essays on perishable inventory management
TRANSCRIPT
Essays on Perishable InventoryManagement
by
Borga Deniz
Submitted to the Tepper School of Businessin Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
at the
CARNEGIE MELLON UNIVERSITYc© Carnegie Mellon University 2007
Thesis Committee
Professor Alan Scheller-Wolf (chair)Professor Itır Z. Karaesmen-AydınProfessor Laurens DeboProfessor Peter Boatwright
1
Abstract
My dissertation addresses a number of challenges in perishable inventory managementin supply chains, providing both theoretical results and managerial insights.
Essay 1: In this essay, we review the literature on supply chain management ofperishable products that have a fixed or random lifetime, primarily focusing on chal-lenges introduced by aging of such products. We start with a review of the literatureon capacity and production planning. We then review the literature on inventorymanagement at a single location. The research on single-location models providesa basis for research on multi-echelon/multi-location models, that are also reviewedin this chapter, along with transshipment and distribution models. The chapter alsocovers research involving modeling novelties, in terms of demand-fulfillment, informa-tion sharing, and centralized vs. decentralized planning. We provide several directionsfor future research by identifying the gaps in the literature and the needs of industrieswhere perishable products are commonly stocked.
Essay 2: We consider a discrete-time supply chain for perishable goods where thereare separate demand streams for items of different ages. We compare two practicalreplenishment policies: replenishing inventory according to order-up-to level policiesbased on either (i) total inventory in system or (ii) new items only. Given thesepolicies, we concentrate on four different ways of fulfilling demand: (1) demand foran item can only be satisfied by an item of that age (No-Substitution); (2) demandfor new items can only be satisfied by new ones, but excess demand for old itemscan be satisfied by new (Downward-Substitution); (3) demand for old items canonly be satisfied by old, but excess demand for new items can be satisfied by old(Upward-Substitution); (4) both downward and upward substitution are employed(Full-Substitution). We compare these substitution options analytically in terms oftheir infinite horizon time-average costs, providing conditions on cost parameters thatdetermine when (if at all) one substitution option is more profitable than the others,for an item with a two-period lifetime. We also prove that inventory is “fresher”whenever downward substitution is employed. Our results are based on sample-pathanalysis, and as such make no assumptions on the joint distribution of demand, savefor ergodicity. We complement our results with numerical experiments exploring theeffect of problem parameters on performance.
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Essay 3: In this paper we study replenishment policies for perishable goods whensubstitution is possible between items of different ages. We analyze different order-up-to level type replenishment policies and compare their performance with each otheras well as the optimal policy. We provide analytical results on optimality of ourproposed policies for some special cases. Also structural properties of our policiesare investigated. A comprehensive computational study is conducted using a variousdemand distributions and parameter settings. Based on our study we find that NIS(order-up-to inventory replenishment policy based on new items only) is closer tooptimal compared to TIS (order-up-to inventory replenishment policy based on totalinventory). We also observe that when items are significantly more valuable thanolder items NIS is more robust than TIS against demand fluctuations.
To my family.
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Acknowledgements
I would like to thank my family for everything they have done for me throughout my
life.
I would like to thank my advisors Dr. Alan Scheller-Wolf and Dr. Itır Karaesmen-
Aydın for their invaluable guidance and support. I am truly grateful to them.
I would like to thank Dr. Laurens Debo and Dr. Peter Boatwright for agreeing to serve
on my committee by sparing their valuable time and for providing helpful comments
and suggestions.
Last, but certainly not least, I would like to thank all my friends. They have made
my life very enjoyable during my studies.
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Contents
1 Introduction 1
2 Managing Perishable Inventory in Supply Chains 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Challenges in Production Planning . . . . . . . . . . . . . . . . . . . 8
2.3 Managing Inventories at a Single Location . . . . . . . . . . . . . . . 10
2.3.1 Discrete Review Models . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Continuous Review Models . . . . . . . . . . . . . . . . . . . . 18
2.4 Managing Multi-Echelon and Multi-Location Systems . . . . . . . . . 29
2.4.1 Research on Replenishment and Allocation Decisions . . . . . 30
2.4.2 Logistics: Transshipments, Distribution, and Routing . . . . . 39
2.4.3 Information Sharing and Centralized/Decentralized Planning . 41
2.5 Modeling Novelties: Demand and Product Characteristics, Substitu-
tion, Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 Single Product and Age-Based Substitution . . . . . . . . . . 46
2.5.2 Multiple Products . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.3 Pricing of Perishables . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Effect of Substitution on Management of Perishable Goods 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Problem Definition and Formulation . . . . . . . . . . . . . . . . . . 54
3.3 Related Work in the Literature . . . . . . . . . . . . . . . . . . . . . 59
3.4 Effect of Substitution: Analysis of Economic Benefit and Freshness . 64
3.4.1 The economic benefit of substitution: Overview of results . . . 65
6
3.4.2 Economic Benefit of Substitution under TIS . . . . . . . . . . 68
3.4.3 Economic Benefit of Substitution under NIS . . . . . . . . . . 75
3.4.4 A note on substitution not being a recourse . . . . . . . . . . 77
3.4.5 Freshness of Inventory . . . . . . . . . . . . . . . . . . . . . . 77
3.5 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.1 Non-convexity/Multi-modality of the Cost Function . . . . . . 79
3.5.2 Making downward substitution more attractive . . . . . . . . 81
3.5.3 Costs, Service Levels and Freshness of Inventory . . . . . . . . 83
3.5.4 Effect of customer behavior: Proportional acceptance of sub-
stitutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.5 Summary of Computational Results . . . . . . . . . . . . . . . 87
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Analysis of Inventory Replenishment Policies for Perishable Goods
Under Substitution 89
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Computational Results: NIS vs. TIS . . . . . . . . . . . . . . . . . . 91
4.3.1 A Preliminary Study . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 Comparison of TIS and NIS while changing variance of demand 93
4.3.3 NIS vs. TIS when new and old item demands are negatively
correlated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.4 TIS vs. NIS when new and old item demands are positively
correlated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.5 NIS vs. TIS when new item demand is higher than old item
demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.6 NIS vs. TIS when new item demand is lower than old item
demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.7 Comparison of TIS and NIS when substitution is not always
accepted: Partial vs. Probabilistic Acceptance . . . . . . . . . 103
4.3.8 Comparison of TIS and NIS when acceptance probability of
substitution depends on substitution costs . . . . . . . . . . . 104
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4.3.9 Search for a good NIS order-up-to level: News-vendor-type
heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.10 Summary of the Computational Study for Comparison of NIS
and TIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Conclusion and Future Research 110
References 115
Technical Appendix 132
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List of Figures
3.1 Time-average costs of substitution policies in Example-1A. . . . . . . 80
3.2 Time-average costs of substitution policies in Example-1B. . . . . . . 80
3.3 Time-average costs of substitution policies in Example-2. . . . . . . . 81
3.4 Minimum expected cost of policies as a function of downward substi-
tution cost (αD) in Example-3. . . . . . . . . . . . . . . . . . . . . . 82
3.5 The effect of proportional acceptance of downward substitution on ex-
pected total cost in Example-4. . . . . . . . . . . . . . . . . . . . . . 87
3.6 The effect of proportional acceptance of upward substitution on ex-
pected total cost in Example-5. . . . . . . . . . . . . . . . . . . . . . 88
4.1 Computation times for the best NIS, the best TIS and the optimal
solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Optimal policies when CV is 0.58 . . . . . . . . . . . . . . . . . . . . 96
4.3 Optimal policies when CV is 1.41 . . . . . . . . . . . . . . . . . . . . 96
4.4 Optimal policies when CV is 0.12 . . . . . . . . . . . . . . . . . . . . 96
4.5 Optimal policies when CV is 0.28 . . . . . . . . . . . . . . . . . . . . 96
4.6 The metric shows where the downward substitution cost (αD) lays
compared to upper and lower bounds . . . . . . . . . . . . . . . . . . 97
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Chapter 1
Introduction
Most products lose their market value (outdate) over time. Some products lose value
faster than others; these are known as perishable products. Traditionally, perishables
outdate due to their chemical structure. Examples of such perishable products are
fresh produce, blood products, dairy products, meat, drugs, vitamins etc. Today
many products that are not perishable in the traditional sense (i.e. products that do
not decay chemically over time) can still be considered as perishable. Such prod-
ucts, mostly high-tech, outdate because of changes in market conditions. Personal
computers, computer components (such as micro-processors, memory, data storage
units), cellular phones, digital cameras, digital music players, personal digital as-
sistants (PDAs) are examples of high-tech products that rapidly lose market value.
The life cycles of such products are getting shorter every year due to technological
advances. The co-founder of Intel, Gordon Moore, stated (Moore, 1965) “the number
of transistors on an integrated circuit for minimum component cost doubles every 24
months.” This statement was later coined as Moore’s Law. In other words Moore’s
Law implies that every two years consumers can expect to buy personal computers
that are twice faster, with no price increase. Data storage technology advanced at
even a faster rate than processing technology. According to Mark Kryder, founder
and director of Carnegie Mellon University’s Data Storage Systems Center and chief
technology officer at hard-drive manufacturer Seagate Technology, since the introduc-
tion of the disk drive in 1956, the density of data it can record saw a 50-million-fold
increase (Walter, 2005). Because of these reasons, for a certain configuration, the cost
of personal computers are expected to drop by fifty percent every 18 to 24 months.
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In addition to developments in technology, an important factor on shrinking life
cycles is competition. High-tech companies compete to introduce their newest prod-
ucts earlier to improve their market share. In the past many companies were not
under the intense competition that exists today. Therefore they did not feel pres-
sure to introduce their newest products until the market was saturated with existing
models. However at the present time there is cutthroat competition in almost all
high-tech consumer product markets. Customers benefit from this competition be-
tween companies. They are becoming more sophisticated than ever and demand more
from products that they buy.
One can argue that high-tech products are not perishable products. With the
acceleration of product life cycles, the line between ‘perishable’ and ‘durable’ prod-
ucts continues to blur. Strictly speaking, goods such as computers and cell phones
obsolesce rather than perish. By definition perishability occurs at a predictable rate
as products change; and obsolescence happens less predictably as the market environ-
ment changes. But the time frame for obsolescence in the traditional sense is longer
than the time frame for high-tech products’ obsolescence; thus we feel the need to
make a distinction between traditional obsolescence and the current situation. We
maintain that rapid obsolescence of high-tech products can be considered as perisha-
bility, as high-tech products are becoming obsolete at an increasingly faster pace than
ever.
Because of the importance of managing perishables we identified several interesting
research problems by carefully studying the literature. A comprehensive literature
study of perishable inventory research is the next chapter, in which we classify the
research on perishables that has been done focusing on the research on perishables
since Nahmias’ (1982) excellent review. This chapter fills the need for a comprehensive
and up-to-date review of research on managing perishable inventory in the area of
operations management, especially a review that can show the recent trends and
point out important future research directions from the perspective of operations
management and supply chain management. We concentrate on the research done
mainly on stochastic inventory management and on those papers which, in our view,
are important and lay the foundation for future work in one of the directions we detail.
We also refer to some papers on non-perishable goods in supply chain management
literature to put the research in perspective.
2
In the third chapter we study the effect of substitution between products at dif-
ferent ages. Managing perishable inventory in supply chains is challenging especially
when products of different “ages” co-exist in the market at the same time and there
is a possibility of substitution between old and new products. In our problem, there
is a single product with a lifetime of two periods; the value of the product decreases
deterministically as it ages, and there may be random demand for both new and
old items. A single supplier replenishes new items periodically, with zero lead time.
At the end of each period, any remaining old items are outdated, while any unused
new items become old. This setup captures the essence of the problem for perishable
goods such as fresh produce and blood as well as durable goods where a new product
is introduced into the market while the old one is phased-out periodically.
In our model all demand for new (old) items is fulfilled from the inventory of
new (old) unless there is a stock-out. In case of stock-outs, the excess stock of new
(old) items may be used to satisfy the excess demand of old (new). We use the
term downward substitution to denote the case where a new product is sold to a
customer that demands old; upward substitution is the reverse. We assume any such
substitution occurs only as a recourse. Our model captures the situation in which the
supplier is the one making the substitute offer; this is supplier-driven substitution as
is common in business-to-business situations. On the other hand, our model likewise
captures the scenario in which the consumer is the one considering the substitute
option without explicit intervention (except product placement) by the supplier. This
is the same as inventory-driven substitution where the demand of one product shifts
to another in case of a stock-out.
In terms of customer choice we do not have any restrictions on upward substi-
tution in our analysis, that is a customer can refuse the upward substitution and
our analytical results will not be affected. However for downward substitution, we
assume that customer and seller will always find a price that both can agree upon,
and the downward substitution will take place. This price would probably be between
the original selling prices of old and new items. In our model we have a downward
substitution cost that could be considered the result of such an agreement between
the parties.
Our work is first in the literature on substitution to introduce the term “dynamic
quality”. In the literature on substitution, the quality of a product is static. For our
3
problem the quality of a product is age (one dimensional). That is, a new product
has a higher quality than an old one. Since our model is an infinite horizon model,
the quality of the product is dynamic. If a product is not sold its quality will change
in the next period (i.e. from old to new) making our model unique in the literature
on substitution.
Our model best applies to traditional perishable products such as fresh produce
or blood rather than the non-traditional perishable products such as computer chips.
The main reason for this is due to the fact that the supply of traditional perishable
products is mostly done with the newest products. However for computer chips old
and new products can be supplied simultaneously. Nevertheless, insights from our
model can still be of use in the non-traditional setting.
Our model also encompasses other phenomenon: If inventory is depleted starting
from the newest items (e.g. buyers prefer longer shelf-life items as in grocery and the
freshest produce/bread), this corresponds to having demand for new items only and
fulfilling the demand by upward substitution (i.e. customers consider buying less fresh
products only after freshest is sold). If instead inventory is depleted in a FIFO fashion
(oldest item issued first) as is the assumption in classical research on perishable goods
(see the review in the next section), then there is demand for only old items and
demand is fulfilled using downward substitution.
For two different order-up-to level type replenishment policies we compare the
effect of substitution on system costs. We find conditions on cost parameters that
would guarantee cost dominance between policies over the infinite horizon. We also
provide results on freshness and service levels of policies as well.
In the fourth chapter we study replenishment policies for perishable goods under
substitution. We analyze inventory replenishment policies within the class of base-
stock policies. Two policies that bring “total” on-hand inventory to an order-up-to
level and “new” on-hand inventory to an order-up-level are compared, presenting the
results of a comprehensive numerical study. We compare our two heuristics with the
globally optimal policy and provide analytical results on optimality of our heuristics
for special cases.
We test our heuristic policies under various demand schemes, including four differ-
ent variance settings. In the third chapter we do not have any restrictions on demand
process in terms of correlation for our analytical results. That is, old and new item
4
demand streams can correlated with each other. Therefore we provide computational
experiments for negatively and positively correlated demand. We also test a rule
of thumb which is developed based on our observations for choosing a substitution
policy.
Our numerical experiments include a comparison of our heuristics when substitu-
tion is not always accepted; we do experiments with partial acceptance and probabilis-
tic acceptance. In the partial acceptance model when substitution is offered a certain
fraction of it is accepted by customer. Under probabilistic acceptance all of the offered
substitution is either accepted or rejected (as a whole) by the customer. We assume
each substitution acceptance is independent. We also develop and compare a num-
ber of NIS order-up-to level heuristics using newsvendor-type logic as well as TIS and
NIS at their optimal order-up-to levels (found by doing a line search using simulation)
versus the globally optimal state-dependent policy found via dynamic programming.
We conduct our numerical studies using simulation and dynamic programming.
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Chapter 2
Managing Perishable Inventory in
Supply Chains
2.1 Introduction
In this chapter, we provide an overview of research in supply chain management of
products that are perishable, in particular, and that outdate, in general. As this
definition is quite broad, we focus on research done mainly on inventory manage-
ment in which a product ages over time, thus largely excluding single-period models
which are commonly used to represent perishable items (with an explicit cost attached
to expected future outdates). While these newsvendor-type models can convey im-
portant insights, they are often too simple to provide answers to some of the more
complex questions that arise as inventory levels, product characteristics, markets and
customer behaviors change over time. Furthermore, we exclude research that models
decay or deterioration of inventories assuming certain functional forms (e.g., exponen-
tial decay). The reader can refer to Goyal and Giri (2001) and Raafat (1991), which is
supplemented by Dave (1991), for a bibliography and classification of research on that
topic. Finally, we also exclude the research on management, planning or allocation
of capacity which is commonly referred as perishable inventory, for instance, in the
airline revenue management literature. Summarizing then, our emphasis is on models
where the inventory level of a product must be controlled over a horizon taking into
6
account demand, supply and a finite shelf-life (which may be fixed or random).
Over the years, several companies have emerged as exemplary of “best practices”
in supply chain management; for example, Wal-Mart is frequently cited as using
unique strategies to lead its market. One significant challenge for Wal-Mart is man-
aging inventories of products that frequently outdate: A significant portion of Wal-
Mart’s product portfolio consists of perishable products such as food items (varying
from fresh produce to dairy to bakery products), pharmaceuticals (e.g. drugs, vita-
mins, cosmetics), chemicals (e.g. household cleaning products), and cut flowers.
Of course Wal-Mart’s supply chain is not alone in its exposure to outdating risks
– to better appreciate the impact of perishability and outdating in society at large,
consider these figures: In a 2003 survey, overall unsalable costs at distributors to
supermarkets and drug stores within consumer packaged goods alone were estimated
at $2.57 billion, and 22% of these costs, over 500 million dollars, were due to expiration
in only the branded segment (Grocery Manufacturers of America, 2004). Within the
produce sector, the $1.7 billion U.S. apple industry is estimated to lose $300 million
annually to spoilage (Webb, 2006).
Note also that perishability and outdating are a concern not only for these con-
sumer goods, but for industrial products (for instance, Chen, 2006, mentions that
adhesive materials used for plywood lose strength within 7 days of production), mil-
itary ordnance, and blood - one of the most critical resources in health care supply
chains. According to a nationwide survey on blood collection and utilization, 5.8% of
all components of blood processed for transfusion were outdated in 2004 in the U.S.
(AABB, 2005).
Modeling in such an environment implies that at least one or both of the following
holds: First, demand for the product may change over time as the product ages; this
could be due to a decrease in the utility of the product because of the reduced lifetime,
lessened quality and/or changing market conditions. Second, operational decisions
can be made more than once (e.g., inventory can be replenished by ordering fresher
products, or prices can be marked down) during the lifetime of the product. Either
of these factors make the analysis of such systems a challenge, owing to the expanded
problem state space corresponding to the additional information that is both available
and valuable. Traditionally, it has required appreciable effort to gather and use such
information, but as we will see, with the advent of increasingly powerful information
7
and control technologies, new and exciting possibilities are emerging.
Our goal in this chapter is not to replicate surveys of past work (such as Nahmias,
1982, Prastacos, 1984, and Pierskalla, 2004; in fact we refer to them as needed in the
remainder of this chapter) but to discuss in more detail directions for future research.
In order to do that, we provide a selective review of the existing research and focus
on those papers which, in our view, constitute crucial stepping stones or point to
promising directions for future work. The reader will also notice that we refer to
several papers in the supply chain management literature that do not specifically
study perishable inventories. We do so in order to highlight analogous potential
research areas in the supply chain management of perishable goods.
The chapter is organized as follows: We first provide a discussion of common
challenges in production planning of perishables in Section 2.2. Then we review
the research on single product, single location models in inventory management of
perishables in Section 2.3. We next focus on multi-echelon and multi-location models
in Section 2.4. Research with novel features such as multiple products, multiple-types
of customers and different demand models are reviewed in Section 2.5. We conclude
by detailing a number of open research problems in Section 2.6.
2.2 Challenges in Production Planning
One of the most comprehensive studies on production planning and perishable goods
is the doctoral dissertation of Lutke Entrup (2005), which focuses on the use of leading
advance planning and scheduling (APS) systems (such as PeopleSoft’s EnterpriseOne
and SAP’s APO) to manage products with short shelf-lives. According to Lutke En-
trup (2005), the use of APS in perishable supply chains remains low in contrast to the
supply chains of non-perishable goods. He then describes how shelf-life is integrated
into the current APS, and identifies particular weaknesses of APS systems for perish-
ables by carefully studying the characteristics and requirements of the supply chains
of three different products. Based on these case studies, he proposes customized solu-
tions which would enable APS systems to better match the needs of the fresh produce
industry. We refer the reader to this resource for more information on practical issues
in production planning for perishable goods. In the remainder of this section, we
8
provide an overview of the analytical research in production planning, mainly focus-
ing on the general body of knowledge in operations management and management
science disciplines.
Capacity planning is one of the key decisions in production of perishables. Re-
search on this topic is primarily focused on agricultural products: Kazaz (2004)
describes the challenges in production planning by focusing on long-term capacity
investments, yield uncertainty (which is a common problem in the industries that in-
volve perishables) and demand uncertainty. He develops a two-stage problem where
the first stage involves determining the capacity investment and the second-stage
involves the production quantity decision. Jones et al. (2001) analyze a production
planning problem where after the initial production there is a second chance to pro-
duce, still facing yield uncertainty; Jones et al. (2003) describe the real-life capacity
management problem for a grower in more detail. Allen and Schuster (2004) present a
model for agricultural harvest risk. Although these papers do not model the aging of
the agricultural product once it is produced, they highlight the long term investment
and planning challenges. We refer the reader to Kazaz (2004) for earlier references on
managing yield uncertainty and to Lowe and Preckel (2004) for research directions
within the general domain of agribusiness.
In addition to capacity planning and managing yield uncertainty, product-mix
decisions are integral to the management of perishables. Different companies in the
supply chain (producer, processor, distributor, retailer) face this problem in slightly
different ways. For example, again in agribusiness, a producer decides on the use of
farm land for different produce, hence deciding on the capacity-mix. A supermarket
can offer a fresh fruit or vegetable as- is or as an ingredient in one of their ready-to-eat
products (e.g., pre-packed fruit salad). While the choice of the product-mix affects
the useful lifetime of a product, it can also help reduce the waste at the retailer if, for
example, produce that is not salable due to problems in its appearance or packaging
is used in preparing the ready-to-eat product. Similarly, fresh produce at a processing
facility can be used in ready-to-eat, cooked products (e.g. pre-cooked, frozen dishes)
or it can be sold as an uncooked, frozen product. These two types of products have
different shelf-lives. Product-mix decisions - whole blood vs. blood components - are
also crucial in blood supply chains. While quite common, such planning problems
have not been studied extensively in the operations management and management
9
science literature; owing to their prevalence in practice the strategic management of
products with explicitly different shelf-lives stands as an important open problem.
We discuss the limited research in this area in Section 2.5.
Note that for perishables, both capacity planning and product-mix decisions are
typically driven by target levels of supply for a final product. In that respect, pro-
duction plans are tightly linked to inventory control models. We therefore discuss
research in inventory control for perishables in the rest of this chapter, treating single
location models in Section 2.3, the multi-location and multi-echelon models in Sec-
tion 2.4, and modeling novelties in Section 2.5. Once again, we attempt to emphasize
those areas most in need for further research.
2.3 Managing Inventories at a Single Location
Single-location inventory models form the basic building blocks for more complex
models – possibly comprised of multiple locations and/or echelons, potentially in-
corporating information flows, knowledge of market or customer behavior, and/or
additional logistics options. This work was pioneered by Veinott (1960), Bulinskaya
(1964) and Van Zyl (1964), who consider discrete review problems without fixed cost
under, respectively, deterministic demand, stochastic demand for items with a one-
period lifetime, and stochastic demand for items with a two-period lifetime. We
will discuss work in this discrete review setting first, before expanding our scope to
continuous review systems. There is a single product in all the models reviewed in
this section, and inventory is depleted starting from the oldest units in stock, i.e.,
first-in-first-out (FIFO) inventory issuance is used.
2.3.1 Discrete Review Models
We provide an overview of research by classifying the work based on modeling as-
sumptions: Research assuming no fixed ordering costs or lead times is reviewed in
Section 2.3.1, followed by research on models with fixed ordering costs but no lead
times in Section 2.3.1. Research on models with positive lead times, which complicate
the problem appreciably, is reviewed in Section 2.3.1.
10
Table 2.1 provides a high-level overview of some of the key papers within the
discrete review arena. Categorization of the papers and the notation used in Table 2.1
are described below:
• Replenishment policy: Papers have considered the optimal control policy (Opt),
base stock policies that order to keep the Total Items in System (TIS) - summed
over all ages - constant or that order to keep the New Items in System (NIS)
constant (see Section 2.3.1), other heuristics (H), and, when fixed ordering costs
are present, the (s, S) policy. When (s, S) policy is annotated with a ‡ this shows
(s, S) policy is implied to be the optimal replenishment policy based on earlier
research (however a formal proof of optimality of (s, S) has not appeared in
print for the model under concern).
• Excess demand: Excess demand is either assumed to be backlogged (B), or
modeled as lost sales (L).
• Problem horizon: Planning horizon is either finite (F) or infinite (I).
• Replenishment lead time: All papers, save for one, assume zero lead time. The
exception is a paper that allows a deterministic (D) replenishment lead time.
• Product lifetime: Most papers assume deterministic lifetimes of general length
(D) although two papers assume the lifetime is exactly two time periods (2).
One paper allows for general discrete phase-type lifetimes, (PH), and assumes
that all items perish at the same time, i.e., a disaster model. This latter is
denoted by 2 in Table 2.1. Another permits general discrete lifetimes, but
assumes items perish in the same sequence as they were ordered. This is denoted
by 7 in the table.
• Demand distribution: Models often assume general demand in each period hav-
ing a continuous density (cont). Others assume discrete distribution (disc),
Erlang – which may include exponential – (Ek) or batch demand with either
geometric (geo) or general discrete phase-type renewal arrivals, (PH). Often-
times the continuous demand is assumed for convenience – results appear to
generalize to general demand distributions.
11
• Costs: Costs include unit costs for ordering (c), unit per unit time holding (h),
perishing (m), unit per unit time shortage (p), one-time costs for shortage (b),
and fixed cost for ordering (K). The annotation 1 in Table 2.1 indicates papers
that allow the unit holding and shortage costs to be generalized to convex
functions. A paper that does not list cost parameters is concerned with the
general properties of the model, such as expected outdates, without attaching
specific costs to these.
Discrete Review Models without Fixed Ordering Cost or Lead Time
Early research in discrete time models without fixed ordering costs focused on char-
acterizing optimal policies. Fundamental characterizations of the (state dependent)
optimal ordering policy were provided by Nahmias and Pierskalla (1973) for the two
period lifetime problem, when penalty costs per unit short per unit time, and unit
outdating costs are present. Shortly thereafter Fries (1975) and Nahmias (1975a) in-
dependently characterized the optimal policy for the general lifetime problem. Both
of these papers had per unit outdating costs, per unit ordering costs, and per unit per
period holding costs. In addition, Fries (1975) had per unit shortage costs, and Nah-
mias (1975a) per unit per period shortage costs; this difference arises because Fries
(1975) assumes lost sales and Nahmias (1975a) backlogging of unsatisfied demand.
The cost structure in Nahmias (1975a) quickly became the “standard” for discrete
time models without fixed ordering cost, and we will refer to it as such within the
rest of this section, although in later papers the per unit per period penalty and hold-
ing costs were extended to general convex functions. The incorporation of separate
ordering and outdating costs allows modeling flexibility to account for salvage value;
use of only one or the other of these costs is essentially interchangeable.
Treatment of the costs due to outdating is the crucial differentiating element
within the perishable setting. In fact, while Fries (1975) and Nahmias (1975a) took
alternate approaches for modeling the costs of expiration – the former paper charges
a cost in the period items expire, while the latter charges an expected outdating cost
in the period items are ordered – Nahmias (1977b) showed that these two approaches
were essentially identical, modulo end of horizon effects. The policy structures out-
lined in Fries (1975) and Nahmias (1975a) were quite complex; perishability destroys
12
Rep
lenis
hm
ent
Exce
ssP
lannin
gLea
dLife-
Dem
and
Art
icle
policy
dem
and
Hori
zon
tim
eti
me
dis
trib
uti
on
Cost
s
Nahm
ias
and
Pie
rskalla
(1973)
Opt
B/L
F/I
02
cont
p,m
Fri
es(1
975)
Opt
LF/I
0D
cont
c,h,b
,m
Nahm
ias
(1975a)
Opt
BF
0D
cont
c,h,p
,m
Cohen
(1976)
TIS
BI
0D
cont
c,h,p
,m
Bro
dhei
met
al.
(1975)
NIS
LI
0D
dis
c
Chaza
nand
Gal(1
977)
TIS
LI
0D
dis
c
Nahm
ias
(1975b,1
975c)
HB
F0
DE
kc,
h,p
,m
Nahm
ias
(1976)
TIS
BF
0D
cont
c,h,p
,m1
Nahm
ias
(1977a)
HB
F0
Dco
nt
c,h,p
,m1
Nahm
ias
(1977c)
TIS
BF
0dis
c7co
nt
c,h,p
,m1
Nandakum
ar
and
Mort
on
(1993)
HL
I0
Dco
nt
c,h,b
,m
Nahm
ias
(1978)
(s,S
)B
F0
Dco
nt
c,h,p
,m,K
1
Lia
nand
Liu
(1999)
(s,S
)‡B
I0
Dbatc
hgeo
h,p
,b,m
,K
Lia
net
al.
(2005)
(s,S
)‡B
I0
PH
2batc
hP
Hh,p
,b,m
,K
William
sand
Patu
wo
(1999,2
004)
HL
FD
2co
nt
c,h,b
,m
Tab
le2.
1:A
sum
mary
of
dis
cre
teti
me
single
item
invento
rym
odels
:1
denote
spapers
inw
hic
hhold
ing
and
short
age
cost
scan
be
genera
lconvex
functi
ons,
2denote
suse
of
adis
ast
er
modelin
whic
hall
unit
speri
shat
once,7
denote
sth
eass
um
pti
on
that
item
s,even
though
they
have
random
life
tim
es,
willperi
shin
the
sam
ese
quence
as
they
were
ord
ere
d,and‡
denote
sth
ecase
when
(s,S)
policy
isim
plied
tobe
the
opti
malre
ple
nis
hm
ent
policy
base
don
earl
ier
rese
arc
h(h
ow
ever
afo
rmalpro
ofofopti
mality
of(s
,S)
has
not
appeare
din
pri
nt
for
the
modelunder
concern
).
13
the simple base-stock structure of optimal policies for discrete review models without
fixed ordering costs in the absence of perishability.
This complexity of the optimal policy was reinforced by Cohen (1976), who char-
acterized the stationary distribution of inventory for the two-period problem with the
standard costs, showing that the optimal policy for even this simple case was quite
complex, requiring state-dependent ordering. To all practical extents this ended acad-
emic study of optimal policies for the discrete review problem under FIFO issuance; to
find an optimal policy dynamic programming would be required, and implementation
of such a policy would be difficult owing to its complex, state dependent nature (with
a state vector tracking the amount of inventory in system of each age). Therefore
many researchers turned to the more practical question of seeking effective heuristic
policies that would be (i) easy to define, (ii) easy to implement, and (iii) close to
optimal.
Very soon after the publication of Nahmias (1975a), a series of approximations
appeared for the backorder and lost sales versions of the discrete review zero fixed
ordering cost problem. Initial works (Brodheim et al., 1975, Nahmias, 1975b) ex-
amined the use of different heuristic control policies; Brodheim et al. (1975) propose
a fascinating simplification of the problem – making order decisions based only on
the amount of new items in the system, what we call is the NIS heuristic, as it or-
ders based only on new items in the system. They realized that if a new order size
was constant, Markov chain techniques could be used to derive exact expressions,
or alternately very simple bounds, on key system statistics (which is the focus of
their paper). Nahmias (1975b) used simulation to compare multiple heuristics for the
problem with standard costs, including the “optimal” TIS policy, a piecewise linear
function of the optimal policy for the non-perishable problem, and a hybrid of this
with NIS ordering. He found that the first two policies outperform the third, which
effectively ended consideration of NIS and its variants for several decades.
Three other papers in sequence, Nahmias (1975c, 1976, 1977a), explicitly treat the
question of deriving good approximation policies (and parameters) for the problem
with the standard costs; the final of these three can be considered as the culmination
of these initial heuristic efforts. It computationally compares the heuristic that keeps
only two states in the inventory vector, new inventory and inventory over one day
old, with both the globally optimal policy (keeping the entire inventory vector) and
14
the optimal TIS policy. (This optimal TIS policy is easily approximated using tech-
niques in Nahmias, 1976.) This comparison showed that for Erlang or exponential
demand and a lifetime of three periods the performance of both heuristics is excep-
tional – always within 1% of optimal, with the reduced state-space heuristic uniformly
outperforming the optimal TIS policy. Note also that using the two-state approxi-
mation eliminates any need to track the age of inventory other than new versus old,
significantly reducing both the computational load and complexity of the policy.
All of these heuristics consider backorder models. The first heuristic policy for
discrete time lost sales models was provided by Nandakumar and Morton (1993), who
began with properties and bounds on the expected outdates under TIS in lost sales
systems, provided by Chazan and Gal (1977). These Nandakumar and Morton (1993)
incorporated into heuristics following Nahmias’ (1976) framework for the backorder
problem. Nandakumar and Morton (1993) compared these heuristics with alternate
“near-myopic” heuristics they developed using newsvendor-type logic, for the problem
with standard costs. They found that all of the heuristics performed within half a
percentage of optimal, with the near myopic heuristics showing the best performance,
typically within 0.1% of optimal. (These heuristics could likely be improved even
further, with the use of even tighter bounds on expected outdates derived by Cooper,
2001.)
Note that both the backorder and lost sales heuristics should be quite robust with
respect to items having even longer lifetimes than those considered in the papers:
Both Nahmias (1975a) and Fries (1975) observed that the impact of newer items
on ordering decisions is greater than the impact of older items. (This, in fact, is
one of the factors motivating the heuristic strategies in Nahmias, 1977a.) Thus, the
standard discrete time, fixed lifetime perishable inventory model, with backorders or
lost sales has for all practical purposes been solved – highly effective heuristics exist
that are well within our computational power to calculate. One potential extension
would be to allow items to have random lifetimes; if items perish in the same sequence
as they were ordered, many of the fixed lifetime results continue to hold (Nahmias,
1977c), but a discrete time model with random lifetimes that explicitly permits items
to perish in a different sequence than they were ordered remains an open, and likely
challenging question. The challenge arises from the enlarged state space required to
capture the problem characteristics – in this case the entire random lifetime vector
15
– although techniques from continuous time models discussed in Section 2.3.2 could
prove useful in this endeavor.
Discrete Review Models without Lead Times Having Positive Fixed Or-
dering Cost
Nahmias (1978) was the first to analyze the perishable inventory problem with no
lead time but positive fixed ordering cost in addition to the standard costs described
in Section 2.3.1. In his paper, for the one-period problem, Nahmias (1978) established
that the structure of the optimal policy is (s, S) only when the lifetime of the object
is two; lifetimes of more than two periods have a more complex, non-linear structure.
Extensions of the two-period (s, S) result to the multi-period problem appeared quite
difficult, given the analytical techniques of the time: The fact that the costs exclud-
ing the fixed ordering cost are not convex appears to render hopes of extending the
K-convexity of Scarf (1960) to the perishable domain to be in vain. Nevertheless, Nah-
mias (1978) reports extensive computational experiments supporting the conjecture
that the general ((s, S) or non-linear) optimal policy structure holds for the multi-
period problem. One possible avenue to prove such structure could be through the
application of recent proof techniques involving decomposition ideas, such as those in
Muharremoglu and Tsitsiklis (2003).
Lian and Liu (1999) consider the discrete review model with fixed ordering cost,
per unit per period holding cost, per unit and per unit per unit time shortage costs,
and per unit outdating costs. Crucially, their model is comprised of discrete time
epochs where demand is realized or units in inventory expire (hence discrete time
refers to distinct points in time where change in the inventory levels occur) as opposed
to distinct time periods in Nahmias (1978). Lian and Liu (1999) analyze an (s, S)
policy which was shown to be optimal under continuous review by Weiss (1980) for
Poisson demand. The instantaneous replenishment assumption combined with the
discrete time model of Lian and Liu (1999) ensures that the optimal reorder level s
will be no greater than -1 because any value larger than -1 will add holding costs
without incurring any shortage cost in their model. The zero lead time assumption
and cost structure of Lian and Liu (1999) are common in the continuous review
framework for perishable problems (see Section 2.3.2); it was a desire to develop a
16
discrete time model to approximate the continuous review that motivated Lian and
Liu (1999). In the paper they use matrix analytic methods to analyze the discrete
time Markov chain capturing the system evolution, and establish numerically that
the discrete review model is indeed a good approximation for the continuous review
model, especially as the length of the time intervals gets smaller.
Lian et al. (2005) follow Lian and Liu (1999), sharing similar costs and replenish-
ment assumptions. This later paper allows demands to be batch with discrete phase
type interdemand times, and lifetimes to be general discrete time phase type, but
requires that all items of the same batch perish at the same time. They again use
matrix analytic methods to derive cost expressions, and numerically show that the
variability in the lifetime distribution can have a significant effect on system perfor-
mance. Theirs is a quite general framework, and provides a powerful method for the
evaluation of discrete time problems in the future, provided the assumption of the
instantaneous replenishments is reasonable in the problem setting.
Of great practical importance again is the question of effective heuristic policies.
Nahmias (1978) establishes, computationally, that while not strictly optimal, (s, S)
type policies perform very close to optimal. Moreover, he also presents two methods of
approximating the optimal (s, S) parameters. Both of these are effective: on average
their costs are within 1% of the globally optimal cost for the cases considered, and
in all cases within 3%. Thus, while there certainly is room for further refinements of
these heuristic policies (along the lines of the zero fixed cost case for example), the
results in Nahmias (1978) are already compelling.
Discrete Review Models with Positive Lead Time
As optimal solutions to the zero lead time case require the use of dynamic program-
ming, problems with lead times are in some sense no more difficult, likewise requiring
dynamic programming, albeit of a higher dimension. Thus problems considering op-
timal policies for discrete review problems with lead times are also often considered
in the context of other generalizations to the model: for example different selling
prices depending on age (Adachi et al., 1999). Interestingly, there is very little work
on extending discrete review heuristics for the zero lead time model to the case of
positive lead times, although many of the methods for the zero lead time case should
17
in principle be applicable, by expanding the vector of ages of inventory kept to include
those items on order, but which have not yet arrived.
One possible direction is provided in the work of Williams and Patuwo (1999, 2004)
who derive expressions for optimal ordering quantities based on system recursions for
a one-period problem with fixed lead time, lost sales, and the following costs: per unit
shortage costs and per unit outdating costs, per unit ordering costs and per unit per
period holding costs. Williams and Patuwo (2004) in fact state that their methods
can be extended to finite horizons, but such an extension has not yet appeared.
Development of such positive lead time heuristics, no matter what their genesis, would
prove valuable both practically and theoretically, making this a potentially attractive
avenue for future research. The trick, essentially, is to keep enough information so as
to make the policy effective, without keeping so much as to make the policy overly
cumbersome. More complex heuristics have been proposed for significantly more
complex models with lead times, such as in Haijema et al. (2005a, 2005b), but these
likely are more involved than necessary for standard problem settings.
2.3.2 Continuous Review Models
We now turn our attention to continuous review models. These are becoming in-
creasingly important with the advent of improved communication technology (such
as radio frequency identification, RFID) and automated inventory management and
ordering systems (as a part of common enterprise resource planning, ERP, systems).
These two technologies may eventually enable management of perishables in real time,
potentially reducing outdating costs significantly. We will first consider continuous
review models without fixed ordering costs or lead times in Section 2.3.2, models
without fixed ordering costs but positive lead times in Section 2.3.2, and finally the
models which incorporate fixed ordering costs in Section 2.3.2.
Tables 2.2 and 2.3 provide a high-level overview of some of the key papers within
the continuous review framework, segmented by whether a fixed ordering cost is
present in the model (Table 2.3) or not (Table 2.2). Categorization of the papers and
the notation used in these tables are described below:
• Replenishment policy: Some papers in Table 2.2 assume a base stock policy,
denoted by a z. Some papers in the same table concern themselves only with
18
characterizing system performance; these have a blank in the replenishment
policy column. In Table 2.3, all papers either use the (s, S) policy, annotated
with a ∗ when proved to be optimal, with a ‡ when only implied to be the
optimal policy based on earlier research (however a formal proof of optimality
of (s, S) has not appeared in print for the model under concern) or when batch
sizes are fixed the (Q, r) or (Q, r, T ) policies, where the latter policy orders when
inventory is depleted below r or when items exceed T units of age.
• Excess demand: Papers may assume simple backlogging (B), some sort of gen-
eralized backlogging in which not all backlogged customers will wait indefinitely
(Bg), or lost sales (L). Two papers assumes all demand must be satisfied; this
field is blank in that case.
• Problem horizon: Planning horizon is either finite (F) or infinite(I).
• Replenishment lead time: In Table 2.2, many papers assume that items arrive
according to a Poisson process, outside of the control of the system manager;
these are denoted by (M). This may be generalized to the case of batch replen-
ishments (batch), the case when the replenishment rates can be controlled, (M3)
or are state dependent (M4). Other papers in the no fixed ordering cost regime
assume either exponential (exp) with a single or ample (ample) servers, renewal
(renewal), continuous (cont) or fixed (D) lead times. Most papers in Table 2.3
assume either zero lead time or a deterministic (D) lead time, although some
allow exponential (exp), or general continuously distributed (cont) lead times.
• Product lifetime: Most lifetimes are deterministic (D), possibly with the as-
sumption that the item can be used for a secondary product after it perishes
(D0), that all perish at once, a disaster model, indicated by (D2), or that items
within a lot only begin to age after all items from the previous lot have left
inventory (D6). Item lifetimes may also be exponentially distributed (exp),
generally distributed (gen), Erlang (Ek) and in one case they decay (decay).
• Demand distribution: As we are within a continuous model, this column de-
scribes the assumptions on the demand interarrival distribution. Most are Pois-
son (M), although in some cases they may have rates that can be controlled by
19
the system manager (M3) or which are state-dependent (M4). Selected papers
allow independent and identically distributed arrivals (iid), continuously dis-
tributed interarrivals (cont), some allow arrivals in batches (batch), and others
have renewal (renewal), or general (general) demand interarrival distributions.
• Costs: Costs include unit costs for ordering (c), per unit time for holding (h),
unit costs for perishing (m), per unit time for shortage (p), one-time unit costs
for shortage (b), and fixed cost for ordering (K). When demand and interarrival
rates can be controlled, there is a cost (s) for adjusting these rates. Some models
calculate profits, via using a unit revenue (r). One paper not only charges the
standard per unit time holding and penalty costs (h) and (p), but also per unit
time costs based on the average age of the items being held (h’) or backlogged
(p’). Two papers optimize subject to constraints (possibly with other costs), and
several (but not all) that concern themselves only with characterizing system
performance have this column left blank. In addition, one paper utilizes a
Brownian control model with two barriers; it defines shortage and outdate costs
slightly differently to account for the infinite number of hits of the barriers. This
is denoted by 8. One paper calculates actuarial valuations of costs and expected
future revenues, where the latter depend on the age of an item when it is sold.
This is denoted with a 9. Finally, when annotated with a 1 a paper allows the
unit holding and shortage costs to be generalized to convex functions.
Continuous Review without Fixed Ordering Cost or Lead Time
The continuous review problem without fixed ordering costs is unique in that under
certain modeling assumptions, direct parallels can be drawn between the perishable
inventory problem and stochastic storage processes in general, and queueing theory
in particular. These parallels provide structural results as well as powerful analytical
tools.
One of the earliest papers to make these connections was written by Graves (1982).
Like many of the papers in this area, Graves (1982) was concerned with character-
izing the system behavior, and as such his paper lacks an explicit cost structure or
replenishment policy. In this work Graves (1982) shows that the virtual waiting time
20
Rep
lenis
hm
ent
Exce
ssP
lannin
gLea
dLife-
Dem
and
Art
icle
policy
dem
and
hori
zon
tim
eti
me
dis
trib
uti
on
Cost
s
Gra
ves
(1982)
B/L
IM
Dbatc
hM
Kasp
iand
Per
ry(1
983)
LI
MD
M
Kasp
iand
Per
ry(1
984)
Bg/L
Ire
new
al
DM
Per
ryand
Posn
er(1
990)
LI
M3
DM
3c,
h,b
,s
Per
ry(1
997)
LI
M3
DM
3c,
h,b
,s8
Per
ryand
Sta
dje
(1999)
Bg/L
IM
4ex
p/D
M4
c,h,p
,m,r
,h’,p’
Per
ryand
Sta
dje
(2000a)
LI
MD
0M
Per
ryand
Sta
dje
(2000b)
LI
Mgen
M
Per
ryand
Sta
dje
(2001)
LI
MD
/M
2M
Nahm
ias
etal.
(2004a)
LI
M4
DM
4
Nahm
ias
etal.
(2004b)
LI
batc
hM
Dre
new
al
c,b,m
,r9
Pal(1
989)
zB
Iex
pex
pM
h,p
,b,m
Liu
and
Cheu
ng
(1997)
zB
g/L
Iex
p/am
ple
exp
Mco
nst
rain
ts
Kalp
akam
and
Sapna
(1996)
zL
Iex
pex
pre
new
al
c,h,b
,m
Kalp
akam
and
Shanth
i(2
000)
zB
g/L
IM
4ex
pM
c,h,p
,b,m
Kalp
akam
and
Shanth
i(2
001)
zL
Ico
nt
exp
Mc,
h,b
,m
Sch
mid
tand
Nahm
ias
(1985)
zL
ID
DM
c,h,b
,m
Per
ryand
Posn
er(1
998)
zB
g/L
ID
DM
Tab
le2.
2:A
sum
mary
ofconti
nuous
tim
esi
ngle
item
invento
rym
odels
wit
hout
fixed
invento
rycost
:g
denote
sa
genera
lized
backord
er
model,
0denote
sth
echara
cte
rist
icth
at
aft
er
item
speri
sh
from
afirs
tsy
stem
they
have
use
ina
second
syst
em
,2
denote
suse
of
adis
ast
er
model
inw
hic
hall
unit
speri
shat
once,
3denote
sth
eability
tocontr
ol
the
arr
ival
and/or
dem
and
rate
s,4
denote
s
state
-dependent
arr
ivaland/or
dem
and,8
denote
ssl
ightl
ym
odifi
ed
cost
definit
ions
tofit
wit
hin
the
conte
xt
ofa
Bro
wnia
ncontr
olfr
am
ew
ork
,and
9denote
sactu
ari
alvalu
ati
ons
base
don
expecte
dcost
s
and
age-d
ependent
revenues.
21
Rep
lenis
hm
ent
Exce
ssP
lannin
gLea
dLife-
Dem
and
Art
icle
policy
dem
and
hori
zon
tim
eti
me
dis
trib
uti
on
Cost
s
Wei
ss(1
980)
(s,S
)*B
/L
I0
DM
c,h,p
,m,r
,K1
Kalp
akam
(s,S
)‡L
I0
exp
Mc,
h,m
,K
and
Ari
vari
gnan
(1988)
Liu
(1990)
(s,S
)‡B
I0
exp
Mh,p
,b,m
,K
Moort
hy
etal.
(1992)
(s,S
)‡L
F/I
0E
kbatc
hiid
Liu
and
Shi(1
999)
(s,S
)‡B
I0
exp
renew
al
h,p
,b,m
,K
Liu
and
Lia
n(1
999)
(s,S
)‡B
I0
Dre
new
al
h,p
,b,m
,K
Lia
nand
Liu
(2001),
(s,S
)‡/
(s,S
)B
I0
/D
Dbatc
hre
new
al
h,p
,b,m
,K
Gurl
erand
Ozk
aya
(2003)
Gurl
erand
Ozk
aya
(2006)
(s,S
)B
I0/D
gen
2batc
hre
new
al
h,p
,b,m
,K
Kalp
akam
and
Sapna
(1994)
(s,S
)L
Iex
pex
pM
c,h,b
,m,K
Ravic
handra
n(1
995)
(s,S
)L
Ico
nt5
D/D
6M
c,h,b
,m,K
Liu
and
Yang
(1999)
(s,S
)B
Iex
pex
pM
h,p
,b,m
,K
Nahm
ias
and
Wang
(1979)
(Q,r
)B
ID
dec
ay
cont
h,b
,m,K
Chiu
(1995)
(Q,r
)B
ID
Dgen
eral
c,h,b
,m,K
Tek
inet
al.(2
001)
(Q,r
,T)
LI
DD
6M
h,m
,K,c
onst
rain
t
Ber
kand
Gurl
er(2
006)
(Q,r
)L
ID
D2
Mh,b
,m,K
Tab
le2.
3:C
onti
nuous
tim
esi
ngle
item
invento
rym
odels
wit
hfixed
ord
eri
ng
cost
:1
denote
spapers
inw
hic
hhold
ing
and
short
age
cost
scan
be
genera
lconvex
functi
ons,
2denote
suse
ofa
dis
ast
er
modelin
whic
hall
unit
sfr
om
the
sam
eord
er
peri
shat
once,5
that
only
one
outs
tandin
gord
er
isallow
ed
at
ati
me,6
the
ass
um
pti
on
that
pro
duct
agin
gst
art
sonly
once
all
invento
ryfr
om
the
pre
vio
us
lot
has
been
exhaust
ed
or
expir
ed,∗
denote
sth
ecase
when
(s,S)
policy
ispro
ved
tobe
opti
mal,
and‡
denote
sth
ecase
when
(s,S)
policy
isim
plied
tobe
the
opti
malre
ple
nis
hm
ent
policy
base
don
earl
ier
rese
arc
h(h
ow
ever
afo
rmalpro
ofofopti
mality
of(s
,S)
has
not
appeare
din
pri
nt
for
the
modelunder
concern
).
22
in an M/M/1 queue with impatient customers and a M/D/1 with finite buffer can
be used to model the inventory in a perishable inventory system with Poisson de-
mands of exponential, or unit size, respectively, with either lost sales or backlogging.
A crucial assumption is that items are replenished according to what Graves (1982)
terms “continuous production;” the arrival of items into inventory is modeled as a
Poisson process, essentially out of the control of the inventory manager. Under this
convention, Graves (1982) notes that the key piece of information is the age of the
oldest item currently in stock, an observation that would be used by many subse-
quent researchers. For example, Kaspi and Perry (1983, 1984) model systems with
Poisson demand and Poisson or renewal supply, as might be the case, for example
in blood banks that rely on donations for stock. For their analysis Kaspi and Perry
(1983, 1984) track what they call the virtual death process, which is the time until
the next death (outdate) if there would be no more demand. This, of course, is just
a reformulation of the age of the oldest item in stock, as used by Graves (1982).
These papers mark the start of a significant body of work by Perry (sometimes
in conjunction with others) on continuous review perishable inventory systems with
no fixed ordering cost, some sort of Poisson replenishment, and nearly all using the
virtual death process. Only Perry and Posner (1990) and Perry and Stadje (1999)
contain explicit cost functions; the rest of the papers concern themselves solely with
performance characteristics, as in Graves (1982). Perry and Posner (1990) develop
level crossing arguments for storage processes to capture the effects of being able
to control supply or demand rates within the model of Kaspi and Perry (1983);
the limiting behavior of this system was analyzed by Perry (1997) using a diffusion
model and martingale techniques. Perry and Stadje (1999) depart from the virtual
death process, instead using partial differential equations to capture the stationary
law of a system which now may have state-dependent arrival and departure rates
with deterministic or exponential lifetimes and/or maximum waiting times, as well
as finite storage space. This work is generalized in Nahmias et al. (2004a) using the
virtual death process. Likewise using the virtual death process, Perry and Stadje
(2000a, 2000b, 2001) evaluate systems where after perishing the item can be used for
a secondary product (such as juice for expired apples); items may randomly perish
before the expiration date; or, in addition to their fixed lifetime, items may all perish
before their expiration date (due to disasters, or obsolescence). Still using the virtual
23
death process, Nahmias et al. (2004b) provide actuarial valuations of the items in
system and future sales, when item values are dependent on age. Recently Perry and
Stadje (2006) solved a modified M/G/1 queue and showed how it related, again, to
the virtual death process in a perishable system, this time with lost sales. Thus work
on extensions to what can reasonably be called the Perry model continues.
Continuous Review without Fixed Ordering Cost having Positive Lead
Time
As mentioned above, a defining characteristic of the Perry model is that the supply
of perishable items arrives according to a (possibly state dependent) Poisson process.
When replenishment decisions and lead times must be included in a more explicit
manner, researchers have had to develop other analytical methods.
When lifetimes and lead times are exponentially distributed the problem is simpli-
fied somewhat, as this allows the application of renewal theory, transform methods,
and Markov or semi-Markov techniques, often on more complex versions of the prob-
lem. Pal (1989) looks at the problem with exponential lead times and lifetimes,
Kalpakam and Sapna (1996), allow renewal demands with lost sales, Kalpakam and
Shanthi (2000, 2001) consider state dependent Poisson lead times and then general
continuous lead times, and Liu and Cheung (1997) are unique in that they consider
fill rate and waiting time constraints. All of these papers consider base-stock, or
(S−1, S) inventory control; Kalpakam and Sapna (1996) and Kalpakam and Shanthi
(2001) consider per unit shortage costs and per unit outdating costs, per unit or-
dering costs and per unit per period holding costs. To these Kalpakam and Shanthi
(2000) add a per unit per period shortage cost, while Pal (1989) also adds the per
unit per unit time shortage cost, but disregards the per unit ordering cost. Liu and
Cheung (1997) take a different approach, seeking to minimize inventory subject to
a service level constraint. In all of cases the cost function appears to be unimodal
in S, but no formal proofs have been provided, owing to the difficulty in proving
unimodality. Furthermore, nowhere has the performance of base-stock policies been
formally benchmarked against the optimal, possibly due to the difficulty in dealing
with a continuous state space – time – within the dynamic programming framework.
This remains an open question.
24
The case of fixed lead times and lifetimes is arguably both more realistic and
more difficult analytically. Schmidt and Nahmias (1985) consider a system operating
under a base-stock policy with parameter S, lost sales, Poisson demand, and per unit
shortage costs and per unit outdating costs, per unit ordering costs and per unit
per period holding costs. They define and solve partial differential equations for the
S-dimensional stochastic process tracking the time since the last S replenishment
orders. Numerical work shows that again cost appears to be monotonic in S (in
fact convex), although surprisingly, the optimal value of S is not monotonic in item
lifetime.
Some thirteen years later, Perry and Posner (1998) generalized this work to allow
for general types of customer impatience behavior, in this case using level crossing
arguments to derive the stationary distribution of the vector of times until each of
the S items in system will outdate (reminiscent of their virtual death process). They
also show that the distribution of the differences between the elements of this vector
follows that of uniform order statistics, which enables them to derive expressions
for general customer behavior. These expressions may, as a rule, require numerical
evaluation.
Perry and Posner (1998) are concerned with general system characteristics; they
do not include explicit costs in their paper. Thus, while Perry and Posner (1998) pro-
vides rich material for future research – for example exploring how different customer
behavior patterns affect different echelons of a supply chain for perishable products
– there is also still a need for research following the work of Schmidt and Nahmias
(1985), with the aim of minimizing costs in the continuous review setting under fixed
lead times and fixed lifetimes.
Continuous Review with Fixed Ordering Cost
Within the continuous review fixed ordering cost model we make a distinction between
those models that assume fixed batch size ordering, leading to (Q,R) type models,
and those that assume batch sizes can vary, leading to (s, S) type models. We consider
the (s, S) models first.
Initial work in this setting assumed zero lead time, which simplifies the problem
considerably, as there is no need to order until all the items are depleted. In this case
25
when considering fixed ordering costs, unit revenues, unit ordering and outdate costs,
and convex holding and penalty costs per unit time, the optimal policy structure
under Poisson demand was found by Weiss (1980) who showed that for the fixed
lifetime problem over an infinite horizon, with lost sales or back ordering, an (s, S)
policy is optimal. Weiss (1980) also established that the optimal s value is zero in
the lost sales case, and in the backorder case no larger than -1 (you never order if
you have items in stock). Thus the optimal policy structure was established, but the
question of efficiently finding optimal parameters open.
The publication of Weiss (1980) initiated a series of related papers, all having
in common the assumption of immediate supply. Kalpakam and Arivarignan (1988)
treat the lost sales model as in Weiss (1980), but assume exponential lifetimes, con-
sider only costs (not revenues), and of these costs disregard the shortage costs as
they are irrelevant, as Weiss showed that the optimal s = 0. They also show that in
this case the cost, assuming an optimal s value, is convex in S. Liu (1990) takes the
setting of Weiss (1980), assumes exponential lifetimes, disregards the unit ordering
costs and revenues, but considers penalty costs per unit and per unit time. His focus
is on providing closed-form expressions for system performance, based on transform
analysis. Moorthy et al. (1992) perform a similar analysis using Markov chain theory
under Erlang lifetimes, assuming no shortage is permitted - or equivalently lost sales,
as in this case Weiss (1980) showed that it is optimal not to allow any shortage. Liu
and Shi (1999) follow Liu (1990), but now allow for general renewal demand. They
focus their analysis on the reorder cycle length, using it as a vehicle to prove various
structural properties of the costs with respect to the parameters. The assumption
of exponential lifetimes is crucial here – the results would be very unlikely to hold
under fixed lifetimes. Liu and Lian (1999) consider this problem, the same problem
as Liu and Shi (1999) but with fixed, rather than exponential lifetimes. They per-
mit renewal demands, and derive closed-form cost expressions, prove unimodality of
costs with respect to parameters, and show that the distribution of inventory level is
uniform over (s, S) (as in the non-perishable case).
All of the previous papers make the special zero lead time assumption of Weiss
(1980), which simplifies the problem. A few papers include positive fixed lead times,
Lian and Liu (2001) treat the model of Lian and Liu (1999) incorporating batch
demands to provide a heuristic for the fixed lead time case, but a proof in that paper
26
contained a flaw, which was fixed by Gurler and Ozkaya (2003). These papers show
how to efficiently find good (s, S) parameters for the fixed lead time problem, but
do not provide any benchmark against optimal. Thus while efficient heuristics exist
for the fixed lead time case, they have not as yet been benchmarked against more
complex control policies.
Random lead times have appeared in several papers within the continuous review,
fixed ordering cost framework. Kalpakam and Sapna (1994) allow exponential lead
times, while also assuming exponential lifetimes. As in the fixed lead time case, this
implies that the (s, S) policy is not longer necessarily optimal. They nevertheless
assume this structure. In addition to fixed ordering costs, they account for unit
purchasing, outdate and shortage costs, as well as holding costs per unit per unit time.
Under the assumption of only one outstanding order at a time, they derive properties
of the inventory process and costs. Ravichandran (1995) permits general continuous
lead times, deriving closed-form expressions for costs under the assumption that only
one order is outstanding at a time, and items in an order only begin to perish after all
items from the previous order have left the system. Liu and Yang (1999) generalize
Kalpakam and Sapna (1994) to allow for backlogs and multiple outstanding orders,
using matrix analytical methods to generate numerical insights under the assumption
of an (s, S) policy. For random lifetimes, still within the (s, S) structure, the most
comprehensive work is by Gurler and Ozkaya (2006), who allow a general lifetime
distribution, batch renewal demand, and zero lead time with a heuristic for positive
lead time. Their cost structure follows Lian and Liu (2001); Gurler and Ozkaya
(2006) can be thought of as generalizing Lian and Liu (2001) to the random lifetime
case. Gurler and Ozkaya (2006) argue that the random lifetime model is important
for modeling lifetimes at lower echelons of a supply chain, as items arriving there
will have already begun to perish. To this end they demonstrate the importance of
modeling the variability in the lead time distribution on costs, including a comparison
to the fixed lifetime heuristic of Lian and Liu (2001).
In general, within the fixed ordering cost model with variable lot sizes, for the zero
lead time case research is quite mature, but for positive lead times and/or batch de-
mands there are still opportunities for research into both the optimal policy structure
and effective heuristics. These models are both complex and practically applicable,
making this yet another problem that is both challenging and important.
27
If lot sizes must be fixed, initial work for “decaying” goods by Nahmias and Wang
(1979), using a (Q, r) policy, was followed two decades later by Chiu (1995), who
explicitly considers perishable (rather than decaying) items by approximating the
outdating and inventory costs to get heuristic (Q, r) values. Nahmias and Wang
(1979) consider unit ordering and shortage costs, and holding costs per unit per unit
time, as well as unit outdate costs. To these Chiu (1995) adds unit ordering costs.
Tekin et al. (2001) take a slightly different approach; they disregard unit ordering
costs and consider a service level constraint, rather than a shortage cost. They also
simplify the problem by assuming that a lot only starts aging after it is put into use,
for example moved from a deep freezer. To combat the effects of perishability, they
advocate placing an order for Q units whenever the inventory level reaches r or when
T time units have elapsed since the last time a new lot was unpacked, whichever
comes first, giving rise to a (Q, r, T ) policy. Not surprisingly, they find that inclusion
of the T parameter is most important when service levels are required to be high
or lifetimes are short. Berk and Gurler (2006) define the “effective shelf life” of a
lot: The distribution of the remaining life at epochs when the inventory level hits
Q. They show that this constitutes an embedded Markov process (as they assume
Poisson demand), and thus via analysis of this process they are able to derive optimal
(Q, r) parameters, when facing unit outdate and penalty costs, holding costs per unit
per unit time, and fixed ordering costs. They compare the performance of their policy
with that of Chiu (1995) and also with the modified policy of Tekin et al. (2001).
Not surprisingly, the optimal (Q, r) policy outperforms the heuristic of Chiu (1995),
sometimes significantly, and is outperformed by the more general modified policy of
Tekin et al. (2001). Nevertheless, this latter gap is typically small, implying that use
of the more simple (Q, r) policy is often sufficient in this setting.
Note that throughout these papers the (Q, r) structure has only been assumed, and
once again there does not appear to be any benchmarking of the performance of the
(Q, r) policy against the optimal, as optimal policies are difficult to establish given the
increased problem complexity the continuous time setting with perishability causes. If
such benchmarking were done, it would help identify those problem settings for which
the (Q, r) or (Q, r, T ) policy is adequate, and those which would most benefit from
further research into more complex ordering schemes. Essentially, the underlying
question of how valuable lifetime information is, in what degree of specificity, and
28
when it is most valuable, remains.
2.4 Managing Multi-Echelon and Multi-Location
Systems
Analysis of multi-echelon inventory systems dates back to the seminal work of Clark
and Scarf (1960); the reader can refer to Axsater (2000) for a unified treatment
of the research in that area, and Axsater (2003) for a survey of research on serial
and distribution systems. In serial and distribution systems, each inventory location
has one supplier. In contrast, multi-location models consider flow of products from
various sources to a particular location (possibly including transshipments). See, for
instance, Karmarkar (1981) for a description of the latter problem. Managing multi-
echelon and/or multi-location systems1 with aging products is a challenge because of
the added complexity in:
• replenishment and allocation decisions, where the age of goods replenished at
each location affects the age-composition of inventory and the system perfor-
mance, and the age of goods supplied/allocated downstream may be as impor-
tant as the amount supplied,
• logistics-related decisions such as transshipment, distribution, collection, which
are complicated by the fact that products at different locations may have dif-
ferent remaining lifetimes,
• centralized vs. decentralized planning, where different echelons/locations may
be managed by different decision-makers with conflicting objectives, operat-
ing rules may have different consequences for different decision-makers (e.g.,
retailers may want to receive LIFO shipments but suppliers may prefer to issue
their inventory according to FIFO), and system-wide optimal solutions need not
necessarily improve the performance at each location.
1Note that multi-location models we review here are different from the two-warehouse problem described in Section
10 of Goyal and Giri (2001); that model considers a decision-maker who has the option of renting a second storage
facility if he/she uses up the capacity of his/her own storage.
29
Given the complexity in obtaining or characterizing optimal decision structures,
analytical research in multi-echelon and multi-location systems has mainly focused
on particular applications (modeling novelties) and heuristic methods. We review
analysis of replenishment and allocation decisions in Section 2.4.1, logistics and dis-
tribution related decision in Section 2.4.2, and centralized vs. decentralized planning
in Section 2.4.3.
2.4.1 Research on Replenishment and Allocation Decisions
Research on multi-echelon systems has been confined to two-echelons2 except in the
simulation-based work (e.g., van der Vorst et al., 2000). Motivated by food supply
chains and blood banking, the upstream location(s) in the two-echelon models typi-
cally involve the supplier(s), the distribution center(s) (DC), or the blood banks, and
the downstream location(s) involve the retailer(s), the warehouse(s), or the hospi-
tal(s). In this section, we use the terms supplier and retailer to denote the upstream
and downstream parties, respectively. We call the inventory at the retail locations
the field inventory.
We first classify the research by focusing on the nature of the decisions and the
modeling assumptions. Table 2.4 provides a summary of the analytical work that
determines heuristic replenishment policies for a supplier and retailer(s), and/or ef-
fective allocation rules to ship the goods from the supplier to the retailers. As we can
see, analytical research has been rather limited, modeling assumptions have varied,
and some problems (such as replenishment with fixed costs or decentralized planning)
have received very little attention.
Notice that the centralized, multi-echelon models in this literature are no dif-
ferent than the serial or distribution systems studied in classical inventory theory;
the stochastic models involve the one-supplier, multi-retailer structure reminiscent
of Eppen and Schrage (1981) or the serial system of Clark and Scarf (1960). For
2The model of Lin and Chen (2003) has three echelons in its design: A cross-docking facility (central decision
maker) orders from multiple suppliers according to the demand at the retailers and the system constraints, and
allocates the perishable goods to retailers. However, the replenishment decisions are made for a single echelon: The
authors propose a genetic algorithm to solve for the single-period optimal decisions that minimize the total system
cost.
30
Rep
lenis
hm
ent
policy
Inven
tory
Exce
ssA
lloca
tion
Tra
ns-
No.
of
Fix
edC
entr
alize
d
Art
icle
Supplier
Ret
ailer
issu
ance
dem
and
rule
ship
men
tsre
tailer
(s)
cost
spla
nnin
g
Yen
(1975)
TIS
TIS
FIF
Oback
log
pro
port
ional,
−−2
−−yes
fixed
Cohen
etal.
TIS
TIS
FIF
Oback
log
pro
port
ional,
−−2
−−yes
(1981a)
fixed
Lyst
ad
etal.
(2006)
TIS
TIS
FIF
Oback
log
myopic
−−>
2−−
yes
Fujiw
ara
TIS
TIS
FIF
Olo
stat
supplier
,−−
−−2
−−yes
etal.
(1997)
(per
cycl
e)(p
erper
iod)
=LIF
Oex
ped
itin
gfo
rre
tailer
Kanch
anasu
nto
rnand
(s,S
)T
ISFIF
Oback
log
at
supplier
,FIF
O−−
2su
pplier
yes
Tec
hanit
isaw
ad
(2006)
lost
at
reta
iler
Pra
staco
s(1
978)†
−−−−
FIF
Olo
stat
reta
iler
myopic
rota
tion
>2
−−yes
Pra
staco
s(1
981)
−−−−
FIF
Olo
stat
reta
iler
myopic
rota
tion
>2
−−yes
Pra
staco
s(1
979)†
−−−−
LIF
Olo
stat
reta
iler
segre
gati
on
rota
tion
>2
−−yes
Nose
etal.
(1983)†
−−−−
LIF
Olo
stat
reta
iler
base
don
convex
rota
tion
>2
−−yes
pro
gra
mm
ing
Fed
ergru
en−−
−−FIF
Olo
stat
reta
iler
base
don
convex
rota
tion
>2
−−yes
etal.
(1986)†
pro
gra
mm
ing
Abdel
-Male
kand
EO
QE
OQ
−−−−
−−−−
1re
tailer
,yes
Zie
gle
r(1
988)
supplier
Ket
zenber
gand
ord
er0
ord
er0
FIF
Oex
ped
itin
gfo
rsu
pplier
,−−
−−1
−−yes
,
Fer
guso
n(2
006)
or
Qor
Qlo
stat
reta
iler
no
Tab
le2.
4:A
sum
mary
ofth
eanaly
ticalm
odels
on
reple
nis
hm
ent
and/or
allocati
on
decis
ions
inm
ult
i-echelo
nand
mult
i-lo
cati
on
syst
em
s(†
indic
ate
ssi
ngle
-peri
od
decis
ion
makin
g).
31
the one-supplier, multi-retailer system with non-perishables, it is known that order-
up-to policies are optimal under the ‘balance assumption’ (when the warehouse has
insufficient stock to satisfy the demand of the retailers in a period, available stocks
are allocated such that retailers achieve uniform shortage levels across the system;
this might involve ‘negative shipments’ from the warehouse, i.e., transshipments be-
tween retailers at the end of that period to achieve system balance), and optimal
policy parameters can be determined by decomposing the system and solving a se-
ries of one-dimensional problems (see for e.g., Diks and de Kok, 1998). There is no
equivalent of this analysis with perishable inventories, mainly due to the complexity
of the optimal ordering policy at a single location. Similarly, there is no work that
investigates continuous replenishment policies with perishables for multi-echelons or
multi-locations, although this is a well-studied problem for single-location models with
perishables (see Section 2.3.2) and non-perishables in multi-echelon supply chains.
Analytical Research on Allocation Decisions
All the models listed in Table 2.4 involve no capacity restrictions, and have a single
supplier who receives the freshest goods upon replenishment (although the goods
may not have the maximal lifetime at the time of arrival when lead time is positive).
However, the shipments from the supplier to the retailers can involve stock of any age
depending on the supplier’s inventory. Allocation and/or transshipment decisions
are simplified, for instance, when there is a single retailer (see Table 2.4) or when
the supplier’s inventory consists of goods of the same age. The latter happens in
the following two cases: (i) The goods start perishing at the retailer but not at the
supplier, i.e. all retailers are guaranteed to receive fresh goods from the supplier (see
Fujiwara et al., 1997, and Abdel-Malek and Ziegler, 1988). (ii) The lifetime of the
product is equal to the length of the periodic review interval. In that case, all the
goods at any location are of the same age and the goods perish at the end of one
cycle, as is the case for the supplier in the model of Fujiwara et al. (1997).
The focus on multi-echelon problems is on allocation decisions when the supplier
is assumed to receive a random amount of supply at the beginning of every period
(Prastacos, 1978, 1979, 1981) - we refer to this as the Prastacos model. The random
supply assumption is motivated by the application area – blood products which rely
32
on donations for supply. Two special distribution systems are considered in these
papers: (i) A rotation (or recycling) system where all unsold units that have not
expired at the retailers are returned to the supplier at the end of each period - these
units are distributed among the retailers along with the new supply of freshest goods
at the beginning of the next period. (ii) A retention system where each retailer keeps
all the inventory allocated. In the Prastacos model, the supplier does not stock any
goods, i.e. all the inventory is allocated and shipped to the retailers at the beginning
of a period, there are no inventory holding costs at any location, and the goal is to
minimize shortage and outdating costs that are uniform across all the retailers.
In a rotation system, the total number of units to outdate in a period depend
only on how the units with only one period of lifetime remaining are allocated in the
previous period, and the total amount of shortage depends only on how the inventory
is allocated, regardless of the age. Based on these observations, Prastacos (1978)
proposes the following myopic allocation policy that minimizes one-period system-
wide outdate and shortage costs: Starting with the oldest, the stocks of a given age
are allocated across the retailers so that the probability that the demand at each
location exceeds the total amount allocated to that retailer up to that point in the
algorithm are equalized, and this is repeated iteratively for items of all ages. Prastacos
(1978) also analyzes a retention system where the supplier only ships the fresh supply
to the retailers at the beginning of each period. In the one-period analysis of the
retention system, the amount to outdate at the end of a period depends on the
amount of oldest goods in stock and demand at each retailer, but does not depend
on the supply allocated in that period. Based on this observation, Prastacos (1978)
suggests a myopic allocation rule that equalizes the one-period shortage probability
at each retailer to minimize the one-period system-wide shortage and outdate costs.
Prastacos (1981) extends the analysis of the Prastacos model to the multi-period
setting and shows that the myopic allocation policy preserves some of the properties
of the optimal allocation that minimizes expected long-run average shortage and
outdating costs, and is, in fact, optimal in numerical examples with 2 retailers and
product lifetime of 2 periods. Since the cost parameters are the same for all the
retailers, the allocation resulting from the myopic rule is independent of the unit
costs of shortage and outdating.
Prastacos (1979) analyzes essentially the same single-period model assuming last-
33
in-first-out (LIFO) issuance, as opposed to FIFO in Prastacos (1978). In case of LIFO,
the optimal myopic allocation policies in both rotation and retention systems depend
on the unit costs of shortage and outdating. In addition, the optimal allocation
policies segregate the field inventory by age as opposed to a fair allocation where each
retailer receives goods of each age category. Under segregation, some retailers have
only newer goods and some only older goods so that system-wide expected outdates
are minimized. The optimal myopic allocation policy under LIFO for both rotation
and retention systems can be determined by solving a dynamic program with stages
corresponding to retail locations. Prastacos (1979) obtains the optimal allocation
rule for specific demand distributions. For rotation, he proposes a heuristic: First,
allocate the stock in order to equalize the probability of shortage at each retailer and
then swap the inventory among retailers to obtain field inventories that are segregated
by age.3
There are several practical extensions of the Prastacos model: The assumption on
uniform outdating and shortage costs can be relaxed, shipments/transportation costs
can be added, penalties can be incurred on leftover inventory (e.g., end of period
holding costs), transshipments among retailers can be enabled, and the supplier may
keep inventory as opposed to shipping all units downstream. The first three of these
issues have been addressed in the literature for only single-period decision-making:
Nose et al. (1983) and Federgruen et al. (1986) generalize the FIFO model of Prasta-
cos (1978) by assuming that there is a unit transportation cost for each item shipped
from the supplier to the retailers, and the unit outdating, shortage and transportation
costs are retailer-specific. They both develop convex programming formulations for
the single-period inventory allocation problem and propose algorithms based on the
Lagrangean relaxation to determine the optimal allocation. Their models rely on the
observation that the costs are only a function of the amount of old vs. fresh (i.e.,
stock that will outdate in one period vs. the inventory that has more than one period
of lifetime remaining) goods allocated to each retailer. In the rotation system consid-
3While they don’t provide a model nor the analysis, Huq et al. (2005) introduce a motivating example for allocating
inventories by segregation. They discuss that outdates and revenue loss due to markdowns or cost of returns can be
reduced by allocating goods to retailers based on the remaining lifetime of the goods and the expected depletion rates
at the retailers.
34
ered by Nose et al. (1983), the retailers are also charged per unit inventory returned
to the supplier at the end of each period (i.e., there is an end-of-period penalty on
leftover inventory at each location).
In the model of Yen (1975) and Cohen et al. (1981a), the supplier uses the FIFO
rule to determine which goods are to be shipped downstream and then uses one of
the following two allocation rules to determine the age-composition of the shipments
to the retailers in each period: (i) proportional allocation and (ii) fixed allocation.
In proportional-allocation, each retailer receives a proportion of goods of each age
category based on their share of the total demand. In fixed allocation, each retailer
receives a pre-determined fraction of goods in each period. Both of these allocation
rules are fair in that the shipments to retailers involve goods of each age category. Yen
(1975) and Cohen et al. (1981a) explore the optimality conditions for the parameters
associated with these allocation rules. They show that, under certain conditions,
there exists a fixed allocation rule that yields the same expected shortage, outdating
and holdings costs as a system operating under the optimal proportional allocation
rule. Their analysis can be extended to include multiple (> 2) retailers. Prastacos
(1981) shows that his myopic allocation rule is the same as proportional allocation
under certain probability distributions of demand.
Analytical Research on Replenishment and Allocation Decisions
Analysis of optimal replenishment policies for a serial, two-echelon system is presented
in Abdel-Malek and Ziegler (1988) assuming deterministic demand, zero lead times,
and price that linearly decreases with the age of the product. They determine the
economic order quantities (EOQs) for the retailer and the supplier by restricting the
order cycle lengths to be no more than the product lifetime.
Under demand uncertainty, several heuristic, discrete review replenishment poli-
cies have been considered and the focus has been on determining the optimal para-
meters for these restricted policies. These heuristic replenishment policies include the
TIS policies (Yen, 1975, Cohen et al., 1981a, Lystad et al., 2006, Fujiwara et al.,1997,
and Kanchanasuntorn and Techanitisawad, 2006), a “zero-or-fixed quantity” ordering
policy in Ketzenberg and Ferguson (2006) - denoted as ‘0 or Q’ in Table 2.4, - and
(s,S) policy for the retailers in Kanchanasuntorn and Techanitisawad (2006); analysis
35
in this latter is restricted to the case where the retailers’ demand is Normal. The re-
plenishment lead times are no longer than one period (i.e. goods are received no later
than the beginning of the next period) with exceptions being Lystad et al. (2006) and
Kanchanasuntorn and Techanitisawad (2006); the latter assumes that the lifetime of
the product is a multiple of the replenishment cycle lengths of the retailers and the
supplier, and that the supplier responds to retailers’ orders in a FIFO fashion (hence
allocation decisions are trivial).
Research that analyzes replenishment and allocation decisions jointly is confined to
the work of Yen (1975), its extension in Cohen et al. (1981a), and Lystad et al. (2006).
Yen (1975) and Cohen et al. (1981a) explore the structural properties of the expected
total cost function that includes expected holding, outdating and shortage costs,
when both the supplier and the retailer use TIS policies to replenish inventory. Fixed
and proportional allocation rules are investigated, and conditions on the existence
of unique target inventory levels under these rules are identified. Yen (1975) also
identifies conditions for the optimality of the proportional allocation rule for this sys-
tem. These conditions are satisfied, for instance, when the lifetime of the product
is restricted to two or three periods, or when the demand at each location and each
period is independent and identically distributed and the target inventory levels of
the retailers are the same. However, the analysis relies on one simplification: Inven-
tory recursions do not explicitly take into account the finite lifetime of the perishable
product, and goods that remain in inventory beyond their lifetime can be used to sat-
isfy excess demand, but are charged a unit outdating cost. Analysis of replenishment
and allocation policies that explicitly take these factors into account remains an open
problem of theoretical interest.
Lystad et al. (2006) use the myopic allocation rule of Prastacos (1981), and propose
heuristic echelon-based TIS policies. For a given system, they first determine the
best TIS policy via simulation. Then, they do a regression analysis to establish
the relationship between the order-up-to levels of this best policy and two heuristic
order-up-to levels: One heuristic is based on the newsvendor-based, approximate
echelon-stock policies for non-perishables and the other is the single-location heuristic
of Nahmias (1976) for perishables. The resulting regression model is then used in
computational experiments to study the effect of lifetime of product on system costs,
and to compare the performance of the approximation against policies that are derived
36
assuming the product is non-perishable. Thus Lystad et al. (2006) take a first step in
developing approximate echelon-based policies for perishables, and this topic deserves
more attention.
Notice that the effectiveness of the proposed allocation rules combined with good
replenishment policies have not been benchmarked in any of these studies, and various
simplifying assumptions have been made to derive the policies. There is a need for
further research on the analysis of replenishment and allocation decisions in multi-
echelon systems. Interesting research directions include investigation of the “balanc-
ing” of echelon inventories for perishables, similar to the balancing assumption in
Eppen and Schrage (1981), analysis of systems without making simplifying assump-
tions on inventory recursions (as in Yen, 1975), analysis of different system designs
(e.g. rotation and retention systems have been studied to some extent), or incorpo-
ration of different cost parameters to the models (e.g., the Prastacos model excludes
holding costs).
Simulation Models of Multi-Echelon Inventory Systems
In addition to the analytical research, simulation models have also been used in ana-
lyzing multi-echelon, multi-location systems with perishable goods; simulation models
present more opportunities in terms of model richness for this complex problem. The
earlier research (e.g., Yen, 1975, Cohen and Pierskalla, 1979, Cohen et al., 1981b) is
motivated mainly by managing regional blood centers and is reviewed in Prastacos
(1984). The simulation model of the single supplier, multiple-retailer system in Yen
(1975) includes returns of unused units from retailers to the supplier, variations of the
fixed and proportional allocation rules, expedited shipments to retailers, transship-
ments between retailers, and limited supply at the supplier. In addition to analyzing
the impact of inventory levels, allocation and transshipment rules on system costs,
Yen (1975) also looks at the impact of magnitude of demand at the retailers, and
observes that the system cost in his centralized model is more sensitive to the sizes
of the retailers rather than the number of retailers.
More recently, van der Vorst et al. (2000) describe a discrete-event simulation
model that can be used in the design/reconfiguration of supply chains of perishable
products. Their simulation model is used to analyze a fresh produce supply chain
37
with three echelons. Among other factors, van der Vorst et al. (2000), test the system
performance - measured in terms of inventory levels at the retailers and distribution
centers, and product freshness - using several scenarios. They report that supply
chain costs improve when production lead times are shorter, ordering and shipment
frequencies are higher, and new information systems are in place.
Katsaliaki and Brailsford (2007) present results of a project to improve procedures
and outcomes by modeling the entire supply chain for blood products in the UK. Their
simulation model includes a serial supply chain with the product flow that includes
collection of supply, processing/testing and storage at a service center, and shipment
to a hospital where blood is crossmatched/transfused4 for patients use. The model
includes multiple products with different shelf-lives. Six different policies vary in (i)
the type of products that are stocked at the hospital, (ii) the target inventory levels,
(iii) the time between crossmatching and release which can influence the amount of
unused and still usable inventory that is returned, (iv) the order trigger points for
expedited deliveries, (v) the inventory issuance rules for releases and returns, (vi)
the order and delivery lead times, and (vii) the number of daily deliveries to the
hospital. Performance is measured in terms of number of expired units, mismatched
units, amount of shortage, and number of routine and expedited deliveries. Final
recommendations include decreasing target inventory levels, having two routine de-
liveries per day as opposed to only one, a trigger point of 35% of the target inventory
level for expedited deliveries and reduction of crossmatch to release time. Note that
allocation decisions are not included in Katsaliaki and Brailsford (2007) because they
model a serial supply chain. Mustafee et al. (2006) provide the technical details of
the simulation model and the distributed simulation environment used in this latter
project.
While reporting the results of a simulation study, Cohen et al. (1981b) discuss
three practical ways of allocating inventory in a centralized system when inventory is
4One common practice in managing blood inventories is cross-matching, which is assigning units of blood from
inventory to particular patients. Jagannathan and Sen (1991) report that more than 50% of blood products held
for patients are not eventually transfused (i.e., used by the patient). The release of products that are cross-matched
enable re-distribution of inventories in a blood supply chain. See Prastacos (1984), Pierskalla (2004) and Jagannathan
and Sen (1991) for more information on cross-matching.
38
issued in a FIFO manner: In the first one, the supplier chooses a retailer and fills its
demand and goes on to fill the demand of the next retailer until all stock is depleted
or all demand is satisfied. In the second one, the supplier uses the proportional
allocation rule, and in the third one, the myopic allocation rule. Cohen et al. (1981b)
suggest using the second method in a practical setting because it has less information
needs (i.e. does not need the probability distribution of demand at each retailer like
the third method) and advise against using the first method in a centralized system
because it will lead to an imbalanced distribution of aging inventory. In addition,
the outdate probabilities of the retailers will vary significantly with the first method;
this increases the possibility of costly transshipments which are discussed in the next
section.
2.4.2 Logistics: Transshipments, Distribution, and Routing
Other than replenishment and allocation decisions, three of the critical logistics ac-
tivities in managing perishables in multi-location systems are transshipments, distri-
bution and collection (particularly for blood). Research that focuses on these three
decisions has been limited. Within the analytical work cited in Table 2.4, the ro-
tation system is the only form in which excess inventory is exchanged among the
retailers, and the exchange happens with a one period delay. Rotation systems are
also the basis for the goal programming model developed by Kendall and Lee (1980).
Prastacos and Brodheim (1980) develop a mathematical programming model for a
hybrid rotation-retention system to efficiently distribute perishables in a centralized
system. Both of these papers are motivated by operations of regional blood centers;
see Prastacos (1984) for a review of these and other earlier work on the distribution
and transshipment problems.
Note that rotation encompasses only indirect transshipments among the retailers.
The simulation model of Yen (1975) includes transshipments between retailers after
each location satisfies its own demand and serves as a guideline for the practical in-
ventory control and distribution system described in Cohen et al. (1981b). Cohen et
al. (1981b) suggest using transshipments in this practical setting if (i) the supplier is
out-of-stock, one retailer has an emergency need, and transshipping units from other
retailers do not significantly increase the probabilities of shortage at those retailers,
39
(ii) the difference between the shortage probabilities of two retailers when a unit is
transshipped from one to the other is greater than the ratio of the unit transporta-
tion cost to the shortage cost, or (iii) the difference between outdate probabilities of
retailers when a unit is shipped from one to the other is greater than the ratio of
transportation cost to the outdate cost. They use the terms emergency, shortage-
anticipating and outdate-anticipating transshipments, respectively, to denote these
three cases. Cohen et al. (1981b) point out that when all the retailers use optimal
TIS policies to manage their inventories, the amount of transshipments is insignifi-
cant based on simulation results. This emphasizes the need for effective replenishment
policies in multi-echelon and multi-location systems.
Federgruen et al. (1986), in addition to their analysis of the allocation decision,
consider the distribution of goods from the supplier to the retailers by formulating a
combined routing and inventory allocation problem. The decisions involve assigning
each location to a vehicle in the fleet and allocating fresh vs. old products among the
locations. The allocations do not affect the transportation costs and the route of
vehicles do not affect shortage and outdating costs. They propose exact and heuristic
solution methods. They also compare their combined routing and allocation approach
to a more hierarchical one where the allocation problem is solved first, and its solution
is used as an input to the distribution problem. Based on computational experiments,
the combined approach provides significant savings in terms of total transportation
costs, although these savings may not lead to a significant decrease in total costs
depending on the magnitude of the inventory related costs. However, the combined
approach has significant benefits when the number of vehicles used is few (where
the hierarchical approach may yield an infeasible solution). In addition to inventory
levels and allocation, the research conducted by Gregor et al. (1982) also examines the
impact of number of vehicles used in distribution on system-wide costs. Simulation
is used in the latter paper.
Or and Pierskalla (1979) consider daily vehicle routing decisions as a part of a
regional location-allocation problem where they also determine the optimal number
and location of blood centers, and the assignment of hospitals to the blood centers
that supply the hospitals on a periodic basis. They develop integer programming
models and propose heuristic solution methods. However, their model is designed
at an aggregate level and age of inventory is not considered. A similar problem is
40
studied by Hemmelmayr et al. (2006) where periodic delivery schedules and vehicle
routes are determined to distribute blood across a region.
Recent research on supply chain scheduling has addressed the need to effectively
distribute time-sensitive goods; Chen (2006) provides a survey of research in this
growing area. However, perishability is not modeled explicitly in this literature, rather
production orders are assumed to come from customers along with information on
delivery time windows and delivery due dates. Similarly, there are several articles
that model and solve real-life distribution problems of perishable products such as
dairy products, or food (e.g., Adenso-Diaz et al., 1998, Golden et al., 2001) where
aging or perishability is not explicitly modeled but is implicit in the time-windows.
More recently, Yi (2003) developed a model for daily vehicle routing decisions to bring
back supply (blood) from collection sites to a central location in order to meet the
daily target level of platelets (that can only be extracted within eight hours of blood
donation); this is a vehicle routing problem with time windows and time-dependent
rewards.
Notice that the research in this area has been confined to a single product, or
multiple products without age considerations. Interestingly, the distribution problem
posed by Prastacos (1984) still remains open: How can a distribution plan for a
centralized system be created to include shipments for multiple products, each with
a different lifetime and supply-demand pattern? Katsaliaki and Brailsford’s (2007)
simulation model provides only a partial answer to this question; their model involves
only one supplier and one retailer. Although challenging, analysis of the centralized
problem with multiple retailers and multiple products definitely deserves attention.
2.4.3 Information Sharing and Centralized/Decentralized Plan-
ning
Information technology paved the way for various industry-wide initiatives includ-
ing Efficient Consumer Response in the grocery industry; these initiatives aim to
decrease total system costs and inventories while improving availability of products
and customer satisfaction. A critical component of these initiatives is the sharing
of demand and product flow information among the suppliers, distributors and re-
41
tailers. Fransoo and Wouters (2000) discuss the benefit of sharing electronic point
of sale (EPOS) information for supply chains of two perishable products (salads and
ready-made pasteurized meals). Their empirical analysis suggests that the benefit of
EPOS would be higher for the supply chain of salad because of the magnitude of the
bullwhip effect observed. The reason for the higher bullwhip effect appears to be the
larger fluctuations in the demand for salad (e.g. when there is a sudden increase in
temperature, there is a spike in demand), associated shortage-gaming by the retail
franchisees, and the additional order amplification by the DC.
Information sharing and the value of information has been widely studied in the
general supply chain literature. Chen (2002) provides a survey of research on this
topic by focusing on where the information is coming from (such as the point-of-sale
data from downstream, or capacity information from upstream in the supply chain),
quantity, accuracy and speed of information, and centralized vs. decentralized plan-
ning in the supply chain (specifically he considers incentives for sharing information
and whether the environment is competitive or not). However, this rich literature
studies non-perishable goods or single-period models; the unique characteristics of
perishables are ignored, except by Ferguson and Ketzenberg (2006) and Ketzenberg
and Ferguson (2006).
Ferguson and Ketzenberg (2006), motivated by the grocery industry, investigate
the value of information for a retailer managing inventory. They focus on the re-
tailer’s replenishment problem and consider an infinite-horizon periodic review inven-
tory model for a single product with finite lifetime, lost sales, one-period lead time
and no outdating cost (see Section 2.3.1). The age of all units in a replenishment are
the same. The age of stock at the supplier is a random variable, and its distribution
is known to the retailer. In case of information sharing, the retailer knows exactly the
age of stock prior to giving an order. Ferguson and Ketzenberg (2006) quantify the
value of information on the age of stock under heuristic replenishment policies with
FIFO, LIFO or random issuing of inventory. Numerical experiments reveal that shar-
ing of product-age information enables the retailer to purchase fresher product and
the average percentage increase in profits via information-sharing is 4.4% and 3.4%,
and average percentage change in outdating is -51.4% and -6.3% for FIFO and LIFO
issuing, respectively. The experimental setup is further used to investigate the value
of other investments for the retailer to influence the issuance policy or the lifetime
42
of the product. An interesting finding is that investments that extend the product
lifetime provide a greater benefit than information sharing.
There are complications associated with different parties operating by different
rules in managing supply chains of perishables: For instance, the supplier can pre-
sumably induce the retailers to order more frequently by adapting an issuance and/or
replenishment policy that leads to more frequent outdates (e.g. using FIFO issuance
and/or having older stock in its inventory will enable the supplier to sell goods that
have a smaller shelf-life to the retailer). This is only mentioned in Ketzenberg and Fer-
guson (2006) - but not analyzed - and is ignored by other researchers. In that paper,
Ketzenberg and Ferguson (2006) study the value of information in a two-echelon set-
ting with one retailer and one supplier. Both parties replenish inventory heuristically,
the retailer’s order quantity in each period is either 0 or Q (which is an exogenous
fixed batch size) and issues inventory in a FIFO fashion, whereas the supplier uses the
same quantity Q in giving orders but need not place an order every period. In fact,
the supplier determines the timing of his replenishments by considering a safety lead
time. The retailer knows the supplier’s inventory state - including the age of items
in stock - and the supplier knows the retailer’s replenishment policy. Ketzenberg and
Ferguson (2006) quantify (i) the value of information regarding the inventory state
and replenishment policies in a decentralized system via numerical examples, and (ii)
the value of centralized planning. Based on computational experiments, they reach
the conclusion that the value of information for perishables can be significant, 70% of
the benefit of centralized planning comes from information sharing, and the increase
in supply chain profit is not always Pareto improving for both parties.
Note that all the papers we introduced so far focus on a single decision maker.
Likewise, Hahn et al. (2004) derive the optimal parameters of a TIS policy for a re-
tailer under two different contracts offered by the supplier; however, the supplier’s
optimal decisions are disregarded. Among the few studies that mention decentralized
decision-making, Popp and Vollert (1981) provide a numerical comparison of cen-
tralized vs. decentralized planning for regional blood banking. Problems that involve
multiple decision-makers, decentralized planning (vs. centralized) and coordination of
supply chains have been widely studied for non-perishables and/or using single-period
models (see, Chen, 2002 and Cachon, 2003). In practice, perishable products share
the same supply chain structure as many non-perishables, and decentralized planning
43
and/or coordination issues are just as critical. Furthermore, there are more challenges
for perishables due to cost of outdating, and possibly declining revenues due to aging.
However, research in this area has been scarce, and this issue remains as one of the
important future research directions. 5
2.5 Modeling Novelties: Demand and Product Char-
acteristics, Substitution, Pricing
The research we have reviewed so far includes models where the inventory of a single
product is depleted either in a LIFO or FIFO manner. Analysis of single location
models in Section 2.3 is confined to FIFO. Earlier research on single location models
has shown the difficulty in characterizing stationary distribution of stock levels under
LIFO even when the lifetime of the product is only two periods (see the comments in
Nahmias, 1982). Nahmias (1982) mentions that when the lifetime is two periods, the
order up to level in a TIS policy is insensitive to the choice of FIFO vs. LIFO despite
the difference in total system costs. Therefore, the replenishment policies/heuristics
developed under FIFO can also be used effectively for LIFO. However, replenishment
and issuance decisions may be interconnected - hence a more a careful analysis is
needed - under more general demand models.
Typically, excess demand is treated via backlogging or lost sales, with some papers
incorporating expedited delivery in their models (e.g., Fujiwara et al., 1997, Ketzen-
berg and Ferguson, 2006, Yen, 1975, Bar-Lev et al., 2005, and Zhou and Pierskalla,
2006). In practice, there is another way to fulfill the excess demand for a product:
Substitution. In the case of perishables, products of different ages often co-exist in the
market place, and inventory can be issued using rules more complicated than FIFO or
LIFO, allowing items of different ages or shelf-lives to be used as substitutes for each
other. This idea first appeared in the perishable inventory literature in Pierskalla
5There is research on coordination issues in supply chains with deteriorating goods: A permissible delay in
payment agreement between a retailer and a supplier is proposed in the deterministic model of Yang and Wee (2006)
to coordinate the supply chain. Chen and Chen (2005) studies centralized and decentralized planning for the joint
replenishment problem with multiple deteriorating goods.
44
and Roach (1972) who assume there is demand for any category (age) and that the
demand of a particular category can be satisfied from the stocks of that category or
using items that are fresher. Pierskalla and Roach (1972) show that FIFO is optimal
in this model with respect to two objectives: FIFO minimizes total backlog/lost sales
and minimizes outdates. The model has one simplification, though: The demand and
supply (replenishment) are assumed to be independent of the issuing policy. Since
issuing can potentially affect demand - fresher goods could lead to more loyal cus-
tomers - the study of models in which this assumption of independence is relaxed will
be important.
The motivation for many of the papers that involve substitution and age-dependent
demand streams come from health care. Cohen et al. (1981a) mention hospitals do-
ing special surgeries get higher priority for fresh blood. Haijema et al. (2005a, 2005b)
mention that platelets have 4-6 days of effective shelf-life, and 70% of the patients
requiring platelets suffer from platelet function disorder and need a fresh supply of
platelets (no older than 3 days) on a regular basis whereas the remaining 30% of
the patients who may lack platelets temporarily due to major trauma or surgery
do not have a strong preference with respect to the age of the platelet up to the
maximal shelf-life. For supply chains involving perishable goods other than blood,
substitution usually depends on customers’ choice and/or a retailer/supplier’s ability
to influence customers’ purchasing decisions. In their empirical research, Tsiros and
Heilman (2005) study the effect of expiration dates on the purchasing behavior of
grocery store customers. They conducted surveys to investigate consumer behavior
across different perishable product categories. They find that consumers check the
expiration dates more frequently if their perceived risk (of spoilage or health issues)
is greater. They also determine that consumers’ willingness to pay decreases as the
expiration date nears for all the products in this study; again finding that the de-
crease varies across categories in accordance with customer perceptions. Tsiros and
Heilman’s (2005) findings support the common practice of discounting grocery goods
that are aging in order to induce a purchase. However, they find that promotions
should differ across categories and across customer groups in order to exploit the
differences in customers’ tendencies to check the expiration dates and the differences
in their perceived risks across categories. In light of these motivating examples and
empirical findings, we provide below an overview of analytical research that considers
45
substitution, multiple products and pricing decisions.
2.5.1 Single Product and Age-Based Substitution
Research like that of Pierskalla and Roach (1972), where products of different shelf-
lives are explicitly modeled is limited. For a single product with a limited shelf-
life, substitution has been considered to sell goods of different ages: Parlar (1985)
analyzes the single-period problem for a perishable product that has two periods
of lifetime, where a fixed proportion of unmet demand for new items is fulfilled by
unsold old items and vice-versa, but his results do not extend to longer horizons.
Goh et al. (1993) consider a two-stage perishable inventory problem. Their model
has random supply and separate, Poisson-distributed demand streams for new and
old items. They computationally compare a restricted policy (where no substitution
takes place) and an unrestricted policy (where stocks of new items are used to fulfill
excess demand for old). Considering only shortage and outdating costs they conclude
that the unrestricted policy is less costly, unless the shortage cost for fresh units is
very high. Ferguson and Koenigsberg (2006) study a problem in a two-period setting
with pricing and internal competition/substitution. In their model, the demand for
each product in the second period is given by a linear price-response curve which is
a function of the price of both products as well as the quality deterioration factor of
the old product, and their decisions include the prices of both products as well as
the number of leftover units of old product to keep in the market. They investigate
whether a company is better off by carrying both or only the new product in the
second period.
Ishii (1993) models two types of customers (high and low priority) that demand
only the freshest products or products of any age, respectively, and obtains the optimal
target inventory level that maximizes the expected profits in a single period for a
product with finite lifetime. The demand of high priority customers is satisfied from
the freshest stock first, and then inventory is issued using FIFO in this model. Ishii
and Nose (1996) analyze the same model under a warehouse capacity constraint.
More recently, Haijema et al. (2005a, 2005b) study a finite horizon problem for blood
platelet production with a demand model of two types of customers similar to Ishii
(1993) and Ishii and Nose (1996). Haijema et al. (2005a, 2005b) formulate a Markov
46
Decision Process (MDP) model to minimize costs associated with holding, shortage,
outdating and substitution (incurred when the demand for a ‘fresh’ item is fulfilled
by older stock) costs. They assume inventory for the any-age demand is issued in
a FIFO manner from the oldest stock and fresh-demand is issued using LIFO from
the freshest stock. Haijema et al. (2005a, 2005b) propose a TIS and a combined
TIS-NIS heuristic, i.e., there is a daily target inventory level for total inventory in
stock and also the new items in stock. Computational experiments show that the
hybrid TIS-NIS policy is an improvement over TIS and that these heuristics provide
near-optimal inventory (production) policies.
For the sake of completeness of this chapter we should mention where our following
chapters stand in the literature. In the next two chapters we provide a detailed analy-
sis of the interplay between replenishment policies and substitution for perishables.
Our infinite horizon, periodic review formulation for a product with two periods of
lifetime includes lost sales, holding, outdating as well as substitution (both new-to-
old and old-to-new) costs. We assume two separate demand streams for new and
old items; demand can be correlated across time or across products of different ages.
We study different substitution options: The excess demand for a new (old) item is
satisfied from the excess stock of old (new), or is lost; the latter is the no-substution
case. Both LIFO and FIFO inventory issuance, as is common in the literature, can
be represented using our substitution model.
In our model the inventory is replenished using either TIS or NIS, and we identify
conditions for the cost parameters under which the supplier would indeed benefit
from restricted (only old-to-new, only new-to-old, or no substitution) or unrestricted
forms of substitution while using a practical replenishment policy. We show that
even when substitution costs are zero, substitution can be economically inferior to
no-substitution for a supplier using a TIS policy. Alternately, even when substitution
costs are very high, no-substitution is not guaranteed to be superior for a supplier
using TIS. These counter-intuitive properties are the side-effect of the TIS policy
which constrains reordering behavior. In contrast, more intuitive results under the
NIS policy exist as do conditions on cost parameters that establish the economic
benefit of substitution for this replenishment policy.
We do extensive computational experiments to quantify the benefits of substitu-
tion and to compare TIS and NIS. Given the literature up to this point, it is surprising
47
that NIS proves more effective than TIS and provides lower long-run average costs
except when the demand for new items is negligible. This is likely due to the pres-
ence of customers who pay a premium for newer product in our model. Similar to the
observation of Cohen et al. (1981b) on the limited need for transshipments (see our
discussion in Section 2.4.2), we show that the amount of substitution is small when
inventory is replenished using effective policies. Our work on perishables really only
begins the consideration of substitution for perishable items - items with longer life-
time, or substitution between different perishable products within the same category
(e.g., different types of fruit) remain to be investigated. Note that substitution is only
modeled as a recourse in the following chapters, and dynamic substitution where one
strategically sells a product of a different age than requested before the stocks of the
requested item are depleted, has not been studied.
2.5.2 Multiple Products
Nahmias and Pierskalla (1976) study the optimal ordering policies for an inventory
system with two products, one with a fixed, finite shelf-life and the other with an
infinite lifetime. The problem is motivated by the operation of a blood bank storing
frozen packed red cells.6 Demand is satisfied from the inventory of perishable product
first in a FIFO manner, any remaining demand is fulfilled from the inventory of non-
perishable product. Nahmias and Pierskalla (1976) analyze the structural properties
of the expected cost function in a finite-horizon, dynamic, discrete review system
and show that the optimal ordering policy in each period is characterized by three
choices: Do not order, order only product with the finite lifetime, or order both
products. Their results include monotonicity of order-up-to level of the perishable
product, e.g, the decrease in order up-to-level is higher with the increase in the stock
levels of newer items as opposed to old ones. They also show that if it is optimal to
order both products in a given period, then it is optimal to bring the total system-
wide inventory up to a level that does not vary with the on-hand inventory levels,
but with the time remaining until the end of the planning horizon.
6We refer the reader to Prastacos (1984) for earlier, simulation-based research on the effect of
freezing blood products on inventory management.
48
Multiple perishable products are also considered by Deuermeyer (1979, 1980).
Deuermeyer (1979) determines the one-period optimal order-up-to-levels for two prod-
ucts. In his model, the products are produced by two processes, one of which yields
both products and the other yields only one product. Deuermeyer (1980) determines
the single-period, optimal order-up-to levels for multiple perishable products, each
with a different lifetime. A critical assumption in the latter is the economic sub-
stitution assumption where the marginal total cost of a product is assumed to be
nondecreasing in the inventory levels of other products. Using the resulting prop-
erties of the single-period expected total cost function, Deuermeyer (1980) is able
to obtain the monotonicity results on order-up-to-levels for the single-period, multi-
product problem. His results mimic that of Fries (1975) and Nahmias (1975a) for
the single product problem. Specifically, these results show that the optimal order-
up-to level of a product is more sensitive to changes in stock levels of newer items
(as discussed above for Nahmias and Pierskalla, 1976), and that the optimal order
quantity decreases with an increase in the on-hand stock levels, while the optimal
target inventory level remains nondecreasing.
2.5.3 Pricing of Perishables
Pricing, in general, has become one of the most widely studied topics in the operations
management literature in the last decade. There is a significant body of research
on dynamic pricing and markdown optimization for ‘perishables.’ One well-studied
research problem in that domain involves determining the optimal price path for a
product that is sold over a finite horizon given an initial replenishment opportunity
and various assumptions about the nature of the demand (arrival processes, price-
demand relationship, whether customers expect discounts, whether customers’ utility
functions decrease over time etc.). We refer the reader to the book by Talluri and
van Ryzin (2004) and survey papers by Elmaghraby and Keskinocak (2003), and
Bitran and Caldentey (2003), for more information on pricing of perishable products.
In that stream of research, all the items in stock at any point in time are of the
same age because there is only one replenishment opportunity. In contrast, Konda
et al. (2003), Chandrashekar et al. (2003), and Chande et al. (2004, 2005) combine
pricing decisions with periodic replenishment of a perishable commodity that has a
49
fixed lifetime, and their models include items of different ages in stock in any period.
They provide MDP formulations where the state vector includes the inventory level of
goods of each age. Their pricing decisions are simplified, i.e., they only decide whether
to promote all the goods in stock in a period or not. Chande et al. (2004) suggest
reducing the size of the state of space by aggregating information of fresher goods
(as opposed to aggregation of information on older goods as in Nahmias, 1977a).
Performance of this approximation is discussed via numerical examples in Chande
et al. (2004, 2005), and sample look-up tables for optimal promotion decisions are
presented for given inventory vectors.7
Based on Tsiros and Heilman’s (2005) observations on customers’ preferences and
close-substitutability of products in fresh-produce supply chains, it is important to
analyze periodic promotion/pricing decisions across age-groups of products. There
are several research opportunities in this area in terms of demand management via
pricing to minimize outdates and shortages across age-groups of products and product
categories.
2.6 Summary
We presented a review of research on inventory management of perishable and aging
products, covering single-location inventory control, multi-echelon and multi-location
models, logistics decisions and modeling novelties regarding demand and product
characteristics. We identified and/or re-emphasized some of the important research
directions in Sections 2.2 to 2.5, ranging from practical issues such as product-mix
decisions and managing inventories of multiple, perishable products, to technical ones
such as the structure of optimal replenishment policies with fixed costs in the single-
product, single-location problem. We will now discuss the effect of substitution on
management of perishable goods and analyze inventory replenishment policies for
perishables under substitution in the following two chapters.
7Another paper that considers prices of perishable products is by Adachi et al. (1999). Items of each age generate
a different revenue in this model, demand is independent of the price, and the inventory is issued in a FIFO manner.
The work entails obtaining a replenishment policy via computation of a profit function given a price vector.
50
Chapter 3
Effect of Substitution on
Management of Perishable Goods
3.1 Introduction
Effective management of a supply chain calls for getting the right product at the right
time to the right place and in the right condition. This is never a simple task, and is
particularly difficult (and often costly) for supply chains containing perishable items.
Historically, product perishability, i.e. products changing chemically, losing value as
they age, has only been a major concern for items such as blood products, fresh
produce and pharmaceuticals (drugs, vitamins, cosmetics). Recently though, devel-
opments have expanded the scope of this problem: Increasing numbers of products
(e.g. in high-tech industries) are “aging” and losing value over time as newer prod-
ucts are introduced while older products are being phased out (products obsolesce).
Strictly speaking perishability and obsolescence are different phenomena - the former
happens at a predictable rate as products change, and the latter less predictably as
the market environment changes. Nevertheless, in both settings the same fundamen-
tal question needs to be addressed - how to manage inventory of a product that grows
increasingly less valuable over time. Answering this question incorrectly can be quite
costly: In a 2003 survey of distributors to supermarkets and drug stores, overall un-
salable costs within consumer packaged goods alone were estimated at $2.57 billion,
51
for the branded segment 22% of these costs were attributed to expiration (Grocery
Manufacturers of America, 2004). The $1.7 billion U.S. apple industry alone loses
$300 million annually to spoilage, according to Redi Ripe, an Albuquerque based
company that focuses on decreasing spoilage in the produce market (Webb, 2006).
And, Accenture estimates the average obsolescence of hand-sets and other accessories
amounts to 2.5-3% of total inventory purchase (about $550-675 million) annually for
the wireless industry in the United States (Colomina et al., 2006).
Further complicating the management of supply chains in general, and those for
perishables in particular, is the fact that increasingly, products of different “ages” - at
different phases in their life cycles - co-exist in the market at the same time. Differ-
ences in consumer preferences, price points, functionality of the products, and/or rate
of adoption have made overlapping of product life cycles not only viable, but often
advisable for many industries. A survey of electronics goods manufacturers in North
America, Europe and Asia Pacific revealed that the share of revenue that comes from
products introduced in the last 12 months was 31% to 37% while the average life of
a product was 44 months (Mendelson and Pillai, 1999). In the auto industry in the
United States, new models are introduced in the third quarter each year (e.g. 2006
Chevrolet Malibu Maxx SS is in the market in Fall 2005). Usually, these new models
- with only slightly upgraded designs - replace the previous year’s models that are
phased-out. Dealers and car manufacturers use various tactics to clear the inventory
of older models by the end of a year while increasing the sales of new ones. And,
we are all familiar with the practice of day-old vs. fresh bagels at bagel shops. In all
these examples, the predominant challenge for a seller is managing demand streams
for different-aged, substitutable products.
Typically it is the supplier who makes a substitute product available through an
explicit offer or via product placement, but the customer’s decision as to whether to
accept the substitution, possibly with some sort of compensation. One observes this
in several different contexts. For instance, certain critical surgeries and treatments
(trauma surgery and platelet transfusions for chemotheraphy patients) specifically ask
for blood products that have maximal freshness, thus creating demand for products
of different ages (Angle, 2003, Haijema et al, 2005a, 2005b, Cohen et al., 1981).
In case of shortage, a doctor may accept a slightly aged product if offered by the
blood bank. In the PC industry, where larger hard disks drive smaller hard disks
52
toward obsolescence, a PC manufacturer that sells direct to customers may install a
80-GB hard disk into a PC although the original request is for a 60-GB hard disk,
when the smaller hard disk is out of stock. In such a setting using a new product
as a substitute for an old (called manufacturer-driven substitution as in Chopra and
Meindl, 2001) is acceptable, but not vice-versa (at least without the consent of the
buyer). Sales incentives can entice a customer who is interested in a new model of a
car that is on backorder to buy an old one that is in stock. Mark-downs on grocery
goods close to their expiration dates can lead to a shift in demand of customers
that otherwise prefer longer shelf-life items. In fact, Tsiros and Heilman (2005) -
in empirical research - determine that consumers’s willingness to pay decreases as
the expiration date nears for various grocery products and their findings support the
common practice of discounting grocery goods that are aging in order to induce a
purchase.
In this chapter, we focus on managing inventories of goods that lose value over
time where there is demand for products of different ages. We focus on the supplier’s
decision of replenishing the inventory and investigate whether or not he/she benefits
from substitution. A supplier can often leverage some of the operational advantages of
substitution, such as risk pooling and higher inventory turnovers. Yet, despite these
advantages, it is not straightforward to analyze the overall effect of substitution in
a general setting over an extended horizon. What complicates this analysis is that
substituting items of different ages may entail direct substitution costs for the supplier
(e.g. rebates), and have additional, possibly unforeseen, future effects on the inventory
of goods on hand (depending on the supplier’s replenishment policy). Thus evaluation
of the long term, overall effect of substitution for products that lose value over time
remains an almost entirely open question. We focus on this question, illuminating
the factors that influence supplier’s performance in terms of the cost of inventory and
the freshness of goods in stock. We investigate when and how the supplier can benefit
from substitution, possibly at additional cost. Since the effect of substitution cannot
be assessed without considering the policy the supplier uses to replenish inventory,
we consider two simple and practical replenishment policies: Order-up-to policies
based on either the total amount of inventory of all ages, or the amount of new items
only. Our goal is to shed light on what cost parameters and demand characteristics
influence a supplier’s performance.
53
Our contributions include the following: (i) We study two practical replenishment
policies, featured in literature and practice. (ii) Under these policies, we show that
substitution may not always be economically beneficial for the supplier even if sub-
stitution costs are zero. Likewise, we show that no substitution may not always be
economically beneficial even if substitution costs are very high. (iii) We determine
the conditions under which a supplier will indeed benefit from substitution. (iv) We
study how the average age of goods in inventory is affected by substitution. (v) We
perform computational work quantifying the sensitivity of our results to problem pa-
rameters. (vi) All our analytical results are based on sample-path analysis, hence
no assumptions, save for ergodicity, are made regarding demand: We allow demand
to be correlated over time, across products, or both, as is common in practical settings
involving substitutable products (if not the literature).
We first define the supplier’s problem and introduce the notation and the formu-
lation in Section 3.2. A brief review of related literature is presented in Section 3.3.
We analyze the supplier’s performance in terms of costs and freshness of goods in Sec-
tion 3.4. To complement our analytical results, we provide several numerical examples
in Section 3.5. Conclusions are discussed in Section 3.6.
3.2 Problem Definition and Formulation
In our problem, there is a single product with a lifetime of two periods; the value of
the product decreases deterministically as it ages, and there may be random demand
for both new and old items. A single supplier replenishes new items periodically, with
zero lead time. At the end of each period, any remaining old items are outdated, while
any unused new items become old. This setup captures the essence of the problem
for perishable goods such as produce and blood as well as durable goods where a new
product is introduced into the market while the old one is phased-out periodically.
We denote by Xni (i = 1, 2) the amount of product with i periods of lifetime remaining
at the beginning of period n. The different demand streams for the items of different
ages are denoted Dni (i = 1, 2). Demand is stochastic and may have arbitrary joint
distribution; dependence between demand for different ages and over time is allowed,
but demand is assumed ergodic and independent of supplier decisions (we discuss this
54
latter assumption further below).
In any period, demand is fulfilled as follows: All demand for new (old) items is
fulfilled from the inventory of new (old) unless there is a stock-out. In case of stock-
outs, the excess stock of new (old) items may be used to satisfy the excess demand of
old (new). We use the term downward substitution to denote the case where a new
product is sold to a customer that demands old; upward substitution is the reverse.
We assume only a fraction, denoted 0 ≤ πD ≤ 1 and 0 ≤ πU ≤ 1, of customers accept
downward and upward substitution, respectively. Throughout the analysis, we assume
any such substitution occurs only as a recourse.1 Our model captures the situation
in which the supplier is the one making the substitute offer; this is supplier-driven
substitution as is common in business-to-business situations and is well-studied in
the literature on non-perishables (see Bitran et al., 2006, and the references therein).
On the other hand, our model likewise captures the scenario in which the consumer
is the one considering the substitute option without explicit intervention (except
product placement) by the supplier. This is the same as inventory-driven substitution
where the demand of one product shifts to another in case of a stock-out. This form
of substitution is common in retail. For more information on the classification of
substitution and related research in inventory management, see Bitran et al. (2006).
Note that the examples mentioned in the previous section (blood bank, PC man-
ufacturer, retail car sales, bagels) fit into this model. This model also captures other
phenomenon: If inventory is depleted starting from the newest items (e.g. buyers
prefer longer shelf-life items as in grocery and the freshest produce/bread), this cor-
responds to having demand for new items only and fulfilling the demand by upward
substitution (i.e. customers consider buying less fresh products only after freshest is
sold). If instead inventory is depleted in a FIFO fashion (oldest item issued first)
as is the assumption in classical research on perishable goods (see the review in the
next section), then there is demand for only old items and demand is fulfilled using
downward substitution. Thus, our model subsumes many of the previous models in
the literature.
Furthermore, we assume any demand unsatisfied at the end of a period is lost (but
backordering new item demand does not change our results). Based on this setting,
1We discuss this assumption further in Section 3.4.4.
55
we can define the following quantities at the end of period n, where x+ denotes
max(x, 0):
dsn = min{πD(Dn1 −Xn
1 )+, (Xn2 −Dn
2 )+} is the downward substitution amount;
usn = min{πU(Dn2 −Xn
2 )+, (Xn1 −Dn
1 )+} is the upward substitution amount;
Ln1 = [(Dn
1 −Xn1 )− dsn]+ is the amount of lost sales for old items;
Ln2 = [(Dn
2 −Xn2 )− usn]+ is the amount of lost sales for new items;
On = [(Xn1 − Dn
1 ) − usn]+ is the amount that outdates at the end of the period
and
Xn+11 = [(Xn
2 −Dn2 )−dsn]+ is the amount of inventory carried to the next period.
We consider two practical replenishment policies for the supplier:
TIS (Total-Inventory-to-S): At the beginning of every period the total inventory
level (old plus new items) is brought up to S. That is, for all n,
Xn1 + Xn
2 = S. (3.1)
NIS (New-Inventory-to-S): At the beginning of every period S new items are or-
dered. Hence, for all n,
Xn2 = S. (3.2)
The TIS policy is simple, common and has a long tradition in the literature.
NIS has not attracted as much attention as TIS from a research perspective, mainly
because of its inferior performance in earlier simulation studies (see the next section
for the literature review). However, NIS has remained in use in practice due to its
simplicity (Prastacos, 1984, Angle, 2003). Note that the order-up-to level S can be
determined by optimization or by practical rules of thumb in either replenishment
policy.
In each period, the order of events is as follows: First inventory is replenished,
then demand occurs and is fulfilled, with substitution occuring only as a recourse.
Unsatisfied demand for both new and old items are lost. Stocks are aged; new goods
become old and old goods perish. Finally, profits are assessed and the supplier places
the order that will arrive in the next period.
56
The supplier’s profit function involves revenues associated with sales, inventory
related costs and revenue/cost terms associated with substitution. There is a unit
revenue ri for age i product and we assume r2 ≥ r1; i.e. newer items are at least as
valuable as older ones. These revenues are obtained only if the demand is fulfilled
without substitution. In case of a stock-out of old items, a customer asking for an
old item which is priced at r1 may buy an unsold new item at the price of r1 + α′D
where α′D ≥ 0; a new product made available as a substitute is never priced below
the old one. Similarly, in case of upward substitution, a customer asking for a new
item which is priced at r2 will buy an unsold old item at the price of r2 − α′U where
α′U ≥ 0.
There are also explicit costs associated with upward or downward substitution:
The supplier incurs a downward substitution cost of α′′D per unit and an upward
substitution cost of α′′U per unit. These are in addition to the revenue loss captured by
α′D and α
′U , and may represent additional production or shipping costs for the supplier
(e.g. shipping goods from different locations), the loss of goodwill, or a “demand
diversion” cost for the supplier2. In addition, there is a unit loss of goodwill cost of
p′i for age i product in case of lost sales. There is also an inventory carrying cost of
h per unit, and an outdating cost of m per unit.
Based on these parameters, the long-run time-average profit of the supplier as a
function of the order-up-to level S (either using TIS or NIS) when the system starts
with 0 ≤ X12 ≤ S new and 0 ≤ X1
1 ≤ S old items is :
R(S) = limT→∞
1
T
T∑
n=1
r2 min(Dn2 , Xn
2 ) + r1 min(Dn1 , Xn
1 ) + (r2 − α′U) usn + (r1 + α
′D)dsn
− hXn+11 − p
′1L
n1 − p
′2L
n2 −mOn − α
′′U usn − α
′′D dsn.
In the above formulation, the only difference between TIS and NIS is in the inventory
recursions regarding the inventory levels of new items (see equations (3.1) and (3.2)).
2Demand diversion takes place in the following way: Customers may request old items in hopes
of receiving new ones, if they realize the supplier is practicing downward substitution. Thus, the
supplier’s substitution decisions may affect demand, in this case, causing some demand shift from
new demand to old demand, which is likely to decrease the supplier’s profits. While our work does
not explicitly model this (or other related) phenomena linking substitution decisions with future
demand, our substitution costs may be taken as a high-level proxy for this.
57
Rearranging the terms, the profit function can be simplified:
R(S) = limT→∞
1
T
T∑
n=1
r2(Dn2 − Ln
2 ) + r1(Dn1 − Ln
1 ) (3.3)
− hXn+11 − p
′1L
n1 − p
′2L
n2 −mOn − (α
′U + α
′′U) usn − (α
′′D − α
′D) dsn
= limT→∞
1
T
T∑
n=1
r2Dn2 + r1D
n1
− hXn+11 − p1L
n1 − p2L
n2 −mOn − αU usn − αD dsn
where αU = α′U + α
′′U is the net cost of upward substitution, αD = α
′′D − α
′D (which
can be negative) is the net cost of downward substitution and pi = ri + p′i is the net
cost of lost sales for i = 1, 2.
Note that analysis of the above profit function only requires analysis of an appro-
priately defined cost function, as the first two terms in (3.4) are fixed. We define C(S)
to be the long-run time-average cost of the supplier as a function of the order-up-to
level S:
C(S) = limT→∞
1
T
T∑
n=1
hXn+11 + p1L
n1 + p2L
n2 + mOn + αU usn + αD dsn . (3.4)
In the rest of the chapter, we analyze the benefit of substitution based on the cost
function C(S). We use time-average costs to avoid issues which may arise in taking
expectations, for example should stationary distributions fail to exist due to demand
characteristics. Note though that, the inventory position (Xn2 , Xn
1 ), under TIS or
NIS, forms an ergodic process (as demand is assumed ergodic); thus all costs can
be expressed as functions of the time-average distribution of the inventory, which
converges. In addition to the economic benefit, we also study the effect of substitution
on the freshness of goods in stock. Freshness is defined as the long-run time average
age of goods in stock which can easily be determined using the inventory levels Xn1
and Xn2 ; discussion on this topic appears in Section 3.4.5.
Note that the formulation above allows for both upward and downward substitu-
tion modified by the fraction of customers accepting the substitution offers. However,
as discussed in the previous section, different forms of substitution are observed in dif-
ferent industries. While downward substitution is very common (e.g. a newer product
is usually an acceptable choice if one needs hard-drives, cars, blood, bakery products,
fresh produce etc.), upward substitution can be practiced only in certain cases and
58
then possibly only by using sales incentives (e.g. end of year clearance of old models in
car dealers, mark-downs of grocery goods closer to their expiration dates). In order to
investigate the relative benefit of different substitution patterns for the supplier, we
consider four specific cases: The No-Substitution case (denoted N ) is the base case
where items of a certain age are used to fulfill the demand for products of only that
age. The inventory levels and total cost (or profit) of this policy are easily computed
in our model by setting πU = πD = 0. On the other extreme is Full-Substitution, (de-
noted F), where both downward and upward substitution can take place; this is the
general model introduced above with πD > 0, πU > 0. There are two restricted cases
which consider only one-way substitution: The Downward-Substitution case, denoted
D, is when πU = 0 and Upward-Substitution case, denoted U , is when πD = 0.3
Our main research question is: Under what conditions one would prefer one form
of substitution - coupled with a practical replenishment policy - over the others? We
first provide partial answers to this question in Section 3.4. We identify conditions on
cost parameters such that supplier-driven or inventory-driven substitution results in
(i) lower costs for the supplier and (ii) fresher goods for the customers; our analytical
results are based on pairwise comparison of substitution options given a replenishment
policy. In Section 3.5, we present results of extensive computational experiments and
provide guidelines on when a replenishment policy and substitution option dominates
the others based on demand characteristics and values of cost parameters.
3.3 Related Work in the Literature
Most of the work on inventory problems for perishable goods focuses on optimal or
near-optimal ordering policies to minimize operating costs when the demand is in-
different to the age of the product and is fulfilled according to FIFO (oldest items
issued first), hence ignoring substitution issues. We refer to this specific model as
single demand stream as opposed to the two demand streams in our model. Even
in the simplified single demand stream setting, the problem is very difficult: Unlike
3Throughout the paper “Upward-Substitution” (or U) and “Downward-Substitution” (or D)
denote the specific case as introduced here. By “downward substitution” or “upward substitution”
we mean the substitution event itself.
59
standard inventory control theory, where generally the only information needed is
the inventory position, the optimal ordering policy for perishables requires informa-
tion about the amount of inventory of every age. Therefore the state space (and
problem complexity) grows with the lifetime of the product. For two-period lifetime
problem, Nahmias and Pierskalla (1973) showed that the optimal order quantity is a
non-linear decreasing function of the state (on hand inventory); this was extended by
Fries (1975) and Nahmias (1975a), independently, to the general m-period lifetime
problem. The fact that the optimal policy requires solution of a multi-dimensional
dynamic program, which is computationally difficult, motivated exploration of heuris-
tic methods. Nahmias (1976) and Chazan and Gal (1977) proposed the fixed-critical
number (order-up-to) policy, in which orders are placed at the end of each period to
bring the total inventory summed across all ages to a specific level. A closed-form
method for the optimal order-up-to level of two-period lifetime problem was found
by Van Zyl (1964). (This corresponds to a special case of our model: There is no
demand for new items, demand for old is fulfilled with downward substitution, and
inventory is replenished according to the TIS policy.)
Still for a single demand stream, Nahmias (1976, 1977) and Nandakumar and
Morton (1993) show that order-up-to policies (TIS) perform very well compared to
other methods, including optimal policies; they develop and analyze heuristics to
choose the best order-up-to level. Cooper (2001) provides further analysis of the
properties of the TIS policy, while Nahmias (1978) shows that when there is a fixed
ordering cost, an (s, S) type heuristic is better than order-up-to policies. Tekin et
al. (2001) consider a continuous review model for perishables and study a modified
lotsize-reorder policy. More recent papers on inventory management of perishables
are by Ketzenberg and Ferguson (2006), and Ferguson and Ketzenberg (2006) - they
focus on information sharing in a supply chain. A summary focused on blood bank
supply chains is provided by Pierskella (2004). All works cited above consider a
single demand stream being fulfilled according to FIFO; FIFO is known to be optimal
for many fixed-life inventory problems with a single demand stream (Pierskalla and
Roach, 1972).
TIS has been shown to have good performance compared to optimal when inven-
tory is depleted using oldest items first (Nahmias, 1976, Nandakumar and Morton,
1993). Haijema et al. (2005a, 2005b) use a Markov Decision Process approach to de-
60
rive heuristics for replenishment of blood platelets under Full-Substitution. They find
that base-stock heuristics that replenish based on total stock in system do well (TIS),
and those that simultaneously take into account total stock and new stock in system
(a TIS-NIS hybrid) do even better in a multi-period lifetime (5 to 7 days) setting.
First proposed by Brodheim et al. (1975), NIS was shown to have inferior performance
compared to TIS, with the earlier demand models and inventory replenishment rules.
Nahmias (1975b) compared multiple heuristics, including the “optimal” TIS policy, a
piecewise linear function of the optimal policy for the non-perishable problem, and a
hybrid of this with NIS ordering. He found that the first two policies outperform the
third. In another simulation study, the cost of the NIS policy was found to be 2-5%
higher than that of the TIS (see Prastacos, 1984, and the references therein). These
observations practically ended the research consideration of NIS and its variants.
There are a few papers that model multiple types of customers or demand streams
for perishables. Nahmias and Pierskalla (1976) examine a system consisting of two
products: one with a lifetime of m periods, and one with an infinite lifetime. There
is a one-way substitution of the second product (non-perishable) for the first (perish-
able). The problem was motivated by the operation of a blood bank storing frozen
packed red cells. Parlar (1985) considers a perishable product that has two-periods of
lifetime, where a fixed proportion of unmet demand for new items is fulfilled by unsold
old items and vice-versa (in our framework this is Full-Substitution), but only for a
single period. His results do not extend to longer horizons. Goh et al. (1993) consider
a two-stage perishable inventory problem. Their model has random supply and sep-
arate demand streams modelled as Poisson process. They computationally compare
a restricted policy (similar to our No-Substitution option) and an unrestricted pol-
icy (similar to our Downward-Substitution option). Considering only shortage and
outdating costs they conclude that the unrestricted policy is less costly, unless the
shortage cost for fresh units is very high. Ishii et al. (1996) focus on two types of
customers (high and low priority), items of m different ages with different prices and
only a single-period decision horizon. High priority customers only buy the freshest
products, so the freshest products are first sold to the high priority customers and
the remaining items are issued according to a FIFO policy. They provide the optimal
ordering policy under a warehouse capacity constraint. Ferguson and Koenigsberg
(2004) study a problem in a two-period setting with pricing and internal competition
61
between new and old items.
There is another line of research related to our work: Inventory management of
substitutable products. Substitution and demand fulfillment features of our model
- specifically substitution taking place as a recourse and a fraction of customers ac-
cepting substitutes - are common assumptions in this literature.
McGillivray and Silver(1978) provide analytical results for two products with simi-
lar unit and penalty costs and unmet demand can be fulfilled from the other product’s
inventory (if there is any inventory left). Parlar and Goyal (1984) study two differ-
entiated products with a fixed substitution fraction and analyze structural properties
of the expected profit function. Their work is extended by Pasternack and Drezner
(1991) by adding revenue earned from a substitution (different from the original sell-
ing price), shortage cost, and salvage value. Rajaram and Tang (2001) subsumes the
existing models for the single-period two-product problem. They show that demand
substitution between products always leads to higher expected profits than the no-
substitution case, and provide a heuristic to determine the order quantities. We show
such a dominance relation need not hold in an infinite-horizon, perishable setting.
There are several other papers looking at single-period inventory problems with mul-
tiple products that are substitutable: Parlar (1988), Wang and Parlar (1994), Bassok
et al. (1999), Ernst and Kouvelis (1999) Smith and Agrawal (2000), Mahajan and
van Ryzin (2001), Avsar et al. (2002), Eynan and Fouque (2003). The reader can
refer to Netessine and Rudi (2003) or Bitran et al. (2006) and the references therein.
Although it is a crucial element in our model, the age of a product is irrelevant from
a substitution standpoint in this particular literature.
Moorthy (1984) developed a theory of market segmentation based on consumer self
selection; prior his work the theory on market segmentation assumed a firm decides
which market segment to sell without incorporating customer choice. Furthermore
his consumer self-selection approach enabled to model “cannibalization” and compe-
tition among firms. In our problem we, similarly, model different market segments
(demand streams) for old and new products but we do not model cannibalization or
competition explicitly (although substitution costs might be considered as a proxy for
such consumer based dynamics in a very general way). In terms of consumer choice
our model does not have any restrictions on upward substitution. That is, consumer
may refuse the upward substitution and our analytical results will not be affected.
62
However for downward substitution, we assume that customer and seller will always
find a price that both can agree upon, and the downward substitution will take place.
This price would probably be between the original selling prices of old and new items.
In our model we have a downward substitution cost that could be considered as a
result of the agreement between the parties.
In our work we do not directly deal with management of product variety however it
can be considered as a related research area as we have two substitutable products in
our problem. Chen et al. (1998) present a framework that helps a firm to position and
price a line of products capturing a variety of delivery cost models. They discuss how
the optimal product line can be obtained by finding a set of cross points. These cross
points determine how the heterogeneous customers are divided among the products.
In our model however, pricing of old and new products is not studied explicitly.
Yano et al. (1998) provide a comprehensive review of mathematical models on design
and pricing for a line of products. In this review the models are are divided into
categories based on manufacturing costs. Manufacturing costs nor pricing are not
the focus of our study; our focus is on other operational costs such as outdating or
substitution costs. Reader can refer to Ho and Tang (1998) and references therein for
more information on management of product variety.
Our study is first in the literature on substitution to introduce the term “dynamic
quality” . In the literature on substitution, quality of a product is static. For our
problem quality of a product is age (one dimensional). That is, a new product has
a higher quality than an old one. Since our model is an infinite horizon model, the
quality of product is dynamic. If a product is not sold its quality will change in the
next period (i.e. from old to new) making our model unique in the literature on sub-
stitution. We have a general cost structure (crucially, we include substitution costs)
and demand fulfillment structure, in which different forms of substitution are mod-
eled. Further, we analytically provide conditions guaranteeing dominance between
different forms of substitution over an infinite horizon. Our work has several unique
aspects: (i) We formally define and evaluate four substitution options for perishable
products (and, in general, for products that lose value over time). (ii) We compare
these different forms of substitution based on two measures (total cost and average age
of inventory in the system), providing analytical proofs or counter-examples for dom-
inance relations. (iii) Our results are free of distributional assumptions on demand,
63
save for ergodicity; demand can be auto-correlated or correlated with the product of
different age. (iv) We consider an infinite horizon problem as opposed to single period
or single replenishment models common in analysis of substitutable products. (iv)
The model with a single stream of demand and FIFO (or LIFO) inventory depletion
– standard in the literature on perishable goods – remains a special case of ours.
(v) We analyze two distinct, practical replenishment policies, and show that the NIS
policy which was known to have inferior performance compared to TIS, can in fact
be better with more general demand models and leads to more intuitive results under
substitution.
3.4 Effect of Substitution: Analysis of Economic
Benefit and Freshness
In this section, we identify conditions on cost parameters that guarantee that a specific
form of substitution is superior to others. The main question here is not which form
of substitution is best but when the practice of upward, downward or both forms of
substitution would indeed be beneficial. We show that reasonable parameter settings
exist that lead to one option being superior than another, in an almost sure sense.
We also identify parameter regions where no such dominance conditions can exist.
Throughout, we use the notation  to denote dominance between two substitution
options: F Â D means that for any ordering level S the time-average cost of F is no
more than that of D in an almost sure sense, where these costs are defined according
to equation (3.4). All of our cost comparisons are based on key results regarding
inventory levels of different substitution options, when a common order-up-to level S
is used. Since the pairwise cost-dominance results do not change if optimal order-up-
to levels are used, the choice of S does not influence our cost comparison results.
Our analytical results on the economic benefit of substitution rely on two condi-
tions. First, the dominance results are valid for p2 > p1. This assumption is not
restrictive: We defined the net cost of lost sales as pi = ri + p′i and assumed r2 ≥ r1
in our model. In many practical cases, the newer item is more valuable (it is the pre-
mium product), hence it is priced higher and bears a higher loss of goodwill. Second,
all the customers accept downward substitution whenever offered, i.e. πD = 1 for
64
options F and D. Although this is a special case of our model, it is not unrealistic:
It is highly likely that the substitute will be accepted when a supplier offers a newer
item (the premium product) with a discount as a substitute for the old. We later
provide numerical results on cases where this latter condition is relaxed. None of
these conditions are required for our results on freshness in Section 3.4.5.
3.4.1 The economic benefit of substitution: Overview of re-
sults
We first provide an overview of our results on cost comparisons before going into
a detailed mathematical analysis. We seek answers to two main questions: Given
an order-up-to level, inventory related costs, substitution related parameters, and
under any demand stream, “can the supplier be better off without substitution?” and
“can the supplier be better off by practicing full- or restricted forms of substitution?”
Pairwise comparison of substitution options provide the answers.
Consider N vs. U : Note that upward substitution uses an unsold old item that
would otherwise outdate, to satisfy the excess demand of a new item. For every item
substituted, the supplier saves outdating (m) and net lost sale (p2) costs, but incurs
the net substitution cost (αU). Therefore, we prove that upward substitution provides
value when:
αU ≤ m + p2. (3.5)
This result is intuitive; the supplier benefits from not allowing substitution (i.e. N ÂU) regardless of πU when the substitution cost is higher than a critical level. Note
that this observation is true for both replenishment policies TIS and NIS.
One would expect a similar result for N vs. D. In downward substitution, an
excess new item is substituted to satisfy the excess demand for old items. Intuitively,
if the substitution cost is low enough, D should be beneficial, and if it is too high,
supplier should be better off with N . However, our analysis below shows that this
need not be the case when supplier replenishes inventory using TIS: Even if αD →∞,
N can be worse than D under TIS, and if αD = 0, D can be worse than N . This is
a significant result since, based on our discussion in the previous sections, D is the
most common form of substitution that takes place in practice.
65
Our results for all pairwise comparisons are summarized in Table 3.1 for the sake of
completeness. As discussed above, under NIS and TIS, αU ≤ m+ p2 is necessary and
sufficient for upward substitution to result in lower costs but downward substitution
is different: The benefit of downward substitution under general demand scenarios
often cannot be established for TIS, while intuitive conditions on αU and αD do exist
under NIS. For instance, if the net downward substitution cost exceeds the sum of
holding, outdating and lost sales cost for old items (the worst thing that can happen
if you choose not to substitute), D can be ruled out under NIS but not under TIS.
Note also that in many, but not all cases, the conditions for policy dominance are
subsets of each other, for example those for D to dominate N and U to dominate Ntogether form the conditions for F to dominate N under NIS.
We summarize some of the other key takeaways from our results: (i) If the supplier
is using TIS, downward substitution reduces costs only if αD ≤ p1− p2, requiring the
downward substitution cost to be negative (which is possible by definition of αD);
even if the downward substitution cost is 0, it is not guaranteed that the supplier
will be better off offering downward substitutes. (ii) On the other hand, even if
downward or upward substitution costs are very high (αD → ∞ and/or αU → ∞),
avoiding substitution is not guaranteed to be the best action for the supplier under
TIS – i.e. conditions do not exist to guarantee the dominance of the no-substition
policy (N ) over the full-substition policy (F) no matter how large the substitution
costs may be. (iii) If the supplier is using NIS, downward substitution reduces costs if
the downward substitution cost is less than the holding cost (αD ≤ h); and downward
substitution should be avoided if the downward substitution cost is more than the
sum of the penalty cost for old items, the outdating cost and the holding cost (αD >
m+h+p1). In case of supplier-driven substitution, these results show when a supplier
benefits from offering a substitute product to a customer. In case of inventory-driven
substitution as in retail, these results can be interpreted as to when a supplier should
make choices (e.g. product placement, pricing) to encourage or discourage customers
to practice substitution.
TIS has attracted nearly all the attention in the literature on perishable goods, and
it is known to be an effective replenishment policy for a special case of our model (only
old items are demanded and downward substitution is employed with no substitution
costs). However, our analysis reveals that it may have counter-intuitive performance
66
Dominance Condition(s) under TIS Condition(s) under NIS
F Â D, U Â N αU ≤ m + p2 αU ≤ m + p2
D Â F , N Â U αU ≥ m + p2 αU ≥ m + p2
F Â N αU ≤ m + p2 αU ≤ m + p2
αD ≤ p1 − p2 αD ≤ h
N Â F Does Not Exist αU ≥ m + p2
αD ≥ m + h + p1
F Â U αD ≤ p1 − p2 αD ≤ min{h, h + αU − (p2 − p1)}U Â F Does Not Exist αU ≤ m + p2
αD ≥ m + h + p1
D Â N αD ≤ p1 − p2 αD ≤ h
N Â D Does Not Exist αD ≥ m + h + p1
U Â D Does Not Exist αU ≤ m + p2
αD ≥ m + h + p1
D Â U αU ≥ m + p2 αU ≥ m + p2
αD ≤ p1 − p2 αD ≤ h
Table 3.1: Sufficient conditions on substitution costs that ensure dominance relations
between substitution options
when demand is differentiated by the age of the product, and substitution takes place
with additional costs. In contrast, NIS - which was deemed inferior in earlier studies
- has more predictable behavior under substitution.
All the conditions listed in Table 3.1 hold for 0 ≤ πU ≤ 1. Note that some of the
bounds we obtain are quite “conservative” because we are seeking strong, almost sure
results, over the entire region of πU values. For instance, the bound αD ≤ p1−p2 that
is needed for F to dominate U and N under TIS, is restrictive; a weaker condition
on αD exists when πU = 1 (see Lemma 5 and 8 in Section 3.4.2). Likewise, when αD
67
satisfies h < αD < m + h + p1, no dominance relation between D and N is provided
under NIS. In Section 3.5, we use computational experiments to investigate the effect
of values of cost parameters for which dominance relations are not established, as well
as values of πD < 1 (recall that the above results rely on the property that downward
substitution is always accepted; i.e. πD = 1 in F and D).
Next, we show how the conditions in Table 3.1 are obtained. We provide the
results on TIS in Section 3.4.2, and on NIS in Section 3.4.3.
3.4.2 Economic Benefit of Substitution under TIS
A deeper understanding of the effect of substitution on stock levels is needed to
establish the cost-dominance conditions. The first key observation is the following
proposition: Along any sample path of demands within either TIS or NIS, F and Dhave equal amounts of stock in any period for a given S, and so do U and N . We
provide this and other important results below.
Preliminary Results under TIS
Proposition 1 For TIS or NIS, F and D have identical inventory levels in any
period for a given S under the same sample path of demands, so do U and N .
For any given S, the order quantities (TIS or NIS) are independent of the upward
substitution decision under identical demand. The reason is as follows: In any period
if upward substitution is done, it will not affect the ordering decision in the next
period as the old item inventory level for the next period will be zero. In case upward
substitution is not done any unsold old items will perish. Therefore the choice of
practicing upward substitution does not affect the inventory levels for the next period
as long as the downward substitution policy and the ordering policy (TIS or NIS) are
the same. Thus F and D are identical in terms of inventory levels; so are U and N .
Proof
XnN = S − (Xn−1
N −Dn−12 )+ > S − [(Xn−1
F −Dn−12 )+ − (Dn−1
1 − S + Xn−1F )+]+ = Xn
F
68
(Xn−1N −Dn−1
2 )+ < [(Xn−1F −Dn−1
2 )+ − (Dn−11 − S + Xn−1
F )+]+
< (Xn−1F −Dn−1
2 )+.
(Xn−1F − Dn−1
2 )+ must be greater than zero in order to satisfy the last inequality
above, implying Xn−1F > Xn−1
N . •
When TIS is employed, we have Xn1 = S −Xn
2 in each period n. This simplifies
the analysis of the cost function as it is sufficient to track only Xn2 . We use the
following notation for ease of exposition: XnI denotes the stock level of new items at
the beginning of period n, i.e. Xn2 under option I for I = F ,N ,D,U .
Proposition 2 Along any sample path of demand and for the same S, if XnI < Xn
J ,
then Xn−1I > Xn−1
J for I = F ,D, J = U ,N .
Proof We only provide the proof for I = F and J = N ; the others follow from
the equivalence of the inventory recursions. XnF and Xn
N are both non-negative by
definition.
XnN = S − (Xn−1
N −Dn−12 )+ > S − [(Xn−1
F −Dn−12 )+ − (Dn−1
1 − S + Xn−1F )+]+ = Xn
F
(Xn−1N −Dn−1
2 )+ < [(Xn−1F −Dn−1
2 )+ − (Dn−11 − S + Xn−1
F )+]+
< (Xn−1F −Dn−1
2 )+.
(Xn−1F − Dn−1
2 )+ must be greater than zero in order to satisfy the last inequality
above, implying Xn−1F > Xn−1
N . •
Proposition 2 states that along any sample path, if the inventory of new items
under option F (or D) is less than the inventory of new items under option N (or U),
then in the previous period the new items in F (or D) must have been more than the
new items in N (or U). Using this fact we can divide the horizon into two classes of
periods: Pairs of periods n− 1 and n for the case when XnF < Xn
N ; and periods not in
these pairs (i.e. runs of consecutive periods in which XkF ≥ Xk
N , for k, k + 1, ...). In
the remainder of the paper, we use the term pair to denote two consecutive periods
with the above property. We will refer to a period with more new items under no-
substitution as N , with more new items under full substitution as F , and otherwise
69
(when they are equal) as E. Therefore FN denotes a pair. A sequence of periods
could look like this, with the “pairs” bracketed:
...FEF︷︸︸︷FN
︷︸︸︷FN FF
︷︸︸︷FN FFEEF...
Note that due to Proposition 2 there are never two consecutive Ns. For future
reference we call the periods between two E periods a cycle. If an E period follows
another one, then it is called a trivial cycle. A non-trivial cycle, starts with an E
period in which downward substitution takes place (making the next period an F ) and
ends with another E. An E period is not only end of a cycle but also the beginning
of the next cycle –trivial or not.
We next provide two more results which are used later in our comparative analysis.
See the Appendix for the proofs.
Proposition 3 If an F period follows another F period, then in the first one there
must be downward substitution.
Proposition 4 Suppose an F is followed by another F in periods n and n + 1.
Let ∆n be the amount of downward substitution in period n. Then, Xn+1F −Xn+1
N =
∆n − (XnF −Xn
N).
Note that the notion and results on FN -pairs easily extend to DN -pairs where
D denotes a period for which D has more new items compared to N . This is because
of the equivalence in inventory recursions of F and D.
Pairwise comparison of substitution options under TIS
We first investigate the (additional) value of upward substitution. Note both F and
U employ upward substitution whereas D and N do not. The next result is a formal
statement of our observation from Section 3.4.1.
Lemma 1 On every sample path and for any πU ∈ [0, 1], F Â D and U Â N , if
and only if condition (3.5) holds.
70
Proof Inventory levels of D and F are defined by the same recursions. Thus their
costs that are independent of upward substitution (i.e. those related to h, p1 and αD)
are equal. Therefore F has a lower cost than D if and only if upward substitution is
profitable, i.e. when αU ≤ m + p2. This remains true for all πU ∈ [0, 1]. The same
argument holds for U vs. N . •
The next question is whether downward substitution – the most common form
of substitution in practice – is beneficial. The following inequality ensures that D is
more profitable than N :
αD ≤ p1 − p2. (3.6)
Below is the formal statement of this result. Choice of πU is not relevant here since
neither D nor N practice upward substitution. See the discussion in Appendix for
the proof.
Lemma 2 If condition (3.6) holds, D Â N .
Condition (3.6) is obtained via case-by-case analysis of inventory levels and cor-
responding costs. The analysis makes use of the inventory recursions under TIS and
the notion of pairs. The condition requires the net substitution cost to be negative
since p1 < p2. Although restrictive, it is satisfied when the customer is willing to
pay the full price for the substitute, and the additional cost of substitution (including
ill-will due to substitution) and lost sales costs are negligible (i.e. α′D = r2 − r1 and
α′′D = p
′1 = p
′2 = 0).
The above condition is sufficient, but not necessary, for D to dominate N . We now
explore necessity: What happens when this condition is violated; is there a critical
value of αD, beyond which N dominates D? Our next result provides the answer.
Lemma 3 When m + p1 > 0, there is no condition on αD that guarantees N Â D.
Proof We provide a counter-example. There are no old items in stock, and the
inventory state (Xn2 , Xn
1 ) for both D and N is initially (S, 0). Consider a sam-
ple path of demand for new and old items {(Dn2 , Dn
1 ), n = 1, 2, ...}, where the se-
quence {(S − ε, S), (0, 0), (0, S), (0, 0), (0, S), (S, 0)} repeats every six periods for any
71
0 ≤ ε ≤ S. The resulting inventory levels at the beginning of each of the next six pe-
riods are {(S, 0), (0, S), (S, 0), (0, S), (S, 0), (S, 0)} for D and {(S−ε, ε), (ε, S−ε), (S−ε, ε), (ε, S − ε), (S − ε, ε), (S, 0)} for N . Note that the six periods starting with the
initial period, when both policies are at (S, 0), form an EDNDND sequence, such
that E indicates equal amount of new items in a period under both D and N , D
(N) indicates there are more new items under D (N ). In this sequence, there are
two pairs, and the inventory in the seventh period is the same as in the initial period
(an E). Let Ci, i = D,N denote the total cost of policy i in this cycle. The cost
difference is CD − CN = ε[αD − (m + p1)(J + 1) − h − p2], where J is the number
of pairs in the cycle (J = 2 in this example). By appropriately selecting a demand
stream, one can construct a sample path where J is arbitrarily large. Thus CD −CN
can always be made non-positive, independent of the magnitude of αD and πU , so
long as m + p1 > 0. •
Lemma 3 is counter-intuitive, it says that even when the downward substitution
cost is very high (αD → ∞) there can be a benefit to using substitution (or more
precisely substitution cannot be ruled out almost surely). While extreme, this exam-
ple illustrates real potential dangers of using the strict replenishment rule TIS with
policy N when demand is intermittent or periodic: these factors can result in a chain
reaction of inventory carrying, outdating and lost sales.
The next question is whether the supplier benefits from offering both upward and
downward substitution. The overall benefit of F over N , depends on both αD and
αU . Combining our observations so far, we have the next result. The proof is in the
Appendix.
Lemma 4 If (3.5) and (3.6) hold, F Â N for any πU ∈ [0, 1].
A weaker sufficient condition exists for F Â N considering the special case πU = 1,
i.e. upward substitution is always accepted. The proof is in Appendix.
Lemma 5 If (3.5) and
αD ≤ min{m + 2p1 − p2, m + p1 − αU ,h + p1
2} (3.7)
hold, then F Â N for πU = 1.
72
We provide an intuitive explanation for condition (3.7): Using downward substi-
tution, a vendor saves h + p1 and incurs αD for every item substituted in the period
preceding a pair. (Such a substitution is necessary to form a pair.) Based on this
fact, conditions
αD ≤ m + 2p1 − p2 (3.8)
and
αD ≤ m + p1 − αU (3.9)
arise as follows: If demand is low in the first period of the pair, N incurs m−h more
cost than F per item substituted upward, because N has fewer new items in the first
period of the pair. In the second period of the pair, N has more new items, which may
give N a benefit of p2− p1 per item (if F cannot substitute upward), or αU per item
(if F can substitute upward). Therefore if h + p1−αD + m− h− (p2− p1) is positive
(corresponding to the first case) and h+p1−αD+m−h−αU is positive (corresponding
to the second case), then N is more costly than F under either situation. Note that
(3.5) is redundant when (3.9) holds, as we assume p1 < p2.
Given the immediate cost difference due to downward substitution, if αD ≤ h+p1
holds then - intuitively - F should be less costly. However,
αD ≤ h + p1
2(3.10)
is in fact required. The explanation for this is as follows: Downward substitution
causes F to have fewer old items (and more new items) than N in the subsequent
period, which may lead F to substitute downward again in this next period, incurring
upward substitution cost αD twice in the same pair.
Again, the above conditions are sufficient but not necessary for F Â N . However,
given that F and D and have the same inventory recursions, our next result is not
surprising.
Lemma 6 When m + p1 > 0, for any value of πU ∈ [0, 1], there is no value of αD
or αU that guarantees N Â F .
73
Proof Follows from the example provided in the proof of Lemma 3. •Based on this last result, full substitution may be better than no substitution re-
gardless of how high the upward and downward substitution costs are. Again, this
counter intuitive behavior is partly due to the use of TIS, which constrains reordering
behavior.
Next, we explore the additional value of downward substitution when combined
with upward substitution: When is F better than the restricted form U? The proof
of the next result is in Appendix.
Lemma 7 If condition (3.6) holds then F Â U for any πU ∈ [0, 1].
For the special case where πU = 1, we have a different result:
Lemma 8 If conditions (3.5) and (3.7) hold, then F Â U for πU = 1.
Proof This follows from Lemmas 5 and 7. •
Similar to our analysis of F vs. N , the above results provide only sufficiency
conditions; there is no guarantee that U will dominate F even when they fail to hold.
Lemma 9 When m + p1 > 0, for any value of πU ∈ [0, 1], there is no condition on
αD or αU that guarantees U Â F .
Proof The proof is identical to the proof of Lemma 3 because U and N have the
same cost (as the recursions that define their inventory levels are the same) and so
do F and D. •
Finally, one-way dominance relation between D and U is as follows:
Lemma 10 The conditions αU > m + p2 and (3.6) guarantees D Â U . However
when m + p1 > 0, there is no condition on αD that guarantees U Â D for any value
of αU .
Proof Follows from results in Lemma 1 and Lemma 2. •
74
3.4.3 Economic Benefit of Substitution under NIS
In this section we continue our analysis, comparing substitution policies under the
NIS policy. We begin again with a preliminary result showing that there are fewer old
items in stock if downward substitution is practiced (Proposition 5), before moving to
pairwise comparison of the policies (Lemmas 11 - 15). Since Xn2 = S for all n under
NIS, we track Y nI , denoting the stock level of old items at the beginning of period n
(i.e. Xn1 ) under policy I, for I = F ,N ,D,U .
Proposition 5 For a given S, Y nF = Y n
D ≤ Y nU = Y n
N for all n along any sample
path. (This result also holds for general πD ∈ [0, 1].)
Proof Due to the recursions that define inventory levels, we know Y nF = Y n
D and
Y nU = Y n
N . If there is no downward substitution in period n, then Y nF will be equal
to Y nN . In case of any downward substitution, there will be fewer old items carried to
the next period under F (or D), i.e. Y nF < Y n
N . Therefore Y nF = Y n
D ≤ Y nU = Y n
N . •
The result is intuitive: under NIS, no substitution or only upward substitution
result in higher levels of old item inventory, since inventory of new is never used to
satisfy excess demand of old. Next, we look at the effect of upward substitution. Just
as in the case for TIS, upward substitution is profitable if and only if the upward
substitution cost, αU , is less than m + p2, which is the cost if upward substitution is
possible but does not take place in a given period.
Lemma 11 On every sample path and for any πU ∈ [0, 1], F Â D and U Â N , if
and only if condition (3.5) holds.
Proof We provide the proof for F vs. D only; the same idea applies to U vs.
N . Due to inventory recursions, we know Y nF = Y n
D for all n. F and D have
the same cost for every period except for the periods where F has (accepted) up-
ward substitution. In any period n, if Dn1 < Y n
F and Dn2 > S, then upward sub-
stitution is offered in the amount of us = min{L, K}, and is accepted in some
amount usn ≤ us, where L = Dn2 − S and K = Y n
F − Dn1 . Let Cn
I denote the
cost of policy I in period n, for I = F ,D,U ,N . Then, we have CnD = mK + p2L ,
75
CnF = m(K−usn)+p2(L−usn)+αU usn and Cn
D−CnF = (m+p2−αU)usn. Therefore
D is costlier than F if and only if αU < m + p2. •
Next, we look at the (additional) value of downward substitution. Unlike TIS, the
decision to use an excess new item in one period does not affect the number of new
items in the next period under NIS. Therefore, the effect of substitution is easier to
determine.
Consider two consecutive periods; in the first downward substitution occurs. For
each new item substituted for old, there is a cost αD. In the same period, the lost
sales cost p1 is saved, and one fewer item is carried to the next period, saving h. The
greatest benefit that can be accrued from the downward substitution occurs if the
demand for old items in the next period is low – one fewer item outdates, saving m.
Thus, when
αD ≥ m + h + p1 (3.11)
holds, downward substitution is never beneficial.
Now suppose, αU > m + p2 holds (that is upward substitution is unprofitable,
by Lemma 11). The worst thing that could happen after substitution is that an
additional old item might be needed later, incurring a cost of p1. If in this case
downward substitution is profitable, then it is always beneficial. This holds under the
following condition:
αD ≤ h. (3.12)
Finally, if upward substitution is viable (i.e. αU ≤ m + p2), in addition to the
previous case (3.12), it may happen that in the next period an upward substitu-
tion is desired but cannot be enacted because of our previous period’s downward
substitution. This will save a substitution cost, αU , but cost p2. If even in this
case substitution is profitable, then downward substitution is always beneficial, i.e.
provided:
αD ≤ min{h, h + αU − (p2 − p1)}. (3.13)
Formal statement of these results follow. Proofs are available in Appendix.
Lemma 12 (F vs. N ) If conditions (3.5) and (3.12) hold, then F Â N . If condi-
tion (3.5) fails to hold but condition (3.11) holds, then N Â F .
76
Lemma 13 (F vs. U) If condition (3.13) holds, then F Â U . If conditions (3.5)
and (3.11) hold, then U Â F .
Lemma 14 (D vs. N ) If condition (3.12) holds, then D Â N . If condition (3.11)
holds, then N Â D.
Lemma 15 (D vs. U) If conditions (3.5) and (3.11) hold, then U Â D. If condition
(3.5) fails and condition (3.12) holds, then D Â U . If condition (3.5) holds, there is
no condition that guarantees D Â U .
3.4.4 A note on substitution not being a recourse
We assume in our analysis that substitution is only used as a “recourse” action – new
items are used to satisfy new item demand and old items to satisfy old item demand
before any substitution takes place. As we assume p2 > p1, this will always be the
cost-optimal action with respect to new item inventory; i.e. new items should be used
to satisfy the demand for new items first. Likewise, if αU > p2 − p1, it is optimal to
use old items to satisfy demand of old items before offering them as substitutes. If
αU < p2−p1, then it is better to satisfy new item demand first using all the inventory.
Such an option does not change the sample path of the inventory in the system, so all
of our other results remain unchanged except that all of our comparison results with
respect to upward substitution (i.e. F vs. D, U vs. N ) would include this condition.
In fact, combining conditions αU ≤ m + p2 in equation (3.5) and αU ≤ p2 − p1, we
get the following: Using upward substitution to satisfy new item demand before old
item demand is beneficial if and only if αU ≤ p2 − p1.
3.4.5 Freshness of Inventory
Our focus so far has been on the economic benefit of substitution. However, the overall
benefit of a substitution - especially for the customers - cannot be assessed unless
service issues are also taken into account. In this section we show that downward
substitution leads to fresher inventory, which may help the supplier in a number of
less tangible ways: These items have not lost their value through aging, customers
77
are more likely to prefer them, and having newer items in stock decreases the risk
of obsolescence. Note that we showed in Sections 3.4.2 and 3.4.3 that this does not
necessarily lead to cost-wise superiority due to substitution costs.
We first define what we mean by freshness: We look at the expected age of goods
in stock – the greater the average remaining lifetime, the fresher the goods. No
assumptions on problem parameters are needed for our results on freshness. For NIS,
pairwise comparison of substitution policies with respect to freshness immediately
follows from our earlier results.
Lemma 16 For the same S, and any πD, πU ∈ [0, 1], average age of inventory
is lower (i.e. goods are fresher) under F and D than U and N when inventory is
replenished using NIS.
Proof Follows from Proposition 5 (there are less old items due to downward sub-
stitution). •
When inventory is replenished using TIS, we compare the average age of goods
in stock for different substitution policies by studying the amount of new items in
stock. This leads to our next result. Detailed discussion and the proof of the result
are provided in Appendix.
For an arbitrary period n, we know we can have XnF ≥ Xn
N or XnF < Xn
N based
on our previous results; we also know XnF = Xn
D and XnN = Xn
U (almost surely) for
all n. Below, we show in a time average sense that the inventory level of new items
is greater, i.e. items are fresher under F . The definition of a pair is the same as in
Section 3.4.2. Outside the pairs it is obvious that there are more newer items under
F . The proposition below analyzes the situation inside a pair.
Proposition 6 Suppose inventory is replenished using TIS. Let periods n− 1 and
n constitute a pair, and define dsn−1 to be the amount downward substituted in period
n− 1. Then, in period n− 1, (i) Dn−11 + dsn−1 < Xn−1
F , (ii) if there is substitution,
it must be downward substitution, and (iii) dsn−1 < ∆.
Proof See Appendix. •
78
Based on Proposition 6, we can show that the difference Xn−1F −Xn−1
N > 0 is no less
than XnN −Xn
F > 0 for a pair in periods n− 1 and n.
Lemma 17 For the same S, and any πD, πU ∈ [0, 1], goods are fresher under Fand D than N and U , when inventory is replenished using TIS.
Downward substitution is clearly superior in this respect: Under both replenish-
ment policies, it leads to fresher inventory. This confirms the value of this practice in
many industries; downward substitution will, at least indirectly, lead to benefits for
the suppliers and customers.
3.5 Computational Results
In this section, we present a series of experiments used to explore the performance of
our substitution policies for a number of parameter settings. We discuss properties
of the cost functions of different substitution policies, under both TIS and NIS in
Section 3.5.1. In Section 3.5.2 we discuss the effect of downward substitution cost
on the optimal substitution policy. We then study costs, service levels and freshness
in Section 3.5.3, and policy comparisons under proportional acceptance of substitute
goods in Section 3.5.4.
3.5.1 Non-convexity/Multi-modality of the Cost Function
In Examples 1A&1B we show the non-convexity and multi-modality of NIS and TIS.
In these examples common costs are h=1, p1=3, p2=9, αU=7, αD=6 and the demand
for new and old items are both discrete uniform distributed between 0 and 25. We
vary m; m = 5 in 1A and m = 2 in 1B. Note that condition αU < m + p2 is satisfied
in both examples; therefore F has lower cost than D, and U has lower cost than N ,
under both TIS and NIS. However αD does not satisfy any of the sufficient conditions
in Table 3.1 – there is no other TIS dominance relation among the substitution policies
holding for all S, as we see in Figures 3.1 and 3.2 for 1A and 1B, respectively.
We also see that the cost function under TIS for policies F and D are non-convex
in 1A; in 1B these functions become non-convex in the NIS case, and multi-modal
79
TIS
40
50
60
70
80
90
100
1 6 11 16 21 26 31 36 41
Order-up-to level (S)
Co
st
F D U N
NIS
40
50
60
70
80
90
100
1 6 11 16 21 26 31 36 41
Order-up-to level (S)
Co
st
F D U N
Figure 3.1: Time-average costs of substitution policies in Example-1A.
in the TIS case. In addition, while N and U have higher optimal order-up-to levels
compared to F and D under TIS, the opposite is true under NIS. Reducing the
outdating cost m lowers the overall costs of all policies, but leaves the optimal S
values relatively unchanged.
Overall, in both examples policies that do not downward substitute, i.e. U and Nperform best, owing to the high substitution costs and p2 >> p1. The optimal costs
of these policies are lower under NIS than TIS, although for any given S the cost of a
substitution policy under NIS is not necessarily lower than that of TIS. In addition,
these policy’s costs are convex with respect to S, raising the possibility that optimal
substitution policies possess convexity. Thus far we have been unable to prove such
a result.
TIS
20
30
40
50
60
70
1 6 11 16 21 2 31 3 41
Order-up-to level (S)
Co
st
F D U N
NIS
20
30
40
50
60
70
1 6 11 16 21 26 31 36 41
Order-up-to level (S)
Co
st
F D U N
Figure 3.2: Time-average costs of substitution policies in Example-1B.
80
TIS
24
28
32
36
40
1 6 11 16 21 26 31
Order-up-to level (S)
Co
st
F D U N
NIS
24
28
32
36
40
1 6 11 16 21 26 31
Order-up-to level (S)
Co
st
F D U N
Figure 3.3: Time-average costs of substitution policies in Example-2.
3.5.2 Making downward substitution more attractive
In Example-2 we make the downward substitution more attractive; that is we set the
old-item demand greater in magnitude than new item demand, and reduce substi-
tution costs: Specifically, we choose h=1, m=2, p1=3, p2=9, αU=3, αD=2. Again
αU < m + p2 is satisfied, while αD violates the sufficiency conditions in Table 3.1.
The demand for new items is discrete uniform between 0 and 6, demand for old items
is discrete uniform between 0 and 25. These changes make downward substitution
more attractive – it is both less costly, and can occur more frequently.
The expected cost functions for Example-2 are presented in Figure 3.3. (Note
that under NIS the costs of F and D and those of U and N coincide.) As we would
expect, the lowest cost is achieved by policies that downward substitute, F and D.
In addition, given the increase in downward substitution, TIS now provides both
lower cost than NIS, and greater robustness to choice of S. Our observations are
consistent with the literature; TIS is effective when there is only one demand stream
and demand is fulfilled in a FIFO fashion (i.e. downward substitution is practiced).
Example-3 (Effect of αD). We fix parameters h=1, m=5, p1=3, p2=9, αU=7,
and vary αD between zero and seven, plotting the cost at optimal values of S for each
value of αD . The demand for both new and old items is discrete uniform distributed
between 0 and 25. Note that upward substitution is beneficial since αU < m + p2.
Figure 3.4 shows the minimum expected cost of each policy as a function of αD; as
αD increases, the cost of policies F and D increase almost linearly under both TIS
and NIS.
81
NIS
20
30
40
50
60
70
0 1 2 3 4 5 6 7Downward substitution cost ( )
Min
imu
m C
os
t
F D U N
TIS
20
30
40
50
60
70
0 1 2 3 4 5 6 7
Downward substitution cost ( )
Min
imu
m C
os
t
F D U N
Figure 3.4: Minimum expected cost of policies as a function of downward substitution
cost (αD) in Example-3.
From Table 3.1, under both NIS and TIS, αD < 1 guarantees F Â N and F Â U .
Under NIS, αD > 9 guarantees N Â F and U Â F . Example-3 illustrates the
behavior of substitution policies in the parameter range where our sufficient conditions
do not hold. In this example, we observe a critical value of αD = 4 under NIS; for
higher values of αD, U andN have lower expected costs. This critical value is between
five and six for TIS. In either case, these values are in the mid-to-high portion of the
parameter ranges identified in our sufficient conditions. Thus, while our analytical
conditions of policies do not cover the entire αD range for dominance relations, they
can provide guidance, especially when αD is closer to one sufficient condition than
the other.
In all of the previous examples, when NIS is used to replenish inventory there
is almost no additional benefit to using upward substitution (over no substitution),
but when TIS is used this difference is pronounced. This is due to the fact that very
little upward substitution is taking place under NIS as compared to TIS, a fact borne
out in the next section. This may be due to the fact that the optimal base-stock
policy of NIS accounts directly for new item demands, reducing the need for upward
substitution. In contrast, in TIS upward substitution is needed more frequently;
because new items are ordered less, depending on the amount of old items.
82
D ∼ {0, 25} D ∼ U[0, 25] D ∼ P(12.5) D ∼ U[10, 15]
Policy NIS TIS NIS TIS NIS TIS NIS TIS
F 38.5 47.4 26.1 28.7 14.5 13.5 9.0 9.2
D 38.5 48.7 26.2 29.1 14.5 13.5 9.0 9.2
U 46.8 64.4 32.0 42.8 18.7 25.5 15.2 20.4
N 46.8 66.6 32.1 44.2 18.7 25.8 15.2 20.5
Table 3.2: Average cost comparison between TIS and NIS under four different demand
models.
3.5.3 Costs, Service Levels and Freshness of Inventory
In this section we present the results of our numerical study on the costs, service levels
and freshness of the policies. For each cost parameter and demand distribution, we
first compute the optimal order-up-to level S for each substitution and reordering
policy. Then, using this order-up-to level, we compute average performance measures
via simulation.
For the costs comparison the experiment is designed as follows: We set h = 1,
m ∈ {2, 5}, p1 ∈ {1, 3}, p2 ∈ {4, 9}, αU ∈ {3, 7, 11, 15} and αD ∈ {−1, 0, 2, 6} which
results in 128 instances based on cost parameters. Note, only some of these instances
satisfy the conditions identified in Section 3.4.
In our experiments, we evaluate TIS and NIS using their respective optimal order-
up-to levels. While the long-term average cost function C(S) in (3.4) is non-convex
in S for either of these replenishment policies (see the examples in Section 3.5.1), the
optimal order-up-to levels are easily determined using line search.
We first discuss the effectiveness of TIS and NIS. We present a high-level cost
comparison of TIS and NIS here. In Table 4.1, we report the average percentage
difference between TIS and NIS for each substitution scenario and demand distrib-
ution. The four demand distributions were: (i) Equal point mass at 0 and 25; (ii)
Discrete uniform demand between zero and 25; Poisson demand with rate 12.5; and
(iv) Discrete Uniform demand between 10 and 15. In all cases the demand for old
and new items are independent.
83
D ∼ {0, 25} D ∼ U[0, 25] D ∼ P(12.5) D ∼ U[10, 15]
Policy NIS TIS NIS TIS NIS TIS NIS TIS
F 17.8% 28.9% 18.6% 35.0% 22.5% 47.7% 40.8% 55.0%
D 17.8% 26.9% 18.4% 34.0% 22.5% 47.7% 40.8% 55.0%
U 0.2% 3.3% 0.4% 3.1% 0.0% 1.4% 0.0% 0.5%
Table 3.3: Average percentage improvement in costs over N .
From Table 4.1 we can see that in fourteen of the sixteen combinations of policies
and demand distributions, NIS, the policy that ignores the information on the old
item inventory, outperforms TIS, the policy that uses this information, often by a
considerable amount (the average difference can be as high as 30%). Moreover, for
the two instances in which TIS was superior, (i.e. for Poisson demand scenario), the
difference was only slight (less than 8 %).
Next we study the policies in terms of service levels, conducting 32 experiments
with the following cost parameters: h = 1, m ∈ {2, 5}, p1 ∈ {1, 3}, p2 ∈ {4, 9},αU ∈ {3, 7} and αD ∈ {2, 6}. Demand for new and old items is independent, and
both are discrete uniform distributed between 0 and 25. For each cost parameter
and demand distribution, we first compute the optimal order-up-to level S for each
substitution and reordering policy. Then, using this order-up-to level, we compute
average performance measures via simulation.
We study two aspects of service level: (i) the percentage of demand lost and
(ii) the percentage of demand satisfied via substitution. Table 3.4 summarizes our
results; the figures represent the averages across all 32 instances. Employing NIS as
the replenishment policy increases the service level and decreases substitution: NIS
not only tends to keep more inventory, but also better keeps the appropriate inventory
of new items, reducing substitution. This effect is significant because the demand of
old and new are both relatively high as compared to the optimal stocking levels in
this experiment.
In the same experiment, we also looked at freshness of inventory. We know that
policies that use downward substitution have fresher inventory for a given S; however,
when policies use different order-up-to levels (which is the case in our experiment),
84
Type of % lost % substituted Avg. Freshness Average Cost
Policy demand TIS NIS TIS NIS TIS NIS TIS NIS
F new 7.0% 4.6% 2.3% 0.3% 1.85 1.82 37.0 39.9
old 41.0% 34.6% 42.6% 35.8%
D new 10.2% 5.0% 0.0% 0.0% 1.85 1.82 37.2 41.4
old 44.9% 34.4% 40.4% 35.8%
U new 13.9% 3.8% 3.1% 0.4% 1.70 1.71 31.5 38.9
old 55.4% 50.9% 0.0% 0.0%
N new 17.7% 4.1% 0.0% 0.0% 1.70 1.71 32.1 44.2
old 57.2% 50.5% 0.0% 0.0%
Table 3.4: Service levels, inventory age and costs averaged across all experiments.
this result may no longer hold. We observed that F and D have fresher goods than
U and N in 31 out of 32 of the experiments, when inventory is replenished by TIS.
When NIS is used, this statistic falls to 29 out of 32. However, downward substitution
is only slightly worse in these four instances; the average age of inventory of F and Dis within 1% of U and N . Average freshness numbers are also provided in Table 3.4.
3.5.4 Effect of customer behavior: Proportional acceptance
of substitutes
In this section we present numerical examples where we vary πD and πU (0 ≤ πD, πU ≤1) in order to illustrate the effect of acceptance proportions.
In all the examples, the demand is discrete uniform distributed between 0 and
25 for both new and old items. The examples below are analyzed for the case when
TIS is used in replenishment. Our observations are similar for NIS, and hence are
omitted.
Example-4 (Proportional Downward acceptance). In this example we use
h = 1, m = 2, p1 = 3, p2 = 4, αD = 6, αU = 3. We study the effect of proportional
85
acceptance of downward substitution by varying πD, assuming πU = 1 in this case.
Thus, πD = 0 represents policy U and πD = 1 represents policy F . The total cost as a
function of πD is presented in Figure 3.5. This graph shows how different components
of the total cost change as we move from policy U to F as πD increases. In all
experiments we use the optimal value of S for the given parameter values (including
πD).
In this particular example, the expected total cost increases almost linearly with
πD (i.e. U is less costly than F). As πD increases, the amount of downward substitu-
tion increases (hence so too does its cost component) while the lost sales cost of old
items decrease. Downward substitution under TIS increases the inventory turnover
for new items; as the amount of new items used to satisfy the demand for old in-
creases, TIS leads to higher number of newer items being replenished. Therefore, as
πD increases, the penalty for lost sales of new items also decreases. In this exam-
ple, only a slight increase and decrease are observed in holding and outdating costs,
respectively.
Example-5 (Proportional Upward acceptance). The costs are the same
as in Example-1A. Here we look at the effect of acceptance proportion for upward
substitution (πU) and observe the changes as we move from policy D to F . Again, in
all experiments we use the optimal value of S for the given parameter values, including
πU . As we see in Figure 3.6, the effect of πU on the costs is minimal; the total cost
decreases slightly as πU increases (i.e. F is less costly than D). This is not surprising
because the difference between the lowest cost of D and F in Example-1A is very
small. (This holds true for all the examples presented in Figures 3.1-3.3). There
are examples where this difference may be slightly more (as in Example -4), but our
main observation does not change: There is not a systematic effect of πU on the cost
components except the upward substitution cost. Thus it appears that the expected
costs are typically more sensitive to πD. This is similar to our analytical results
on dominance conditions. Finding dominance conditions for upward substitution
is easier as upward substitution does not affect future inventory levels. Therefore
myopic substitution policies are optimal and we have if-and-only-if type dominance
conditions for upward substitution. However downward substitution changes future
inventory levels and dynamic relationships more complicated. Therefore we cannot
have tight dominance conditions for infinite horizon. Thus the system costs fluctuate
86
0
10
20
30
40
50
60
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1D
Ex
pec
ted
Co
st UpwSubs
DwnSubst
Outdating
Penalty(Old)
Penalty(New)
Holding
Figure 3.5: The effect of proportional acceptance of downward substitution on ex-
pected total cost in Example-4.
more as πD varies.
3.5.5 Summary of Computational Results
The numerical study presented above implies that downward substitution is an impor-
tant lever for improving service levels and freshness of inventory, and that downward
substitution has a more significant effect on the system performance than upward
substitution (the value of πD is more important than πU). In contrast to what has
appeared in the bulk of the literature, we see examples where NIS appears to be a
more suitable replenishment policy with respect to costs, service levels and inventory
freshness, especially when there is significant demand for new items and substitution
costs are appreciable. We are also able to segment substitution policies by cost: The
relative differences in expected costs of substitution policies are highest when we com-
pare policies that use upward substitution to ones utilizing downward substitution.
Hence, a supplier can significantly benefit from using N or U over F or D, and vice
versa.
87
0
10
20
30
40
50
60
70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
U
Exp
ecte
d C
ost
UpwSubs
DwnSubst
Outdating
Penalty(Old)
Penalty(New)
Holding
Figure 3.6: The effect of proportional acceptance of upward substitution on expected
total cost in Example-5.
3.6 Conclusion
In this study, we focused on inventory management of perishable goods and goods
that lose their value over time. In our model, demand is differentiated with respect to
the age of the product, and some customers accept substitutes in case of stock-outs.
Our goal was to find out under what conditions substitution provides lower costs
for the supplier and fresher goods for the customers. We considered two different
base-stock type inventory replenishment policies, TIS and NIS, for the supplier and
formalized four different substitution options. We identified conditions on cost para-
meters that guarantee one form of substitution being economically superior (almost
surely) compared to others. We also show that downward substitution policies are
always beneficial with respect to average age of goods in stock. Our analysis revealed
that TIS, which is advocated as a good replenishment policy for perishables, can lead
to counter-intuitive performance when the demand is differentiated by the age of the
product and substitution takes place. On the other hand, NIS is very promising as
a replenishment policy. NIS is more robust against new item’s demand fluctuations
than TIS. It is simple, leads to lower costs under many demand scenarios and has
more predictable behavior under substitution.
88
Chapter 4
Analysis of Inventory
Replenishment Policies for
Perishable Goods Under
Substitution
4.1 Introduction
In the previous chapter we analyzed the effect of substitution on operational costs. In
this chapter we focus on the replenishment policies. Using the same model as in the
previous chapter we allow demand for both new and old products, and we consider
the possibility of substitution between old and new products. We refer the reader
to Chapter 3 for the details of the model. In Section 4.2 first we present analytical
results on the optimality of our heuristic policies for special cases. We also provide
our findings on structural properties of the policies. In Section 4.3 we provide the
results of a comprehensive numerical study. We study the policies under various
demand schemes; and we also study them when substitution is not always accepted.
In Section 4.3.9 we develop and compare a number of NIS order-up-to level heuristics
using newsvendor-type logic. We only develop heuristics for NIS, as we have observed
89
in the previous chapter that NIS appears to be outperforming TIS.
We also conduct a numerical study using simulation and dynamic programming.
We compare TIS and NIS at their optimal order-up-to levels (found by doing a line
search using simulation) to the globally optimal state-dependent policy found via
dynamic programming.
4.2 Analytical Results
In this section we present analytical results establishing the optimality of NIS for
special cases based on the substitution type that is allowed or the demand type (for
old or new items) that exists. Demand is assumed to be i.i.d. for these results.
Lemma 18 NIS is the optimal replenishment policy, under N when demand is i.i.d.
Proof : The idea behind the proof is as follows: If there is no substitution allowed,
each ordering decision becomes independent of each other. The ordered new items
in any period are sold, or they perish in two periods. Therefore this case boils down
to a “2-period Newsvendor” problem. The NIS amount, which is the solution to the
“2-period Newsvendor”, is the optimal solution. The formal proof is as follows:
In a period n, let the ordering decision be qn. In period n the following cost is
incurred due to this decision:
p2[Dn2 − qn]+ + h[qn −Dn
2 ]+. (4.1)
In the the following period, n + 1, the cost of the decision is :
p1[Dn+11 − (qn −Dn
2 )+]+ + m[(qn −Dn2 )+ −Dn+1
1 ]+. (4.2)
Therefore the ordering decision qn+1 does not affect the cost of decision qn in period
n + 1; nor is the decision qn+1 affected by the decision qn as (qn − Dn2 )+ does not
affect Dn+11 , due to the i.i.d. assumption.
Hence each ordering decision for every period is separable from the others; and
the problem reduces to the “2-period” newsvendor problem, finding the quantity qn
that minimizes (4.1) + (4.2). Since demand over the periods is i.i.d. NIS is optimal.
90
Lemma 19 If old item demand is zero (with no upward substitution) then NIS is
optimal, when demand is i.i.d.
Proof : The proof is immediate from the proof of Lemma 4.2. The problem reduces
to the 1-period classical newsvendor problem in this case.
Lemma 20 If new item demand is zero (with no downward substitution) then NIS
is optimal, when demand is i.i.d.
Proof : The proof is immediate from the proof of Lemma 4.2. The problem reduces
to the newsvendor problem with “lead-time” of one period. The lead-time does not
affect the solution.
If new item demand is zero (with downward substitution) then the problem be-
comes the single demand stream FIFO problem. In the next section we numerically
show that TIS is a good heuristic but not optimal, NIS is comparable.
We also provide the following analytical results on structural properties of optimal
ordering functions:
Lemma 21 The optimal ordering function is independent of on-hand inventory when
the optimal substitution policy (among the four policies) is N and demand is i.i.d.
Proof : The proof is immediate from Lemma 4.2; as q∗(x) = S for any x due to
Lemma 4.2.
4.3 Computational Results: NIS vs. TIS
First we show that closed-form solutions for the two-period lifetime case (TIS) for
a given demand distribution is quite complicated, in section 4.3.1. In the remaining
sub-sections we do a computational study to shed more light on the performances of
the NIS and TIS policies. In Section 4.3.2 we present comparisons of TIS and NIS for
various demand distributions. We present comparison of TIS and NIS when old and
new item demands are positively correlated and negatively correlated in sections 4.3.3
and 4.3.4 respectively. The results of our computations with high new item demand
91
is in Section 4.3.5, and with low new item demand is in Section 4.3.6. In sections
4.3.7 and 4.3.8 we present our results when substitution is not always accepted. And
finally in section 4.3.9 we develop and analyze a number of NIS order-up-to level
heuristics using newsvendor-type logic.
4.3.1 A Preliminary Study
We did a preliminary computational study to try to determine whether the steady
state distribution of the TIS levels has a closed form. Brodheim et al. (1975) study
NIS in a similar way using a Markov chain model, when demand is present only
for the oldest items. They realized that Markov chain techniques could be used to
obtain exact expressions, or very simple bounds, on key system measures such as
probability of shortage, the average age of the inventory, and the average quantity
outdated products per period.
In this section we assume total demand has a Poisson distribution with mean λ.
Demand goes to new items with probability p and to old items with probability 1− p
.
For no-substitution policy, the transition probability, Pkl, for Xn2 is :
Pkl =
0 If k + l < S for 0 ≤ l ≤ S − 1,e−λp(λp)k+l−S
(k+l−S)!If k ≥ S − l for l 6= S,
1−∑k−1m=0
e−λp(λp)m
m!If l = S.
Stationary distribution for S = 2 case is as follows:
π0 =e−λp − e−2λp
1 + e−λpλp− e−2λp,
π1 =e−λpλp
1 + e−λpλp− e−2λp,
π2 =1− e−λp
1 + e−λpλp− e−2λp.
For full-substitution, the transition probability, Pkl, for Xn2 is as follows:
92
Pkl =
0 If k + l < S,
e−λ λ−l
l!If k = S, 0 ≤ l ≤ S − 1,
1−∑S−1m=0
e−λλm
m!If k = l = S ,
e−λp ∑lm=0
e−λ(1−p)(λ(1−p))m
m!Ifk + l = S,k 6= 0,l 6= S,
1 If k = 0 and l = S,
1−∑k−1m=0
e−λp(λp)m
m!
+∑k−1
m=0(e−λp(λp)m
m!− (1−∑S−m−1
n=0e−λ(1−p)(λ(1−p))n
n!)) If l = S,1 ≤ k ≤ S − 1,
e−λp (λp)k−S+l
k−S+l!
∑S−km=0
e−λ(1−p)(λ(1−p))m
m!
+∑k−S+l−1
m=0e−λλlpm
m!(l−m)!(1−p)m Otherwise.
Stationary distribution for S = 2 case is:
π0 =e−λ − e−2λ(1 + λ− λp)
1− e−2λ(1 + λ− λp) + e−λλp,
π1 =e−λλ
1− e−2λ(1 + λ− λp) + e−λλp,
π2 =1− e−λ(1 + λ− λp)
1− e−2λ(1 + λ− λp) + e−λλp.
The stationary distribution of the inventory levels does not seem to have a simple
closed form; therefore in this study we focus on comparison of our heuristics.
4.3.2 Comparison of TIS and NIS while changing variance of
demand
In order to observe performances of TIS and NIS under different demand variance
we did a numerical study with four different demand distributions with different
coefficients of variation (CV). In Table 4.1 we present this cost comparison of TIS
and NIS based on simulation (400,000 periods after initial warm-up). We conducted
four sets of 32 experiments with the following cost parameters: h = 1, m ∈ {2, 5},p1 ∈ {1, 3}, p2 ∈ {4, 9}, αU ∈ {3, 7} and αD ∈ {2, 6}. Each of the four sets used a
different demand distribution, where we varied the characteristics of the demand as
well as the coefficient of variation. The demand distributions were: (i) Equal point
mass at 0 and 25 (CV = 1.41); (ii) Discrete uniform demand between zero and 25
(CV = 0.58); Poisson demand with rate 12.5 (CV = 0.28); and (iv) Discrete Uniform
93
demand between 10 and 15 (CV = 0.12). In all cases the demand for old and new
items are independent.
Table 4.1 shows the average total costs for the sixteen combinations of policies
and demand distributions.
D ∼ {0, 25} D ∼ U[0, 25] D ∼ Poisson(12.5) D ∼ U[10, 15]
Policy Opt. NIS TIS Opt. NIS TIS Opt. NIS TIS Opt. NIS TIS
F 47.3 47.9 53.7 36.6 37.0 39.9 24.7 25.8 25.6 18.6 19.4 21.1
D 47.9 47.9 60.0 36.9 37.2 41.4 24.7 25.8 25.6 18.6 19.4 21.2
U 46.1 46.7 56.1 31.3 31.5 38.9 18.7 18.7 24.4 15.2 15.2 20.0
N 46.8 46.8 66.6 32.1 32.1 44.2 18.7 18.7 25.8 15.2 15.2 20.5
Table 4.1: TIS vs. NIS in terms of cost (avg. over 32 instances) under four different
demand models, all with mean 12.5.
We find the optimal cost via dynamic programming 1. NIS, the policy that ignores
the information on the old item inventory, outperforms TIS, the policy that uses this
information, often by a considerable amount (up to 42.2 %). Moreover, for the two
instances in which TIS was superior, for Poisson demand and full or downward substi-
tution, the difference was only slight (less than 1 %). These results give preliminary
support to the industry practice of utilizing NIS (Angle, 2003) in practice, but con-
trast sharply with virtually all of the literature on control of perishable inventories.
Interestingly, similar results have recently been seen in some studies of continuous
time lost-sales systems without substitution (Reiman, 2005).
The table above compares average costs of each policy over all instances, specifi-
cally it considers the costs of a policy whether it is the optimal policy or not. Now
we look a little more closely at our data, focusing only on the optimal policy for each
instance. According to our computational studies when demand is highly variant
(CV is 0.58 or 1.41) NIS outperforms TIS no matter what the optimal substitution
1In Figure 4.1 we see computation times for NIS, TIS and dynamic programming as maximum
demand grows. Computations are done with a 1.2 GHz Intel Pentium.
94
0
10
20
30
40
50
60
70
80
90
100
5 25 35 50
Max Demand
Min
ute
s TIS
NIS
Opt
Figure 4.1: Computation times for the best NIS, the best TIS and the optimal solu-
tions.
policy is (in all four quadrants of Figures 4.2 and 4.3)2. For example when demand is
Uniform [0, 25] (i.e. Figure 4.2) only for 1 instance (out of 32) TIS (Full-substitution)
is better than NIS (Full-substitution). Why is NIS so close to optimal? New items
are more valuable and NIS is more robust against new item’s demand fluctuations.
As being able to fulfill new item demand has primary importance regardless of the
optimal substitution policy, and the high variability of demand makes it best to ig-
nore old item information. Not ignoring old item information (TIS) would lead us to
under-order when demand fluctuates a lot from one period to another. For example,
if demand is low in a period, the old item inventory level will be high in the next
period. And if TIS is used the order quantity will be small leaving the new item
inventory level low. This would be costly if new item demand is high in the next
period.
If the demand has a low variance (CV is 0.12 or 0.28) NIS is better if downward
substitution is not viable (see the upper and lower right quadrants in Figures 4.4 and
4.5). The reason to this as follows: When downward substitution is not viable the
problem boils down to the newsvendor problem (as upward substitution is not done
frequently), which is optimally solved by NIS. However if downward substitution is
viable and if demand uncertainty is not high TIS outperforms NIS (see the upper and
2The percentages indicate percent cost differences between the policy and the optimal.
95
D
U
2%10%
NISTIS
0%29%
NISTIS
2%9%
NISTIS
1%24%
NISTIS
Full Subs.
Downward Subs. No Subs.
Upward Subs.
•Best substitution and replenishment policies based on the
m+p2
1 instance
12 instances 16 instances
3 instances
Uniform [0, 25]
Figure 4.2: Optimal policies when CV is 0.58
D
U
0%20%
NISTIS
0%33%
NISTIS
3%12%
NISTIS
1%20%
NISTIS
Full Subs.
Downward Subs. No Subs.
Upward Subs.
•Best substitution and replenishment policies based on the
m+p2
14 instances
2 instances2 instances
14 instances
{0, 25}
Figure 4.3: Optimal policies when CV is 1.41
D
U
7%4%
NISTIS
0%25%
NISTIS
14%2%
NISTIS
0%26%
NISTIS
Full Subs.
Downward Subs. No Subs.
Upward Subs.
•Best substitution and replenishment policies based on the
m+p2
1 instance
7 instances 21 instances
3 instances
Uniform [10, 15]
Figure 4.4: Optimal policies when CV is 0.12
D
U
8%2%
NISTIS
0%29%
NISTIS
9%1%
NISTIS
0%30%
NISTIS
Full Subs.
Downward Subs. No Subs.
Upward Subs.
•Best substitution and replenishment policies based on the
m+p2
21 instances
3 instances1 instance
7 instances
Poisson (12.5)
Figure 4.5: Optimal policies when CV is 0.28
lower left quadrants of these same figures). Under a low-variance demand the effect
of keeping extra new inventory in order to fulfill new item demand is marginal. Being
efficient is more important and using old item inventory information for replenishment
decisions reduces costs, this makes TIS better than NIS in such an environment.
This raises the question of how a manager could know which “quadrant” their op-
erations lay in. For the upper versus the lower half it is simple, as we have necessary
and sufficient conditions for upward substitution to be beneficial, namely αU < m+p2.
Whether or not downward substitution is advisable is more complex, and arguably
more important, because this also has an impact on which ordering policy is best. For-
tunately, our computational studies shed light on this question: Another observation
96
h h+m+p1
D
Figure 4.6: The metric shows where the downward substitution cost (αD) lays com-
pared to upper and lower bounds
due to our computational studies is that for high-variance demand downward substi-
tution is viable for relatively large downward substitution costs. This is observed by
using the following metric M (see Fig. 4.6) under NIS:
M =αD − h
m + p1
(4.3)
Our experiments show that under highly variant demand downward substitution is
viable as long as M is not larger than 0.5. For low-variance demand downward
substitution is only allowed for low downward substitution costs (the metric cannot
be greater than 0.2).
4.3.3 NIS vs. TIS when new and old item demands are neg-
atively correlated
In this section we compare NIS and TIS when the demand streams for new and
old items are negatively correlated. We model the negative correlation of demand
streams by making the sum of new and old item demand constant each period. In
our simulation, new item demand is uniformly distributed between 0 and 25, and old
item demand is 25 minus the new item demand. Therefore in each period new and old
item demands add up to 25. We run our simulations for 400,000 periods. Cost figures
(per period) for all parameter settings (i.e. h = 1, m ∈ {2, 5}, p1 ∈ {1, 3}, p2 ∈ {4, 9},αU ∈ {3, 7} and αD ∈ {2, 6}) under NIS and TIS are presented in Table 4.3. In Table
4.3 we observe that for 12 out of 32 parameter settings F and D provide the lowest
cost and in 17 instances U is the least costly and in the remaining 3 parameter settings
N has the lowest cost for NIS. Similarly, for TIS U provides the lowest cost in 17 out
97
F D U NNIS 34.66 34.66 30.29 32.10
TIS 34.17 34.17 33.46 44.16
% difference -1.43% -1.43% 10.46% 37.56%
Table 4.2: NIS vs. TIS when old and new item demand streams are negatively
correlated.
of 32 instances; for 14 out of 32 parameter settings F and D are both the least costly
(i.e. upward substitution is not done). For 1 instance N has the lowest cost.
We present the average (averaged over the 32 parameter settings) NIS and TIS
cost figures in Table 4.2. TIS is slightly better than NIS when downward substitution
is allowed. It is observed that when downward substitution is allowed no upward
substitution occurs, that is F and D become identical.
In Section 4.3.2 we indicated that, in general, if the metric M (see the Equation
4.3) is not larger than 0.2, downward substitution is viable. In this experiment there
are 12 parameter settings where M is less than or equal to 0.2. We observe that in
10 out of the 12 settings downward substitution is viable (under the better heuristic
ordering policy, TIS or NIS). Among the remaining 22 instances, in which M is greater
than 0.2, in 20 instances downward substitution is not viable. We should also mention
that, the rule of thumb works with 100% accuracy if NIS is always practiced as the
replenishment policy for this experiment.
4.3.4 TIS vs. NIS when new and old item demands are pos-
itively correlated
In order to see the effect of positively correlated demand we ran simulations with
uniform demand between 0 and 25. In the model we have old and new items exactly
the same. Therefore the demand streams in our simulation are perfectly positively
correlated. Cost figures (per period) for all parameter settings under NIS and TIS
are presented in Table 4.5. In Table 4.5 we observe that for 12 out of 32 parameter
settings F and D provide the lowest cost and in 20 instances U and N are the least
98
NIS TIS
F D U N F D U Nh=1, αD=2, αU=3, p2=4, p1=1, m=2 22.87 22.87 20.11 21.61 22.45 22.45 21.68 28.20
h=1, αD=2, αU=3, p2=4, p1=1, m=5 22.91 22.91 22.40 26.63 22.86 22.86 21.69 32.10
h=1, αD=2, αU=3, p2=4, p1=3, m=2 22.90 22.90 33.41 33.70 22.45 22.45 36.52 43.00
h=1, αD=2, αU=3, p2=4, p1=3, m=5 25.02 25.02 38.67 41.80 23.73 23.73 36.49 49.53
h=1, αD=2, αU=3, p2=9, p1=1, m=2 22.95 22.95 21.45 23.51 22.45 22.45 21.68 39.80
h=1, αD=2, αU=3, p2=9, p1=1, m=5 24.18 24.18 25.90 31.39 23.68 23.68 21.71 46.33
h=1, αD=2, αU=3, p2=9, p1=3, m=2 22.92 22.92 33.51 34.00 22.47 22.47 36.48 52.45
h=1, αD=2, αU=3, p2=9, p1=3, m=5 25.00 25.00 40.48 44.14 23.69 23.69 36.52 61.68
h=1, αD=2, αU=7, p2=4, p1=1, m=2 22.88 22.88 22.02 21.63 22.44 22.44 29.68 28.16
h=1, αD=2, αU=7, p2=4, p1=1, m=5 22.88 22.88 25.24 26.64 22.89 22.89 29.70 32.11
h=1, αD=2, αU=7, p2=4, p1=3, m=2 22.93 22.93 33.79 33.70 22.44 22.44 45.01 43.11
h=1, αD=2, αU=7, p2=4, p1=3, m=5 25.02 25.02 40.89 41.87 23.69 23.69 45.72 49.40
h=1, αD=2, αU=7, p2=9, p1=1, m=2 22.92 22.92 22.72 23.59 22.48 22.48 30.89 39.87
h=1, αD=2, αU=7, p2=9, p1=1, m=5 24.15 24.15 28.14 31.36 23.72 23.72 30.85 46.58
h=1, αD=2, αU=7, p2=9, p1=3, m=2 22.93 22.93 33.87 34.01 22.47 22.47 45.07 52.64
h=1, αD=2, αU=7, p2=9, p1=3, m=5 25.04 25.04 42.05 44.04 23.72 23.72 45.68 61.63
h=1, αD=6, αU=3, p2=4, p1=1, m=2 41.31 41.31 20.13 21.63 41.33 41.33 21.72 28.19
h=1, αD=6, αU=3, p2=4, p1=1, m=5 41.34 41.34 22.43 26.62 41.34 41.34 21.67 32.05
h=1, αD=6, αU=3, p2=4, p1=3, m=2 41.86 41.86 33.38 33.71 42.29 42.29 36.49 42.96
h=1, αD=6, αU=3, p2=4, p1=3, m=5 52.50 52.50 38.66 41.80 51.10 51.10 36.48 49.54
h=1, αD=6, αU=3, p2=9, p1=1, m=2 41.81 41.81 21.45 23.60 42.22 42.22 21.69 39.86
h=1, αD=6, αU=3, p2=9, p1=1, m=5 52.52 52.52 25.98 31.41 51.15 51.15 21.71 46.40
h=1, αD=6, αU=3, p2=9, p1=3, m=2 41.84 41.84 33.56 33.97 42.20 42.20 36.47 52.65
h=1, αD=6, αU=3, p2=9, p1=3, m=5 52.57 52.57 40.47 44.04 51.16 51.16 36.46 61.47
h=1, αD=6, αU=7, p2=4, p1=1, m=2 41.37 41.37 21.99 21.60 41.38 41.38 29.65 28.20
h=1, αD=6, αU=7, p2=4, p1=1, m=5 41.36 41.36 25.24 26.66 41.36 41.36 29.67 32.05
h=1, αD=6, αU=7, p2=4, p1=3, m=2 41.87 41.87 33.83 33.73 42.27 42.27 45.12 43.05
h=1, αD=6, αU=7, p2=4, p1=3, m=5 52.55 52.55 40.82 41.79 51.18 51.18 45.63 49.46
h=1, αD=6, αU=7, p2=9, p1=1, m=2 41.84 41.84 22.71 23.61 42.23 42.23 30.90 39.93
h=1, αD=6, αU=7, p2=9, p1=1, m=5 52.52 52.52 28.14 31.32 51.09 51.09 30.83 46.39
h=1, αD=6, αU=7, p2=9, p1=3, m=2 41.86 41.86 33.85 33.99 42.28 42.28 45.08 52.68
h=1, αD=6, αU=7, p2=9, p1=3, m=5 52.57 52.57 42.03 44.04 51.13 51.13 45.74 61.52
Average Cost: 34.66 34.66 30.29 32.10 34.17 34.17 33.46 44.16
Table 4.3: NIS and TIS cost figures for all parameter settings when old and new item
demand streams are negatively correlated.
99
F D U NNIS 35.83 35.83 32.11 32.11
TIS 43.34 44.97 41.67 44.20
% difference 20.96% 25.49% 29.80% 37.66%
Table 4.4: NIS vs. TIS when old and new item demand streams are positively corre-
lated.
costly for NIS; and for TIS the table shows that F provides the lowest cost in 14
out of 32 instances. For 14 out of 32 parameter settings U is the least costly, for 2
instances D has the lowest cost and for 2 instances N is the least costly.
Table 4.4 summarizes the computational study’s results for . The costs presented
in the table are averaged over the 32 parameter settings (i.e. h = 1, m ∈ {2, 5},p1 ∈ {1, 3}, p2 ∈ {4, 9}, αU ∈ {3, 7} and αD ∈ {2, 6}).
Table 4.4 shows that NIS outperforms TIS at least by 20% for any substitution
policy. Under positively correlated demand TIS does not benefit from the pooling
effect unlike the negatively correlated demand case.
In this experiment we observe that for all 12 parameter settings in which M (see
the Equation 4.3) is less than or equal to 0.2 downward substitution is viable. And
for all 22 instances in which M is greater than 0.2 downward substitution is not
viable. Our rule of thumb is working with 100% accuracy for NIS and TIS for this
experiment. This is in line with the observation that NIS outperforming TIS by a
greater margin under positively correlated demand. Therefore we can say that our
rule of thumb would be more accurate when demand steams are positively correlated
than when demand streams are negatively correlated.
4.3.5 NIS vs. TIS when new item demand is higher than old
item demand
In this section we compared NIS and TIS when the demand for new item demand is
greater than the old demand. We did experiments when new item demand distribution
100
NIS TIS
F D U N F D U Nh=1, αD=2, αU=3, p2=4, p1=1, m=2 22.17 22.17 21.61 21.61 26.13 26.88 27.07 28.16
h=1, αD=2, αU=3, p2=4, p1=1, m=5 25.95 25.95 26.64 26.64 28.32 29.49 30.63 32.10
h=1, αD=2, αU=3, p2=4, p1=3, m=2 32.35 32.35 33.78 33.78 36.19 38.04 41.29 43.05
h=1, αD=2, αU=3, p2=4, p1=3, m=5 39.25 39.25 41.77 41.77 41.27 43.43 47.34 49.54
h=1, αD=2, αU=3, p2=9, p1=1, m=2 24.01 24.01 23.58 23.58 33.63 36.44 34.96 39.84
h=1, αD=2, αU=3, p2=9, p1=1, m=5 30.02 30.02 31.37 31.37 37.00 40.41 41.95 46.41
h=1, αD=2, αU=3, p2=9, p1=3, m=2 32.48 32.48 33.97 33.97 39.06 45.39 46.02 52.59
h=1, αD=2, αU=3, p2=9, p1=3, m=5 40.70 40.70 44.11 44.11 47.33 52.13 56.01 61.75
h=1, αD=2, αU=7, p2=4, p1=1, m=2 22.18 22.18 21.61 21.61 27.09 26.86 28.51 28.17
h=1, αD=2, αU=7, p2=4, p1=1, m=5 25.91 25.91 26.61 26.61 29.11 29.49 31.63 32.10
h=1, αD=2, αU=7, p2=4, p1=3, m=2 32.33 32.33 33.77 33.77 38.57 38.04 43.65 43.11
h=1, αD=2, αU=7, p2=4, p1=3, m=5 39.21 39.21 41.83 41.83 42.79 43.44 48.77 49.48
h=1, αD=2, αU=7, p2=9, p1=1, m=2 23.99 23.99 23.55 23.55 35.03 36.37 37.61 39.92
h=1, αD=2, αU=7, p2=9, p1=1, m=5 29.91 29.91 31.29 31.29 38.38 40.40 43.65 46.54
h=1, αD=2, αU=7, p2=9, p1=3, m=2 32.43 32.43 34.01 34.01 42.38 45.35 49.52 52.69
h=1, αD=2, αU=7, p2=9, p1=3, m=5 40.67 40.67 44.04 44.04 49.31 52.15 58.27 61.80
h=1, αD=6, αU=3, p2=4, p1=1, m=2 30.78 30.78 21.63 21.63 35.32 35.66 27.03 28.17
h=1, αD=6, αU=3, p2=4, p1=1, m=5 33.04 33.04 26.64 26.64 36.33 37.00 30.69 32.12
h=1, αD=6, αU=3, p2=4, p1=3, m=2 43.78 43.78 33.77 33.77 51.49 52.23 41.31 43.02
h=1, αD=6, αU=3, p2=4, p1=3, m=5 48.72 48.72 41.84 41.84 53.53 54.60 47.29 49.51
h=1, αD=6, αU=3, p2=9, p1=1, m=2 34.70 34.70 23.54 23.54 46.70 48.70 34.90 39.89
h=1, αD=6, αU=3, p2=9, p1=1, m=5 39.51 39.51 31.39 31.39 49.02 51.55 41.91 46.37
h=1, αD=6, αU=3, p2=9, p1=3, m=2 44.49 44.49 34.02 34.02 59.21 61.97 46.03 52.70
h=1, αD=6, αU=3, p2=9, p1=3, m=5 51.54 51.54 44.09 44.09 62.61 65.68 55.97 61.76
h=1, αD=6, αU=7, p2=4, p1=1, m=2 30.74 30.74 21.64 21.64 35.76 35.64 28.51 28.16
h=1, αD=6, αU=7, p2=4, p1=1, m=5 33.00 33.00 26.61 26.61 36.75 36.94 31.59 32.05
h=1, αD=6, αU=7, p2=4, p1=3, m=2 43.83 43.83 33.79 33.79 52.41 52.18 43.63 43.05
h=1, αD=6, αU=7, p2=4, p1=3, m=5 48.65 48.65 41.81 41.81 54.31 54.67 48.80 49.51
h=1, αD=6, αU=7, p2=9, p1=1, m=2 34.71 34.71 23.55 23.55 47.68 48.69 37.60 39.89
h=1, αD=6, αU=7, p2=9, p1=1, m=5 39.57 39.57 31.38 31.38 49.91 51.40 43.66 46.42
h=1, αD=6, αU=7, p2=9, p1=3, m=2 44.49 44.49 34.08 34.08 60.58 61.94 49.55 52.67
h=1, αD=6, αU=7, p2=9, p1=3, m=5 51.54 51.54 44.07 44.07 63.83 65.77 58.25 61.79
Average Cost for NIS: 35.83 35.83 32.11 32.11 43.34 44.97 41.67 44.20
Table 4.5: NIS and TIS cost figures for all parameter settings when old and new item
demand streams are positively correlated.
101
is Uniform between 0 and 25 while old item demand demand distribution is Uniform
between 0 and 6.
We ran our simulation for 400,000 periods with the following parameters: h = 1,
m ∈ {2, 5}, p1 ∈ {1, 3}, p2 ∈ {4, 9}, αU ∈ {3, 7} and αD ∈ {2, 6}. On average TIS is
costlier than NIS by the following percentages: F by 5%; D by 20%; U by 9%; N by
28%. There are some instances (under F) where NIS is costlier than TIS with high
m, h, and low αU , αD,but overall NIS is superior.
4.3.6 NIS vs. TIS when new item demand is lower than old
item demand
In this section we compared NIS and TIS when the demand for new items is lower than
old items. We again run our simulation for 400,000 periods with the same parameters
(i.e. h = 1, m ∈ {2, 5}, p1 ∈ {1, 3}, p2 ∈ {4, 9}, αU ∈ {3, 7} and αD ∈ {2, 6}).We first did experiments when new item demand distribution is Uniform between
0 and 6 while old item demand demand distribution is Uniform between 0 and 25.
On average TIS outperforms NIS slightly under F and D, but is significantly
worse under U and N . The differences in average costs are: F by -1%; D by -1%; Uby 10%; N by 11%.
We observe that under low demand for new items F and D perform almost the
same (so do U and N ) as upward substitution is required less. In some instances (for
all four policies but mostly under F and D ) NIS is costlier than TIS with high m,
h, and low p1.
We then did experiments when there is no demand for new items, while old item
demand demand distribution is Uniform between 0 and 25 with zero downward sub-
stitution cost (i.e. FIFO). On average NIS is costlier than TIS by the following
percentages: F by 11%; D by 11%; U by 7%; N by 7%. In some instances (when p1
is high) U and N are costlier under TIS than NIS.
Thus, as new item becomes less important, TIS begins to outperform NIS.
102
4.3.7 Comparison of TIS and NIS when substitution is not
always accepted: Partial vs. Probabilistic Acceptance
In this section we numerically compare partial and probabilistic acceptance models.
In the partial acceptance model when substitution is offered a certain fraction of it is
accepted by customers. We use πD and πU to denote the fraction of substitution that
is accepted for downward and upward substitutions respectively. Under probabilistic
acceptance all of the offered substitution is either accepted or rejected (as a whole)
by the customer. Acceptance probabilities are denoted by φD and φU for downward
or upward substitution, respectively. We assume each substitution acceptance is
independent.
Table 4.6 shows the cost comparison of the policies under partial acceptance.
The simulations were done for 400,000 periods and only for one parameter setting
(p1 = 5, p2 = 9, h = 1, m = 5, αU = 7, αD = 6). Proportions for acceptance of
substitution (πD and πU ) are varied over (0, 0.1, 0.2, ..., 1). The cost (per period)
figures in the table is averaged across all proportions. Cost of N is given for the sake
of completeness and as a reference point.
F D U NNIS 48.86 48.96 43.83 44.10
TIS 58.80 61.18 57.72 61.75
% difference 20.34% 24.96% 31.68% 40.01%
Table 4.6: NIS vs. TIS when a proportion of substitution is accepted
Table 4.7 shows the cost comparison of the policies under probabilistic acceptance.
Similarly, the simulations were done for 400,000 periods and only for one parameter
setting (p1 = 5, p2 = 9, h = 1, m = 5, αU = 7, αD = 6). Probabilities for acceptance
of substitution (φD and φU ) are varied over (0, 0.1, 0.2, ..., 1) whenever there is a
chance of substitution. The cost (per period) figures in the table is averaged across
all probabilities. Again, cost of N is given for reference.
Tables 4.6 and 4.7 show that the models produce very similar results. We observe
that NIS outperforms TIS especially when substitution is offered/allowed less. The
103
F D U NNIS 48.86 49.08 43.71 44.07
TIS 60.26 62.84 57.90 61.73
% difference 23.34% 28.04% 32.46% 40.09%
Table 4.7: NIS vs. TIS under probabilistic acceptance of substitution
reason for this is as follows: New item inventory level is lower under TIS than under
NIS. As new items are more valuable, penalty costs will be higher if substitution is
not allowed. This would cause NIS to outperform TIS, particularly when substitution
is allowed or offered less.
4.3.8 Comparison of TIS and NIS when acceptance proba-
bility of substitution depends on substitution costs
It is quite possible that the acceptance probability of a substitution offer depends on
substitution costs. We conducted experiments and observed performances of TIS and
NIS by modelling acceptance probability of substitution as a function of substitution
cost as follows:
φD(αD) =αD
h + 2×m + p1
, (4.4)
φU(αU) =αU
m + p2
, (4.5)
where φD and φU are the acceptance probabilities for upward and downward substi-
tution, respectively.
Table 4.8 summarizes the cost figures under TIS and NIS with probabilistic accep-
tance when the acceptance probability is a function of substitution. In the experiment
we conducted, the cost parameter settings were as follows: p1 = 5, p2 = 9, h = 1,
m = 5, αU ∈ {0, ..., 14}, αD ∈ {0, ..., 14}. The cost numbers in the table are averages
across substitution costs and demand distribution is uniform between 0 and 25. Cost
of N is given for the sake of completeness and as a reference point. The table shows
that NIS outperforms TIS by at least 20%.
104
For this example, the optimal substitution costs for NIS are found as αD = 2 and
αU = 10 (i.e. φD(αD) = 0.14 and φU(αU) = 0.71) providing an average cost of 42.83.
Under TIS the optimal substitution costs are αD = 3 and αU = 7 (i.e. φD(αD) = 0.21
and φU(αU) = 0.50) with an average cost of 55.97.
F D U NNIS 57.36 57.49 43.86 44.07
TIS 69.46 70.89 59.38 61.72
% difference 21.10% 23.32% 35.37% 40.06%
Table 4.8: NIS vs. TIS when acceptance probability depends on substitution cost
4.3.9 Search for a good NIS order-up-to level: News-vendor-
type heuristic
In this section we propose and evaluate six news-vendor-type heuristics for NIS’s order
quantity. We did the numerical study with the 32 parameter settings. The demand
has the uniform distribution between between 0 and 25. The results are averaged
over these 32 cost parameters.
Heuristic 1 (H1):
In this heuristic we use p2 + p1 as the cost of understocking and h + m as the cost
of overstocking in the traditional news-vendor setting. Therefore the critical fractile
(CF) is as follows:
CF1 =p2 + p1
p2 + p1 + h + m(4.6)
The results, which are averaged across 32 parameter settings, are summarized in
the following table (Table 4.9):
105
F D U NCost S Cost S Cost S Cost S
H1 42.65 15.75 42.94 15.75 35.77 15.75 36.87 15.75
Opt. NIS 36.99 21.69 37.17 21.84 31.47 20.50 32.10 20.78
% difference 15.31% -27.38% 15.53% -27.90% 13.67% -23.17% 14.85% -24.21%
Table 4.9: Heuristic 1 vs Optimal NIS
Heuristic 2 (H2):
In this heuristic we use p2 as the cost of understocking and h + m− p1 as the cost of
overstocking in the traditional news-vendor setting. Therefore the critical fractile is:
CF2 =p2
p2 + h + m− p1
(4.7)
The results, which are averaged across 32 parameter settings, are summarized in
Table 4.10.
F D U NCost S Cost S Cost S Cost S
H2 39.98 18.13 40.23 18.13 33.62 18.13 34.51 18.13
Opt. NIS 36.99 21.69 37.17 21.84 31.47 20.50 32.10 20.78
% difference 8.10% -16.43% 8.24% -17.02% 6.84% -11.59% 7.51% -12.78%
Table 4.10: Heuristic 2 vs Optimal NIS
Heuristic 3 (H3):
In this heuristic we use p2 +p1 +αU as the cost of understocking and h+m as the cost
of overstocking in the traditional news-vendor setting. Therefore the critical fractile
is:
106
CF3 =p2 + p1 + αU
p2 + p1 + αU + h + m(4.8)
The results, which are averaged across 32 parameter settings, are summarized in
Table 4.11.
Heuristic 4 (H4):
In this heuristic we use p2 +p1 as the cost of understocking and h+m+αD as the cost
of overstocking in the traditional news-vendor setting. Therefore the critical fractile
is:
CF4 =p2 + p1
p2 + p1 + h + m + αD
(4.9)
Heuristic 5 (H5):
In this heuristic we use p2 + p1 + αU as the cost of understocking and h + m + αD as
the cost of overstocking in the traditional news-vendor setting. Therefore the critical
fractile is:
CF5 =p2 + p1 + αU
p2 + p1 + αU + h + m + αD
(4.10)
F D U NCost S Cost S Cost S Cost S
H3 40.08 18.19 40.41 18.19 32.83 18.19 33.87 18.19
Opt. NIS 36.99 21.69 37.17 21.84 31.47 20.50 32.10 20.78
% difference 8.37% -16.14% 8.70% -16.74% 4.33% -11.28% 5.52% -12.48%
Table 4.11: Heuristic 3 vs Optimal NIS
107
F D U NCost S Cost S Cost S Cost S
H4 49.62 11.94 49.83 11.94 45.07 11.94 46.11 11.94
Opt. NIS 36.99 21.69 37.17 21.84 31.47 20.50 32.10 20.78
% difference 34.15% -44.96% 34.05% -45.35% 43.22% -41.77% 43.62% -42.56%
Table 4.12: Heuristic 4 vs Optimal NIS
Heuristic 6 (H6):
In this heuristic we use p2 +αU as the cost of understocking and h+m−p1 as the cost
of overstocking in the traditional news-vendor setting. Therefore the critical fractile
is:
CF6 =p2 + αU
p2 + αU + h + m− p1
(4.11)
Comparison of all six heuristics:
Based on the analysis above H6 is the best heuristic among all six, producing results
within 4% of the optimal. H2 and H3 are within 9% of the optimal; H3 is better
than H2 when downward substitution is not allowed. H1 and H5 fourth and fifth best
heuristic respectively. H4 is the worst performing heuristic, its cost is within 35% to
45% of the optimal cost.
F D U NCost S Cost S Cost S Cost S
H5 44.05 14.81 44.32 14.81 38.32 14.81 39.46 14.81
Opt. NIS 36.99 21.69 37.17 21.84 31.47 20.50 32.10 20.78
% difference 19.09% -31.70% 19.23% -32.19% 21.77% -27.74% 22.93% -28.72%
Table 4.13: Heuristic 5 vs Optimal NIS
108
F D U NCost S Cost S Cost S Cost S
H6 38.17 20.44 38.42 20.44 31.77 20.44 32.52 20.44
Opt. NIS 36.99 21.69 37.17 21.84 31.47 20.50 32.10 20.78
% difference 3.21% -5.76% 3.37% -6.44% 0.97% -0.30% 1.32% -1.65%
Table 4.14: Heuristic 6 vs Optimal NIS
4.3.10 Summary of the Computational Study for Compari-
son of NIS and TIS
Based on our computations NIS outperforms TIS in most of the cases. First we
used various demand distributions while comparing TIS and NIS. We observed that
if downward substitution is viable and demand uncertainty is low than TIS outper-
forms NIS. When old item demand and new item demands are negatively correlated
TIS outperforms NIS if downward substitution is viable; however when old and new
demands are positively correlated NIS has at least 20% lower costs than TIS.
In order to incorporate customer behavior in terms of acceptance of substitution
we did two sets of experiments. First we assumed that customers accept only a
proportion of the substitution offer (i.e. partial acceptance of substitution). Second
we modeled the system such that acceptance of substitution is probabilistic in nature;
that is, based on a probability a substitution offer is rejected or accepted as a whole.
Our results show that both models yield similar results. We also present the TIS vs.
NIS comparison when acceptance probability depends on substitution cost.
As NIS is generally a better replenishment policy for our problem compared to
TIS; finding a good NIS order-up-to level without using computational methods is
important especially for the cases computations are time consuming. Therefore we
developed six heuristics to find a good NIS order-up-level based Newsvendor-type
logic. Based on the comparison of these heuristics we see that one of them performs
within 4% of the optimal NIS which looks promising in case computational solutions
are not readily available.
109
Chapter 5
Conclusion and Future Research
In Chapter 3 we formalize and compare four different fulfillment policies for perish-
able goods, Full-Substitution (F), Upward-Substitution (U), Downward-Substitution
(D), and No-Substitution (N ) under two different base-stock type inventory replen-
ishment policies, TIS and NIS. In this chapter we show that substitution may or may
not be beneficial with respect to operational costs, and provide conditions on cost
parameters that characterize the regions in which different substitution strategies are
most profitable, almost surely. We likewise show that downward substitution policies
are always beneficial with respect to average freshness of products.
In light of our analysis, we can consider the substitution strategies of some of the
examples in the introduction. Fresh produce suppliers are likely to benefit from sub-
stitution, mainly depending on the downward and upward substitution costs. On the
other hand, computer chip manufacturers typically cannot use upward substitution –
if a customer needs a fast chip they often will not be satisfied with a slow one (i.e. αU
is high). They do practice downward substitution after fusing the chip – to control
demand diversion (reducing αD).
In Chapter 4 we study the replenishment problem. We show that NIS is provably
optimal when no substitution is allowed or substitution is not possible. This is based
on the fact that every order decision is independent when substitution of any kind
does not occur when demand is i.i.d. Therefore NIS is optimal when substitution
policy is N or there is no demand for old items or there is no demand for new items.
We did a preliminary study and observed that the steady state distribution of on-hand
inventory does not appear to have a closed form, restricting our analytical studies to
110
policy comparisons rather than finding the optimal order-up-to levels analytically.
Based on our computational study on replenishment policies with a variety of
demand types, patterns, distributions and substitution choices we conclude that NIS
outperforms TIS in most cases. We also find that when new items are significantly
more valuable than older items NIS is more robust against demand fluctuations.
TIS might perform well when new and old item demands are negatively correlated
and downward substitution cost is low; or when demand has a low variance and
downward substitution cost is low. We also showed that a good heuristic (in order
get a near-optimal NIS order-up-to level) can be developed based on newsvendor-type
logic. Such heuristics might be beneficial in case determining the optimal NIS level
computationally is difficult.
The research within the perishable domain has largely been confined to inventory
management of a single product as the survey in Chapter 2 shows. However, grocery
or blood supply chains involve multiple perishable products with possibly differing
lifetimes. Joint replenishment is a typical practice in these industries, and analytical
research that studies the interaction between multiple items in ordering decisions -
focusing on economies of scale, or substitution/complementarity effects of products
with different lifetimes and in different categories - has not been studied. These
interactions provide opportunities for more complex control policies, which make
such problems both more challenging analytically, and potentially more rewarding
practically.
Considering multiple products, another problem that has not attracted much at-
tention from the research community is determining the optimal product-mix when
one type of perishable product can be used as a raw material for a second type of
product, possibly with a different lifetime, that we mentioned in Section 2.2. Deci-
sions regarding when and how much of a base product to sell/stock as is, versus how
much to process in order to obtain a final product with a different lifetime or different
potential value/revenue are quite common in blood and fresh produce supply chains.
A significant majority of the research on inventory management or distribution
of perishable goods disregards capacity constraints. While increasing the complexity
in the analysis, consideration of limited capacity will lead to more realistic models.
Consequently, more realistic models will enable design of heuristic policies that are
possibly more effective in practice.
111
The practical decision of when (if at all) to dispose of the aging inventory has
not received much attention even in single location models, possibly because capacity
is assumed to be unlimited and/or demand is assumed to be satisfied with FIFO
inventory issuance. However, disposal decisions are especially critical when capacity
is constrained (e.g., a retailer has limited shelf-space to display the products), and
customers choose the products based on their (perceived) freshness/quality. Veinott
(1960), in his deterministic model, included disposal decisions for a retailer of per-
ishable products with fixed lifetime. Martin (1986) studied optimal disposal policies
for a perishable product where demand is stochastic. His queueing model considers
the trade-off between retaining a unit in inventory for potential sales vs. salvaging
the unit at a constant value. Vaughan (1994) models an environment where a re-
tailer decides on the optimal parameter of a TIS policy and also a ‘sell-by’ date that
establishes an effective lifetime for the product with a random shelf life; this may
be considered a joint ordering and outdating policy. Vaughan (1994) discusses that
his model would be useful for retailers if they were to select suppliers based on their
potential for ordering and outdating, but no analysis is provided.
When customers prefer fresher goods, disposal and outdating are key decisions
that affect the age-composition (freshness) of inventory, and can influence the de-
mand. Analysis of simple and effective disposal and outdating policies, coordination
of disposal with replenishment policies, and analysis of inventory models where cus-
tomers (retailers) choose among suppliers and/or consider risk of supply/freshness
remain among the understudied research problems.
The majority of research on perishables assumes demand for a product is either
independent of its age, or that the freshest items are preferred. These typical as-
sumptions motivate the primary use of FIFO and LIFO issuance in inventory control
models. However, one can question how realistic these issuance policies would be,
especially in a business-to-business (B2B) setting. A service level agreement between
a supplier (blood center) and its retailer (hospital) may not be as strict as ‘freshest
items must be supplied’ (motivating LIFO) or as loose as ‘items of any age can be
supplied’ (motivating FIFO), but rather ‘items that will not expire within a specified
time-window must be supplied’. Faced with such a demand model, and possibly with
multiple retailers, a supplier can choose his/her optimal issuance policy which need
not necessarily be LIFO or FIFO. To the best of knowledge, Ishii (1993), Ishii and
112
Nose (1996), and Haijema et al. (2005a, 2005b) are the only ones that have shown
some awareness of this important issue; see our discussion in Section 2.5. For future
research on perishables to be of more practical use, we need demand models that are
representative of the more general business rules and policies today.
While problems that involve competition (among retailers, suppliers, or supply
chains) have received a lot of attention in the last decade (see, for e.g., Cachon,
1998), models that include competition involving perishable and aging products have
not appeared in the literature. One distinct feature of competition in a perishable
commodity supply chain is that suppliers (retailers) may compete not only on avail-
ability and/or price but also on freshness.
In the produce industry, a close look at the relationship between suppliers and
buyers reveals several practical challenges. Perosio et al. (2001) present survey results
that indicate that about 9% of the produce in the U.S. is sold through spot markets,
and about 87.5% of product purchases are made under contracts with suppliers. This
motivates Perosio et al. (2001) to make the following observation, which is essen-
tially a call for further research: “Despite a number of considerable disadvantages, in
general, todays buyers and sellers alike appear to be won over by the greater price cer-
tainty that contracting makes possible...However, high degrees of product perishability,
weather uncertainty and resulting price volatility, and structural differences between
and among produce buyers and sellers create significant challenges to the design of
the produce contract.”
Recently, Burer et al. (2006) introduced different types of contracts used in the
agricultural seed industry and investigated - via single-period models - whether the
supply chain can be coordinated using these contracts. In their ongoing work, Boya-
batli and Kleindorfer (2006) study the implications of a proportional product model
(where one unit of input is processed to produce proportional amounts of multiple
agricultural outputs) on the optimal mix of long-term and short-term (spot) contract-
ing decisions. We believe further analysis of supplier-retailer relations, and the design
of contracts to improve the performance of a supply chain that involves perishable
products remain fruitful research topics.
Pricing was mentioned as one of the important research directions by Prastacos
(1984) to encourage collaboration between hospitals and blood banks/centers; this
is also echoed in Pierskalla (2004). There seems to be almost no research in this
113
direction to date. According to a recent survey in the US, the mean cost of 250ml. of
fresh frozen plasma to a hospital varied from $20 to $259.77, average costs of blood
components were higher in Northeastern states compared to the national average, and
hospitals with higher surgical volume typically paid less than the national average for
blood components in 2004 (AABB, 2005). Given the importance of health care both
for the general welfare and the economy, there is a pressing need for further research
to understand what causes such variability in this environment, and whether pricing
can be combined with inventory management to better match demand for perishable
blood components with the supply. The potential relevance of such work reaches well
beyond the health care industry.
Advances in technology have increased the efficiency of conventional supply chains
significantly; for perishable goods, technology can potentially have an even greater
impact. Not only is there the potential to enable information flow among different
parties in a supply chain, as has proved to be valuable in conventional chains, but there
is also the possibility of detecting and recording the age of the products in stock (e.g.
when RFID is implemented). This information can be used to affect pricing decisions,
especially of products nearing their usable lifetimes. Moreover, advances in technology
can potentially increase the freshness and extend the lifetime of products (e.g., when
better storage facilities or packaging equipment is used). The relative magnitudes of
these benefits calibrated to different product and market characteristics remains an
important open problem.
The majority of the work on the analysis of inventory management policies assume
that the state of the system is known completely; i.e., inventory levels of each age of
product at each location are known. However, this may not be the case in practice.
Cohen et al. (1981a) discuss the need for detailed demand and inventory informa-
tion to apply shortage or outdate anticipating transshipment rules in a centralized
system, and argue that the system would be better without transshipments between
the retailers if accurate information is not available. Chande et al. (2005) presents
an RFID architecture for managing inventories of perishable goods in a supply chain.
They describe how the profile of current on-hand inventory, including the age, can
be captured on a real-time basis, and conclude by stating that “there is a need for
measures and indicators ... to determine ... (a) whether such development would be
beneficial, and (b) when implemented, how the performance of the system compares
114
to the performance without auto ID enhancements.”
In Chapters 3 and 4 our focus was on the effect of substitution given practical
inventory policies and on the analysis of various replenishment policies, respectively.
There are a number of possible future directions for research closely related to the re-
search these two chapters. A challenge lies in the case where a supplier does not have
to commit to a single substitution option and substitution does not take place only as
a recourse (i.e. substitution decision need not be “static”). Although such a policy is
more difficult to implement (and to analyze), it provides more flexibility in satisfying
the demand. Hence, investigation of static versus dynamic substitution decisions in
fulfilling the demand warrants further attention. Likewise, evaluating substitution
policies for a product that has a lifetime of more than two periods is intriguing; even
for three-period lifetime case, substitution from new to old (in presence of products
with medium age) may not be justified. In a related direction, using a multiple pe-
riod lifetime with zero null demand for the first several periods of lifetime offers the
possibility of modelling positive lead-times in replenishing perishable goods. Another
worthwhile research area lies in investigating demand models based on customer-
choice (where the choice can be price-driven or shelf-life driven). Adding customer
behavior to the existing model or developing new models with customer gaming issues
are also promising lines of future research. In terms of future research for compu-
tational study on management of perishable inventory, one can conduct numerical
experiments with various demand distributions to test our heuristics or any other
possible heuristics.
115
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APPENDIX
The proofs of the analytical results are given below.
A.1. Cost Comparison Results under TIS
To prove the results on TIS, we use the definition of a pair introduced in Section 3.4.2.
CnI is the cost of substitution policy I in period n including holding, spoilage and
penalty costs, XnI and Y n
I are the amount of new and old items in stock, respectively,
for policy I, I = F ,N ,U ,D. Recall that we define a period as E (for equal) if
XnN = Xn
F ; we call the periods between two E periods a cycle and J is the number of
pairs in a cycle. If an E period follows another one, then it is called a trivial cycle.
A non-trivial cycle, starts with an E period in which downward substitution takes
place and ends with another E.
Proof of Proposition 3
Proof : Assume period n is an F period with XnF = Xn
N + ∆ for some ∆ > 0. If
demand for new items in this period is greater than or equal to the number of new
items under N (i.e. Dn2 ≥ Xn
N) then N will be at the inventory state (S, 0) in the next
period which implies that the period n + 1 is not an F period. Therefore Dn2 must
be less than XnN . That is, F has more than ∆ unsold new items, which are available
for substitution. If there is no need for the unsold new items (Dn1 ≤ S − Xn
N − ∆)
then all of these items would be carried over to the next period. However this would
make the period n + 1 an N period because under policy N , less new items would be
carried. Hence if there are two consecutive F periods, then there must be downward
substitution in the first one.
Proof of Proposition 4
Proof : Continuing from the proof of Proposition 3 let Dn1 = S − Xn
N − ∆ + k1
and Dn2 = Xn
N − k2 where k1, k2 > 0. Therefore Xn+1N = S − k2 and Xn+1
F =
S − (∆ + k2 − k1)+ as F has ∆ + k2 to substitute and k1 is needed. Since period
n+ 1 is an F period k2 must be greater than (∆ + k2− k1)+ thus k1 > ∆. (Note that
k1 > ∆ implies demand for old items in period n must be more than old item inventory
132
under N , i.e. Dn1 > S − Xn
N .) The substitution amount is ∆n = min{∆ + k2, k1}and Xn+1
F − Xn+1N = k2 − (∆ + k2 − k1)
+ hence Xn+1F − Xn+1
N = ∆n − ∆ (i.e. the
substitution amount minus ∆).
Proof of Lemma 4
We divide the proof of Lemma 4 into two Propositions.
Proposition 7 If conditions (3.5) and (3.10) hold, then F has lower cost than Noutside the pairs.
Proof : We look at a series of F periods starting or ending either with an E period
or a pair. Let period n be any F period in the series with ∆n = XnF − Xn
N . As
was pointed out in the proof of Proposition 4 to have another F in period n + 1,
Dn2 = Xn
N −k2 < XnN and Dn
1 = S−XnN +k1 > S−Xn
N . Then Xn+1F = S− (k2−k1)
+
and Xn+1N = S − k2 (i.e. ∆n+1 = min(k1, k2)). Also:
CnN = hk2 + p1k1, Cn
F = h(k2 − k1)+ + p1(k1 − k2)
+ + αD(∆n + ∆n+1),
CnN − Cn
F = (h + p1)∆n+1 − αD(∆n + ∆n+1).
Let the series start at period 1 and end at period M − 1 (i.e. ∆i > 0 ∀ i =
1, ..., M − 1). In periods 0 and M there are 2 possibilities each:
• If period 0 is an E (i.e. ∆0 = 0), then:
C0N − C0
F = (h + p1)∆1 − αD∆1.
• If period 0 is an N (i.e. end of a pair ), then: In this case C0N −C0
F would be an
amount carried over from the previous pair (the one that just ended in period
0). In other words an amount of (h + p1)∆1 − αD∆1, which is saved from the
previous pair (this is explained later in Proposition 8), will be used as C0N −C0
F .
Thus,
C0N − C0
F = (h + p1)∆1 − αD∆1. (5.1)
133
• If period M is an E (i.e. ∆M = 0) then:
CM−1N − CM−1
F ≥ −αD∆M−1. (5.2)
The proof of (5.2) is as follows; if period M − 1 is F there are six demand
scenarios that lead to an E period immediately after. We analyze these possible
cases:
Case 1. DM−12 = XM−1
N − k2 and DM−11 = S − XM−1
N + k1 with k1, k2 ≥ 0.
These demands result in XMF = S−(k2−k1)
+ and XMN = S−k2. F is short
of ∆M−1 + k1 old items while it has ∆M−1 + k2 unsold new items. Thus
minimum of these is the downward substitution amount. Then we have
CM−1N = hk2+p1k1 and CM−1
F = h(k2−k1)++p1(k1−k2)
++αD min(∆M−1+
k2, ∆M−1 + k1). In order for period M be an E, k1 or k2 (or both) must be
0; hence CM−1N − CM−1
F = −αD∆M−1.
Case 2. DM−12 = XM−1
N + k2 and D1 = S −XM−1N −∆M−1 + k1 with 0 ≤ ki <
∆M−1 for i ∈ {1, 2} so that XMF = S − (∆M−1 − k2 − k1)
+ and XMN = S.
The downward substitution amount is ∆M−1 − k2 because in order for
period M be an E, k1 + k2 ≥ ∆M−1 must hold. Then we have CM−1N =
p2k2 + m(∆M−1 − k1) and CM−1F = p1(k2 + k1 −∆M−1) + αD(∆M−1 − k2);
hence CM−1N − CM−1
F = (p2 − p1)k2 + p1(∆M−1 − k1) + m(∆M−1 − k1) −αD(∆M−1 − k2) > −αD∆M−1.
Case 3. DM−12 = XM−1
N +k2 and DM−11 = S−XM−1
N +k1 with k1, k2 ≥ 0 result
in XMF = XM
N = S. F is short of ∆M−1+k1 old items while it has ∆M−1−k2
unsold new items. Thus the downward substitution amount is ∆M−1− k2.
Then CM−1N = p2k2 +p1k1 and CM−1
F = p1(k1 +k2)+αD(∆M−1−k2); hence
CM−1N − CM−1
F = (p2 − p1)k2 − αD(∆M−1 − k2) > −αD∆M−1.
Case 4. DM−12 = XM−1
N + ∆M−1 + k2 and DM−11 = S − XM−1
N − ∆M−1 − k1
with k1, k2 ≥ 0. F is short of k2 old items while it has k1 unsold old
items. Then, there is an upward substitution in the amount of min(k2, k1)
resulting in XMF = XM
N = S. Thus CM−1N = p2(∆M−1+k2)+m(∆M−1+k1)
and CM−1F = p2(k2− k1)
+ + αU min(k1, k2) + m(k1− k2)+. If k1 < k2, then
CM−1N − CM−1
F = (m + p2)(∆M−1 + k1) − αUk1 > −αD∆M−1; otherwise
CM−1N −CM−1
F = (m+p2)(∆M−1+k2)−αUk2 > −αD∆M−1 as αU < m+p2.
134
Case 5. DM−12 = XM−1
N +∆M−1 +k2 and DM−11 = S−XM−1
N −∆M−1 +k1 with
∆M−1 > k1 ≥ 0, k2 ≥ 0 result in XMF = XM
N = S. No substitution takes
place. Thus, CM−1N = p2(∆M−1 + k2) + m(∆M−1− k1) and CM−1
F = p2k2 +
p1k1. Hence CM−1N −CM−1
F = m(∆M−1−k1)+p2∆M−1−p1k1 > −αD∆M−1.
Case 6. DM−12 = XM−1
N + ∆M−1 + k2 and DM−11 = S − XM−1
N + k1 with
k1, k2 ≥ 0 result in XMF = XM
N = S. There is no substitution, and CM−1N =
p2(∆M−1 + k2) + p1k1 and CM−1F = p2k2 + p1(∆M−1 + k1); hence CM−1
N −CM−1
F = (p2 − p1)∆M−1 > −αD∆M−1.
Therefore (5.2) is correct.
• If in period M a pair starts (i.e. the period M and M + 1 are F and N respec-
tively) then:
CM−1N − CM−1
F = (h + p1)∆M − αD(∆M−1 + ∆M).
We reduce this amount by (h + p1 − αD)∆M ; we save this (h + p1 − αD)∆M
as the “starting cost” of N for the forthcoming pair in period M and add it to
the cost of N for the cost comparison in the pairs (Proposition 8). Therefore
in this case:
CM−1N − CM−1
F = −αD∆M−1. (5.3)
Thus we have the following cost structure:
C0N − C0
F = (h + p1)∆1 − αD∆1,
C1N − C1
F = (h + p1)∆2 − αD(∆1 + ∆2),
C2N − C2
F = (h + p1)∆3 − αD(∆2 + ∆3),
· · ·CM−2
N − CM−2F = (h + p1)∆M−1 − αD(∆M−2 + ∆M−1),
CM−1N − CM−1
F ≥ −αD∆M−1.
Hence,M−1∑
i=0
CiN − Ci
F ≥ (h + p1 − 2αD)∆, (5.4)
where
∆ =M−1∑
i=1
∆i.
135
Then left term in (5.4) is positive (i.e. F is less costly) as αD < (h+p1)/2 is assumed.
Proposition 8 If conditions (3.8) and (3.9) hold, then F has lower cost than Ninside the pairs.
Proof : Both N and F start at an E state at the beginning. Until the period in
which downward substitution occurs both have same inventory state, they are E’s.
The period after the substitution takes place becomes an F period (i.e. F has more
new items than N ). Due to this substitution F incurs a substitution cost αD but
saves h + p1 as a penalty cost is avoided by fulfilling a demand for old product and
that item is sold instead of being held in inventory. Therefore, F incurs h + p1 − αD
less cost than N per substitute item. There might be a number of F periods until
an N period; using the accounting explained preceding equation (5.3), we say that at
the beginning of a pair, F has a cost lower by an amount equal to the starting cost
(h + p1 − αD per substitution).
Let the first pair start in period M with XMF XM
N + ∆M and XM+1N = XM+1
F + ∆.
Intuitively this means under F more items (in the amount of ∆) have been aged and
∆ less items perished at the end of period M resulting a cost difference of (m− h)∆
in favor of F . (If these items did not perish then the ∆ items would not have aged,
they would have been substituted.)
We first prove that “the first part of the pair cost” until period M + 1 is at least
(m + p1−αD)∆. To have an N period in period M + 1 there are four possibilities for
the demand in period M :
Case i. DM2 = XM
N −k2 and DM1 = S−XM
N −∆M−k1 with k1, k2 ≥ 0. No substitution
takes place and the resulting inventory levels are XM+1F = S − ∆M − k2 and
XM+1N = S − k2 (i.e. ∆ = ∆M). Then CM
N = hk2 + m(∆M + k1) and CMF =
h(∆M +k2)+mk1. Hence CMN −CM
F = (m−h)∆M . After adding (h+p1−αD)∆M
we have (m + p1 − αD)∆ as the first part of the pair cost.
Case ii. DM2 = XM
N − k2 and DM1 = S −XM
N −∆M + k1 with ∆M > k1 ≥ 0, k2 ≥ 0.
There is downward substitution in the amount of k1 so that XM+1F = S−(∆M +
k2−k1) and XM+1N = S−k2 (i.e. ∆ = ∆M−k1). Then CM
N = hk2 +m(∆M−k1)
136
and CMF = h(∆M +k2−k1)+αDk1. Hence CM
N −CMF = (m−h)(∆M−k1)−αDk1.
After adding (h + p1−αD)∆M , we have (m + p1−αD)∆ + (h + p1− 2αD)k1 as
the first part of the pair cost.
Case iii. DM2 = XM
N + k2 and DM1 = S−XM
N −∆M − k1 with ∆M > k2 ≥ 0, k1 ≥ 0.
There is no substitution then XM+1F = S − (∆M − k2) and XM+1
N = S (i.e.
∆ = ∆M−k2). Therefore CMN = p2k2+m(∆M+k1) and CM
F = h(∆M−k2)+mk1.
Hence CMN −CM
F = (m−h)(∆M−k2)+(m+p2)k2. After adding (h+p1−αD)∆M
we have (m + p1−αD)∆ + (h + p1−αD + m + p2)k2 as the first part of the pair
cost.
Case iv. DM2 = XM
N + k2 and DM1 = S − XM
N − ∆M + k1 with 0 ≤ ki < ∆M
for i ∈ {1, 2}. Under F there are ∆M − k2 unsold new items and k1 more
old items are needed and in order to have a pair k1 < ∆M − k2 must hold.
Therefore XM+1F = S − (∆M − k2 − k1) and XM+1
N = S (i.e. ∆ = ∆M − k2 − k1
). Then CMN = p2k2 + m(∆M − k1) and CM
F = h(∆M − k2 − k1) + αDk1.
Hence CMN −CM
F = (m− h)(∆M − k2 − k1) + (m + p2)k2 − αDk1. After adding
(h+p1−αD)∆M we have (m+p1−αD)∆+(h+p1−αD+m+p2)k2+(h+p1−2αD)k1
as the first part of the pair cost.
In period M + 1 if high demand for new items is seen this would benefit N since
there is more new item under N in this period. We do a case-by-case analysis to
show that this benefit is not greater than its cost, (m + p1 − αD)∆. The proof is by
induction, that is we first show that the claim is true for J = 1 (J is number of pairs
in the cycle) and 2, then assume it is true for J = j and prove it stays correct for
J = j + 1. For J = 1 there are 3 possibilities. By examining these possible cases:
Case 1. DM+12 = XM+1
F + ∆ + k2 and DM+11 = S −XM+1
F + k1 where k1 and k2 ≥ 0:
CM+1F = p1k1 + p2(∆ + k2), CM+1
N = p1(∆ + k1) + p2k2
CM+1F − CM+1
N = (p2 − p1)∆ ≤ (m + p1 − αD)∆
as we assume p2 − p1 ≤ m + p1 − αD.
Case 2. DM+12 = XM+1
F +∆+k2 and DM+11 = S−XM+1
F −∆+k1 where ∆ ≥ k1 ≥ 0
and k2 ≥ 0:
CM+1F = p2(∆ + k2 − (∆− k1)) + αU(∆− k1) = p2(k2 + k1) + αU(∆− k1),
137
CM+1N = p2k2 + p1k1
CM+1F − CM+1
N = (p2 − p1)k1 + αU(∆− k1) ≤ (m + p1 − αD)∆
as we assume αU + αD ≤ m + p1.
Case 3. DM+12 = XM+1
F + ∆ + k2 and DM+11 = S − XM+1
F − ∆ − k1 where k1 and
k2 ≥ 0:
CM+1F = p2(∆ + k2 − (∆ + k1))
+ + m(∆ + k1 − (∆ + k2))+
= p2(k2 − k1)+ + m(k1 − k2)
+ + αU(∆ + k1)
CM+1N = p2k2 + mk1
If k2 ≥ k1, then CM+1F −CM+1
N = −(p2 +m)k1 +αU(∆+ k1) ≤ (m+ p1−αD)∆.
If k2 < k1, then CM+1F −CM+1
N = −(p2 +m)k2 +αU(∆+ k2) < (m+ p1−αD)∆.
For all three cases above, in the next period F and N reach the same state, an E, as
DM+12 = XM+1
F + ∆ + k2. If there are more F periods from period n until the cycle
closure the claim is true due to Proposition 3.
For J = 2, there are three cases (of the original nine) with DM+12 = XM+1
F + k2
where ∆ > k2 ≥ 0, and the inventory state for F is (S, 0) while the state for Nis (S − (∆ − k2), ∆ − k2). This implies the period M + 2 is an F period with
∆M+2 = ∆− k2.
Note that while proving for J = 2, in order to make a correct cost analysis, we
subtract (h + p1 − αD)(∆M+2) from CM+1N for the first pair. We are “saving” this
amount to use in the second pair in this cycle (either a series of F s and then a pair
or an immediate pair) as we mentioned in the proof of Proposition 7; see equation
(5.1). This makes the J = 2 case identical to J = 1 case from that period on:
Case 4. DM+12 = XM+1
F + k2 and DM+11 = S − XM+1
F + k1 where k1 ≥ 0 and
∆ > k2 ≥ 0:
CM+1F = p2k2 + p1k1, CM+1
N = h(∆− k2) + p1(∆ + k1)− (h + p1 − αD)(∆− k2)
CM+1F − CM+1
N = (p2 − p1)k2 − αD(∆− k2) < (m + p1 − αD)∆
138
Case 5. DM+12 = XM+1
F + k2 and DM+11 = S − XM+1
F −∆ + k1 where ∆ ≥ k1 ≥ 0
and ∆ > k2 ≥ 0:
CM+1F = p2[k2 − (∆− k1)]
+ + m[∆− k1 − k2]+ + αU min(k2, ∆− k1)
CM+1N = h(∆− k2) + p1k1 − (h + p1 − αD)(∆− k2).
If k1 + k2 ≤ ∆, then
CM+1F − CM+1
N = m∆−mk1 −mk2 − h∆ + hk2 − p1k1 + αUk2
+ (h + p1 − αD)(∆− k2)
= (m + p1)(∆− k1 − k2) + αUk2 − αD(∆− k2)
= (m + p1 − αD)(∆− k2) + αUk2 − (m + p1)k1 < (m + p1 − αD)∆.
If k1 + k2 > ∆, then
CM+1F − CM+1
N = p2k2 + p2k1 − p2∆− h∆ + hk2 − p1k1 + αU(∆− k1)
+ (h + p1 − αD)(∆− k2)
= p2(k2 + k1 −∆) + p1(∆− k2 − k1) + h(−∆ + k2 + ∆− k2)
+ αU(∆− k1)− αD(∆− k2)
= (p2 − p1)(k1 − (∆− k2)) + αU(∆− k1)− αD(∆− k2)
< (m + p1 − αD)∆.
Case 6. DM+12 = XM+1
F +k2 and DM+11 = S−XF −∆−k1 where k1 and ∆ > k2 ≥ 0:
CM+1F = m(∆ + k1 − k2) + αUk2,
CM+1N = h(∆− k2) + mk1 − (h + p1 − αD)(∆− k2)
CM+1F − CM+1
N = m(∆ + k1 − k2)− h(∆− k2)−mk1 + (h + p1 − αD)(∆− k2) + αUk2
= (m + p1 − αD)(∆− k2) + αUk2 < (m + p1 − αD)∆.
We also have the cases where DM+12 = XM+1
F − k2 (i.e. new items are abundant) and
there is no opportunity for the N policy to cash in on the extra new items that it
has in period n. We also show that for these cases CM+1F −CM+1
N < (m + p1 − αD)∆
holds again, saving (h + p1 − αD)∆M+2:
139
Case 7. DM+12 = XM+1
F − k2 and DM+11 = S −XM+1
F + k1 where k1 ≥ 0 and k2 ≥ 0:
XM+2F = S − (k2 − k1)
+, XM+2N = S − (∆ + k2) then ∆M+2 = ∆ + min(k1, k2).
Costs are as follows:
CM+1F = h(k2 − k1)
+ + p1(k1 − k2)+ + αD min(k1, k2),
CM+1N = h(∆ + k2) + p1(∆ + k1)− (h + p1 − αD)∆M+2.
If k1 < k2:
CM+1F − CM+1
N = −αD∆ < (m + p1 − αD)∆.
Case 8. DM+12 = XM+1
F − k2 and DM+11 = S − XM+1
F −∆ + k1 where ∆ ≥ k1 ≥ 0
and k2 ≥ 0:
XM+2F = S − k2, XM+2
N = S − (∆ + k2) then ∆M+2 = ∆. Costs are follows:
CM+1F = hk2 + m(∆− k1), CM+1
N = h(∆ + k2) + p1k1 − (h + p1 − αD)∆M+2.
If k1 < k2:
CM+1F − CM+1
N = (m + p1)(∆− k1)− αD∆, ≤ (m + p1 − αD)∆.
Case 9. DM+12 = XM+1
F − k2 and DM+11 = S −XM+1
F −∆− k1 where k1 and k2 ≥ 0:
XM+2F = S − k2, XM+2
N = S − (∆ + k2) then ∆M+2 = ∆. Costs are follows:
CM+1F = hk2 + m(∆ + k1), CM+1
N = h(∆ + k2) + mk1 − (h + p1 − αD)∆M+2.
If k1 < k2:
CM+1F − CM+1
N = (m + p1 − αD)∆.
Hence F has lower cost than N in the pairs as well, and from then on we have a
J = 1 case.
Assume the claim is true for J = j and F has extra (h + p1 − αD)(∆− k2) at the
end of the last pair as in case J = 2. If the (j + 1)st pair is immediately after the
140
jth one; then F with the extra (h + p1 − αD)(∆ − k2) benefit, will be less costly if
m + 2p1 > p2 + αD holds. If there is no pair as an immediate successor of the jth
pair, then there will be a number of F periods before the pair j + 1 and F will have
a h + p1 − αD benefit before this next pair as discussed in Proposition 7. Then the
proof is identical to the J = 1 case. Hence the claim is true.
Proof of Lemma 7
Proof : For cases in which there is no upward substitution, U is identical to N and
we know that F is better than N under conditions (3.5) and (3.7). So we look at
cycles where there is upward substitution.
For E periods, if there is any upward substitution, then F and U have the same
costs and the next period is also an E period.
If there is upward substitution by U or F in an F period, then the next period
is either a U period (i.e. there is a pair) or an E (specifically both policies reach the
inventory state (S, 0) ). Let n − 1 be such an F period with Xn−1F = Xn−1
U + ∆′.
To have an upward substitution, Dn−12 must be greater than Xn−1
U which makes
XnU = S. Let ∆ = Xn
U − XnF ; then it must hold that ∆
′> ∆ ≥ 0. This implies
Dn−12 = Xn−1
U + ∆′ − ∆; and we also see that when ∆ = 0 then period n is E with
state of (S,0).
We compare U with N : Xn−1N = Xn−1
U + ∆′ − ∆ and in period n, both U and
N are at state (S, 0). The cost of having the ∆′difference is (h + p1 − αD)∆
′more
for U as compared to F . As Xn−1F − Xn−1
N = ∆, cost of U is greater than N by
(h + p1 − αD)(∆′ − ∆) for the difference of new items because fewer items were
substituted downward to get a net difference of ∆. U incurs the cost of upward
substitution, which is αU per item. Thus, if Dn−11 ≤ S − Xn−1
U − (∆′ − ∆) then U
is more costly than policy N by (h + p1 − αD)(∆′ − ∆) + αU(∆
′ − ∆) until period
n. Otherwise (i.e. if Dn−11 > S − Xn−1
U − (∆′ − ∆)), U will have to substitute less
than ∆′ − ∆ and will incur p2 for every new item demand that it cannot fulfill by
substitution while N incurs p1 as penalty costs. Therefore, for every state in U , we
can find a counterpart in N that has a lower cost than U . As N has higher cost than
F , U is also more costly than F .
141
A.2. Cost Comparison Results under NIS
Similar to what we did in Section 3.4, we define the pairs, and divide the horizon
into different classes of periods. However, in this case a pair is defined based on the
inventory level of old items: If the amount of old goods in stock in period n is higher
under policy F , then we call that period an F period. We call a period E if the
amount of old items in stock for two policies is equal. We define cycle, trivial cycle,
non-trivial cycle, and CnI for I = F ,N ,U ,D as previously.
Proof of Lemma 12
Proof : By Proposition 5, in any given period n Y nN = Y n
F + ∆ where ∆ ≥ 0 is the
downward substitution amount in period n− 1. In an E period, ∆ = 0; otherwise it
is N. F and N have the same cost in a trivial cycle. We look at a non-trivial cycle.
For each substitution, F incurs a substitution cost αD but saves h + p1; F incurs
h + p1 − αD less cost than N per substitute item. In this N period (let this period
be the nth period) there are three possibilities for F :
i. No substitution takes place: Let the demand for old items be Dn1 = Y n
F + K.
As Y nN = Y n
F + ∆ we have the cost difference between the policies for the cycle
given below. (Note that we do not include the costs that are incurred in cycle-
ending E periods in our “cycle cost”. We account for that cost in the subsequent
cycle.)
n∑
i=n−1
CiN−Ci
F = (h+p1−αD)∆+
−p1∆ if K ≥ ∆, Dn2 ≥ S,
m(∆−K)− p1K if ∆ > K ≥ 0, Dn2 ≥ S,
m∆ if 0 > K ≥ −XnF , Dn
2 ≤ S.
The next period is an E and the cycle ends. Thus, for this case, the claim is
true.
ii. Upward substitution takes place: To have upward substitution Dn2 must be
greater than S and Dn1 must be less than Y n
F , hence Dn2 = S+L and Dn
1 = YF−K
where L,K > 0. The cost difference between F andN for the cycle is as follows:n∑
i=n−1
CiN − Ci
F = (h + p1 − αD + m)∆ + (m + p2 − αU)us
where us = min{K, L} is the upward substitution amount. As the next period
is an E, the cycle ends. Therefore the claim is true for this case.
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iii. Downward substitution takes place: A downward substitution in period n
causes the period n + 1 to be an N, therefore the cycle does not end in period
n + 1. There can be a number of Ns in a cycle, we define J as this number.
For J = 2, let Dn2 = S − L and Dn
1 = Y nF + K where S > L ≥ 0 and K > 0.
Downward substitution takes place twice in this cycle, in periods n− 1 and n.
We compare the costs of F and N through the periods n− 1, n and n+1. The
cost difference between N and F for the first two periods of the cycle (n − 1
and n) is as follows:
n∑
i=n−1
CiN−Ci
F = (h+p1−αD)∆+(h+p1−αD)dsn−p1K+m(∆−K)++p1(K−∆)+
where dsn = min{K, L} is the downward substitution amount in period n. Note
that ∆ = dsn−1. Thus:
n∑
i=n−1
CiN−Ci
F = (h+p1−αD)∆+(h+p1−αD)dsn−p1K+
{m(∆−K) if K < ∆,p1(K −∆) if K ≥ ∆.
= (h+p1−αD)dsn +
{(h− αD)∆ + (m + p1)(∆−K) if K < ∆,(h− αD)∆ if K ≥ ∆.
However the (h + p1 − αD)dsn part of the expression above will be accounted
for the n + 1st period’s cost difference. The reasoning is as follows: The next
period (n+1) is an N. As there is no downward substitution in period n+1, the
cycle ends in period n+2, which is an E. For the cost difference between F and
N in period n + 1 we refer to part i or ii of the proof as there is no downward
substitution. Therefore for correct cost accounting (h+ p1−αD)dsn is included
in cost comparison in period n + 1. Thus the claim is proved for J = 2.
The general J = j case is depicted Table 5.1, and the following expression is
the cost difference between N and F for the J = j case:
n+j−2∑
i=n−1
CiN − Ci
F =
n+j−3∑
i=n−1
[(h + p1 − αD)dsi − p1K
i+1 + m(dsi −K i+1)+ + p1(Ki+1 − dsi)+
]
+ (h + p1 − αD)dsn+j−2.
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Period n-1 n n+1 ... n+j-2 n+j-1 n+j
Down. Subs. dsn−1 dsn dsn+1 ... dsn+j−2 dsn+j−1 = 0 dsn+j
X iN −X i
F 0 dsn−1 dsn ... dsn+j−4 dsn+j−2 dsn+j−1 = 0
Type E N N ... N N E
Table 5.1: A sequence of N periods under NIS.
where dsn = min{Kn, Ln}, Ln = S −Dn2 and Kn = Dn
1 − Y nF . Thus,
n+j−2∑
i=n−1
CiN − Ci
F = (h + p1 − αD)dsn+j−2 +n+j−3∑
i=n−1
(h + p1 − αD)dsi − p1Ki+1
+
{m(dsi −K i+1) if Ki+1 < dsi,p1(K
i+1 − dsi+1) if Ki+1 ≥ dsi.
= (h + p1 − αD)dsn+j−2
+n+j−3∑
i=n−1
{(h− αD)dsi + (m + p1)(dsi −K i+1) if K i+1 < dsi,(h− αD)dsi if K i+1 ≥ dsi.
Similar to the J = 2 case, (h + p1 − αD)dsn+j−2 is included for the cost com-
parison in period n + j − 1 referring part i or ii. Thus the claim is proved.
Proof of Lemma 13
Proof : First, as we assume upward substitution is sensible, αU < m+p2. Comparing
F and U is similar to the comparison of F and N as in any given period n Y nN = Y n
U =
Y nF + ∆ for some ∆ ≥ 0, by Proposition 5. Unless there is an upward substitution,
N and U have the same cost, therefore part i (when K ≥ ∆ ) and iii of the proof of
Lemma 12 provide the condition αD < h that is needed for F to be less costly than
U ; and the condition αD > m + h + p1 so that U is less costly than F . We analyze
the remaining cases:
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Upward substitution under both policies: Dn2 = S +L and Dn
1 = Y nF −K leads
to upward substitution under both F and U , where L > 0, Y nF > K > 0.
CnU = (h + p1 − αD)∆ + p2(L− usU) + m(∆ + K − usU) + αU usU
where usU = min{∆ + K, L}.
CnF = p2(L− usF ) + m(K − usF ) + αU usF
where usF = min{K, L}.
CnU − Cn
F = (h + p1 − αD)∆ + m∆− (m + p2 − αU)(usU − usF ),
= (m + h + p1 − αD)(∆− (usU − usF ))
+ (h− αD + αU − (p2 − p1))(usU − usF ). (5.5)
(5.5) is positive if αD < h + αU − (p2 − p1), as min{∆ + K, L} ≥ min{K,L}and ∆ ≥ L−K (when L > K). (5.5) is negative if αD > m + h + p1. Thus the
claim is true for this case.
Upward substitution under U , no substitution under F : In this case Dn2 =
S + L and Dn1 = Y n
F + K with L > 0, ∆ > K > 0.
CnU = (h + p1 − αD)∆ + p2(L− us) + m(∆−K − us) + αU us
where us = min{∆−K, L}.
CnF = p2L + p1K.
CnU − Cn
F = (h + p1 − αD)∆ + m(∆−K − us) + (αU − p2)us− p1K,
= (h− αD)∆ +
{(αU − p2 + p1)(∆−K) if K + L > ∆,(m + p1)(∆−K)− (m + p2 − αU)L if K + L ≤ ∆,
=
(h− αD)K + [h− αD + αU − (p2 − p1)](∆−K) if K + L > ∆,
(h− αD)(∆− L) + [h− αD + αU − (p2 − p1)]L+ (m + p1)(∆−K − L) if K + L ≤ ∆.
= (h− αD)K +{[h− αD + αU − (p2 − p1)](∆−K) if K + L > ∆,(h− αD + m + p1)(∆− L−K) + [h− αD + αU − (p2 − p1)]L if K + L ≤ ∆.
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Thus if αD < min{h, h+αU−(p2−p1)} F is less costly than U . If αD > h+m+p1
U is less costly than F ; as h + m + p1 > max{h, h + αU − (p2 − p1)} when
m + p2 > αU . Thus the claim is proved.
Proof of Lemma 15
Proof : Comparison of D to U is identical to the comparison of D with N except
when U does upward substitution. We analyze that case.
Y nD +∆ = Y n
U by Proposition 5 and Dn2 = S+L, Dn
1 = Y nD +∆−K, (S ≥ Y n
D +∆ >
K > 0 and L > 0) leads to upward substitution under U and no substitution under
D.
CnU = p2(L− us) + m(K − us) + αUus,
where us is the upward substitution amount and is equal to min{K,L}. Then
CnD = p2L + m(K −∆)+ + p1(∆−K)+.
There are four demand scenarios:
Case 1. When K ≤ L and K ≤ ∆; us = K and
CnU = (h + p1 − αD)∆ + p2(L−K) + αUK, Cn
D = p2L + p1(∆−K),
CnU − Cn
D = (h− αD)∆ + (αU − (p2 − p1))K,
= (h− αD)(∆−K) + (h− αD + αU − (p2 − p1))K.
Case 2. When K ≤ L and K > ∆; us = K and
CnU = (h + p1 − αD)∆ + p2(L−K) + αUK, Cn
D = p2L + m(K −∆),
CnU − Cn
D = (m + h + p1 − αD)∆− (m + p2 − αU)K,
= (h− αD + αU − (p2 − p1))∆− (m + p2 − αU)(K −∆).
Case 3. When K > L and K ≤ ∆; us = L and
CnU = (h + p1 − αD)∆ + m(K − L) + αUL, Cn
D = p2L + p1(∆−K),
CnU − Cn
D = (h− αD)∆ + (m + p1)K − (m + p2 − αU)L,
= (h− αD)∆ + (m + p1)(K − L) + (αU − (p2 − p1))L.
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Case 4. When K > L and K > ∆; us = L and
CnU = (h + p1 − αD)∆ + m(K − L) + αUL, Cn
D = p2L + m(K −∆),
CnU − Cn
D = (m + h + p1 − αD)∆− (m + p2 − αU)L.
Thus although the condition αD < min{h, h + αU − (p2 − p1)} is sufficient for D to
be less costly than U in most of the demand scenarios (including no substitution and
downward substitution cases), the analyses of cases 2 and 4 show that there is no
condition that guarantees D to be less costly than U in general, unless αU > m + p2,
which in itself guarantees that upward substitution is an unattractive decision.
However, if αD > h+m+p1, then U is guaranteed to be less costly than D. (Note
αD > m + h + p1 implies αD > h + αU − (p2 − p1) as αU < m + p2.)
Proof of Lemma 14
Proof : Unless there is an upward substitution D and F have the same cost because
Y nF = Y n
D = Y nN − ∆ for some ∆ ≥ 0 by Proposition 5. Therefore parts i (no
substitution) and iii (downward substitution) of the proof of Lemma 12 provide the
conditions αD < h that is required for D to be less costly than N . We also need
to analyze the following case (where there is no substitution): Dn2 = S + L and
Dn1 = Y n
D −K where L > 0 and XnD > K > 0. Then,
CnN = (h + p1 − αD)∆ + p2L + m(∆ + K), Cn
D = p2L + mK,
CnN − Cn
D = (m + h + p1 − αD)∆.
Thus the claim is true.
A.3. Results on Freshness of Inventory
Proof of Proposition 6
Proof : Inside a pair, we define ∆′
= XnN − Xn
F > 0 and ∆ = Xn−1F − Xn−1
N > 0.
To prove part (i), it is enough to observe: By definition of a pair XnF = Xn
N −∆′ ≤
S −∆′< S. But if Dn−1
2 ≥ Xn−1F , then from the inventory recursions under TIS, we
have XnF = S, which is contradicting. Part (ii) follows from the inventory recursions
147
and substitution from old items to new items implies XnF = S. For part (iii), assume
Dn−11 = S − Xn−1
N + k where k > 0. Since old item inventory under F in period
n− 1 is S−Xn−1N −∆, there is excess demand of ∆ + k for old items in period n− 1.
There must be some new items to fulfill this shortage by downward substitution due
to part (i). (Otherwise both F and N will be at inventory state (S, 0).) Thus if ds is
the amount of accepted downward substitution, old item inventory in period n will
be [Xn−1N − ds − Dn−1
2 ]+ and [Xn−1N − Dn−1
2 ]+ for F and N respectively. This is a
contradiction since in period n there must be less old items under N (i.e. there must
be more new items under N ) by definition of a pair. Therefore the claim is true.
Proof of Lemma 17
Proof : Equalities follow from inventory recursions (i.e. XnF = Xn
D and XnU = Xn
N for
all n). We show the inequality below.
Suppose there is a pair in periods n − 1 and n. Inside a pair, we define ∆′
=
XnN −Xn
F > 0 and ∆ = Xn−1F −Xn−1
N > 0. There are two possible cases inside a pair
due to Proposition 6.
Case 1. Dn−11 and Dn−1
2 are less than the inventories under policy F ; so no shortage
or substitution takes place under F . Dn−12 = Xn−1
N + ∆ − k (S > k ≥ 0) and
Dn−11 = S −Xn−1
N −∆− l (l ≥ 0). Then,
XnF = S − (Xn−1
N + ∆−Xn−1N −∆− k) = S − k
and XnN = S + ∆− k. Thus ∆
′= ∆.
Case 2. There is downward substitution. Demands are: Dn−12 = Xn−1
N + ∆ − k
(S > k ≥ 0) and Dn−11 = S − Xn−1
N − ∆ + l, but the downward substitution
amount dsn−1 is constrained such that min{∆, l} > dsn−1 > 0. Then
XnF = S−(Xn−1
N +∆−Xn−1N −∆+k−dsn−1)
+ = S−k+dsn−1; XnN = S+∆−k.
Thus ∆′= S + ∆− k − (S − k + dsn−1) = ∆− dsn−1.
In both cases, we observe that ∆′ ≤ ∆. This means on the average there are always
more new items under F , in the pairs. By definition XkF ≥ Xk
N for a period k that is
outside the pairs.
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