after eachsolving round. The safety stock level is an increasing function of the target servicelevel, which we measure by the fill rate, i.e. the long-run fraction of demand satisfiedroutinely from the shelf (Silver et al. 1998). Based on the stored backorder quantitiesand the target service levels, safety stock levels can be determined.

The proposed approach is suitable for companies that have implemented an APS.APS systems are planning systems that are based on cost minimization models thatensure that, given the resource and material availability constraints of the produc-tion system and given certain service level constraints, the best possible quantity of acertain item is released at the lowest value of the objective function. These planningsystems are based on mathematical programming models that are solved in a rollinghorizon setting (Spitter et al. 2005).

The proposed approach has been applied successfully at Organon, a worldwideoperating biopharmaceutical company with production sites, warehouses, and distri-bution centers spread all over the world.

The remainder of this paper is organized as follows. Section 2 discusses a literaturereview on this topic. Next, Sect. 3 discusses the problem definition and thereafter,Sect. 4 discusses the proposed approach. The approach has been applied in a real-life

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situation, which is discussed in Sect. 5. Finally, Sect. 6 draws some conclusions aboutthe approach.

2 Literature review

There is an extensive amount of literature available on inventory control models inmulti-stage or multi-echelon inventory systems incorporating uncertainties. We referthe reader to survey articles by Van Houtum et al. (1996) and Diks et al. (1996).Our research is within the field of Supply Chain Operations Planning (De Kok andFransoo 2003). The objective of Supply Chain Operations Planning is to coordinatethe release of materials and resources in a supply chain network such that customerservice constraints are met at minimal costs (De Kok and Fransoo 2003). Two differentapproaches exist for modelling the Supply Chain Operations Planning problem.

One approach is based on multi-echelon stochastic inventory theory. In this ap-proach, demand that is faced by the supply chain is modelled as a stochastic variable.The key decisions of this approach are the inventory positioning at the various stock-points in the supply chain, the allocation of quantities at inventory points where theproduct flow diverges, and the determination of safety stock levels at the several stock-points. Therefore, the determination of safety stocks is defined as part of the problem.Lead times are (deterministic) input variables to the model and capacity is assumed tobe controlled through a combination of order acceptance in the demand managementfunction and a workload control function in the production department. The logic isbased on a line of research that has been initiated by Clark and Scarf (1960).

The alternative approach is based on mathematical programming principles. In thisapproach, demand is inserted into the model as forecasts for every period in the plan-ning horizon. Safety stocks are input parameters to the model and the key decisionsare the allocation of inventory quantities at the stockpoints in the supply chain. Leadtimes are either modelled as deterministic input variables (e.g., Spitter et al. 2005)or are observed as output variables of the model (e.g., Stadtler 2003). Capacity con-straints are modelled explicitly as aggregate constraints. The principles are based onresearch stemming from advanced MRP modelling (Billington et al. 1983) or frommulti-period lot sizing problems (Tempelmeier and Derstroff 1996; Stadtler 2003).The principles have been implemented in commercial software, mostly using CPLEXsolving logic (See also Stadtler and Kilger 2005).

The two approaches differ also from a safety stock perspective. In this first approach,safety stocks are defined as part of the problem, whereas in the second approachsafety stocks are input parameters to the planning model, which have to be determinedexternally.

This paper focuses on the determination of safety stocks for the latter type of plan-ning approaches. A lot of papers appeared in the last decades on determining safetystocks in multi-stage or multi-echelon inventory models for covering demand uncer-tainties. We mention a set of papers that are related to our work.

Inderfurth and Minner (1998) propose a dynamic programming approach to treatthe problem of determining safety stocks in multi-stage inventory systems, assum-ing normally distributed demand and periodic review base stock control policies.

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Furthermore, they also assume that no internal delays occur and that each stockpointis satisfying a service level constraint. More approaches that are based on dynamicprogramming algorithms can be found in Inderfurth (1991) and Minner (1997).

Graves and Willems (2000) discuss the so-called guaranteed-service model for set-ting safety stocks in a multi-stage setting to cover demand uncertainties. They developa model for positioning safety stocks in a supply chain where each stage is controlledby a base-stock policy, assuming an upper bound for the (customer) demand level.Therefore, the safety stocks set by their approach cover demand realizations belowthe upper bounds. This assumption is necessary to model guaranteed service timesbetween each stage in the supply chain and its customers.

There are also papers on determining safety stocks in multi-stage inventory systemswhere the approach is based on simulation studies. Optimization methodologies basedon simulation of inventory systems are discussed in Kleijnen and Wan (2006).

Eilon and Elmaleh (1968) perform simulation studies to compare the performance offive alternative inventory control policies given wide fluctuating and seasonal demandpatterns. The results of the simulations are several non-linear curves showing the rela-tion between the fill rate and mean stock level. Three of these five control policiesinclude safety stocks, but the authors do not discuss how they determined the safetystock parameters.

Wemmerlöv and Whybark (1984) also perform simulation experiments to evaluateseveral single-stage lot sizing procedures under demand uncertainty. Cost compari-sons of the procedures are made with a service level of at least 99.999%. The safetystocks needed to achieve these service levels are determined by a search routine, i.e.repeating the simulations until the target service levels are reached.

De Bodt and Van Wassenhove (1983) present a case study at a company, which usesMRP in a dynamic environment, i.e. the company faces substantial demand uncertain-ties. The safety stock setting is analysed by a simulation study. Several strategies weredefined (combinations of safety stock and safety time) and analysed which resultedin graphs relating average inventory level to service levels. They provide managerialinsight by showing that considerable savings can be made at this company, but do notdiscuss how the safety stocks should be determined.

In the studies of Callarman and Hamrin (1984) the performance of three lot sizingrules in MRP systems is compared, given an uncertain demand process. The cost com-parisons have been made by introducing safety stocks at each run to keep the servicelevels at 95 and 98%. The required safety stocks are determined by using the so-calledService Level Decision Rule (SLDR), which has been developed by Callarman andMabert (1978). The SLDR is based on linear regression analysis on simulated valuesof the following set of factors: forecast errors, coefficient of variation of demand, andthe expected time between orders. In order to achieve the target service level, theSLDR is used with a search routine.

Our work is closely related to Kohler-Gudum and De Kok (2001) who propose aso-called Safety Stock Adjustment Procedure (SSAP) to obtain target service levelsin simulation models. The technique is based on the assumption that a Time PhasedOrder Point (TPOP) policy is applied. Their simulation study aims to determine thediscrete probability density function of the net stock process. Based on this probabilitydistribution, the safety stock is adjusted to ensure the specified target service level.

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Our approach differs from Kohler-Gudum and De Kok (2001) on two aspects.First, our approach determines the empirical distribution of the backorder quantitiesfor setting safety stocks instead of determining the probability density function of theinventory process. Based on a set of assumptions (independence of the demand andreplenishment process of safety stock levels), the necessary amount of safety stock isdetermined by adding the adjustment quantity to the initial safety stock that can be anarbitrary value. In our approach, the initial safety stock is set equal to zero and after-wards, the safety stock level is determined based on backordered quantities. Second,the model that is used in our approach is a planning model that is solved in rollinghorizon setting, and therefore, the planning process is imitated as much as possible.Kohler-Gudum and De Kok (2001) do not discuss the simulation model extensively.

Although there exists a large body of literature on determining safety stock levels,to our knowledge, this literature does not provide an effective methodology that canaddress supply chains that face several constraints like capacity constraints, productionin batch sizes, and non-stationary forecast process. Most approaches make restrictiveassumptions about the demand process (Inderfurth and Minner 1998; Graves andWillems 2000) or do not explicitly discuss how they set the safety stock levels (Eilonand Elmaleh 1968; De Bodt and Van Wassenhove 1983). Our approach is closelyrelated to Kohler-Gudum and De Kok (2001), but we extend the approach by usingan empirical supply chain planning model in the simulation study and that makes theresults of the approach more reflecting the (planning) practice.

3 Problem definition

We consider a supply chain that consists of an arbitrary number of stages and stock-points in which a product passes through multiple production sites before it is finallydelivered to outside customers. This supply chain is planned and controlled by a cen-tral decision authority that has access to all relevant status information (like inventorylevels and work-in-process quantities) at all production sites and makes release deci-sions for the entire supply chain. The release decisions result from a deterministicmathematical programming model that is solved in a rolling horizon setting (Stadtlerand Kilger 2005), which has been implemented in an Advanced Planning and Sched-uling system. For these kinds of planning models, safety stocks are input parametersthat have to be determined externally.

Formulation of the planning problem by a mathematical programming modelassumes a deterministic view of supply chain planning by considering all model param-eters, as demand, lead times, production rates to be known with complete certainty.This assumption of complete and deterministic information is desirable from a modelcomplexity point of view, but given the dynamic and uncertain nature of most supplychains, this assumption is violating reality. Demand uncertainty is an important factorto be considered in supply chain planning, and therefore, safety stocks are kept tocover part of the demand uncertainties.

The core function of supply chain planning models is to coordinate material andresource release decisions in the supply chain such that predefined customer ser-vice levels are achieved with minimal costs. Safety stocks are kept to deal with

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demand uncertainties and consequently to increase service levels. The service levelis an increasing function of the safety stock level. Therefore, more safety stocks areneeded to increase the service level, which results in increased inventory holding costs.

From the other side, demand uncertainties can cause stock outs that result in lostsales, emergency shipments, or loss of goodwill. Since we assume that all excessdemand that is not directly satisfied from inventory is backordered, costs that arerelated with a backorder are backorder costs, which are harder to quantify than inven-tory holding costs. The problem of setting safety stocks is mainly a trade-off betweeninventory holding costs and backorder costs. Section 4 discusses the modelling ofthese costs and discusses also the considered supply chain planning model in detail.

4 The approach

We consider a supply chain that is planned and controlled by a central decision author-ity, which may be supported by an Advanced Planning and Scheduling system. Weassume that the supply chain planning model is based on a mathematical programmingmodel that is solved in a rolling horizon setting, where the forecasts may be updatedwhen the planning horizon is shifted. We also assume that the demand process andreplenishment decisions are independent of the safety stock level. Furthermore, weassume that all excess demand at all stages in the supply chain is backordered. Wedo not make any assumption about the demand and forecast process, which makesthis approach less restrictive to a certain probability density function of the demandprocess.

Based on the discussed assumptions, a simulation experiment is performed in thefollowing way. The planning horizon is divided into a fixed number of time buckets,which are filled by forecasts of the demand generated by a demand generator, whichgenerates a series of forecasts based on historical demand and forecasts data. Then,the planning model with demand forecasts is solved given all kinds of materials andresources constraints. The planning model may be based on linear programming mod-els or mixed-integer programming models if some decisions require integer variables.Such discrete decisions can, among others, regard lotsizing in production or transpor-tation. At the end of the first time bucket (planning cycle), the state of the system (e.g.the inventory levels and forecasts) is updated and the planning cycle is repeated withthe horizon shifted by one period.

Figure 1a illustrates the inventory development of a certain product and the result-ing backorder process that is output of 100 simulation runs. After the simulation runs,the horizontal axis is shifted (see Fig. 1b) such that the number of backorders is lim-ited, i.e. the customer service level is increased to a certain predefined level. Figure 1bshows that increasing the safety stock level decreases the number of backorders, andtherefore, increases the service level.

Thus, by solving the supply chain planning problem very frequently where eachtime the forecasts are updated, long-run backorder quantities indicate the amount ofsafety stocks that was needed to prevent the backorders partially, i.e. to achieve acertain customer service level. Note that the customer service level is externally deter-mined for all products at all stages in the supply chain. The customer service level has

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A Inventory Inventory

Safety stockBackorders

00

0‘

levelInventory

level

time

B

Fig. 1 The inventory development of a certain product; a shows the results of simulation runs and b showshow the horizontal axis is shifted to limit the number of backorders, i.e. to achieve a certain customer servicelevel

Demand generator

SC Planning model

Backorder quantities

horizon shift

f(d), µd,i, d,i

Optimal solution

(t), t 1 = ,...,Tˆid

Safety stock levels

Service level setting

Explanation of used symbols:

f(d) probability density function of the demand processµd,i Expected (exogenous) demand of end-item i

d,i Standard deviation of forecast errors of end-iem iT Planning horizon

Forecast of demand of end-item i in period t (t=1,...,T)( )ˆid t

Fig. 2 Several steps of the approach

been defined as the long-run fraction of demand satisfied directly from shelf (fill ratemeasure). Having discussed the theoretical idea behind the approach, the steps of theapproach (see Fig. 2) will be discussed in the following sections in more detail.

4.1 Demand generator

The first step of the approach is the generation of a series of forecasts which areinput to the supply chain planning model. We do not make any assumption about thedistribution of the demand process. Historical data about demand and forecasts maybe (statistically) fitted into the best fitting probability distribution function. Havingchosen the most suitable probability density function for the demand distribution, the

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first two moments of the distribution can be derived to determine the parameters ofthe demand distribution. Figure 2 shows that these parameters (µd,i , σd,i ) are inputfor the demand generator where µd,i is the expected (exogenous) demand of end-itemi and σd,i is the standard deviation of forecast errors of end-item i .

Suppose that we have historical sales data of n time periods, then µd,i can becalculated by

µd,i = 1

n

−1∑

s=−n

di (t + s) (1)

where di (t) is the demand for item i in period t . Forecast errors can be determinedby several measures (Silver et al. 1998). The Mean Absolute Deviation (MAD) isrecommended for its computational simplicity. The MAD for item i as function of theforecast horizon h can be calculated by

MADi (h) = 1

T

−1∑

s=−T

(di (t + s)− di (t + s − h, t + s)

)(2)

where T is the length of the planning horizon, di (t) the demand for item i in periodt , and di (t − h, t) the forecast made in period t − h for the demand in period t . It isreasonable to assume that the MAD is an increasing function of the forecast horizonh (Heath and Jackson 1994). The conversion of MADi (h) to σi (h) is extensively dis-cussed in Silver et al. (1998). Having determined the parameters of the demand distri-bution, the random generator can generate a series of forecasts of the demand di (t) fort = 1, . . . , T . The generated forecasts are input for the supply chain planning model.

Several extensions are possible. For example, the demand process may not be sta-tionary which makes the µd,i a function of time (demand process follows a trend orhas a seasonal effect). Furthermore, the standard deviation of the forecast errors maybe not a function of the forecast horizon h. Several kinds of adaptations are possiblein order to imitate the demand and forecast process as much as possible.

4.2 Supply chain planning model

One approach to supply chain planning models is based on deterministic mathemat-ical programming principles (De Kok and Fransoo 2003). The advantage of usingthe supply chain planning model (implemented in an APS system) for the simulationstudy is that it already contains the network structure(s), the item list, bill-of-mate-rials structure, batch sizes, and the routings. Using the supply chain planning modelfor this approach is highly recommended, as these models are reflecting the planningpractice. Furthermore, for those companies that have implemented an APS system,little modelling effort is required for this approach.

The mathematical programming model that is used to determine the safety stocklevels is a stand-alone model, but derived from the supply chain planning model. Thesupply chain planning model may have to be adapted, as safety stocks have to be set

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Fig. 3 A three-stage supply chain considered in the supply chain planning model

equal to zero in the supply chain planning models or replenishment decisions shouldnot consider safety stock levels. Then, the planning problem should be solved withoutconsidering safety stocks, i.e. backorders are planned when demand exceeds availableinventories. Depending on the supply chain planning model, the planning problem hasto be solved such that all relevant cost factors have to be considered except costs asso-ciated with consumption of safety stocks. The solution of the supply chain planningmodel contains order releases for the production system and planned inventory levels.The order releases within the length of the lead time of a certain item at a certain stage(frozen horizon) are stored, as they are not allowed to be changed in the next solvinground. Below, we discuss the considered supply chain planning model in detail, whichis used to determine the safety stock levels. Figure 3 shows a rough outline of thethree-stage supply chain that is considered in the supply chain planning model.

4.2.1 Objective function

Equation (3) is the objective functions of the supply chain planning model. The objec-tive function minimizes the total costs (TC), which consist of several cost factors thatare assigned to several stages in the supply chain. We consider a (pharmaceutical) sup-ply chain with three stages. Stage 3 is the most upstream stage where the raw materials(active ingredients) are stored. The active ingredients are processed to tablets, whichare stored at the second stage. Thereafter, the tablets are packaged and stored at themost downstream stage (stage 1). N j is the total number of items at stage j withj ∈ {1, 2, 3}, n j is a certain item that belongs to stage j , t is a certain (discrete) timeperiod, and T is the planning horizon.

Min TC =∑

n1

T∑

t=1

c1 · UDn1(t)+∑

n1

T∑

t=1

c2 · OPn1 · BMn1(t)

+∑

n1

T∑

t=1

c3 · EIn1(t)+∑

n2

T∑

t=1

c4 · UDn2(t)

+∑

n2

T∑

t=1

c5 · UDDn2(t)+∑

n2

T∑

t=1

c6 · BCn2(t)

+∑

n2

T∑

t=1

c7 · EIn2(t)+∑

n3

T∑

t=1

c8 · UDn3(t)

+∑

n3

T∑

t=1

c9 · UDDn3(t)+∑

n3

T∑

t=1

c10 · EIn3(t) (3)

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The first three terms of the objective function are related to the first stage in thesupply chain. For this stage, we consider three cost factors that have to be minimized.The first terms are costs associated with unsatisfied demand (backorders) c1 ·UDn1(t)where UDn1(t) is the backorder quantity for item n1 in period t . The second termdeals with costs for replenishing a quantity that deviates from the (minimum) replen-ishment quantity c2 · OPn1 · BMn1(t). OPn1 is the period order quantity for item n1and BMn1(t) the deviation from the minimum replenishment quantity for item n1 inperiod t . Campaign sizes are determined based on a trade-off between ordering costsand inventory holding costs, whereas batch sizes are quantities that are determined bylegislative authorities. Therefore, producing in fixed batch sizes is required, whereasdeviating from the campaign size is undesired. The third term is the total inventoryholding cost at this stage c3 ·EIn1(t) where EIn1(t) is the inventory level of item n1 atthe end of period t .

For the second stage of the supply chain, four cost factors are considered. The firstterms sum backorders that result from exogenous demand at this stage. So, c4 ·UDn2(t)is unsatisfied demand (backorder) costs for item n2 in time period t , whereas the sec-ond term c5 · UDDn2(t) considers unsatisfied demand (backorders) that result fromendogenous (derived) demand from the first stage of the supply chain. Since the sec-ond stage in the supply chain considers the production of tablets in campaigns (a fixedmultiple of batch sizes), the third term c6 · BCn2(t) considers costs associated withdeviating from the fixed campaign size BCn2(t) for item n2 in period t . The fourthterm considers the total inventory holding costs for item n2.

The third stage in the supply chain considers three cost factors: costs associated withunsatisfied demand (backorders) of exogenous demand c8 ·UDn3(t), costs associatedwith unsatisfied demand (backorders) that result from endogenous demand (from thesecond stage of the supply chain) c9 · UDDn3(t), and total inventory holding costsc10 · E In3(t) of item n3. For confidentially reasons, we cannot show the values of thecost parameters, except that c1 > c2 > · · · > c10. The determination of these costparameters was not part of this study, as they can be taken over from the objectivefunction of the supply chain planning model.

4.2.2 Stage 1 model

The objective function (3) is minimized subject to several constraints, which are dis-cussed below per stage in the supply chain. Equations (4) are materials balance equa-tions with EIn1(t) is the inventory level of item n1 at the end of period t , T Rn1(t) thereplenishment quantity of item n1 in period t , and TDn1(t) the (exogenous) demandfor item n1 in period t . The latter parameter contains data that are input to the planningmodel. Further, EIn1(0) is the initial inventory level.

EIn1(t) = EIn1(t − 1)+TRn1(t)−TDn1(t), n1 = 1, . . . , N1, t=1, . . . , T (4)

Equation (5) determine the minimum replenishment quantity for item n1 in periodt , as the replenishments are based on periodic order quantity (OP). I Dn1(t) is the(forecast of) independent demand for item n1 in period t .

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MRn1(t) =OP∑

i=1

I Dn1(t + i), n1 = 1, . . . , N1, t = 1, . . . , T (5)

Having determined the minimum replenishment quantity, Eq. (6) determine the realreplenishment quantities SMn1(t) for item n1 in period t . BMn1(t) is then the deviationfrom minimum replenishment quantity for item n1 in period t that is considered in theobjective function.

SMn1(t) = MR n1(t)− BMn1(t), n1 = 1, . . . , N1, t = 1, . . . , T (6)

The total replenishment quantity for item n1 for the entire planning horizon (TR n1

(t)) is determined by two parts: SMn1(t) which we have just discussed and FPn1(t)which are fixed replenishment quantities of item n1 in period t determined in previ-ous solving rounds. The binary parameter α regulates that within the lead time of theplanning horizon no new decisions are taken.

TRn1(t) = α · SMn1(t)+ (1− α) · FPn1(t), n1 = 1, . . . , N1,

t = 1, . . . , T, α ={

0 i f t � L1 i f L < t � T

(7)

Equation (8) determine which part of the exogenous demand IDn1(t) for item n1 inperiod t is satisfied (SDn1(t)). The unsatisfied demand quantity UDn1(t) for item n1in period t is punished in the objective function.

SDn1(t) = IDn1(t)− UDn1(t), n1 = 1, . . . , N1, t = 1, . . . , T (8)

Equation (9) are equations for TDn1(t) which is determined by SDn1(t) that resultfrom Eq. (8) plus UDn1(t − 1) which is the unsatisfied demand in t − 1, i.e. backorderquantity for item n1 from period t .

TDn1(t) = SDn1(t)+ UDn1(t − 1), n1 = 1, . . . , N1, t = 1, . . . , T (9)

4.2.3 Stage 2 model

Several constraints apply to stage 2 which will be discussed now. Like in stage 1,Eq. (10) are the balance equations for the materials flow. The symbols have thesame meaning as in stage 1, except that indices show that the equations apply to thisparticular stage.

EIn2(t) = EIn2(t−1)+TRn2(t)−TDn2(t), n2=1, . . . , N2, t=1, . . . , T (10)

Equation (11) determine the total replenishment quantity for item n2 in period twhere α is the same binary parameter that is used in Eq. (7). PPn2(t) is the production

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quantity to be produced of item n2 in t and FPn2(t) are firmed production quantitiesthat are determined in previous solving rounds.

T Rn2(t) = α · PPn2(t)+(1−α) · F Pn2(t), n2=1, . . . , N2, t=1, . . . , T (11)

Equation (12) require that the production quantity of item n2 to be produced inperiod t must be an integer multiple of Qn2 , the batch size of item n2 multiplied byyn2 , the yield factor of the production process that produces item n2.

PPn2(t) = Qn2 · yn2 · NBn2(t), n2 = 1, ..., N2, t = 1, . . . , T, (12)

with N Bn2(t) ∈ N0.Equation (13) determine the derived (endogenous) demand at stage 2. This is the

multiplication of the (with lead time L shifted) replenishment quantities of items n1with the BOM factor.

DDn2(t) = BOMn2,n1 ·∑

n1

SMn1(t − L), n2 = 1, . . . , N2, t = 1, . . . , T (13)

Equation (14) determine the costs associated with going below the campaign sizeBCn2(t), which is punished in the objective function. CSn2 is the campaign size (acertain number of batches of n2) of item n2.

BCn2(t) = CSn2 − NBn2(t), n2 = 1, . . . , N2, t = 1, . . . , T, (14)

with BCn2(t) ∈ N0.Unsatisfied demand from t−1 (resulting from either exogenous demand UDn2(t−1)

or endogenous demand UDDn2(t − 1) determine the backorder quantity BOn2(t) ofitem n2 in period t .

BOn2(t) = UDDn2(t − 1)+ UDn2(t − 1), n2 = 1, . . . , N2, t = 1, . . . , T (15)

Equation (16) show that the satisfied part of demand for item n2 in period t SDn2(t)is equal to the exogenous demand IDn2(t) for item n2 in period t minus unsatisfieddemand quantity UDn2(t) for item n2 in period t , which is punished in the objectivefunction.


Equation (17) are the application of the same idea (as Eq. 16) to the dependent(endogenous) demand for item n2 in period t .

SDDn2(t) = DDn2(t)− UDDn2(t), n2 = 1, . . . , N2, t = 1, . . . , T (17)

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The sum of SDn2(t), SDDn2(t), and the backorders for item n2 in period t BOn2(t)are equal to TDn2(t), total demand for item n2 in period t .

TDn2(t)=SDDn2(t)+SDn2(t)+BOn2(t), n2=1, . . . , N2, t=1, . . . , T (18)

4.2.4 Stage 3 model

Constraints (19) till (26) apply to the third stage of the supply chain. Equation (19) arethe balance equations for this stage. EIn3(t) is the inventory level of item n3 at the endof period t , TRn3(t) is the replenishment quantity of item n3 in period t , and TDn3(t)is the total demand of item n3 in period t .

EIn3(t) = EIn3(t − 1)+TRn3(t)−TDn3(t), n3=1, . . . , N3, t=1, . . . , T (19)

The replenishment quantity TRn3(t) is partly determined in the previous solvingrounds (FPn3(t), firm planned replenishment orders for item n3 in period t) and newreleased orders On3(t) to be determined for item n3 in period t . The orders are sent to(external) supplier(s).

TRn3(t) = (1− α) · FPn3(t)+ α · On3(t), n3 = 1, . . . , N3, t = 1, . . . , T (20)

Furthermore, constraints (21) require that the ordered items are (a) integer multi-ple(s) of Qn3 , batch sizes for item n3.

On3(t) = NBn3(t) · Qn3, n3 = 1, . . . , N3, t = 1, . . . , T, (21)

with NBn3(t) ∈ N0.The total demand for item n3 is determined by adding the satisfied parts of the

dependent (endogenous), independent (exogenous) demand plus the backorders foritem n3 in period t .

TDn3(t) = SDDn3(t)+SDn3(t)+BOn3(t), n3=1, . . . , N3, t=1, . . . , T (22)

Equations (23) and (24) show how the satisfied parts of the dependent SDDn3(t)and independent demand SDn3(t) for item n3 in period t are determined. IDn3(t) isthe independent demand for item n3 in period t and DDn3(t) is the dependent demandfor item n3 in period t .


SDDn3(t) = DDn3(t)− UDDn3(t), n3 = 1, . . . , N3, t = 1, . . . , T (24)

The dependent demand DDn3(t) is determined by multiplying the BOM-factor withTPn2(t − L) with L is the lead time of second stage of the supply chain.

DDn3(t) = BOMn3,n2

∑

n2

TPn2(t − L), n3 = 1, . . . , N3, t = 1, . . . , T (25)

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The backorder quantity for item n3 in period t BOn3(t) is the summation of theunsatisfied dependent and independent demand for item n3 in period t − 1.

BOn3(t)=UDDn3(t − 1)+ UDn3(t − 1), n3=1, . . . , N3, t=1, . . . , T (26)

Finally, non-negativity constraints have to be considered.

EIn1(t), TDn1(t), TRn1(t), SMn1(t), BMn1(t), SDn1(t), UDn1(t), EIn2(t),

TDn2(t), TRn2(t), PPn2(t), UDn2(t), UDDn2(t), SDn2(t), SDDn2(t), EIn3(t),

TDn2(t), TRn2(t), On2(t), UDn2(t), UDDn2(t), SDn2(t), SDDn2(t) � 0 (27)

4.3 Backorders and safety stocks

After solving the mathematical programming model that we discussed in the previoussection, the planning horizon is shifted by one period after which the demand gen-erator generates a new series of forecasts for the shifted horizon. The order releaseswithin the frozen horizon determined in the previous solving round are not allowed tobe changed, as these orders are assumed to be scheduled in a more detailed planninglevel or already taken in process. The supply chain planning model is solved again,but since a frozen horizon, fixed order releases and an update of the forecasts are takeninto consideration, backorders may occur if the available inventories are no longersufficient to satisfy the updated required quantities.

The planned backorder quantities after each solving round are stored. A large num-ber of replications is necessary to draw valid conclusions on the empirical distributionof the backorders. Furthermore, the results of the first couple of runs have to be ignored,as the system has to reach a state that is independent of the initial conditions. The rela-tion between the backorder quantities and the determination of safety stock levels willbe explained in the following. Suppose that the safety stock levels were set before-hand equal to the maximum measured backorder quantities at all stages in the supplychain, a service level of 100% would have been achieved in the supply chain, given thegenerated forecasts of the demand process. Therefore, the last step of the approach isto set a target customer service level for the several items at the several stages. Basedon these target customer service levels, safety stock levels can be determined for theitems.

The service level for the most downstream stage (stage 1) in the supply chain isdetermined by Eq. (28) where βn1 is the fill rate for item n1, i.e. the long-run fractionof independent demand IDn1(t) satisfied directly from shelf (without backordering).

βn1 =∑

t

(1− UDn1(t)

IDn1(t)

), n1 = 1, . . . , N1 (28)

For stages 2 and 3 of the supply chain, Eq. (29) determine the service level, as thesestages face exogenous (independent) demand and endogenous (dependent) demandfrom the next (downstream) stage of the supply chain.

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βn j =∑

t

(1− UDn j (t)+ UDDn j (t)

IDn j (t)+ DDn j (t)

), j = {2, 3}, n j = 1, . . . , N j (29)

5 Application of the approach

The discussed approach has been implemented at Organon, a worldwide operatingbiopharmaceutical company with an annual turnover of more than 2.4 billion Euros.The company consists of more than 10 production sites and about 60 national distri-bution centres spread all over the world. Organon has more than 30 branded productsin its portfolio and markets only prescription medicines for improving both the healthand quality of human life.

Figure 4 shows a rough outline of one of Organon’s tablet supply chains with themain production processes and stockpoints. Active ingredients form input to the tab-lets production process. Some additional materials may be needed for this productionprocess. The packaging process blisters the tablets, packs the blistered tablets in car-tons and instructions for use are added. Next, the finished products are shipped tomore than 60 national warehouses (which are owned by Organon) spread all over theworld. From these national warehouses, finished products are sold and distributed tocustomers like hospitals, pharmacists, and wholesalers.

This supply chain is planned and controlled by an APS, which was implementeda couple of years ago. The APS is a planning system that controls the supply chainby calculating high-level production plans for the several stages in the supply chain.The forecasts which are input to the planning problem are provided by the forecastingsystem which calculates statistical forecasts of the expected demand on SKU levelbased on historical demand information. Having implemented the Advanced Plan-ning and Scheduling system, Organon was facing the question how to determine thesafety stock parameters (which are input to the planning models) that cover (partially)demand uncertainties such that the entire supply chain is considered and total inventoryholding costs are minimized given certain customer service levels.

In the following sections, we discuss the results of a project performed withinOrganon to determine the safety stock levels using the discussed approach. For con-fidentially reasons, the product names for which this approach has been implementedwill not be mentioned. Further, the numbers do not reflect the real numbers, as theyare divided by an arbitrary factor. With respect to the simulation experiment, the runlength was set equal to 100 periods and the number of replications was five, fol-lowing the approach proposed by Law and Kelton (2000). The average CPU time isabout 2 mins. The supply chain planning model has been implemented in a standardAdvanced Planning System that uses CPLEX as solver.

Fig. 4 A rough outline of the supply chain of Organon

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5.1 The supply chain of product A

Figure 5 shows the supply chain of product A. The most upstream stage in this supplychain is the active ingredient (AI 1). The tablet production is performed at two produc-tion sites of Organon, and therefore, active ingredients 1 (AI 1) are shipped to anotherproduction site (AI 2). After tablets production 2 (TB 2), the tablets are packaged andshipped to warehouses 1 till 11, which supply Organon’s end customers. Productionsite 1 is supplying warehouses 12 till 26. Warehouses 27 till 34 are supplied by localsubcontractors who get the active ingredients from Organon. Therefore, a direct linkhas been made between the stockpoints AI 1 and warehouses 27 till 34.

We applied the proposed approach to this supply chain to determine the safetystocks levels of each item at each stockpoint. We note that safety stocks cannot bepooled, since the products in each warehouse are different due to the fact that they arecountry-specific. The lead times, batch sizes, bill-of materials structure and all othercharacteristics of this supply chain have been taken over from the supply chain plan-ning model. The mathematical formulation of the mixed-integer programming modelthat is solved in a rolling horizon setting is the one discussed in Sect. 4.2.

Based on stored historical demand and forecasts data, we found that the normaldistribution is statistically fitting the forecasts and sales data the best. A goodness-of-fit test has been used to find the suitable distribution that fits the best to the data. Theresult was that the average demand is time-independent, but forecast errors showed astrong correlation with the forecast age, i.e. the number of periods between the momentthe forecast was made and the moment the demand is realized. The demand genera-tor randomly generates a series of forecasts based on the parameters of the normallydistributed demand (µd,i , σd,i (h)).

Fig. 5 The supply chain ofproduct A

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0

10

20

30

40

50

60

70

80

90

100

Time

titnauqredrokca

By

Fig. 6 An example of the backorders that result from a simulation study for a certain item

Table 1 Safety stock levelsbased on the proposed approachand the current safety stocklevels

Current situation Model suggestion

National warehouses 100 89

Tablets stockpoint 51 47

Active ingredients 49 21

As discussed in the previous section, the outputs of the approach are series ofplanned backorders that are stored. Figure 6 shows the development of backorders ofa certain item. As this figure shows, the backorder quantities are mostly equal to zero,which means that there is mostly enough inventories available to satisfy the requiredquantities. Whenever the required quantity (either dependent or independent demand)is not (fully) satisfied, a backorder is planned for the next period.

It is not possible to show here all results that we obtained from the implementationof the discussed approach. However, the results of our approach for this particularsupply chain are presented in Table 1. The second column of Table 1 shows the cur-rent safety stock levels and the third column shows the safety stock levels that resultfrom our approach based on a target service level of 99%. It was not our intention todecrease current safety stock quantities, but Table 1 shows that in this case, substantialsavings may be achieved by implementing the approach. However, the comparisonis not completely fair, as in the current situation also other types of uncertainties arecovered.

5.2 The supply chain of product B

The supply chain of product B has also been used for the validation of the proposedapproach. Figure 7 shows the supply chain of product B. Contrary to product A; twoactive ingredients (AI 1 and AI 2) are required for the tablet production. After theproduction process, the tablets are shipped to two packaging sites where the tabletsare also stored (TB2 and TB3). After packaging of TB2, the products are shipped to 19national warehouses all over the world, whereas TB3 is shipped to only one national

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Fig. 7 The supply chain ofproduct B

AI 1

AI 2

TB 1

TB 2

TB 3 20

19

1

Table 2 Safety stock levelsbased on the proposed approachand the current safety stocklevels

Current situation Model suggestion

National warehouses 162 136

Tablets stockpoint 71 46

Active ingredients 29 22

warehouse. The same approach has been applied for this supply chain to determine thesafety stock levels. As we mentioned in the case of supply chain of product A, safetystocks can not be pooled due to the fact that each product is country-specific. Basedon historical demand and forecasts data, the demand generator generates a series offorecasts that are input to the mathematical programming model that is presented inSect. 4.2.

Table 2 presents the results of the proposed approach (to obtain a service level of99%) and the current safety stock levels. The results show that substantial savingscan be made, but even more important, the approach turns out to give satisfying andreasonable results.

6 Conclusions

In this paper, we introduced an approach to determine safety stock levels in multi-itemmulti-stage inventory systems that face demand uncertainties. The problem of deter-mining safety stock levels in a supply chain to meet certain predefined target customerservice levels is based on a simulation study where the supply chain planning problemis solved in a rolling horizon setting. We assume that the supply chain is planned andcontrolled by a central authority that sets releases to the production system based onmathematical programming models. Combining the long run backorder quantities thatresult from the simulation study with predefined target customer service levels, theapproach allows for determining safety stock levels in the supply chain.

The approach does not make any assumption about the demand process. Further-more, all kinds of constraints can be included that are also considered in the supplychain planning model. The approach is based on two main assumptions. The firstassumption is that the requirement process and replenishment decisions are completelyindependent from the safety stock levels. The second assumption is that all unsatisfieddemand is backordered. As a form of validation, we discussed an application of theapproach to two supply chains at Organon, a worldwide operating pharmaceutical

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company. The approach helped the company to determine the safety stocks that coverdemand uncertainties.

A shortcoming of our approach is that we assume that any upstream unavailabilityof stock leads to an order delay at the next stage, which affects the performance of theinventory system. This is not necessarily what happens in practice. A short study thatwe performed showed that usually protection against a shortage is not only achievedthrough the use of safety stocks, but also by using the slack in the lead times or byreprioritizing the orders such that a higher customer service level is achieved than ini-tially planned. This effect can be compensated by setting the target customer servicelevel lower than the ‘real’ target customer service level and this could be an object offurther study.

Acknowledgments The authors would like to extend their word of thanks to Organon, especially to JohnKoelink and Joop Wijdeven for initiating and supporting this project. Further, we would like to thank twoanonymous referees for their valuable and constructive comments that helped to improve the clarity of thispaper.

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analysis in production/distribution systems. IIE Trans 26(3):17–30Inderfurth K (1991) Safety stock optimization in multi-stage inventory systems. Int J Prod Econ 24:103–113Inderfurth K, Minner S (1998) Safety stocks in multi-stage inventory systems under different service

measures. Eur J Oper Res 106:57–73Kleijnen JPC, Wan J (2006) Optimization of simulated inventory systems: OpQuest and alternatives. Tilburg

University: CentER Discussion Paper, no. 2006-75Kohler-Gudum CK, De Kok AG (2002) A safety stock adjustment procedure to enable target service lev-

els in simulation of generic inventory systems. Technische Universiteit Eindhoven: BETA WorkingPaper 71

Law AM, Kelton WD (2000) Simulation modelling and analysis, 3rd edn. McGraw-Hill, New YorkMinner S (1997) Dynamic programming algorithms for multi-stage safety stock optimization. OR Spektrum

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Spitter JM, Hurkens CAJ, De Kok AG, Negenman EG, Lenstra JK (2005) Linear programming modelswith planned lead times. Eur J Oper Res 163:706–720

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Supplier managed inventory in the OEM supply chain:the impact of relationship types on total costs and costdistribution

P. L. M. Van Nyen · J. W. M. Bertrand ·H. P. G. Van Ooijen · N. J. Vandaele

Abstract We investigate the impact of four variants of supplier managed inventoryon total costs and cost distribution in a capital goods supply chain consisting of a partssupplier who delivers parts to an original equipment manufacturer’s assembly plant.The four supplier managed inventory variants differ in the components of inventorycosts that the supplier has to carry. The performance of the supplier managed inventoryrelationships is benchmarked with the situation where the assembly plant manages theinventories. Interesting managerial insights follow from this comparison.

Keywords Supply chain relationships · Vendor managed inventory · Coordination ·Production–inventory system · Lotsizing

P. L. M. Van Nyen (B)OM Partners, Koralenhoeve 23, 2160 Wommelgem, Belgiume-mail: [email protected]

J. W. M. BertrandTechnische Universiteit Eindhoven, Faculty of Technology Management,Den Dolech 2, P.O. Box 513, Paviljoen F08, 5600 MB Eindhoven, The Netherlandse-mail: [email protected]

H. P. G. Van OoijenTechnische Universiteit Eindhoven, Faculty of Technology Management,Den Dolech 2, P.O. Box 513, Paviljoen F01, 5600 MB Eindhoven, The Netherlandse-mail: [email protected]

N. J. VandaeleKatholieke Universiteit Leuven, Campus Kortrijk,E. Sabbelaan 53, 8500 Kortrijk, Belgiume-mail: [email protected]


DOI 10.1007/s00291-007-0105-4


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OR Spectrum (2009) 31:167–194


P. L. M. Van Nyen et al.

1 Introduction

Original equipment manufacturers (OEMs) often subcontract production of parts tospecialized firms that, by working for various customers, can operate at a scale thatallows for the economic operation of their technologies. Such parts manufacturingshops generally use a number of different technologies grouped into work centers tomanufacture a wide range of parts, with varying routings in the shop and each parthaving low to medium demand. These shops are characterized in the literature as jobshops. In this paper we study whether it can be advantageous for a parts manufacturingshop to engage in a supplier managed inventory (SMI) relationship with its customers.In such a relationship, it is the supplier who manages the inventory of customer-specificparts that it produces for the OEMs. For the supplier, a main advantage of engagingin the SMI relationship is the possibility to optimize production batch sizes, resultingin lower system-wide costs.

Substantial savings can be obtained from optimizing production batch sizes, bothfor the parts supplier from reduced work-in-process and setup costs and for the OEMsfrom reduced delivery times and lower safety stocks. However, this requires optimiz-ing batch sizes from a system-wide perspective, which in turn requires centralizeddecision making and implies specific organizational arrangements. An organizationalarrangement that would enable centralized decision making about batch sizes is SMI.Under SMI, the parts manufacturer, acting as the supplier, would be responsible foravailability of parts at the OEMs assembly plants, and would be free to set a productionand delivery batch size for each of the parts in order to optimize system-wide costsunder a service constraint. In this paper we investigate for which situations it can beadvantageous for a parts supplier who runs a job shop like production system, to engagein a SMI relationship with his OEM customers. In particular we focus on the advan-tages that can be obtained from being able to optimize batch sizes. We thus neglectother advantages that might result from applying SMI, such as reduced transportationcosts or mitigation of the bull whip-effect or improved shop floor scheduling. SMIalso allows for improving the coordination of inventory and transportation decisions,which may result in considerable cost savings. We will not take into account this effectin our research. This problem is called the inventory routing problem and is discussedin a.o. Bell et al. (1983), Campbell et al. (1998) and Bertazzi et al. (2002). SMI canalso contribute to mitigating the bullwhip effect in supply chains. Lee et al. (1997)analyze the relationship between supplier and customer and identify four mechanismsthat contribute to the amplification of demand variations. One of these mechanismsoccurs if a customer is uncertain about the supplier’s lead time and adapts its reorderpoint in response to realized lead times. This mechanism is eliminated if the suppliermanages the inventory. Finally, SMI can also lead to improved shop floor scheduling.Research of Zheng and Zipkin (1990) has shown that substantial inventory cost reduc-tions can be obtained from having the production priorities depend on the inventoryposition. In this research, we do not consider this effect and assume FCFS sequencingof production orders.

Our research approach is as follows. We first conceptually analyze the performanceof OEM-managed inventory and four variants of SMI. The conceptual analysis givesinsights into the advantages and disadvantages of the proposed SMI relationship types,

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for both the supplier and the OEMs. After the conceptual analysis, a numerical studyis performed. The numerical study is designed such that it allows us to determine thecharacteristics of the situations for which it would be beneficial for the supplier toengage in a SMI relationship, and what relationship type is needed.

The analysis and numerical study is carried out as follows. We have defined fivefactors that we expect to influence the advantages to be obtained from system-widebatch size optimization, and have selected low and high values for each of these fac-tors. Then we have randomly selected five instances of a 10-products–5-machine jobshop production system that we use as a research tool. For each instance and eachsetting of the factors, we have first calculated the batch sizes that would result if eachOEM would optimize its own inventory and ordering costs, and then calculated thesystem-wide costs and the distribution of these costs over supplier and OEMs thatwould result from the application of these batch sizes. This non-SMI case serves as areference for calculating the benefits obtained from SMI. Next we have defined fourvariants of SMI, each variant differing in the inventory costs elements incurred by thesupplier. For each of these variants we have calculated optimal supplier determinedbatch sizes, and the corresponding system-wide costs and their distribution over thesupplier and OEMs. In this set of experiments, we assume that ordering costs andinventory holding costs are identical for the OEM and supplier. Optimization of batchsizes is achieved with a heuristic that incorporates an approximate queuing modelthat has been shown to give accurate results (Van Nyen et al. 2005). The numericalresults obtained are analyzed to identify the conditions for which system batch sizeoptimization is highly advantageous, and the type of supply relationships for which itis attractive for the partners to engage in such a relationship.

The rest of this paper is organized as follows. In Sect. 2 we review literature on batchsize coordination and on Supplier Managed Inventory. Section 3 presents the supplychain studied in this paper. Section 4 presents the relationship types and cost modelsper relationship type. Section 5 gives a conceptual analysis of the generic effects ofeach of the four SMI relationships on the distribution of costs. A numerical study ofthe effects of the relationship types on total costs and their distribution is given inSect. 6, followed by an analysis of the data. Conclusions are given in Sect. 7.

2 Literature review

Supplier managed inventory is quite common in real life. Especially in the retailindustry this management policy is frequently applied. Most often it is referred toas vendor managed inventory (VMI). Several case studies on VMI are described inthe literature. Well known VMI implementations include Campbell Soup (Clark andMcKenney 1994), Barilla SpA (Hammond 1994; Simchi-Levi et al. 2000) and theagreement between Wal-Mart and Procter & Gamble (Cottrill 1997). Other casescan be found in the automotive industry (Valentini and Zavanella 2003), the foodindustry (Tyan and Wee 2003), the retail industry and the health care sector (Gerber1991). More examples can be found in Andel (1996), Burke (1996), Cottrill (1997),Holmström (1998) and Waller et al. (1999). Many authors mention the coordinationof production and inventory decisions as a main advantage of VMI. However, to the

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best of our knowledge, there is little research on how to coordinate production andinventory decisions in VMI relationships. Two contributions can be mentioned. Fryet al. (2001) study the savings due to better coordination of production, delivery andinventory facilitated by a VMI contract. They focus on a supply chain that consistsof a single retailer and a single supplier, which have a (z, Z ) type VMI agreement.The (z, Z ) levels correspond to minimum and maximum allowed inventory levels atthe retailer. Bertazzi et al. (2005) study a production–distribution system in which oneitem is produced. Two different types of VMI policies are investigated. Both types aimto determine the production policy, retailer replenishment policies and transportationpolicy so as to minimize total system costs. The computational results show that theVMI policies significantly reduce the average costs compared to the traditional retailermanaged inventory policy. The cost reduction mainly results from improvements inthe transportation costs.

Similar to the work mentioned above, we model supplier–OEM relationships inorder to determine the benefits to be obtained from better coordination of productionand inventory. Specifically we consider the batching decisions. However, our researchis different from the previous work since we focus on a multi-product situation inwhich production orders for different products compete for limited production capac-ity in a production system consisting of multiple work centers. Moreover, we explicitlyconsider setup times and setup costs at the work centers. Unlike Fry et al., we do notallow the supplier to outsource some of its production. Similar to Fry et al., but unlikeBertazzi et al., we allow the decision maker to act in its own interest, i.e., we model adecentralized decision maker. Unlike Bertazzi et al., we focus on a production–inven-tory system, and we do not include transportation issues. Finally, we introduce a widerspectrum of VMI relationship types than those studied in previous work.

Other research on VMI uses analytical models to analyze the gain of using VMIin a two-echelon inventory system and to support the vendor with the tactical inven-tory control decisions. Unlike our approach, these models do not explicitly includemanufacturing operations. In this category, we mention the contributions of Aviv andFedergruen (1998), Achabal et al. (2000) and Kaipia et al. (2002). Also, analyticalmodels have been developed to analyze the synchronization of inventory and trans-portation decisions, see Bell et al. (1983), Campbell et al. (1998), Çetinkaya and Lee(2000), Axsäter (2001), Cachon (2001), Bertazzi et al. (2002), Cheung and Lee (2002)and Disney et al. (2003). Furthermore, some analytical models have been developedto understand the role of VMI in a supply chain channel, see Dong and Xu (2002) andDisney and Towill (2002). Mishra and Raghunathan (2004) investigate the impact ofVMI on brand competition. Their research shows that VMI intensifies the competitionbetween competing brands because of brand substitution. Finally, the research on VMIand other supply chain relationships is strongly related to the research on informationsharing and supply chain coordination. An overview of research on information shar-ing can be found in Chen (2003) and Huang et al. (2003). For a review of the researchon supply chain coordination, we refer the reader to Thomas and Griffin (1996). Sahinand Robinson (2002) cover both information sharing and supply chain coordination.

In this paper we study the benefits of determining ordering and production batchsizes under a SMI relationship, where the suppliers run a job shop like productionsystem. There is quite some literature on the selection of batch sizes in stochastic job

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shop production systems. Contributions in this area are Bertrand (1985), Karmarkaret al. (1985) and Lambrecht et al. (1998). Vandaele et al. (2000, 2007) present real-lifeapplications of this kind of batch size optimization models. However, all contributionsfocus on the performance of the job shop and ignore inventory holding costs. Van Nyenet al. (2005) extend the method developed in Lambrecht et al. (1998) to incorporateboth production and inventory costs. In this paper we will use this extended methodto numerically investigate the different relationship types.

3 The supply chain model

We study a supply chain that consists of a parts supplier and OEMs that order parts atthe supplier. The supplier runs a multi work center job shop that produces the parts forthe OEMs. After finishing the production process at the supplier, the parts are kept instock at the OEMs. The OEMs take the parts out of the stock when they need them inthe assembly processes. This kind of supply chain can be modeled as a multi-product,multi-machine production-inventory (PI) system with a single inventory echelon, seeFig. 1.

We consider K parts (k = 1, . . . , K ) that are kept on stock. The demand for eachproduct is a stationary renewal process. The demand interarrival times Ak are stochas-tic variables with a known expectation E[Ak] and squared coefficient of variation (scv)c2[Ak]. The demand size (number of product units requested per demand) is equal toone. Demand that cannot be satisfied directly from stock is backordered. The stockmanagement has to ensure that a target fill rate βk is attained. This type of servicelevel agreement is common in the parts supply business. The target fill rate βk is setduring contract negotiations between the supplier and the OEM. We assume that thistarget fill rate is not renegotiated when changing the relationship type.

The inventory management generates replenishment orders for the different prod-ucts in order to satisfy the demand according to a (bk, Qk) continuous review reorderpoint inventory policy. Each time the inventory position drops below the reorder pointbk , a replenishment order with batch size Qk is generated. The reorder points bk and

Fig. 1 Supplier–OEMs relationship modeled as a multi-product, multi-machine production–inventorysystem

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the batch sizes Qk for k = 1, . . . , K are the decision variables. When an order isgenerated, a fixed cost ok is incurred. For the items of product k in stock a carryingcharge is incurred. The carrying charge consists of a financial inventory holding cost,hf

k and a physical inventory holding cost, hpk . The replenishment orders are made-to-

order by the production system. Therefore, the replenishment orders are equivalent toproduction orders and production batch sizes are equal to replenishment batch sizes.

The production orders are manufactured in a production system that consists of Mfunctionally organized work centers. We assume there is ample supply of raw material.Each of the products requires a specific serial sequence of production steps, whichresults in a job shop routing structure. The production orders for different productscompete for capacity at the different work centers. Before the production of an orderfor product k at a work center j can start, a machine setup has to be performed. Thismachine setup takes a certain time L jk and cost s jk . The setup costs over the entirerouting of product k are denoted as sk . The setup times and costs are sequence andbatch size independent. After the setup, the processing of the production order starts.The production time for one unit of product k on work center j is given by Pjk . Themanufacturing process is subject to variability: setup times and processing times arestochastic variables with a known expectation and scv. When the production of theentire batch is completed, the batch is transferred to the next work center on the routingof the product. These transfer batch sizes are equal to the production batch sizes. Acarrying charge hwip

k is incurred for the work-in-process inventory for each item ofproduct k that is in process per unit of time. After finishing the last production step, theentire batch is transferred to the inventory at the OEMs. We assume that the transfertimes are negligible.

4 Relationship types

In this section we present the five supply relationship types, the availability of infor-mation at the parties as a function of the supply relationships and we introduce thedifferent cost elements.

4.1 Relationship types

The choice of relationship is a strategic decision since it influences the responsibilitiesof the parties, the cost division over the parties and the access to information by theparties. We consider one non-SMI and four SMI relationship types, based on threeresponsibilities that are related to the management of inventories. Each relationshiptype defines which party (OEMs or supplier) (1) controls the inventory by setting thereorder points bk and batch sizes Qk and incurs the ordering costs; (2) incurs the phys-ical inventory costs; (3) incurs the financial inventory costs. The physical inventorycosts consist of all costs related to the inventory storage, e.g., expenses incurred inrunning a warehouse, handling and counting costs, etc. The financial holding costsconsist of all the costs that come with the ownership of inventory, e.g., the opportu-nity cost of capital, insurance, obsolescence and damage (Silver et al. 1998 Chap. 3;

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Table 1 Characterization of supplier–OEM relationships

Non-SMI SMI-NC SMI-S SMI-C SMI-F

Inventory control O S S S S

Inventory storage O O S O S

Inventory ownership O O O S S

O original equipment manufacturer (OEM), S supplier

Valentini and Zavanella 2003). This leads to the definition of the following relationshiptypes:

1. Non-SMI. The OEMs are responsible for ordering and storing their own parts.2. SMI–Non consignment (SMI-NC). The supplier is responsible for the inventory

management at the OEM’s premises, but does not incur any inventory costs.3. SMI–storage (SMI-S). The supplier is responsible for the management and storage

of the OEM’s parts.4. SMI–consignment (SMI-C). The supplier is responsible for the inventory man-

agement at the OEM’s premises. Moreover, he owns the products until they aretaken from the stock.

5. SMI–full (SMI-F). The supplier is responsible for the management and the storageof the OEM’s parts. Additionally, he owns the products until they are taken fromthe stock.

Table 1 summarizes the five relationship types and indicates the responsibilities ofeach party. In the literature we encountered the SMI-NC and SMI-C relationship(Simchi-Levi et al. 2000). We introduce SMI-S and SMI-F because they correspondto actual practices in industry and because they have interesting analytical properties.

4.2 Information availability

The access to information is dependent on the relationship type. We make two assump-tions. Firstly, the party that is responsible for the inventory control has informationabout the demand process and the ordering and inventory holding costs. The litera-ture on information sharing describes that the availability of demand information isa major benefit for the supplier, who can use the additional information to improvehis production and inventory control decisions. See, e.g. Chen (2003) and referencestherein. Secondly, only the supplier has access to production-related information (pro-cessing times, setup times, routing structure, work-in-process costs and setup costs).However, we observed that in numerous real-life cases the setup costs are signaled (oreven transferred) to the OEM during supply contract negotiations.

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4.3 Cost definition

In this subsection we define the relevant costs. We introduce and model the differentcost elements and we present cost objective functions for the different relationshiptypes.

4.3.1 Cost elements

We consider five cost elements: ordering costs (OC), financial inventory holding costs(FIC), physical inventory holding costs (PIC), production setup costs (SC) and work-in-process holding costs (WIPC). Table 2 presents the allocation of these cost elementsover the parties under the various relationships. The supplier always carries the setupcosts and work-in-process holding costs. In the case of NON-SMI, the OEMs carrythe ordering costs and the physical and financial inventory holding costs. A shift fromNON-SMI to other relationship types implies that the supplier carries more and morecost components, so that under SMI-F the supplier carries all the costs considered inthis model.

4.3.2 Cost models

We consider five cost components. Table 3 introduces some additional notation. Inthis paper, we assume that ok, hf

k and hpk are identical for the OEMs and supplier.

1. Ordering costs.

Under a continuous review, fixed order quantity (bk, Qk) policy, a new replenish-ment order for product k is generated when Qk demands have arrived after the previousreplenishment order was generated. The average time between two successive demandarrivals is E[Ak]. Therefore, in a (bk, Qk) policy, a new replenishment order is gen-erated every Qk E[Ak] time units, on average. Since each time when a new order is

Table 2 Allocation of costs over OEMs and supplier

Cost element NON-SMI SMI-NC SMI-S SMI-C SMI-F

OC O S S S S

FIC O O O S S

PIC O O S O S

SC S S S S S

WIPC S S S S S

O OEM, S supplier

Table 3 Some additional notation

� (Q1, . . . , Qk , . . . , QK )

ssk (�) safety stock for product k

Tk (�) throughput time of production orders for product k (stochastic variable)

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generated a fixed cost ok is incurred, the ordering costs per unit of time are:

OCk(Qk) = ok

Qk E[Ak]

2. Financial inventory holding costs.

The average amount of inventory that is available at the stock points is given byQk2 + ss(�), see e.g., Silver et al. (1998, Chap. 7, p. 258). Therefore, the financial

inventory holding costs are given by:

FICk(�) = h fk

[Qk

2+ ssk(�)

]

3. Physical inventory holding costs.

Similarly to the financial holding costs, the physical inventory holding costs are:

PICk(�) = h pk

[Qk

2+ ssk(�)

]

4. Setup costs.

In a make-to-order model every replenishment order generates a production order.Therefore the derivation of the setup costs is similar to the derivation of the orderingcosts. Then, the setup costs are given by:

SCk(Qk) = sk

Qk E[Ak]

5. Work-in-process inventory holding costs.

Using Little’s law, we compute the expected amount of work-in-process inventoryin the production system as the average order throughput time multiplied by the orderarrival rate. In our supply chain, the order arrival rate equals 1

Qk E[Ak ] and the averagetime spent in the production system is E[Tk(�)]. Then, the work-in-process holdingcosts can be written as:

WIPCk(�) = hwi pk

E[Tk(�)]

Qk E[Ak]

Above, expressions for the relevant cost components of the supply chain are pre-sented. In our numerical experiments, we would like to compute these costs for dif-ferent problem instances. It can be seen from the above formulae that the supply chaincosts depend strongly on the selection of batch sizes � = Q1, . . . , Qk, . . . , QK . Thisdependence can be either directly, in case of setup and ordering costs, or indirectly,through the dependence of safety stocks ssk(�) and throughput times E[Tk(�)] onthe batch sizes �. For the numerical analysis, we use an approximate analytical modeldeveloped by Van Nyen et al. (2005) to obtain estimates for ssk(�) and E[Tk(�)].

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The analytical model is summarized in Appendix 1. Using these estimates, the vectorof batch sizes � can be optimized and the corresponding costs can be computed forspecific problem instances.

4.3.3 Cost objectives

The party that is responsible for making the inventory and production control deci-sions tries to minimize its own costs. Therefore, the cost objective that is optimizeddepends on the relationship type. Below, we define a cost objective function for eachrelationship type.

NON-SMI In the NON-SMI relationship, the OEMs determine the batch sizes so thattheir own costs are minimized, without taking into account the impact on the supplier.Typically, the batch sizes and reorder points are set sequentially; see e.g. Silver et al.(1998, Chap. 7, p. 254). In general, we expect the OEMs to be unaware of the costparameters at the supplier’s side. However, during contract negotiations the suppliermay signal setup costs (e.g., via batch size dependent prices). Since we do not want toinclude pricing mechanisms into the model because this would unnecessarily compli-cate the analysis, we mimic this effect by assuming that the OEMs determine economicbatch sizes based on inventory carrying costs and the sum of his own ordering costsand set-up costs at the supplier. In formula:

QNON-SMIk =

√√√√2(ok + sk)

E[Ak](

h fk + h p

k

)

Supplier managed inventory Under the SMI relationships, the OEMs transfer theirpower for managing the inventories to the supplier. The supplier has access to detaileddemand and process information so that he can coordinate batch sizes with the objec-tive of minimizing his costs under a service level constraint. The cost components thatthe supplier carries under the different SMI variants are listed in Table 2. We modeledeach cost component in Sect. 4.3.2. This leads to the following cost objective functionsfor the different SMI relationship types.

1. SMI-NC

K∑

k=1

[ok + sk

Qk E[Ak]+ hwip

kE[Tk(�)]

E[Ak]

+[(

hfk + hp

k

)(Qk

2+ ss(�)

)− FICNSMI

k − PICNSMIk

]+]

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2. SMI-S

K∑

k=1

[ok + sk

Qk E[Ak]+ hwip

kE[Tk(�)]

E[Ak]+ hp

k

(Qk

2+ ss(�)

)

+[

hfk

(Qk

2+ ss(�)

)− FICNSMI

k

]+]

3. SMI-C

K∑

k=1

[ok + sk

Qk E[Ak]+ hwip

kE[Tk(�)]

E[Ak]+ hf

k

(Qk

2+ ss(�)

)

+[

hpk

(Qk

2+ ss(�)

)− PICNSMI

k

]+]

4. SMI-F

K∑

k=1

[ok + sk

Qk E[Ak]+ hwip

kE[Tk(�)]

E[Ak]+

(hf

k + hpk

)(Qk

2+ ss(�)

)]

These cost functions can be interpreted as follows. The sum is taken of all relevantcost components over all products. The first term within the summation representsthe ordering and setup costs. Under all SMI relationships, the supplier controls theinventories and incurs the ordering costs. Thus ordering costs are always transferred tothe supplier. The second term is the work-in-process cost. Both terms follow directlyfrom the cost definition above. The terms related to the financial and physical inven-tory costs depend on the relationship type and need some more explanation. Under theSMI relationships, the supplier’s batch size decisions fully determine the inventorycosts. Under the SMI-NC, -S, and -C relationship, the OEMs still carry all or someof the inventory costs (see Table 2). As a result, the OEM’s inventory costs are deter-mined by the decisions of the supplier. It seems natural to assume that the OEMs donot allow increases in their inventory costs after changing from NON-SMI to a SMIrelationship. Therefore, the supplier must ensure that the inventory costs of the OEMsdo not increase in the SMI relationship. We have encountered this kind of arrangementin real life cases (Kikkert 2006), where an OEM allowed its supplier to control theinventories at the site of the OEM under a service level constraint and a constrainton the maximum physical inventory per item. However, imposing a hard constraintmay unnecessarily restrict the search space for the optimization of the batch sizes.Therefore, we propose to use a soft constraint by assigning a cost when the averageinventory under NON-SMI is exceeded. A similar arrangement was studied in Fry et al.(2001), where a penalty cost b+ is charged when the inventory position of product kexceeds the maximum inventory level Zk . The proposed policy can be interpreted as acompensation mechanism: if the inventory costs under SMI are higher than the inven-tory costs under NON-SMI, the supplier compensates the OEMs for the cost increase.

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This policy gives the supplier sufficient flexibility to optimize his costs by setting thebatch sizes, while the OEMs do not face any increase in their inventory holding costs.The compensation mechanism is modeled in the objective functions for the SMI-NC,-S and -C relationship with the terms that contain the operator [a]+, which denotesthat the maximum of 0 and a is taken. Under SMI-F the supplier carries all inventorycosts, so no compensation mechanism is needed.

5 Conceptual analysis

This section presents a conceptual analysis of the OEM–supplier relationships, underthe assumption that ok, hf

k and hpk are identical for the OEM and supplier. First we

discuss the relationship that leads to the lowest system-wide costs. After this we ana-lyze the division of costs over the different parties. We introduce two cost effects, thetransfer effect and the coordination effect.

5.1 Supply chain’s optimal cost

In the case of SMI-F the supplier carries all the relevant costs. Therefore, the suppliercan optimize all the costs simultaneously (system-wide optimization), while in theother relationships only a subset of the costs are optimized (partial optimization). Asa consequence, the total supply chain costs of the SMI-F relationship are lower thanor equal to those of the other relationships and the cost of SMI-F is the supply chain’soptimal cost:

TRCSMI−FSC = TRC∗SC

This observation is useful in situations where the achievement of the lowest system-wide costs is more important than the division of costs over the different parties. Thismay occur, for example, when the OEMs and supplier are subdivisions of the samecompany. In this case, it is more important to obtain the lowest cost for the wholecompany, than to locally optimize the costs of the subdivisions.

5.2 Distribution of costs over the parties: transfer effect and coordination effect

Suppose that initially the supplier and OEMs have a NON-SMI relationship. Fur-ther suppose that the supplier now wants to change the relationship into a SMI typerelationship. The impact of this change is twofold. Firstly, cost components are trans-ferred from the OEMs to the supplier. This cost transfer is called here the transfereffect. Together with the costs, however, also the power for controlling the inventoryis transferred from the OEMs to the supplier. This allows the supplier to determinethe batch sizes, which may results in cost reductions that we refer to as the coordina-tion effect. In the remainder of this section, the coordination effect and transfer effectare investigated in more detail. First, we introduce some additional notation. TRCx

ydenotes the total relevant costs of a SMI relationship x for a certain party y that result

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from minimizing the objective functions in Sect. 4.3.3. OCNSMIk denotes the ordering

cost for product k in the NON-SMI relationship.

5.2.1 Transfer effect

The OEM’s transfer effect TExO can be computed for the different relationships x :

TESMI - NCO = −

K∑

k=1

OCNSMIk

TESMI - SO = −

K∑

k=1

(OCNSMI

k + PICNSMIk

)

TESMI - CO = −

K∑

k=1

(OCNSMI

k + FICNSMIk

)

TESMI - FO = −

K∑

k=1

(OCNSMI

k + FICNSMIk + PICNSMI

k

)

Note that TExO ≤ 0, since it is a cost reduction. On the other hand, the supplier’s

transfer effect TExS is a cost increase that consists of all the costs that are transferred

from the OEMs to the supplier:

TExS = −TEx

O ≥ 0

The transfer effect of the supply chain TExSC is the sum of the transfer effect of the

OEMs and the supplier, so that the transfer effect for the supply chain is zero:

TExSC = TEx

S + TExO = 0

5.2.2 Coordination effect

The supplier can reduce his costs in a SMI relationship by coordinating batch sizes.His coordination effect is characterized by:

CExS ≤ 0

The OEMs are compensated for increases in the inventory holding costs (see Sect. 4.3.3).Therefore, the OEMs can only benefit from the integration of production and inventorycontrol decisions, so also their coordination effect is smaller than or equal to zero:

CExO ≤ 0

The coordination of production and inventory decisions results in cost decreases, bothfor the OEMs and the supplier. Therefore, the coordination effect always results inlower supply chain costs:

CExSC = CEx

S + CExO ≤ 0

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5.2.3 Total costs

The OEMs’ cost in the SMI relationship x equals their cost under NON-SMI plus thetransfer effect and the coordination effect index. This implies that the OEMs have aguaranteed benefit of a change in the relationship, which is at least as high as the trans-fer effect TEx

O . Therefore, the OEMs have a clear incentive to change the relationship.

TRCxO = TRCNSMI

O + TExO + CEx

O ≤ TRCNSMIO + TEx

O

The supplier’s cost can be computed in a similar way. However, for the supplierthe transfer effect and the coordination effect are opposing and their magnitude isunknown. Therefore, the total cost effect of a change in the relationship is uncertain.

TRCxS = TRCNSMI

S + TExS + CEx

S>

<TRCNSMI

S

Finally, we discuss the effect of a change in relationship on the supply chain costs.Since the transfer effect is zero and the coordination effect is always non-positive, achange from NON-SMI to any SMI relationship will always result in equal or lowersupply chain costs.

TRCxSC = TRCNSMI

SC + TExSC + CEx

SC ≤ TRCNSMISC

From the point of view of the OEMs and the total supply chain, a transition fromNON-SMI to SMI always results in lower costs. Therefore, the only uncertainty is thecost impact of a change on the supplier’s costs. This cost impact will be quantified forspecific instances in the next section.

6 Numerical analysis

A crucial element in the conceptual analysis in the previous section is the magnitude ofthe coordination effect. In this section we use numerical analysis of a selected numberof case situations to gain insights into the magnitude of this effect as a function thecase characteristics. This allows us to generate insights into the situations where theuse of SMI type relationships is beneficial.

All cases in the numerical study are variants of the supply chain with 10 productsthat are produced to stock in a job shop consisting of 5 work centers with one machineeach. We consider this 10 products, 5 machine supply chain to be sufficiently complexto capture all effects that occur in larger scale real life systems, while still being suffi-ciently small to allow for numerical optimization of the batch sizes and safety stocks.The analysis of integrated production and inventory systems is not straightforward;see e.g. Zipkin (1986) for an overview of the complexities. We use an approximateanalytical model to estimate the expected order throughput times, the work in processand the safety stock as a function of the batch sizes for the different products �. Thismodel is summarized in Appendix 1, and extensively described in Van Nyen (2005)

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and Van Nyen et al. (2005). We refer the reader to these contributions for technicaldetails.

For the NON-SMI relationship, the approximate analytical model is used to esti-mate the replenishment order lead time distribution that emerges when the OEMs settheir batch sizes according to Sect. 4.3.3. Next, safety stocks are set to achieve thetarget service level, given this lead time distribution. For the SMI relationships, theapproximate analytical model is used in combination with a heuristic search procedureto find the vector of batch sizes that minimizes the supplier’s cost objective functions(as defined in Sect. 4.3.3) and to estimate the corresponding throughput times, safetystocks and costs.

For the numerical analysis we have generated five instances of our 10 products, 5machines job shop system. The input data are based on real-life data obtained fromtwo component suppliers in the OEM market and on the experience of the authors inthe OEM component supply market. Each product always requires processing onceon each machine and the routing of each product is generated randomly. The expectedprocessing time per item on a machine is randomly generated from the set of values5, 10, 15, 20 and 25 min. Actual setup and processing times per item on a machine areexponentially distributed. The products are identical in terms of expectation and scv ofinterarrival times and setup times. For each of the five randomly generated instanceswe have numerically investigated the supply chain costs for 32 different scenariosregarding factors that we expect to affect cost effects as we move from NON-SMI tofully supplier managed inventory. These factors are: the net utilization of the machines,the variability of demand, the ordering and setup costs, the setup time and the target fillrate. Each of these factors has been varied over two values: high and low. Specifically,the following values have been used:

Low HighA. Net utilization of machines (without setup times) 0.70 0.85

B. Scv of interarrival times of demand 0.5 2.0

C. Sum of ordering costs and setup costs ok + sk 100 300

D. Expectation of setup times 150 450

E. Target fill rate 0.90 0.99

This leads to 5 × 32 = 160 different case situations. For all cases, the inventoryholding costs are set as follows: hf

k = 1.67, hpk = 0.83 and hwip

k = 1.67e/item peryear. The sum of orderings costs and setup costs, ok + sk is a factor in our design(factor C), but their split is fixed: ok

sk= 1

2 . The cost parameters ok, hfk and hp

k areidentical for the OEM and supplier.

6.1 Total supply chain cost

For each of the five relationships, we have computed the total supply chain costs foreach of the 32 different scenarios and for each of the 5 randomly generated routingstructures. From the conceptual analysis we know that SMI-F results in the lowest sup-ply chain costs among the five relationship types. We have calculated for each SMI

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Table 4 Percentage cost savings of SMI over NON-SMI

SMI-NC SMI-S SMI-C SMI-F

min �1 (%) 17.7 23.2 25.2 25.7

avg �1(%) 7.4 10.9 12.6 13.0

max �1 (%) 1.4 3.7 5.1 5.4

Table 5 Results of ANOVA

Source Sum of squares d f Mean square F-ratio P value

A. Utilization 0.559152 1 0.559152 2185.39 0.0000∗B. Scv of interarrival times 4.6092E − 06 1 4.6092E − 06 0.02 0.8934

C. Fixed costs 0.00388724 1 0.00388724 15.19 0.0001∗D. Expectation of setup times 0.00115679 1 0.00115679 4.52 0.0352∗E. Target fill rate 0.241044 1 0.241044 942.1 0.0000∗AB 3.10978E−05 1 3.10978E−05 0.12 0.7279

AC 0.00262453 1 0.00262453 10.26 0.0017∗AD 0.000315344 1 0.000315344 1.23 0.2688

AE 0.0127712 1 0.0127712 49.92 0.0000∗BC 0.000598789 1 0.000598789 2.34 0.1283

BD 0.000299138 1 0.000299138 1.17 0.2814

BE 0.000403185 1 0.000403185 1.58 0.2115

CD 0.00194649 1 0.00194649 7.61 0.0066∗CE 0.000667867 1 0.000667867 2.61 0.1084

DE 0.000678567 1 0.000678567 2.65 0.1057

Blocks 0.000805844 4 0.000201461 0.79 0.5352

Total error 0.0358203 140 0.000255859

Total (corr.) 0.862207 159

relationship the percentage cost savings �1 of SMI over NON-SMI for each of the 160cases. These percentages are summarized in Table 4, which shows per relationship theminimum, the average and the maximum relative cost savings.

The data in Table 4 suggest that the benefits resulting from each SMI relationshiptype can be quite substantial. The cost savings are the highest for the SMI-F relation-ship. For this relationship type, we observe average cost savings over the total costunder NON-SMI of 13%, but the savings can be as high as 25.7%. Moreover, we seethat the simplest SMI-type SMI-NC already results in half of the cost savings obtainedwith the full-blown variant SMI-F. The SMI-C relationship achieves almost minimalcosts: on average only 0.5% higher than SM-F. As mentioned before, SMI-F gives theminimal costs.

Using the randomly generated process configurations as randomizing factor, westatistically analyzed the effect of the five factors on the cost difference between NON-SMI and SMI-F. The ANOVA reveals that seven factors have P values less than 0.05,

234


Fig. 2 Standardized Pareto chart for factors in the ANOVA

indicating that they are significantly different from zero at the 95.0% confidence level(see Table 5). However, two factors account for a major part of the variance in results,as can be seen from Fig. 2. These are the net utilization of the machines and the targetfill rate of the products. These results imply that a large reduction in total costs can beobtained from engaging in a SMI relationship if utilizations are high and/or target fillrates are high. This can be explained as follows. It is well known that the sensitivity ofthroughput times for batch size decisions is high when the utilization of the productionsystem is high. This effect is illustrated in the much cited paper on lot sizing and leadtimes by Karmarkar (1987). Therefore it makes sense that SMI performs much betterthan NON-SMI in situations with high-capacity utilization. Moreover, the throughputtime reductions that result from the SMI-F, lead to stronger decreases in the safetystock costs when the target fill rates are high. In real life, parts suppliers need to operateat high levels of capacity utilization in order to remain profitable in a market in whichsales prices are continuously under pressure. Moreover, the OEMs are imposing everincreasing service levels on their parts suppliers. Based on the numerical results, weexpect that supply chains operating in such competitive markets may greatly benefitfrom engaging in SMI type relationships.

6.2 Distribution of costs over the parties: transfer effects and coordination effects

The numerical data show that for high machine utilizations and/or high target fill rates,the supply chain cost savings from implementing SMI can be substantial. However,when going from a NON-SMI relationship to a SMI relationship, one or more costcomponents are transferred from the OEMs to the supplier. The question therefore iswhether it is financially attractive for the supplier to engage in a SMI relationship.This will be the case if the coordination effect leads to a reduction in the supplier’scosts that is larger than the costs transferred to him. To check this, we have computedfor each of the 32 different scenarios the decrease in costs relative to NON-SMI undereach of the four SMI relationships.

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First, the magnitude of the transfer effect TExy and the coordination effect CEx

yare calculated. We present the transfer and coordination effect relative to the costsin the NON-SMI setting. The relative transfer and coordination effect for party y forrelationship x is defined as:

RTExy =

TExy

TRCNSMIy

× 100% RCExy =

CExy

TRCNSMIy

× 100%.

We also computed the total relative cost effect of each SMI relationship compared tothe NON-SMI relationship:

�xy =

TExy + CEx

y

TRCNSMIy

× 100%

Table 6 summarizes the minimum, average and maximum over the 160 instances ofRTEx

y, RCExy and �x

y for the OEMs, supplier and the total supply chain in the differentSMI relationship types.

The numerical results in Table 6 contradict a common belief in the literature, namelythat the availability of additional (demand) information and the power to integrate pro-duction and inventory decisions are sufficient incentives for a supplier to engage inSMI (or VMI) type relationship. In this set of experiments, it appears that in all casesfor the supplier the transfer effect dominates the benefits of the coordination effect.This implies that the supplier always faces a cost increase going from NON-SMI toSMI on this set of experiments. Additional incentives for the supplier that could resultfrom applying SMI, such as reduced transportation costs or mitigation of the bullwhip-effect or improved shop floor scheduling, are not studied here. The inclusion ofthese effects would lead to a better performance of SMI for the supplier. We also seethat under SMI-NC, the simplest type of SMI, the coordination effect for the OEM’sis already quite high (on average 11.3%), whereas for the supplier the coordinationeffect is still quite low (on average 3.7%). For the supplier the coordination effectstrongly increases under more advanced type of SMI, up to on average 26.1% for fullSMI. However, for the supplier also the transfer costs increase, resulting in a stronglynegative total cost effect (up to an average 70.6% increase in costs). This suggeststhat the OEMs are the main benefactors of engaging in SMI relationships. In our setof experiments, the supplier does not have a direct cost benefit of entering into a SMItype relationship and additional incentives are therefore necessary for making sucha relationship financially attractive. In the next sub-section we will investigate thecoordination effect in more detail.

6.3 Mechanisms behind the coordination effect

In this sub-section, we investigate the mechanisms through which the supplier achievesthe coordination effect on his own costs and on the costs of the OEMs. Both work-in-process and safety stock are for a large part determined by the throughput times of the

236


Table 6 Relative transfer effect, coordination effect and total cost effect

x → SMI-NC SMI-S SMI-C SMI-F

RTExO

Min −25.0 −50.0 −75.0 −100.0

Avg −18.1 −45.4 −72.7 −100.0

Max −11.7 −41.1 −70.6 −100.0

RCExO

Min −22.1 −22.8 −13.7 0.0

Avg −11.3 −13.1 −8.8 0.0

Max −2.9 −5.8 −4.9 0.0

�xO

Min −39.8 −64.1 −84.4 −100.0

Avg −29.4 −58.5 −81.5 −100.0

Max −19.2 −53.1 −79.4 −100.0

RTExS

Min 13.6 32.8 52.0 71.2

Avg 16.7 43.4 70.0 96.7

Max 19.7 53.1 86.7 120.3

RCExS

Min −12.5 −23.7 −38.6 −55.7

Avg −3.7 −8.9 −16.6 −26.1

Max −0.2 −1.7 −5.1 −9.6

�xSMin 1.7 17.8 30.8 42.4

Avg 12.9 34.5 53.5 70.6

Max 19.3 48.9 74.3 97.1

RCExSC

Min 0.0 0.0 0.0 0.0

Avg 0.0 0.0 0.0 0.0

Max 0.0 0.0 0.0 0.0

RCExSC

Min −17.7 −23.2 −25.2 −25.7

Avg −7.4 −10.9 −12.6 −13.0

Max −1.4 −3.7 −5.1 −5.4

�xSCMin −17.7 −23.2 −25.2 −25.7

Avg −7.4 −10.9 −12.6 −13.0

Max −1.4 −3.7 −5.1 −5.4

production orders in the shop. The throughput times in turn are to a large extent deter-mined by the batch sizes. The relation between the throughput times and batch sizesin the multi-product production–inventory system can be modeled by the approximate

237


queueing model presented in Appendix 1. From elementary results from queueingtheory, we learn that there are four main elements that affect the throughput times inthe shop: (1) machine utilization ρ; (2) variation of the interarrival times c2

a ; (3) theexpectation of the processing time E[P]; and (4) the variation of processing times c2

p.This is represented in the Kingman approximation (1961) for the expectation of thethroughput times in a GI/G/1 queue:

E[T ] =(

c2a + c2

p

2

)(ρ

1− ρ

)E[P]+ E[P]

Batch size decisions affect all these four factors simultaneously. To minimize work-in-process costs, the supplier should select batch sizes that minimize the averagethroughput time. However, under most SMI relationships the supplier also (partly)incurs costs of keeping finished goods inventory. Since safety stocks are stronglydependent on the variability of replenishment lead times, the supplier should also takeinto account the standard deviation of the throughput times when optimizing the batchsizes.

Now we present numerical results that illustrate how the batch sizes depend onthe supplier–OEM relationship type. Table 7 gives for each of the five relationshipsthe batch sizes and the expectation and standard deviation of the throughput timesfor one specific problem instance. E[Tk] and σ [Tk] are expressed in days. In the caseof NON-SMI, all products have the same batch size because they are symmetrical interms of their costs and arrival rates and because the differences between products interms of processing requirements are ignored. Going from NON-SMI to SMI-F, weobserve that the average of the batch sizes and the average and standard deviation in thethroughput time decline systematically. Moreover, for all SMI relationships the batchsizes differ between products in order to take into account the differences in processingrequirements. As a result of the optimization of the batch sizes, the supplier incursmore ordering and setup costs. This cost increase is compensated by decreases in thethroughput time-related costs (work-in-process and safety stock costs), that followfrom the substantial reductions in the average and variation of the throughput times.

The data in Table 7 illustrate how SMI relationships achieve reductions in sys-tem-wide costs. It is the supplier who is in a position to realize these cost reductionsby cleverly setting the batch sizes. However, our numerical results indicate that undereach SMI relationship type, the supplier’s cost increase makes it unattractive for him toengage in such a relationship. It is in the interest of the OEMs to let the supplier makethe batching decisions, since the numerical data in Table 6 show that this substantiallyreduces their costs. Therefore, it is in the proper interest of the OEMs to engage in sup-ply contracts in which the supplier is made responsible for setting the batch sizes. Inorder to make such a contract attractive for both parties, the supplier should probablybe compensated for the costs that are transferred. Costs that can be identified easilyby the supplier and agreed upon are ordering costs and financial inventory costs. Thephysical inventory cost per part at the OEMs is much more difficult to establish and tobe compensated for. Our numerical results in Table 4 show that the performance underSMI-C, where ordering costs and financial inventory costs are transferred, is very close

238


Table 7 Relation between batch sizes and average and standard deviation of throughput times for oneproblem instance

k 1 2 3 4 5 6 7 8 9 10 Avg

NON-SMI

Qk 442 442 442 442 442 442 442 442 442 442 442

E[Tk

]49 55 49 53 61 58 64 60 57 60 57

σ[Tk

]19 19 19 19 19 19 19 19 19 19 19

SMI-NC

Qk 490 440 490 475 390 430 330 390 430 370 425

E[Tk

]48 52 48 52 55 55 54 54 53 53 53

σ[Tk

]17 17 17 17 17 17 17 17 17 17 17

SMI-S

Qk 485 370 510 390 325 360 280 330 360 325 375

E[Tk

]44 46 45 45 48 48 47 47 46 47 46

σ[Tk

]15 15 15 15 15 15 15 15 15 15 15

SMI-C

Qk 410 315 430 330 280 310 245 285 310 280 320

E[Tk

]39 40 40 40 42 42 42 42 41 41 41

σ[Tk

]13 13 13 13 13 13 13 13 13 13 13

SMI-F

Qk 365 280 380 295 250 275 215 255 275 250 285

E[Tk

]36 37 36 36 39 38 38 38 37 38 37

σ[Tk

]12 12 12 12 12 12 12 12 12 12 12

to the performance under SMI-F. Thus SMI-C seems to be a realistic candidate formodification by including a compensation for taking over ordering cost and financialinventory costs.

A supplier who considers offering SMI-C to an OEM can, given the demand levelsper part Di indicated by the OEM, use the queueing model in Appendix 1 to calculateoptimal batch sizes Qi and estimate the magnitude of the transfer effect and coordi-nation effect that would result from applying these optimal batch sizes. Sophisticatedmodels are available nowadays to perform these calculations for real life production–inventory systems (see, e.g., Lambrecht et al. 1998 or Van Nyen et al. 2005). Let thetransfer effect TESMI-C

y and coordination effect CESMI-Cy for party y be defined as in

Sect. 5.2. Further let �pi be the price increase per part demanded by the supplierunder SMI-C. Our numerical results indicate that for the supplier the cost impactof going from NON-SMI to the SMI-C relationship is always disadvantageous, i.e.TESMI-C

S + CESMI-CS > 0. Then the SMI-C offer would be financially attractive for

the supplier only if there is a price compensation: �pi ≥ TESMI-CS +CESMI-C

SDi

.The OEM is the benefactor of changing the relationship: his costs decrease by∣∣TESMI-C

O + CESMI-CO

∣∣. In order to convince the supplier to enter into the SMI-C

239


relationship, the OEM may share some of his benefits with the supplier by payinga higher price. The OEM would never want to pay more than what he expects to

gain from entering in the SMI-C relationship, i.e., �pi ≤∣∣TESMI-C

O +CESMI-CO

∣∣Di

. In thisexpression, the benefits of the coordination effect for the OEM are shared between thesupplier and the OEM. However, during real-life negotiations the magnitude of thecoordination effect is not yet known by the OEM, so we may expect that the OEM isnot willing to share these benefits. Therefore, the price increase is typically bounded

by: �pi ≤∣∣TESMI-C

O

∣∣Di

.

It follows that this negotiation game has a solution if there exists a price increase:

TESMI-CS + CESMI-C

S

Di≤ �pi ≤

∣∣TESMI-CO

∣∣Di

Since TESMI-CS = ∣∣TESMI-C

O

∣∣ and CESMI-CS ≤ 0, this is always the case and we may

conclude that the supplier can always find a �pi that is attractive both for the OEMand for himself.

7 Conclusions

In this paper we have modeled and numerically analyzed a supply chain consisting of aparts manufacturer and his OEM customers in order to study the conditions for whichit can be advantageous for the supplier to manage the inventories of his customers. Weconsider setup costs, setup time and work-in-process costs at the supplier, and orderingcosts, inventory costs and a service constraint at the OEMs. We proposed five typesof relationships between the supplier and OEMs. In the first relationship the OEMsmanage their inventories and place replenishment orders at the supplier. The other fourrelationships are variants of supplier managed inventory (SMI). In these relationshipsthe supplier manages the inventories at the OEMs and carries one or more compo-nents of the ordering and inventory costs. If the supplier manages inventories, he cancoordinate batch sizes so as to minimize his own costs. Costs reductions, both at thesupplier and at the OEMs, that result from coordinated batch sizes are referred to asthe coordination effect. We have investigated four SMI variants, one in which only theordering costs are transferred to the supplier, one in which either financial or physicalinventory costs are transferred and one full-SMI, in which all costs are transferred tothe supplier.

Numerical analysis of a set of problem instances revealed that substantial system-wide cost saving can be achieved under all SMI variants, in particular if the shopoperates under high-capacity utilization and/or the OEMs require high service levels.We have shown that these savings are due to the strongly reduced order through-put times that are possible if batch sizes can be coordinated. As a result, inventorycosts always decrease under SMI, making SMI attractive for the OEMs. However, forall problem instances studied, the supplier costs increased under SMI, because SMIimplies the transfer of one or more cost components from the OEM to the supplier.The OEMs always are better off under SMI. This suggests that a supplier should not

240


offer SMI to the OEMs unless there is some compensation for his net increase in costs.We have shown that, since system-wide costs always decrease under SMI, there existsunder each SMI relationship a range of product price increases that at least compensatethe supplier for his increased costs and still make it financially attractive for the OEMsto engage in the SMI relationship.

In this paper, we assumed that inventory holding costs are identical for supplierand OEMs. In real life supply chains, this is often not the case. Cost differences maybe caused by differences in the cost of capital (interest rate, required return on invest-ments, etc.) or differences in the costs of labor and space (labor contracts, location ofwarehouses, etc.). In future research, it may be worthwhile to investigate how thesecost differences have an impact on the choice for a certain supply chain relationshiptype.

Acknowledgements The authors would like to thank the editors and the referees for their detailed andvery helpful comments.

Appendix I: Modeling the physical supply chain

In this appendix, we present a model to compute the characteristics of the physicalsupply chain for a given vector of batch sizes for all products �. We determine thefollowing characteristics of the physical supply chain: (1) the expectation and scv ofthe batch interarrival times and batch production times; (2) the expectation and vari-ance of order throughput times in the production system; (3) the reorder points for thestock points.

Characteristics of production batches

In the production–inventory system studied in this paper, the generation of a replenish-ment order results in a production batch. Therefore, we can derive the characteristicsof the production batches by analyzing the characteristics of the replenishment orders.In a (bk, Qk) policy, a replenishment order of size Qk is placed every time the inven-tory position hits the reorder level bk . Consequently, the expected time between twoproduction batches for product k arriving to the shop is given by: E

[AB

0k

] = Qk E[Ak].Demand interarrival times are assumed to be i.i.d., so the variance of the interarrivaltimes of production batches is: σ 2

[AB

0k

] = Qkσ2[Ak].

Production batches are of fixed size Qk . Consequently, the expected processing

time of a production batch of product k at work center j is given by: E[

P Bjk

]=

E[Pjk

]Qk + E

[L jk

]. Since processing times of single units are assumed to be i.i.d.

and independent from the setup times, the variance of the processing time of a batch

is: σ 2[

P Bjk

]= σ 2

[Pjk

]Qk + σ 2

[L jk

].

241


Throughput times in job shop

In this section, we compute the expectation and variance of the throughput timesthrough the job shop. In general, the arrival and production processes are non-Markovian processes. This implies that standard queueing theory, e.g., on productform networks, cannot be used to find performance measures. Instead, we use approx-imative techniques that were developed by Whitt (1983) to analyze general openqueueing networks.

Interaction between different work centers

In this first step, we analyze the interaction between the different work centers in thejob shop production system. Due to the conservation of flow property, the expectedinterarrival time of production batches to each work center j in the routing of productk is given by:

E[

ABjk

]= E

[AB

0k

]

Next, the expected aggregate batch interarrival time of production orders to work

center j can be computed from:(

E[

ABj

])−1 =K∑

k=1

(E

[AB

jk

])−1.

The expected aggregate batch production time at a work center is given by:

E[

P Bj

]=

K∑

k=1

E[

ABj

]

E[

ABjk

] E[

P Bjk

]

The scv of the aggregate production time is given by:

c2[

P Bj

]=

E[

ABj

]

E2[

P Bj

]K∑

k=1

⎡

⎣E2

[P B

jk

]

E[

ABjk

](

c2[

P Bjk

]+ 1

)⎤

⎦− 1

The utilization of work center j is:

ρ j =E

[P B

j

]

E[

ABj

]

In the queueing network, the arrival process to a work center is constituted by arrivalsof new batches and by the departure process of batches leaving the previous work cen-ter in the routing of a product. Therefore, the scv of the interarrival times of batchesof product k at machine i is given by:

242


c2[

ABik

]= c2

[AB

k

]rk(0, i)+

M∑

j=1

c2[

DBjk

]rk( j, i)

In this expression, c2[

DBjk

]is the scv of the departure process of batches of product

k leaving work center j , rk(0, i) is a 0/1 variable that equals 1 if work center i is thefirst work center in the routing of product k and rk( j, i) is a 0/1 variable that equals1 if work center i is the successor of work center j in the routing of product k. Whitt(1994) presents an approximation:

c2[

DBjk

]≈ ρ2

jkc2[

P Bjk

]+ f jk

∑

l �=k

ρ2jl f −1

jl

(c2

[AB

jl

]+ c2

[P B

jl

])+

(1− 2ρ jkρ j + ρ2

jk

)c2

[AB

jk

]

where

f jk = E[A j

]

E[A jk

] ρ jk =E

[P B

jk

]

E[

ABjk

]

By combining these expressions, we get a system of linear equations. By solving thislinear system, we obtain the scv of interarrival times for all products and work centers.After this, an approximation for the scv of the aggregate arrival process at work centerj can be obtained with:

c2[

ABj

]≈ w j

K∑k=1

f jkc2[

ABjk

]+1−w j . In this formula, w j is a weighting function:

w j =[1+ 4

(1− ρ j

)2(

n∗j − 1)]−1

with n∗j =[

K∑k=1

f 2jk

]−1

.

Performance measures for individual work centers

Now, the network of interrelated work centers is decomposed into individual workcenters. We obtain approximations for the expected waiting times E

[W j

], using an

adaptation of the Kraemer and Langenbach-Belz formula proposed by Whitt (1983):

E[W j

] ≈c2

[AB

j

]+ c2

[P B

j

]

2

ρ j

1− ρ jE

[P B

j

]g j

where

g j =

⎧⎪⎪⎨

⎪⎪⎩

exp

[− 2(1−ρ j)

3ρ j

(1−c2

[AB

j

])2

c2[

ABj

]+c2

[P B

j

]

], c2

[AB

j

]< 1

1, c2[

ABj

]≥ 1

An approximate expression for σ 2[W j

], the variance of the waiting times, is due to

Whitt (1983). This approximation is omitted for reasons of brevity. The expectationand variance of the throughput time of a production batch of product k at work centerj can be approximately computed as:

243


E[Tjk

] ≈ E[W j

]+ E[

P Bjk

]

σ 2[Tjk] ≈ σ 2[W j

]+ σ 2[

P Bjk

]

Performance measures for the combined network

We approximate the throughput times for the complete production system by consid-ering every work center to be independent of the others. Then, the expectation andvariance of the throughput times through the job shop are:

E[Tk] ≈M∑

i=0

M∑

j=1

E[Tjk

]rk(i, j) σ 2[Tk] ≈

M∑

i=0

M∑

j=1

σ 2[Tjk]rk(i, j).

The assumption that work centers behave independent of each other is not valid in thegeneral case since this only holds for product-form networks. However, it is commonthat queueing network analyzers make this assumption (Whitt 1983).

Reorder points and safety stocks

In this section, we compute the reorder points so that the target fill rate βk is satis-fied. The computation is based on standard inventory theory, see, e.g., Silver et al.(1998, Chapter 7, p. 253). First, we characterize the average demand during the order

throughput time: E[

X Tkk

]= E[Tk ]

E[Ak ] .

Van Nyen (2005) derives an approximation for the variance of the demand duringthe order throughput time for the case of Poisson demand, using approximative resultsfrom renewal theory presented in De Kok (1991):

σ 2[

X Tkk

]= σ 2[Tk]

(E[Ak])2 +E[Tk]

E[Ak]

Now we can fit a distribution function on the two moments of the demand during theorder throughput time. In this research, we use the normal distribution to compute the

reorder points. The reorder point can be determined by: bk = E[

X Tkk

]+ ssk . The

safety stock can be computed as ssk = zkσ[

X Tkk

]. In this formula, zk is the so-called

safety factor. Silver et al. (1998, p. 736) present an accurate approximation methodfor zk for a given target fill rate βk .

Optimization of objective functions

The approximate analytical model presented here can be used to compute the value ofthe objective functions for a given vector of batch sizes �. Several search algorithmscould be used to optimize the objective functions. In our research, we use a search

244


algorithm that modifies the batch size Qk of one product at a time. The algorithmsearches in a single direction until no further improvement is possible, while the val-ues of the other batch sizes are kept fixed. Then, the batch size of another product ischanged until no further improvement is possible. This is repeated until there is noproduct for which a further improvement is possible. This final solution cannot beimproved in any direction and is the (local) optimum �∗. The search procedure hasbeen extensively tested and its performance proved to be satisfactory. See Van Nyen(2005) for numerical results on the tests.

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246

Vendor-managed inventory and the effect of channelpower

Bogdan C. Bichescu · Michael J. Fry

Abstract We analyze decentralized supply chains that follow general continuousreview (Q, R) inventory policies subject to vendor-managed inventory agreementswhere the supplier chooses the order quantity Q, and the retailer chooses the reor-der point R. Within the VMI scenario, we explore the effect of divisions of channelpower on supply chain and individual agent performance by examining different gametheoretic models. Optimal policies and analytical results, including existence and uni-queness proofs for equilibrium solutions under VMI, are derived. Numerical resultsare provided to compare the effectiveness of VMI and to analyze different channelpower relationships under a variety of environmental conditions. We find that VMIcan result in considerable supply chain savings over traditional relationships and thatthe relative division of channel power can significantly effect the performance of VMI.Interestingly, we find that the greatest system benefits from VMI arise in asymmetricchannel power relationships, but that individual agents lack the incentive to assume aleadership role.

Keywords Inventory · Game theory · Vendor-managed inventory · Channel power

1 Introduction

Increasing competition and the rapid adoption of advanced information technologyhas prompted retailers and suppliers to reengineer their supply chains and examine

B. C. Bichescu (B)Department of Statistics, Operations and Management Science, College of Business Administration,The University of Tennessee, Knoxville, TN 37996, USAe-mail: [email protected]

M. J. FryDepartment of Quantitative Analysis and Operations Management, College of Business,University of Cincinnati, Cincinnati, OH 45221-0130, USA



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OR Spectrum (2009) 31:195–228DOI 10.1007/s00291-007-0102-7


B. C. Bichescu, M. J. Fry

collaborative supply chain efforts to reduce costs and improve efficiency. Retailers’sharing of point-of-sale (POS) data using electronic data interchange (EDI) systemshave become common practice. Vendor-managed inventory (VMI) has emerged inthis context as an initiative that takes the collaborative efforts beyond informationsharing and allows the supplier to exercise some amount of control on the actualinventory levels at the retailer. Under a typical VMI agreement, the supplier controlsthe order quantities delivered to the retailer, possibly subject to contractual limitationsspecifying minimum service level requirements, etc. (see Fry et al. 2001). Wal-Martand Procter & Gamble (P&G) represent one of the first large-scale successes of suchVMI agreements. Their partnership began in 1985 and significantly improved P&G’son-time deliveries to Wal-Mart while increasing inventory turns (Buzzel and Ortmeyer1995). VMI projects, including those implemented at Dillard Department Stores,JCPenney and Wal-Mart have shown sales increases of 20–25 and 30% inventoryturnover improvements (Buzzel and Ortmeyer 1995).

Our goal in this paper is to examine the effect of channel power in vendor-managedinventory agreements in a supply chain. Channel power refers to an agent’s ability tocontrol the decision making process; it can be a function of the agent’s relative size,market presence, customer loyalty, etc.

In this paper, we analyze a VMI agreement between a supplier who delivers a singleproduct to a retailer. The supply chain follows a continuous review (Q, R) inventorypolicy, according to which the retailer decides the reorder point R, and the supplierdetermines the replenishment quantity Q (a similar model is used in Corbett 2001). Inthis way, the supplier controls the delivery amount (Q) and the retailer retains somecontrol over service levels (R). The retailer incurs an inventory holding cost per item,per unit time. The supplier incurs an inventory holding cost of his own as well as a fixedreplenishment cost per order. Penalty costs from stockouts are split between supplierand retailer; this represents the situation where a retailer stockout leads to a loss ofcustomer goodwill (and possible lost future sales) at both the retailer and supplier.Production at the supplier occurs at a deterministic linear rate. In cases where thesupplier’s on-hand inventory is insufficient to cover the retailer’s order, we assumethat the supplier can outsource the shortfall from a third party. There exists a positivedelivery lead time and customer demand during the lead time is normally distributed.

The effect of channel power on VMI performance has not been previously exami-ned. In this paper, we analyze three distinct power relationships within VMI: a power-ful supplier, a powerful retailer and equally powerful supplier and retailer. Industryexamples exist for all three VMI power scenarios (e.g., Wal-Mart as a powerful retailer,Barilla SpA as a powerful supplier, etc.—see Buzzel and Ortmeyer 1995; Hammond1994). We wish to analyze what effect these different channel power relationships haveon channel performance under VMI, as well as examining the effect on the individualagent decisions and performance. We model channel power similar to Netessine andRudi (2004) as the ability of an agent to control the decision-making process in thesupply chain. Specifically, the more powerful firm moves first in a Stackelberg game.These scenarios are modeled using a game-theoretic approach, with the equal-powerscenario being analyzed as a simultaneous-decision game and the powerful agent(retailer or supplier) scenarios being analyzed as Stackelberg games. This representsan important contribution of our paper, as we provide one of the few explicit formu-

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Vendor-managed inventory and the effect of channel power

lations of channel power in the operations management literature, and the first, to ourknowledge, within the framework of VMI.

To better capture the overall benefits of the VMI agreement, we compare supplychain performance under a VMI contract with the performance achieved under acentralized supply chain and under a traditional retailer-managed inventory (RMI)supply chain. In a centralized supply chain, there exists a single decision maker whodecides on both reorder point and replenishment quantity, while under RMI the retaileralone determines both the reorder point and the replenishment quantity, but ignoresthe supplier’s cost function.

We present analytical results for optimal policies in each scenario, including exis-tence and uniqueness results for equilibrium solutions under VMI. We then perform anumerical study to compare supply chain performance and generate insights. We findthat VMI results in considerable savings over RMI scenarios, in general, and that therelative distribution of channel power can significantly affect the performance of VMI.We examine the effect of scenario parameters on both the overall performance of VMIand on the effect of channel power within VMI through a comprehensive, full-factorialexperiment. We also find that the lowest costs at the supply chain level result fromasymmetric channel power relationships, while individual agents lack incentive to takea leadership role. We discuss implications and possible remedies for this finding.

The remainder of this paper is organized as follows. To better motivate our analysis,the next section provides a review of the literature relevant to our work. We start ouranalysis in Sect. 3, where we formulate the centralized model. Then, in Sects. 4 and 5we present models for the VMI and the RMI scenarios. Section 6 presents analyticalmodels for the simultaneous and sequential gaming scenarios under VMI; Sect. 7presents numerical results. Finally, we present conclusions and managerial insights inSect. 8.

2 Literature review

Our current work studies the impact of channel power on the performance of a supplychain following a continuous review inventory policy under a general VMI agreement.Thus, our work relates to several existing research streams, including papers thatexamine VMI contracts, models for general (Q, R) inventory policies and works thatexplore the implications of channel power.

We formulate models that relate to general continuous review inventory models.Our approach closely follows the approximate cost formulation proposed by Hadleyand Whitin (1963) for a centralized system. This approach is amenable to analyticalsolutions of the agents’ best response functions, which are crucial for our purpose ofanalyzing the impact of channel power on performance. For this reason, we favor theapproximate model of Hadley and Whitin (1963) over the exact formulation proposedby Zheng (1992) for a centralized supply chain. For recent developments on continuousreview models, we direct the reader to Federgruen (1993), Hopp et al.(1997), Zipkin(2000) and Hill and Omar (2006). We build on these existing models by including thegame theoretic framework to account for channel power and extending the models toboth the retailer managed and VMI scenarios.

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VMI programs, pioneered by companies such as Wal-Mart, P&G, Campbell Soup(Clark and McKenney 1994) and Barilla SpA (Hammond 1994), etc. have becomepopular in practice due to continued advances in information technology and increasingsupply chain competition. Correspondingly, there are many recent academic papersthat examine vendor-managed inventory. Cachon and Fisher (1997) and Clark andHammond 1997) use empirical evidence to compare the benefits of VMI and infor-mation sharing. They conclude that most of the benefits of VMI could be achievedby information sharing alone. Narayanan and Raman (1998) formulate a model thatcompares RMI with a specific VMI agreement where the retailer basically rents spaceto the supplier. Aviv and Federgruen (1998) investigate the benefits of VMI in a supplychain with one supplier and N retailers under periodic review. The authors assumethat under VMI the timing and magnitude of shipments to the retailers is decided bythe supplier. They find that VMI is always more beneficial than information sharingalone. Bernstein and Federgruen (2003) study a VMI agreement where, similar to ourpaper, decision rights are split between retailer and supplier. However, Bernstein andFedergruen model a scenario where the supplier controls inventory replenishment andthe retailer determines the product price, whereas we assume price to be exogenousand we allow the supplier to determine replenishment quantity Q, and the retailerto choose service level z. Bernstein and Federgruen also assume that the suppliercovers all holding costs, i.e., a consignment-type agreement, where we assume thatthe supplier and retailer pay separate holding costs.

Fry et al. (2001) examine a specific type of VMI agreement called a (z, Z) contract,between a supplier and retailer. In this setting, the supplier controls the inventory reple-nishment policy at the retailer subject to limits on service level and maximum inventorychosen by the retailer (the z, Z quantities). The authors contrast the performance ofthe VMI contract with RMI and identify the scenarios where VMI performs best incomparison to RMI. Cachon (2001) examines coordination in two-echelon supplychain with one supplier and N retailers. The supplier and retailers follow continuousreview (Q, R) policies, and, as in this research, the total backorder penalty cost is splitbetween supplier and retailer. Under VMI, the supplier is responsible for choosing thepolicies at each of the retailers in the supply chain. Cachon shows that VMI achievesthe optimal solution only if the supplier and the retailers make fixed transfer paymentsto participate in the VMI agreement.

Nagarajan and Rajagopalan (2004) also compare the performance of RMI andVMI under continuous and periodic review policies. As part of the VMI agreement, theauthors explore various subsidizing schemes, e.g., the supplier subsidizes the retailer’spenalty and holding costs or the supplier subsidizes retailer’s holding cost and theretailer subsidizes supplier’s replenishment cost. These two parameter contracts areshown to coordinate the channel under certain conditions. Wu et al. (2005) propose an(α, h) VMI contract that coordinates a supply chain with one retailer and one supplier.According to this contract, the supplier acts as a Stackelberg leader, manages theretailer’s inventory and bears inventory carrying cost; the retailer decides the targetedsales at the start of the selling season. The contract variable α determines how salesand penalty costs are split between supplier and retailer; h indicates a holding costsubsidy paid by the retailer to the supplier for each unit left unsold at the end of theseason.

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In a related study, Corbett (2001) examines the impact of incentive conflicts andinformation asymmetry on performance in a two-player decentralized supply chain,which follows a continuous review (Q, R) policy. The author uses a principal-agentapproach to model scenarios where the principal, either the buyer or the supplier, actsas the leader in the supply chain, but lacks full information on the agent’s costs, whilethe agent follows and possesses full information. Corbett finds that in the absence of acentral planner with full information, no party can induce jointly optimal behavior forall agents in the supply chain without sacrificing his own profits. This conclusion isrelated to an important finding in our work, which shows that while the system prefersone of the agents to lead, neither party has sufficient incentive to exercise channelpower on their own.

Consignment arrangements are often closely related to VMI agreements. Underconsignment, the responsibility of inventory decision making is transferred to the sup-plier as in VMI, but ownership of goods is retained by the supplier until the momentof sale (see Bolen 1988; Narayanan and Raman 1998 could also be considered aconsignment-type contract). These contracts are popular in a variety of industries invarious countries (Valentini and Zavanella 2003). Due to their characteristics, suchcontracts are often appealing to and initiated by the retailers, who are typically repre-sented as the powerful players in the supply chain (e.g., see Wang et al. 2004). Thus,in addition to VMI, consignment agreements may represent a potential environmentwhere unequal splits of power are likely to exist.

An important contribution of our research is that it explores the impact of channelpower in a VMI agreement. The topic of channel power, defined as the ability of a firmto influence the intentions and actions of another firm (see Emerson 1962) has seensome examination in the social, political and marketing literature. However, to the bestof our knowledge, this topic has seen relatively little development in the operationsmanagement literature and, as Cachon (2003) notes, additional research is needed onthis issue. Cachon (2003) states that the use of a profit reservation level and first moverapproaches are possible ways to model power. Here, we adopt the latter approach anduse a game-theoretic framework to characterize players’ actions. For empirical worksthat study the issue of power within an operations management framework, we directthe reader to Maloni and Benton (2000) and Benton and Maloni (2004). For papersin the marketing literature that study the issue of power defined as the proportion ofchannel profits that accrue to each of the channel members and use a game theoreticapproach to determine price, we direct the reader to Choi (1991) and Kadiyali et al.(2000).

Netessine and Rudi (2004) compare a vertically integrated supply chain, a decentra-lized supply chain and a drop-shipping supply chain. Under a drop-shipping contract,the wholesaler, or the manufacturer, ships the product directly to the end customer;thus, the retailer is relieved of any inventory responsibility and inventory-related costs.Similar to the models presented here, various channel power structures are analyzedfor the drop-shipping model: a powerful wholesaler, a powerful retailer and equallypowerful wholesaler and retailer. The authors find that drop-shipping is most attractivewhen the supply chain has a powerful retailer and least attractive when channel poweris equally split. The implications of channel power are also explored by Bichescu andFry (2007) in a supply chain setting where order quantity and shipping frequency are

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decision variables and decision rights are split between a retailer and a supplier. Usinggame-theoretic concepts, the authors find that a supplier-dominated channel not onlyoutperforms an equal split of power, but also approaches closely the performance ofthe integrated channel. For a more comprehensive discussion on game theory appliedto supply chain models, we direct the reader to the excellent review by Cachon andNetessine (2004).

3 The centralized scenario

In a centralized scenario a single decision maker makes all relevant decisions to mini-mize total supply chain cost. Here we formulate a continuous review model for thecentralized scenario to find the optimal reorder point R, and optimal order quantityQ, that minimize overall system costs. We assume that customer demand follows anormal distribution, with mean µ and standard deviation σ . There exists a positivedelivery lead time L , between supplier and retailer. Thus, the reorder point is definedby µL + zσ

√L . Note that we use the security level, z, to represent service level and

to serve as a proxy for the reorder point. For analytical tractability we will generallyrequire z ≥ 0, but we will later discuss the case of z unrestricted.

Per-unit holding costs for on-hand inventory are incurred at both the retailer andat the supplier, and are denoted as h R and hS , respectively. The retailer experiencesa linear inventory depletion rate (which in expectation is µ) and the supplier has adeterministic linear production rate (for tractability of results this is set at µ). Cus-tomer demand that cannot be satisfied from the available inventory at the retailer isbackordered, which results in a penalty cost of p per unit. We assume that when, dueto the temporal uncertainty of retailer orders, the supplier has not produced enoughto cover the retailer‘s order of Q units, the supplier can outsource the shortfall froman uncapacitated third party at premium unit price b with negligible lead time. Thus,the retailer always receives her orders in full.1 Each order shipped from the supplierto the retailer results in a fixed order cost of S. Thus, the elements of the expectedcentralized cost function per time unit corresponding to our model are detailed below.

• Holding costs (h R + hS)Q2 + h Rzσ

√L;

• Penalty costs µQ pσ√

L�(z), where �(z) = {φ(z)− z[1−�(z)]}; φ(·) and �(·)represent the standard normal pdf and cdf, respectively;

• Shipping costs µQ S.

• Shortfall costs b µQ E

[(Q − T µ)+

], where T represents the random variable for

time between orders at the supplier.

Lemma 1 The shortfall cost per period, b µQ E

[(Q − T µ)+

], can be expressed as

bµK, where K > 0 is a scalar value, independent of Q.

Proof See Appendix B. ��

1 This assumption is common in both the related literature and in practice. Fry et al. (2001), Lee et al.(2000), and Gavirneni et al. (1999) contain similar assumptions. This reflects the reality in many scenarioswhere supplier will go to extraordinary lengths to insure full delivery to retailer such as in automotiveindustry, electronics industry, etc.

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Lemma 1 proves that the expected shortfall at the supplier is invariant to our deci-sion variables, Q and z. Thus, no matter what ordering policies are followed, theexpected shortfall costs at the supplier are determined only by the demand distributionparameters, µ and σ . Therefore, the expected centralized cost function is

KC (z, Q) = µ

Qpσ√

L�(z)+ µ

QS + Q

2(hS + h R)+ h Rzσ

√L + bµK. (1)

Next, we explore the analytical properties of the centralized cost function. We useLemma 2 in the proofs of Propositions 1 and 4.

Lemma 2 θ(z) = 2φ(z)�(z)− [1−�(z)]2 ≥ 0.

Proof Note that θ(0) > 0 and limz→∞ θ(z) = 0. Further,

∂θ(z)

∂z= −2zφ(z)�(z)− 2φ(z)[1−�(z)] + 2φ(z)[1−�(z)]= −2zφ(z)�(z) ≤ 0, ∀z ≥ 0.

Thus, θ(z) is a nonincreasing function bounded below by 0 for z ∈ [0,∞). Therefore,θ(z) ≥ 0, ∀z ≥ 0. ��

Proposition 1 KC (z, Q) is convex in z and Q and jointly convex.

Proof In order to build the Hessian matrix, note that

∂KC (z, Q)

∂z= µ

Qpσ√

L [−1+�(z)]+ h Rσ√

L,

∂2 KC (z, Q)

∂z2 = µpσ√

Lφ(z)

Q≥ 0.

∂KC (z, Q)

∂ Q= − µ

Q2 pσ√

L�(z)− µ

Q2 S + hS + h R

2,

∂2 KC (z, Q)

∂ Q2 = 2µ[pσ√

L�(z)+ S]Q3 ≥ 0.

∂2 KC (z, Q)

∂ Q∂z= µ

Q2 pσ√

L [1−�(z)] .

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Then,

∣∣∣∣∣∣

∂2 KC (z,Q)

∂z2∂2 KC (z,Q)

∂ Q∂z

∂2 KC (z,Q)∂ Q∂z

∂2 KC (z,Q)

∂ Q2

∣∣∣∣∣∣=

∣∣∣∣∣∣

µpσ√

Lφ(z)Q

µpσ√

L[1−�(z)]Q2

µpσ√

L[1−�(z)]Q2

2µ[pσ√

L�(z)+S]Q3

∣∣∣∣∣∣

= µ2

Q4

{p2σ 2L

{2φ(z)�(z)− [1−�(z)]2

}

+ 2pSσ√

Lφ(z)}

= µ2

Q4

{p2σ 2Lθ(z)+ 2pSσ

√Lφ(z)

}≥ 0,

∀z ≥ 0 from Lemma 2.

��Corollary 1 In the centralized scenario, the optimal service level is z∗C (Q) =�−1

(µp−h R Q

µp

)and the optimal order quantity is Q∗C (z) =

√2µS+2µpσ

√L�(z)

h R+hS. In

addition, Q∗C (z) monotonically decreases in service level z, and z∗C (Q) decreases inorder quantity Q.

Proof The expressions for z∗C (Q) and Q∗C (z) are obtained by solving ∂KC (z,Q)∂z = 0

and ∂KC (z,Q)∂ Q = 0 for z and Q, respectively. The interdependencies between the two

decision variables are found using the Implicit Function Theorem (IFT) as follows:

∂ Q∗C (z)

∂z= −

∂2 KC (z,Q)∂ Q∂z

∂2 KC (z,Q)

∂ Q2

= −µpσ√

L[1−�(z)]Q2 × Q3

2µ[pσ√

L�(z)+ S]

= −Qpσ√

L[1−�(z)]2[pσ√

L�(z)+ S] ≤ 0 and

∂z∗C (Q)

∂ Q= −

∂2 KC (z,Q)∂ Q∂z

∂2 KC (z,Q)

∂z2

= −µpσ√

L[1−�(z)]Q2 × Q

µpσ√

Lφ(z)

= −1−�(z)

Qφ(z)≤ 0.

��

4 The VMI agreement

Here we model a VMI agreement, according to which the retailer is responsible forchoosing the optimal service level z, and the supplier chooses the optimal order quan-tity, Q. We assume that penalty costs from out-of-stocks are split between the retailerand supplier through a parameter α ∈ [0, 1] so that the retailer’s share of the penalty

254


cost per unit is αp and the supplier’s share is (1−α)p. This represents the reality wherea stock-out at the retailer results in a penalty to both the retailer and the supplier—they both incur a loss of goodwill (see Cachon and Zipkin 1999, Nagarajan andRajagopalan 2004, among others for similar modeling assumptions). Note that accor-ding to our VMI setting, the retailer does not transfer his inventory decision rightsentirely to the supplier, but retains a certain degree of autonomy by reserving theright to choose the service level z. This is different from many previous VMI works,which assume that the supplier has full control of the retailer’s inventory policy (e.g.,Aviv and Federgruen 1998; Cachon 2001). Our model is reflective of reality where theretailer retains some ability to control customer service levels even under VMI (seeFry et al. 2001 and references therein for supporting evidence) and is closely relatedto the split of decision rights modeled in Corbett (2001).

4.1 Retailer’s costs under VMI

In a decentralized supply chain, the retailer and supplier incur separate costs and, thus,each faces a different objective function. According to the split of decision rights des-

cribed above, the retailer’s costs are represented by holding costs(

h R

(Q2 + zσ

√L))

and penalty costs(

µQ αpσ

√L�(z)

). Thus, the retailer’s cost function is

K RVMI(z) =

µ

Qαpσ√

L�(z)+ h R

(Q

2+ zσ√

L

). (2)

Proposition 2 demonstrates that there is an optimal z (z∗VMI(Q)) for the retailer tochoose when the supplier sets the order quantity, Q.

Proposition 2 K RVMI(z) is convex in z and is minimized at z = z∗VMI(Q).

Proof We have

∂K RVMI(z)

∂z= µ

Qαpσ√

L {−zφ(z)− 1+�(z)+ zφ(z)} + h Rσ√

L

= µ

Qαpσ√

L [−1+�(z)]+ h Rσ√

L and

∂2 K RVMI(z)

∂z2 = µ

Qαpσ√

Lφ(z) ≥ 0.

Thus, K RVMI(z) is convex and there exists z = z∗VMI(Q) that minimizes K R

VMI(z).

Solving∂K R

VMI(z)∂z = 0,

z∗VMI(Q) = �−1(

µαp − h R Q

µαp

). (3)

��

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Observation 1 K RVMI(z) is convex in Q and is increasing in Q when

Q ≥√

2µαpσ√

L�(z)h R

and decreasing otherwise.

The above result is easily proven from the first and second order conditions for K RVMI(z)

with respect to Q.

4.2 Supplier’s costs under VMI

The costs incurred by the supplier under VMI are holding costs(

Q2 hS

), shipping costs

(µQ S

), penalty costs

(µQ (1− α)pσ

√L�(z)

), and shortfall costs (bµK). Thus, the

supplier’s cost function is defined as

K SVMI(Q) = µ

Q(1− α)pσ

√L�(z)+ µ

QS + Q

2hS + bµK. (4)

Proposition 3 K SVMI(Q) is convex in Q and is minimized at Q = Q∗VMI(z).

Proof We have

∂K SVMI(Q)

∂ Q= − µ

Q2 (1− α)pσ√

L�(z)− µ

Q2 S + hS

2and

∂2 K SVMI(Q)

∂ Q2 = 2µ

Q3 (1− α)pσ√

L�(z)+ 2µ

Q3 S ≥ 0.

Setting∂K S

VMI(Q)

∂ Q = 0 and solving for Q,

Q∗VMI(z) =√

2µS + 2µ(1− α)pσ√

L�(z)

hS. (5)

��From (5), Q∗VMI > QE O Q for α < 1 and z < ∞, where QE O Q =

√2µShS

isthe standard economic order quantity (first developed by Harris 1915). An intuitiveexplanation for this, also noted by Hadley and Whitin (1963) for the centralized case,is that expected backorders depend on z but are independent of Q. Therefore, for

α < 1 the supplier will order an additional√

2µS+2µ(1−α)pσ√

L�(z)−√2µS√hS

units dueto uncertainty in demand.

Observation 2 K SVMI(Q) is decreasing in z.

Proof It is straightforward to note that

∂K SVMI(Q)

∂z= −µ

Q(1− α)pσ

√L [1−�(z)] ≤ 0.

��

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5 Optimal policies under RMI

Here we analyze a traditional RMI scenario where the retailer decides both the orderquantity Q, and the service level z. Such a scenario represents a situation where theretailer holds extensive channel power such that she can control all decisions relatedto order delivery. Therefore, the retailer’s costs under RMI are composed of penaltyand holding costs and are identical to Eq. (2), except that the costs are now a functionof z and Q

(K R

RMI(z, Q))

as both are decision variables for the retailer. Proposition 4and Corollary 2 detail the retailer’s optimal policy under RMI.

Proposition 4 K RRMI(z, Q) is convex in z and Q and jointly convex.

Proof Note that

∂K RRMI(z, Q)

∂z= µαpσ

√L [−1+�(z)]

Q+ h Rσ

√L

∂2 K RRMI(z, Q)

∂z2 = µαpσ√

Lφ(z)

Q≥ 0,

∂K RRMI(z, Q)

∂ Q= −µαpσ

√L�(z)

Q2 + h R

2,

∂2 K RRMI(z, Q)

∂ Q2 = 2µαpσ√

L�(z)

Q3 ≥ 0 and

∂2 K RRMI(z, Q)

∂ Q∂z= µαpσ

√L [1−�(z)]

Q2 .

Then,

∣∣∣∣∣∣

∂2 K RRMI(z,Q)

∂z2∂2 K R

RMI(z,Q)

∂ Q∂z∂2 K R

RMI(z,Q)

∂ Q∂z∂2 K R

RMI(z,Q)

∂ Q2

∣∣∣∣∣∣=

∣∣∣∣∣∣

µαpσ√

Lφ(z)Q

µαpσ√

L[1−�(z)]Q2

µαpσ√

L[1−�(z)]Q2

2µαpσ√

L�(z)Q3

∣∣∣∣∣∣

= µ2α2 p2σ 2 L

Q4

{2φ(z)�(z)− [1−�(z)]2

}

= µ2α2 p2σ 2 L

Q4 θ(z) ≥ 0, ∀z ≥ 0 from Lemma 2.

��

Corollary 2 In the RMI scenario, the optimal service level chosen by the retai-

ler is z∗RMI(Q) = �−1(

µαp−h R Qµαp

)and the optimal order quantity is Q∗RMI(z) =√

2µαpσ√

L�(z)h R

. In addition, Q∗RMI(z) monotonically decreases in service level, z, andz∗RMI(Q) decreases in order quantity, Q.

257


Proof The optimal solution under RMI is obtained by solving∂K R

RMI(z,Q)

∂z = 0 and∂K R

RMI(z,Q)

∂ Q = 0 for z and Q, respectively. Using the IFT,

∂ Q∗RMI(z)

∂z= −

∂2 K RRMI

∂z∂ Q

∂2 K RRMI

∂ Q2

= −µαpσ√

L [1−�(z)]

Q2 × Q3

2µαpσ√

L�(z)

= −Q[1−�(z)]2

≤ 0 and

∂z∗RMI(Q)

∂ Q= −

∂2 K RRMI

∂ Q∂z

∂2 K RRMI

∂z2

= −µαpσ√

L [1−�(z)]

Q2 × Q

µαpσ√

Lφ(z)

= −1−�(z)

Qφ(z)≤ 0.

��The supplier’s costs under RMI are completely determined by the retailer’s choice

of Q∗RMI and are given by K SRMI(Q∗RMI), where

K SRMI(Q∗RMI) =

µ

Q∗RMI(1− α)pσ

√L�(z)+ µ

Q∗RMIS + Q∗RMI

2hS + bµK.

Note that the optimal policy under RMI can be obtained using an iterative schemethat cycles between z∗RMI(Q) and Q∗RMI(z) until convergence is achieved.

6 Effect of channel power under VMI

In this section, we explore the implications of the distribution of channel power bet-ween supply chain agents. We consider three distinct channel power relationships:a powerful retailer, a powerful supplier and equally powerful retailer and supplier.Recall that our definition of channel power relates to an agent’s ability to controlthe decision-making process in the supply chain. Thus, the powerful agent scenariosassume that one of the agents, either the supplier or the retailer, has greater bargainingpower and therefore can control the decision making process by making his decisionfirst; the other agent follows and makes her decision subject to the leader’s choice.These scenarios are modeled using a Stackelberg game, where the powerful playeracts as the Stackelberg leader. The equally powerful agent scenario assumes that nei-ther agent has sufficient channel power to control supply chain decisions. In theseconditions, the supplier and the retailer make the inventory replenishment and servicelevel decisions simultaneously. This scenario is modeled using a simultaneous gamewhere the solution is characterized by a Nash equilibrium. Full information is assu-med in all scenarios. This modeling approach has been used by others in the supplychain research literature to describe channel power, e.g., Choi (1991) and Netessineand Rudi (2004).

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6.1 The case of equally powerful retailer and supplier

Here we assume that the retailer and the supplier simultaneously choose their optimalstrategies under full information to represent a shared power scenario. We first identifywhen an equilibrium exists and then when the equilibrium is unique. Note first that theretailer’s and supplier’s best-response functions are defined by relations (3) and (5),respectively. From (3), define Qinv(z) as

Qinv(z) = µαp[1−�(z)]h R

. (6)

From (5), and because z ≥ 0, the feasible range of order quantities is limited toQ ∈ (QE O Q, Q∗VMI(0)). Furthermore, from (6) and z ≥ 0, Q ∈ (0, Qinv(0)). BecauseQinv(z) and Q∗VMI(z) are monotonically decreasing in z, if Qinv(0) < Q∗VMI(0), theretailer’s and supplier’s response functions do not intersect, therefore, no equilibriumexists in the feasible domain (see Fig. 1). However, if Qinv(0) ≥ Q∗VMI(0), thereexists a unique interior Nash Equilibrium (see Fig. 2). Proposition 5 formally statesthis result.

Proposition 5 For an equally powerful retailer and supplier, there exists a uniqueNash equilibrium in the feasible domain if Qinv(0) ≥ Q∗VMI(0).

Proof Note that limz→∞ Qinv(z) = 0 < limz→∞ Q∗VMI(z) = QE O Q . BecauseQinv(z) and Q∗VMI(z) are both strictly decreasing functions, it follows that if Qinv(0) ≥Q∗VMI(0) then Qinv(z) and Q∗VMI(z) must intersect exactly once. ��

It can be shown that when z < 0 and Qinv(0) < Q∗VMI(0), the two responsefunctions represented by Qinv(z) and Q∗VMI(z) may not intersect or may intersecttwice, depending on the values of the environmental parameters. However, due to thecomplexity of these response functions, we cannot obtain closed form conditions thatprecisely characterize the existence of an equilibrium. Therefore, we require z ≥ 0

Fig. 1 No Nash equilibrium in the decentralized scenario µ = 20, σ = 12, p = 10, hS = 0.02, h R = 0.1,S = 100, α = 0.5, L = 10

259


Fig. 2 Existence of a Nash equilibrium in the decentralized scenario µ = 20, σ = 12, p = 10, hS = 0.02,h R = 0.05, S = 100, α = 0.6, L = 10

and Qinv(0) ≥ Q∗VMI(0) for mathematical tractability in our analysis. We expect thatz ≥ 0 will hold in almost all practical scenarios; however, we note that, as suggestedin Zipkin (2000), when the unit backorder cost is sufficiently close to the unit holdingcost, a company may find it efficient to hold an amount of inventory that is less thanthe mean demand during the leadtime.

6.2 The case of the powerful retailer

We model the scenario of a powerful retailer by allowing the retailer to act as aStackelberg leader and to choose z first. Assuming full information, the retailer willtake into account the supplier’s optimal strategy when choosing z. Thus, under thisscenario, the retailer will seek to minimize (2), where Q = Q∗VMI(z). The followinganalytical results describe the properties of the Stackelberg equilibrium.

Proposition 6 When the retailer is the Stackelberg leader, there exists a uniqueStackelberg equilibrium, if S > 0.1575(1− α)pσ

√L.

Proof See Appendix B. ��Note that for reasonable values of penalty cost p, demand standard deviation σ , and

lead time L , the condition stated in Proposition 6 holds, especially as α approaches1 and thus (1 − α), approaches 0. In all our numerical trials the condition fromProposition 6 is satisfied.

6.3 The case of the powerful supplier

In this section, we assume that the supplier is the Stackelberg leader and, thus, chooseshis optimal order quantity Q, first, knowing the retailer’s response function is repre-sented by (3). The supplier’s optimal strategy is obtained by minimizing (4), whenz = z∗VMI(Q).

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Proposition 7 When the supplier is the Stackelberg leader, there exists a uniqueStackelberg equilibrium.

Proof See Appendix B. ��

7 Numerical results

In this section, we conduct a numerical study that provides insights on the performanceof VMI and how channel power affects supply chain performance under VMI. Weexamine the effect of various environmental factors on VMI performance and the effectof channel power through a comprehensive full-factorial experiment. The factors arecoefficient of variation of customer demand (CV), ratio of holding cost at the retailerto holding cost at the supplier (HR), shipping cost (SC), unit penalty cost for stockouts(P), shipping lead time (L) and retailer’s share of the penalty cost (α). Each of thesefactors is assumed to take low, medium and high values as shown in Table 1 below andare representative of practical values in the United States electronics and automotiveindustries. The mean customer demand per period µ, and the unit holding cost perperiod at the supplier hS , are held constant at 20 and 0.02, respectively, throughoutthis study. Note that a consignment-type system could be considered by setting HR≤ 1.0 (i.e., allowing for hS ≥ h R). We have considered such situations in extendednumerical results and have found no differences in insights; thus, we present onlycases where HR > 1.0 here. Further, we assume that outsourcing cost b, is negligiblesince the shortfall cost term, bµK is independent of both z and Q and, thus, invariantto our decision making.

To capture how the performance of the centralized, VMI and RMI scenarios com-pare, we use two measures, Decentralized Performance Gap (ϒ) and VMI Cost Reduc-tion Factor (), where

ϒ = VMI system cost− centralized system cost

VMI system costand

= RMI system cost− VMI system cost

RMI system cost.

Table 1 Data for theexperimental design

Factor Level

Low Medium High

CV 0.10 0.30 0.60

HR 1.50 2.50 5.00

P 10 15 20

SC 50 75 100

L 3 5 10

α 0.60 0.75 0.90

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Thus, ϒ shows the performance gap between the centralized and VMI scenarios,while measures the cost savings of moving from RMI to VMI. Note that our defi-nition of ϒ is similar to the concept of price of anarchy, which is used in economicsand computer science to measure the extent to which uncoordinated selfish decisionsdegrade the performance of a system compared to the global optimum.2 The effectsof channel power are specifically measured by the Stackelberg Cost Improvementindicator, which captures, within the VMI model, the percentage cost reduction ofthe powerful retailer (δ) and powerful supplier (�) scenarios over the equal powerscenario. This cost improvement is measured at both the agent and system level. Spe-cifically, let δR , δS and δ, respectively, represent the retailer’s, supplier’s and supplychain’s percentage cost improvements when moving from an equal power scenario to

a retailer-led VMI scenario

(e.g.,δR = K R

VMI(Nash)−K RVMI(Stackelberg, Retailer−led)

K RVMI(Nash)

); let

�R , �S and � represent similar cost improvements for a supplier-led supply chain(e.g., �R = K R

VMI(Nash)−K RVMI(Stackelberg, Supplier−led)

K RVMI(Nash)

).

7.1 Performance comparison of centralized, VMI and RMI scenarios

Tables 4 and 5 in Appendix A display the results of our numerical study concerninghow the performance of the centralized and decentralized scenarios compare. Thecolumns showing the Decentralized Performance Gap ϒ , and the VMI Cost ReductionFactor , in Table 4 illustrate the relative performance of the centralized, VMI andRMI scenarios. System costs are computed for each of the three different VMI channelpower scenarios, thus separate ϒ and values are reported for each scenario in Table 4.Table 5 contains results on the optimal policies, represented by stocking factor z, andorder quantity Q, and the corresponding supply chain cost under the centralized, VMIand RMI scenarios. To build a better understanding of the performance differences,Table 4 displays the average, minimum and maximum cost improvement values ateach level of the six environmental factors. This allows for a clearer identificationof the range of savings offered by moving from RMI to VMI () and from VMI tocentralized control (ϒ).

The results in Table 4 show that the performance penalty resulting from decentrali-zed decision-making is significantly influenced by environmental factors, the perfor-mance gap ϒ averaging 16.60% over all channel power scenarios and ranging froma minimum of 6.43% to a maximum of 31.85%. Further, note that this performancegap is consistently lower in the asymmetric power cases and lowest when the supplieracts as the Stackelberg leader. Also, the performance gap in the supplier-led cases isconsistently lower then in the retailer-led cases (see Sect. 7.2 for a detailed discussionof the impact of channel power on performance). In addition, according to Table 5,the centralized scenario achieves customer service levels that outperform VMI in allcases; however, these values are lower than those obtained in the RMI scenario. This

2 The price of anarchy is defined as the ratio of the Nash equilibrium solution to the system optimum. Foradditional details see Papadimitriou (2001) and references therein.

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is attributable to the position of absolute power that the retailer enjoys in the RMIcase, such that the retailer can obtain high service levels that minimize her costs whileignoring supplier’s costs.

Table 4 shows that VMI leads to significant savings over RMI, regardless of thechannel power relationship. The VMI Cost Reduction Factor , averages 86.02%over all scenarios. However, ranges from a minimum of 29.60% to a maximumof 98.65%, indicating that the magnitude of savings offered from moving from RMIto VMI are also highly dependent on scenario parameters. Table 5 indicates that theretailer’s power to completely control the decision-making process under RMI leadsto much smaller order quantities (Q) than under any of the VMI scenarios or undercentralized control. Because the retailer has full power under RMI and disregards anycosts incurred by the supplier, she sets Q quite low resulting in high shipping costsfor the supplier and, hence, excessive system costs. These findings are consistent withexisting evidence in the literature, e.g., Fry et al. (2001), Nagarajan and Rajagopalan(2004), which shows that a well-designed VMI contract outperforms traditional RMIin many realistic scenarios.

Table 5 also shows that VMI scenarios tend to lead to order quantity values thatare higher than a centralized solution. Again, this reflects the greater power of thesupplier in all VMI scenarios (compared to RMI or to centralized control) since thesupplier controls the order quantity solution and he prefers larger Q values to reduceshipping costs. However, the split of channel power between retailer and supplier inVMI prevents the extreme solutions seen under RMI so that VMI leads to considerablesystem cost savings. We also note that Q values remain fairly constant across thechannel power scenarios under VMI. This is because the supplier’s choice of Q issomewhat insensitive to changes in z under VMI.3 Thus, the order quantity changesvery little across the different power scenarios under VMI, but there are significantdifferences in the safety stock parameter z.

In other words, inventory policies will have similar order quantities across the dif-ferent VMI power scenarios, but the policies will differ mainly in customer servicelevels through different reorder points. Not surprisingly, the highest customer servicelevels (z) are offered when the retailer has greater channel power; however, these ser-vice levels are still less than those preferred in the centralized scenario. As mentionedpreviously, both ϒ and vary significantly with the environmental factors. To betterunderstand when the penalty from decentralized decision-making is largest and whenmoving from a traditional RMI setting to a VMI agreement would provide the highestbenefit, we perform a full-factorial experimental design whose main results regardingϒ and are summarized in Table 2, where strong positive (negative) direct effectsare represented by “↑” (“↓”), weak positive (negative) direct effects are representedby “↗” (“↘”) and cases where a significant direct effect could not be identified arerepresented by “−”.

3 This can be verified by examining∂ Q∗C (z)

∂z in the proof of Corollary 1 and noticing that as �(z) → 1,

∂ Q∗C (z)∂z → 0. Because z values in equilibrium are relatively large (z > 1.2 in all VMI scenarios in Table 5,

hence �(z) > 0.88), we expect∂ Q∗C (z)

∂z to be near zero.

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Table 2 shows that holding cost ratio HR, demand uncertainty CV and deliverylead time L, are the factors that have the strongest effect on the performance gapbetween centralized and VMI scenarios. When holding costs at the retailer increase, thecentralized system responds with lower order quantities and more frequent shipmentsto the retailer, while the VMI scenarios adopt an opposite strategy which leads tohigher holding costs and, thus, lower performance. Further, the larger order quantitiescharacteristic to the VMI scenarios allow for a better response under VMI to increasesin demand uncertainty relative to the centralized scenario. Thus, we find that theperformance of the VMI scenarios is close to first-best solutions represented by thecentralized case. The performance gap is largest when the holding cost ratio, shippingcost and the split of penalty costs α, are high, and demand uncertainty and delivery leadtime are low. Under such environmental conditions, a coordination contract would bemost beneficial.

Figure 3 displays the significant direct effects on VMI Cost Reduction Factor .According to Fig. 3, demand uncertainty CV, delivery lead time L, and holding costratioHR, are factors with a strong negative impact on the cost savings of VMI. As thesefactors increase RMI system costs decrease while VMI system costs (in all channelpower scenarios) increase. Thus, the VMI Cost Reduction Factor , decreases. Ourone-way analysis of variance (ANOVA) results show that the unit penalty cost forstockouts P, and the retailer’s share of the penalty cost α, are not significant at the5% level. However, while not explicitly shown here, we can report that multiple-way

Table 2 Direct effects on ϒFactor Decentralized performance gap (ϒ)

Equal power Retailer—Led Supplier—Led

CV ↓ ↓ ↓HR ↑ ↑ ↑P ↘ − −SC ↗ ↗ ↗L ↘ ↓ ↓α ↓ ↘ −

70%

75%

80%

85%

90%

95%

100%

Low Med High

Factor Level

Perc

enta

ge C

hang

e in

CV HR SC LFactors

Fig. 3 Significant direct effects on

264


ANOVA shows significant interactions between P and CV and HR and between α

and CV and HR. Thus, these cross-effects may mask the direct effects of P and α on, explaining why P and α are non-significant. Note also that higher penalty costsresult in higher system costs under all scenarios. Thus, we notice that system costsincrease proportionally in the VMI and RMI scenarios as P increases, representingan alternative explanation as to why P has little influence on the overall variance of. Increasing shipping cost SC, clearly leads to higher RMI system costs because theretailer is completely insensitive to shipping costs, thus Q and z remain unchanged asSC increases, resulting in higher supplier, and system, costs. Therefore, the effect ofhigher shipping costs is stronger under RMI than in the VMI scenarios, so increaseswith SC.

In conclusion, we find that VMI consistently leads to savings over RMI, regardlessof channel power allocations; however, relative cost savings are highest when demanduncertainty, holding cost ratio and delivery lead time are low and shipping cost is high.Here we must point out that while we believe our RMI model to be an appropriateapproximation of system behavior for comparison purposes, the actual cost valuesincurred under RMI in our model are most likely worst-case estimates. In our modelwe assume that the retailer acts with complete disregard to the supplier’s costs underRMI. Such a strategy may be optimal for the short-term, but obviously would notbe sustainable over the long-term. In reality, a retailer may implicitly increase orderquantities to help suppliers offset excess shipping costs (which is why, in reality, manysuppliers require minimum shipping quantities, which are also not part of our model).

7.2 Channel power analysis

Our main results regarding the effect of channel power on supply chain performanceunder VMI are captured in Table 6 in Appendix A. All values in Table 6 are positive,indicating that unequal channel power VMI scenarios are always at least as efficientas the equally powerful VMI scenario both at the supply chain and individual agentlevels. This result is consistent with findings in the economics literature, e.g., Boyerand Moreaux (1987), Amir and Stepanova (2006) and in operations, e.g., Netessineand Rudi (2004). An intuition for this result can be developed by comparing the cen-tralized and decentralized solutions and the agents’ best response functions. We knowthat the best response functions are monotonically decreasing and that the centrali-zed system achieves high service levels with relatively low order quantities. Further,previous work has shown that the Stackelberg leader is better off than under the Nashequilibrium (see for example Netessine and Rudi 2004). Therefore, the leader willselect a solution closer to the integrated solution. Thus, if the retailer is the leader, shewill select a higher z compared to the Nash solution. This leads to lower costs for thesupplier as well, as its cost function is decreasing in the leader’s choice (see Obser-vation 2). Similarly, when the supplier leads, the order quantity Q, will be smallercompared to the Nash equilibrium. Given that the retailer’s cost function is increa-sing in the supplier’s decision in all our numerical trials (specifically, the condition inObservation 1 is satisfied for optimal ranges of z and Q in all cases), the supplier willbe better off following in the Stackelberg game.

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More specifically, our results show that the cost improvement of the retailer-ledVMI scenario over simultaneous gaming ranges from ∼ 0 to 11.17% and the costimprovement under the supplier-led scenario ranges from∼ 0 to 16.72%. Furthermore,� ≥ δ in each parameter scenario, thus the supply chain consistently performs bestwhen the supplier acts as the Stackelberg leader.

We are interested next in identifying which scenario parameters have the greatesteffect on determining when channel power has the greatest impact on supply chainsavings under VMI. The effects of channel power are shown in Table 6 in the columnslabeled δ and �. According to the results in Table 6 we find that the largest costimprovement over the equal power scenario is 16.72% which is achieved when thesupplier is the Stackelberg leader. Note that this is also the case where a retailer-ledsupply chain achieves the greatest cost reduction, but the savings are only 11.18%. Tofurther explore the effect of scenario parameters on channel power savings, we againperform a full-factorial experiment, this time measuring Stackelberg Cost Improve-ment (δ and �). The direct effects of the factors are shown in Table 3. We find thatholding cost ratio HR, and coefficient of variation CV, have the strongest effect onStackelberg Cost Improvement and that this applies to both the retailer-led (δ) andsupplier-led (�) scenarios. Lead time L, has a moderate positive impact, while α hasa significant negative impact and shipping cost SC, appears to be insignificant. Again,the parameters that are significant and their effects are consistent across supplier-ledand retailer-led VMI scenarios and at the agent and system level (with the exceptionof penalty cost being insignificant in its effect on δS). Thus, it appears that movingfrom an equal power VMI scenario to an asymmetric channel power relation is mostbeneficial when demand variability, holding costs, and lead times are high, penaltycosts are low and the supplier’s share of penalty costs are large (small α). Furthermore,this applies equally to whether the system is moving to greater retailer channel poweror greater supplier power; however, system cost savings are somewhat greater whenchannel power is concentrated at the supplier (as can be seen in Table 5).

Interestingly, Table 6 also shows that δR < �R and �S < δS for each parameterscenario. In other words, the agent incurs lower cost when s/he is the follower thanwhen s/he is the leader in the Stackelberg game. Proposition 8 somewhat formalizesthis result from the supplier’s perspective. Note that in all our numerical results (seeTable 5) service levels and order quantities are ordered such that zR ≥ zS ≥ zN

and QN ≥ Q R ≥ QS , where the subscript indicates the channel power distribution

Table 3 Direct effects onStackelberg cost improvement

Factor Stackelberg cost improvement

δR δS δ �R �S �

CV ↑ ↑ ↑ ↑ ↑ ↑HR ↑ ↑ ↑ ↑ ↑ ↑P ↘ − ↘ ↘ ↘ ↘SC − − − − − −L ↗ ↗ ↗ ↗ ↗ ↗α ↓ ↓ ↓ ↓ ↓ ↓

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under VMI (N = Nash, S = powerful supplier, R = powerful retailer). Proposition 8analytically demonstrates that if this condition holds, then the supplier always prefersto be the follower.

Proposition 8 If zR ≥ zS ≥ zN and QN ≥ Q R ≥ QS, then

K SVMI(zN , QN ) ≥ K S

VMI(zS, QS) ≥ K SVMI(zR, Q R).

Proof Note that

∂K SVMI(Q)

∂z= µ(1− α)pσ

√L [−1+�(z)]

Q≤ 0. (7)

Combining (7) with zS ≥ zN , we have K SVMI(zN , QN ) ≥ K S

VMI(zS, QN ). Giventhat QS represents the supplier’s best response to a given service level zS , we haveK S

VMI(zS, QN ) ≥ K SVMI(zS, QS). Further, because of (7) and zR ≥ zS ,

K SVMI(zS, QS) ≥ K S

VMI(zR, QS). Since Q R is the supplier’s best response when theretailer chooses zR , K S

VMI(zR, QS) ≥ K SVMI(zR, Q R). Therefore, K S

VMI(zN , QN ) ≥K S

VMI(zS, QS) ≥ K SVMI(zR, Q R). ��

Our numerical results indicate that we can make an even more general statementhere since in all results

K RVMI(zN , QN ) ≥ K R

VMI(zR, Q R) ≥ K RVMI(zS, QS) and

KVMI(zN , QN ) ≥ KVMI(zR, Q R) ≥ KVMI(zS, QS),

where KVMI represents the cost to the supply chain as a whole. Thus, our results implythat agents prefer to be the follower rather than the leader in a Stackelberg game, thus,there exists a first-mover disadvantage under VMI. This finding is consistent with theresults shown in Gal-Or (1985) and Amir and Stepanova (2006) for their duopolymodels, Netessine and Rudi (2004) for their drop-shipping model and Corbett (2001)for his analysis of information asymmetry on performance. Desiraju and Moorthy(1997) also report cases of a first-mover disadvantage in a price leadership scenario,e.g., a retailer who acts as the Stackelberg leader announces a retail price; the manu-facturer (the follower in the game) will charge a wholesale price as close as possibleto the retail price, leaving just enough retail profit to keep the retailer interested. Wenotice a similar behavior in our setting, where the Stackelberg follower extracts themajority of the cost savings resulting from asymmetric channel power, leaving theleader with just enough cost benefits to motivate him to move from equal power.

The above result leads to the interesting conclusion that at the agent level, channelpower does not seem to translate into lower costs. Thus, agents are not motivatedto exercise channel power even when they have it. However, at the system level,costs are clearly lowest when there is some form of asymmetric channel relationship,regardless of whether power is concentrated at the retailer or the supplier. Therefore,in order to achieve the greatest supply chain efficiency, one of the agents must beenticed to exercise channel power (i.e., assume Stackelberg leadership). This can be

267


accomplished through some sort of transfer payment between agents (see Cachon 2001for a discussion of VMI and side payments to achieve channel coordination) or perhapsa pairing of VMI with additional collaborative efforts such as holding cost subsidies(see Nagarajan and Rajagopalan 2004 for an example of such an arrangement).

8 Conclusions

In this paper we analyze a decentralized supply chain operating according to a vendor-managed inventory (VMI) agreement and a continuous review (Q, R) inventory policy.Under VMI, the supplier determines the order quantity amount Q, to send to theretailer, while the retailer retains control of the reorder point R. To capture the effectof channel power in the supply chain on the performance of VMI, we develop threedifferent models for VMI corresponding to (1) a supply chain with a powerful supplierthat can lead the decision making process; (2) a supply chain with a powerful retailer;and (3) a scenario where both agents have approximately equal channel power. Thepowerful agent scenarios are modeled as Stackelberg games with the powerful agentas the Stackelberg leader; the equal-power scenario is modeled as a simultaneousdecision game. Optimal policies are found for each scenario in the form of equilibriumsolutions for Q and R. Models are also developed for the centralized scenario whereone decision maker chooses both Q and R and a traditional retailer-managed inventory(RMI) scenario for comparison purposes.

Our analysis shows that VMI leads to supply chain savings in many scenarios,regardless of the channel power relationship. We find that VMI produces the greatestsavings over RMI when demand uncertainty, holding cost ratio and delivery lead timeare low and shipping cost is high. However, we find that the amount of savings canbe significantly affected by channel power in the supply chain. In all cases systemcosts are lower when there is an asymmetric power relationship than under an equalpower scenario. Interestingly, however, the lowest costs are incurred at the agent levelby being a follower and not a leader in the Stackelberg game. Thus, while the systemprefers either the retailer or supplier to lead, neither has incentive to exercise channelpower on their own. Therefore, some additional incentive must be provided to an agentto accept the leadership role in order to reduce system costs. This additional incentivecould take the form of side payments or additional collaborative mechanisms. We alsofind that the lowest system costs are incurred when the supplier is the Stackelbergleader. This appears to be contrary to the current trend of channel power shifting fromsuppliers to retailers, such as is described in a Wall Street Journal article discussingthe shift in channel power from suppliers such as Levi Strauss to retailers such asWal-Mart (Wall Street Journal, June 17, 2004). Based on our findings, such powerconcentration at the retailer may, at least in VMI systems, lead to costs that are higherthan a system where power is concentrated at the supplier, although such a system willstill be more efficient than a supply chain where power is equally distributed betweensupplier and retailer.

This research represents an initial step to better understand channel power. However,we believe that, due to its far reaching implications, this topic deserves additionalattention in the operations literature. Thus, future research can investigate settings

268


where multiple competing retailers sell the product of one supplier to explore whethercompetition can mitigate the first-mover disadvantage that we note in our work (e.g.,see Bernstein and Federgruen 2003, 2004, 2005). Alternatively, one may consider thecase of multiple suppliers selling substitutable or complementary products through acommon retailer (see for instance Netessine and Rudi 2003, Wang 2006) or may allowcustomer demand to be price sensitive, as explained in Petruzzi and Dada (1999).Another possible extension is to relax the assumption of full information and analyzecases where the supply chain agents may hold asymmetric information on each other’scost parameters (e.g., Corbett 2001; Corbett et al. 2004). In all cases, it is expectedthat these extensions will lead to considerable modeling challenges. However, theseextensions incorporate many of the complications encountered in practice and shouldlead to models that better represent the actual influence that channel power exerts onsupply chain agents’ decision making.

Appendix A

Table 4 System performance comparison of centralized and decentralized scenarios (%)

Factor Factor Measure Decentralized performance gap (ϒ) VMI cost reduction factor ()

value Nash Ret–Led Sup–Led Nash Ret–Led Sup–Led

Avg 18.3 18.1 17.9 96.44 96.45 96.46

0.1 Min 9.1 9.0 9.0 90.93 91.12 91.25

Max 30.1 29.4 29.0 98.66 98.66 98.66

Avg 17.3 16.7 16.1 87.73 87.87 87.99

CV 0.3 Min 8.0 8.0 7.8 67.21 69.17 70.36

Max 30.6 28.2 27.2 95.56 95.56 95.56

Avg 16.1 15.0 13.9 73.21 73.77 74.24

0.6 Min 6.7 6.7 6.4 29.60 37.46 41.37

Max 31.9 26.5 24.6 90.35 90.35 90.36

Avg 17.4 16.7 15.9 85.37 85.66 85.90

10 Min 6.7 6.7 6.4 29.60 37.46 41.37

Max 31.9 29.4 29.0 98.60 98.60 98.60

Avg 17.2 16.6 16.0 85.85 86.08 86.27

P 15 Min 6.7 6.7 6.4 33.23 39.34 42.46

Max 29.9 29.3 29.0 98.63 98.63 98.63

Avg 17.1 16.5 16.0 86.15 86.35 86.52

20 Min 6.7 6.7 6.4 35.20 40.51 43.24

Max 29.8 29.3 29.0 98.66 98.66 98.66

Avg 8.9 8.7 8.5 90.72 90.75 90.78

0.03 Min 6.7 6.7 6.4 69.34 69.75 70.00

Max 9.6 9.5 9.4 98.66 98.66 98.66

Avg 15.3 14.9 14.4 87.43 87.54 87.63

269


Table 4 continued

Factor Factor Measure Decentralized performance gap (ϒ) VMI cost reduction factor ()

value Nash Ret–Led Sup–Led Nash Ret–Led Sup–Led

HR 0.05 Min 11.2 11.1 10.7 58.26 59.60 60.40

Max 16.8 16.5 16.4 98.21 98.21 98.21

Avg 27.6 26.2 25.0 79.22 79.80 80.28

0.1 Min 19.4 19.1 16.7 29.60 37.46 41.37

Max 31.9 29.4 29.0 97.08 97.08 97.09

Avg 17.6 17.1 16.6 90.07 90.19 90.30

3 Min 7.9 7.8 7.7 63.55 65.92 67.33

Max 30.7 29.4 29.0 98.66 98.66 98.66

Avg 17.3 16.7 16.1 86.75 86.95 87.12

L 5 Min 7.4 7.4 7.2 51.25 55.24 57.47

Max 31.0 29.2 28.8 98.23 98.23 98.23

Avg 17.1 16.2 15.4 80.24 80.64 80.98

10 Min 6.7 6.7 6.4 29.60 37.46 41.37

Max 31.9 28.9 28.3 97.42 97.42 97.43

Avg 16.9 16.2 15.5 83.03 83.33 83.58

50 Min 6.7 6.7 6.4 29.60 37.46 41.37

Max 30.6 29.0 28.6 98.09 98.09 98.09

Avg 17.3 16.6 16.0 86.22 86.44 86.64

SC 75 Min 7.1 7.1 6.9 41.15 47.34 50.60

Max 31.3 29.2 28.9 98.45 98.45 98.45

Avg 17.5 16.9 16.3 88.13 88.31 88.47

100 Min 7.4 7.4 7.2 48.60 53.82 56.68

Max 31.9 29.4 29.0 98.66 98.66 98.66

Avg 18.3 16.9 15.9 85.14 85.68 85.99

0.6 Min 8.5 7.4 6.8 29.60 37.46 41.37

Max 31.9 29.4 28.9 98.62 98.62 98.62

Avg 17.1 16.6 15.9 85.89 86.05 86.25

α 0.75 Min 7.3 7.0 6.5 37.44 39.54 42.07

Max 29.5 29.3 29.0 98.64 98.64 98.64

Avg 16.4 16.3 16.0 86.34 86.36 86.45

0.9 Min 6.7 6.7 6.4 41.35 41.61 42.67

Max 29.2 29.2 29.0 98.66 98.66 98.66

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Table 5 Optimal scenario policies

Factor VMI

Factor Centralized Equal power Powerful retailer Powerful supplier RMI

value z Q Cost z Q Cost z Q Cost z Q Cost z Q Cost

0.10 1.785 202.8 15.7 1.279 386.3 19.8 1.512 384.9 19.7 1.285 382.9 19.7 3.226 2.6 629.5

CV 0.30 1.778 205.6 16.9 1.269 392.1 21.1 1.506 387.6 20.9 1.287 381.5 20.7 2.868 8.4 195.0

0.60 1.768 209.9 18.7 1.253 401.2 23.0 1.497 391.7 22.5 1.291 379.6 22.2 2.617 18.0 95.7

10 1.605 206.4 17.0 1.045 394.3 21.2 1.312 388.4 20.9 1.072 380.7 20.7 2.777 10.0 297.9

P 15 1.799 206.1 17.1 1.297 393.0 21.3 1.531 388.0 21.1 1.317 381.5 20.9 2.919 9.6 307.8

20 1.927 205.9 17.2 1.458 392.4 21.4 1.674 387.8 21.2 1.474 381.8 21.0 3.016 9.4 314.6

0.03 1.950 246.2 13.2 1.601 387.4 14.5 1.716 386.3 14.5 1.607 382.9 14.5 3.099 9.1 318.8

HR 0.05 1.795 209.6 16.1 1.327 390.8 19.0 1.524 387.7 18.9 1.340 382.1 18.8 2.930 9.6 308.0

0.10 1.586 162.4 22.0 0.872 401.5 30.3 1.277 390.2 29.8 0.915 378.9 29.3 2.683 10.3 293.5

3 1.780 204.8 16.6 1.271 390.5 20.7 1.508 386.8 20.5 1.286 381.9 20.4 3.002 6.8 401.5

L 5 1.778 205.8 17.0 1.268 392.6 21.2 1.506 387.8 20.9 1.287 381.4 20.7 2.916 9.0 306.2

10 1.773 206.2 17.9 1.255 396.9 22.3 1.501 389.7 21.9 1.284 380.5 21.7 2.794 13.2 212.6

50 1.859 170.6 14.5 1.371 325.5 18.0 1.598 320.6 17.8 1.393 314.3 17.6 2.904 9.7 205.6

SC 75 1.769 208.1 17.2 1.257 397.0 21.5 1.497 391.8 21.2 1.278 385.0 21.0 2.904 9.7 306.8

100 1.703 239.6 19.5 1.171 457.2 24.4 1.421 451.8 24.2 1.191 444.7 23.9 2.904 9.7 407.9

0.60 1.777 206.1 17.1 1.127 401.7 21.7 1.532 390.4 21.1 1.168 379.2 20.8 2.831 9.9 301.8

α 0.75 1.777 206.1 17.1 1.280 391.8 21.2 1.507 388.2 21.1 1.297 381.7 20.8 2.909 9.6 307.1

0.90 1.777 206.1 17.1 1.393 386.1 21.0 1.477 385.6 21.0 1.398 383.0 20.9 2.971 9.5 311.4

Table 6 Impact of channel power (%) on system and agent cost

Factor Factor Measure Ret-Led.cost improve.(δ) Sup-Led. cost improve.(�)

value δR δS δ �R �S �

Avg 0.23 0.38 0.28 0.85 0.01 0.55

0.1 Min 0.00 0.00 0.00 0.06 0.00 0.02

Max 1.85 2.87 2.13 4.92 0.14 3.60

Avg 0.63 1.13 0.79 2.28 0.08 1.53

CV 0.3 Min 0.01 0.01 0.01 0.16 0.00 0.07

Max 5.04 8.62 5.98 12.59 1.25 9.61

Avg 1.12 2.24 1.47 3.92 0.31 2.76

0.6 Min 0.01 0.03 0.02 0.28 0.00 0.13

Max 9.13 17.29 11.17 20.70 4.79 16.72

Avg 0.77 1.41 0.97 2.70 0.17 1.86

10 Min 0.00 0.01 0.00 0.07 0.00 0.03

Max 9.13 17.29 11.17 20.70 4.79 16.72

Avg 0.64 1.22 0.82 2.28 0.12 1.57

P 15 Min 0.00 0.00 0.00 0.06 0.00 0.03

271


Table 6 continued

Factor Factor Measure Ret-Led.cost improve.(δ) Sup-Led. cost improve.(�)

value δR δS δ �R �S �

Max 7.38 14.73 9.15 17.17 3.32 13.83

Avg 0.58 1.12 0.75 2.07 0.10 1.42

20 Min 0.00 0.00 0.00 0.06 0.00 0.02

Max 6.55 13.49 8.19 15.39 2.71 12.40

Avg 0.13 0.30 0.22 0.90 0.01 0.44

0.03 Min 0.00 0.00 0.00 0.06 0.00 0.02

Max 0.72 1.98 1.33 3.96 0.20 2.14

Avg 0.37 0.78 0.53 1.73 0.05 1.05

HR 0.05 Min 0.01 0.01 0.01 0.10 0.00 0.06

Max 2.10 5.09 3.22 7.77 0.72 5.13

Avg 1.48 2.67 1.79 4.42 0.33 3.35

0.1 Min 0.03 0.05 0.04 0.25 0.00 0.18

Max 9.13 17.29 11.17 20.70 4.79 16.72

Avg 0.51 0.91 0.64 1.83 0.07 1.24

3 Min 0.00 0.00 0.00 0.06 0.00 0.02

Max 5.46 9.45 6.50 13.51 1.49 10.37

Avg 0.63 1.18 0.81 2.26 0.11 1.54

L 5 Min 0.00 0.01 0.00 0.07 0.00 0.03

Max 6.80 12.21 8.19 16.33 2.45 12.77

Avg 0.87 1.71 1.13 3.02 0.22 2.12

10 Min 0.00 0.01 0.01 0.10 0.00 0.04

Max 9.13 17.29 11.17 20.70 4.79 16.72

Avg 0.71 1.40 0.93 2.52 0.16 1.75

50 Min 0.00 0.01 0.00 0.07 0.00 0.03

Max 9.13 17.29 11.17 20.70 4.79 16.72

Avg 0.65 1.23 0.84 2.33 0.13 1.60

SC 75 Min 0.00 0.01 0.00 0.06 0.00 0.03

Max 8.75 15.60 10.52 20.22 4.11 16.06

Avg 0.62 1.12 0.78 2.20 0.11 1.50

100 Min 0.00 0.00 0.00 0.06 0.00 0.02

Max 8.55 14.65 10.16 19.99 3.79 15.72

Avg 1.49 2.71 1.88 4.38 0.32 3.03

0.6 Min 0.07 0.12 0.10 0.37 0.00 0.16

Max 9.13 17.29 11.17 20.70 4.79 16.72

Avg 0.44 0.91 0.58 2.03 0.07 1.38

α 0.75 Min 0.02 0.04 0.03 0.18 0.00 0.08

Max 2.55 5.95 3.36 9.39 1.01 7.40

Avg 0.06 0.13 0.08 0.64 0.01 0.43

0.9 Min 0.00 0.00 0.00 0.06 0.00 0.02

Max 0.32 0.89 0.45 2.91 0.10 2.26

272


Appendix B

Proof of Lemma 1 The retailer places an order to the supplier once every Q units ofcustomer demand. Thus, the time between orders at the supplier is a random variable,T defined as Q/D, where D represents customer demand rate with mean Q/µ.4 Then,applying Theorem 2.1.2 in Casella and Berger (1990) for D ∈ (0,∞), the probabilitydistribution function of T is

fT (t) = 1

σ√

2πe−

(Qt −µ

)

2σ2Q

t2 . (B.1)

Using (B.1) and recalling that the supplier’s production rate is µ, the expectedshortfall per cycle at the supplier, E

[(Q − T µ)+

], becomes

E[(Q − T µ)+

] =Qµ∫

0

(Q − t) fT (t) dt = QFT(

Q

µ

)−

Qµ∫

0

t fT (t) dt. (B.2)

Using the identity

FT (t) = P(T ≤ t) = P

(Q

D ≤ t

)= 1− P

(D ≤ Q

t

)= 1− FD

(Q

t

),

Eq. (B.2) becomes

E[(Q − T µ)+

] = Q [1− FD(µ)]−Qµ∫

0

Q

σ√

2π

1

te−

(Qt −µ

)

2σ2 dt.

Substituting ξ = Q/t−µσ

and noting that dt = − Qσ

(ξσ+µ)2 dξ , gives

E[(Q − T µ)+

] = Q

⎛

⎝1

2−∞∫

0

µ

σξ + µφ(ξ) dξ

⎞

⎠ . (B.3)

So, b µQ E

[(Q − T µ)+

] = bµ(

12 −

∫∞0

µσξ+µ

φ(ξ) dξ)= bµK. ��

Proof of Proposition 6 The existence of the Stackelberg equilibrium when the retaileris the leader follows from the continuity of the retailer’s cost function. Uniqueness of

4 Given that T is not continuous at D = 0, we require, for tractability, that D > 0 For reasonable valuesof the coefficient of variation, the probability of a negative demand is sufficiently small to satisfy thisassumption.

273


the equilibrium is satisfied if we can show that the retailer’s cost function is quasi-convex. The retailer wishes to minimize

K RVMI(z) =

αpσ√

µLhS√2

�(z)√S + (1− α)pσ

√L�(z)

+h R√

µ√2hS

√S + (1− α)pσ

√L�(z)+ h Rzσ

√L.

We proceed next to get the first and second order conditions for K RVMI. For ease of

exposition, define

A = �(z)√S + (1− α)pσ

√L�(z)

and B =√

S + (1− α)pσ√

L�(z),

so that

K RVMI(z) = A

αpσ√

µLhS√2

+Bh R√

µ√2hS+ h Rzσ

√L.

Then,

∂A

∂z=−[1−�(z)]

√S + (1− α)pσ

√L�(z)+ (1−α)pσ

√L�(z)[1−�(z)]

2√

S+(1−α)pσ√

L�(z)

S + (1− α)pσ√

L�(z)

= −[1−�(z)][2S + (1− α)pσ√

L�(z)]2[S + (1− α)pσ

√L�(z)]3/2

and

∂2A

∂z2 =2

{φ(z)[2S+(1−α)pσ

√L�(z)]+(1−α)pσ

√L[1−�(z)]2

}[S+(1−α)pσ

√L�(z)]

4[S+(1−α)pσ√

L�(z)]5/2

−3(1− α)pσ√

L[1−�(z)]2[2S + (1− α)pσ√

L�(z)]4[S + (1− α)pσ

√L�(z)]5/2

= (1− α)pσ√

L[S + (1− α)pσ√

L�(z)] {2φ(z)�(z)− [1−�(z)]2}

4[S + (1− α)pσ√

L�(z)]5/2

+4φ(z)S[S + (1− α)pσ√

L�(z)] − 3(1− α)pσ√

L S[1−�(z)]24[S + (1− α)pσ

√L�(z)]5/2

.

274


Furthermore,

∂B

∂z= − (1− α)pσ

√L[1−�(z)]

2√

S + (1− α)pσ√

L�(z)and

∂2B

∂z2 =2(1− α)pσ

√Lφ(z)

√S + (1− α)pσ

√L�(z)−

{(1−α)pσ

√L[1−�(z)]

}2

√S+(1−α)pσ

√L�(z)

4[S + (1− α)pσ√

L�(z)]

=[(1− α)pσ

√L]2 {

2φ(z)�(z)− [1−�(z)]2}+ 2(1− α)pσ√

Lφ(z)S

4[S + (1− α)pσ√

L�(z)]3/2

≥ 0.

After recombining terms and algebraic manipulation, the numerator of∂2 K R

VMI(z)∂z2

becomes

αpσhS

√Lµ

[S + (1− α)pσ

√L�(z)

]

×[(1− α)pσ

√L

{2φ(z)�(z)− [1−�(z)]2

}+ 4φ(z)S

]

−3α(1− α)p2σ 2L√

µShS[1−�(z)]2

+h R√

µ[S + (1− α)pσ√

L�(z)]{[

(1− α)pσ√

L]2

×{

2φ(z)�(z)− [1−�(z)]2}+ 2(1− α)pσ

√Lφ(z)S

}

=[

S + (1− α)pσ√

L�(z)] {

2φ(z)�(z)− [1−�(z)]2}

(1− α)pσ√

L

×[

pασhS

√Lµ+ h R

√µ(1− α)pσ

√L]

−3α(1− α)p2σ 2L√

µShS[1−�(z)]2+

[S+ (1−α)pσ

√L�(z)

] [4φ(z)SpασhS

√Lµ+2h R

√Lµ(1−α)pσφ(z)S

]

= (1− α)p2σ 2L√

µ{

2φ(z)�(z)− [1−�(z)]2} [

S + (1− α)pσ√

L�(z)]

× [αhS + (1− α)h R]

+2pσ√

LµSφ(z)[

S + (1− α)pσ√

L�(z)]

[2αhS + (1− α)h R]

−3α(1− α)p2σ 2L√

µShS[1−�(z)]2, (B.4)

and the denominator of∂2 K R

VMI(z)∂z2 is 4

√2hS[S + (1− α)pσ

√L�(z)]5/2 ≥ 0. Thus,

in search of a condition for∂2 K R

VMI(z)∂z2 ≥ 0, we focus on (B.4) and impose that it be

positive. A sufficient condition is that the sum of its last two terms be positive, which

275


translates into

S ≥ (1− α)pσ√

L{3αhS[1−�(z)]2 − 2φ(z)�(z)[2αhS + (1− α)h R]

}

2φ(z)[2αhS + (1− α)h R] . (B.5)

However, for mathematical tractability, we do not use (B.5) directly but rather anupper bound for (B.5) obtained from the inequality 3[1 − �(z)]2 − 2φ(z)�(z) ≤2[1−�(z)]2. Thus, we have

S ≥ (1− α)pσ√

L[1−�(z)]2[αhS − (1− α)h R]2φ(z)[2αhS + (1− α)h R] , (B.6)

as a sufficient condition.It is straightforward to note that if α

(1−α)≤ h R

hS, then αhS − (1− α)h R ≤ 0. In this

case, condition (B.6) is satisfied ∀S ≥ 0. If, however, α(1−α)

≥ h RhS

, it can be shown that

terms [1−�(z)]2φ(z) and αhS−(1−α)h R

2αhS+(1−α)h Rare bounded above by 0.63 and 0.5, respectively.

Thus, (B.6) is equivalent to

S ≥ (1− α)pσ√

L0.63

20.5 = 0.1575(1− α)pσ

√L . (B.7)

��

Proof of Proposition 7 The supplier’s cost function is continuous and thus there existsa Stackelberg equilibrium for the case of the powerful supplier. The supplier’s costfunction is

K SVMI(Q) = µ

Q(1− α)pσ

√L�(z∗VMI(Q))+ µ

QS + Q

2hS + bµK.

We focus next on showing that the equilibrium is unique. Using the IFT,

∂φ(z∗VMI(Q))

∂ Q= ∂φ(z)

∂z

∂z∗VMI(Q)

∂ Q= z∗VMI(Q))[1−�(z∗VMI(Q))]

Q

= z∗VMI(Q))h R

µαp, and

∂�(z∗VMI(Q))

∂ Q= ∂�(z)

∂z

∂z∗VMI(Q)

∂ Q= [1−�(z∗VMI(Q))]2

Qz∗VMI(Q)

= h2R Q

(µαp)2φ(z∗VMI(Q)).

276


Then,

∂K SVMI(Q)

∂ Q=

µ(1− α)pσ√

Lh2

R Q2

(µαp)2φ(z∗VMI(Q))− µ(1− α)pσ

√L�(z∗VMI(Q))

Q2

−µS

Q2 +hS

2

= (1− α)σ√

Lh2R Q2 − (µαp)2(1− α)σ

√L�(z∗VMI(Q))φ(z∗VMI(Q))

Q2α2µpφ(z∗VMI(Q))

−µS

Q2 +hS

2,

∂2 K SVMI(Q)

∂ Q2 =[(1− α)σ

√Lh2

R Q − µα(1− α)pσ√

L�(z∗VMI(Q))z∗VMI(Q)h R

]µpQ2α2φ(z∗VMI(Q))

[µpφ(z∗VMI(Q))]2(Qα)4

−[(1− α)σ

√Lh2

R Q2 − (µαp)2(1− α)σ√

L

×�(z∗VMI(Q))φ(z∗VMI(Q))]

×[2Qµα2 pφ(z∗VMI(Q))+ αQ2(z∗VMI(Q))h R

]

[µpφ(z∗VMI(Q))]2(Qα)4

+ 2Sp2φ(z∗VMI(Q))2α3µ3

[µpφ(z∗VMI(Q))]2(Qα)4 .

We want∂2 K S

VMI(Q)

∂ Q2 ≥ 0. Note that∂2 K S

VMI(Q)

∂ Q2 ≥ 0 if its numerator is positive. Thus,

we now focus on C(Q), where∂2 K S

VMI(Q)

∂ Q2 = C(Q)

∂ Q2 . After algebraic manipulation,

C(Q) = α(1− α)σ√

L{

2(µαp)3�(z∗VMI(Q))φ(z∗VMI(Q))2

−h2R Q2µαpφ(z∗VMI(Q))− h3

R Q3z∗VMI(Q)}+ 2Sp2φ(z∗VMI(Q))2α3µ3.

Due to the existence conditions for z∗VMI(Q), Q ∈ (0,µαp2h R

). Now,

limQ→0

φ(z∗VMI(Q)) = limQ→0

�(z∗VMI(Q)) = 0, so limQ→0

C(Q) = 0. Furthermore,

limQ→ µαp

2h R

z∗VMI(Q) = 0, limQ→ µαp

2h R

φ(z∗VMI(Q)) = limQ→ µαp

2h R

�(z∗VMI(Q)) = δ > 0. Thus

limQ→ µαp

2h R

C(Q) = α(1− α)σ√

L(µαp)3(

2× δ3 − δ

4

)+ 2φ(0)2Sµ3α3 p2 > 0.

277


Note that δ ≈ 0.39894 > 0. Thus, the extreme points of C(Q) are positive. Then,

dC(Q)

d Q= 4h Rz∗VMI(Q)α(1− α)σ

√L

{4(µαp)2�(z∗VMI(Q))φ(z∗VMI(Q))− h2

R Q2

+ h3R Q3

4µαpφ(z∗VMI(Q))z∗VMI(Q)+ Spα2µ2φ(z∗VMI(Q))

α(1− α)σ√

L

}.

Let,

D(Q) = 4(µαp)2�(z∗VMI(Q))φ(z∗VMI(Q))− h2R Q2.

Then,

limQ→0

D(Q) = 0 and limQ→ µαp

2h R

D(Q) = (µαp)2(

4× δ2 − 1

4

)> 0.

Furthermore,

dD(Q)

d Q= 2h2

R Q + 4h Rµαpz∗VMI(Q)�(z∗VMI(Q)) ≥ 0.

Thus, D(Q) ≥ 0 ∀Q ∈ (0,µαp2h R

), which implies dC(Q)d Q ≥ 0. Hence, all critical points

of C(Q) are≥ 0; therefore, C(Q) ≥ 0 and∂2 K S

VMI(Q)

∂ Q2 ≥ 0. It follows that K SVMI(Q) is

convex and a unique equilibrium exists. ��

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Part IVApplications in the Chemical Industry

Value chain management for commodities: a case studyfrom the chemical industry

M. Kannegiesser · H.-O. Günther · P. van Beek ·M. Grunow · C. Habla

Abstract We present a planning model for chemical commodities related to anindustry case. Commodities are standard chemicals characterized by sales and supplyvolatility in volume and value. Increasing and volatile prices of crude oil-dependentraw materials require coordination of sales and supply decisions by volume and valuethroughout the value chain to ensure profitability. Contract and spot demand differen-tiation with volatile and uncertain spot prices, spot sales quantity flexibility, spot salesprice–quantity functions and variable raw material consumption rates in productionare problem specifics to be considered. Existing chemical industry planning modelsare limited to production and distribution decisions to minimize costs or makespan.Demand-oriented models focus on uncertainty in demand quantities not in prices. Wedevelop an integrated model to optimize profit by coordinating sales quantity, price

M. Kannegiesser · H.-O. Günther (B)Department of Production Management, Technical University of Berlin,Wilmersdorfer Str. 148, 10585 Berlin, Germanye-mail: [email protected]

P. van BeekManagement Studies Group and Operations Research and Logistics Group,Wageningen University, Hollandseweg 1, 6706 KN Wageningen, The Netherlandse-mail: [email protected]

M. GrunowDepartment of Manufacturing Engineering and Management, Technical University of Denmark,Building 425, 2800 Kgs. Lyngby, Denmarke-mail: [email protected]

C. HablaDepartment of Enterprise-Wide Software Systems, The Fern Universität in Hagen,Universitätsstr. 1, 58097 Hagen, Germanye-mail: [email protected]



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OR Spectrum (2009) 31:63–93DOI 10.1007/s00291-008-0124-9


M. Kannegiesser et al.

and supply decisions throughout the value chain. A two-phase optimization approachsupports robust planning ensuring minimum profitability even in case of worst-casespot sales price scenarios. Model evaluations with industry case data demonstrate theimpact of elasticities, variable raw material consumption rates and price uncertaintieson planned profit and volumes.

Keywords Value chain management · Sales and supply network planning ·Demanduncertainty · Commodities · Chemical industry

1 Introduction

The chemical industry is one of the key global industries with product sales of e1,776 billion globally in 2004 (CEFIC 2005). In this article, we focus on the segmentof chemical commodities. Commodities are mass products produced and sold in highvolumes with standardized quality and few variants. Price is the key buying criterionfor customers. Examples are standard polymers, certain types of intermediate productsor basic chemicals. Sales prices for theses commodities are volatile and can changeregularly, e.g., weekly or monthly based on negotiations between the company and itscustomers.

Prices for raw materials can also change regularly. Specifically, many key rawmaterials in the chemical industry showed a severe rise in prices due to the increaseof the crude oil price over the last years. Raw price volatility and increases have to beconsidered in sales and supply planning of commodity products to ensure profitabilityof the business. Therefore, the focus on demand and supply volume planning alone isnot sufficient since a feasible volume plan might not be profitable for the company dueto the volatility of supply costs and sales prices. The monthly planning process needsto support integrated decisions on volume and values, specifically on sales quantitiesand prices considering available supply volumes and raw material costs. In this paper,an integrated planning model related to a real-life case from the European chemicalindustry is presented.

In our investigation, we consider a simplified intra-organizational value chain net-work of a company producing chemical commodities. The industry context of thiscase is a company operating a complex, multi-stage value chain network producingpolymers that also require several intermediate products as raw material. The companyis operating at several production sites and is serving different sales locations. Thebusiness is a commodity business where raw materials and finished products are char-acterized by market price and volume volatility. Annual production volumes exceed 1Mio. tons. In this study we focus on the monthly sales and operations planning processfor the entire value chain network for a planning horizon of 6–12 months.

Figure 1 shows a section of the network. The company has grouped multiple cus-tomers in regional or industry-specific sales locations. Two production resources arelocated in one production location, from where sales locations are served. One market-facing multi-purpose resource produces multiple finished commodity products. Thesecond single-purpose resource produces the intermediate product for the multi-purpose resource in continuous production mode. The intermediate product produced

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Value chain management for commodities

Production location 1

Procurement location 1 R1 R2

continuous multi-purpose

Production

Sales location 1Sales location 1


Procurement Sales


raw material

...

intermediate finished

= production locationLl ∈

= procurement & sales location

Legend:

= resource Rr ∈

= material flow

Tt ∈periods: monthly planning bucket

= product Pp ∈

Fig. 1 Section of the considered value chain network

on resource R1 requires a raw material product procured from an external procurementlocation.

The planning problem at hand shows a number of characteristics that are typical ofthe chemical industry.

• Spot and contract business differentiation is an important issue in the chemicalindustry specifically in commodity business.

• Price and volume volatility for chemical commodities in sales and procurement ismore significant than in other industries, e.g., in discrete parts manufacturing.

• The entire production system is organized as a multi-stage network with multi-purpose and continuously operated production resources.

• Material flows are predominantly divergent with intermediates used in multiplesubsequent products.

• Raw material consumption rates in production are variable depending on the degreeof capacity utilization.

These characteristics can be found, for example, in basic chemicals and/or polymerproduction, while fine chemical and pharmaceutical production can be seen as a spe-cialty type of business relying on smaller quantities and complex batch productionmode. The simplified network as shown in Fig. 1 focuses on the interaction betweenprocurement, production and sales. The problem at hand is an excerpt from the globalvalue chain planning problem of a polymer producing company. In our investigationwe focus on the interaction between key business functions in the global value chaincontext. The model developed represents a prototype which is used by the plannersto better understand volume and value dynamics from sales to procurement and theirimpact on profit in a value chain network. In a later stage, the company intends tointroduce the Supply Network Planning module of an advanced planning software

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system for operative planning (cf. Dickersbach 2006). To reduce the complexity ofthe prototype model, several standard features such as inventory records and transpor-tation are excluded mainly because they do not have a high profit impact comparedto sales and procurement issues. Exchange rates and risk hedging inventories are alsoexcluded here though they do represent further important issues in the investigatedglobal value chain network which will be included in the final implementation of thevalue chain planning model.

Traditionally, supply network planning models focus on the flow of goods in thenetwork while assuming sales and procurement prices as being fixed. Revenue man-agement, however, represents a topic which has recently gained considerable interestboth in practice and in academia. For an application in the iron and steel industryand a discussion of dynamic pricing in the US automotive industry, cf. Spengler et al.(2007) and Biller et al. (2005), respectively. Key issues of revenue management aredynamic pricing strategies as well as accept and reject decisions to make more effec-tive use of resources. Booking and pricing systems of airlines, hotels, car rentals,telecommunication systems and cargo transportation are just a few popular examplesof revenue management, cf. Gosavi et al. (2007), Bartodziej et al. (2007), Lee et al.(2007), Defregger and Kuhn (2007), Reiner and Natter (2007). These papers focuson revenue maximization based on pricing and decisions to influence the demand forservices such as airline seats, rental car capacity or hotel rooms which are in limitedsupply. Active sales and pricing decisions investigated in revenue management areprincipally relevant for the industrial planning problem considered in our paper. How-ever, in contrast to service industries we deal with physical products and the complexdecision-making process in a global chemical value chain.

This paper aims at integrating ideas of revenue management into supply networkplanning to optimize profit throughout the entire intra-organizational value chain net-work. We choose “value chain management” as an overall term for the integration ofdemand-oriented management concepts such as revenue management as well as sup-ply-oriented logistics management concepts which primarily focus on material flows.Specifically, our modeling approach reflects the following key issues:

• For chemical commodities as well as for many other industrial products (e.g., fer-tilizers or animal feed products), contract and spot demand can be distinguished.While sales prices and quantities are fixed for contract demand, spot market salescan be highly variable with regard to both price and quantity. We develop a valuechain planning model that, in addition to production and distribution planning, alsosupports pricing and sales decisions for spot demand.

• Similar to sales commodity markets, raw materials can be procured either based onfixed contracts with suppliers or on the spot market. In the latter case, the companyhas to decide on the procurement quantity taking the volatility of procurementprices into account. Our modeling approach also reflects these issues which are ofincreasing importance in many industries specifically confronted with increasingraw material prices.

• Empirical investigations have shown that both spot sales prices for commoditiesas well as procurement prices for raw materials are characterized by high uncer-tainty. Modeling these prices as independent random variables, as it is assumed in a

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large number of academic contributions, is not always realistic because the driversbehind the market development, e.g., development of crude oil prices, are ignored.Hence, our approach is based on scenario analysis which utilizes human exper-tise to forecast market developments in combination with subjective probabilitymeasures.

• Finally, it is shown how technological flexibility with respect to consumption ratesof raw material and feasible processing modes of the chemical production equip-ment can be used in order to balance sales market demands and procurementopportunities.

The overall objective of the proposed optimization model is to maximize profit by coor-dinating sales turnover with quantity and prices as well as supply decisions throughoutthe value chain. Model evaluations with industry case data demonstrate clearly theapplicability of the value chain optimization model. The model has been developedand implemented together with the company proving the industry case and also theproblem requirements and assumptions such as contract and spot demand.

The remainder of this article is organized as follows. The next section providesan overview of the relevant literature. In Sect. 3, a mixed-integer linear optimizationmodel for sales and supply planning in intra-organizational value chain networks isdeveloped. Section 4 presents a case study evaluation based on a real application fromthe European chemical industry.

2 Literature review

In the academic literature a wealth of papers dealing with demand and supply networkmanagement has been published. For an overview and classification, see, e.g., Thomasand Griffin (1996), Stadtler (2005) and Tang (2006). Some of these papers focus ondemand, others on supply aspects of the problem. Among the demand-focused papersemphasis is given either on demand forecasting, demand uncertainty, or pricing deci-sions.

The objective of demand forecasting is to predict future demand quantities as accu-rate as possible based on historical data. For an overview of demand forecasting withinsupply chain management see Kilger and Wagner (2008) and Meyr (2008). The clas-sical approach towards demand forecasting does not apply to the considered chemicalcommodity business, where contract demand is certain and spot demand does notneed to be fulfilled. In addition, the development of demand does not follow historicaldemand patterns, but is rather influenced by future raw material prices as investigatedby Asche et al. (2003) for crude-oil related products.

The paper by Gupta and Maranas (2003) represents one example for dealing withdemand uncertainty in the chemical industry. The authors propose a demand and sup-ply network planning model to minimize costs. Production decisions are made “hereand now” and demand uncertainty is balanced with inventories independently incorpo-rating penalties for safety stock and demand violations. Demand quantity uncertaintyis modeled as a normally distributed continuous random variable with known mean andstandard deviation and penalty costs are charged for unfilled demand. This approach,however, is not suitable in our commodity case, since spot demand and factors such as

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demand price uncertainty for chemical commodities or fluctuating raw material andcrude oil prices have to be considered. Another example from the chemical engineer-ing literature has been given by Chen and Lee (2004). They develop a multi-companydemand and supply network planning model to maximize profit under demand uncer-tainty and pricing decisions. Demand uncertainty is modeled with quantity scenariosand probabilities. A two-phase optimization strategy is developed to reach robustplans. Pricing decisions are modeled with fuzzy logic considering satisfaction levelsof buyer and seller assuming collaboration and preference transparency between bothparties. This assumption, however, is not valid in the spot sales commodity businessconsidered in our investigation.

Chakravarty (2005) develops an optimization model for global network designdecisions incorporating sales quantity and price decisions. Chakravarty uses demandcurves, where demand quantity is a function of price, and sales turnover is decidedusing quadratic optimization. The model scope of profit optimization incorporatingvariable sales prices and supply quantities as well as costs is similar to the consideredproblem more on a macro network design level rather than on a monthly planninglevel for a chemical industry value chain. In addition the assumption of a monopo-listic market constellation, where the company is able to influence demand by pricesetting reflected in the demand curves is not valid in the considered case.

In contrast to demand planning, the supply side of chemical industry value chainshas been widely investigated especially with focus on production planning and sched-uling. Examples of papers dealing with industrial applications are Blömer and Günther(2000), Neumann et al. (2002), Kallrath (2002a,b) or on multi-site supply networkplanning with given demand, cf. Timpe and Kallrath (2000), Grunow (2001),Grunow et al. (2003), and Berning et al. (2002). Production scheduling for batchand campaign production and synchronization of production plans across plants con-sidering sequence and production mode constraints are major subjects in this field ofresearch. The specific aspect of variable raw material consumption, which is essentialin the industrial application considered in our investigation, has not sufficiently beenaddressed in the literature so far.

Procurement planning in general and spot and contract procurement planning in thechemical industry particularly have recently been investigated in a number of papers.For instance, Stadtler (2008) discusses general tasks of purchase planning integratedin overall supply chain management at the order level. Recent papers discuss pro-curement strategies for spot and contract markets. Reiner and Jammernegg (2005)develop a risk-hedging model and compare different procurement strategies includingspeculation inventories. Marquez and Blanchar (2004) present extended procurementstrategies based on real-options to optimize contract portfolios considering in-tran-sit and warehouse inventories. Seifert et al. (2004) underline the importance of spotprocurement next to contract procurement and show the advantage, if a fraction ofdemand is based on spot market procurement.

So far, models presented in the academic literature focus either on demand or onsupply aspects. In the academic literature we did not find any realistic value chainplanning model that integrates sales and supply decisions by volume and value ina price-volatile chemical commodity business, although this planning problem is ofhigh importance not only in the chemical commodity industry.

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3 Sales and supply planning model

To support decision making in the considered intra-organizational value chain networka mixed-integer linear programming (MILP) model is proposed. Maximizing profitthroughout the entire value chain is seen as the overall objective function. Principally,the value chain profit consists of the following constituents:

Profit = spot sales turnover depending on variable sales prices and quantities+ fixed contract sales turnover− spot procurement costs depending on variable procurement prices and

quantities− fixed procurement costs− variable production costs depending on variable raw material consumption

and processing mode

The first two elements of the profit function and related constraints are reflected bythe sales model introduced in Sect. 3.1. The supply model presented in Sect. 3.2 con-siders all other issues related to procurement and production. To solve the model, twodifferent optimization strategies are proposed (see Sect. 3.3).

3.1 Sales model

3.1.1 Demand and sales planning for chemical commodities

In the industrial application considered, the central task is to plan monthly sales vol-umes and values in the network for 6–12 months. The planning process starts with amonthly demand forecast of quantities and prices submitted by the “Sales and Market-ing” department of the company. The forecast aggregates demand of single customersat the sales location level resulting in a cumulated demand quantity and a weightedaverage price. The planning objective is to maximize profit considering available pro-duction and procurement capabilities, sales prices and supply costs. The planningresult is a tactical sales and operations plan with sales quantities and prices as well asproduction and procurement quantities per month. The planning problem shows somespecifics as described in the following.

Contract and spot sales quantity managementContract and spot demand can be distinguished in chemical commodities markets.Contract demand is based on agreements between the company and customers withsales quantities and prices being fixed for a defined period. Contract demand quanti-ties and prices are fulfilled as forecasted and are deterministic. Spot demand is alsoforecasted by quantity and price. However, spot demand does not need to be fulfilledcompletely since the company can make active sales decisions on the acceptance orrejection of spot sales requests. The spot price can be bilaterally negotiated, requestedby the customer directly or set by the company. In the latter case the customer reactswith a quantity bid. In any case prices are negotiated bilaterally between company andcustomer. Double auction mechanisms with multiple buyers and sellers submittingoffers and bids cleared in one market price are not considered in this context.

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push

Sales

spot demand quantity

Demand

1 2 3

contractdemand quantity

contractdemand price

spot demand price

contractsales quantity

contractsales price

Periods

spot sales price

spot sales quantity

1 2 3 Periods

- - -

↓↑

cut

-

sales quantity deviation to demand quantityLegend:

Available total supply

Price effects: demand price = sales price demand price ≥ sales price demand price ≤ sales price- ↓ ↑

Fig. 2 Principle of contract and spot demand and sales

Spot sales quantities are flexible and can be lower or higher than the forecastedquantities for various reasons as shown in Fig. 2. Firstly spot sales are lower than thedemand quantity if the spot demand quantity exceeds the available supply and thecompany needs to make monthly volume quotation decisions cutting volumes first onan overall sales location level and then also on a detailed, individual customer level.Merely for illustration reasons, the available supply is shown to be constant in Fig. 2.Of course in the real application the available supply can vary, e.g., due to variationsin procurement quantities and production capacity. Secondly spot sales are reduced ifspot demand prices are too low compared to raw material costs forcing the companyto make a loss when supplying the customer. Hence the spot demand forecast has abid character as in single-sided auctions competing for limited supply. The bid can besuccessful and is fully supplied or it can be partly or even fully rejected depending onthe available supply quantity and the bid price. Like in stock markets and exchanges,bids need not be necessarily executed in the marketplace if the bid volume and pricecannot be cleared with a suitable offer.

Note that there are no penalties in spot business as it can be found in supply networkplanning, where an artificial penalty is applied if demand cannot be met. These oftensubjective penalties are not related to actual business agreements or actual monetarypenalties negotiated between the company and the customer. In our case customerseither have a fixed contract or they do flexible spot business on a tactical level. Thisflexibility, however, does not destabilize the respective value chain operations since itis limited to the tactical planning level and does not impact the operational order level.Customers have a very early information and commitment on a monthly level whetherthey receive the requested spot quantities or not. If a spot customer has received a con-firmation, the supplier delivers the related orders accurately and with high reliability.To summarize, in the considered industrial application demand is not regarded as agiven monolithic quantity to be fulfilled in the traditional supply chain managementsense but is defined as a mix of fixed contract demand and flexible spot demand.

290


Spot sales price–quantity functions and elasticitiesSpot sales decisions have intuitive price effects as shown in Fig. 2: higher averageprices are achieved when cutting spot sales quantities or lower prices are requiredwhen pushing additional quantities into the market. However, we do not assume amonopolistic market situation, where the company is able to influence or dominatemarket prices. In our case, price is a result of the spot sales quantity decision madeby the company. Since the price is an average price across several customer forecastsgrouped into one sales location, it is intuitive that the average price increases when salesquantities are lower than the demand forecast quantity. It is assumed that customerswith lowest prices are cut first. Hence the average price across the remaining customersincreases. In this context, competitor behavior has no influence on this price–quantityfunction. Competitors have influence on the overall market prices and the availablesupply. However, the model focuses on bilateral negotiations between the companyand its customers. This business relationship is confidential, i.e., the competitor doesnot know about spot quantities ordered by the customers and the corresponding spotsales prices. The competitor does not even know to what extent a customer is suppliedon contract or spot basis. Since the business relationships are kept confidential thecompetitor is not able to take specific reactions.

Spot sales price uncertaintySpot demand quantity and prices are uncertain in the commodity business for the con-sidered planning horizon. Since price is the main buying criterion, mid-term demandquantity is mainly influenced by the price level. Additionally, commodity suppliersoften make supply volume decisions before sales prices are finally fixed due to com-plex multi-stage production systems, large lead times in production and raw materialsupply, lack of change-over flexibility in production with production plans fixed forone month, planned shut-downs for maintenance as well as long transportation leadtimes specifically in global value chain networks. Therefore, supply volumes in com-modity business are fixed prior to sales prices in the market. Hence, the spot sales priceremains an uncertain parameter. In our investigation, the spot sales price is consid-ered as uncertain leading to different price and sales turnover scenarios for the samesales quantity. Therefore, contract demand quantity and price as well as spot demandquantities are treated deterministic while spot sales prices are considered stochastic.

3.1.2 Derivation of price–quantity functions for spot demand

In the value chain network investigated the following entities have to be considered inthe formulation of the sales model:

• Products include finished products sold on the market, intermediate products pro-duced and raw materials procured.

• Locations represent the nodes of the value chain network such as sales, productionor procurement locations.

• The planning horizon is divided into discrete time buckets (periods), months bydefault.

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Demand and sales are planned for all valid product–sales location combinations{p, l} ∈ P L S and a medium-term planning horizon covering periods t ∈ T .

Given the simplifying assumption that transit times between production and saleslocations and thus inventory balances can be neglected, the model can be separated bytime periods and periods could even be neglected. The same is true for locations. How-ever, from an industry-practice perspective, it is important to show the monthly devel-opment and interrelationships of sales and operation figures thus making the dynamicsin the value chain across business functions and the volatility in profits, prices andvolumes more transparent. Thus, instead of performing single period experiments amodel formulation is suggested that integrates all activities within the entire planninghorizon. Locations are essential as reference points for the aggregation of demand andthe determination of the price–quantity functions. In the real application, transit timesas well as intermediate inventories, safety stocks, etc. can easily be embedded into themodel formulation.

Demand input data comprise the demand forecast provided by the “Sales and Mar-keting” organization of the company. Demand forecasts are aggregated from a singlecustomer level to an aggregated sales location level. The contract demand forecastindicates the total demand quantities of all relevant products for each period and prod-uct–location combination. In addition, the corresponding average sales price can bederived from the customer contracts. Owing to the usual contract terms, total salesturnover achieved from contract sales is fixed.

In contrast, total sales turnover achieved from spot sales depends on the decisions ofthe company on spot prices and sales quantities for each period and product-locationcombination. As explained in the previous subsection, the company receives quantityand price bids from its spot market customers. In addition, the local “Sales and Market-ing” units forecast expected bids for future periods. It is important that all spot salesopportunities are forecasted as total demand bids regardless of whether productioncapacity needed to fulfill this demand is available or not. It should be noted that theresulting average spot sales price increases if the spot demand exceeds the productioncapacity and the company selects the spot demand bids with the best spot sales prices.This relationship is expressed by the elasticity ε defined as −ε = (�p/p) : (�x/x).Here, the elasticity can be interpreted as the change of the average spot price p withrespect to the change of the spot sales quantity x . Forecasting individual customerspot demand for 6–12 months is more difficult than forecasting the overall spot mar-ket demand. The latter is essential in order to evaluate if spot demand exceeds ownsupply. To model the relationship between spot price and quantities, we show howadequate price–quantity functions can be derived from the forecasted customer bids.

The derivation of price–quantity functions is based on the following major assump-tions:

• The relationship between spot sales price and spot sales quantity can be modeled asa linear function within the feasible minimum and maximum quantities defined bythe management of the company. Of course, the price–quantity relationship couldalso be modeled using a non-linear function depending on the actual price–quantitybids the company receives. In our case we found that the linear function showed asufficient statistical fit based on the real data provided by the company.

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• External factors affecting the spot demand quantity, e.g., competitor actions, arenot considered, i.e., spot sales demand only depends on spot sales price for eachperiod and product–location combination.

The detailed steps of the algorithm for determining the price–quantity function forspot sales demand and a numerical example are provided in Table 1. To keep thepresentation simple, we assume one single period and one individual product–loca-tion combination, i.e., the corresponding indices are omitted. Given are spot demandquantity qc and price forecast pc for individual customers c ∈ C (step 1). The spotdemand forecast by price and quantity represent future sales opportunities defined bythe “Sales and Marketing” unit of the company. Historical sales can provide someguidance. However, anticipating future price trends in the market also depends onfuture demand and the raw material price development. Note that the forecast does nothave to be necessarily discussed with the customer but can and should be based on themarket knowledge of the “Sales and Marketing” organization also reflecting targetsand new sales opportunities which “Sales and Marketing” wants to actively pursue inthe market. Next, all price forecasts are sorted in non-increasing order giving ranksr = 1, . . . , R (step 2). In step 3, demand forecast quantities qc are summed up to acumulated spot demand quantity Qr for each rank r = 1, . . . , R with Q R being thetotal demand quantity across all forecasts In step 4, the corresponding average spotdemand price forecast Pr for each rank r = 1, . . . , R is determined with PR being theaverage price across all forecasts. In the following steps 5 and 6, the quantity shareQr/Q R and the average price ratio Pr/PR of each rank r = 1, . . . , R are determined.

Table 1 Algorithm to determine the price–quantity function of spot demand and numerical example

Algorithmic steps Customer

A B C D

1. List individual customers c ∈ C with spot demand quantity qc and price forecast pc

Quantity (t) 100 200 100 200

Price (e/t) 100 90 80 70

2. Sort forecasts in non-increasing order of price using ranks r = 1, . . . , R

Rank 1 2 3 4

3. Determine cumulated spot demand quantity Qr for rank r = 1, . . . , R

� Quantity (t) 100 300 400 600

4. Determine average spot demand price Pr for rank r = 1, . . . , R

Ø Price (e/t) 100 93.3 90 83.3

5. Determine quantity share Qr /Q R of rank r = 1, . . . , R

� Quantity (%) 17 50 67 100

6. Determine average price ratio Pr /PR of rank r = 1, . . . , R

� Price (%) 120 112 108 100

7. Perform linear regression for price ratios and quantity shares

Regression y = −0.2407·x + 1.2408; R2=1.00

8. Determine price elasticity

Elasticity ε = 0.2407

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In step 7 a linear regression for price ratios with respect to quantity shares is carriedout giving the price–quantity function. Finally, the spot demand elasticity is obtainedas the negative slope of the regression function (step 8). Note that the elasticity isdetermined based on linear regression considering the quantity shares and averageprice shares and not the absolute quantity and average prices in the price–quantityfunction.

This proposed algorithm requires a sufficient number of individual customer bidsor forecasts within one sales location and thus relies on effective support by the local“Sales and Marketing” units. If the number of price–quantity bids is not sufficient andthe regression is not accurate enough, elasticities cannot be directly used for decisionmaking. In this case, elasticity is assumed to be 0 meaning no price effects are includedin the model and calculated profits are lower and more cautious than in reality. If allcustomers have the same spot prices, the average price is equal to the individual pricesand the elasticity is equal to 0 meaning that no average price effects occur in case of vol-ume reductions. In the investigated example from the chemical industry, we observedthat price elasticities were volatile and ranked mainly between 0.1 and 0.5 different bymonth, product and locations analyzed for 12 months. The number of customers forone product and one location varied each month between 10 and 36. The R-squaredvalue for the linear regression varied monthly between 0.4 and 0.99. Without havingconducted a full elasticity analysis across the entire portfolio, the analysis helps toprove market perceptions such as a higher elasticity exists in one market compared toanother market or comparing elasticity between products being perceived to have adifferent elasticity. The statistical quality of the linear regression analysis in selectedmonths was considered as good in terms of the number of customers involved and theR-squared value proving the applicability of the approach. Alternatively, a quadraticregression of the sales turnover curve could be applied. This concept, however, doesnot create the same basis for understanding in the “Sales and Marketing” organizationof the company since elasticity is the parameter known in “Sales and Marketing” todiscuss and understand price–quantity dynamics in the market rather than discussingquadratic regression parameters that cannot be well understood and translated intodirect price-quantity-relations.

Another issue of considerable practical importance in commodity markets is theuncertainty of market prices arising from a great number of external factors. In ourcase study investigation, price uncertainty is reflected by alternative price scenarioss ∈ S. In the real application, scenarios have to be defined for each product–loca-tion combination. To keep the presentation simple, we again consider only one singleproduct–location combination.

To model the volatility of market prices, a price factor δs for spot demand price,e.g., 0.8, 1.0 and 1.2, and a corresponding subjective scenario probability ωs valid forthe entire planning horizon have to be defined by management. Typically three sce-narios “worst”, “best” and “average” are used in order to limit the complexity andkeep the scenario planning pragmatic. The price scenario philosophy of the companyis to have only one single sales plan with quantity x0 that is executed in the market atdifferent price levels ps . In addition we assume identical price–quantity functions, i.e.,identical spot demand elasticity for all price scenarios meaning that the price factorδs is impacting all customers homogenously not changing their spot demand volume.

294


sp

0x

„worst“

„average“

„best“

)(xp

„worst“

„average“

„best“

x

2=s

1=s

3=s

minx maxx

0xx

xxp ⋅)(

minx maxx

s S∈Elasticity

0

s

xp

x pε ∆− = ⋅

∆

x∆

p∆

Fig. 3 Price–quantity function and sales turnover curve for individual price scenarios

Figure 3 illustrates the concept of scenario-based price–quantity functions, whichbasically describe the dependency of sales price p on quantity x . With price–quantityfunction p(x) the resulting sales turnover is given as p(x) · x . The scenario-basedprice–quantity functions have different slopes but the elasticity considering relativeaverage price shares and relative quantity shares is identical in each scenario. In addi-tion to given input data, sales control data are defined by the planner executing salesand marketing business rules to set the boundaries for spot sales quantities. Controlparameters xmin and xmax indicate the minimum and maximum spot demand thatneeds to be fulfilled as shown in Fig. 3.

The concept of scenario-dependent sales turnover functions represents a signifi-cant advantage of demand price scenarios compared to demand quantity scenarios,since the company does not have to manage different volume scenarios creating highcomplexity in all areas of planning from sales to procurement. Moreover, the scenarioprice factors can be directly applied to model the sales turnover in the objective func-tion of the optimization model without affecting quantity constraints of the model.This advantage might change the perspective on demand uncertainty from quantityscenarios towards price scenarios related to a defined sales quantity. This is even morepracticable, since prices can be changed faster in practice compared to productionvolumes and material flows. In particular in the production of chemical commodities,considerable changeover times of the processing equipment have to be considered.Moreover, transportation lead times and limitations on transit stock often reduce theflexibility to adjust production quantities and redirect material flows on short notice.

3.1.3 Linear approximation of spot sales turnover

Since spot price and quantity depend on each other according to the linear price–quantity function, the profit function is quadratic. In the following, we show how apiecewise linear approximation of the sales turnover function can be achieved. This

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approximated turnover

add additional partial quantity points to improve approximation

turnover gradient

actual turnover curve

x

Ni ∈

y

1~y

3~y

2~y

4~y

1~q 4

~q

minxmaxx

3~q2

~q

Spot sales quantity

Fig. 4 Linear sales turnover approximation approach

approach is based on the concavity property of the sales turnover function and thelimited region of sales quantity flexibility to be considered. For a review of linearapproximation techniques for non-linear functions see Kallrath (2002b). For the opti-mization problem investigated in this paper, some application-specific features haveto be considered which are described in the following. As in the previous subsection,we skip the indices for periods and product–location combinations in order to improvethe understandability of the presentation.

The sales turnover approximation approach illustrated in Fig. 4 is based on partialquantity points subdividing the sales turnover curve into multiple sections, for whichsales turnover is linearly approximated. As explained in the previous subsection, xmin

and xmax are given as management-defined control parameters, which indicate theminimum and maximum spot demand that needs to be fulfilled, respectively. The setof partial quantity points i ∈ N has four elements by default: 0, xmin, Q R and xmax,where Q R indicates the total quantity of all forecasted customer quantities (see thealgorithm for determining the price–quantity function in the previous subsection).Note that xmax > Q R expresses the possibility of gaining additional spot marketqauantity at lower sales prices. In the case of xmax = Q R only forecasted orders areconsidered. The three non-zero points are fixed and indexed by imin for xmin, imid

for Q R and imax for xmax. The approximation can be improved by adding additionalpartial quantity points i+ between imin, imid and imid, imax, respectively. Partial spotsales quantities qi are determined at each partial quantity point i ∈ N . Correspondingpartial spot sales turnover yi values are calculated for each partial spot sales quantityqi using the exact sales turnover function. Partial spot sales turnover between twopartial quantity points is approximated based on the spot sales turnover gradient ofthe linear connection for the partial quantity section j = 1, . . . , N − 1 between twopartial quantity points.

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Since the sales turnover curve is concave and the linear sales turnover gradientsdecrease monotonically, no integer variables are required to decide which partial quan-tity section is filled first. The objective function to maximize sales turnover will ensureto fill the partial quantity sections from left to right. It should be noted that the linearsales turnover approximation does not depend on the individual sales price scenario.

3.1.4 Constraints of the sales model

Before the constraints that make up the sales model are presented, the respective nota-tion has to be defined. Note that some variables, e.g., for modeling aggregate salesfigures, are introduced to improve the readability of the model formulation. Thesevariables could be replaced by the corresponding expressions.

Indices, index setsp ∈ P productsl ∈ L locationsl ∈ L S sales locationsi ∈ N partial quantity pointsj = 1, . . . , N − 1 partial quantity sectionst ∈ T periods{p, l} ∈ P LSal valid product–sales location combinations

for sales productsParametersq Sc

plt contract demand quantity forecast for product–sales locationcombination {p, l} and period t

X Ssplt , X

Ssplt minimum and maximum spot sales quantity for product–sales

location combination {p, l} and period t , respectivelyτ Ss

jplt spot sales turnover gradient of the linear sales turnoverapproximation for partial quantity section j , product–saleslocation combination {p, l} and period t

q Ssiplt partial spot sales quantity at partial quantity point i for

product-sales location combination {p, l} and period tpSc

plt contract sales price for product–sales locationcombination {p, l} and period t

Decision variablesx S

plt total sales quantity for product–sales locationcombination {p, l} and period t

x Ssplt spot sales quantity for product–sales location

combination {p, l} and period tx Ss

jplt partial spot sales quantity for partial quantity section jfor product–sales location combination {p, l} andperiod t

ySsplt spot sales turnover for product–sales location

combination {p, l} and period t

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ySsjplt partial spot sales turnover for the partial quantity

section j for product–sales locationcombination {p, l} and period t

In the following, only the constraints of the sales model are presented. Additionalconstraints of the supply model are presented in Sect. 3.2.3. Finally, the objectivefunction for maximizing the profit for the entire value chain network is defined inSect. 3.3. It should be noted that constraints of the sales model do not depend on theindividual spot sales scenario.

The total sales quantity is obtained as the sum of contract sales and spot salesquantity:

x Splt = x Ss

plt + q Scplt ∀{p, l} ∈ P LSal, t ∈ T (1)

The spot sales quantity is limited between minimum and maximum boundaries:

X Ssplt ≤ x Ss

plt ≤ XSsplt ∀{p, l} ∈ P LSal, t ∈ T (2)

The total spot sales quantity equals the sum of the partial spot sales quantities:

x Ssplt =

N−1∑

j=1

x Ssjplt ∀{p, l} ∈ P LSal, t ∈ T (3)

Partial spot sales quantities need to fit in the respective section between consecutivepartial spot sales quantities:

x Ssjplt ≤ q Ss

iplt − q Ssi−1,plt ∀{p, l} ∈ P LSal, i ∈ N , i > 1, j = 1, . . . , N − 1, t ∈ T

(4)

Partial spot sales turnover is given as the product of partial quantity and partial salesturnover gradient:

ySsjplt = τ Ss

jplt · x Ssjplt ∀{p, l} ∈ P LSal, j = 1, . . . , N − 1, t ∈ T (5)

The spot sales turnover equals the sum of the partial spot sales turnovers:

ySsplt =

N−1∑

j=1

ySsjplt ∀{p, l} ∈ P LSal, t ∈ T (6)

Further constraints, for example, on sales contract quantity rules and flexibility arepossible but excluded here.

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3.2 Supply model

3.2.1 Procurement and consumption of raw materials for chemical commodities

Variable raw material consumption rates in productionRaw material consumption in production is traditionally treated as constant basedon given recipe factors. Recipe in the chemical industry is a synonym for the bill-of-material in discrete parts manufacturing and includes all input products with theirrespective input fraction required to produce one unit of one or several output productsin a production process. However, in chemical production the degree of raw materialconsumption rates and hence the recipe factors often depend on the processing mode ofthe equipment, which can be employed at different utilization or throughput levels. Inthis case the recipe is not composed of static input factors but of recipe functions, whichexpress the relationship between the input consumption and the output quantity pro-duced. Hence the problem of how to decide on raw material consumption and how tobalance volatile raw material costs with sales quantities and prices needs to be solved.

Spot and contract procurementRaw materials are procured either based on fixed contracts or on the spot market.Differences in spot and contract prices have been observed in many business sectors,cf., Reiner and Jammernegg (2005). In analogy to the demand side, procurement con-tracts are fixed by quantity and price with the objective to ensure a basic supply of rawmaterials. Spot procurement supports the requirements of the company for flexibilityin supply and sales planning facing uncertain market prices. By utilizing spot procure-ment the company can decide the actual procurement quantity with certain flexibilityaround the offered quantity. Price levels for contracts and spot business differ andare volatile.

3.2.2 Modeling flexible recipes

Key issues of the supply model are to decide on the variable raw material consump-tion rates in production and on spot procurement quantities. Both issues are highlyinterrelated, i.e., high production rates determine the amount of raw material that hasto be supplied. Moreover, raw material costs per output ton produced can grow withhigher production utilization and throughput rates. In the overall context of value chainoptimization, production rates have to comply with decisions reflected by the salesmodel, e.g., on spot sales quantities and prices.

In the following, the basic principle of flexible recipes is presented. To keep theexplanations simple, we consider only one single type of finished product that is pro-duced from one single raw material on one resource at a specific location during agiven period, i.e., indices for input and output products, resources, locations, and peri-ods are omitted. In the real application, however, there are multiple input products.Typically, one input product represents the main feed into the production process whilethe others are auxiliary substances which can be procured on short notice.

Let C denote the production capacity of the resource measured in tons of outputper period and let x in and xout indicate the input of raw material and output of finished

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products, respectively. Capacity utilization is defined as U = xout/C . Minimum utili-zation rates and the capacity as maximum utilization rate have to be maintained. Evenin periods with extremely low demand, production processes must run at a minimumutilization rate to ensure process stability and product quality. A complete shut-downof an asset is technically feasible, for example, in the case of planned maintenanceor in emergency cases but not considered as a planning option in regular operations.In many types of chemical mass production, raw material consumption depends onthe utilization rate of the equipment employed. Hence, linear recipe functions canbe derived, which indicate the input of raw material required to produce the desiredamount of output.

In Table 2 the derivation of linear recipe functions is explained using a numericalexample. Utilization rates are given in steps of 20% assuming that all rates are usedwith equal probability. Capacity is given at 1,440 tons per day. The next two rowsindicate pairs of input and output quantities for each utilization rate. These figures canbe derived from the technological parameters of the production equipment. The recipefactor is defined as the ratio of input to output quantities. Note that recipe factors onlyrefer to the main raw material and do not include other input materials. This explainsthe value of the recipe factor of less than 1.0 for U = 20%. Finally, linear regression isapplied with respect to the recipe factors. As a result, a variable consumption factor ofa = 1.3 and a constant factor of b = −144 are obtained based on the given utilizationrates and the underlying technological parameters.

The special case of a static recipe is given for b = 0. In this case, the raw materialconsumption does not change with capacity utilization. Otherwise the recipe factorgrows with increasing resource utilization. The linear recipe function for the exampleof Table 2 is illustrated in Fig. 5. As a reference case, the static recipe is shown. Linearrecipe functions are one type of recipe found in chemical industry. Of course otherforms of recipe functions are possible depending on the consumption pattern analyzedfor a specific resource.

In the supply model production input and output quantities depend on each otheraccording to the recipe functions. Output and utilization decisions determine the rawmaterial quantities to be supplied at each location. As mentioned before, raw materi-als are procured in both spot and contract mode. While contract procurement needsto be executed as agreed, spot procurement is flexible with minimum and maximumquantities for each type of raw material and each product–location combination. Thecompany decides on spot procurement quantities within these intervals.

Table 2 Linear recipe function example

Utilization rate U 20% 40% 60% 80% 100%

Capacity C 1,440 1,440 1,440 1,440 1,440

Raw material input quantity x in 230 605 979 1,354 1,728

Production output quantity xout 288 576 864 1,152 1,440

Recipe factor (x in/xout) 0.80 1.05 1.13 1.18 1.20

Linear regression w.r.t. recipe factors x in = 1.3 · xout − 144; a = 1.3, b = −144

R2 = 0.99

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Recipefactor

Staticrecipe

Linearrecipe

0

1.0

1.5

0% 20% 40% 60% 80% 100%

Utilization

0,5

Fig. 5 Static recipe and linear recipe function

3.2.3 Constraints of the supply model

Before the constraints which make up the supply model are presented, the respectivenotation has to be defined. Note again that some aggregate (redundant) variables areintroduced to improve the readability of the model formulation.

Indices, index setsp ∈ P productsl ∈ L locationsr ∈ R resourcest ∈ T periodsl ∈ L P production locationsl ∈ L O procurement locationsl ∈ L S sales locations{p, r} ∈ P Rin, P Rout valid input/output product–resource combinations{p, l} ∈ P L in, P Lout valid input/output product–production

location combinations{p, l} ∈ P LProc valid product–procurement location

combinations for procured products{p, l} ∈ P LSal valid product–sales location combinations

for sales productsParametersC P

rt capacity of production resource r in period tU P min

r minimum utilization rate of production resource rapr , bpr parameters of the linear recipe function for

product–resource combination {p, r}dt number of production days in period tcPvar

pr variable production cost per unit for product–resourcecombination {p, r}

cPspotplt average cost rate per unit for spot procurement for

product–procurement location combination {p, l} in period t

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cPconplt average cost rate per unit for contract procurement of

product–procurement location combination {p, l} in period tq Pcon

plt contract procurement quantity for product–procurementlocation combination {p, l} in period t

X Pspotplt , X

Pspotplt minimum and maximum spot procurement quantity for

product–procurement location combination {p, l}in period t , respectively

Decision variablesx Pout

prt production output quantity for product–resource combination{p, r} in period t

x Pinprt production input quantity for product–resource combination

{p, r} in period tx Pout

plt production quantity for product–production location combination{p, l} in period t

x Pinplt secondary demand in product–production location combination

{p, l} in period t

x Pspotplt procurement spot quantity for product–procurement location

combination {p, l} in period txProc

plt procurement quantity for product–procurement location combination{p, l} in period t

x Splt total sales quantity for product–sales location

combination {p, l} and period tvPvar

prt variable production costs for product–production locationcombination {p, l} in period t

vProcplt procurement costs for product–production location combination

{p, l} in period tIn the following, the constraints of the supply model are presented.Capacity and minimum utilization rate limit the total production quantities of all

products produced on the resource in a specific period:

U P minr · C P

rt ≤∑

{p,r ′}∈P Rout :r ′=r

x Poutpr ′t ≤ C P

rt ∀r ∈ R, t ∈ T (7)

The input quantity of intermediate or raw material products required depends onthe production rate of the resource and the linear recipe function which is determinedon a tons per day basis. Hence the number of production days needs to be consideredin constraint (8).

x Pinprt =

⎛

⎝apr ·∑

{p′,r ′}∈P Rout

x Poutp′r ′t

⎞

⎠+ (bpr · dt

) ∀{p, r} ∈ P Rin, t ∈ T (8)

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Production output and input quantities are aggregated at the location level.

x Poutplt =

∑

{p′,r}∈P Rout :p′=p

x Poutp′r t ∀{p, l} ∈ P Lout, t ∈ T (9)

x Pinplt =

∑

{p′,r}∈P Rin:p′=p

x Pinp′r t ∀{p, l} ∈ P L in, t ∈ T (10)

Given the simplified network with two dedicated resources at a single productionlocation these constraints are not required. However, for practical reasons it is impor-tant to keep locations and resources separated, since key effects such as flexible recipesare related to specific resources and their technology rather than to an entire productionlocation.

The total variable production costs are obtained as product of production quantityand variable production cost rate:

vPvarprt = cPvar

pr · x Poutprt ∀{p, r} ∈ P Rout, t ∈ T (11)

Total procurement costs are calculated based on variable spot procurement quanti-ties and fixed contract procurement quantities:

vPvarplt = (x Pspot

plt · cPspotplt )+ (q Pcon

plt · cPconplt ) ∀{p, l} ∈ P LProc, t ∈ T (12)

Total procurement quantities are obtained by summing up spot and contract pro-curement quantities:

xProcplt = x Pspot

plt + q Pconplt ∀{p, l} ∈ P LProc, t ∈ T (13)

Total spot procurement quantity is limited between the minimum and maximumboundaries:

X Pspotplt ≤ x Pspot

plt ≤ XPspotplt ∀{p, l} ∈ P LProc, t ∈ T (14)

The following equation balances total supply quantities consisting of productionand procurement quantities with total demand consisting of total sales quantity and sec-ondary demand of production based on the assumption of single sourcing. In practice,material balances also include inventories and transportation quantities which areimportant in global networks with several weeks lead times and considerable transitinventories. Since we focus on the integration of business functions in value chains,these issues are beyond the scope of this paper.

∑

l ′∈L P :l ′=l

x Poutpl ′t +

∑

l ′∈L O :l ′=l

xProcpl ′t

=∑

l ′∈L P :l ′=l

x Pinpl ′t +

∑

l ′∈L S :l ′=l

x Spl ′t∀{p, l} ∈ P Lout, P L in,

P LProc, P LSal, t ∈ T(15)

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3.3 Optimization strategies

The objective of the proposed modeling approach is to maximize profit for the entirevalue chain network. It is assumed that the company behaves risk-averse in face ofthe price uncertainty and seeks to ensure minimum profits. Two optimization strate-gies can be applied incorporating spot sales price scenarios to reflect price uncertainty(cf. Chen and Lee 2004):

• One-phase optimization: maximize expected profit across one or multiple pricescenarios. This approach corresponds to the classical “expected value” maximiza-tion known from decision theory.

• Two-phase optimization: maximize expected profit across multiple price scenariostaking into account the constraint that a given minimum profit value is reached.From a practical point of view, this approach seems to be more appropriate insituations where a high variability of profit can be expected and the risk of lowprofit outcomes shall be minimized.

The one-phase optimization strategy considers one or multiple spot price scenarios.Each scenario (see Sect. 3.1.2) is characterized by the spot price factor δplst , whichexpresses possible spot price levels, e.g., 0.8, 1.0, and 1.2, for each relevant product–location combination {p, l}, period t and scenario s ∈ S. Each scenario is assigned asubjective probability ωs . While supply decisions remain unchanged, the various spotprice scenarios lead to multiple sales turnover scenarios that are realized with the samespot sales quantity. Since price scenarios are represented by specific price factors, theycan be directly applied to model spot sales turnover in the objective function.

The expected profit determines the average profit across all price scenarios weightedwith their scenario probability ωs . With the notation defined in Sects. 3.1.4 and 3.2.3the expected profit function can be defined as follows:

(16)

The expected profit across multiple scenarios provides a more realistic picture ofthe future profit situation compared to one single scenario. However scenarios areconsolidated and expressed as a single value based on their probability weights. Theplanner would have no information about potential worst case profits and might liketo sacrifice expected profit opportunities for safety in exchange. This is addressed bythe two-phase optimization approach.

The two-phase optimization strategy (see Figure 6) first maximizes the minimumscenario profit zmin, which is lower or equal to all single scenario profits zs , where zs

304


Phase 1 Phase 2

expmax zminmax z

min*sz z≥

subject to subject to

Sszzs ∈∀≥ ,min

min*min zz →

and all other constraints and all other constraints

Phase 1 Phase 2

expmax zminmax z

min*sz z≥

subject to subject to

Sszzs ∈∀≥ ,min

min*min zz →

and all other constraints and all other constraints

Fig. 6 Two-phase optimization strategy

is defined as follows:

zs =∑

t∈T

⎡

⎣∑

{p,l}∈P L S

ySsplt · δplst +

∑

{p,l}∈P L S

pScplt · q Sc

plt

−∑

{p,r}∈P Rout

vPvarprt −

∑

{p,l}∈P LProc

vProcplt

⎤

⎦ (17)

This first phase determines the best minimum profit zmin from all scenarios. zmin isthen fixed as baseline profit zmin ∗ for the second phase of the optimization, where theexpected profit zexp is maximized across all scenarios given the condition that eachscenario profit reaches the minimum scenario zmin ∗. This concept aims to obtain morerobust solutions considering probabilistic demand quantity scenarios.

4 Case study evaluation

The optimization model presented in the previous section was implemented in ILOGOPL Studio 3.71 using CPLEX 9.1 as solver and was tested with industry case dataon an Intel Pentium 4 PC with 1598 MHz and 256 MB RAM. Table 3 indicates thenumber of entities included in the case study evaluation.

For confidentiality reasons data from the company are sanitized in a way that dataused for the case study evaluation are generated reflecting realistic dimensions of theinvestigated business application. However, data used in the simulation show the samescale. Several numerical experiments were carried out in order to analyze the impactof integrating sales and supply decisions by volumes and values based on the devel-oped value chain planning model. Numerical results are presented in the followingsubsections.

4.1 Price scenario experiments

In the first experiment we compare the optimization strategies introduced in subsec-tion 3.3 for different spot price scenarios. Two alternative demand spot price scenarios“best case” and “worst case” with equal probability of 0.25 are defined in addition to

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Table 3 Number of entities inthe case study evaluation

Basic elements Number of entities

Products 50

- Finished 48

- Intermehdiate 1

- Raw material 1

Locations 11

- Sales 9

- Production 1

- Procurement 1

Resources 2

- Continuous 1

- Multi-purpose 1

Periods 6

1-phase optimization

0

100

200

1 2 3 4 5 6Period

Index

Best profit indexExpected profit indexWorst profit indexSales quantity index

2-phase optimization

0

100

200

1 2 3 4 5 6Period

Index

Best profit indexExpected profit indexWorst profit indexSales quantity index

more robust,

less extreme solutions

lessrobust,more

extremesolutions

Fig. 7 Comparison of the 1-phase and 2-phase optimization strategies

the standard scenario with probability 0.5. The best case assumes a continuous priceincrease while the worst case assumes a continuous price decrease. Consequently theexpected profit is the average of the best and worst case scenario results and equiva-lent to the standard scenario result in this special case. Numerical results are shown inFigure 7 for a planning horizon of six periods. Results of the one-phase optimizationstrategy show relatively constant sales quantities and expected profits slightly belowthe index value of 100. The results of the first period are indexed at 100 in order tocompare the results of the subsequent periods with the first period.

Executing this sales plan can lead to very positive best-case scenario profitsbut also to very poor profits, if the worst-case price scenario occurs. Less extremeplans can be reached with the two-phase optimization strategy: scenario profits are lessvariable and the worst case scenario results are comparatively better than in the

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Fig. 8 Elasticity model reactiontest results

89 100125

155187

220

0

100

200

300

400

0.0 0.2 0.4 0.6 0.8 1.0 Elasticity

Profit index

70

80

90

100

110

120

Profit indexSales indexProduction indexProcurement index

Quantity indices

one-phase-optimization strategy. The overall value chain plan in sales, productionand procurement is more cautious with lower sales quantities and lower expectedprofits as the pay-off for better minimum profits.

To conclude, the two-phase optimization results in lower average profits. In the realapplication, planners might also vary the subjective weights for the different scenar-ios or set alternative minimum profit levels. This way additional information on therobustness of the obtained solution and a better understanding of the complex rela-tionships between volumes and values in a price-volatile commodity business couldbe gained.

4.2 Spot price elasticity test

The price elasticity of spot demand for all finished products is varied in multiplescenarios from 0 to 1. The base plan has the elasticity 0.2 and results for the elas-ticity value of 0.2 are indexed at 100. Elasticity of 0.2 means that the average salesprice increases by 2%, if sales volumes are decreased by 10% and vice versa. Experi-ments are conducted applying the sales turnover approximation method with 24 partialquantity points to reach a high accuracy of the approximation as it will be evaluated insubsection 4.4. Experimental results shown in Figure 8 reveal that different elasticitieslead to different optimal profits and quantities in sales, production and procurement,since sales volume-dependent average price effects are considered in the model.

In the specific case the base plan with an elasticity of 0.2 leads to a situation ofunder-utilization of production capacity, since high raw material costs can not alwaysbe compensated by sales prices. Higher elasticities lead to higher sales volumes, capac-ity utilization and profit increase since the relative sales volume increase can be realized

307


with a lower relative average sales price decrease. In this situation it is profit-optimal toincrease production and push additional sales volume into the market with lower salesprices. Full utilization is reached for elasticities of 0.6. Consequently an elasticity of0.0 leads to lower profits, lower sales volumes and production utilization, since theinsufficient sales price level does not change with sales quantity decisions.

Note that higher elasticity leads to higher sales and production volumes in thisspecific case of under-utilization due to the specific raw material prices and recipefunctions. In case of full-utilization and different raw material prices and recipe func-tions, higher elasticities can also lead to a reduction of sales and production volumesif reduction of sales volumes and the respective increase of the average price is profit-optimal compared to supply costs. To conclude, the test demonstrates the influenceelasticity can have on commodity sales and supply decisions and resulting profits.

Our numerical results reveal that considering average price effects reflected byelasticities can have significant influence on the overall volume plan. Hence, a profit-optimal supply and production plan does not necessarily maximize capacity utiliza-tion. Therefore, firms should not try to change or reduce elasticities but consider themin their sales and production planning taking the profit impact of price effects intoaccount. Focusing on volumes alone and not considering existing price elasticity willlead to suboptimal plans and reduced profits.

4.3 Raw material price experiments

The influence of raw material prices on profit and utilization is investigated in thethird experiment. Production capacity appears to be a bottleneck not sufficient to servedemand with full spot sales flexibility and elasticity of 0.2. Prices for the raw materialrequired in the intermediate production process are varied around a basis index of 100from 80 to 140. Two raw material recipe scenarios are considered: a static recipe fac-tor of 1.2 and a linear recipe function, where raw material consumption rates increasefrom 0.8 at 50% utilization to 1.2 at 100% utilization. Figure 9 shows the results ofthe raw material price scenario experiments.

It is obvious from Fig. 9 that profit decreases in all cases, the more procurementprices increase. However the static recipe leads to comparatively higher sales and pro-duction volumes and lower profits compared to the case with the linear recipe function.The reason is that the static recipe represents the maximum factor of the recipe func-tion that does not change. In comparison raw material consumption rates and costscan be decreased in the case of linear recipe functions by reducing the productionutilization. Therefore, all volume indices are reduced in the case of linear raw materialconsumption, since raw material costs due to higher prices can be saved loweringproduction and raw material consumption. The opposite effect occurs if the maximumvalue of the recipe function values is higher than the static recipe factor. In both casesraw material unit costs cannot be directly allocated to production output as basis forproduct profitability and contribution margin analysis since raw material quantitiesand costs depend on overall value chain planning decisions. Our numerical resultsreveal that a recipe function with different raw material consumption rates dependingon production utilization has a major impact on the optimal profit and on capacity

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Linear recipe function

142120

10083 70 60 53

0

100

200

300

400

80 90 100 110 120 130 140 Rawmaterial

price index

Profit index

-100

-50

0

50

100


Quantity indices

Static recipe

143121

10079

5942 28

0

100

200

300

400

80 90 100 110 120 130 140 Raw material

price index

Profit index

-100

-50

0

50

100


Quantity indices

Fig. 9 Raw material price model reaction test results

Table 4 Sales turnover approximation model performance test

Partial quantity points 4 6 8 14 24 44 64

Constraints (thou.) 27 37 48 79 131 234 338

Variables (thou.) 24 35 45 76 128 232 335

Solution time (s) 1 2 2 5 28 118 276

Profit gap (%) 10.79 2.35 1.05 0.25 0.07 0.01 Basis

utilization. Not considering these dynamics would endanger a company’s profitabilitywhen focusing only on maximizing production utilization.

4.4 Accuracy of the sales turnover approximation

Finally, the accuracy of the piecewise linear sales turnover approximation methodis tested using the industry test data set and elasticities of 0.2. The number of partialquantity points is varied from 4 to 64 as shown in Table 4. Numerical results reveal thatalready 24 partial quantity points are sufficient to reach 99.93% of the objective func-tion value obtained for the very accurate approximation based on 64 partial quantitypoints. The approximation is even more accurate if sales quantity flexibility is close tothe forecast point. The approximation is less accurate, if spot sales quantities can becut entirely as in the test data, since the sales turnover curve has highest gradients nearthe point of origin, where the gap between actual and approximated sales turnover ishighest.

Considering the tactical planning purpose, run times of 1 min or less are acceptablein practice. These short run times even enable a planner to evaluate different scenarios,i.e., running the model with different parameter settings. To utilize the scenario mode

309


of the optimization model, a small number of partial quantity points would sufficethus permitting solutions within only a few seconds.

4.5 Industrial application

As mentioned before, the presented optimization model has been developed as a pro-totype model to support the introduction of the Supply Network Planning module ofan advanced planning system. In particular, our model helps to determine the requiredscope of the Supply Network Planning module implementation, to reveal the necessityof customizing the standard advanced planning software, and to evaluate the possiblebenefits for the company. Hence, the focus of our model formulation was to reflect thecompany’s key optimization problem, namely balancing the consumption of a basicprice-volatile raw material which is processed in continuous production mode withoutput volumes of more than 1 Mio. tons per year and coordinating the respectivesales, production and procurement activities.

Prior to the implementation of an enhanced optimization model, the industrial com-pany started a major business reorganization project in order to improve the coordina-tion of business functions from procurement to sales for their global production sitesand sales representations. In the course of this project several of the key instrumentsincluded in our model formulation were put into practice, in particular, the conceptof spot and contract demand management and the instruments of demand elasticitiesand turnover functions as well as linear raw material recipe functions. Major effectsof their application are the following.

• Changing the planning philosophy from pure demand fulfilment, which can beseen as the traditional supply chain management orientation, towards focussing onthe global value chain profit by introducing demand management concepts basedon the differentiation between spot and contract demand with active spot salesdecisions helped the company to turn around the loss-making business unit into ahighly profitable one.

• Recognizing the effects of spot demand elasticity and applying them in an inte-grated value chain planning effort provided the company additional insights intothe dynamics of the global markets they are operating in. In fact, several marketsshow very high elasticities with significant price differences for the same productwhile demand on other markets is fairly insensitive to sales prices. These insightshelped the company to make better sales decisions. Specifically, in the case ofsupply shortages decisions on cutting spot sales volumes could directly be derivedfrom the model calculations.

• Incorporating linear raw material consumption functions and variable prices intothe value chain planning model directly identified potential cost savings of severalMio. $ per year. Formerly, the company used to fully utilize production capac-ity. After gaining insights from the model application into the interdependenciesbetween procurement and sales volume and prices and the use of different produc-tion modes, the company recognized that this is not necessarily the profit-optimalproduction strategy. Now managers seek to determine differentiated profit-optimalutilization levels for their key production assets at three global sites. As a result,

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production volumes are shifted from less resource-efficient assets to the moreefficient ones in the global network.

• The basic model is used by three global value chain planners for monthly plan-ning of a global business unit. The model helped them to better understand theprofitability levers in the value chain network from procurement to sales.

Further benefits are expected from introducing the Supply Network Planning moduleof an advanced planning software system to be used jointly with the Demand Planningand ATP/CTP modules which have already been implemented.

5 Summary and outlook

In this paper a model is presented to coordinate sales and supply decisions for com-modities in a chemical industry value chain. Price–quantity functions, volatile anduncertain prices, flexible quantities in sales, procurement and production as well as uti-lization-dependent recipes create complex interdependencies which make it extremelydifficult for the human planner to determine profit-optimal network-wide sales andsupply plans even for small-sized value chain networks. Price–quantity function elas-ticities support decisions toward sales volume reductions or increases considering theeffect of increasing or decreasing average prices. We evaluated the piecewise linearapproximation approach to decide on sales turnover with sales price and volumesas variables. The approximation delivered very accurate results within short solutiontimes and thus can be seen as an efficient approach to solve the underlying quadraticoptimization problem. Variable raw material recipes have a direct impact on volumesand values, if raw material prices cannot be compensated by sales prices. Applyingtwo-phase optimization strategies for sales price scenarios leads to more robust plansensuring target profitability even in case of worst-case prices with the pay-off of morecautious and lower expected profits.

The model presented in this paper has been implemented by the company as basisfor numerical investigations. The company has extended the basic model with furtherfeatures such as inventory balances and transportation activities as well as exchangerates and further specifics of chemical commodity production such as throughputsmoothing. Contract and spot sales planning has been implemented in their APS-based demand planning system and procurement planning for key raw materials havebeen established by their global purchasing department. Implementing these integratedsales and supply planning tools has shown major effects on the overall profitabilityof the business unit. Specifically the spot price mechanism used to better coordinatesales and supply decisions showed a major impact for the company.

Integrating sales and supply decisions throughout the value chain poses newinterdisciplinary research questions as an outlook. Neither supply network planningminimizing costs to fulfill given demand nor revenue management maximizing rev-enue based on a given supply adequately addresses the problem of managing anindustrial value chain end-to-end by volumes and value. Business rules for sellingproduction output profit-optimal in contract or spot business as well as alternativemethods to model price–quantity functions considering the impact on supply andprofit are potential further areas of research. The overall research focus may shift from

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supply flexibility and cost minimization towards end-to-end supply, sales and pricingdecisions to utilize the value chain in the most profitable way.

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MILP-based campaign scheduling in a specialtychemicals plant: a case study

Marcus Brandenburg · Franz-Josef Tölle

Abstract Supply chain management in chemical process industry focuses onproduction planning and scheduling to reduce production cost and inventories andsimultaneously increase the utilization of production capacities and the service level.These objectives and the specific characteristics of chemical production processesresult in complex planning problems. To handle this complexity, advanced planningsystems (APS) are implemented and often enhanced by tailor-made optimization algo-rithms. In this article, we focus on a real-world problem of production planning arisingfrom a specialty chemicals plant. Formulations for finished products comprise severalproduction and refinement processes which result in all types of material flows. Mostprocesses cannot be operated on only one multi-purpose facility, but on a choice ofdifferent facilities. Due to sequence dependencies, several batches of identical pro-cesses are grouped together to form production campaigns. We describe a methodfor multicriteria optimization of short- and mid-term production campaign schedulingwhich is based on a time-continuous MILP formulation. In a preparatory step, deter-ministic algorithms calculate the structures of the formulations and solve the bills ofmaterial for each primary demand. The facility selection for each production campaignis done in a first MILP step. Optimized campaign scheduling is performed in a secondstep, which again is based on MILP. We show how this method can be successfullyadapted to compute optimized schedules even for problem instances of real-worldsize, and we furthermore outline implementation issues including integration with anAPS.

M. Brandenburg (B)Beiersdorf AG, Unnastr. 48, 20245 Hamburg, Germanye-mail: [email protected]

F.-J. TölleBayer Business Services GmbH, 51368 Leverkusen, Germanye-mail: [email protected]



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OR Spectrum (2009) 31:141–166DOI 10.1007/s00291-007-0084-5



Keywords Supply chain management · Campaign planning and scheduling ·Chemical process industry ·MILP · Advanced planning systems

A list of symbols

Master data/problem parameters

F Set of facilitiesf ∈ F FacilityP Set of productsp ∈ P ProductA Set of processesa ∈ A ProcessA f ⊆ A Set of processes that can be operated on a facility f ∈ FFa ⊆ F Set of facilities that can operate process a ∈ Aza, f Cycle time of process a ∈ A operated on facility f ∈ Fba, f Batch size of process a ∈ A operated on facility f ∈ Fap ∈ A Unique process (apart from refinement) that produces product

p ∈ PP−a , P+a ⊆ P Set of input, output products of process a ∈ Aδ−a,p and δ+a,p Input, output amounts of product p ∈ P for process a ∈ A

(fractions of batch size ba, f )Bp = (Ap, Pp, Fp) Formulation of product p ∈ PPp = ∪p∈Ap P+a Set of output products of the processes in Ap

Ap Processes that have to be operated to produce product p ∈ PFp = ∪a∈Ap Fa Set of facilities on which processes of Ap can be operatedsta,a′ Duration of the set up activity required between processes

a, a′ ∈ AR ⊆ A Set of processes which have the refinement propertyp∗ Off spec producta∗ Refinement processR∗ Set of refinement processes

Transactional data/instance parameters

E Set of order elementsε ∈ E Order elementπε Product ordered by order element ε ∈ Eqε Order quantity of order element ε ∈ Etε Due date of order element ε ∈ Ec Production campaignts Starting time of a campaigntc Completion time of a campaignn Number of batches of a campaign (“campaign size”)v Set up indicator of a campaign (v = 1 ⇔ set up activity is

performed before campaign c starts)

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MILP-based campaign scheduling in a specialty chemicals plant: a case study

Cε Set of all campaigns linked to order element ε ∈ E (“campaign chain”)C∗ε Set of all possible campaign chains for order element ε ∈ Ei(p, t) Inventory level for product p ∈ P at time t ≥ 0t f Earliest availability date of facility f ∈ F

Feasible solution

T Vector of earliest availability times (t f ) f ∈F

I Set of initial inventory levels {i(p, 0) | p ∈ P}S Schedule

Algorithm 1

Bπ Formulation for product π ∈ Plp Manufacturing level of product p ∈ Pla Manufacturing level of process a ∈ Almax Maximum manufacturing level of all processes a ∈ Aπ

Fε Set of all possible facility combinations for order ε ∈ EFi

ε Set of i-th possible facility combination for order ε ∈ E

Algorithm 2

i p Algorithm parameterimin

p Algorithm parameterαn Algorithm parameterna Algorithm parameter

MILP 1

x > (Ci ) Selection indicator for campaign Ci ∈ Cε

w Workload variable

Algorithm 3

wc,c′ Minimum delay times between campaigns c, c′ ∈ Cε

γk Algorithm parameter

MILP 2

ts, t ′s ≥ 0 Starting times of campaigns c, c′ ∈ ∪ε∈E Cε

tc, t ′c ≥ 0 Completion times of campaigns c, c′ ∈ ∪ε∈E Cε

αε ≥ 0 Lateness of order ε ∈ Exc,c′ ∈ {0, 1} Sequence indicatorm ≥ 0 Timespanhc(X) Holding cost for products in X ⊆ Ptc(ε) Lateness cost of demand element ε ∈ Esca,a′ Cost for set up activity between processes a, a′ ∈ Amc Timespan costT Duration of all campaigns in the schedule T =∑

ε∈E∑

c∈Cεn · za, f

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1 Introduction

Supply chain management (SCM) has turned out to be not only a good lever forcost cutting and cash generating, but also provides a competitive advantage withindifferent industries. This has made most companies focus on SCM and has resulted ina higher orientation to the SCM core processes plan, source, make, deliver and return,as described by Supply Chain Council (2004), which comprise strategic, tactical andoperational levels. Many disciplines help to improve opportunities for SCM excellence.In particular, IT progress enabled the development of advanced planning systems(APS) which enhanced transactional enterprise resource planning (ERP) systems andreplaced classical MRP II systems.

Different characteristics of process industry push production planning and sche-duling into the forefront of tactical and operational supply chain planning resultingin special requirements for APS and, often, in tailor-made models for decision sup-port. The large variety of industry-specific characteristics in chemical production andthe huge number of combinations occurring in the real world result in many differentplanning problems of NP-hard complexity. These can be solved by applyingeither standardized optimization methods with relatively limited optimization resultsor problem-specific algorithms involving a relatively high development, implementa-tion and maintenance effort. The latter are differentiated between exact or deterministicmethods of mathematical optimization (incl. MILP, MINLP, graph theory, constraintprogramming) and meta-heuristics (incl. evolutionary strategies, tabu search, simula-ted annealing), both divided into off-line and on-line algorithms.

It is beyond the scope of this article to review all the literature on production plan-ning and scheduling in a process industry; therefore, we focus on only a small selectionof publications. A thorough introduction to the main concepts of APS is given by Drexlet al. (1994) or Stadtler and Kilger (2005). Günther (2005) gives a good overview ofthe architecture and applications of APS, Tempelmeier (2001) or Meyr et al. (2005a)analyze the structure of a typical APS. Meyr et al. (2005b) compares selected commer-cial APS, details on SAP-based SCM are given by Knolmayer et al. (2002), Bartschand Bickenbach (2001) or Kallrath and Maindl (2006). Reklaitis (1996) or Kallrath(2003a) give good introductions to planning and scheduling in the process industry,Neumann et al. (2003) describe resulting requirements to APS. Different case studies,e.g. Altrichter and Caillet (2005), Reuter (2005) or Richter and Stockrahm (2005),show the application of SAP APO in the process industry. Grossmann (2005) shows anew perspective on enterprise-wide optimization. An overview of different optimiza-tion concepts for scheduling problems is given by Drexl and Kimms (1997), Kolischand Padman (2001), Shah (1998), Pinto and Grossmann (1998), Grossmann et al.(2002) and Mendez et al. (2006). Complexity issues are considered by Monma andPotts (1989) and Pekny and Reklaitis (1998).

In this article, we describe an optimization method for a real-world problem ofshort- and mid-term production planning and scheduling arising from a specialty che-micals plant. The optimization method is based on a time-continuous MILP modelformulation. Section 2 provides the problem statement and outlines the solutionapproach, Section 3 explains the model formulation and the optimization method.

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Section 4 contains computation results and implementation issues, while Sect. 5details the conclusions.

2 Problem statement

2.1 Application environment

The problem considered in this article arises from a specialty chemicals plant pro-ducing fragrances, flavors and aroma chemicals. The plant comprises a number ofnon-identical multi-purpose single-activity facilities on which production processescan be operated in non-pre-emptive batch mode. To produce a finished good, differentproducts, processes and facilities have to be considered, all depending on the formu-lation for the finished product. Apart from refinement, the process that produces aproduct is unique, and each process and each product can be assigned to a uniqueformulation (“formulation encapsulation”). The number of processes and manufactu-ring levels are formulation-dependent, and both differ greatly from one formulation toanother. The facilities cannot be assigned to a certain process, formulation or manu-facturing level, and most processes can be performed on a choice of different facilities,resulting in batch sizes and cycle times that depend on the process as well as on theselected facility. Some processes have a recycling property, i.e. an input product isnot transformed completely but a fraction of this amount is discharged unchangedwhen the process ends. Other processes run in joint production and release more thanone output product. Some of these joint products—so-called off spec products—canbe upgraded in a refinement process to achieve a product that fully meets the qualityrequirements of the output of the main process. These characteristics result in all typesof material flows (linear, convergent, divergent, cyclic).

On the basis of the state task network (STN) representation proposed by Kondiliet al. (1993), an example of a typical formulation scheme is depicted in Fig. 1. A setup activity has to be performed between two processes scheduled on the same facilityif and only if these two processes are not part of the same formulation. To reducethese set up activities, several batches of identical processes are grouped togetherto form (production) campaigns, i.e. they are operated on the same facility withoutinterruptions or idle times in between. Storage constraints for products do not have tobe considered for either shelf-life or storage capacity.

2.2 Formal description of the problem

The problem addressed here can roughly be stated as follows: Given

1. a set of production facilities with corresponding capacities,2. a set of products that can be processed on these facilities, associated constraints,

production parameters, and bills of material (BOM),3. a set of demand elements (both forecasts and customer orders), and4. a set of penalties and cost functions.

Provide a feasible schedule which simultaneously ensures that

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Fig. 1 Scheme of a typical formulation

1. enough material at each manufacturing level is produced to satisfy all primary andsecondary demands,

2. all production constraints, resource availabilities, and business requirements arerespected,

3. good solutions in a practical sense with regard to the multiple conflicting objectivesare obtained, and

4. identical batches are grouped to form campaigns.

2.2.1 Master data/problem parameters

Let F denote the set of facilities and P the set of products. Furthermore, let A be theset of processes with subsets A f ⊆ A of processes that can be operated on a facilityf ∈ F and subsets Fa ⊆ F of facilities that can operate process a ∈ A. A processa ∈ A operated on facility f ∈ F has a fixed cycle time za, f and a fixed batch sizeba, f , both depending on the specific process and the chosen facility. Let ap ∈ A bethe unique process (apart from refinement) that produces product p ∈ P with inputand output products given by the subsets P−a and P+a ⊆ P respectively in amountsδ−a,p and δ+a,p respectively (fractions of batch size ba, f ). A formulation for finishedproduct p ∈ P is denoted by Bp = (Ap, Pp, Fp) with a set Ap ⊆ A of processesthat have to be operated to produce p ∈ P , a set Pp = ∪p∈Ap P+a of output productsof the processes in Ap and a set Fp = ∪a∈Ap Fa of facilities on which the processesof Ap can be operated. For each pair of processes a, a′ ∈ A, let sta,a′ denote theduration of the set up activity required between a and a′. Let R ⊆ A denote the setof processes which have the refinement property, i.e. for a ∈ R there exist outputproducts p, p∗ ∈ P+a \P−a (p∗ off spec product) and a refinement process a∗ ∈ Awith the properties p ∈ P+a∗\P−a∗ and p∗ ∈ P−a∗ . Let R∗ denote the set of all refinementprocesses.

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2.2.2 Transactional data/instance parameters

The demand elements are reflected by a set E = {(πε, qε, tε) |πε ∈ P; qε > 0; tε > 0}of order elements ε = (πε, qε, tε) requiring a quantity qε > 0 of product πε ∈ Pat due date tε > 0. A campaign c = (ts, tc, n, v, a, f ) consists of n ∈ N batches ofprocess a ∈ A f operated without interruption or idle time on facility f ∈ Fa betweenthe starting time ts and the completion time tc and is sometimes preceded by a set upactivity (v = 1 ⇔ set up activity is performed before campaign c starts). For orderelement ε = (πε, qε, tε) ∈ E , let Cε = {c = (ts, tc, n, v, a, f ) | ts > 0, tc > 0, n ∈N, v ∈ {0, 1}, a ∈ Aπ , f ∈ Fa} be the set of all campaigns linked to the demandelement ε ∈ E , in the following such a set will be called a campaign chain. Due to thefact that |Fa | > 1 for most of all a ∈ Ap, there might be more than only one possiblecampaign chain for a demand element ε ∈ E . Let C∗ε denote the set of all possiblecampaign chains for ε ∈ E . Let i(p, t) denote the inventory level for product p ∈ Pat time t ≥ 0 and let t f denote the earliest availability date for facility f ∈ F .

A campaign c′ ∈ S with processes a′ ∈ A is called formulation successor of acampaign c ∈ S with processes a ∈ A if c′ consumes products that are produced byc, i.e. if P−a′ ∩ P+a = ∅. A campaign c′ ∈ S is defined as the schedule successor of acampaign c ∈ S if c and c′ are scheduled on the same facility f ∈ F in such a waythat (i) ts ≤ t ′s and (ii) t∗e < ts or t ′e < t∗s for all other campaigns c∗ scheduled on f .

2.2.3 Properties of a feasible solution

Given an instance (E, T, I ) with a set E of order elements ε ∈ E , a vector T =(t f ) f ∈F of earliest availability dates for the facilities f ∈ F and a set I = {i(p, 0) | p ∈P} of initial inventory levels, a schedule S = {(ts, tc, n, v, a, f ) | ts > 0, tc > 0, n ∈N, v ∈ {0, 1}, a ∈ A, f ∈ Fa} consisting of campaign elements c ∈ S is calledfeasible if the following conditions hold:

1. Primary demands are satisfied regarding quantity produced allowing backloggingfor each product, i.e. for each π ⊆ ε, ε ∈ E :

i(π, 0)+∑

c∈S

δ+a,π · ba, f · n −∑

c∈S

δ−a,π · ba, f · n ≥∑

ε∈E,π⊆ε

qε

2. Inventory levels are always non-negative, i.e. for all p ∈ P, t ≥ 0:

i(p, t) ≥ 0

3. At each facility f ∈ F , at most one activity (process or set up) is scheduled at atime, i.e. for each pair of campaigns c, c′ ∈ S scheduled on f ∈ F with ts ≤ t ′s :

tc ≤ t ′s − v′ · sta,a′

4. Set up activities are scheduled if required, i.e. for each campaign c and its schedulesuccessor c′ scheduled on f ∈ F with ts ≤ t ′s:

v′ = 1⇔ {a, a′} ⊆ Ap ∀p ∈ P (a and a′ do not belong to the same formulation)

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5. The earliest availability date is respected for each facility, i.e. for each f ∈ F andeach c ∈ S scheduled on f :

ts − v · sta′,a ≥ t f ∀a′ ∈ A

Besides the mathematical aspect of solution quality, which can be easily measured byan objective function, the acceptance of the solution by the supply chain planner inthe plant is a decisive quality criterion. Some key user requirements could be consi-dered easily by constraints or specific properties of the solution approach, e.g. therequirement to have at most one campaign for each reaction within a campaign chain.Other requirements were realized by fine tuning the solution approach, e.g. definingappropriate parameter settings for the objective function in accordance with the keyusers.

2.3 Solution methodology

Numerous approaches to solve such problems exist, and these apply different mathe-matical optimization methods and representations of the production process and theplanning horizon. Benchmark problems are described by e.g. Kondili et al. (1993),Shah et al. (1993), Papageorgiou and Pantelides (1993) or Kallrath (2003a). A sur-vey of time-continuous versus time-discrete approaches is given by Floudas and Lin(2004). A process representation based on state task networks (STN) is introduced byKondili et al. (1993), Pantelides (1994) proposed the resource task network (RTN).Applications of STN or RTN are evaluated by e.g. Shah et al. (1993), Schilling andPantelides (1996), Dimitriadis et al. (1998) and Giannelos and Georgiadis (2002).

MILP-based mathematical optimization concepts are reviewed by Floudas and Lin(2005), applications are shown by e.g. by Blömer and Günther (1998, 2000), Burkardet al. (1998a,b), Hui et al. (2000), Gupta and Karimi (2003), Yi and Reklaitis (2003)and Burkard and Hatzl (2005, 2006). Kallrath (2003b) presents an MILP-based methodfor combined strategic and operational planning, Karimi and McDonald (1997a,b)apply MILP for integrated mid-term planning and short-term scheduling. Meyr (2004)introduces a MILP-based approach for combined lot sizing and sequencing.

Decomposition techniques are introduced by Harjunkoski and Grossmann (2002),Maravelias (2006) or Castro et al. (2005). Maravelias and Grossmann (2004a,b) orTimpe (2003) present hybrid MIP/CP algorithms, Schulz et al. (1998) or Alle et al.(2003) apply MINLP-based methods. Campaign planning and scheduling is discussedin detail by Papageorgiou and Pantelides (1996a,b) or Oh and Karimi (2001a,b), in-dustrial applications are described by Berning et al. (2003) applying genetic algorithmtechniques or by Grunow et al. (2003a,b) based on MILP formulations.

We will present a time-continuous MILP-based solution approach for campaignscheduling. The basic idea is as follows:

1. Identical batches for the same primary or secondary demand are grouped to forma campaign.

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2. For each primary demand ε ∈ E , a campaign chain Cε , i.e. a set of campaigns, isdetermined in such a way that– Cε contains at most one campaign for each process,– including initial inventory, the amounts produced by these campaigns are suf-

ficient to satisfy the primary demand qε and all secondary demands arisingfrom ε, and

– no campaign of Cε feeds other campaigns that do not belong to Cε .In the following, the property described by the last two bullet points will be calledorder encapsulation of a campaign chain. The formulation encapsulation propertyof the considered problem facilitates the generation of order specific campaignchains. In the event that some of the processes of Aπ can be operated on a choice ofdifferent facilities, different campaign chains are calculated independently fromeach other, one chain for each possible facility combination. For each possiblechain, this calculation determines the number of batches of which each campaignconsists. The two possible campaign chains for the example formulation shownabove are depicted in Fig. 2 (demand quantity q = 20 t, no initial inventoriesi(p, 0) = 0 for all p ∈ P).

3. For each primary demand ε ∈ E , exactly one set Cε of campaigns is selected tosatisfy ε. For each a ∈ Aπ this determines the facility f ∈ Fa on which a isoperated.

4. Starting times ts, completion times tc and set up activities v are determined for allcampaigns in ∪ε∈E Cε in such a way that the resulting schedule is feasible.

Steps 1 and 2 are realized by solving BOM for all orders and all possible facilitycombinations; steps 3 and 4 are realized by solving MILPs. Complexity reduction isthe main reason for dividing the facility selection and the campaign scheduling intotwo MILPs. Solving both problems simultaneously would significantly increase the

Fig. 2 Two possible campaign chains for the example formulation

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number of variables and constraints or result in non-linearities requiring a completelydifferent solution approach. Furthermore, it would be even more difficult to apply thesolution method to real-world instances, which was already a challenging requirement.

3 Model formulation and solution approach

3.1 Campaign chain creation and selection

At first, all possible campaign chains C∗ε for each demand element ε ∈ E are calcula-ted. On the basis of the structure of each formulation, i.e. its products and processes andtheir corresponding manufacturing levels, the sizes of all campaigns, i.e. the numberof batches of which each campaign consists, are calculated. This calculation is perfor-med for all possible facility combinations taking into account the available inventory.Having determined the sizes of all possible campaigns, just one set of campaigns isselected for each demand element by solving MILP 1.

3.1.1 Campaign chain creation

Algorithm 1 is performed for each demand element ε = (π, q, te) ∈ E in order todetermine the structure of the formulation Bπ = (Aπ , Pπ ,∪a∈Aπ Fa) and the manu-facturing levels l p and la for each product p ∈ Pπ respectively for each a ∈ Aπ :

Algorithm 1 (Determination of formulation structures)

Pπ = {π}, Aπ = {aπ }, P ′ = {π}, lπ = 0WHILE P ′ = 0 DO

P ′ = P ′\{p}Aπ = Aπ ∪ {ap}IF ap ∈ R THEN

Aπ = Aπ ∪ {a∗p}Pπ = Pπ ∪ P+a∗p

FOR p′ ∈ P−apDO

P ′ = P ′ ∪ {p′}l p = l p + 1Pπ = Pπ ∪ {p′}

le = max{l p | p ∈ Pπ }FOR a ∈ Aπ DO

la = max{l p | p ∈ P+a }FOR k = 1 TO le DO

Ak = {a | a ∈ Aπ , la = k}lmax = max{la | a ∈ Ak}Having performed Algorithm 1 for each demand element ε ∈ E , the set Fε = {Fi

ε | 1 ≤i ≤ �a∈Aπ |Fa |} of all possible facility combinations can now easily be determined bythe Cartesian product Fε = ×a∈Aπ Fa . Each of these facility combinations Fi

ε results

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in just one campaign chain Cε which can satisfy the primary demand and all of itsresulting secondary demands.

Algorithm 2 is performed for each ε ∈ E to determine the required sizes of thesecampaigns. To keep them small, Algorithm 2 will ensure that possibilities for off specproduct refinement and consumption of available initial inventory are both used to themaximum extent. For that reason, it is required to update the initial inventory i(p, 0)

after each iteration of Algorithm 2 on ε ∈ E . In the event that different chains withdifferent batch sizes are possible for the same demand element, this update will bedone with the minimum amount imin

p of inventory that is left when scheduling one ofthe possible chains.

ALGORITHM 2 (Calculation of campaign sizes)

FOR p ∈ Pπ DOimin

p = i(p, 0)

FOR Fiε ∈ Fε DOFOR p ∈ Pπ DO

i p = i(p, 0)

qp = 0qπ = qε

FOR k = 0 TO lmax DOFOR a ∈ Ak DO

FOR p ∈ P+a DOIF i p < qp THEN

qp = qp − i p

i p = 0ELSE

i p = i p − qp

qp = 0na = max{ qp

δ+a,p ·ba, fa� | p ∈ P+a , fa given by Fi

ε }n∗a = 0IF a ∈ R THEN

na = min{n ∈ N | δ+a,p · ba, fa · n + αn · ba, fa ≥ qp}with αn = � n·δ+a,p∗ ·ba, fa+i p∗

δ−a∗,p∗ ·ba∗, fa∗�

na∗ = qp−na ·δ+a,p ·ba, fa

δ+a∗,p ·ba∗, fa∗�

i p∗ = i p∗ + δ+a,p∗ · ba, fa · na − δ−a∗,p∗ · ba∗, fa∗ · na∗FOR p ∈ P−a DO

qp = qp + (δ−a,p − δ+a,p) · ba, fa · na + δ+a,p · ba, fa

FOR p ∈ Pπ DOimin

p = min{iminp , i p}

FOR p ∈ Pπ DOi(p, 0) = imin

p

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3.1.2 Campaign chain selection

For each demand element, just one campaign chain fulfilling the primary demand aswell as all arising secondary demands has to be selected. The objective of this selec-tion is to ensure balanced use of the available facilities instead of putting most of theworkload on only a few facilities. The selection is done by solving the MILP 1 thatminimizes the maximum facility-specific workload.

MILP 1 (Campaign Selection)

Minimize

w

Subject to

1. Just one campaign chain for each order element selected:

∑

Ci∈Cε

x(Ci ) = 1 ∀ε ∈ E

2. Workload constraint considered:

∑

ε∈E

∑

Ci∈Cε

∑

c∈Ci

n · za, f · x(Ci ) ≤ w ∀ f ∈ F

Domains

w ≥ 0 Workloadx(Ci ) ∈ {0, 1} Selection indicator.

Minimizing the workload has shown to be a reasonable objective for two reasons:

– A consideration of cost parameters in the MILP 1 is not necessary, because thefacilities have a comparable cost structure. In case that major differences in coststructures occur, the constraints of type 2 could be weighted by a facility-specificcost parameter.

– Balancing the workload allows to reduce the timespan of the campaign scheduleobtained by MILP 2. It furthermore allows a better synchronization of subsequentcampaigns resulting in less inventory and enables earlier order fulfilment whichhelps reduce order tardiness.

After solving MILP 1 the selected facility combination Fiε is determined for each

order ε ∈ E . Performing Algorithm 2 again for only the fixed facility combination Fiε

ensures optimized use of available initial inventory because iminp does not have to be

considered in this second algorithm run.

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3.2 Campaign scheduling

Having fixed the sizes and facilities for each campaign, the last step is to determinestarting and completion times for each campaign in such a way that the resultingschedule is feasible. One way to ensure inventory feasibility is to formulate massbalances explicitly by hard constraints in the MILP-based step. To reduce the numberof constraints in MILP 2, we chose a different approach: for each demand elementε ∈ E , the chain Cε has the order encapsulation property. Due to this, the requirednon-negativity of inventory levels is ensured implicitly if appropriate minimum delaysbetween a campaign c ∈ Cε and all of the formulation successors in Cε are respected,inventory-based interdependencies between campaigns in different chains do not exist.These delays are calculated using Algorithm 3 and then introduced, as well as capacitylimitations and set up requirements, into MILP 2 with hard constraints.

3.2.1 Calculation of minimum delay times

Algorithm 3 is performed for each demand element ε ∈ E and each campaign c ∈ Cε

to determine minimum delays between starting times of c and each of its formulationsuccessors in Cε . This calculation follows the assumption that, in the event that refi-nement processes have to be considered in a formulation, these refinement processesfeed the last batches of their formulation successors.

Algorithm 3 (Minimum delay times)

Unless indicated otherwise, c and c′ indicate campaign objects (ts, tc, n, v, a, f ) and(t ′s, t ′c, n′, v′, a′, f ′) respectively.FOR c′ formulation successor of c DO

IF f = f ′ THENwc,c′ = n · za, f

ELSEIFa ∈ R∗ with n∗ batches of main process a∗ ∈ A on f ∗ ∈ F THEN

i p = i(p, 0)+ δ+a∗,p · ba∗, f ∗ · n∗ELSE

i p = i(p, 0)

γk = �(k−1)·δ+a,p ·ba, f+i p−δ+

a′,p′ ·ba′, f ′(δ−

a′,p−δ+

a′,p)·ba′, f ′

�wc,c′ = max{za, f · k − za′, f ′ · min{γk, n′} | 0 ≤ k < n}IF γk < n THEN

IF wc,c′ < n · za, f − γn+1 · za′, f ′ THENwc,c′ = n · za, f − γn+1 · za, f ′

3.2.2 Campaign scheduling

MILP 2 determines starting and completion times ts and tc and also the campaignsequence on each facility modeled by binary sequence indicators xc,c′ which requirevirtual first and last dummy campaigns f d( f ) and ld( f ) on each facility f ∈ F .

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Weighted by appropriate cost parameters, the objective function simultaneously mi-nimizes inventory (implicitly by reducing delays between formulation predecessorsand successors), due date violations, set up activities and the timespan of the totalschedule. Different types of constraints ensure the feasibility of the schedule: Types 1and 2 focus on timespan and lateness for the objective function, type 3 ensures that allbatches of a campaign are scheduled without idle time in a campaign. Types 4–6 en-sure feasibility of the schedule with regard to delays, resource availabilities and set uprequirements. Types 7–10 ensure that the sequence indicators are set appropriately, i.e.that each campaign has a unique schedule predecessor and a unique schedule successor.

MILP 2 (Campaign Scheduling)

Unless indicated otherwise, c and c′ indicate campaign objects (ts, tc, n, v, a, f ) and(t ′s, t ′c, n′, v′, a′, f ′), respectively.

Minimize

∑

ε∈E

∑

c∈Cε

∑

c′∈Cε

hc(P+a ∩ P−a′ ) · (t ′s − ts)

+∑

ε∈E

tc(ε) · αε

+∑

ε∈E

∑

c∈Cε

∑

ε′∈E

∑

c′∈Cε′sca,a′ · xc,c′

+mc · m

Subject to

1. Lateness of order ε ∈ E is at least as big as the difference between the completiontime of the last campaign of Cε and the due date for ε ∈ E :

tc − tε ≤ αε ∀ε ∈ E, c ∈ Cε with a = aπ

2. Timespan m ends after completion of all campaigns:

tc ≤ m ∀ε ∈ E, c ∈ Cε

3. No idle time within a campaign c ∈ Cε—completion time tc of campaign c equalsstarting time ts of c plus duration of all n batches of c:

tc = ts + n · za, f ∀ε ∈ E, c ∈ Cε

4. Minimum delays wc,c′ between campaign c ∈ Cε and its formulation successorc′ ∈ Cε have to be respected:

ts + wc,c′ ≤ t ′s ∀ε ∈ E, c, c′ ∈ Cε with P+a ∩ P−a′ = ∅

328


5. Earliest availability time has to be respected for each facility:

t f ≤ ts ∀ε ∈ E, c ∈ Cε, f ∈ F

6. Campaigns c ∈ Cε and its schedule successor c′ ∈ Cε′ must respect set upactivities:

tc + sta,a′ − (1− xc,c′) · T ≤ t ′s ∀ε ∈ E, c ∈ Cε, ε′ ∈ E, c′ ∈ Cε′ with f = f ′

7. Each campaign c has just one schedule predecessor and just one schedule succes-sor. Each first dummy campaign f d( f ) has just one schedule successor, each lastdummy campaign ld( f ) has just one schedule predecessor:

Predecessor∑

ε∈E

∑

c∈Cε

xc,c′ + x f d( f ′),c′ = 1 ∀ε′ ∈ E, c′ ∈ Cε′

∑

ε∈E

∑

c∈Cε

xc,ld( f ′) + x f d( f ′),ld( f ′) = 1 ∀ f ′ ∈ F

Successor∑

ε′∈E

∑

c′∈Cε′xc,c′ + xc,ld( f ′) = 1 ∀ε ∈ E, c ∈ Cε

∑

ε′∈E

∑

c′∈Cε′x f d( f ′),c′ + x f d( f ′),ld( f ′) = 1 ∀ f ′ ∈ F

8. No campaign c is the schedule successor of itself:

xc,c = 0 ∀ε ∈ E, c ∈ Cε

9. No schedule predecessor for the first dummy campaign f d( f ), no schedule suc-cessor for the last dummy campaign ld( f ):

xc, f d( f ′) = 0 ∀ε ∈ E, c ∈ Cε, f ′ ∈ F

xld( f ′),c = 0 ∀ε ∈ E, c ∈ Cε, f ′ ∈ F

x f d( f ), f d( f ′) = 0 ∀ f, f ′ ∈ F

xld( f ), f d( f ′) = 0 ∀ f, f ′ ∈ F

xld( f ),ld( f ′) = 0 ∀ f, f ′ ∈ F

329


10. Schedule predecessor and schedule successor (incl. first or last dummy campaigns)have to be operated on the same facility:

xc,c′ = 0 ∀ε ∈ E, c ∈ Cε, ε′ ∈ E, c′ ∈ Cε′ with f = f ′

x f d( f ′),c = 0 ∀ε ∈ E, c ∈ Cε, f ′ ∈ F with f = f ′

xc,ld( f ′) = 0 ∀ε ∈ E, c ∈ Cε, f ′ ∈ F with f = f ′.

Domains

ts, t ′s ≥ 0 Starting times of campaigns c, c′ ∈ ∪ε∈E Cε

tc, t ′c ≥ 0 Completion times of campaigns c, c′ ∈ ∪ε∈E Cε

αε ≥ 0 Lateness of order ε ∈ E

xc,c′ ∈ {0, 1} Sequence indicator of campaigns c, c′ ∈ ∪ε∈E Cε

and dummy campaigns c, c′ ∈ { f d( f ), ld( f ) | f ∈ F}

m ≥ 0 Timespan.

Parameters

hc(X) Holding cost for products in X ⊆ P

tc(ε) Lateness cost of demand element ε ∈ E

sca,a′ Cost for set up activity between processes a, a′ ∈ A

mc Timespan cost

T Duration of all campaigns in the schedule T =∑ε∈E

∑c∈Cε

n · za, f .

4 Realization and empirical results

4.1 Size of problem instances and computational limitations

Real-world instances of the planning problem considered are very large; some keyfigures listed in Table 1 give an impression of their size. Although the solution methoddescribed so far is straightforward, each MILP and each algorithm is calculated in onestep, it cannot be applied to such large instances without modifications.

Both MILP steps are complexity drivers, because their underlying problems Cam-paign Selection and Campaign Scheduling are NP-hard (proofs base on reductionto Partition rsp. Hamilton Path and are available from the authors on request). Real-world instances result in not more than 200–300 binary variables for MILP 1 whichcan therefore still be solved exactly within an acceptable time. On the contrary, com-putation analysis has shown that a heuristic approach is required for MILP 2: Small

330


Table 1 Characteristics of real-world problem instances

Master data Transactional data

Min. Avg. Max. Min. Avg. Max.

|A| 154 |E | 85 92 100

|F | 26 Horizon 300 days

|P| 226 za, f 8 h 28.8 h 84 h

|A p | 1 2.5 6 sta,a′ 8 h

|Fa | 1 1.4 4 |C∗ε | 1 3.6 54

|A f | 1 8.7 26

|B| 62

instances were identified for which MILP 2, although comprising less than 1,000binaries, could not be solved exactly in reasonable time. Real-world problem instancesresult in 10,000–13,000 binary variables and therefore cannot be solved exactly.

4.2 Adaption of solution method

Due to the fact that the size of real-world instances does not allow MILP 2 to be solvedexactly, a solution heuristics for the campaign scheduling problem is required. Threespecific characteristics of the problem and of the model formulation enable a simplebut effective solution approach:

– The demand elements are distributed uniformly across the planning horizon. Thisallows to partition the demand elements set E = E1 ∪ · · · ∪ En (Ei ∩ E j = ∅) insuch a way that the order subsets are sorted by increasing due dates of their orders,i.e. for ε ∈ Ei , ε

′ ∈ E j the following relationship holds: i < j ⇒ tε ≤ tε′ .– Due to order encapsulation, the mapping of campaigns to demand elements is

unique. This allows to iteratively construct a schedule for all Ei , because campaignsthat belong to a chain Cε, ε ∈ Ei do not interact with campaigns from other ordersε′ ∈ E j , j = i .

– Finalizing the production for a demand element ε ∈ E significantly earlier doesnot result in higher penalties.

Due to these characteristics, it is possible to solve MILP 2 iteratively without losing toomuch solution quality: The schedule is determined by partitioning the set E of demandelements to subsets E1, . . . , En (Ei ∩ E j = ∅) and then solving MILP 2 iterativelyfor the respective subsets E j . During these iterations it is ensured that on each facilitythe sequence of the scheduled campaigns follows the chronological sequence of thepartitioned demand subsets. For two campaigns c and c′ assigned to demand elementsε ∈ Ei and ε′ ∈ E j and scheduled on facility f ∈ F , we have i < j ⇒ ts < t ′s . Theeffects of this iterative approach of constructing an optimized schedule are outlinedin Fig. 3. After the last iteration, the whole MILP 2 is solved with fixed sequence

331


indicators xc,c′ , i.e. without changing the campaign sequence obtained before by theiterations.

The described method is one possible way to solve the scheduling problem, but it isnot the only possibility for reducing the complexity of MILP 2. Two other approachesbased on rounding strategies for LP relaxations were examined: The strategy was tosolve the LP relaxation of MILP 2 iteratively and round just one sequence variable to 1for each facility in each iteration. The variable to be rounded was selected either by thelowest starting times of the campaigns or by the highest value of the binary sequenceindicators. The results from these approaches were not promising. Other solutionapproaches, e.g. priority rule based top down scheduling, have not been evaluatedalthough they might lead to good results.

4.3 Computation analysis and results

The main target of the computation analysis was to ensure the applicability of theoptimization method to real-world instances of the problem and to customize it tothe specific requirements of the key users. A thorough numerical investigation bycomparing the proposed method with other optimization approaches based on thisspecific problem or even other (academic) benchmarking problems has not been infocus. Nevertheless, the efficiency of the proposed method has been evaluated andempirical evidence of its applicability to real-world problem instances is given.

Within the computation analysis, 18 small instances comprising only a limitednumber of demand elements and five large instances consisting of a number of demand

Fig. 3 Iterative approach for schedule construction

332


elements comparable to real-world requirements were evaluated. Characteristics ofthe instances are shown in Table 2. The method was performed to all instances withdifferent parameter settings for the objective function of MILP 2—four settings eachfocusing on only one of the optimization objectives (inventory reduction, due dateachievement, timespan minimization, set up optimization), and a fifth setting focusingon all objectives in a balanced way.

An analysis of the results for the small problem instances achieved by solving MILP2 exactly allows to estimate the number of iterations of MILP 2 required to solve real-world problem instances: On the one hand, MILP 2 can be solved exactly for smallproblem instances comprising up to 8 campaigns scheduled on the same facility, buton the other hand, MILP 2 cannot be solved exactly for some instances with not morethan seven campaigns on the same facility. This observation indicates that MILP 2 canbe solved exactly if not more than 6–7 campaigns are scheduled on one facility, andreal-world instances comprise max. 20–30 campaigns scheduled on the same facility.Therefore it should be sufficient to iterate the heuristics for MILP 2 only 4–5 timeswith equally large partitions of the demand element set E to determine an optimized

Table 2 Characteristics of evaluated problem instances

Instance Inst. # Camp. |E | # Due |C| # Binaries MILP 2size no. per fac. dates solved exactly

Small 1 5 14 12 31 207 Yes

2 5 14 12 32 185 Yes

3 6 15 11 33 236 Yes

4 6 15 13 33 279 Yes

5 6 21 17 54 490 Yes

6 6 23 15 55 824 Yes

7 7 15 13 39 540 Yes

8 7 20 16 48 629 Yes

9 7 23 14 48 607 Yes

10 8 14 11 24 206 Yes

11 7 24 17 57 952 No

12 7 23 17 55 824 No

13 7 24 13 57 952 No

14 7 31 19 64 1,000 No

15 7 22 15 61 1,095 No

16 9 15 17 39 540 No

17 10 18 14 47 825 No

18 14 26 17 64 1,733 No

Large 1 21 87 27 196 10,059 No

2 23 95 28 227 12,571 No

3 26 94 28 215 12,816 No

4 27 99 26 221 13,204 No

5 29 85 28 213 12,117 No

333


solution for real-world problem instances. Similar conclusions based on the number oforder elements instead of campaigns on one facility indicate 6–7 required iterations.Tests on different large problem instances have shown that 4–6 iterations were alwayssufficient to determine optimized schedules within 350–540 s. A total calculation timeof 30 min (incl. data transfer to the APS and visualization) was considered acceptable.

A cost comparison between the schedules that were generated manually by theSC Planners and the ones that were calculated by the optimization method was notperformed. Nevertheless, all schedules obtained by optimization were highly acceptedby the key users who were enabled to reduce the work spent for routine planning andto focus more strongly on planning exceptions. Furthermore, the APS provides a setof standard algorithms for schedule construction and optimization. Different ways ofapplying these algorithms were also tested, but in most cases it was not possible togenerate a feasible schedule of comparable quality within an acceptable time. Somekey figures characterizing size, properties and complexity of optimized schedules aregiven in Table 3.

To evaluate the efficiency of the proposed method, the obtained calculation resultsfor both exact solution of MILP 2 and the iteration heuristics are analyzed and com-pared with lower bounds obtained by the LP relaxation of MILP 2. For the timespanvariant of the problem, i.e. the parameter setting of the objective function which onlyfocuses on timespan minimization, a second lower bound is given by the maximumworkload calculated by MILP 1 within the campaign selection step. The results ofthese comparisons are shown in Table 4.

The comparison with the optimal solutions of MILP 2 shows that for balancedobjectives the iteration heuristics has an optimality gap of 4% on average and 12%in the worst case. Focusing only on set up optimization or timespan minimizationresults in higher optimality gaps. Focusing only on inventory reduction resultsin optimal solutions, which is easy to explain because scheduling the campaign withoptimal synchronization is simple if no other objectives have to be considered. Anoptimal schedule for the inventory reduction variant of the problem can be obtainedas follows: start with an arbitrary campaign chain and schedule the campaigns of thischain by decreasing manufacturing level in such a way that minimum delay times areachieved. Perform a right shift of campaigns to obtain an optimal synchronization andcontinue with the next chain until all campaigns are scheduled.

The comparison between the iterative approach and the LP relaxation of MILP2 does neither confirm nor contradict the observations made so far. A comparison

Table 3 Characteristics of optimized schedules for real-world problem instances

Min. Avg. Max.

# of campaigns in total 196 214 227

# of campaigns on one facility 1 8.2 29

# batches in total approx. 1,800

# batches in one campaign 3 9 45

Timespan approx. 250 days

# free binaries 10,059 12,103 13,204

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Table 4 Gaps of the iterative heuristics approach

Instance size Bound Gap Optimization objective

Balanced Timespan Inventory Due date Set up(%) (%) (%) viol. (%) (%)

Small MILP 2 opt. Avg. 4 15 0 3 10

Max. 12 39 0 24 28

LP relax. Avg. 11 28 1 6 26

Max. 19 60 3 34 93

MILP 1 opt. Avg. − 31 − − −Max. − 88 − − −

Large LP relax. Avg. 62 84 1 2, 640 48

Max. 217 116 2 4, 915 184

MILP 1 opt. Avg. − 50 − − −Max. − 128 − − −

with optimal solutions for MILP 2 has shown that especially for the minimization oftimespan or due date violations the LP relaxation does not deliver a good lower bound.This might explain why the approach of rounding based on LP relaxation does notlead to good results. These effects can already be observed at the small instances andthey are stronger visible at the large instances.

Comparing the results of the iterative approach for the timespan problem with theworkload calculated by MILP 1 shows a gap of 31% on average and 88% in the worstcase. For the large instances, the gap is 50% on average and 128% in the worst case.At a first glance, these observations do not seem to be promising, but a comparisonwith the optimal solutions for MILP 2 has shown that the workload obtained by MILP1 unfortunately is not a good bound: even the optimal solution has high gaps to theworkload bound—27% on average and 88% in the worst case for small probleminstances.

The facility workload obtained by MILP 1 or the solutions of the LP relaxationof MILP 2 are not appropriate bounds to estimate the optimality gap of the iterationheuristics. Nevertheless, a comparison of results obtained by the iterative approachand optimal solutions of MILP 2 has shown that the partition approach does notresult in a large loss of optimality and solution quality. An explanation is that differentdemand elements for the same finished product are often put into different partitions,and therefore the chronological sequence of demand elements induces a chronologicalsorting of the corresponding production campaigns. Furthermore, each partition stillcomprises approximately 20–30 demand elements, and the resulting (smaller) decisionproblem still has resource conflicts with a considerable optimization potential. Withinreasonable time trial runs with problem instances of real-world size resulted in solu-tions of good quality—as shown especially by key user acceptance and comparisonto standard algorithms of the APS—and therefore give evidence of the applicabilityof the solution approach in daily business.

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4.4 Integration in APS

For a variety of reasons, it has to be possible to integrate the optimization method intoan APS to gain maximum benefit in business applications:

– APS offer possibilities for graphical depiction of the resulting production schedulesand give quick access to additional information on the schedule elements.

– APS facilitate manual modification of existing production schedules, especiallyfor management by exceptions.

– APS enable integration with ERP systems, other planning modules or processcontrol systems.

A more detailed description of benefits of APS is given by e.g. Kolisch et al. (2000).Siletti and Petrides (2003) show the necessity for the interaction between batch processscheduling and systems for production planning and process control. Experiences andexpectations of the application of SAP R/3 in process industry are described by e.g.Schumann (1997).

The described solution method is implemented in C code and bases on a CPLEXOptimizer for solving the MILP steps. This method is integrated into a standard APScontaining master and transactional data as well as additional parameters (e.g. costparameters for objective functions). All parameters can be exported quickly and easilyby Tcl routines via an open interface.

An example of such an optimized schedule depicting single batch process ordersof the scheduled campaign objects in an APS is shown in Fig. 4. The APS enabledthe key user to get detailed information on the elements and inventory levels of anoptimized schedule or to modify it by mouse clicks. The schedule itself has a timespan

Fig. 4 Gantt chart of an optimized schedule

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of more than 8 months and shows high variances in resource occupation although abalanced workload was targeted in MILP 1.

5 Conclusion

In this paper, we present a real-world problem of campaign scheduling for a specialtychemicals plant. The plant comprises multi-mode production facilities to operate batchprocesses with joint production, refinement steps, sequence dependencies and otherindustry-specific characteristics. Multi-level production formulations include all typesof material flows (incl. recycling) across more than 200 products. We have describeda solution methodology on an MILP-based time-continuous model formulation tosolve problem instances of real-world size. The feasibility of this approach is provedempirically, the results obtained exhibit reasonable computation time and solutionquality. The possibility of integration into APS is outlined and demonstrated using astandard tool.

The paper leaves room for further research activities, for instance, applying othersolution techniques to the considered problem. The possibilities of embedding alldecisions in one mathematical model and the analysis of results achieved by suchapproaches in comparison to the solutions obtained by the method described in thispaper are of special interest. The adaptation of the described solution approach toother industry specific problem characteristics—frequency-dependent set ups, cyclicdowntimes of production facilities, variable batch sizes or storage restrictions just toname a few—could be relevant for the industrial practice. Such an analysis could bebased on different well-known benchmark problems which have already been usedfor evaluations of other solution approaches.

Acknowledgment The authors would like to thank Hans-Heinrich Böther, Heinrich Schuchard and thereferees for their constructive comments and suggestions that contributed to the improvement of this paper.

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Part VApplications in the Automotive Industry

Supply chain planningin the German automotive industry∗

Herbert Meyr


Abstract Following the evolution in the computer industry, quite a lot of car man-ufacturers currently intend to move from a built-to-stock oriented production ofstandardized cars towards a customized built-to-order (BTO) production. In the pre-mium segment of Germany’s automotive industry, the share of customized BTO carstraditionally is comparatively high. Nevertheless, German car manufacturers have

Surprisingly, comprehensive overviews of the short- and mid-term planning land-

planning system and for the respective planning methods. In this way, challenges for

Keywords Supply chain planning · Operations research · Automotive industry

1 Introduction

Mass customization [42] that aims at offering customized products in a high varietybut for still low prices and within short delivery times gains increasing importancein various branches of business and, in the meantime, also captivates the automotiveindustry. The BMW Group, for example, spent $55 million on its new Euro-pean online-ordering system [24] to cut order-to-delivery times by 20 days on the

H. MeyrDepartment of Production and Supply Chain Management, Technical University of Darmstadt,Hochschulstr. 1, 64289 Darmstadt, Germanye-mail: [email protected]

∗ This article was originally published in OR Spectrum 26/4 (2004), pp. 447–470, and reflects thesituation in the automotive industry during the years prior to this publication date.


Operations Research (OR) that are able to support the various planning tasks involved.In the second part, the major change in strategy, currently to be observed in the Ger-

the premium segment of the German automotive industry, and reviews methods of

scape of car manufacturers cannot be found in the scientific literature. Thus, the first

a future application of OR methods in the automotive industry can be identified.

spent a lot of efforts in recent years to further increase this share in order to realize

man automotive industry, is briefly summarized in order to derive its impacts for the

part of the paper discusses supply chain planning, as traditionally established in

short delivery times, high delivery reliability and a fast responsiveness.

(B)

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OR Spectrum (2004) 26:447–470DOI 10.1007/s00291-004-0168-4


H. Meyr

average. At the same time, BMW offers up to 1032 variants (at least theoretically),several thousands of them actually being demanded [51, p. 42]. Other manufacturersalso declared their intention to decrease order-to-delivery times from an average of40 days to about 15 days [22] and try to make the transition from “build-to-stock“(BTS) to “build-to-order” (BTO) that has successfully been demonstrated by thecomputer industry, and first and foremost by its paragon Dell.

The transition to BTO in the computer industry caused a reorganization of plan-ning processes and led to an increased use of “Advanced Planning Systems” (APS,[29]), i.e. of computer-based decision support systems, which – at least partly – relyon sophisticated methods of Operations Research (OR). Thus the questions arise,whether and how the transition of the automotive industry changes their respectiveplanning tasks and planning processes, and to what extent planning and OR meth-ods are and will be affected. Since mutual interrelations are particularly importantfor operational planning tasks, the discussion will concentrate on mid- and short-term supply chain planning, and here especially focusing on the car manufacturers’point of view. But before discussing changes it has to be shown what the planninglandscape of automotive industries traditionally looks like. There are, of course,discussions of various individual planning tasks (see Sect. 3) and some overviewsof the order-to-delivery process (see e.g. [23, 51]). However, to the author’s knowl-edge, in scientific literature no comprehensive overviews of the short- and mid-termplanning landscape of car manufacturers can be found.

Due to this lack of literature and since common scientific approaches like ques-tionnaires and structured interviews did not seem to be very promising because quitea lot of confidence is needed to get such a sensitive information, the following char-acterization of the planning system of car manufacturers mainly builds on variousjoint projects with German car manufacturers and communication with their respon-sible planners and with employees of automotive consultancies. In order to verifythe conclusions drawn, a working paper has been written, sent to skilled people inthese companies and they have been asked for statements about its validity. The re-sults of this process are presented in the following. To sum up, the contribution ofthis paper is

– first, that the planning systems of German car manufacturers are analyzed, de-scribed and thus made available to the academic literature,

– secondly, that OR methods suitable for planning within the automotive industriesare reviewed, categorized with respect to the planning tasks of (German) carmanufacturers and that insufficiently supported planning tasks are disclosed, and

– thirdly, that the challenges of the managerial changes from BTS to BTO areoutlined that arise for the planning tasks, the planning systems and for the ORmodels/methods involved.

Due to this broad scope of the paper, a review of OR methods – even thoughrestricted to short- and mid-term planning – cannot be comprehensive. This paperrather intends to give an idea where (i.e. at which subsection within the overallplanning system of a car manufacturer) OR methods already contribute or may con-tribute in the future.

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Supply chain planning in the German automotive industry

Long-term, strategic planning provides potentials, which mid-term planning hasto further develop and short-term planning has to implement. Of course, also long-term planning tasks are supported by OR methods. Concerning the product design,for example, the optimal commonality of automotive components (e.g. wire har-nesses) is determined [53] or the impact of product variety on the performance ofmixed model assembly lines is analyzed [6, 16]. It is even worth to include assem-bly sequencing issues into product design decisions [52]. Analytical and simulationmodels provide general hints (“chaining strategies’’) how to assign products to man-ufacturing plants so that high process flexibility is achieved for both single stage[28] and multi stage [5, 19] automotive supply chains. Linear Programming (LP) orMixed-Integer linear Programming (MIP) models are, for instance, used by APS todesign the inbound system of assembly plants [20] or car distribution networks [2](see also [36] without use of APS). Concerning the inside of assembly plants, theplanning of the physical layout and of buffer sizes of assembly shops, in general,and of body shops [41, 49, p. 20 ff. and 73 ff.], in particular, can be supported bysimulative, analytical and combinatorial optimization methods. A comprehensiveoverview of OR methods for the well-known assembly line balancing, which is arather strategic than mid-term task in the automotive industry, is given by [47]. Arecent survey of heuristic methods for cost-oriented assembly line balancing can befound in [1].

In order to understand why automotive planning systems are organized the waythey are, Sect. 2 describes the characteristics of automotive supply chains. Thesevary substantially for car manufacturers in different parts of the world (NorthAmerica, Japan/Korea, Europe/Germany), operating on different market segments.

but not luxury cars (like Rolls-Royce, Maybach, Bentley) and on the German au-tomotive industry. Nevertheless, quite a lot of the statements and findings of this

related product segments because (even though beginning with different starting

presents the traditional short- and mid-term planning system and – after introduc-

(Sect. 6).

2 Automotive supply chains

Cars are sold to final customers either directly via sales subsidiaries of the car man-ufacturer or indirectly via legally separate retailers. The bill-of-material (BOM) isstrictly convergent, i.e. assembly processes are dominant. Cars often are thought tobe standard products. However, in the premium segment of this line of business,there is a high degree of customization. This allows the customer to specify oblig-atory features like the color of the car and type of upholstery or optional features

paper can be transferred to car manufacturers in other parts of the world tackling

This paper mainly concentrates on premium brands (like BMW, Mercedes, Audi)

methods can be derived and challenges for future research can finally be identified

ing the respective planning tasks – points to appropriate planning and OR methods.

implemented in the German automotive industry (Sect. 4), their impact on the plan-

points) many of them similarly intend to change to a BTO production. Section 3 then

After briefly summarizing the measures to improve BTO assembly currently being

ning system is discussed in Sect. 5. Thus changed requirements for planning

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H. Meyr

like air conditioning or a navigation system, to name only a few. In the follow-ing both obligatory and optional features are just referred to as “options”. A carmanufacturer usually offers several types of cars (e.g. the E-class or C-class ofDaimlerChrysler), which again differ in several body-in-white variants (coupe, con-vertibles, etc.). Not every customer needs his car immediately. According to [51,p. 38] the order lead time desired by a final customer is normally distributed with amean value of 4–6 weeks.

The sales organization and distribution network of a car manufacturer have a di-

of the manufacturer, sales persons responsible for different world regions (also at theheadquarter), sales companies in different countries or local areas and a rather highnumber of further retailers and sales subsidiaries. This type of customized premium

either by the final customer, a retailer or a sales department of the manufacturer–

of retailers’ and sales departments’ orders (see Sect. 4.1).

rounds: In the first one, a retailer sends his mid-term requests for cars to the manu-

during the next year. Usually, this “negotiation” process is clearly dominated by themanufacturer so that – due to the preferences of the manufacturer – the agreed quotamay be less or even higher than the original requests. Since these quotas are, for ex-ample, defined for the next year on a monthly basis, only body-in-white variants

but the options are not specified at this point in time.In a second round, about three to five weeks before planned production, the re-

tailer has to specify the options for all cars of his quota, which are due and havenot been assigned to final customer orders that had arrived in the meantime. Froma retailer’s point of view, these cars are “built to stock” (BTS-cars), based on a sortof forecasting process for options. From the manufacturer’s point of view, an orderof the retailer exists, thus justifying the term “built to order”.

Figure 1 illustrates the different states of demand information that are impliedby these two interaction rounds. The curve (I) shows the cumulated share of fullyspecified orders of final customers with respect to the overall number of orders(incl. forecasts) considered by planning. It can be computed by calculating the distri-bution function of the order lead times, that are desired by final customers (see [51,p. 38]). This distribution function is drawn backward in time, starting with the de-livery of cars to final customers.

For the section above the curve, no information about the preferences of finalcustomers is available. This lack of information has to be replaced with forecasts.Concerning the options of BTS-cars, this is done by retailers with a lead time of3–5 weeks before production (II). Beforehand, with a lead time of one year at amaximum, only the retailers’ requests for models are known (III), which also arethe result of a forecasting process of the retailer. For yet earlier planning tasks, a carmanufacturer has to rely on his own pure forecasts for models (IV).

industry try to increase the share of final customers’ orders and to decrease the share

cars can only be assembled “to order”, i.e. there has to be an “order” available –

Commonly, manufacturer and retailers communicate in two types of interaction

vergent structure, which comprises several stages like the central sales department

facturer. Both “negotiate” the number of cars (so-called “quota”) the retailer will get

and the type of engines (referred to as “models” in the following) are considered,

that specifies the options of the car. Current SCM initiatives in the automotive

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pureforecast

for models(IV)

requestsfor modelsby retailers

(III)

sending requestsfor models

optionsspecified

byretailers

(II)

specification of optionsby retailers

share of final customers‘ orders

100 %

time

50 %

delivery tofinal customer

optionsspecified by

final customer(I)

„built-to-stock“of retailers

Fig. 1 Demand information available to a car manufacturer

The production system in a car assembly plant usually comprises the four stagespressing of metal or aluminium sheets, welding the body-in-white from the mouldedsheets in the body shop, painting it in the paint shop and final assembly, wherepainted body, engine, transmission and the further equipment are brought togetheror built in. For the final assembly one or several production lines are used. A pro-duction line consists of quite a lot of serially arranged assembly stations, betweenwhich cars are conveyed with a fixed belt rate. The processing time at an assemblystation depends on the option chosen for the car to be assembled. Therefore, theoverall utilization of a station is determined by the sequence in which cars/ordersare assembled on a line (the so-called “model mix”). If too many cars requiring thesame options are following one another, some of the stations may be overloadedwhereas others are underloaded. Thus a “balanced” model mix has to be found,almost equally utilizing the various stations of an assembly line.

Because of the convergent BOM and ten thousands of components to be pur-chased, a procurement network with several hundreds direct and an enormousnumber of indirect suppliers has to be coordinated. For the delivery of incoming

components are – as far as possible – delivered “just in time” (JIT) at the day of as-sembly, partly even directly to the assembly line and thus arranged in the sequenceof planned assembly (“sequence-in-line supply”, SILS). The remaining incominggoods are collected by regional carriers, consolidated and brought to intermediate

The structure of an automotive supply chain is characterized by a convergentflow of material upstream of the assembly plants of the car manufacturer and adivergent flow of finished cars downstream. An automotive SC is difficult to co-

bottlenecks, but also incoming goods. For reasons of flexibility, high-volume carmodels can sometimes be produced at several assembly plants. Because of these

the final customers. Furthermore, final customers need a reliable delivery date because

warehouses of the car manufacturers, which are close to their assembly sites.

constraints, the promised delivery dates cannot always satisfy the expectations of

ordinate, because not only production capacity and manpower may turn out to be

goods normally several transport modes are applied. Voluminous and expensive

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important further activities (like selling the old car, making money available) haveto be synchronized with the arrival of the new car. Thus, “order promising” notonly has to aim at setting a delivery date close to the customer’s wishes, but also atpromising a reliable date considering as many of the above constraints as possible.

Besides a significant intra-organizational information flow between differentplanning units/departments of the car manufacturer itself (as will be discussed inSect. 3), there is also a vital inter-organizational exchange of information betweenthe different members of the SC. Commonly, car manufacturers prepare a roughmid-term supply plan of the next year for their (first-tier) suppliers in order to drawearly attention to potential capacity bottlenecks. In the short term, daily supply plansare sent to the suppliers. These include binding orders for the next day, but also quitereliable “forecasts” for the next days/weeks and even rough forecasts for the nextmonths.

3 Traditional planning processes

To cope with the various planning tasks of automotive supply chains, quite a lot ofplanning units/departments have to be involved. These planning tasks and the re-spective decisions can be assigned to several planning levels (e.g. strategic, tactical,operational) comprising different planning horizons (e.g. long-, mid-, short-term).Depending on the planning horizon and the lead time necessary to make a certain

ent state of knowledge about actual customer demand is available. Therefore, froma manufacturer’s point of view, one may distinguish between forecast-driven long-and mid-term planning (phases (III) and (IV) of Fig. 1) and order-driven short-termplanning (phases (I) and (II)). In Sect. 3.1 forecast-driven mid-term planning tasksand their information flows, which are more or less common for the German au-tomotive industry, will be discussed first. Order-driven planning will then be theconcern of Sect. 3.2. Within both sections organizational issues are left aside. Thiswill be covered in Sect. 3.3.

3.1 Forecast-driven planning

Figure 2 summarizes the forecast-driven planning activities. Planning tasks aremarked by rectangles, arcs illustrate the information flows in between. From thebottom to the top, the level of aggregation and the planning horizon are increasing,the frequency of planning is decreasing, however. The planning tasks are roughlyassigned to the logistical functions procurement, production, distribution and sales,again. Of course, not all of the mid-term planning tasks of a car manufacturer willbe discussed. Only the most important ones which show a close interrelation havebeen selected.

The annual budget planning determines the overall monetary budgets of the carmanufacturer’s departments and assembly plants for the next year. For this, produc-tion plans for the respective plants and the sales plans for the respective sales regionshave to be calculated, too. This is done once per year, for the next year, by deciding

348

decision, different phases of the time axis of Fig. 1 are relevant and thus a differ-


retailerssuppliersplants

demandplanningMRP

allocationplanning

take rates

forecasts,take rates

requestsfor models

requests ofregions

volume goals,earning goals,

annualworking time

productionplans

(weekly)production

plans,overtime

(weekly)aggregate quotas

detailed quotas

supplyplans

procurement production distribution sales

budget planning

pro-duction

sales

master production pl.

pro-duction

sales

(monthly)production& salesplans

(monthly)aggregate

quotas

Fig. 2 Overview of (mainly) forecast-driven planning

about production and sales quantities of car models (per plant and world region,for example) on a monthly basis. The overall yearly quantities can be consideredas “volume goals” of the next year for both sales and production. From these, theexpected production costs and earnings can be derived (“earning goals”).

A further result of the annual budget planning is the usage or reservation of addi-tional capacities, as far as these can still be influenced on a mid-term basis. Becauseof the long lead times (e.g. two years or more to install an assembly line or a plant),usually capacities of production resources are adapted to customer demand in thelong term and thus are a concern of strategic planning. However, agreements aboutthe extent and flexibility of the yearly working time, for example, are also a task ofmid-term planning. A lot of further constraints have to be respected like potentialbottlenecks of suppliers, model mix restrictions (capacities of crucial options, min-imum utilization) and upper or lower bounds of the sales in certain markets. Lowerbounds, for example, result from strategic directives about the presence in impor-tant markets, upper bounds may be due to marketing analyses about final customerdemand.

Input data for the annual budget planning mainly are forecasts for final cus-tomers’ demand (see also Fig. 1), which result from the demand planning. These

ified orders from final customers (e.g. car rentals), of the retailers’ annual requestsfor models, of the sales companies’ decentral knowledge about the local prefer-ences of their customers (“requests of regions”) and on basis of information aboutmarketing capabilities to influence final customer demand.

Since budget planning has to decide about car models on the one hand and to an-ticipate potential bottlenecks of suppliers on the other hand, the component demandneeds to be estimated, too. One way to do this is to forecast take rates directly, i.e. tocalculate the probability that a certain option or even component is demanded (in

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are made on basis of historical sales data, of the few already known and fully spec-

H. Meyr

a specific customer region) and to multiply it with the total number of car modelsplanned (for this region).

The task master production planning is similar to the annual budget planning.Again, production and sales plans have to be determined and coordinated. However,both now require a higher level of detail (e.g. weekly instead of monthly quanti-ties) and they are not used to derive budget goals any further. The planning horizonof a monthly rolling horizon planning varies between three months and one year.Nevertheless, only the weekly quantities of the first month or the first two months(depending on the lead times of planning) are put into practice.

Input data (see Fig. 2) are the already mentioned sales forecasts for models andforecasts for take rates. Because of the high share of final customers’ orders, thatis available for this shorter planning period (see Fig. 1), these monthly forecastsare more reliable than the annual forecasts used for budget planning. Further inputdata are the production and sales quantities per month that have been agreed uponin the budget planning, or the respective volume and earning goals (e.g. per year).One objective of the master production planning is to meet these targets as close aspossible in the short term. Constraints to be respected are quite the same as wererelevant for the budget planning. However, again a higher level of detail is necessary.

Results of the master production planning are the updated and more detailed(e.g. weekly) production plans of the assembly plants and sales plans. The lat-ter ones include the quotas for the different sales regions. Because of the abovementioned constraints, these quotas may exceed or fall below the requests for carmodels, originally demanded by the regions. A similar setting of (monthly insteadof weekly) quotas for sales regions may possibly also be part of the annual bud-

spreadsheet modeling.

which are known as a result of the budget planning on a monthly basis and as aresult of the master production planning on a weekly basis, to the lower levels of

quotas of world regions to different countries, and afterwards an allocation of thesemore detailed quotas to the countries’ respective retailers and sales subsidiaries.As an example, in the following only the relation “world region → countries” isconsidered: After the annual budget planning, the respective monthly quotas (sales

get planning. For both budget and master production planning LP or MIP models

not used in practice at the moment. Planning usually is only supported by simple

master production planning, are the basis to derive the component demand in a fur-

forecast horizon is.

communicated to the first-tier suppliers as a preview of the quantities to be deli-

On the sales side, the allocation planning has to allocate the aggregate quotas,

vered within the next months. As the options of the cars are just specified for the3–5 weeks before production (see Fig. 1, phases (I) and (II)), and since the share of

The production plans for car models, which are a result of the annual budget and

final customers’ orders decreases rapidly for longer lead times (phases (III) and

ther material requirements planning (MRP) procedure. The component demand is

this planning task may occur on several hierarchical levels, e.g. first an allocation of

(IV)), this component demand becomes more and more unreliable, the longer the

seem to be appropriate. However, for reasons to be explained in Sect. 3.3, they are

the sales system. Depending on the organizational structure of the car manufacturer,

350


plans) of the world regions have to be allocated to the countries with respect totheir original requests. If the requests cannot all be satisfied, it has to be decided,whose demand will only be fulfilled partly. This “shortage planning” may followsome predefined rules (so-called “fair share rules”, see e.g. [30, p. 169 f.]), which,for example, might reflect the purchase behavior of a country in the past, or moreor less be based on “negotiations” between representatives of the world regions andof the respective countries. Furthermore, a region has to balance the deviations ofthe countries’ actual demands from their former requests between all the differentcountries assigned to the region. For this purpose, the region may also (call for and)hold a “regional” pool of cars, originally not having been requested by one of thecountries.

3.2 Order-driven planning

Until now planning tasks have been discussed, which mainly build on forecasts foroptions. In other words, only a few fully specified orders are known at the time ofplanning. In this section planning tasks will be considered which are exclusivelytriggered by fully specified orders, either of final customers or of sales subsidiariesand retailers. Figure 3 gives an overview of these order-driven planning tasks andtheir interrelations.

Direct buying of cars via the Internet is not (yet) worth mentioning. Normallyprivate customers order their cars via the sales subsidiaries or retailers of the carmanufacturer. The respective sales personnel tells the final customers the expecteddelivery dates of their desired cars. Usually, a granularity of weeks is sufficientfor the customer, who e.g. has to provide the money on time and to synchronizethe delivery with the selling of his used car. Thus order promising, i.e. promising

finalcustomerssuppliers

allocationplanning

order promisingline assignment &

model mix planning

sequencing

plant assignment

MRP &lot-sizing

MRP distribution

customerorders

customer &retailer orders

specified orders

with due dates

(weekly)production ordersper plant

daily buckets

daily buckets

procurementlot-sizes

JIT-callsSILS car

sequence cars

specificationchanges

(weekly) production plans,

overtime

aggregate quotas

promiseddates

detailedquotas

procurement production distribution sales

Fig. 3 Overview of order-driven planning

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reliable delivery dates to the customer, is an important task. If a free quota of thesales subsidiary or retailer is available, the final customer gets his desired deliverydate promised. Otherwise, the next free quota is recommended or a standard deliverytime is proposed (if quotas are not available in sufficient detail). The customer mayaccept the promised date, change the options of his desired car or even the modeltype (in order to get an earlier delivery date), or try his luck with another retailer.

Furthermore, retailers and sales subsidiaries have to specify the options for thatpart of their quotas that has not been filled up with final customers’ orders until theagreed date of specification (see phase (II) of Fig. 1). In order to reduce inventoriesat the retailers’ sites, the desired options of potential customers have to be antici-pated as precisely as possible. Because of the rather small number of customers andlarge number of options, this is an almost unsolvable problem for a single retailer.Thus, Stautner [51] suggests central support of the manufacturer for these decen-tral forecasts of the retailers (see Sect. 5.1) and Holweg and Pil [24] even propose acentral pool of BTS cars.

Traditionally, these fully specified orders are collected by the respective salesorganization, responsible for a certain retailer, and sent in bulk (e.g. all orders of aweek) to the next higher level of the sales hierarchy. A central order managementdepartment of the car manufacturer finally has to select an assembly plant, able toproduce the car model requested by a certain order. This plant assignment has toconsider the production quantities and capacities per plant, that have been agreedupon in the master production planning. If the actually requested car options signifi-cantly deviate from the ones assumed within master production planning (e.g. whenanticipating bottlenecks of components or model mix constraints), some orders haveto be fulfilled earlier and others have to be delayed, thus resulting in a re-assignmentof orders to weeks.

The selection of an assembly plant was not a big problem so far because tradi-tionally car manufacturers had little flexibility in assigning cars to plants and thusthis task has (up to the author’s knowledge) not directly been addressed in the ORliterature. However, recently body shop and assembly have become flexible enoughto allow model swap and thus the degrees of freedom and the need for intelligentplanning methods grow. In [17], the more important assignment of customer ordersto discrete time buckets with respect to promised due dates and to material/capacityconstraints is introduced as a planning task called “demand supply matching” andcorresponding LP/MIP models are formulated. However, the specific requirementsof the automotive industry (e.g. several assembly plants, model mix constraints) arenot considered. Lovgren and Racer [33] make a first step towards mixed model as-sembly line sequencing with respect to given due dates of orders. They calculatedetailed sequences of cars for a single assembly line. Thus, their model is ratherdesigned for the short-term line sequencing (see below) than the more aggregateplant assignment. However, the problem of early or late demand fulfillment in theautomotive industry is at least generally addressed.

After this assignment, the decentral short-term production planning departmentsof the assembly plants have production orders available, that ought to be assembledwithin (or up to) their pre-defined week of production (ideally still the promisedweek minus a standard lead time for delivery to the respective customer). The

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shorter the planning horizon is, the more restrictive the model mix constraints are.Thus, the line assignment & model mix planning have to distribute the productionorders among the possibly parallel assembly lines and to assign days of produc-tion to the orders. Doing this, the most important model mix constraints (e.g. “at amaximum 300 air conditionings per day”) have to be considered, but the assemblysequence of a day is not yet determined. Scholl [47, p. 108 f.] denotes this task as“Master Sequencing” and suggests, for reasons of complexity, a further aggregationof individual orders to families of cars. Again, this planning task has not adequatelybeen tackled in the literature. Only Mergenthaler et al. [35] and Ding and Tolani[11] address the single line (sub-)problem directly. The former ones try to smooththe daily workload of a week by modifying a bin packing algorithm in order tominimize the quadratic model mix deviation in a greedy manner, whereas the latterones apply simple neighborhood operations like “switching models of differentlyutilized days” in a two-phase greedy algorithm.

As compared to mid-term planning, car options are now known with a high re-liability. Since the daily assembly buckets are also known as a result of the lineassignment & model mix planning, the daily demand of components can directlybe derived. For components and material, that are collected by regional carriers and

procedure is appropriate that balances the trade off between inventory holding costsand degressive transportation costs of the regional carriers and determines adequatesupply frequencies.

The daily buckets of the line assignment & model mix planning are also guide-lines for the daily sequencing of the assembly lines. Here, the sequences of theproduction orders on the final assembly lines are determined on a rolling horizonbasis with a planning horizon of one to two weeks. The level of detail again is higherthan in model mix planning. Now all potential bottlenecks have to be considered,for example, the availability of all of the components and “distance” restrictions ofthe lines like “no two cars with air-conditioning are allowed to follow each other ”.For this reason, sometimes the earlier assignment to days of production cannot bemaintained. However, it should be avoided to postpone an order to another weekthan the planned (and promised) one. To use flexible workforce or to work duringlunch breaks are short-term measures to extend capacity.

Undoubtedly, most scientific research on planning aspects of automotive supplychains has been done in the fields of balancing and sequencing mixed-model assem-bly lines. In the sequencing literature, usually it is assumed that orders have alreadybeen assigned to a certain period (e.g. a day) of production, so that promised duedates need not to be considered any further. The various sequencing approachesdiffer with respect to their different objectives. Besides cost-oriented objectives,mainly time related or JIT-objectives and combinations thereof are pursued (seee.g. [32, p. 44 ff.] and [47, p. 98 ff.]). A comprehensive literature review of modelsand exact/heuristic solution methods is given by Scholl [47]. Summing up, prioritybased (greedy) heuristics [47, p. 205 ff.] are – for reasons of complexity – clearlyfavored over exact (mainly branch and bound) methods [47, p. 199 ff.]. Newerheuristic approaches also apply multi agent systems [9] or local search methods like

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temporarily stored in an intermediate warehouse (see p. 347), an MRP & lot-sizing

simulated annealing or genetic algorithms (see e.g. [25, 35, 43]; [46, p. 40]).

H. Meyr

For a review of models and methods with respect to the different objectives,the reader is referred to Lochmann [32]. Models with time related objectives[32, p. 58 ff.] try to smooth the work load and minimize the overload of the var-ious stations of a line. For this, usually MIP models are formulated. JIT-objectivesattempt to smooth the material supply at the stations in order to keep the inventoryof components constantly low. The usage rates of components are either leveleddirectly [32, p. 81 ff.] or, in case the cars require a similar number and mix of compo-nents, the mix of cars is leveled instead [32, p. 86 ff.]. The latter “level scheduling”was introduced by Miltenburg [37] and commonly pursues nonlinear goals. Thusboth time related objectives and JIT-objectives directly address the model mix con-straints discussed so far.

The car sequencing problem (CSP), originally introduced by Parretto et al. [40],allows to model the above mentioned minimum distances between orders with thesame options and further separation rules like a “maximum number of identicaloptions within a car sequence of predefined length”. The CSP in not widely knownwithin the OR community, but one of the classical problems in the literature onconstraint satisfaction problems [7]. Brailsford et al. [7] review this kind of literature,showing that these “soft” constraints also pursue time-related objectives and thatJIT- and some further objectives can also be modeled as soft constraints of a CSP.They report that – by using a hybrid approach combining simulated annealing andconstraint logic programming – David and Chew [10]are able to obtain good solutionsfor a practical problem at Renault involving 7,500 cars with 50–100 options each.

Recent approaches of Drexl et al. combine the classical CSP with level schedul-ing [12] and solve it in a two stage approach [13]. Also Monden [38, Chap. 17]extends his JIT-oriented “goal-chasing” heuristics in order to respect CSP distanceobjectives (denoted as “continuation control” and “interval control”). Zeramdini etal. [55] propose a two-stage approach, smoothing the components’ usage first andthe workload secondly, to optimize the bicriteria sequencing problem. Korkmazeland Meral [31] reformulate the same combined problem as an assignment prob-lem with weighted objectives and develop heuristics for it. Hyun et al. [25] andlater on Ponnambalam et al. [43] consider “minimization of setup costs” as a thirdstriving objective and find (near-) Pareto optimal solutions for the multi-objectiveproblem by using a genetic algorithm. A further overview of models and methodsfor combined objectives is given in [32, p. 92 ff.]. Concluding this brief discussionof sequencing, it can be stated that there is a trend in recent literature on mixed-model assembly line sequencing to consider several objectives, simultaneously.

The frozen car sequence is then the basis to derive the component demand forJIT calls and SIL supply. This short-term material requirements planning (MRP) isnot a “real” planning task because there is nothing left to be decided about. Just theBOM has to be exploded as late as possible before the scheduled delivery (usuallyseveral times per day). It is just mentioned to provide a complete picture of supplierrelationships.

If final customers do not pick up their cars at the assembly sites directly, the fin-ished cars have to be brought to the customers or their respective retailers and salessubsidiaries. There again are some decisions to be made concerning the distributionof the finished cars. For example, the actual carrier has to be chosen, and transport

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domain of logistic service providers [8].

3.3 Organizational issues

One has to be aware that in the preceding sections only “abstract” planning tasksof car manufacturers have been described, but organizational issues have been leftaside. In reality, often several different planning departments are involved in a single

mon plan. Within each coordination round, a single department has to contribute itsown (locally “optimal”) partial plan until some predefined date. Such a (temporar-ily valid) partial plan is a sort of self-commitment of the respective department andprovides input for the next planning activity of another department. This procedureiterates until the common plan hopefully respects all relevant constraints and fulfillsthe various and sometimes conflicting objectives of the different departments to anacceptable level.

The mutual arcs between production and sales in the budget planning and masterproduction planning boxes of Fig. 2 ought to indicate that in practice the respectiveplanning task usually is not tackled in a single, simultaneous planning procedure,but in the above mentioned coordination rounds. This is one reason why LP andMIP models are not used for a simultaneous budget planning or a simultaneousmaster production planning as it is common practice in other types of industrieslike consumer goods manufacturing, for example [45]. Wahl [54] proposes appro-priate models to – at least individually – optimize the planning decisions of the salesdepartment in this way. But even such a local application of LP and MIP has notbeen implemented in practice for reasons like missing (IT) infrastructures, inappro-priate forms of organization or mostly a lack of acceptance and understanding ofOR methods.

4 Current trends in the German automotive industry

As Fig. 1 shows, “to move from BTS to BTO” is a somewhat imprecise formulation.The task is rather to increase the share of final customers’ orders. Further strategicgoals, currently pursued in the German automotive industry, are to shorten customerorder delivery times of customized cars, to keep promised delivery dates with ahigh reliability and to allow customers to change their car options also in the veryshort term [51, p. 31 ff.]. In order to reach these goals in addition to supply chaincollaboration (see e.g. [18]) two major bundles of measures, online ordering andlate order assignment, have been and are still being implemented.

4.1 Online ordering

The total order-to-delivery lead time (OTD) can be shortened by reducing the leadtimes of all individual processes (like order entry and processing, manufacturing,

retailers within a tour) have to be determined. Some of these tasks are in the planning

planning task. Then there are several “coordination rounds“ whose result is a com-

frequencies (how often to deliver to a retailer) and vehicle routes (sequence of

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Fig. 4 Example of lead time reduction by online ordering

distribution) involved. Since manufacturing and distribution only comprise a verysmall percentage of the OTD (about 16% according to Holweg and Jones [22,Fig. 3]), the highest potential can be found in order entry and processing. “On-line ordering” initiatives aim at simplifying and accelerating the circumstantial andtimely collecting and (weekly) bulk processing of orders within the multi-stage saleshierarchy. Thus retailers send fully specified ordering requests of final customers viathe Extranet or Internet directly to a central order processing system, where the re-quests are online (i.e. within seconds or minutes) checked for technical feasibilityand provided with a promised delivery date. In case of final customer’s acceptanceof the promised date, the final order is processed with the same speed on the sameroute. By implementing such a system, the car manufacturer BMW tries to reducethe lead time of order entry from 13–17 days to a single day [44], for instance.Figure 4 graphically illustrates how online ordering reduces demand uncertainty.In this (fictitious) example, cutting the lead times of order entry in half triples theshare of final customers’ orders known. Thus the forecast-based BTS inventory ofretailers (phase (II), see also Fig. 1) can be reduced significantly.

4.2 Late order assignment

Traditionally each body-in-white, physically processed within the body shop, is al-ready assigned to a customer order (“order assignment”) and a re-assignment toanother order is only rarely practicable. Following the pull-principles of the just-in-time philosophy the final assembly as the last production stage has to be plannedfirst and synchronizes all direct suppliers and upstream production stages, especiallythe paint shop and the body shop. In the light of “lean thinking” the work-in-processbuffers (body store and painted body store) should be small and thus body and paintshop ideally produce in the same sequence of customer orders as is planned for thefinal assembly. However, these buffers are still necessary because process failures inthe body and paint shops occur frequently [49, p. 29 f.]. According to Holweg [21]

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decreasing lead timesof order entry increasesshare of final customers

(IV) (III) (II)

optionsspecified by

final customers(I)

100 %

accumulatedcustomershare

30 %

10 %time

average lead times of

orderentry

pro-duc-tion

distributioninven-tory

arrivalretailer

handing overto customer


the rework rate is even up to 40–50%. For this reason, a planned assembly sequencecan only be considered to be reliable, when the respective orders’ painted bodieshave left the paint shop. Thus the sequence can only be transmitted to the SIL sup-pliers a few hours before planned assembly, depending on the assembly station andthe respective component.

In order to guarantee more reliable assembly plans, which can be fixed for alonger time interval (about 4–6 days), the order assignment nowadays is postponedto the final assembly stage (“late order assignment” or “late order tagging”, see[21]), i.e. the bodies in the body and paint shops are no longer identified by customerorders. Body and paint shops still get the information about the customer orders tobe assembled, but are free to deviate from the planned assembly sequence. Althoughthere is no demand uncertainty, safety stocks have to be installed for each body-in-white variant and paint color. These safety stocks exclusively hedge against theprocess failures in the body and paint shops. In order to limit the total amount ofsafety stock required and to restrict buffer sizes, the number of body-in-white vari-ations and paints (the so-called “internal complexity”, [21]) should be low. For thisreason, BMW reduced the number of body-in-white variations from 40,000 to 16for their new three series when introducing late order assignment [21]. The higherstability of assembly plans is expected to increase the radius of JIT/SIL delivery andthe share of JIT/SIL-suppliers significantly.

5 Impacts on planning

Online ordering and late order assignment have been and still are being introducedby BMW (project title “Kundenorientierter Vertriebs- und Produktionsprozess”[44]) and DaimlerChrysler (project titles “Global Ordering” and “Perlenkette”[18]). Further car manufacturers intend to follow. These two types of measures con-siderably influence the traditional planning landscape as discussed in Sect. 3. Thusit is necessary to check how planning requirements and information flows change(some planning tasks may loose importance whereas others win) and which newplanning tasks arise.

5.1 Impacts of online ordering

Online ordering and online order promising require extremely short response timesfor incoming customer requests. If highly reliable promised delivery dates shallbe achieved, the capacities of all potential bottlenecks (material or production re-sources) have to be checked. Thus the formerly decentral order promising has tobe automated and centralized. The changes in the planning landscape depend onthe level of delivery reliability aspired. In the following only two extreme scenar-ios, denoted as quota-available-to-promise (QATP) and capable-to-promise (CTP)scenario, are discussed as examples. Of course, there are various intermediates con-ceivable between these extremes.

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promiseddates

line assignment &model mix planning

plant assignment

finalcustomers

allocationplanning

onlineorder promising

specified orderswith due dates

aggregate quotas

detailedquotas(QATP)

production ordersper plant


daily buckets

dailybuckets

retailers

requests,orders,


promiseddates

promiseddates

salesproduction

(weekly) production plans,

overtime

Fig. 5 QATP scenario

5.1.1 QATP scenario

The QATP scenario is more or less an automation of already existing processes. AsFig. 5 shows, the general logic of planning stays the same. The quotas for retail-ers and sales subsidiaries, which have been determined on a weekly basis anywayand have been synchronized with capacities in the medium term, are (as far as theyhave not yet been assigned to final customers’ orders) considered to be “availableto be promised”. Incoming customer requests and customer orders, respectively,are checked for technical feasibility [26], first, and according to simple precedencerules [30] for free quotas, secondly. Such a proceeding is known from material con-strained industries like the computer industry and successfully applied there [29]. Incontrast, however, material availability is not yet checked in the simple QATP sce-nario. The installation of an online ordering system (OOS) is technically lavish andcostly, but hardly changes the planning logic. When comparing Fig. 5 with Fig. 3,the major differences are that specified orders (and their due dates) are directlytransmitted to the plant assignment instead of using the multi-stage sales hierarchyand that specification changes can be sent faster (and thus later) to the model mixplanning.

However, because the mid-term capacity check, on which free quotas (QATP)are based, had no detailed information about the customers’ choice of car options,there is a high probability that the promised delivery dates do not fit the model mixconstraints and thus cannot be kept on the short-term.

5.1.2 CTP scenario

In order to achieve a higher delivery reliability, a shorter-term and more detailed ca-pacity check is necessary, which motivates the other, more challenging extreme,the capable-to-promise (CTP) scenario. When accepting orders and confirming

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delivery dates, the customer orders are directly booked [22] to a day of production orweek of production (if a late delivery is desired by the final customer) of an adequateassembly plant. In contrary to the QATP scenario, all or at least the most crucialconstraints, relevant for model mix planning (like options of the orders, materialrequired, production capacity, quotas of the respective sales hierarchy), are consid-ered. The order promising is extended such that production orders can automaticallybe generated. Thus the online order promising takes on planning tasks of the short-term production planning or – at least – limits its scope. Furthermore, also the plantassignment has to be integrated into such a comprehensive online ordering.

Questions, which have to be answered online, are for example: Is a BTS carphysically available somewhere in the supply chain, which fits the requirements ofthe new customer order to a very high degree? Is a similar BTS car planned and canits options be changed so that the order still can be assigned to it? Which plant hasto be chosen if a new production order has to be generated? Should be producedearlier or later than the desired date, if this is already (over)booked? If model mixconstraints are limiting, which car specifications should a customer change in orderto still get his desired delivery date promised?

However, one has to keep in mind that the computational burden to update allthe necessary data and the desired response times of the OOS are conflicting. Themajor problem is to find the right trade off between modeling capacities as detailedas necessary (increases delivery reliability) and updating as few data as possible (inorder to guarantee short response times).

Figure 6 shows the embedding of a CTP online order promising into the plan-ning landscape. The online order promising needs free quotas (QATP) and not yetassigned net capacities of material (“material-available-to-promise”, MATP) andassembly resources (CTP), e.g. expressed by a maximum number of cars with a spe-cific (combination of) option(s) per day, as inputs. The results of the order promisingare weekly delivery dates, which are promised to the customers, and “production

line assignment &model mix planning

netting

finalcustomers

allocationplanning

sequencing

production ordersper plant

and day/week


aggregate quotas

detailed quotas(QATP)

retailers

onlineorder promising

(incl. plant assignm.)

promiseddates

promiseddates

promiseddates

MATP, CTP

dailybuckets

dailybuckets

salesproduction

(weekly) productionplans, overtime

requests,orders,


Fig. 6 CTP scenario

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orders” with a promised delivery date and a planned day (or week) of production,which are sent to the line assignment & model mix planning of the respective plants.

A decentral model mix planning is still necessary for several reasons. For exam-ple, the preliminary production plans of order promising have to be updated withrespect to (for complexity reasons) still unconsidered capacity constraints and a lineassignment has to be made. Production orders, which have only been allocated toa week of production because of rather long customer order lead times being de-sired, have to be assigned to a day of production. The more detailed the capacityconstraints of order promising are, the less changes of its plans should be necessarylater on in the model mix planning because the most crucial potential bottleneckshave already been anticipated. However, short-term failures in supply and produc-tion can never be avoided and thus make a re-planning necessary.

The results of the model mix planning are daily buckets, which again are sent tothe sequencing, but are – in a further netting procedure – also used to calculate the(net) MATP and (net) CTP for the online order promising. Further input for the net-ting is up-to-date information about plant capacities and projected material supply,which have been synchronized in the master production planning in the mediumterm (see Fig. 2). Fleischmann and Meyr [17] illustrate the interaction between“order entry” (online order promising) and “MATP/CTP (re)calculation” (nettingand model mix planning) by means of two more detailed workflows and discussthe planning tasks of demand fulfillment for various positions of decoupling points.They also propose LP and MIP models for order (re-) promising, which are usefulif several customer requests/orders can be processed in a batch. However, if eachcustomer request has to be answered immediately, the degree of freedom is ratherlow. Consequently, the importance and impact of previous planning tasks, like mas-ter production and allocation planning, grows. The APS vendor SAP [46] offers asoftware module called Realtime-Positioning, which has especially been designedfor the online order promising in the CTP-scenario, and Ohl [39, p. 207 ff.] dis-cusses the advantages of “code rules”, describing the interrelations between variouscar options, for a capacitated BOM-explosion of online queries. However, formalmodels for the MATP/CTP calculation and search are not presented. Similarly tothe approach of Ohl, Bertrand et al. [4] propose to use a hierarchical pseudobill ofmaterial for the MATP check in case of strong interdependencies between differentoptions (so-called “non-modular products”).

Besides newly arriving customer requests/orders also changes of the specifica-tion of already accepted orders can be processed and checked for capacity usingthe online ordering system. Furthermore, the specification of not yet fulfilled quo-tas by retailers (see phase (II) of Fig. 1) can be checked with respect to model mixconstraints. Online ordering accelerates order processing and increases the share offinal customers’ orders (see Fig. 4), but BTS cars cannot completely be avoided [51,p. 6]. In order to decrease the times in inventory of the remaining BTS cars, finalcustomers’ desired options should be anticipated more precisely. Central statisticsabout the final customers’ preferences and about frequently purchased options cancomfortably be made available to retailers by means of the OOS. They widen thelocal view of the retailers and promise a higher quality of forecasts for BTS spec-ifications [51, p. 176 ff.]. These proposals for BTS options and the more detailed

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MATP/CTP capacity check can be seen as new potentials that arise due to the cen-tralization of order promising and the online connection to retailers.

5.2 Impacts of late order assignment

Late order assignment undoubtedly has its major impacts on strategic planning.Products have to be re-designed so that a high number of options (high externalvariety) can be kept up while simultaneously reducing the number of body-in-whitevariations (low internal complexity [21]). There is a rich OR literature on designfor postponement and modularization (see Sect. 1 and [3], for instance), which triesto support such issues. Furthermore, the re-dimensioning of the (body store and)painted body store is a strategic planning task.

But also for the operational planning of the body and paint shops and their re-spective stores new challenges arise because of the higher degrees of freedom. Thesafety stocks of the body store and the painted body store have to be refilled with re-spect to the failure probability of the respective production processes. Because of therather loose coupling to the assembly sequence and because of the increased buffersizes, lot-sizing issues can now be considered easier in the paint shop. Althoughchangeover times are negligible, batching lots is economically desirable because achange of the paints incurs costs between e10 and e30 [49, p. 30]. Taking cars outof the body store is a Sequential Ordering Problem [15], a special variant of theTraveling Salesman Problem. For paint shops as a practical application, Spiecker-mann [49, p. 126 ff.] proposes a branch-and-bound approach which takes advantageof special knowledge about common structures of body stores in the automotiveindustry (see also [50] for earlier approaches to the same problem). Engel et al.[14] propose a heuristic for workload leveling which can be extended for the batchsequencing of paints of the same color.

Inman and Schmeling [27] prove the operational advantages of late order assign-ment by means of simulation. They compare the traditional irreversible coupling oforders and physical vehicles at the body shop with a flexible assignment procedure(“Agile Assemble-to-Order” (AAO) system) that is able to assign and re-assign or-ders to vehicles before the body shop, paint shop and the final assembly are entered.The objective of the AAO system is a weighted function comprising penalty termsfor violating lead time, paint color, spacing and levelness constraints. Orders areselected by the AAO system in a greedy manner with the weights varying accordingto preferences of the production stage under consideration.

6 Conclusions and outlook

Concerning forecast-driven planning it can be stated that the quotas of the tra-ditional master production and allocation planning had a detrimental effect onmeeting final customers’ demand on time. This gets even worse if the same quotasare directly taken over to an OOS with automatic booking and without the pos-sibility of human intervention (see Sect. 5.1.1). Thus, if still necessary to smooththe workload in the medium term, one has to think about more flexible allocation

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first step in this direction. The choice of adequate aggregation levels, allowing topostpone decisions as long as possible, is crucial.

planning tasks like budget, master production and allocation planning. Wahl [54]has proven that this would also be true for (at least the sales side of) automotiveindustries. The reasons, why the proposals of Wahl have not been put into practice,should have diminished or even vanished in the meantime. Information technologyhas improved dramatically in recent years and there seems to be a higher willingness

to apply LP and MIP methods in practice. In addition, simultaneous optimizationcovering several departments like production, procurement and sales in a singlemodel could exploit further potentials and – at least simulatively – support andaccelerate the lengthy coordination rounds (see Sect. 3.3).

Regarding the traditional order-driven planning it has been shown that ORsupport for the planning tasks plant assignment and line assignment & model mixplanning was very poor. However, these tasks will change their character anywaywhen online ordering and the CTP scenario are installed. On the other hand, there isrich literature on assembly line sequencing and research in this field is an ongoingprocess. Recent OR-related papers tend to pursue several objectives simultaneously,thus becoming more attractive for practical application in the automotive industry.However, scalability of sophisticated methods is still a problem and should be atopic of future research.

As we have seen, the measures to move from BTS to BTO also have signifi-cant impact on planning. The consequences for forecast-driven planning have beensketched above. Further challenges can be identified for the future order-drivenplanning. Due to late order assignment the close coupling of body, paint and assem-bly shops has been decreased now. Thus there remains supplementary freedom forpaint shop sequencing and batching of paints of the same color. However, becauseof still limited buffer sizes, OR models have to take care that paint shop sequencesmay not deviate too far from assembly sequences.

Online ordering is most challenging in the CTP scenario when incoming ordershave to be booked directly into a (capacitated) production plan of a plant. In thiscase, online order promising takes over functionalities of the traditional plant as-signment and the traditional line assignment & model mix planning. The three mostcrucial problems are

– how to model quotas and model mix restrictions as constraints for the onlineorder promising (within the netting procedure, respecting the results of the pre-vious master production and allocation planning),

– which fast algorithms or search rules to use for allocating free QATP, MATP andCTP (within online order promising) and

– how to revise the resulting preliminary production plans in case of still unconsid-ered constraints and unforeseen short-term events (new line assignment & modelmix planning, respecting the already promised due dates).

customers’ demand. Virtual, central car pools, accessible for several retailers, are a

In other lines of business it has been shown that LP and MIP models can support

to make use of OR tools. APS, for example, are a comfortable and user friendly way

mechanisms incorporating the increased knowledge (see Sect. 4.1) about final

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Research has to be done on both OR models/methods for the different planning tasksinvolved and – since responsibilities change – also on the (hierarchical) interrelationof these planning tasks within the overall planning framework. If, above all, carmanufacturers think about customized sales prices, which may vary according to thedelivery times desired by final customers, the relationship to revenue management(see e.g. [34, 48]), as common in airline industries, has to be further investigated.

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Modeling and optimizing of strategic and tacticalproduction planning in the automotive industryunder uncertainty

Ralf Bihlmaier · Achim Koberstein · René Obst

Abstract This work considers the strategic flexibility and capacity planning underuncertain demands in production networks of automobile manufacturers. We presenta deterministic and a stochastic model, which extend existing approaches, especiallyby an anticipation scheme for tactical workforce planning. This scheme is comparedto an extended formulation of the deterministic model, which incorporates workforceplanning via detailed shift models. The stochastic model is efficiently solved by anaccelerated decomposition approach. The solution approach is integrated into a deci-sion support system, which calculates minimum-cost product allocations and capacityplans. Our numerical results show that, in spite of the considerably increased com-plexity, our approach can efficiently handle hundreds of scenarios. Finally, we presentan industrial case study.

Keywords Strategic network design · Anticipation of tactical planning · Stochasticprogramming · Decomposition approach

1 Introduction

Today companies of the automobile industry face a market situation which is par-ticularly characterized by dynamic change and uncertainty. Due to decreasing salesfigures and stagnating market prices especially the companies in the premium

R. Bihlmaier (B)Daimler AG, Group Research and Advanced Engineering, 89073 Ulm, Germanye-mail: [email protected]

A. Koberstein (B) · R. ObstDS & OR Lab, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germanye-mail: [email protected]



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segment increasingly concentrate on new niche markets. To get higher market-sharesin the oligopolistic cutthroat competition, they try to constantly reduce product lifecycles and raise the diversity of their products. One of the consequences is the increaseof over-capacities, since installed capacities of the production facilities are consideredto be short- or medium-term unchangeable (cf. Becker 2005; Jordan and Graves 1995;Friese et al. 2005).

Since the production facilities in the automobile industries require high investments,a low degree of utilization yields to diminishing profit margins in terms of a high fixedcosts risk (see e.g. Roscher 2008, p.39). Therefore highly dynamic and uncertainmarkets pose a high risk to the companies. The adequate answer to these challengesis to make production facilities more flexible within the whole network (Bruynesteyn2003). In general the meaning of flexibility is the ability to react purposefully on unex-pected changes (Schneeweiss 1999). In the research literature, there exist a multiplicityof different flexibility terms. A detailed theoretical taxonomy of the term flexibility isgiven by Sethi and Sethi (1990). More applicable definitions are discussed for exampleby Chandra et al. (2005), Gerwin (1982) and Pibernik (2001). Regarding the strategicscope of our work, we only use the terms product flexibility, volume flexibility andsuccessor flexibility in the following. Product flexibility measures the ability of a pro-duction line to manufacture several different products, volume flexibility denotes itsability to adapt to different output rates in a cost efficient way (see Roscher 2008, p.28).A line is characterized as successor flexible if it can be enabled to produce future prod-uct variants. In a flexible production network a low degree of utilization and high profitrisk can be avoided by exploiting the opportunity to produce one product in differentplants or to produce different products in one plant respectively. Thus market demandscan effectively be assigned to several capacities. Hence, the assignment of products topotential plants and the installation of flexible production capacity are the fundamentaltasks for the strategic network planning. However, the degree of flexibility is fairlyrestricted by high investments, fixed costs or restrictions like the applied manufacturingtechnology or different materials for the varying products (Friese et al. 2005).

Nowadays, deterministic network design problems of considerable complexity canbe solved. However, it was early recognized that deterministic models are not suit-able to represent planning problems in highly dynamic and uncertain environments.Therefore stochastic versions of the deterministic models and specially tailored solu-tion methods were proposed. However, only very recent models, e.g. as proposed bySantoso et al. (2005), are able to solve a stochastic network design problem of realisticcomplexity for a large number of scenarios in acceptable time. In this paper we extendand customize the general model of Santoso et al. to comply with special requirementsof production network planning in the automobile industry. In particular, we incor-porate an anticipation scheme for tactical workforce planning and compare it to anextended formulation of the deterministic model, in which we use detailed shift mod-els to consider the task of workforce planning in a more realistic way. Furthermore weconsider multi-period combined product allocation and capacity initialization withregard to combined investment and cost parameters. These enhancements lead to aconsiderably higher complexity compared to existing models. However, our numer-ical results indicate that it is still possible to solve instances of realistic size, whichinvolve hundreds of scenarios. We incorporated our solution approach into a decision

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support system which is being deployed in the strategic production planning of a majorGerman car manufacturer. Finally we present a real world case study with the goal toevaluate and leverage benefits of flexibility in the manufacturer’s production network.

The rest of the paper is structured as follows. In the next section we describe thechallenges of strategic network planning in the automobile industry, clarify the goals ofthis study and give a literature survey. In Sect. 3 the problem is modeled as a determin-istic and a two-stage stochastic mixed integer program with fixed complete recourse.The deterministic model is extended by a detailed model for tactical workforce plan-ning. Section 4 deals with a brief description of the implemented solution approachbased on an extended version of Benders’ decomposition algorithm. The applicationof model and solution method to real world problems is discussed in Sect. 5 by solvinga complex planning problem of realistic size stemming from the European automobileindustry. Finally, Sect. 6 reflects the conclusions of this work and gives an outlook tofuture studies.

2 Statement of the problem

2.1 Strategic network planning in the automobile industry

In the order-oriented production of the automobile industry, strategic decisions mustmainly concentrate on future markets. In this respect a main difficulty of global networkplanning lies in the consideration of uncertain magnitudes. Santoso (2003) identifiesa variety of such uncertain magnitudes in global production networks, for example,product demands, product life cycles, market prices or production costs and trans-port costs which all are discussed by Vidal and Goetschalckx in detail (2000). Alsoexchange rates can be regarded as such a critical risk factor of uncertainty (Meyer2004), but corporate hedging strategies limit their influence. We therefore concentrateon uncertain demand quantities and represent their magnitudes by forecasted discreteprobability distributions. However, this is just a mild limitation since continuous dis-tributions can be discretely approximated sufficiently close (Boettcher 1989).

The main decisions in strategic network planning include where to close existingor open new production sites, where to manufacture which product and from whereto satisfy the customers demand (Goetschalckx 2002) in every time period. Further-more the capacities of the manufacturing lines have to be fixed. Due to complexityavoidance, decisions related to suppliers or supplier structure are not included in thedecision model in this paper. The planning horizon should cover at least one productlife cycle for each product or two life cycles if successor flexibility potentials could beutilized. The current life cycle duration of automotive products is 5–7 years Barnettet al. (1995), so that the planning horizon of the stated problem will end up in fiveto maximal twenty years. Regarding the long-term character of the involved strategicdecisions, 1 year is a proper representation for one time period in our model.

In the following we distinguish between technical and organizational capacityin order to distinguish the strategic planning task of initialization from the tacticalplanning task of adaptation. Technical capacity is defined as the maximal quantity amanufacturing facility is able to produce, whereas organizational capacity determines

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the actual utilization of a manufacturing facility, e.g., depending on a chosen workingtime model. Regarding the process of capacity dimensioning, it has to be consideredthat installed capacities can be adjusted to changing market circumstances by applyingeither technical or organizational planning options. Technical options are e.g. addingequipment to the production lines, changing production technology or altering thecycle time of a production line. But these technical options are typically linked tohigh costs. A more appropriate approach is to take advantage of the organizationalopportunity of workforce flexibility. Organizational options based on workforce flex-ibility are the variation of shift length, percentage of temporary staff, Saturday shiftsetc. Using these midterm instruments, the management is able to react properly todemand variations and therefore to lower the fixed costs risk. In order to fully exploitthe impact of these additional degrees of freedom of the tactical planning, they haveto be already anticipated in the strategic planning. This anticipation should be done atleast approximately to ensure no or just a sub-proportional increase in planning com-plexity. Jordan and Graves (1995) clarify that flexibility and capacity can representsubstitutable magnitudes. Therefore the modeling of strategic network planning hasto involve decisions for both aspects simultaneously.

In order to determine the primary result of the decision model, the opening andclosing of facilities and the product assignments, even the optimal decisions of theoperational level should be rudimentarily anticipated. The objective of this integratedoptimization is to determine the production and transportation potentials as well asthe shortfalls by minimum costs and under consideration of the uncertain demandsand a given corporate strategy and policy. Hereby several decisions like using existingfacilities and given product allocations for reasons like securing existing job banks orlowering additional investments as well as local content conditions have to be regardedin our model formulation. The secondary result of the model gives information on theexpected utilization of the flexibility and capacity defaults. This result can only bedetermined by anticipation of demand. For each type of product there are degrees offreedom in the outputs of assigned production lines, in the transportation quantitiesto markets and in the shortfalls by unfulfilled demands. Duties or import taxes, whichcould also have an impact on planning global production networks (Arntzen et al.1995), are not considered in detail. We assume that duties can be seen as a static vari-able cost parameter which we include into the term of transportation costs in order toshorten the model formulation later on.

2.2 Literature review

Since the ground breaking work of Geoffrion and Graves (1974) in the mid-seventies,a large variety of literature work has been published in the field of strategic networkplanning (see Beamon 1998 for a survey). The examples given are organized in theorder of increasing complexity and applicability to real world problems.

Geoffrion and Graves (1974) develop one of the first models for strategic networkplanning. The scope of the mixed-integer model is to create a cost-reducing designof a multi-commodity production and distribution network. The solution’s approachis based upon a Benders’ Decomposition which separates the binary decisions (e.g.

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location of distribution centers) from the continuous decisions (e.g. transport flows).However, the model is not a multi-period stochastic formulation.

Arntzen et al. (1995) present a multi-period, mixed-integer model for global supplychain planning. The model includes both a detailed production, inventory and trans-portation planning and strategic decisions as product allocation with related fixedcosts. Investment requirements are not considered. The objective function includesthe minimization of costs as well as weighted production and shipping times. A par-ticular feature of the model is the focus on international aspects such as duties, importtaxes or duty drawbacks. The model is applied to a real-world problem of a computermanufacturer.

Jordan and Graves (1995) concentrate on evaluating product flexibility in order tohedge against uncertain demands. The main target of this work is to find an optimaldegree of product flexibility. They document through numerical studies that a specialpartly-flexible strategy exhibits nearly the same advantages as a fully-flexible strat-egy. This special strategy is called chaining, because products and plants are linkedtogether alternately in a closed chain. Their approach could be seen as trend-settingin the field of flexibility evaluation. Monetary analysis as well as the considerationof capacity decisions are not considered in this approach. Boyer and Leong (1996)expand the model of Jordan and Graves by including diverse setup costs which areincurred by simultaneous production of several products in a flexible plant.

Following the work of Jordan and Graves, Francas et al. (2007) evaluate the impactof demand dynamics caused by product life-cycles. Using a stochastic programmingmodel the authors show that benefits of flexible configurations might be substantiallymisjudged if product life-cycles are not considered. However, their results also indi-cate that prominent flexibility strategies like chaining plants remain robust even whenlife-cycles are included in the analysis.

Chandra et al. (2005) formulate a model for flexibility planning specifically for theautomotive industry. The goal of their work is to investigate the dependencies betweenproduct allocation, commonality of parts and capacity planning. They develop analgorithmic strategy which assumes product allocation and commonality of parts tobe fixed configurations. They then calculate expected demand quantities using MonteCarlo simulation and, lastly, optimize the capacity planning using a combination of agenetic algorithm and linear optimization.

MirHassani et al. (2000) consider a multi-period, mixed-integer, two-stage stochas-tic program to determine an optimal solution of capacity planning for supply chaindesign problems. The first stage comprises of the opening and closing of plants, andsets capacity levels. In the second stage, optimized decisions about production anddistribution costs are made. Additionally, the authors demonstrate how to use previ-ous Wait-And-See analysis results in the solution method. The approach extends thefirst stage problem by involving the second stage decision of one chosen scenario. Bysolving the extended problem using Benders’ Decomposition, they show that whenchoosing a “good” scenario, the solution time is greatly reduced. Using numericalstudies, the authors demonstrate how well this approach can handle hundreds of sce-narios.

Alonso-Ayuso et al. (2003) present a two-stage stochastic program and a cor-responding solution method for supply chain design problems. The implemented

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strategic decisions cover plant openings, product allocation, product selection andraw material supplier selection—all of which to maximize the net profit margin. Thenet profit margin consists of sales, operating costs and depreciation of investmentsfigures. They model uncertain values in the costs of raw materials and production, andincorporate them into a scenario-based approach. In the second stage, the strategicdecisions are evaluated by making tactical decisions on discrete capacity expansion aswell as production, stock and transport volumes. The authors develop a heuristic basedon a branch-and-fix coordination scheme to solve their model efficiently. Numericalstudies are only presented for a very small number of scenarios, thus, for this detailedapproach, the uncertainty is only considered rudimentarily.

Fleischmann et al. (2006) present a detailed multi-period mixed-integer modelbased on experiences at BMW (for detail see Ferber 2005). While it is also based onclassic theoretical modeling approaches, the following issues are pointed out to havea positive impact on acceptance in practice. Firstly, the choice of a cash flow basedobjective, the net present value, allows to compare an optimized solution with man-ually computed strategies. Secondly, the simultaneous optimization of capacity andflexibility strategies fulfills the claim for integrated planning by Jordan and Graves.Furthermore, discrete technical capacity stages are extended with a linear overload torepresent workforce planning instruments.

Santoso et al. (2005) develop a mixed-integer, two-stage stochastic program forplanning realistically scaled supply chain design networks. The first stage of the pro-gram includes decisions on the opening and closing of facilities, capacity levels andproduct allocation among the plants. The second stage includes tactical decisions todetermine optimal production and transportation volumes. The authors consider uncer-tainty in transportation costs, demand and supply quantities as well as discrete stepsof plant capacities. In order to minimize computing time, they integrate an acceler-ated Benders’ Decomposition method utilizing a sampling strategy to handle a greatnumber of scenarios.

All the approaches above show models and methods for special variants of strategicnetwork design problems. Some concentrate on the stochastic character of strategicflexibility planning. Few include realistic capacity optimization. A very important andcrucial aspect of capacity planning is missing in almost all of these approaches: thedivision of capacity planning tasks into technical capacity initialization and organiza-tional capacity adaptation. To cope with complex real-world problems we present atwo-stage stochastic, mixed-integer program (including the anticipation of organiza-tional capacity adaptation) for strategic flexibility and capacity planning of productionnetworks in the automobile industry in the next section. This model deals with the mainstrategic decisions in the first stage and the tactical and operational decisions in thesecond stage by optimizing a net present value of the profits over an extended planninghorizon.

3 Model formulation

Let P be the set of products being produced, transformed or transported in a supplychain network within the planning horizon described by the set of time periods T .

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Table 1 List of indexes

Symbol Definition

P Set of products

F Set of facilities (plants/production lines)

S f Set of capacity-initializing stages for line f ∈ F

M Set of markets

T Set of time-periods

N Set of demand scenarios

W Set of shift models

Table 2 Cost parameters

Symbol Definition Unit

rt Interest rate for the calculation of the capital value in period t (%)

k P Ip f Amount of product specific investment, (MU)

if product p is allocated to facility f

kK Is f Amount of capacity based investment, (MU)

if technical capacity stage s is initialized in facility f

k PVps f t Variable production costs of product p (MU/QU)

in capacity stage s, facility f and period t

k P Fp f t Production based fixed costs of product p in facility f and period t (MU)

kK Fs f t Capacity based fix costs of the initialized capacity stage s, that occur, (MU)

if it is actually deployed in facility f and period t

kT Ip f f ′t Cost rate for internal transport of one unit of product p (MU/QU)

from facility f to facility f ′ in period t

kT Ep f mt Cost rate for external transport of one unit of product p (MU/QU)

from facility f to market m in period t

kSFpmt Opportunity costs for shortfall of one unit of product p (MU/QU)

in market m and period t

k M I Ns f t Cost to reduce the capacity of stage s in facility f (MU/CU)

and period t using organizational instruments by one unit (linear approximation)

k M AXs f t Cost to increase the capacity of stage s in facility f (MU/CU)

and period t using organizational instruments by one unit (linear approximation)

Every element p ∈ P may represent a raw material, an intermediate good or a finalgood. Furthermore let F be the set of production lines and plants. Consequently, anelement f ∈ F characterizes a facility transforming a product p ∈ P into anotherproduct p′ ∈ P . The set of markets M and the set of logistic connections betweenproduction lines or production lines and markets form the production network as adirected non-cyclic graph. In Table 1 we give a complete list of the sets used in ourmodel. Parameters of the model are distinguished in cost-based and quantity-based

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Table 3 Quantity based parameters


dpmt Demand of product p in market m and period t (QU)

cKs f Capacity of stage s in facility f per period in regular working time (CU)

cE F Fp f Factor that reflects the loss of efficiency induced by flexible (%)

production of product p in facility f

cAVp f Factor that reflects the loss of capacity in the first period (%)

of production of product p in facility f

cK Bps f Amount of capacity units of stage s needed to produce one unit (CU/QU)

of product p in facility f

cM Kf Technical capacity per period in facility f in maximal working time (CU)

cB O Mp′ p Number of units of product p′ to produce one unit (QU/QU)

of product p (bill of material)

Table 4 Miscellaneous parameters


ρn Probability of scenario n

d K M I Ns f Minimal relative capacity reduction by organizational instruments (%)

in capacity stage s and facility f

d K M AXs f Maximal relative capacity increase by organizational instruments (%)

in capacity stage s and facility f (%)

d P M I Np f t Lower bound on the amount of product p produced in facility f and period t

d P M AXp f t Upper bound on the amount of product p produced in facility f and period t

d L Ep f t Upper bound on product allocation variable y P A

p f t ,

which indicates the allocation of product p to facility f in period t

(if set to 0, y P Ap f t is fixed to 0)

d L Fp f t Lower bound on product allocation variable y P A

p f t ,

which indicates the allocation of product p to facility f in period t

(if set to 1, y P Ap f t is fixed to 1)

parameters (see Tables 2, 3). Additionally, several parameters exist to represent cor-porate-policy settings (see Table 4). There are cost based parameters regarding bothsingle period payment flows—like investments or investment-based one time costs—and continuously payment flows—like fixed and variable costs. MU represents the unitof measurement for cost parameters and refers to capacity units (CU), quantity units(QU), time-periods (t) or a combination of several units of measurement. Followingthe hierarchical planning process of the considered problem, decision variables of thecorresponding model can be ordered in strategic and tactical variables. Strategic deci-sions involve all ‘Yes/No’-decisions and are characterized by a binary code. In order

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Table 5 Decision variables

Symbol Definition

y P Ip f t Indicator variable: 1, if the allocation of product p

to facility f is initialized in period t , 0 otherwise

y P Ap f t Indicator variable: 1, if the product p is produced

in facility f period t , 0 otherwise

yK Is f t Indicator variable: 1, if the technical capacity stage s

in facility f is initialized in period t , 0 otherwise

yK As f t Indicator variable: 1, if the technical capacity stage s

in facility f is deployed in period t , 0 otherwise

x Pps f t Real nonnegative variable: amount of product p produced

in facility f and period t using capacity stage s

xT Ip f f ′t Real nonnegative variable: amount of product p transported

from facility f to facility f ′ in period t (internal transport)

xT Ep f mt Real nonnegative variable: amount of product p transported

from facility f to market m in period m (external transport)

x M I Ns f t Real nonnegative variable: amount of which the capacity of stage s in facility f

is reduced by organizational instruments

x M AXs f t Real nonnegative variable: amount of which the capacity of stage s in facility f

is increased by organizational instruments

zSFpmt Real nonnegative variable: shortfall of product p on market m in period t

to approximate the real problem, all tactical decisions are represented by continuousreal-valued variables (see Table 5).

3.1 Deterministic model

Below we present the deterministic formulation of the optimization model. As anobjective value of the described model (1)–(17), a monetary ratio in terms of thepresent value of period-based payment flows is determined in formula (1). Referringto Goetschalckx (2001), the net present value represents an adequate objective for stra-tegic network design problems as it reflects both an efficiency principle and temporaryadvantages. The impact of the strategic and tactical decisions on the payment flowsare formulated separately in the functions Z F S(t) and Z FT (t), respectively. Strate-gic decisions involve investments and induce fixed costs. Tactical decisions dependon variable costs and cause running expenses and profits.

min Z F =∑

T

Z F S (t)+ Z FT (t)

(1+ rt )t (1)

with Z F S (t) =∑

P

∑

F

(k P I

p f y P Ip f t + k P F

p f t y P Ap f t

)

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+∑

S

∑

F

(kK I

s f yK Is f t + kK F

s f t yK As f t

)(2)

and Z FT (t) =∑

S

∑

F

(k M I N

s f t x M I Ns f t + k M AX

s f t x M AXs f t

)

+∑

P

∑

S

∑

F

k PVps f t x

Pps f t

+∑

P

∑

F

∑

F ′kT I

p f f ′t xT Ip f f ′t

+∑

P

∑

F

∑

M

kT Ep f mt x

T Ep f mt

+∑

P

∑

M

kSFpmt z

SFpmt (3)

subject to (4)–(17).

Constraints (4) and (5) enforce the indispensable dependencies of the strategic deci-sions. In particular, link and technical capacity decisions can only be used after thecorresponding initialization. Due to the character of the capacity decision only one ofthe given set of options can be utilized for each production facility (6).

y P Ap f t ≤

∑

t ′≤t

y P Ip f t ′ ∀p, f, t (4)

yK As f t ≤

∑

t ′≤t

yK Is f t ′ ∀s, f, t (5)

∑

S

∑

T

yK Is f t ≤ 1 ∀ f (6)

The decisions on the strategic level have a direct influence on the tactical utilizationof the production network. On the one hand, the decision about the links determinesthe disposition of production feasibilities to lines or locations (7). On the other hand,the capacity is already dimensioned on this decision level by construction of buildingsand technical facilities (8–10). Thereby the frame of action for cost-efficient capacityadaptation via organizational instruments is given by a linear approximation schemefor every capacity initialization option. The practicability of the approximation isshown in Sect. 5.

cK Bps f x P

ps f t ≤ cM Kf y P A

p f t ∀p, s, f, t (7)∑

P

∑

S

cK Bps f x P

ps f t ≤∑

S

(cK

s f yK As f t + x M AX

s f t − x M I Ns f t

)∀ f, t (8)

x M AXs f t ≤ d K M AX

s f cKs f yK A

s f t ∀s, f, t (9)

x M I Ns f t ≤

(1− d K M I N

s f

)cK

s f yK As f t ∀s, f, t (10)

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The forecasted demand for final products on the customer markets is playing adecisive role regarding the design and use of optional network structures. Since thesatisfaction of all possible realizations of demand is an economic goal by its ownright, and the objective value only involves expected pay-offs, the relative differencebetween possibly requested demand and provided supply quantities must be compen-sated by unfulfilled demand, the so-called shortfall. To simplify the modeling, weassume that every unfulfilled demand quantity results in lost sales.

zSFpmt +

∑

F

xT Ep f mt ≥ dpmt ∀p, m, t (11)

Since our planning problem is based on a multilevel value-added process, twoadditional equations (12 and 13) have to be included to ensure a closed system of thematerial flow. For every node of our network exactly one incoming and one outgoingequation is implemented to achieve a material balance.

∑

F ′xT I

p f ′ f t =∑

S

∑

P ′cBO M

pp′ x Pp′s f t ∀p, f, t (12)

∑

S

x Pps f t =

∑

F ′xT I

p f f ′t +∑

M

xT Ep f mt ∀p, f, t (13)

Additional constraints are established in order to incorporate project expert knowl-edge or political decisions into the generic model. Constraint (14) shows the factthat some product allocation decisions might be fixed, prohibited or technologicallyimpossible. Furthermore, constraint (15) enforces, if given, a frame for the feasibleoutput of several manufacturing facilities. For example, this output frame could be setto secure the economic future of a production site or to align the output assignmentbetween different sites.

d L Fp f t ≤ y P A

p f t ≤ d L Ep f t ∀p, f, t (14)

d P M I Np f t ≤

∑

S

x Pps f t ≤ d P M AX

p f t ∀p, f, t (15)

An aspect that cannot be neglected in the long-term capacity planning is, that not thewhole capacity in years of product launches is available for each affected productionline. Therefore the capacity will be reduced by a percentage rate cAV

p f in constraint (16).Furthermore, we include potential disadvantages like efficiency losses when producingdifferent products on the same line by a rate cE F F

p f in constraint (17).

cKs f yK A

s f t + x M AXs f t − x M I N

s f t ≤(

1−∑

P

cAVp f y P I

p f t

)cM K

s f ∀s, f, t (16)

cKs f yK A

s f t + x M AXs f t − x M I N

s f t ≤(

1−∑

P

cE F Fp f y P A

p f t

)cM K

s f ∀s, f, t (17)

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By constraints (4) to (17), the solution space for the deterministic model is describedsufficiently. But for a potential reduction of the required solution time, additional validinequalities (18) and (19) are applied to the model. Cordeau et al. (2006) describessuch additional inequalities for logistic network design problems. An analysis of theinfluence to solution time will also be conducted later on in Section 5.

∑

T

y P Ip f t ≤ 1 ∀p, f (18)

∑

T

yK Is f t ≤

∑

T

∑

P

y P Ip f t ∀s, f (19)

3.2 Extension to tactical workforce planning

As mentioned in Sect. 2, tactical workforce planning should already be anticipatedat the strategic level (for a detailed workforce planning model see e.g. Askar andZimmermann 2006). These decisions have an essential impact on the optimal designof the network structure by enabling the production systems to adapt capacities overtime. However, the integration—especially the level of detail—of the tactical work-force planning has to be done carefully because of the crucial influence on requiredsolution time. Hence, the model formulated above integrates the tactical decisions viaa linearized approximation scheme. This guarantees acceptable solution time withouthighly affecting strategic decisions. However, this approximation scheme is not suffi-cient to determine total life-cycle costs. Therefore we extend the above model whichsupports the identification of optimal capacity adaptation paths on the one hand, andthe calculation of life-cycle costs on the other hand for a given network structure anda given demand realization. The additional parameters and decision variables used inthe extended model are described in Table 6.

The original model will be adjusted by substituting the capacity adaptation costterms in the objective function and the capacity adaptation constraints. In the newobjective function, the function Z FT (t) is redefined as shown in Eq. (20).

Z FT (t) =∑

F

∑

W

∑

S

(r SMws f t k

W Ff xW F

ws f t

+ k H W Ff x H W F

f t + k FW Ff x FW F

f t

)

+∑

P

∑

S

∑

F

k PVps f t x

Pps f t

+∑

P

∑

P ′

∑

F

∑

F ′kT I

p f f ′t xT Ipp′ f f ′t

+∑

P

∑

F

∑

M

kT Ep f mt x

T Ep f mt

+∑

P

∑

M

kSFpmt z

SFpmt (20)

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Table 6 Additional parameters and decision variables in workforce planning extension


ySMws f t Indicator variable: 1, if shift model w is chosen

in capacity stage s, facility f and period t , 0 otherwise

xW Fws f t Real nonnegative variable: number of employees deployed in shift model w,

capacity stage s, facility f and period t

x H W Ff t Real nonnegative variable: number of employees hired in facility f in period t

x FW Ff t Real nonnegative variable: number of employees dismissed in facility f in period t

r SMws f t Cost parameter: factor for the shift model bonus of shift model w, capacity stage s, (%)

facility f in period t (it is multiplied with employees’s wage

kW Ff to obtain time, shift model, and capacity stage dependent costs)

kW Ff Cost parameter: wage per employee in facility f (MU)

k H W Ff Cost parameter: hiring costs per employee in facility f (MU)

k FW Ff Cost parameter: dismissal costs per employee in facility f (MU)

cSMws f t Capacity parameter: amount of capacity available in shift model w and stage s, (CU)

facility f and period t

dW Fws f t Workforce parameter: minimal number of employees required to deploy shift model w

in capacity stage s, facility f and period t

Constraints (8), (9), (10), (16) and (17) in the original model are replaced by con-straints (21) to (26). The suitable shift model for a capacity stage s in a facility fand in period t is determined in constraint (21). Constraint (22) ensures, that at mostone shift model is chosen for a capacity stage s deployed in time period t and facil-ity f . Constraint (23) prevents the number of employers xW F

ws f t from falling below therequired workforce for a chosen shift model. Constraint (24) determines, how manyemployees have to be hired or dismissed to meet the required workforce in period tand facility f based on the previous time period. Constraints (25) and (26) are theequivalents to constraints (16) and (17) in the original model.

∑

P

cK Bps f x P

ps f t ≤∑

W

cSMws f t ySM

ws f t ∀s, f, t (21)

∑

W

ySMws f t ≤ yK A

s f t ∀s, f, t (22)

xW Fws f t ≥ dW F

ws f t ySMws f t ∀w, s, f, t (23)

∑

W

∑

S

xW Fws f t = x H W F

f t − x FW Ff t +

∑

W

∑

S

xW Fws f (t−1) ∀ f, t (24)

∑

W

cSMws f t ySM

ws f t ≤(

1−∑

P

cAVp f y P I

p f t

)cM K

s f ∀s, f, t (25)

∑

W

cSMws f t ySM

ws f t ≤(

1−∑

P

cE F Fp f y P A

p f t

)cM K

s f ∀s, f, t (26)

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Note, that the extended model is of much greater computational complexity than theoriginal model. This is mainly due to the additional binary decision variables ySM

ws f t .Because these variables are situated on the tactical stage of the model, the stochasticversion of the extended model is not amenable to the Benders’ decomposition approachpresented in Sect. 4. Therefore, it is not solvable for practical problem dimensions.Nevertheless, the extended deterministic model is solvable and can be deployed, e.g.,to evaluate strategic planning solutions found by the stochastic model, which is pre-sented in the next Section. A typical scenario of how the different model variants canbe combined in practise is presented in the case study in Sect. 5.

3.3 Stochastic model

To extend the deterministic model of the planning problem by stochastic influences,we presume that only demand quantities are uncertain with known probability distribu-tions. Basics of stochastic programming are discussed by Birge and Louveaux (1997)and Kall and Wallace (1994). The extended model, more precisely the determinis-tic equivalent model, is formulated as a two-stage stochastic, mixed-integer programwith fixed complete recourse. The first stage decisions on the strategic level remainunchanged, the second stage decisions are included in the objective function via theexpected value of each scenario’s optimal value Qn . The two stages of the determinis-tic equivalent model are specified below. Problem (27) represents the strategic (first)stage, which is formulated as a pure 0–1 problem, containing the decisions aboutproduct allocation and capacity dimensioning. The tactical (second) stage is shown inproblem (28). Only constraint set (29) differs slightly from the demand constraint (11)by considering a scenario-dependent demand dnpmt . Also, constraints (16) and (17)are not considered in the stochastic formulation, since they were not needed in ourpractical case studies. They can easily be added, if necessary.

min Z F Stoch =∑

T

Z F S(t)

(1+ rt )t+

∑

N

ρn Qn

(y P A, yK A

)(27)

s.t. Constraints (4)–(6), (14), (18)–(19)

y P Ip f t , y P A

p f t , yK Is f t , yK A

s f t ∈ {0, 1} ∀p, f, s, t

with

Qn

(y P A, yK A

)= min

∑

T

Z FT (t)

(1+ rt )t(28)

s.t. zSFpmt +

∑

F

xT Ep f mt ≥ dnpmt ∀n, p, m, t (29)

Constraints (7)–(10), (12), (13), (15)

x M I Ns f t , x M AX

s f t , x Pps f t , xT I

p f f ′t , xT Ep f mt , zSF

pmt ≥ 0 ∀ f, s, p, m, t

Note, that, in order to keep the model computationally tractable, the linear approx-imation of the organizational capacity adaption is used in the stochastic model instead

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of the extension described in Sect. 3.2. Nevertheless, the computation of the formu-lated stochastic model via common MIP solving algorithms will not lead to acceptablesolution times when applying the model to real world cases in the automotive industry.However, since the second stage does not contain any binary variables, the model isamenable to the highly specialized benders decomposition approach, which will bepresented in the next section.

4 Solution method

This section describes an adequate solution method for the formulated two-stage sto-chastic program. A common algorithm for the optimization of two-stage stochasticprograms is the decomposition scheme proposed by Benders in (1962). This decom-position scheme is the basis of the proposed algorithm strategy. Hence, the formulatedmodel is split into a master problem, including all strategic decisions, and into nsub problems, each of which includes the tactical decisions for one demand scenario.The anticipated tactical decisions—regarding organizational capacity adaptation—areincorporated following the linear approximation scheme. This is done, because the lin-ear approximation of the capacity–cost relationship has just or even no influence on thestrategic decisions compared to the detailed mixed integer formulation of the tacticalplanning problem. Still, the detailed calculation of costs and utilization has to be done.Therefore, after solving the main problem, the shift model planning task is performedfor fixed strategic decisions and every demand scenario.

4.1 Abstract design of the solution algorithm

The first set of the algorithmic strategy is the generation of different demand scenarios.The user can influence this process by determining a set of generation parameters likemean value, variance, correlation and life-cycle curves for the products’ demand quan-tities. The different demand scenarios are computed by a Monte Carlo Method whichinvolves the calculation of log-normal distributions for each product’s life-cycle andconsiders also correlations among the different products. The second step starts theinitialization of the strategic variables by running a heuristic procedure based on a sim-ple estimation of distribution algorithm for combinatorial problems (for more detailssee Larranaga and Lozano 2001). The dependencies between the parameter values areestimated via the computation of probabilistic distribution models like Bayesian net-works. The fitness evaluation of each strategic variable realization is done by solvingsmall linear programs which model the production plan and demand fulfillment inan approximated way. Subsequently, the applied Benders’ Decomposition approachoperates in usual iterative way by solving of the master problem, solving the subprob-lems and adding n optimality cuts to the master problem until a stopping criteria isreached. Since the formulated stochastic model is characterized by a complete fixedrecourse, no feasibility cuts have to be added to the master problem. Algorithm 1shows the framework of the implemented algorithmic strategy.

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Algorithm 1: Abstract algorithmic strategyInput: strategic network planning task, predicted demand quantitiesGenerating n demand scenarios via MonteCarlo Simulation;1Initialization of strategic variables;2while Stopping criteria is not met do3

Solve Master problem;4Solve n Subproblems including linear approximation scheme;5Add n Optimality Cuts to Master;6

Store optimal values of strategic decisions;7Solve mixed integer program of tactical shift planning for fixed strategic decisions;8Output: optimal flexibility strategy and capacity plan

4.2 Applied Benders’ decomposition

For the present formulation of the deterministic equivalent model, the two-stage sto-chastic program is separated into two models: the strategic and tactical level. In orderto reduce processing time, the model is implemented in the multi-cut version (Birgeand Louveaux 1988). Following the approach of MirHassani et al. (2000) the mastermodel includes the tactical decisions of one specifically selected scenario to increasethe content of information—especially in the first iteration steps of the decompositionscheme. Thereby, the master or strategic level model is extended only by |N | − 1continuous variables Θn′ that represent a lower bound to the tactical level objectivefor each scenario representation |N | − 1. The |N | − 1 tactical decisions are sepa-rated into |N | − 1 subproblems. The decomposition scheme iterates the solving ofthe extended master model, the solving of the |N | − 1 subproblems, the calculationof the cut coefficients and the addition of the resulting optimality cuts until a definedstopping criteria is met. The stopping criteria applied to the present approach dependson the difference between lower bound and upper bound of the computed objectivevalue in each iteration. The master model in iteration i is given in 30 and 31.

min Z F Stoch =∑

T

Z F S(t)

(1+ rt )t+ (30)

Qn

(y P A, yK A

)+

∑

N\{n}ρnΘn

s.t. Constraints (4)–(6), (14), (18)–(19)

y P Ip f t , y P A

p f t , yK Is f t , yK A

s f t ∈ {0, 1} ∀p, f, s, t

∑

T

∑

F

∑

P

αn,kp f t y P A

p f t +∑

T

∑

F

∑

S

βn,ks f t yK A

s f t

≤ Θn − γ n,k ∀n ∈ (N\n) ∀k = 1, . . . , i (31)

In each iteration the current best solution of the strategic variables is stored and ispassed to the subproblems. On this basis the dual solutions of the remaining |N | − 1

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subproblems are computed. Equations (32) to (34) show the calculation of the coef-ficients α

n,kp f t , β

n,ks f t and γ n,k for the optimality cuts applied to the extended master

problem for the n-th scenario after the k-th iteration. Before the first iteration the coef-ficients are initialized with 0. The calculation makes use of vectors of dual variables δ,ε, ζ , η, κ , λ and µ associated (in this ordering) with the subproblem restrictions (29),(7)–(10) and (15). Constraint (15) is associated with two vectors of dual variables λ

and µ. The dual variables of the constraints (12) and (13) are not used.

αn,kp f t =

∑

S

cM Kf ε

n,k−1ps f t (32)

βn,ks f t =

(1− d K M I N

s f

)cK

s f κn,k−1s f t + d K M AX

s f cKs f η

n,k−1s f t + cK

s f ζn,k−1f t (33)

γ n,k =∑

T

∑

P

(∑

M

dnpmtδn,k−1pmt +

∑

F

(d P M I N

p f t λn,k−1p f t + d P M AX

p f t µn,k−1p f t

))(34)

As proposed by Santoso et al. (2005) and Wentges (1996), acceleration techniquesare applied to the decomposition scheme. Two of the presented techniques, the con-cept of trust region and the upper bounding heuristic, are added because they showedthe greatest positive influence on processing time for our model formulation. In thefirst iterations of the Benders’ Decomposition a bad convergence behavior can appear,because the calculated solutions of the master problem are oscillating wildly in thesolution space. By avoiding this oscillation, a more effective convergence behav-ior can be achieved. The main idea of the trust region method is to limit the gapbetween sequenced solutions explicitly, so that randomness of the search directioncan be reduced (Ruszczynski and Shapiro 2003). In our case the trust region methodcan be further enhanced. This is due to the fact, that on the one hand the strategicdecisions are explicitly split into product allocation and capacity initialization. Onthe other hand, due to the multi-period nature of the problem the trust region con-

cept has to be applied in a different way. Let Y P Ii :=

{(p, f )|∑T y P I

p f t = 1}

and

Y K Ii :=

{(s, f )|∑T yK I

s f t = 1}

be sets of indices of the i-th iteration. By introducing

the constraint

∑

(p, f )∈Y P Ii

(∑

T

(1− y P I

p f t

))+

∑

(p, f )/∈Y P Ii

(∑

T

y P Ip f t

)≤ ∆P I

i (35)

∑

(s, f )∈Y K Ii

(∑

T

(1− yK I

s f t

))+

∑

(s, f )/∈Y K Ii

(∑

T

yK Is f t

)≤ ∆K I

i (36)

in the master problem of the (i + 1)-th iteration, it is guaranteed, that the computedsolution is bounded within a Hamming distance of ∆K I

i and ∆P Ii , respectively, to the

previous solution. To apply a limitation within the first iterations the decompositionalgorithm is started with ∆P I

i < |F | × |P| and ∆K Ii < |F | × |S|. Furthermore, in the

process of the method the ∆’s are raised with a given rate R∆ to

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R. Bihlmaier et al.

∆P Ii+1 = ∆P I

i + (|F | × |P|)R∆ (37)

∆K Ii+1 = ∆K I

i + (|F | × |S|)R∆ (38)

until the constraints (35) and (36) redundant. Thereby a raise of ∆ is performed whenthe current objective function value lbi of the master problem is near to the currentmaximum objective function value lbmax so that for a given rate Rlb

|lbi − lbmax|lbmax < Rlb (39)

is fulfilled. If (39) is fulfilled for the ∆i ’s, the method is defined as stable and thetrust region constraint is enlarged. During the operation of Benders’ decompositionthe value of the upper bound ub decreases with an ever smaller rate. When the opti-mality gap is small, the method performs numerous iterations, in which the upperbound improves only insignificantly. This performance problem can be attributed tothe calculated optimal flexibility decisions y P I inc

p f t of the master problem, which differ

only slightly. Corresponding capacity decisions yK I inc

s f t variate more but have an evensmaller influence on the resulting objective function value. To avoid these inefficientiterations the following heuristic procedure by Santoso et al. (2005) can be deployed,whose problem-specific adaption is described in algorithm (2).The heuristic considers

Algorithm 2: Upper bounding heuristic

Input: parameter N sub where N sub << N

Fix best solution found so far y P I inc

p f t and y P Ainc

p f t ;1

Solve Deterministic equivalent model with N sub scenarios;2

Store optimal solutions yK I uh

s f t and yK Auh

s f t ;3

Compute objective value z = fN

(y P I inc

p f t , y P Ainc

p f t , yK I uh

s f t , yK Auh

s f t

);4

if z < fN

(y P I cur

p f t , y P Acur

p f t , yK I cur

s f t , yK Acur

s f t

)then5

update y P I cur

p f t ← y P I inc

p f t and y P Acur

p f t ← y P Ainc

p f t ;6

update yK I cur

s f t ← yK I uh

s f t and yK Acur

s f t ← yK Auh

s f t ;7

if z < ub then8update ub← z;9

update yK Ainc

s f t ← yK Auh

s f t ;10

a small number N sub of the N scenarios. In the first step the best product allocationfound up to this point is fixed. In the second step the reduced deterministic equivalentmodel will be solved for N sub scenarios, resulting in presumably suboptimal capac-ity decisions yK I uh

s f t and yK Auh

s f t . For the combined fixation of these flexibility- and

capacity decisions, the objective function value fN

(y P I inc

p f t , y P Ainc

p f t , yK I uh

s f t , yK Auh

s f t

)of

the problem for N scenarios is computed in the third step. Furthermore, the objective

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Table 7 Main characteristicsProperty Quantity

Product variants 16

Facilities 8

periods 5

Technical capacity stages 2

Binary variables 1,440

Continuous variables 1,520

Equalities 1,480

Inequalities 13,520

function value fN

(y P I cur

p f t , y P Acur

p f t , yK I cur

s f t , yK Acur

s f t

)of the current solution of the mas-

ter problem is determined. If the solution y P I inc

p f t , y P Ainc

p f t , yK I uh

s f t , yK Auh

s f t is better, thenthis one will be fixed as current solution of the decomposition algorithm. In case thatthis solution is additionally a lowest upper bound, ub is updated and the solution isstored.

5 Numerical results and case study

In this section, we evaluate our modeling and solution approach regarding computa-tional efficiency and applicability to real-world problems. First, the performance ofthe acceleration techniques is measured by applying it to an extended version of anabstract problem formulated by Jordan and Graves (1995). Secondly, we compare thetwo implemented capacity adaptation schemes, the linear approximation scheme andthe detailed workforce planning, for a small near-real problem instance. Finally, theresults of a pilot application to a real-world problem in the automotive industry arediscussed. The outcomes of the solution approach are compared to a flexibility andcapacity strategy developed by planners.

5.1 Performance tests

The performance tests are conducted on an extension of the abstract strategic net-work design problem proposed by (Jordan and Graves 1995, p. 585). It is extended tomultiple time periods with distinctive life-cycles and different cost parameters for pro-duction, transport and shortfall at each facility. In addition, the flexibility options arenot restricted, but associated with different investment levels. Table 7 shows the maincharacteristics of the deterministic formulation of the test problem. Demand quantitiesare computed on the basis of a logarithmic normal distribution with given parameters,i.e., the expected value and deviation ranges.Our solution approach is implementedin JAVA using the concert technology libraries of CPLEX 10.0 for solving the lin-ear and mixed integer programs within the decomposition scheme. To compare theperformance with common solution approaches, we implemented a complete formu-lation of the deterministic equivalent model within this framework. The experiment

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0

5.000

10.000

15.000

20.000

25.000

30.000

35.000

0 50 100 150 200 250

Scenarios

CP

U T

ime

in s

eco

nd

sstandard

decomposition

solution approach

Fig. 1 Solution time

computations were run on a Pentium M 1.7 GHz PC with 1.0 GB RAM running Win-dows XP. To analyse the performance of our approach, we compared the results ofdifferent sample sizes to the results of the complete deterministic equivalent formula-tion. A solution is regarded as optimal if the gap between lower and upper bound is lessthan 0.1%. Figure 1 shows the results of our solution approach (solution approach)compared with a standard Benders’ Decomposition scheme (decomposition) as well aswith the complete deterministic equivalent formulation (standard) with sample sizesfrom 5 to 200 scenarios. The standard Benders’ Decomposition scheme is alreadyadvantageous compared to the deterministic equivalent formulation for a sample sizegreater than 20 scenarios. However, the standard approach is impractical for morethan 50 scenarios, the solution time would be greater than 10 h. In total our solutionapproach shows a significant superiority under the given circumstances.Regarding theimplemented techniques, different combinations were examined to achieve an optimalsetting which performs best on different problem instances. The results in Fig. 2 show,that there is no technique which dominates all of the other techniques with respectto performance. Single deployments of the various acceleration techniques are sig-nificantly outperformed by combinations of them. The best combination for both theabstract problem and the real-world instance seems to be the combination of multi-cutting, trust region and expansion of the master model by a specially chosen subsetof the subproblem.

5.2 Comparison of capacity adaptation schemes

The first case study deals with the anticipation scheme for the tactical workforce plan-ning. It is based on a small real world problem instance. The target is to determinea cost-efficient network structure of two products, three possible plants each with

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5000

7500

10000

12500

15000

17500

20000

22500

25000

220

Scenarios

CP

U T

ime

in s

eco

nd

s

benders decomposition

Expanded Master(EM)

Trust Region(TR)

Upper Bounding Heuristic(UB)

StartValues(SV)

Multi Cut (MC)

EM+TR+MC

20 40 60 80 100 120 140 160 180 200

Fig. 2 Solution techniques

Stage 1 Stage 2 Stage 3 Stage 1 Stage 2 Stage 3 Stage 1 Stage 2 Stage 3Capacity in regular working time (c^K) 700 725 750 400 425 450 700 1000 1400Capacity based invest (k^KI) 4100 4200 4400 4800 4900 5000 3200 3600 6400Capacity based fixed costs (k^KF] 385 390 395 430 440 450 520 630 1000Minimal relative capacity reduction (c^KMIN) 33% 33% 33% 33% 33% 33% 33% 33% 33%Maximal relative capacity increase (c^KMAX) 104% 104% 104% 104% 104% 104% 104% 104% 104%Capacity reduction costs (k^MIN) 0,10 0,10 0,10 0,45 0,43 0,42 0,25 0,23 0,17Capacity increase costs (k^MAX) 0,07 0,07 0,07 0,60 0,60 0,60 0,10 0,09 0,07Number of employees per shift (d^WF) 250 260 270 200 225 250 200 250 400Wage per employee (c^WF)Wage per temporary workerHiring cost per employee (c^HWF)Dissmissial Cost per employee (c^FWF)Shiftmodel bonus (r^SM)maximal amount of temporary workers

Input Data for detailed Workforce planning

0,550,600,010,55

late shift (15%), night shift (20%), Saturday shift (30%), Saturday late shift (45%)30%

Plant 1 Plant 2 Plant 3

Input Data for linear approximation

Demand Quantities

0

200

400

600

800

1.000

1.200

1.400

1.600

15

Period

Dem

and

per

Per

iod

Product A

Product B

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Fig. 3 Input data for workforce planning

three possible capacity expansion stages for one life-cycle with half-year periods. Weonly consider investments and labor fixed costs. Figure 3 shows the possible networksstructure and the related input data, for confidentiality reasons the problem data hasbeen abstracted.

While the linear approximation model needs an ex ante calculation of capacity fixedcosts and adaptation costs, in terms of labor costs, the detailed workforce planningmodel explicitly incorporates decisions of hiring and dismissing workers, considers

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Capacity Load Plant 1 (linear Approximation)

0

100

200

300

400

500

600

700

800

15

Period

Capacity Load Plant 1 (Detailed Workforce Planning)

0

100

200

300

400

500

600

700

800

Period

Production Product BProduction Product ACapacity (Regular WorkTime) Plant 1Capacity (Organizational) Plant 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14

Fig. 4 Comparison of Capacity Adaptation (Example: Plant 1)

Table 8 Comparison of workforce approximation schemes

Linear approximation Detailed workforce planning

Product Allocation Plant 1 A, B A, B

Product Allocation Plant 2 – –

Product Allocation Plant 3 A A

Capacity Plant 1 700 715

Capacity Plant 2 0 0

Capacity Plant 3 1,000 1,020

Product specific invests 3,550 3,550

Capacity specific invests 7,700 8,800

Labor fixed costs 14,211 13,671

different shift models and the option of employing workers just temporarily. The resultsshow that the strategic decisions are mostly identical. Applying the detailed workforceplanning just leads to a minor increase in technical capacity initialization in case ofthe higher shift premium for Saturday. Figure 4 presents the difference in capacityload for the product-flexible Plant No. 1. While the linear approximation model is try-ing to utilize the technical capacity, the detailed workforce planning is visualizing analmost realistic utilization of the employed workforce.If the whole network structureis considered, however, these different planning objectives, as implemented within thedifferent approximation schemes, lead to the same strategic decisions with nearly thesame costs (Table 8).

5.3 Application to real-world problem

The real-world example consists of a network of three existing manufacturing plants.Regarding the abstract problem of the former subsection, it could be seen as a moredetailed planning task on the same network structure. To be more realistic we consideradditional planning restrictions and parameters. In this case, there are no decisionsabout shutdown of existing or opening of new facilities. The production process isdivided into several stages which have to be almost completely carried out at eachlocation due to “just in sequence” concepts and high internal logistics rates. Never-theless, flexibility strategies have to be incorporated since every location consists of

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Table 9 Concept comparison based on certain demand

Performance indicator Planners’ strategy Computed strategy

Additional links in BIW 0 6

Technical capacity utilization (%) 82 87

Demand fullfilment (%) 100 100

Investment costs (GE) 1,080 1,109

Operating costs (GE) 1,210 1,180

Net present value (GE) 988 990

several production lines which can be configured in their degree of flexibility and tech-nical capacity. On the other hand pooling concepts for body in white components likehoods or doors could be implemented. The importance of successor flexibility is dueto the fact that the planning horizon for three car models with up to three main variantseach covers two product life-cycles. The emphasis of the study is on the body shop,since the potential of flexibility in this section is strongly linked to high investmentrequirements and a high degree of automation. In the sections paint shop and finalassembly, the decisions are solely concentrated on the installation of suitable capaci-ties as the flexibility strategies are supposed to be already dignified by the decisions inthe body shop. Reasons for this are the system immanent flexibility (paint shop) andthe high share of manual processes (final assembly). Five possible technical capacitystages with associated investment requirements, as well as cost rates for the estimateddemand quantities, were determined for each production line. The adaptation of capac-ity via organizational actions results from the opportunity to work in ten to seventeenshifts per week. The fixed costs related to varying shift models were approximated byassuming a proportional relationship between organizational capacities and cost rates.Shift model changing costs were neglected due to their minor influence on total costwhich was a result of preceding numerical studies. Finally up to 100 scenarios werecomputed out of the estimated demand quantities. These scenarios were determined bya plain monte carlo scheme. We will compare the solution of the optimization modelto a configuration designed by the company’s planning experts. This configurationresults by calculating the technical capacities on the basis of the estimated demandquantities. The flexibility strategy resulting form this procedure is mainly characterizedby solitary production lines. The fixed costs related to the choice of the shift modelsare computed by a small sized linear program, since this planning exercise itself istoo complex for manual calculation. Comparing the flexibility and capacity strategyof this manual calculation (further on called planners’ strategy) with the solution ofa deterministic optimization approach (computed strategy) based on certain demandscenarios, a slight advantage of the solitary strategy is recognizable (cf. Table 9). Thisadvantage is due to two effects. Firstly, just a slightly higher optimal system capacityis installed for the solitary strategy. Secondly, a minor degree of flexibility to fulfill theestimated demand is implemented, only the several variants for each product modelare manufactured on flexible production lines. These two aspects result in a betternet present value by lower investment requirements. Considering uncertain demand

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Table 10 Concept comparison based on demand deviation

Performance indicator Planners’ strategy Computed strategy

Average technical capacity utilization (%) 82 87

Average demand fullfilment (%) 99.23 99.64

Average shortfall per year 1,473 422

Investment costs (GE) 1,080 1,109

Average operating costs (GE) 1,310 1,241

Average shortfall costs (GE) 85 21

Average net present value (GE) 1,021 1,006

Standard deviation of npv (GE) 2.7 2.3

quantities with deviations up to 40 percent, the computed strategy is getting moreand more advantageous (see Table 10). The computed strategy is able to cover higherdemand quantities due to a higher ‘free’ system capacity. Therefore shortfall costs arequite low. Scenarios where volumes are shifted from one product model to another,lower fixed costs can be achieved by usage of the flexibility potentials. On the otherhand the planners’ strategy is linked to higher shortfall costs in the high volume sce-narios. In the product mix shifting scenarios the planners’ strategy is characterized bylow utilization and therefore by higher fixed costs.

Including uncertainty into the comparison of the two strategies, different approachescan be chosen. Well-tried indicators for the analysis considering uncertainty are, e.g.,the mean value for the estimated returns or the variance for risk measurement. Regard-ing the real-world-problem the average net present value is taken as an equivalent tothe mean value. Now the cost advantage is turning round towards the computed strat-egy. Nevertheless indicators like the mean value or standard deviation are mostly tooaggregated to achieve a perfect reliability in the advantage of one strategy. New waysto visualize the advantages of flexible strategies have to be found.

6 Conclusions

In this paper, we have proposed mathematical formulations of strategic network designproblems under uncertain demands for the automotive industry from a capacity andproduction planning perspective. To emphasize the necessity of anticipating consec-utive stages in a hierarchical planning process, the formulations also include tacticalaspects such as workforce planning. Additionally, customized solution approacheshave been implemented based on Benders’ Decomposition. The numerical resultsshow that the solution approach greatly decreases the solution time in comparison tostandard methods. Furthermore, we have demonstrated that these methods can handlelarge-scale, real-world problems and lead to better decisions than widely acceptedmethods for actual planning problems in the automotive industry. Continued researchis necessary to study in detail the effect of tactical decisions in strategic network designproblems. Moreover, the models must be extended to support calculation of strategicsupply and transport problems which occur in flexible production networks.

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