global supply chain planning for pharmaceuticals

chemical engineering research and design 8 9 ( 2 0 1 1 ) 2396–2409

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

journa l homepage: www.e lsev ier .com/ locate /cherd

Global supply chain planning for pharmaceuticals

Rui T. Sousaa, Songsong Liub, Lazaros G. Papageorgioub, Nilay Shaha,∗

a Centre for Process Systems Engineering, Imperial College London, London SW7 2AZ, UKb Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, LondonWC1E 7JE, UK

a b s t r a c t

The shortening of patent life periods, generic competition and public health policies, among other factors, have

changed the operating context of the pharmaceutical industry. In this work we address a dynamic allocation/planning

problem that optimises the global supply chain planning of a pharmaceutical company, from production stages

at primary and secondary sites to product distribution to markets. The model explores different production and

distribution costs and tax rates at different locations in order to maximise the company’s net profit value (NPV).

Large instances of the model are not solvable in realistic time scales, so two decomposition algorithms were devel-

oped. In the first method, the supply chain is decomposed into independent primary and secondary subproblems, and

each of them is optimised separately. The second algorithm is a temporal decomposition, where the main problem

is separated into several independent subproblems, one per each time period. These algorithms enable the solution

of large instances of the problem in reasonable time with good quality results.

© 2011 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers.

Keywords: Global supply chain; Pharmaceuticals; Mixed integer linear programming; Large-scale optimisation;

Decomposition algorithm

proportional to its sales, general and administrative (SG&A)

1. Introduction

In the past 30 years, the operating context of the pharmaceuti-cal industry has evolved and become much more challenging.The establishment of regulatory authorities and market matu-rity have led to an increase in the costs and time to developnew drugs, decreasing the productivity of the research anddevelopment (R&D) stage and shortening the effective patentlives of new molecules. These two factors, in conjunctionwith the appearance of many substitute drugs in several ther-apeutic areas, have led to the reduction of the exclusivityperiod of new products. Another factor having an impact onthe operation of this industry was the transition of the pay-ing responsibilities from individuals to governmental agenciesand insurance companies, which in association with highdemands for pharmaceuticals, due to aging populations, putstrong pressure on prices and prescription policies (Shah,2004).

From the point of view of manufacturing, the global phar-maceutical industry can be divided into five sub-sectors: large
R&D based multinationals, generic manufacturers operating
∗ Corresponding author. Tel.: +44 0 20 7594 6621; fax: +44 0 20 7594 6606E-mail address: [email protected] (N. Shah).Received 25 November 2010; Received in revised form 23 March 2011;

0263-8762/$ – see front matter © 2011 Published by Elsevier B.V. on behdoi:10.1016/j.cherd.2011.04.005

in the international market, local companies based in onlyone country, contract manufacturers without their own port-folio and biotechnological companies mainly concerned withdrug discovery. The first group, the intensive R&D based indus-tries, is economically the most important and tends to havelarge and complex supply chains due to the global nature of itsactivity. In addition, these companies are the most vulnerableto the global financial, regulatory and social changes so thiswork will focus on their supply chains.

The industry’s preferred mechanism to overcome the pro-ductivity crises has been to increase investment in currentbusiness activities, primarily R&D and sales, the two extremeends of the supply chain. This has been implemented byorganic growth or by mergers and acquisitions (M&A) toexploit economies of scale. However, statistics show thatproductivity continues to decline after a decade of vigorousgrowth in investments on these areas (Coe, 2002). There areno significant economies of scale in sales activities. The rev-enues generated by a pharmaceutical company are directly

.

Accepted 10 April 2011

expenditure, suggesting that two merged companies will not

alf of The Institution of Chemical Engineers.

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dx.doi.org/10.1016/j.cherd.2011.04.005

chemical engineering research and design 8 9 ( 2 0 1 1 ) 2396–2409 2397

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ecessarily be more profitable than they would be separately,.e. there is no improvement on the return rates. Further more,espite the theoretical higher probability of successful productevelopment with greater scale in R&D, in reality it does notranslate into improved pipeline value. Companies will onlymprove their profit margins if they change the relationshipetween volume and costs, which can be achieved throughroductivity gains in the supply chain.

Supply chain optimisation is an excellent way to increaserofit margins and is becoming current practice, not only inharmaceutical industries but also in other areas of business.rntzen et al. (1995) described the restructuring of the sup-ly chain at Digital Equipment Corporation with savings ofver US$ 100 million. They developed a large mixed integerinear programming (MILP) model that incorporates a global,

ulti-product bill of materials for supply chains with arbitrarytructure and a comprehensive model of integrated globalanufacturing and distribution decisions. Camm et al. (1997)

escribed a project related to P&G’s supply chain in Northmerica. The main objective of the study was to streamline

he work processes to eliminate non-value added costs anduplication. The study involved hundreds of suppliers, over 50roduct lines, 60 plant locations, 10 distribution centres andundreds of customer zones. It allowed the company to save200 million before taxes. Kallrath (2000) reported on a projectn BASF where a multi-site, multi-product, multi-period pro-uction/distribution network planning model was developed,iming to determine the production schedule in order to meetgiven demand. Neiro and Pinto (2004) described a petroleum

upply chain planning problem of Petrobras in Brazil, whichomprises 59 petroleum exploration sites, 11 refineries andve terminals, with 20 types of supplied petroleum and 32roducts to local markets. Sousa et al. (2008) considered an

ndustrial global supply chain of a multinational agrochemi-als company, comprising two subsystems: US network andorldwide formulation network.

Supply chain management in the process industries hasong been used as a tool to define production and distribu-ion policies, as well as product allocation. This is the case ofohen and Lee (1988) who described the modelling of a supplyhain composed of raw material vendors, primary and sec-ndary plants (each one with inventories of raw materials andnished products), distribution centres, warehouses and cus-omer areas. Later, Cohen and Moon (1991) used supply chainptimisation to analyse the impact of scale, complexity (theperating costs are a function of the utilisation rates and num-er of products being processed in each facility) and weight ofach cost factor (e.g. production, transportation and alloca-ion costs) on the optimal design and utilisation patterns ofhe supply chain systems.

Timpe and Kallrath (2000) described an MILP model,hich combined production, distribution and marketing and

nvolved plants and sales points, to cover the relevant fea-ures required for the complete supply chain management ofmulti-site production network. Jayaraman and Pirkul (2001)eveloped a Capacitated Plant Location Problem (CPLP) typeodel for planning and coordination of production and dis-

ribution facilities for multiple commodities, comprising rawaterials suppliers, production sites, warehouses and cus-

omer areas. The authors followed a holistic approach to theupply chain, resulting in a deterministic, steady-state, multi-chelon problem.

Park (2005) considered both integrated and decoupled pro-uction and distribution planning problem, consisting of

multiple plants, retailers, items over multiple periods. Theauthor proposed mixed integer optimisation models and atwo-phase heuristic solution to maximise the total net profit.Oh and Karimi (2006) highlighted the importance of dutydrawback regulations in the production-distribution plan-ning problem, and incorporated three main regulatory factors:corporate taxes, import duties and duty drawbacks, in the pro-posed linear programming (LP) model.

Tsiakis and Papageorgiou (2008) considered the optimalconfiguration and operation of multi-product, multi-echelonglobal production and distribution networks, integrating pro-duction, facility location and distribution with financial andbusiness issues such as import duties, plant utilisation,exchange rates and plant maintenance. An MILP model wasformulated and applied to a case study for the coatingsbusiness unit of a global specialty chemicals manufacturer.Verderame and Floudas (2009) proposed a discrete-time multi-site planning with production disaggregation model to providea tight upper bound on the true capacity of daily productionand shipment profiles between production facilities and cus-tomer distribution centres. Salema et al. (2010) presented ageneral dynamic model for the simultaneous design and plan-ning of multiproduct supply chains with reverse flows, wheretime is modelled along a management perspective to deal withthe strategic design and the tactical planning simultaneously.The applicability of the model is proved in an example basedon Portuguese glass industry. You et al. (2010) addressed asimultaneous capacity, production, and distribution planningproblem for a multisite supply chain network including a num-ber of production sites and markets and propose a multiperiodMILP model and two decomposition approaches for solution.

When performing long term process planning, uncertaintyfactors (e.g. in product demand) have to be taken into accountin order to produce robust models whose output decisions willperform well in a variety of scenarios (Verderame et al., 2010).Tsiakis et al. (2001) addressed a strategic problem of stochas-tic planning for multi-echelon (although with rigid structure)supply chains.

Iyer and Grossmann (1998) extended the work of Liu andSahinidis (1996), a specific problem of long-range capacityexpansion planning in the chemical industry. The inputs werea set of available chemical processes, an established produc-tion and distribution network and demand forecasts affectedby uncertainty leading to an MILP, multi-period planningmodel with multiple scenarios for each time period. Ahmedand Sahindis (2003) considered forecast uncertainty parame-ters by specifying a set of scenarios in a stochastic capacityexpansion problem. A multistage stochastic mixed integerprogramming formulation with fixed-charge expansion costswas formulated.

Oh and Karimi (2004) developed a deterministic MILP modelfor the capacity-expansion planning and material sourcing inglobal chemical supply chains with the introduction of twoimportant regulatory factors, corporate tax and import duty.Guillen et al. (2005) proposed a two-stage stochastic optimisa-tion approach to address a multiobjective supply chain designproblem. The Pareto-optimal solution was obtained by the �-constrained method. Puigjaner and Laínez (2008) proposed ascenario-based MILP stochastic model considering both pro-cess operations and finance decisions with an objective ofmaximising the corporate value. A model predictive modelstrategy was integrated with the stochastic model for solution.

Several authors addressed the issue of supply chain optimi-sation and long-term process planning in the pharmaceutical

2398 chemical engineering research and design 8 9 ( 2 0 1 1 ) 2396–2409

industry. Rotstein et al. (1999) started a series of papers dedi-cated to the specific problem of supply chain optimisation inthe pharmaceutical industry. Later, Papageorgiou et al. (2001)published a paper based on the previous one, where the pro-duction stage is formulated with high degree of detail andincluding the trading structure of the company. The proposeddeterministic model considers up to 8 possible products in thecompany’s portfolio. Levis and Papageorgiou (2004) extendedthis work to account for uncertain demand forecasts, depen-dent on the results of the clinical trials for each product.Gatica et al. (2003) proposed an MILP approach for the prob-lem of capacity planning under clinical trials uncertainty,where four clinical trial outcomes for each product are consid-ered as is typical in the pharmaceutical industry. Amaro andBarbosa-Póvoa (2008) considered the integration of planningand scheduling of generalised supply chains with the exis-tence of reverse flows. The developed approach was applied tothe solution of a real pharmaceutical supply chain case study.

So far, most of the problems referred in this review concerndetailed or very detailed descriptions of supply chains of rel-atively small systems (i.e. 1 or 2 sites, and up to 8 products).The long term strategic planning of large pharmaceutical com-panies has not been addressed in any of the previous works.In this work we build a model of the global supply chain ofa large pharmaceutical company, with a long list of productsin its portfolio and an extensive network of manufacturingsites with locations all over the world. The allocation policyof products to sites also differs from previous works. In eachtime period, each product will be produced at a single location(single sourcing policy), however the product/site assignmentmay change along the time horizon reflecting actual practice.This feature increases significantly the binary variables space.

In order to keep the model size within reasonable limits, itcannot be too detailed in its description of the supply chain.Nevertheless, even with this approach, it is necessary to usedecomposition algorithms. The development of decomposi-tion approaches is a promising research direction in the area(Grossmann, 2005; Maravelias and Sung, 2009).

To solve a typical large MILP model, Iyer and Grossmann(1998) used a bi-level decomposition (hierarchical) algorithmto solve the original model. In the first step, the design stage,the capacity expansion variables were aggregated in a newvariable set, time independent, and the processes to developare chosen. In the second level model (operation model), onlythe processes chosen to be developed are subjected to invest-ment. Bok et al. (2000) proposed a bi-level decomposition.The relaxed problem making the decisions for purchasing rawmaterials generates an upper bound to the profit, while thesubproblem yields a lower bound by fixing the delivery fromthe relaxed problem. Levis and Papageorgiou (2004) solved anaggregated model, computationally less expensive, althoughdetailed enough to make the “here-and-now” decisions. In thesecond step, the values of the corresponding variables werefixed and the detailed model is solved. Üster et al. (2007) useda Benders decomposition approach for a multi-product closed-loop supply chain network design problem. Three differentapproaches for adding multiple Benders cuts are proposed.Li and Ierapetritou (2009) formulated the integrated produc-tion planning and scheduling as bilevel optimisation problemswith one planning problem and multiple scheduling prob-lems. A decomposition approach based on convex polyhedralunderestimation was proposed and successfully applied to the
integrated planning and scheduling problem of multipurposemultiproduct batch plants.
Some authors solved the large models resulting fromsupply chain optimisation problems through Lagrangeandecomposition. In their work, Gupta and Maranas (1999)formulated an extension of the “economic-lot-sizing” prob-lem, characterised by determination of the production levelsof multiple products, in multiple sites, with deterministicdemands and multiple time periods. Jayaraman and Pirkul(2001) relaxed three blocks of constraints concerning assign-ment of customers to warehouses, raw materials availabilityand material flows balance. This allowed them to decomposethe original problem in three different sets of subproblems.Maravelias and Grossmann (2001) introduced a good exampleof a model composed of two (or more) independent sub-models with one linking constraint.

Jackson and Grossmann (2003) built a multiperiod optimi-sation model for the planning and coordination of production,transportation and sales for a network of geographicallydistributed multiplant facilities supplying several markets.Two Lagrangean decomposition methods were adopted totackle the problem, spatial and temporal decompositions.In both cases, the authors followed the regular algorithmof Lagrangean decomposition to reach the optimal solutionof the original problem, as described in Reeves (1995). Thenumerical examples show the temporal decomposition towork significantly better than the spatial decomposition.

Eskigun et al. (2005) developed a Lagrangean heuristic forthe proposed large-scale integer linear programming modelfor supply chain network design problem of an automotivecompany with capacity restriction on vehicle distributioncentres. Shen and Qi (2007) embedded Lagrangean relax-ation in branch and bound to solve an integrated stochasticsupply chain design problem which is formulated as a non-linear programming (NLP) model. The Lagrangean relaxationsubproblems are then solved by a low-order polynomial algo-rithm.

Chen and Pinto (2008) proposed several various decompo-sition strategies for the continuous flexible process networkmodel by Bok et al. (2000), including Lagrangean decom-position, Lagrangean relaxation, and Lagrangean/surrogaterelaxation. Among the four decomposition strategies, it wasproved that the solutions generated from Lagrangean relax-ation are better, although its CPU time is lower. Hinojosaet al. (2008) proposed an MILP formulation for a dynamic two-echelon multi-commodity capacitated plant location problemwith inventory and outsourcing aspects. The authors solvedthe resulting, independent subproblems from a Lagrangeanrelaxation scheme and a dual ascent method to find a lowerbound on the optimal objective value. Nishi et al. (2008) usedan augmented Lagrangean decomposition and coordinationapproach for the proposed framework for distributed optimi-sation of supply chain planning for multiple companies. Anaugmented Lagrangean approach with a quadratic penaltyfunction was used to decompose the original problem intoseveral subproblems for each company to eliminate dualitygap.

Puigjaner et al. (2009) used the optimal conditionaldecomposition (OCD), a particular case of Lagrangeandecomposition, for the supply chain design-planning modelextended from the work by Puigjaner and Laínez (2008). In theOCD, the difference from the classic Lagrangean decomposi-tion is its automatic updating process of Lagrange multipliers.You and Grossmann (2010) proposed a spatial decomposition
algorithm based on the integration of Lagrangean relaxationand piecewise linear approximation to solve mixed integer


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onlinear programming (MINLP) model for large-scale jointulti-echelon supply chain design and inventory manage-ent problems.From the literature discussed above, there is a gap in the

esearch on supply chain planning taking both primary andecondary manufacturing in the pharmaceutical industry intoccount. This paper aims to fill this gap.

The rest of our paper is organised as follows. Section 2ntroduces a brief description of the typical supply chain andts components in the pharmaceutical industry. Section 3 isoncerned with the problem description. The mathematicalormulation of the model is presented in section 4. In Sec-ion 5, two decomposition algorithms are conceived to solvehe large model resulting from the problem formulation. Inection 6, the model as well as the performance of the devel-ped algorithm is tested with two illustrative examples. Someoncluding remarks are drawn in Section 7.

. Supply chains in the pharmaceuticalndustry

supply chain may be defined as an integrated process whereeveral business entities work together to produce goods, ser-ices, etc. (Shah, 2005; Barbosa-Póvoa, 2009; Papageorgiou,009) This is a major issue in many industries, as organisationsegin to appreciate the criticality of creating an integratedelationship with their suppliers and customers. Typically, in

anufacturing industries the stages are: raw materials acqui-ition, primary (and secondary) manufacture and distributiono retailers and customers. Each one may comprise one or

ore sub-stages and products may be kept in storage unitse.g. warehouses) between stages (Papageorgiou, 2006).

Many companies, especially multinationals, possess com-lex trading structures where, for tax purposes, the manu-acturing plants, intellectual property and distribution centresre considered to be different entities. This brings more flex-bility to supply chain optimisation as it allows the practicingf several different pricing policies.

Supply chains in the pharmaceutical industry, one typicalndustry of products with a high added value per mass unit,omprise two manufacturing stages: primary manufacturingor active ingredient (AI) production and secondary manu-acturing for formulation and packaging. As very high-valueroducts, AIs are usually produced in low amounts and in fewentralised locations worldwide (Sousa et al., 2007).

.1. Primary manufacturing

he customer-facing end is driven by orders. These ones rep-esent disturbances that propagate backwards in the supplyhain (in the opposite sense of materials flow), being ampli-ed as they get closer to the first stages (bullwhip or Forresterffect). Primary sites are responsible for active AI productionnd may face significant fluctuations in demand. They areharacterised by long cycle times, which make it difficult tonsure end-to-end responsiveness.

The production volumes involved are usually low result-ng in multipurpose plants to spread the capital cost betweenroducts. The changeover activities, that take place when theanufactured product changes, include thorough cleaning to

void cross-contaminations. This usually takes a long time
o it is desirable to keep the planning complexity (numberf different products using the same resource) at a low level
and long campaigns become the norm, otherwise equipmentutilisation is too low.

Since the production volumes are low, transportation costsare not significant at this end of the supply chain, so pri-mary sites may be located anywhere in the world, even distantfrom secondary manufacturers. The factors ruling the choiceof location will be tax rates, existence of skilled working force,political and economic stability etc.

2.2. Secondary manufacturing

Secondary manufacturers prepare and formulate the productsin a suitable form for final consumers. This involves addingthe AI to some “excipient” inert materials along with furtherprocessing and packing.

As it was pointed out above, secondary sites are oftengeographically separate from AI producers. This is frequentlythe result of tax and transfer price optimisation within theenterprise. At this end of the supply chain, due to largeamounts of inert products added to the AI plus the packing,the transportation costs become very significant, so usuallythere are many more secondary than primary locations, serv-ing regional or local markets.

3. Problem description

As referred to before, the aim of this project is the supply chainoptimisation of a large pharmaceutical company. The supplychain components are primary sites (AI manufacturers) andrespective storage facilities, secondary sites and respectivewarehouses and final product market areas. The distributionnetworks within each market area are out of the scope of thiswork.

Each primary site may supply the AI to any of the sec-ondary sites and be located in any place around the world. Forsecondary sites and markets, we consider several geographi-cal areas. The structure of a supply chain example with fivegeographical areas (including Europe, Asia, Africa and Middle
East, North America and South America) is shown in Fig. 1.Since transportation costs are very significant at this end of


the supply chain, material flows between two different geo-graphical areas are not allowed.

We assume the following information is known:

• multi-period demand forecast profile of the company’s port-folio;

• production, transportation and allocation specific costs;• tax rates;• the company’s supply chain structure, with primary and

secondary locations available.

The model aims to make the following supply chain deci-sions:

• where to allocate the manufacture of primary and sec-ondary products and how to manage the available resourcesduring the whole time horizon;

• what production amounts and inventory levels shall be setfor each manufacturing site;

• how to establish the product flows between echelons, i.e.between primary and secondary sites and between sec-ondary sites and markets.

Each geographic area produces and consumes some, butnot all, product families from the company’s secondary(final) products portfolio. Within each geographic area, it wasdecided that a single sourcing policy would be followed, i.e.each product (both primary and secondary) will be producedin one and only one site on each time period. The product/siteassignment may change between different time periods (“allo-cation transfer”), although the number of exchanges is limited.In line with the single sourcing policy, this transfer can onlytake place between sites in the same geographical area.

The model includes the occurrence of changeover activitiesas these significantly affect the equipment availability. How-ever, due to the global scope of the project and the dimensionof the problem, some production details as scale-up times andqualification runs are not considered.

The location of the several members in the SC is anextremely important issue to large multinational enterprises,as the profit after taxes may change significantly due to the dif-ferent tax rates in different countries. So, the objective is setas the maximisation of the enterprise’s net profit value (NPV).The terms of the objective function are listed below. A cost forunfulfilled demand has also been considered. Revenues comefrom the secondary products sales. The cost terms include:

• production costs;• transportation costs;• inventory handling costs;• products allocation costs;• unmet demand costs;• tax costs.

4. Mathematical formulation

4.1. Notation

Indicesc primary sitesi primary products
j geographical areasl primary sites resources (manufacturing equipments)
m market locationsp secondary productsr secondary sites resources (manufacturing equip-

ments)s secondary sitest, tt time periods

SetsMj markets in geographical area jPj secondary products in geographical area jSj secondary sites in geographical area j

ParametersAsrt Availability of resource r in time period t in secondary

site s (hour)APclt Availability of resource l in time period t in primary

site c (hour)CIV Inventory handling costs ($/kg)COT Changeover time in secondary sites resources (hour)COTP Changeover time in primary sites resources (hour)CPPci Production cost of primary product i in site c ($/kg)CPSsp Production cost of secondary product p in site s ($/kg)CTAsp Cost of transferring the allocation of secondary prod-

uct p to secondary site s ($)CTAPci Cost of transferring the allocation of primary product

i to primary site c ($)CTPc Average transportation costs from primary site c to

secondary sites ($/kg)CTSsm Transportation cost of secondary products from sec-

ondary site s to market m ($/kg)CUp Cost of not satisfying the whole demand of sec-

ondary product p ($/kg)Dpmt Demand forecast of product p in market m in time

period t (kg)Kpr Equals 1 if product p uses resource r and 0 otherwiseKil Equals 1 if product i uses resource l and 0 otherwiseMax Upper limit of the production amount in one time

period (kg)MTpr Manufacturing time demand of secondary product p

in resource r (hour/kg)MTPil Manufacturing time demand of primary product i in

resource l (hour/kg)PFip Product p formulation as a function of primary prod-

uct iT Number of time periodsTFp Transformation factor for final product p (yield)TRPc Tax rate on primary site locationTRS Tax rate on secondary site locationV1i Internal price of primary product i ($)V2pm Selling price of product p in market m ($)XPTN Allocation transfer limit of primary products in each

time periodXTNj Allocation transfer limit of secondary products in

geographical area j in each time period

Binary variablesXspt Equals 1 if secondary product p is produced at site s

in time period t, 0 otherwiseXPcit Equals 1 if primary product i is produced at site c, 0

otherwiseXPTcit Allocation transfer decision variable (equals 1 if

XPci, t+1 = 1 and XPcit = 0)
XTspt Allocation transfer decision variable (equals 1 if
Xsp, t+1 = 1 and Xspt = 0)


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ontinuous variablesVspt Inventory of secondary product p in site s at time

period t (kg)VPit Inventory of primary product i at time period t (kg)Rspt Production of product p in site s in time period t (kg)RPcit Production of primary product i in site c at time

period t (kg)Psit Amount of primary product i transported to site s in

time period t (kg)Sspmt Amount of product p transported from site s to mar-

ket m in time period t (kg)Lpmt Sales of product p in market m in time period t (kg)

pmt Unsatisfied demand of product p in market m in timeperiod t (kg)NPV, objective function ($)

.2. Secondary product allocation constraints

s ∈ Sj

Xspt = 1 ∀j, p ∈ Pj, t (1)

Tsp,t−1 ≥ Xspt − Xsp,t−1 ∀s, p, t > 1 (2)

spt ≤ 1 − XTspt ∀s, p, t (3)

s ∈ Sj

∑p ∈ Pj

XTspt ≤ XTNj ∀j, t (4)

Eq. (1) guarantees that, within each geographical area, eachroduct is allocated to one and only one secondary site inach time period. Eqs. (2) and (3) guarantee that each alloca-ion transfer will only take place after the actual decision haseen taken. Eq. (4) limits the number of allocation transfersccurring in each time period.

.3. Primary product allocation constraints

c

XPcit = 1 ∀i, t (5)

PTci,t−1 ≥ XPcit − XPci,t−1 ∀c, i, t > 1 (6)

Pcit ≤ 1 − XTPcit ∀c, i, t (7)

c

∑i

XPTcit ≤ XPTN ∀t (8)

Eqs. (5)–(8) play the same role in primary products alloca-ion as Eqs. (1)–(4) respectively for secondary products.

.4. Capacity constraints

p

MTpr · PRspt ≤ Asrt −(∑

p

Kpr · Xspt − 1

)· COT ∀s, r, t (9)

i

MTPil · PRPcit ≤ APclt −(∑

i

KPil · XPcit − 1

)· COTP ∀c, l, t

(10)

Eqs. (9) and (10) limit the production of secondary andrimary products in accordance with resource availability,
espectively. Note that although more general resources can beonsidered in the proposed model, the resources in the exam-
ples discussed later refer to the manufacturing equipments inthe production sites.

4.5. Mass/flow balance constraints

PRspt ≤ Max · Xspt ∀s, p, t (11)

SLpmt =∑s ∈ Sj

TSspmt∀j, p ∈ Pj, m ∈ Mj, t (12)

SLpmt ≤ Dpmt∀j, p ∈ Pj, m ∈ Mj, t (13)

Upmt = Dpmt − SLpmt∀j, p ∈ Pj, m ∈ Mj, t (14)

Each secondary product will only be produced in the sec-ondary site where it was allocated, as shown in Eq. (11). Eq. (12)is the flow balance between storage facilities (in the secondarysites) and the markets; Eq. (13) ensures that the sales of eachproduct in each time period is less than or equal to the respec-tive demand and Eq. (14) estimates the unmet demand in orderto introduce a penalisation term in the objective function. Thelink between production and outgoing flows from secondarysites (Eqs. (11) and (12)) is provided by the inventory constraint,Eq. (17).

PRPcit ≤ Max · XPcit ∀c, i, t (15)

TPsit =∑

p

PRspt · PFip

TFp∀s, i, t (16)

Eq. (15) gives the production relations in the primary sites.Eq. (16) establishes the primary products flow to secondarysites based on production amounts, product formulations andmanufacturing losses of secondary products.

4.6. Inventory constraints

IVspt = IVsp,t−1 + PRspt −∑

m ∈ Mj

TSspmt∀j, s ∈ Sj, p ∈ Pj, t (17)

IVPit = IVPi,t−1 +∑

c

PRPcit −∑

s

TPsit ∀i, t (18)

Eqs. (17) and (18) are the inventory balances for secondaryand primary sites, respectively.

4.7. Non-negativity constraints

IVspt, IVPit, PRspt, PRPcit, SLpmt, TPsit, TSspmt, Upmt ≥ 0 (19)

4.8. Objective function

The total NPV must include:

• Sales revenue;

˘ =∑

p

∑m

∑t

V2pm · SLpmt (20)

• Primary and secondary products production costs (per site);

∑∑
KPc =
i t

CPPci · PRPcit ∀c (21)


XT X SL IV TS U PR TP PRP IVP XP XPT(1) x(2) x x(3) x x(4) x(11) xx(9) xx(12) x x(13) x(17) x x x(14) x x(16) x x(10) x(15) x x(18) x x x(5) x(6) x x(7) x x(8) x

x

XT X SL IV TS U PR TP PRP IVP XP XPT(1) x(2) x x(3) x x(4) x(11) xx(9) xx(12) x x(13) x(17) x x x(14) x x(16) x x(10) x(15) x x(18) x x x(5) x(6) x x(7) x x(8) x

x

Fig. 2 – Structural matrix. Lines correspond to the model

KPs =∑p ∈ Pj

∑t

CPSsp · PRspt∀j, s ∈ Sj (22)

• Primary and secondary products transportation costs (persite);

KTc = CTPc ·∑

i

∑t

PRPict ∀c (23)

KTs =∑p ∈ Pj

∑m ∈ Mj

∑t

CTSsm · TSspmt∀j, s ∈ Sj (24)

• Primary and secondary sites inventory handling costs;

KIprim = CIV ·∑

i

∑t

IVPit (25)

KIs = CIV ·∑p ∈ Pj

∑t

IVspt∀j, s ∈ Sj (26)

• Primary and secondary products allocation costs (per site);

KAc =∑

i

∑t

CTAPci · XPTcit ∀c (27)

KAs =∑p ∈ Pj

∑t

CTAsp · XTspt∀j, s ∈ Sj (28)

For each site, the total costs before taxes will be as follows.Here, the average inventory costs are considered for primarysites (see below).

Kc = KPc + KTc + KIprim

#c+ KAc ∀c (29)

Ks = KPs + KTs + KIs + KAs ∀s (30)

• Tax costs on primary and secondary site locations;

�c ≥ TRPc ·[(∑

i

∑t

V1i · PRPcit

)− Kc

]∀c (31)

�s ≥ TRSs ·

⎡⎣⎛⎝∑

p ∈ Pj

∑m ∈ Mj

∑t

V2pm · SLpmt

⎞⎠− Ks

⎤⎦∀j, s ∈ Sj

(32)

The NPV is given by Eq. (33):

Z = ˘ −[∑

c

(Kc + �c) +∑

s

(Ks + �s)

]−∑

p

∑m

∑t

Upmt · CU

(33)

As mentioned above, transportation costs at the primaryend of the supply chain are not significant, so average costsfor transportation of primary products between secondary andprimary sites are used, depending on the primary site loca-tion and its distances to the secondary sites. This reducesthe number of variables since it is possible to express the pri-mary products flow variables, TPsit, with one dimension less.
One consequence of this procedure is that the primary prod-uct inventory costs cannot be assigned to a specific primary
constraints and the columns correspond to the variables.

site (because the model formulation does not allow the inclu-sion of the site index in the variable IVPit), which prevents theaccurate calculation of tax costs on these locations. Anotherconsequence is that the product flow from one specific pri-mary site to one specific secondary site is not unknown fromthe model. Thus, the primary product production amount isused to calculate the transportation cost in any specific pri-mary site.

5. Solution methods

With the purpose of testing the model, two sets of data,with different sizes, were generated to simulate hypotheti-cal problems. The smaller problem is solvable in an hour;however, the larger one does not terminate in a reasonabletime (CPU < 50,000 s). This motivated us to develop heuristicprocedures to solve the larger instances of this model, corre-sponding to real-world problems.

According to previous works by other authors, (see Section1) the most suitable approaches for this kind of models aredecomposition and/or aggregation methods and Lagrangeandecomposition procedures. In this paper, we propose twodecomposition methods. The first one consists of separatingthe SC in its two echelons, primary and secondary (spatialdecomposition algorithm). In the second method the model isdecomposed into several independent subproblems, one pereach time period (temporal decomposition algorithm).

5.1. Spatial decomposition algorithm

During the last a few years, computational hardware has expe-rienced enormous improvement. Faster computer processorsallow the solution of ever-larger problems within reasonableCPU times. Nevertheless, the state-of-the-art of optimisationtools always tends to lag the requirements of realistic prob-lems.

The principle underpinning decomposition methods isproblem size reduction, through decomposition into smallersubproblems, deletion of hard constraints, or, as often hap-pens, a combination of both.

Our model’s structural matrix (Fig. 2) shows that it canbe composed of two ready decomposable subproblems, cor-responding to primary and secondary echelons connected by
a linking constraint (Eq. (16)), the flow balance between thetwo areas of the supply chain.


START

Relax secondary binary variables Solve MILP ALLOCATIONS1

W = 0

XT = 0; X = 0;

TSpsmt = SLpst/#Mj

SLpmt = ΣsjSLpst/#Mj

Upmt = Upjt/#Mj

W > n

JC = 1

STOP

JC <= 5

Fix secondary variables j <> JC; (TP, PR, TS, SL, IV, U, X, XT)Fix primary binary variables;

Solve MILP ALLOCATIONS2

Update TP, PR, SL, IV, U, X, XT

JC = JC + 1

Fix secondary binary variables Account for changeover

Solve MILP ALLOCATIONS3

W = W+1

yes

no

yes

no

Fig. 3 – Flowchart of the spatial decomposition algorithm.

rasbi

•

•

•

Fig. 4 – Evolution of Z along the several iterations of the

tion of the allocation constraints, as detailed below. In fact,

In the following description, “primary binary variables”,efers to those variables concerned with primary productsllocation while “secondary binary variables” are related toecondary products allocation. This is a three-step method,ased on the structure of the supply chain model, as illustrated

n Fig. 3.

Step 1: the secondary binary variables are relaxed and aninteger set of primary variables is generated. Simultane-ously, this provides an estimate of the production amounts,sales and inventory for both primary and secondary prod-ucts, as well as an upper bound to the problem.Step 2: the primary binary variables are fixed and the pro-gram optimises each of the secondary geographical areas, j,separately, until a complete set of integer secondary binaryvariables is generated.Step 3: the secondary binary variables are fixed and themodel reallocates the primary products and adjusts the pro-
duction amounts.
spatial decomposition algorithm.

The first step uses an aggregated version of the model,i.e. ALLOCATIONS1, (see Appendix) whose solution processis faster than the original model. Between the first and thesecond steps a set of operations is performed to convert theresults obtained with the aggregated model into values thatcan be used in the detailed model. In the second step, eachgeographical area, j, is optimised separately in the originalmodel with primary binary variables and secondary variablesin other areas fixed (model ALLOCATIONS2), until a completeset of secondary binary variables is generated. These variables,in turn, are fixed in the original model (MILP model ALLO-CATIONS3) in the third step of the procedure, where primaryproducts are reallocated optimally.

The second and third steps may be repeated iterativelyuntil the objective function is no longer improved, or a speci-fied number of iterations are reached. On the one hand, thereis no guarantee that the optimum will be found, since bothsolutions, from the second and third steps, are lower boundsto the problem. On the other hand, in each step, it is possibleto calculate an upper bound to the difference to the optimuminteger solution, given by the difference between the result ofstep 1 and the actual value of Z (Fig. 4).

5.2. Temporal decomposition algorithm

This work is oriented towards long-term planning, whereeach time period has the duration of several months. Underthese circumstances, the inventory relations (constraints) donot significantly influence the final allocation decisions. Apossible approach is to decompose the problem into severalindependent subproblems, one per time period that can beoptimised separately. Two kinds of constraints link the differ-ent periods: allocation transfer and inventory constraints (Eqs.(2), (3), (6), (7), (17) and (18)).

Setting the inventory variables to 0 at the end of each timeperiod leads to Eq. (34) and (35). Each one of the new equa-tions may now be written independently for each time period,i.e. without depending on variables referring to adjacent timeperiods.

0 = PRpst −∑

m ∈ Mj

TSpsmt∀j, p ∈ Pj, s ∈ Sj, t (34)

0 =∑

c

PRPict −∑

s

TPist ∀i, t (35)

Eqs. (2), (3), (6) and (7) need to be reformulated more care-fully; the allocation decisions information has to be passedbetween time periods. This is achieved through modifica-

each sub-model (corresponding to each time period) is not


tt = T

START

STEP 1 ALLOCATIONT1

Optimisation of time period tt

XXSpst = Xps,ttXXPict = XPic,tt

tt = tt - 1

tt = 0

X2pst = X1pstXT2pst = XT1pst XP2ict = XP1ict

XPT2ict = XPT1ict

STEP 2 ALLOCATIONT2

LP recalculation of the continuous variables

STOP

no

yes

Fig. 5 – Flowchart of the temporal decomposition algorithm.Indices 1 and 2 of the binary variables in the 7th box referto ALLOCATIONT1 and ALLOCATIONT2, respectively.

fully independent; it uses some information (allocation deci-sions) from previous (or subsequent) time periods, as there isa limit on the product/site allocation transfers that can occurin each time period. According to this, the optimisation pro-cess has to be performed following either the direct or inversechronological sequence of time periods. In order to reduce therelations between time periods, each allocation transfer deci-sion is assumed to take place in the same time period as thetransfer itself.

Using the variables substitution expressed in Eq. (36), Eqs.(2), (3), (6) and (7) are reformulated for each time period t = tt(using inverse chronological sequence).

Variable substitution

Xsp,t+1 = XXSsp and XPci,t+1 = XXPci ∀s, p, c, l, t

= tt and t < T (36)

Secondary sites

Xspt ≤ 1 − XTspt ∀s, p, t = tt and t < T (37)

XTspt ≥ XXSsp − Xspt ∀s, p, t = tt and t < T (38)

Primary sites

XPcit ≤ 1 − XPTcit ∀c, i, t = tt and t < T (39)

XPTcit ≥ XXPci − XPcit ∀c, i, t = tt and t < T (40)

The inclusion of binary allocation variables in the capac-ity balance constraints (Eqs. (9) and (10)) to account for thechangeover operations may result in the generation of infea-sible capacity balance constraints in the following time periodbeing optimised in the solution process (see relationshipbelow).

Asr,t−1 <

(∑p

KprXspt − 1

)· COT ∀s, r, t (41)

APcl,t−1 <

(∑i

KPilXPclt − 1

)· COTP ∀c, l, t (42)

If the solution of these infeasibilities involves the realloca-tion of more products than the allocation transfer upper limit(Eqs. (4) and (8)), then the model will be infeasible. In order toprevent this, extra constraints are included:

Asr,t−� ≥(∑

p

KprXspt − 1

)· COT ∀s, r, t, � < t (43)

APcl,t−� ≥(∑

i

KPilXPclt − 1

)· COTP ∀s, r, t, � < t (44)

In the first step of the algorithm, each time period isoptimised separately and sequentially. The information con-cerning product/site assignment is passed to the followingtime period and so on, until all time periods have been opti-
mised and a complete set of binary variables is generated.In this step, the objective function is a function of t. In the
second step, the binary variables are fixed and the continu-ous variables are recalculated in order to improve the finalNPV.

ALLOCATIONT1 is the MILP model used to calculate thebinary variables in each time period. It comprises the produc-tion and sales constraints and modified allocation constraints,i.e. Eqs. (36)–(40).

ALLOCATIONT2 is the original model, where all thetime periods are optimised simultaneously, with the binaryvariables fixed at the values calculated on step 1. So ALLOCA-TIONT2 is a LP model.

Fig. 5 shows the flowchart of the temporal decompositionalgorithm, in which the optimisation process in the first stepis performed following the inverse chronological sequence oftime periods.

6. Illustrative examples

Two examples motivated by industrial processes were gener-ated in order to test the model as well as the performance ofthe developed decomposition algorithms. The dimensions of
each example are presented in Table 1.


Table 1 – Example problems.

Problem 1 Problem 2

6 active ingredients 10 active ingredients6 primary sites 10 primary sites5 secondary geographical areas 5 secondary geographical areas30 secondary product families 100 secondary product families33 secondary sites 70 secondary sites10 market areas 10 market areas12 time periods (4 months each) 12 time periods (4 months each)

Table 2 – Comparison between the number of binaryvariables in the full variable space model anddecomposition algorithms.

Problem 1 Problem 2

Full space 12,480 84,096Spatial dec. alg. 1st Step 864 2400

2nd Step (min/max) 1080/3360 9840/24,6263rd Step 864 2,400

Temporal dec. alg. 1040 7008

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

876543210Problem Size

CPU

(s)

(1)

(2)

Fig. 6 – Influence of the problem’s size in the solving time.CPU time increases exponentially with the size of theproblem (1) and (2) – faster processors allow moving thec

mn(

adaFs

Fsrw

Table 3 – Decomposition algorithms’ performance forproblem 1 (COT = 100).

Zopt Gapa (%) CPU (s)

LP 437,995 – –Full space 426,178 2.7 1560Spatial dec. alg. 420,172 4.1 289Temporal dec. alg. 416,208 5.0 9

a With respect to the LP model solution.

Table 4 – Decomposition algorithm’s performance forproblem 2 (COT = 100).

Zopt Gapa (%) CPU (s)

LP 1,008,746 – –Full space 954,454b 5.4 50,022Spatial dec. alg 959,531 4.9 3,165Temporal dec. alg. 974,874 3.4 53

a With respect to the LP model solution.b Terminated by the CPU time limit of 50,000 s.

Table 5 – Statistics for the spatial decompositionalgorithm. The changeover time (COT = 300) representsup to 10% of the availability of each resource.

CPU (s) Optimum Gap (%)

Step 1 923 1,002,529 3.1Step 2a j = 1 414 994,831 0.4

j = 2 1,026 977,326 1.8j = 3 12 976,087 0.1j = 4 4,958 962,594 1.6j = 5 16 956,641 0.7

Step 3 227 985,910 1.0Final 7,576 985,910 1.0

a j refers to secondary geographical areas.
urve to the right.
All the tests were performed on a Windows XP basedachine with 1 GB RAM and 3.4 GHz Pentium 4 processor, run-

ing the GAMS 22.8 (Brooke et al., 2008) with CPLEX 11.1 solverILOG, 2007).

Statistics concerning the number of integer (binary) vari-bles in the full space models and in the different steps of theecomposition algorithm are shown in Table 2. Taking intoccount the relationship between size and CPU expressed inig. 6 and the values in Table 2, it is to be expected that the
um of the CPU times to solve the three steps of the decom-
j=1

j=2

j=3

j=4

j=50

500

1000

1500

2000

2500

3000

3500

8007006005004003002001000

CPU

(s)

Excess Capacity (%)

ig. 7 – CPU dependence on excess capacity of step 2, of thepatial decomposition algorithm, for each geographicalegion, j. CPU and EC values are an average over many runsith different values of COT.

position algorithm will be lower than the time to solve the fullmodel.

Tables 3 and 4 show the results of the decompositionalgorithms. The performance of these methods is highlydependent on the set of data being optimised.

The spatial decomposition method is particularly sensi-tive to the parameter excess capacity as defined in Eq. (45) thatrelates demand and available manufacturing capacity. Thisparameter is affected by the time taken on each changeover
operation, as this affects the manufacturing equipment avail-abilities.
Table 6 – COT influence on the solution time ofdecomposition algorithms for problem 2.

Spatial dec. alg. Temporal dec. alg.

COT CPU (s) Optimum CPU (s) Optimum

100 3165 959,531 53 974,874133 3438 956,851 55 977,816300 7576 985,910 64 971,758500 10,800 967,861 81 971,330


Fig. 8 – Primary product allocations from temporal decomposition algorithm for problem 2 (COT = 100).

0

2000

4000

6000

8000

10000

12000

14000

16000

C10C9C8C7C6C5C4C3C2C1

Prim

ary

Prod

uc�

on (k

g)

Primary Site

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

por
Fig. 9 – The primary products productions from the tem
ECrtj=

⎛⎜⎜⎜⎝

∑s ∈ Sj

Arst

∑∑MTprDpmt

− 1

⎞⎟⎟⎟⎠× 100∀r, t, j = 1, 2, 3, 4, 5 (45)

p ∈ Pjm ∈ Mj

Fig. 10 – Breakdown of the total cost from the temporal

al decomposition algorithm for problem 2 (COT = 100).

Table 5 presents the CPU times for the three steps of thismethod, where the first and third steps represent about 15%of the total solution time when COT = 300. Table 6 shows howthe changeover time (COT) affects the CPU time to solve prob-
lem 2 with the spatial decomposition algorithm, by changingthe available capacity. Fig. 7 relates the average computational
decomposition algorithm for problem 2 (COT = 100).


tc

btirilpt(a

tsofautfemtap

ccasoiaaths

7

Ipwta

yotsTgwla

tptsb

ime to solve each of the secondary areas and their excessapacity, defined in Eq. (45).

The temporal decomposition algorithm has, by far, theest performance of both solution algorithms developed inhis work; its sensitivity to the excess capacity parameters much lower than that for the spatial decomposition algo-ithm. One downside of this algorithm is that the quality ofts results is highly dependent on the demand profile, i.e.arge changes in demand levels between consecutive timeeriods will reinforce the importance of the inventory rela-ions and lead to poorer quality results, while Eqs. (43) and44) will introduce many limitations on the possible productllocation.

Fig. 8 shows the allocation of the primary products (I1–I10)o primary sites (C1–C10) obtained by the temporal decompo-ition algorithm for problem 2, from which we can see notnly the product allocations, but also the allocation trans-er decisions. Primary products I1, I3, I8, I9 and I10 are allllocated to one primary site, while other primary prod-cts make allocation transfers, in which both I4 and I7 havehe maximum number of allocation transfers (two trans-ers). Fig. 9 shows the production at each primary site inach time period. C5 and C8 are the most productive pri-ary sites, while C6, C7 and C10 are the three ones with

he lowest production amounts. Within each site, we canlso see that the productions vary throughout all the timeeriods.

Fig. 10 gives the percentage of each cost term in the totalost given by the temporal decomposition algorithm. The taxost on the secondary sites, the transportation cost (of primarynd secondary products), the production costs (of primary andecondary products) and the unmet demand cost account forver 99% of the total cost, while the other costs, including thenventory handling costs (on primary and secondary sites) andllocation costs (of primary and secondary products), occupytiny portion of the total cost. Particularly, the tax cost on

he primary sites are zero, as the cost on primary sites isigher than the revenue, but the tax cost on the secondaryites accounts for around 40%.

. Concluding remarks

n this work, we have defined a problem important to theharmaceutical industry, reviewed relevant related publishedorks, developed a model to cover the global network alloca-

ions and allocation transfers and investigated two solutionlgorithms to tackle this particular large MILP problem.

In the spatial decomposition method, the sensitivity anal-sis to the changeover time shows that, for this particular setf data, the optimum value is not affected significantly, sincehe bottleneck of the supply chain is the capacity of primaryites. On the other hand, the CPU time increases significantly.his is particularly the case for the “critical” secondary geo-raphical areas, where j = 2 and j = 4. These correspond to areashere the capacity used is close to the limit, mainly due to a

ower excess capacity (Eq. (45)), which becomes more criticals the changeover time increases.

The temporal decomposition method performs well withhese sets of data although the quality of the results may beoor in other cases. Nevertheless, it opens new possibilitieso explore even larger instances of the problem, such as its
tochastic version, where demand and other parameters maye uncertain.
Acknowledgements

The funding for R.T.S. from the Portuguese Science and Tech-nology Foundation (FCT), and for S.L. from the OverseasResearch Student Award Scheme (ORSAS) and the Centre forProcess Systems Engineering (CPSE) is gratefully acknowl-edged.

Appendix A. Appendix: Aggregate Model

The variables space of the detailed model may become verylarge for realistic problems, mainly because of the tetra-dimensional transportation variables of secondary productsfrom sites to markets, TSspmt. We develop an aggregate versionof the model, without transportation variables that, in spite ofbeing less detailed, is more tractable and suitable for the devel-opment of algorithms (the number of variables is reduced by∼30%)

A.1. Parameters

The aggregated model uses all the parameters of the detailedmodel except demands, secondary products’ transportationcosts and secondary products’ selling price that are substi-tuted by aggregated or average parameters.

DJpjt Total demand of secondary product p in area j in timeperiod t, Eq. (A1).

CTSJspj Average secondary product p transportation costfrom site s to markets in area j, Eq. (A2).

V2Jpj Average price of secondary product p in markets inarea j, Eq. (A3).

DJpjt =∑

m ∈ Mj

Dpmt∀t, j, p ∈ Pj (A1)

CTSJspj =

∑m ∈ Mj

(CTSsm

∑t

Dpmt

)

0.1 +∑

t

∑m ∈ Mj

Dpmt

∀j, p ∈ Pj, s ∈ Sj (A2)

V2Jpj =

∑m ∈ Mj

(V2pm

∑t

Dpmt

)

0.1 +∑

t

∑m ∈ Mj

Dpmt

∀j, p ∈ Pj (A3)

A.2. Variables

Almost all the variables from the detailed model are kept inthe aggregated model, but some of them have to be redefinedin order to fit the dimensional changes in the parameters.Variable TSpsmt is simply removed.

SLspt Sales of product p from site s in time period t;
Upjt Unsatisfied demand of product p in area j in time
period t;


A.3. Constraints

Eq. (12) has to be substituted and Eqs. (13), (14) and (17) have tobe modified to be according with the redefined set of variables.

∑t

∑s ∈ Sj

PRspt ≤∑

t

DJpjt∀j, p ∈ Pj (A4)

SLspt ≤ DJpjt∀j, s ∈ Sj, p ∈ Pj, t (A5)

Upjt = DJpjt −∑s ∈ Sj

SLspt∀j, p ∈ Pj, t (A6)

IVspt = IVspt−1 + PRspt − SLspt ∀s, p, t (A7)

A.4. Objective function

• Sales revenues;

˘ =∑

j

∑p ∈ Pj

∑s ∈ Sj

∑t

V2Jpj · SLspt (A8)


KPc =∑

i

∑t

CPPci · PRPcit ∀c (A9)

KPs =∑p ∈ Pj

∑t

CPSsp · PRspt∀j, s ∈ Sj (A10)


KTc = CTPc ·∑

i

∑t

PRPict ∀c (A11)

KTs =∑p ∈ Pj

∑t

CTSJpsj · SLspt∀j, s ∈ Sj (A12)

• Primary and secondary sites inventory handling costs;

KIprim = CIV ·∑

i

∑t

IVPit (A13)

KIs = CIV ·∑p ∈ Pj

∑t

IVspt∀j, s ∈ Sj (A14)

• Primary and secondary products allocation costs (per site);

KAc =∑

i

∑t

CTAPci · XPTcit ∀c (A15)

KAs =∑p ∈ Pj

∑t

CTAsp · XTspt∀j, s ∈ Sj (A16)

For each site, the total costs before taxes will be as follows:

Kc = KPc + KTc + KIprim

#c+ KAc ∀c (A17)

Ks = KPs + KTs + KIs + KAs ∀s (A18)

• Tax costs on primary and secondary site locations;

�c ≥ TRPc ·[(∑

i

∑t

V1i · PRPcit

)− Kc

]∀c (A19)

�s ≥ TRSs ·

⎡⎣⎛⎝∑

p ∈ Pj

∑t

V2Jpj · SLspt

⎞⎠− Ks

⎤⎦ ∀j, s ∈ Sj (A20)

The NPV is given by Eq. (A21):

Z = ˘ −

[∑c

(Kc + �c) +∑

s

(Ks + �s)

]−∑

j

∑p ∈ Pj

∑t

Upjt · CU (A21)

Between the first and second steps of the spatial decompo-sition algorithm, the conversion of the results obtained withthe aggregated model to values that can be used in the detailedmodel is performed as follows:

TSspmt = SLspt

#Mj∀j, s ∈ Sj, m ∈ Mj, p, t (A22)

SLpmt =

∑s ∈ Sj

SLspt

#Mj∀j, m ∈ Mj, p, t (A23)

Upmt = Upjt

#Mj∀j, m ∈ Mj, p, t (A24)

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