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Retail inventory management with lost sales

Curseu - Stefanut, A.

DOI:10.6100/IR721256

Published: 01/01/2012

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Citation for published version (APA):Curseu - Stefanut, A. (2012). Retail inventory management with lost sales Eindhoven: Technische UniversiteitEindhoven DOI: 10.6100/IR721256

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Retail Inventory Management with Lost Sales

This thesis is number D147 of the thesis series of the Beta Research School forOperations Management and Logistics. The Beta Research School is a joint effort ofthe departments of Industrial Engineering & Innovation Sciences, and Mathematicsand Computer Science at Eindhoven University of Technology and the Centre forProduction, Logistics and Operations Management at the University of Twente.

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-3061-8

Printed by Proefschriftmaken.nl, EindhovenCover designed by Paul Versparget


PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voorPromoties in het openbaar te verdedigenop maandag 23 januari 2012 om 16.00 uur

door

Alina Curseu

geboren te Alba-Iulia, Roemenie

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. J.C. Fransooenprof.dr. N. Erkip

Copromotor:prof.dr. T. van Woensel

To my family

Acknowledgements

I would like to take this opportunity to thank all those who, one way or another, haveoffered support over the past few years.

First, I would like to thank my supervisors, prof. Jan Fransoo and prof. Tom vanWoensel from TU/e, for giving me the opportunity to carry out this research. Itcertainly opened the door for many learning experiences. I thank prof. Fransoo forhis advice and insightful discussions, as well as his support and patience over the years.For the many hours and energy invested in this project, I am especially thankful tomy co-promotor Tom van Woensel. I also wish to acknowledge his feedback, valuablecomments and view on retail problems, and his relentless involvement throughout thedifferent phases of this project.

Second, I owe special thanks to my second promoter prof. Nesim Erkip from BilkentUniversity who kindly accepted to join this project. This dissertation benefited a lotfrom his advice and constructive ideas over the years, his prompt and careful feedbackon different parts of this dissertation. Thank you also for hosting my short visit toBilkent University.

For their willingness to serve on my doctoral committee, as well as for their timededicated to evaluate this dissertation, I would like to thank prof. Stefan Minnerfrom University of Vienna, prof. Rene de Koster from Erasmus University Rotterdam,prof. Ton de Kok and prof. Ivo Adan from TU/e.

I am also thankful to the members of the OPAC group for creating a stimulatingacademic environment, to the fellow PhD students for providing peer support, and tothe OPAC secretaries for their continuous assistance. In particular, I wish to thankOla Jabali for many open and friendly discussions, and her lively support over theyears. I am also thankful to my successive officemates Ingrid Vliegen and YoussefBoulaksil for their company, to Gergely Mincsovics for many fruitful discussions onMDPs and not only, and to Ingrid Reijnen for her support and a couple of extremelynice photos to treasure.

On a personal note, I wish to thank all old and new friends over the years. I thank myold friends from home, for their long lasting friendship, support and encouragement.

I thank my new friends in the Netherlands for their trust and support over theyears. Life would have been very different without the Romanian community inthe Netherlands. I thank you all for many familiar discussions, many birthday andchildren parties that energized the work-life balance. I am also fortunate to havingmade wonderful Dutch friendships. Thank you Sandra for your continuous supportand for being part of my family’s life. Thank you Apinya and Carel for our growingfriendship over the years.

Most of all, I am deeply thankful to my family, for their unconditional love andsupport in every way possible throughout the process of this dissertation and beyond.I am grateful to my husband Petru for standing by me in more difficult, and doubtfultimes. This dissertation would not have been possible without his boundless loveand support. Finally, I am deeply grateful for the two wonderful children in my life,Dragos and Antonia. Thank you both for changing my life so beautifully!

Alina Curseu

November 2011, Eindhoven

Contents

1 Introduction 11.1 Motivation and objective . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Scope of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Research questions and contributions of the dissertation . . . . . . . . 8

1.3.1 Modeling handling operations in grocery retail stores . . . . . . 81.3.2 Lost-sales inventory models with batch ordering and handling

costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Retail inventory control with shelf space and backroom consid-

eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.4 Efficient control of lost-sales inventory systems with batch

ordering and setup costs . . . . . . . . . . . . . . . . . . . . . . 111.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Modeling handling operations in grocery retail stores: an empiricalanalysis 152.1 Introduction and related literature . . . . . . . . . . . . . . . . . . . . 152.2 Conceptual model and hypotheses development . . . . . . . . . . . . . 182.3 Study design and data description . . . . . . . . . . . . . . . . . . . . 212.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Sequential regression results . . . . . . . . . . . . . . . . . . . . 242.4.2 Overall regression results . . . . . . . . . . . . . . . . . . . . . 282.4.3 Validation of the results . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Analytical insights and implications for retailers . . . . . . . . . . . . . 322.5.1 Extending the EOQ-model with shelf stacking . . . . . . . . . 322.5.2 Order of magnitude for efficiency gains in stacking . . . . . . . 33

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Appendix A. Shelf stacking activities . . . . . . . . . . . . . . . . . . . . . . 37Appendix B. Descriptive statistics of the empirical datasets . . . . . . . . . 37Appendix C. Validation results for chain B . . . . . . . . . . . . . . . . . . 37

3 Lost-sales inventory models with batch ordering and handling costs 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Old and new heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4.1 The (s, S, nq) and (s,Q, nq) policies . . . . . . . . . . . . . . . 543.4.2 The (s,Q|S, nq) policy . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5.1 On the structure of the optimal policy . . . . . . . . . . . . . . 563.5.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . 583.5.3 Performance of the (s,Q|S, nq) heuristic . . . . . . . . . . . . . 62

3.6 Penalty for not taking handling into account . . . . . . . . . . . . . . 673.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Appendix A. On unit vs. batch costs . . . . . . . . . . . . . . . . . . . . . . 71Appendix B. Computational issues . . . . . . . . . . . . . . . . . . . . . . . 72

4 Retail inventory control with shelf space and backroom considera-tion 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 System under study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Model formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.1 Model with continuous backroom operations . . . . . . . . . . 814.3.2 Model with fixed extra handling costs . . . . . . . . . . . . . . 83

4.4 Numerical study: the model with continuous backroom operations . . 854.4.1 On the structure of the optimal policy . . . . . . . . . . . . . . 854.4.2 Sensitivity analyses: the effect of V , Ks and p . . . . . . . . . . 864.4.3 Managerial insights . . . . . . . . . . . . . . . . . . . . . . . . . 924.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5 Numerical study: the model with fixed extra handling costs . . . . . . 984.5.1 On the structure of the optimal policy . . . . . . . . . . . . . . 994.5.2 Sensitivity analyses: the effect of V , Ke and p . . . . . . . . . . 1014.5.3 Managerial insights . . . . . . . . . . . . . . . . . . . . . . . . . 1064.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Appendix A. Transition probability matrix . . . . . . . . . . . . . . . . . . 110Appendix B. Additional numerical results . . . . . . . . . . . . . . . . . . . 111Appendix C. Related models . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Efficient control of lost-sales inventory systems with batch orderingand setup costs 1155.1 Introduction and related literature . . . . . . . . . . . . . . . . . . . . 1155.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3 On the structure of the optimal policy: K = 0 vs. K > 0 . . . . . . . 122

5.3.1 The case L = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.3.2 The case 0 < L < 1 . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3.3 The case L = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4 The effectiveness of the (s,Q|S, nq) policy . . . . . . . . . . . . . . . . 1325.4.1 The (s,Q|S, nq) policy . . . . . . . . . . . . . . . . . . . . . . . 1325.4.2 Effectiveness of the (s,Q|S, nq) policy . . . . . . . . . . . . . . 134

5.5 Performance of the (s, nq) policy . . . . . . . . . . . . . . . . . . . . . 1395.5.1 Numerical results: L = 0 . . . . . . . . . . . . . . . . . . . . . 1405.5.2 Numerical results: L = 1 . . . . . . . . . . . . . . . . . . . . . 1415.5.3 Numerical results: 0 < L < 1 . . . . . . . . . . . . . . . . . . . 142

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Appendix A. Selected numerical results . . . . . . . . . . . . . . . . . . . . 144Appendix B. Approximate (s,Q|S) policies . . . . . . . . . . . . . . . . . . 148

6 Conclusions 1536.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.2 Future research directions . . . . . . . . . . . . . . . . . . . . . . . . . 155

References 157

Summary 165

About the author 169

1

Chapter 1

Introduction

Faced with the increasing challenge of providing ”the right product in the rightplace at the right time and at the right price” (Fisher et al., 2000), many retailersconcentrate on improving the efficiency of their operations. Efficient management ofstore operations is crucial to the retailer’s own success (Pal and Byron, 2003) and oftencritical for the performance of the entire supply chain. Many believe that ”the last100 feet” of the supply chain from store receipt to the shelf represent both the highestsupply chain cost and the biggest customer service risk (Supply Chain EffectivenessSurvey, 2002).

Typical tactical and operational decisions retailers face in managing their stores referto product assortment and variety (which products to store), location and shelf-spaceallocation in the store (where and how much space should be assigned to each stockkeeping unit (SKU)) and products’ replenishment (when and how much to reorder ofeach SKU). An essential objective for most retailers is to provide a high availabilityof their products at low operational costs. This ultimately challenges retailers toformulate good plans, well executed (Fisher, 2009).

In this dissertation, we focus on the store, the last tier in the retail supply chain.Our research aims to support decisions concerning two essential areas of storeoperations: merchandise handling and inventory management. Decisions regardingthe assortment of products, location and allocation of shelf space are outside ourscope and they have been addressed elsewhere (see e.g., Corstjens and Doyle, 1981,Dreze et al., 1994, Urban, 1998, 2002, Van Ryzin, 1999, or Yang, 2001).

The research motivation, objectives and the outline of the dissertation are presentedfurther on.

2 Chapter 1. Introduction

1.1. Motivation and objective

We place our research in the context of the grocery industry, a sector that contributesconsiderably to the total sales volume in retail (Guptill and Wilkens, 2002). Europeangrocery retailing is an extremely competitive sector with high operating costs and lowprofit margins. One natural way to remain competitive is to reduce and/or managecosts in key areas. Proper control of store operating expenses typically requiresbalancing transportation, inventory, shelf space and handling costs. Currently, modelsthat assess the overall operational costs in retail stores on multiple dimensions arenot available. Much of the academic model-based research in retail operations hasfocused on issues such as inventory, marketing, or planograming decisions separately(see, e.g., Corstjens and Doyle, 1981, Dreze et al., 1994, Urban, 1998, Cachon, 2001).Typically, in these models the handling time and its related costs are not consideredexplicitly. There is also a general lack of understanding of what drives handling costsin retail stores, and only scant evidence exists in the academic literature on this topic.

The goal of this dissertation is twofold. First, we aim to provide a betterunderstanding of the main drivers of handling costs in retail stores. Second, ourobjective is to integrate inventory, handling and shelf space into a single model foranalysis and optimization of replenishment decisions.

Limited published evidence exists on store level processes (Falck, 2005, Smaros et al.,2004) and even more limited research exists on models of in-store processes (Kotzaband Teller, 2005). For most retailers, in-store handling operations are not only labour-intensive processes, but also very costly. An empirical study of Saghir and Jonson(2001) found that 75% of the total handling costs in a grocery retail chain occursin the store. In another study, Broekmeulen et al. (2004) showed that the handlingcosts at the store level clearly dominate the other operational cost components inthe retail supply chain consisting of the retail distribution center and the store (seeFigure 1.1). Statistics from the Food Marketing Institute also suggest that labor costsmake up more than 57% of store operating costs. However, academic approachestowards modeling of in-store handling operations and integration with operationaldecisions such as inventory management are rare. This dissertation explores theseopportunities.

Inventory management remains a key strategic weapon for many retailers. Theacademic research and studies on inventory management systems is rather abundant(see Silver, 2008 for a review). However, much of the research remains rathertheoretical and there is still a gap between theory and practice. For example, manyof the retail inventory management models and methods use assumptions that weredeveloped for application areas other than retailing. For instance, it is often assumedthat unmet demand is backordered (i.e., customers wait for the unavailable stock tobe replenished), while in traditional retailing, unmet demand is typically lost. Theconsumer behavior studies reveal that when a product is out-of-stock, the customertypically buys a substitute product or visits another store. A study by Gruen et al.

1.1 Motivation and objective 3

Inventory costs

in store

7%

Handling costs

in store

38%

Inventory costs

in DC

5%

Handling costs

in DC

28%

Transportation

cost (DC to

store)

22%

Figure 1.1 Cost structure of the retail supply chain (Broekmeulen et al., 2004)

(2002) shows that only a small percentage of consumers (15%) are willing to wait whenconfronted with an out-of-stock situation, whereas the remaining 85% will either buya different product (45%), visit another store (31%), or entirely drop their demand(9%)(see Figure 1.2). The study also reveals that 70-75 percent of out-of-stock are adirect result of inadequate store ordering and shelf restocking practices. Hence, manyopportunities for improvement exist in these areas.

Buy item at

another store

31%

Delay purchase

15%

Substitute-

same brand

19%

Substitute-

different brand

26%

Do not

purchase item

9%

Figure 1.2 Consumer response to out-of-stocks (Gruen et al., 2002)

The inventory control problem of grocery retailers share several other features,additional to lost sales. Demand for products is stochastic, the store orders on aperiodic basis and receives replenishments according to a fixed schedule. For example,some products are ordered daily, others are ordered every second, or third day.Typically, the replenishment lead times are rather short in the grocery sector. Theorders placed in the morning are often received at the end of the day, or the beginningof the next day. In any case, the replenishment lead time is typically shorter than


the length of the review period. Furthermore, the orders are usually constrained tobatches of fixed sizes (the case packs), generally dictated by the manufacturer fromthe need to coordinate inventory and transportation of several products. Upon orderreceipt at the store, the replenishment stock needs to be stacked on the shelves, andthis activity is part of the shelf stacking process at the stores. Shelf space is limited,dictated by marketing constraints, and surplus stock, which does not fit on the shelf,is temporarily stored in the stores’ backroom, often a small place, poorly organized.

Due to the rather complex nature of the replenishment problem, these characteristicsare rarely taken into account in one comprehensive model, in the analysis of lost-salesinventory systems, as our literature review in the following chapters reveals. Thisdissertation also aims to contribute to the literature on single-location, single-itemlost-sales inventory theory. More details about the scope of the dissertation, researchquestions and main contributions are discussed in the following sections.

1.2. Scope of the dissertation

The problems addressed in this dissertation are motivated by, bot not limited to thegrocery retail sector, in particular dry groceries. Our research mainly revolves aroundinventory management and shelf stacking practices at store level. The relationshipbetween the store and the retailer’s distribution center is not taken into account. Thefocus of the dissertation is on developing and solving single-location inventory controlmodels that capture realistic features of a store’s ordering and replenishment process.

A grocery store is a retail store that stocks different kinds of (mostly food) items andsells them to consumers. In doing so, many logistics activities are carried out andneed to be coordinated within a retail outlet from an incoming dock to the checkout counters. A schematic representation of the general flow of goods in a retailstore, from order receipt to check out is presented in Figure 4.1. In this dissertation,we concentrate our concept of handling operations on the shelves stocking processes,which refer to all activities needed to prepare shelves filling, such as break case packsto end-consumer units, shelves (re)filling, or merchandise presentation. We leaveoutside our scope activities related to order receipt and check out. We also distinguishbetween the sales floor and the store backroom, and assume that the retailer uses thebackroom to temporarily stock additional inventory that does not fit in the regularshelves upon delivery. However, questions on how should a retailer best operate hisbackroom are not addressed here.

All store processes depend on stochastic end-customer demand, and their ultimategoal is efficiency, which means to satisfy the amount of items as requested by end-customers at the lowest costs possible. Two types of costs are considered relevantin this dissertation: (i) inventory-related costs (for ordering, for holding products onstock, and penalty costs for not being able to satisfy end-customer demand), and (ii)handling-related costs (for shelves stacking either with directly incoming goods, or

1.2 Scope of the dissertation 5

Check out

Shelves

Backroom

Incoming

stock

Second

replenishment

process

First

replenishment

process

Receipt

Figure 1.3 Generic flow of goods in a retail store

with stock from the backroom). As shown by Broekmeulen et al. (2004), handlingcosts (mostly shelf stacking) are much higher than inventory holding costs at storelevel. The novelty in this dissertation consists in the explicit consideration of handling-related costs in the optimization of inventory decisions. Since we target operationalcosts, assortment or shelf space related costs are not considered. Also, we focus onnon-perishable products, thus obsolescence costs are not taken into account.

To control inventories, we focus on the product level, and the main question understudy is when and how much to order of a particular item to satisfy end-customerdemand at minimum total costs. To address this question, we consider severalinventory control problems, which retain a number of features that are observed in thereplenishment practices of grocery retailers. Products are facing stochastic customerdemand and excess demand is lost, rather than backordered. Two main assumptionsabout the consequences of stock outs prevail in the stochastic inventory control theory:backordering of unmet demand and lost sales, respectively. Models under the secondassumption received far less attention in the literature, mainly because they areanalytically less tractable than the backorder models. For the backorder models,simple classes of replenishment policies are proven to be optimal, but the results donot extend in general to lost-sales models. In this dissertation, we focus on stochasticlost-sales inventory control models in various settings.

The replenishment decisions are taken under periodic review. Although continuousinventory monitoring (i.e. continuous review) could be justified by the emergenceof technologies such as RFID (radio frequency identification) at item level (Metzger,2008), retail stores usually prefer to inspect their inventories periodically at regularintervals. This offers them the opportunity to coordinate the replenishment andtransportation of different products. Our research concentrates on periodic reviewinventory management systems.

Many products are replenished by the store in case packs (batches), each pack


containing several units of the product. The nature and size of the packs areusually determined by the supplier or manufacturer due to storage, transportationand handling rather then inventory considerations. Therefore from an inventorymanagement perspective, the case pack size is a fixed rather then a decision parameterfor the retailer. Depending on the store format and product type, products aredisplayed on the shelves in their original package, or case packs are first brokendown into individual consumer units. The particular application of interest in thisdissertation is where stock is supplied to the stores in pre-packed form, and the retailerdisplays individual units on the shelves in response to end-consumer demand. Thisdistinction between the unit of demand and supply is rarely taken explicitly intoaccount in the development of inventory control policies (Hill, 2006). The packagingdesign problem and its impact on the store’s logistics processes is outside our scope(see, e.g., Hellstrom and Saghir, 2007, Van Stipdonk, 2007).

In the literature, the problem with order quantities that are restricted to be integermultiples of the batch size is typically referred to as batch ordering. We consider thefeature of full batch ordering (partial batches are not allowed, see for example Alp etal., 2009) and investigate the impact of different batch sizes on the performance ofthe inventory systems.

Another characteristic of the inventory replenishment process in the grocery sector isthat the time between placing an order and receiving it, called the replenishment leadtime, is typically shorter than the review period length. This feature is also referredto as fractional lead time. The majority of inventory models in the literature assumethat the lead time is an integral multiple of the review period length (Zipkin, 2000).The exceptions are rare, as our literature review in the following chapters reveals. Inthis dissertation we follow the line of research that considers fractional lead times.

Shelf space is a scarce resource in traditional store-based retailing. The retailer hasto allocate limited shelf space among many different products, and the distributionof appropriate amounts, together with their location, is indicated in the so calledplanogram. The allocation of shelf space is typically decided at a tactical level,considering many marketing variables, and aims at stimulating customer purchasesand maximizing profits (see e.g., Hubner and Kuhn, 2010 for a review on assortmentand shelf space planning models). For inventory replenishment decisions, the shelfcapacities at product level are usually predetermined. Due to insufficient shelfspace, part of the retailer’s assortment is split between the sales floor and thebackroom. Additional handling operations can be expected in transferring stock fromthe backroom to the sales floor. Since the retailer’ costs are sensitive to in-storemerchandise handling, we consider the effect of using the backroom on the combinedcost of ordering, holding, lost sales and handling, in a single-item inventory problem.

Beside the features that we mentioned, other features may be important in controllinga retailer inventory, which we do not address in this dissertation. Among others, thenon-stationarity of the demand process (in particular cyclic demand patterns arecommon for European retailers (Van Donselaar et al., 2009), demand estimation and

1.2 Scope of the dissertation 7

forecasting in the presence of stockouts (Wecker, 1978, Nahmias, 1994, Agrawal andSmith, 1996, Raman and Zotteri, 2000), the efficient measuring of stockouts (Corstenand Gruen, 2004), the accuracy of inventory records (Raman and DeHoratius, 2001,Atali et al., 2009) and implications for inventory management (Kok and Shang, 2007,DeHoratius et al., 2008), the management of promotion items (Huchzermeier et al.,2002), or RFID applications (Lee and Ozer, 2007).

In short, in this dissertation we study stochastic single-item inventory control modelsunder different valid assumptions inspired from the replenishment practice of groceryretailers. Optimal as well several alternative inventory control policies are consideredthroughout the dissertation in a multi-period inventory setting. The main definitionsand notation are introduced next.

Inventory control policies

All inventory policies are studied under periodic-review, with R the fixed reviewperiod length. For notation expediency, we shall omit R in all policy notations. Thewell-known (s,Q) and (s, S) policies (Zipkin, 2000) with fixed (Q) or variable ordersize (S denoting the order-up-to level) are extended to the case of batch ordering,and are denoted by (s,Q, nq) and (s, S, nq), respectively, with q the fixed batch size.New policies are also studied: the (s,Q|S) and the (s,Q|S, nq) policy, respectively.The inventory control policy implemented in automated store ordering systems atgrocery retailers resembles an (s, nq) policy (Van Donselaar et al., 2009), which isalso considered in this dissertation. The policy definitions are as follows.

The (s, S, nq) policy: The (s, S, nq) policy has two parameters, s and S (0 ≤ s ≤ S)and may be described as follows. Whenever the inventory level at a review periodis less than or equal to s, order the largest integer multiple of q which results in aninventory position less than or equal to S.

The (s,Q, nq) policy: The (s,Q, nq) policy has two parameters s ≥ 0 and Q ≥ 0and may be described as follows. Whenever the inventory level at a review periodis less than or equal to s, order Q units such that the order size Q is a nonnegativeinteger multiple of q.

The (s,Q|S, nq) policy: The newly proposed (s,Q|S, nq) policy has three parameterss, S and Q with 0 ≤ maxs,Q ≤ S ≤ s +Q. Under this policy, the order quantityin each period depends on the beginning inventory on hand x and is given by

a(x) =

⌊Q/q⌋q if 0 ≤ x ≤ S − ⌊Q/q⌋q⌊(S − x)/q⌋q if S − ⌊Q/q⌋q < x ≤ s0 if s < x ≤ S,

where ⌊x⌋ denotes the largest integer, smaller or equal to x ∈ R.

The (s, nq) policy: The (s, nq) policy has one parameter s ≥ 0 and works as follows.Whenever the inventory on hand is less than s, an order is placed for the minimuminteger multiple of q such that, after ordering, the inventory will rise at or above s.


The remainder of the chapter presents the main research questions and contributionsof the dissertation, outlined by chapter.

1.3. Research questions and contributions of thedissertation

Traditional store-based retailing heavily relies on two operations: inventory replenish-ment and merchandise handling. Therefore, many opportunities for improvement inthese operations exist, once their most important characteristics and drivers are wellunderstood. In this dissertation, we explore such opportunities in several studies.First, we focus on the shelf stacking process in retail stores, aiming for a betterunderstanding of the specifics of this process. Then, we develop and solve several lost-sales inventory control models, which take into account key characteristics of the retailenvironment: batch ordering, handling costs, shelf space and backroom operations.In the following, we discuss the specific research questions and methodologies for eachindividual study. The presentation closely follows the outline of the dissertation.

1.3.1 Modeling handling operations in grocery retail stores

Shelf stacking represents the daily process of manually refilling the shelves withproducts from new deliveries. Chapter 2 presents an empirical study of the shelfstacking process in grocery retail stores. We examine the complete process at thelevel of individual sub-activities and study the main factors that affect the executiontime of this common operation. In Chapter 2 we address the following researchquestion:

What are the key factors that drive the shelf stacking time in retail stores?

We tackle this question by carrying out a motion and time study and statisticalanalyses in order to construct a conceptual model of the shelf stacking operations.We identify seven shelf stacking subtasks: grabbing/opening a case pack, searching,walking, preparing the shelves, filling new inventory, filling old inventory, and wastedisposal. Further, we find that the three most time-consuming sub-activities arestacking new inventory, grabbing/opening a case pack, and waste disposal.

Based on the insights from the different sub-activities, a prediction model is developedthat allows estimating the total stacking time per order line1, solely on the basis ofthe number of case packs and consumer units. The model is tested and validated

1An order line typically contains a number of consumer units from a specific article, or StockKeeping Unit (SKU)

1.3 Research questions and contributions of the dissertation 9

using real-life data from two European grocery retailers and serves as a useful toolfor evaluating the workload required for the usual shelf stacking operations. Finally,we illustrate the benefits of the model by analytically quantifying the potential timesavings in the stacking process, and present a lot-sizing analysis to demonstrate theopportunities for extending inventory control rules with a handling component. Theseopportunities are further explored in Chapters 3 and 4.

1.3.2 Lost-sales inventory models with batch ordering andhandling costs

Empirical evidence shows that handling costs are relatively high in the grocery retailsupply chain. Van Zelst et al. (2009) reported that the handling of goods in the storeaccounted for 75% of the total logistics store costs, while inventory accounted for theremaining 25% of total costs. However, handling-related costs are rarely acknowledgedin inventory replenishment decisions. In this research, we aim to bridge this gap byexplicitly recognizing the handling costs (for shelf stacking of replenishment orders)at the retailer as a critical cost component, and integrating inventory and handlinginto a single model for analysis and optimization of inventory replenishment decisions.In Chapter 3, we aim to answer the following research questions:

How could the retail inventory control models be adapted to incorporate handling indecision making? And what is the impact of adding this aspect on the overall systemperformance?

In Chapter 2, we showed that we can reliably estimate the handling time per StockKeeping Unit (SKU) required to execute the shelf stacking operation using an additivemodel (fixed plus linear terms), depending on the number of case packs (batches)and the number of consumer units in the replenishment order. In Chapter 3, weassume a similar structure for the shelf stacking costs. This leads to a replenishmentcost structure that allows economies of scale. For example, the retailer could decideto order less frequently but in a larger number of case packs in order to reducethe handling costs. However, less frequent deliveries lead to an increase in theaverage inventory level. We investigate this tradeoff in the setting of one retailer,who periodically manages the inventory of a single item facing stochastic demands,and lost-sales for unmet demand. Additionally, the replenishment order is restrictedto integers multiple of a fixed batch size q, and the lead time is assumed to be less thanthe review period length. The objective of the system is to minimize the long-runaverage costs.

We use stochastic dynamic programming to model and solve the inventory controlproblem. Since optimal policies have a rather complex structure, we propose aheuristic policy, referred to as the (s,Q|S, nq) policy, which partially captures the


structure of optimal policies and shows close-to-optimal performance in many settings.We also benchmark the performance of the heuristic against two reasonable alternativepolicies, the (s, S, nq) and (s,Q, nq) policy, and quantify the overall improvement.Furthermore, we present several insights from sensitivity analyses regarding theimpact of problem parameters, in particular the batch size and material handlingcosts, on the system’s performance. Finally, we show that material handling costsmay substantially affect the overall system’s performance, when ignored, especiallyfor items with low profit margins.

1.3.3 Retail inventory control with shelf space and backroomconsideration

In Chapter 4, we extend the retail setting from Chapter 3 to include anotherrealistic dimension of the retailer’s inventory decision, namely the shelf space. Aswe mentioned earlier, the shelf space allocation for each product is typically dictatedby marketing constraints (Dreze et al., 1981) and comes as a result of planogramingdecisions (Corsten and Doyle, 1981, Yang, 2001). Consequently, for operationaldecisions such as inventory replenishment, the shelf space is often an exogenousparameter. We also make this assumption here, and consider therefore that theavailable shelf space has been predetermined. We study the inventory managementproblem for a single product under similar assumptions to those formulated in Chapter3. We extend the setting to include the following situation: upon arrival at thestore, the replenishment stock has to be stacked on the shelves to serve end-customerdemand. This operation is part of the shelf stacking process at the store (also referredto as the first replenishment process). Since shelves have limited capacity, storemanagers keep surplus stock that did not fit on the shelves temporarily in the store’sbackroom, which creates the need for a second restocking of the shelves (referred to asthe second replenishment process). In turn, this translates into additional handlingcosts for the retailer. The following research questions are addressed in Chapter 4:

How could inventory control models be adapted to account for shelf space limitationsand use of the backroom? And what is the impact of including these features on theperformance of the inventory control models?

In Chapter 4, we adapt the lost-sales inventory model studied in Chapter 3 to includeshelf space constraints and additional costs associated with the second replenishmentprocess. Two models are developed, using stochastic dynamic programming: the firstone assumes linear extra handling costs and continuous backroom operations, whilethe second one assumes that a fixed cost is charged to the system for exceeding theallocated shelf capacity. These costs are charged in addition to a fixed cost for placingan order, which further increases the system complexity. In a numerical study, wediscuss how backroom usage impacts the performance of the inventory control models,

1.3 Research questions and contributions of the dissertation 11

where performance is measured with respect to the optimal ordering decisions andthe associated long-run average total costs. Furthermore, we measure the retailer’sbenefit of accounting for additional handling operations, as the marginal cost decreasethe retailer may achieve relative to the case where neither the shelf space, nor theextra handling costs are included in the optimization of inventory decisions. Severalinteresting managerial insights into the tradeoff between the different cost componentsare also illustrated.

1.3.4 Efficient control of lost-sales inventory systems withbatch ordering and setup costs

In Chapter 5, we consider a variant of the classical periodic-review, lost-salesstochastic inventory problem, which has the following features: batch ordering andfractional lead time. We assume the standard cost structure, with a zero or fixed setupcost for ordering, and the objective is to minimize the long-run average cost of thesystem. Lost-sales inventory models are known to be analytically more challengingthan their backorder counterparts (Hadley and Within, 1963), therefore variousheuristics have been proposed in the literature to address this issue (Zipkin, 2008a,Nahmias, 1979). The features of batch ordering, fractional lead time and setup costhave been studied less frequently in the literature, especially in an all embracingmodel. Our research addresses this deficit and in Chapter 5, we aim to answer thefollowing research questions:

Can we derive an efficient heuristic to control the single-item lost-sales inventoryproblem with batch ordering and setup costs? And how efficient is the (s, nq) policy,a commonly applied heuristic in grocery retailing, in controlling the inventory system?

Using Markov decision processes, we investigate numerically the structure of theoptimal policies, in an extensive computational study. We provide numerical evidenceto support the (s,Q|S, nq) heuristic as a very good alternative to optimal solutions.Our results show that the cost increase from using the heuristic, against the optimalsolution, is at maximum 0.2% (when K = 0) and 1.7% (when K > 0), respectively.The heuristic generalizes both the (s,Q) and (s, S) policy (when q = 1) and isadjusted to account for batch restrictions on the order quantity (when q > 1). Inmany instances, it even captures the true structure of the optimal policies. We alsocompare the performance of the heuristic with those commonly used, and demonstrateits superiority and effectiveness. In particular, we find that the best (s, S) policies areperforming increasingly better and close to optimality as the penalty cost increases,while the best (s,Q) policies may outperform the best (s, S) policies in settings withsmall penalty costs. Finally, we test the performance of the (s, nq) policy, oftenimplemented in automated ordering systems at grocery retailers (Van Donselaar etal., 2009), and show that it may result in substantial cost increase when implemented


in the presence of fixed setup costs, which is often the case in grocery stores due tothe presence of fixed handling costs.

Finally, in Chapter 6, we conclude the dissertation and discuss several future researchdirections.

Concisely, in this dissertation, we develop adapted inventory control models, anddesign new solution approaches which take more effectively into account different retailcharacteristics. First, we build upon an empirical study and derive a formal modelof handling costs, which includes fixed and variable components. Then, we study theperformance of a lost-sales inventory system where we explicitly recognize the handlingcosts as a critical component, in addition to a standard cost structure. We showthat optimal policies have quite complicated structures and propose a new heuristicpolicy, which is intuitive for practitioners, shows close to optimal performance, andproves superior to reasonable alternative policies. We also give several managerialinsights from sensitivity analyses and quantify the added value of handling costs inthe decision making. Next, we extend the model to account for limited shelf space andthe cost of handling backroom stock, which leads to a more realistic system perspectivefor reordering decisions and enhanced cost control capabilities. Finally, we study avariant of the single-item lost-sales inventory model with standard cost structure toreinforce the proposed heuristic. While the inventory control models presented in thisdissertation have been inspired by grocery retailing, the analysis, solution techniquesand insights may be applicable to other settings, where the unsatisfied demand islost, there is a non-unit size of stock transfer and there are economies of scale in thereplenishment cost structure.

1.4. Outline of the dissertation

Table 1.1 summarizes the different characteristics of the retail environment, asconsidered in each chapter of this dissertation. Each chapter of the dissertation isself contained and can be read independently. The research presented in Chapter 2appeared also as Curseu et al. (2009a) and the research described in Chapters 3, 4and 5 is based upon Curseu et al. (2009b) and Curseu et al. (2010a,b), respectively.

1.4 Outline of the dissertation 13

Table 1.1 Dissertation outline: main features by chapter

Chapter Comment Shelfstacking

Inventorycontrol

Shelf spaceand backroom

2 Empirical study X

3 New heuristic testingX X

Impact of handling

4 Numerical optimizationX X X

Impact of shelf space

5 Numerical optimizationX

Heuristics testing

15

Chapter 2

Modeling handling operationsin grocery retail stores: anempirical analysis

Abstract: Shelf stacking represents the daily process of manually refilling the shelveswith products from new deliveries. For most retailers, handling operations are labour-intensive and often very costly. This chapter presents an empirical study of the shelfstacking process in grocery retail stores. We examine the complete process at thelevel of individual sub-activities and study the main factors that affect the executiontime of this common operation. Based on the insights from different sub-activities,a prediction model is developed that allows estimating the total stacking time perorder line, solely on the basis of the number of case packs and consumer units. Themodel is tested and validated using real-life data from two European grocery retailersand serves as a useful tool for evaluating the workload required for the usual shelfstacking operations. Furthermore, we illustrate the benefits of the model by analyticallyquantifying the potential time savings in the stacking process, and present a lot-sizinganalysis to demonstrate the opportunities for extending inventory control rules with ahandling component.

2.1. Introduction and related literature

In today’s highly competitive market environment many retailers are concentratingon controlling costs, as a means of achieving operational excellence and their businesssuccess as a whole. In a recent logistics survey (Butner, 2005), an overwhelming 83%of participants ranked logistics cost reduction as their primary objective, competing

16 Chapter 2. Modeling handling operations in grocery stores

with the permanent strive to provide a high customer service. Typically, retailoperational costs include the cost for inventory, shelf space, transportation, andhandling. Existing research in retail operations mainly concentrates on inventory,marketing, or planograming decisions separately (see e.g., Corstjens and Doyle, 1981,Dreze et al., 1994, Urban, 1998, Cachon, 2001, Hoare and Beasley, 2001). Storehandling operations and associated costs are often not modeled explicitly in thesestudies (see e.g. Themido et al. 2000, where handling costs are treated in an aggregateway). This research focuses exclusively on the handling cost component of retailoperations, an area, which we believe is still largely overlooked.

For most store-based retailers, store handling operations are not only labour-intensive,but also very costly. Broekmeulen et al. (2004) studied the operational costs incurredin the part of the retail supply chain that includes the retailer’s distribution centerand the store (given assortment and shelf space allocation ). They identify inventoryholding, transportation (from the distribution center to the store) and handling(order picking in the warehouse and shelf stacking in the retail store) as relevantcost components, and find that in-store handling costs represent the largest shareof operational costs, accounting for more than one third of total operational costs.Another empirical study by Saghir and Jonson (2001) suggests that 75% of the totalhandling time in a grocery retail chain occurs in the store, and investigates howpackaging evaluation methods may assist in reducing the total handling time.

There is however, in general, a lack of understanding of what drives handling costsin retail stores, and little evidence exists in the academic literature on this topic.In this chapter, we address this shortcoming; we focus on the shelf stacking processin grocery retail stores and study the key factors that drive the execution time ofthis store operation. Shelf stacking represents the daily process of manually refillingthe shelves in the store with products from new deliveries. As with most manualactivities, such processes are often time consuming and costly. Furthermore, unlessclear and reliable work standards are implemented, such activities may well sufferfrom a lot of variation, which will negatively affect the overall store performance.

We conduct an empirical analysis by means of a traditional motion and time study(Barnes, 1968). While such studies are often conducted in the OR field (Niebel, 1993),they are not present in the specific area of retail operations. The main contributionsof this chapter are threefold. First, we examine the shelf stacking process at the levelof individual subtasks and analyze the impact of different drivers (e.g., number ofcase packs and consumer units, etc.) on the individual shelf stacking times, as wellas the total stacking time. For the retail practice, we offer a better understanding ofthe distribution of workload in shelf stacking to the individual sub-activities, whilewe specifically recognize those sub-tasks that are mostly influenced by the key driversidentified, as compared to those for which the variation in workload is potentiallyaffected by other factors. This has further implications for identifying inefficienciesin the entire process.

Secondly, we investigate whether it is possible to derive a reliable estimate of the

2.1 Introduction and related literature 17

shelf stacking time, using a small representative set of key time-drivers. Using multipleregressions, a prediction model is developed, which allows estimating the shelf stackingtime to a large extent only on the basis of the number of case packs per order lineand the number of consumer units (measures which are readily available). In contrastto common assumptions in the literature, we find that an additive, rather than alinear structure is appropriate for describing the specific relationship. Real-life datawas used to test the model and to assure it has face validity, which is relevant for itsgeneral applicability to other settings.

Thirdly, we investigate the potential of the prediction model for a better estimationand control of the overall logistical costs. Closed-form analytical expressions forexpected efficiency gains are investigated to quantify the potential gains that couldbe achieved. Moreover, we present a lot-sizing analysis to illustrate the benefits ofthe model in extending currently available inventory control models with a handlingcomponent. This idea is further explored in Chapters 3 and 4.

The remainder of the chapter is organized as follows: first, we give a brief overviewof related literature; then in Section 4.3, we describe the shelf stacking process andderive a conceptual model for estimating the time required to fulfil this common storeactivity. Section 2.3 introduces the methodology we used to test the proposed modeland describes the datasets supporting our analyses. Section 2.4 presents the results ofour study; the last sections of the chapter are devoted to discussions and conclusions.

Literature review

While warehouse handling operations received considerable attention in the literature(Rouwenhorst et al., 2000, Tompkins et al., 2003), there is still much opportunity forresearch in the field of store handling operations. An early study that considers bothinventory and handling costs comes from 1960s (Chain Store Age, 1963). SLIM (StoreLabor and Inventory Management), a system widely promoted in the mid-1960s,focused on minimizing store handling expense, by reducing backroom inventories andthe double handling of goods (Chain Store Age, 1965). Two other studies carried outby the Swedish group DULOG in 1976 and 1997, measured package handling time inthe store, in order to gather information about the impact of the type of package onhandling efficiency in the grocery retail supply chain (DULOG, 1997).

Time-study approaches are sometimes reported in the warehouse operations researchfor estimating order-picking times. Gray (1992) uses basic multiple regression toderive estimations of the necessary time to pick all items from a pick list for acustomer order, and applies it for establishing labour productivity standards. Grayet al. (1992) consider the general problem of warehouse design and operation, andpropose a model in which order-picking time includes three components: walking,stopping and grabbing. Varila et al. (2007) uses order-picking in a warehouse asa case activity to illustrate, using regression analysis, that a time-based accountingsystem is often suitable in tracing the cost behavior of an activity, especially when this


is directly proportional to time. Their work is similar in objective to the time-drivenABC, a concept recently introduced by Kaplan and Anderson (2004) as a simplerand more accurate alternative to the traditional ABC systems. However, in retailingliterature, time-studies are rarely reported.

Recently, Van Zelst et al. (2009) showed that significant efficiency in terms of shelfstacking time could be gained once the impact of most important drivers is wellunderstood. This chapter supports their main findings. Although both studiesinherently start from the same underlying empirical dataset, a number of significantdifferences in this chapter are identified compared to Van Zelst et al. (2009). First,the current chapter explicitly focuses its analysis on the order line level1, rather thanthe consumer unit level as in Van Zelst et al. (2009). Focusing on the consumer unitsinvolved a non-linear data transformation, which might lead to an estimation bias.By using the original data, measured on the order line level, these potential problemsassociated with the use of ratios as reported in Atchley et al. (1976) and Berges (1997)are avoided. Secondly, we extend the basic analysis in Van Zelst et al. (2009) towardsthe individual sub-activities and we support our results with extensive testing andvalidation. Finally, we follow an analytical approach to illustrate the benefits of ourfindings for the practice of retailers. This latter involves both analytically derivinggains in terms of handling and the consequence of incorporating the handling functioninto a lot sizing decision model.

2.2. Conceptual model and hypotheses development

Generally, a store undertakes the following replenishment process: upon arrival of anew shipment, the truck is unloaded; next, the store clerks move the deliveries intothe store and then restock the shelves with the newly arrived products. The shelfstacking process defined in this research starts after the incoming products are movedinto the store and are taken to the shelves area (usually by rolling containers), in orderto be stored on the shelves. Therefore, neither the walking with the rolling containerin the store, nor the replenishment process from the backroom and the correspondingtime delay are part of the defined shelf stacking process. Furthermore, we focus onproducts that are replenished in pre-packed form but presented to the final consumerin individual units. This situation is typical for a large part of the assortment of mostretailers (we consider here dry groceries, and products which are comparable in termsof the stacking process and productivity).

For each Stock Keeping Unit (SKU), the store clerks unpack the product and stockthe consumer units on the shelves at the assigned location (as indicated in theplanogram2). An important sub-activity in this process is shelf maintenance: thestore clerks need to check the ’best before’ date of the products on the shelf and

1An order line typically contains a number of consumer units from a specific article or stockkeeping unit (SKU)

2The planogram is a diagram of fixtures and products that illustrates where and how every SKU

2.2 Conceptual model and hypotheses development 19

remove old inventory, if necessary, before one can stack new items on the shelves.Also, the oldest consumer units are sometimes shifted in front of the shelf to facilitateFirst-In-First-Out retrieval or proper shelf display. For each SKU, the shelf stackingprocess ends with disposing the empty case packs.

Although inherently not a complex task, the shelf stacking process is manuallyexecuted and thus may suffer from a lot of variation. If time drives costs, then itbecomes valuable to understand what drives time. We are particularly interested inestimating the Total Stacking Time per order line (TST ) (i.e. for each individualSKU), based on a set of underlying factors, given a specific inventory replenishmentrule, assortment, shelf space and package. To better examine the causes and effectsof time variation, we examine the total stacking process at the levels of individualsubtasks. Breaking down the entire operation into small components allows, on theone hand, an assessment of the contribution of each individual sub-activity to theTST , and on the other hand, a better indication of the potential variables affectingthe TST . Therefore, we have divided the shelf stacking activity into seven subtasks:

• grabbing/opening a case pack (G),

• searching for the assigned location (S),

• walking to the assigned location (W),

• preparing the shelf for stacking the new items (P),

• filling new inventory on the shelves (Fn),

• filling the old inventory back on the shelves (Fo) and

• disposing the waste package (D).

We refer to Appendix A for a complete description of the definitions used in thisresearch. Few remarks are worthwhile. Because grabbing or opening a case pack weredifficult to separate, these activities were measured together. Also, the walking sub-activity does not include walking with the rolling container to the right aisle withinthe store or between aisles, but occasionally does include moving the rolling containerto the right shelf location in case of heavy products, for example. The differencebetween filling old versus new inventory is relevant as depending upon the inventorylevel just before filling, the activity filling old inventory will become important forhigher inventory levels.

The total stacking time per order line (TST ) has been divided accordingly into seventime components and the key variables that could logically influence the executiontime of each subtask are identified. It is expected that the time needed to stacknew inventory on the shelves depends on the number of units being handled, while

should be displayed on the shelf in order to increase customer purchases (Levy and Weitz, 2001)


grabbing and unpacking a case pack, traveling within the shelf aisle to and from theright location, or disposing the wasted case packs depend on the number of case packsbeing handled per order line. Lastly, searching for the right shelf location, preparingthe shelf or restacking old inventory if necessary, are normally executed only once, foreach SKU, independent of the number of case packs or consumer units. The set ofpotential time-drivers for each sub-activity are summarized in Table 5.2.

Table 2.1 Potential drivers of time variation, for each sub-activity

Order line information Product information

Sub-activity Number of CP Number of CU Product category

1 Grabbing/Opening (G) x x x2 Searching (S) - - -3 Walking (W) x - x4 Preparing (P) - - x5 Filling New Inventory (Fn) x x x6 Filling Old Inventory (Fo) - - -7 Disposing waste (D) x - -

CP : case packs; CU : consumer units

In reality, there could be many other potential factors affecting the duration of theshelf stacking time (such as SKU volume, weight or type of packaging, the distancetraveled within the aisle, the old inventory position just before new replenishment,the labour, the environment, etc.; recently, Hellstrom and Saghir (2007) investigatedthe relationship between the packaging system and logistics processes in the retailsupply chain). Herein, we concentrate only on order line-related (number of casepacks (CP ) and number of consumer units (CU) per order line,) and product-relatedcharacteristics (product category) as the key drivers of time variation of the shelfstacking process. The product subgroup variable captures any time variation thatcould be attributed to differences in product-related characteristics not measuredspecifically in this study (such as total weight or volume of products being handled).In general, the order line information refers to the number of items (case packs orconsumer units) being handled, and is thus an appropriate cost driver, while theproduct information approximates the difficulty in handling products from differentcategories. These variables are selected as potential predictors in our subsequentanalyses.

The dependent variables are the individual times per sub-activities (T s, with s ∈G,S,W,P, Fn, Fo,D) and the Total Stacking Time (TST ), all expressed in seconds.The explanatory variables are hypothesized to have the following influence on theexecution time of each sub-activity:

Hypothesis 1 The number of case packs (CP ) has a positive effect on the individualtimes TG, TW , TFn, TD and TST .

2.3 Study design and data description 21

Hypothesis 2 The number of consumer units to be stacked (CU) has a positive effecton sub-activities’ execution times TG, TFn and TST .

We expect that CP and CU have no significant effect on TS , TP , and TFo. Underthese hypotheses, the Search, Prepare and Fill Old sub-activities could be regardedas fixed activities, while only the remaining activities are variable, depending on theset of hypothesized factors.

2.3. Study design and data description

Two grocery retail chains (denoted here by A and B) agreed to participate in thisstudy. Empirical data on the stacking process was collected using a motion and timestudy approach (Barnes, 1968). Data from chain A are used to test the hypotheses,and data from chain B are used to validate the results. In four stores (two foreach supermarket chain), nine experienced employees, familiar with the operations,were videotaped during the shelf stacking process. The product subgroups (all drygroceries) were selected such that they contain:

• both fast and slow moving items;

• different case pack sizes;

• SKUs that are replenished in pre-packed form and sold as individual units;

• SKUs for which sufficient shelf space is available to accommodate more thanone case pack in a delivery (see also Broekmeulen et al., 2004);

• items that are comparable in terms of the handling process and productivity(i.e. no soft drinks, beer or diary products);

• the product categories should have a sale pattern as stable as possible (noseasonal changes or promotions)

Finally, we note that the data collection period did not include days with peak ordropping demand, and the stores were consistent in their operations.

The stacking of items on the shelves is observed and recorded for each SKU.Afterwards, the execution time of each individual sub-activity and the Total StackingTime per order line (TST ) was registered using a computerized time registration tool,and results were entered into a database. Additional information necessary to identifythe stacking process for each order line was added as well, such as the SKU type, thenumber of case packs and case pack size per order line, and the product category eachSKU belongs to.


The final dataset contains 1048 observations, for chain A, across nine different productcategories, and 563 observations, for chain B, across five different product categories.Tables B1 to B4 (Appendix B) contains descriptive statistics of the variables usedin this study. The average total time to stack an order line into the shelves is 57.31seconds, ranging from a low 10 seconds per order line (personal care category) to a high334 seconds (coffee), with a standard deviation of 36.6 seconds. This reveals the degreeof variation that exists in the TST between different order lines and this study aims atgaining a better understanding of the factors causing this variation. We further notethat some degree of variation exists also between the TST corresponding to differentproduct categories. The average TST varies between 35.47 seconds (products ofpersonal care) and 80.86 seconds (coffee milk). With reference to the explanatoryvariables of this study, we note that the average number of case packs per order linevaries between 1 CP (all categories) to 9 CP (coffee), with an average of 1.3 CP anda standard deviation of 0.7 CP . The average number of consumer units per order lineexhibits quite some variation, ranging from 3 CU (personal care) to 135 CU (coffee),with an average of 16.78 CU per order line.

Based on this empirical data, we derive the distribution of the Total Stacking Timeand the relative contribution of each individual sub-activity to the TST , as illustratedin Figure 5.1. We note that the most time consuming sub-activity in the shelf stackingprocess is the Stacking of new inventory (Fn) (about 48% of the TST ), followed bythe Grabbing and unpacking the case packs (G) (about 20% of TST ) and Disposingthe waste (D) (about 13% of TST ), respectively. Together, they account for almost81% of the TST . Tables B2 and B4 (Appendix B) provides descriptive statistics of thedependent variables used in this study. The corresponding average times for executionof the three most time consuming sub-activities are 27.32 seconds (Stacking newinventory), 11.65 seconds (Grabbing/opening a case pack) and 7.28 seconds (Disposingwaste), respectively.

2.4. Results

In order to test our hypotheses, we performed several separate regression analyseswith T s (s ∈ G,S,W,P, Fn, Fo,D) and TST as dependent variables, and CP , CUand product category as the independent ones. We adopt two different strategiesfor estimating the Total Stacking Time per order line (TST ), which we refer toas sequential regression and overall regression, respectively. Both approaches allowpredicting the TST as a function of the identified drivers using multiple linearregressions. However, the two approaches have different practical purposes. While thefirst approach reveals detailed information regarding the causes of variation in eachsubtask, as explained by the hypothesized variables, and provides more informationregarding the contribution of each individual sub-activity to the variability of theentire process, the second approach is selected as a simple, less expensive alternativefor practical forecasting of the TST . The sequential regression starts from the

2.4 Results 23

1%

4%

6%

8%

13%

20%

48%

0% 10% 20% 30% 40% 50% 60%

Stack old inventory

Search

Prepare the shelf

Walking

Dispose waste

Grab and unpack case

Stack new inventory

% of total shelf stacking time

Su

b-a

ctiv

ity

Figure 2.1 Distribution of the total shelf stacking time (Chain A)

following functional form:

TST =∑s∈A

T s, A = G,S,W,P, Fn, Fo,D, (2.1)

where the duration of each individual sub-activity, T s per order line is estimated usingthe following general linear regression model:

T s = bs0 + bs1CP + bs2CU +

PC−1∑pc=1

αspcDpc + εs, (2.2)

for every sub-activity s ∈ A, and where PC represents the number of different productcategories considered in the analysis. A set of dummy variables is used to accountfor differences between product categories Dpcpc=1:PC . To avoid perfect multi-collinearity, one category (from the group of product categories) will act as a referencefor the others (Gujarati, 1995). The overall regression has the following functionalform:

TST = b0 + b1CP + b2CU +

PC−1∑pc=1

γpcDpc + ε.

Both approaches allow for an estimation of the expected TST (in seconds). Sequentialregression requires the TST be estimated in two steps: first, an estimation of


individual sub-activities’ times per order line is necessary (based on (2.2)), whichthen add up naturally into the Total Stacking Time according to (2.1). The overallregression on the other hand, allows one to predict the TST directly on the basis ofthe key drivers identified.

Starting from the sequential regression model formulation introduced by equation(2.2), we derived three predictive models to test the effect of each explanatory variableused in this study. We first estimate each model for the first dataset (chain A) andthen validate the results on the second dataset (chain B). The tested models for eachindividual sub-activity are specified next. Similar models are used for the analysis ofthe overall total stacking time, too.

Model 1 : T si = bs01 +

PC−1∑pc=1

αspc1Dpci + εs1i ,

Model 2 : T si = bs02 + bs12CPi + bs22CUi +

PC−1∑pc=1

αspc2Dpci + εs2i ,

Model 3 : T si = bs03 + bs13CPi + bs23CUi + εs3i ,

where s ∈ A, and εs1i , εs2i , εs3i are the error terms for each order line i = 1 : N.

Model 1 is an ANOVA model with only the product category identifier as anexplanatory variable, which is modeled here by the group of dummy variablesDpcpc=1:PC−1. Therefore, this model estimates differences in execution time acrossproducts categories and is used as a reference in our analysis. Model 2 includes themain effects of the number of case packs (CP ) and the number of consumer units (CU)per order line, respectively. This model tests the effect of the explanatory variablesfrom our Hypotheses, while controlling for differences across product categories.Model 3 is a simple regression model with only CP and CU as explanatory variables.Thus, Models 2 and 3 by comparison show if the product grouping has a significanteffect on the execution times.

2.4.1 Sequential regression results

For the derivation of the TST , we carried out a sequential analysis. First, for eachindividual sub-activity, we tested regression Models 1 to 3 and derived estimatesof the execution times T s (s ∈ A) for each sub-activity. Then, these estimatesare used to predict the TST , as indicated by equation (2.1). Separate analyses foreach individual sub-activity correspond to our motivation of identifying which sub-activities are mostly affected by the selected order line- and product-related factors.The final derivation of the TST is in line with our purpose of deriving a predictivemodel for estimating the total time necessary to stack the products from an orderline into the shelves.

2.4 Results 25

Models 1 and 2 are analyzed using hierarchical regression. The group of dummyvariables representing the merchandising category was considered as a control variable,and it was introduced in the first step of hierarchical regressions. The referencecategory was chosen to be the one with the largest number of samples in the dataset.The first empirical dataset contains nine product subgroups and the largest categoryin this dataset is Personal care (see Table B2, Appendix B). In the second step ofhierarchical regression, we added together the main effects CP and CU .

The results of the ordinary least squares estimation for the first data set are presentedin Table 5.3. Relevant collinearity diagnosis (such as coefficient of correlation, varianceinflation factors) indicated no significant problems with respect to multi-collinearity.Table 5.3 gives the standardized coefficient estimates for each individual sub-activity.Overall, results for Model 1 indicate that the product category variable alone explainsonly a small proportion of the total variance in the execution times of correspondingsub-activity. The three largest adjusted R2, obtained for Fill New, Prepare andDispose in this sequence, vary from 10% to almost 17%. We also note that althoughsome product categories dummies are not significant predictors, the group of dummiesis overall significant (as confirmed by the overall F-statistics), and this holds true forevery individual sub-activity.

Results from the second regression step indicate that Model 2 explains a significantlyhigher proportion of the variance in sub-activities’ times. The adjusted R2 rangesfrom .008 (for Search sub-activity) to as high as .679 (for Fill New sub-activity).The three largest proportions of variance in the dependent variable accounted for bythe explanatory variables of Model 2 belong to Fill New (R2

adj equals 67.9%), Grab

and unpack (R2adj of 41.6%) and Dispose (R2

adjof 31.6%) sub-activity, respectively.Recall from Figure 5.1 that these are also the three most influential sub-activitieswith respect to their relative contribution to the Total Stacking Time. The overallF-statistics indicate a significant joint contribution of the variables in predictingthe execution times for all sub-activities (at p ≤ .05). However, we note that theexplanatory variables CP and CU do not contribute significantly in explaining thetime for searching, and have only a marginal contribution in explaining the time forpreparing the shelves, filling old inventory and walking, respectively (R2 change of0.011 and 0.057).

Further, we note that when the subgroup effect is removed from the analysis (Model3), the adjusted R2 for the Fn, G and D drops marginally from the previous modelto 63.3%, 39.8% and 25.4%, respectively. The F-statistics show that the jointcontribution of CP and CU is statistically significant for Fn, G and D and theirstandardized coefficients are both positive, thus showing support for our hypothesesfor these sub-activities. Note that these results are also consistent between Models 2and 3. While CU has a larger contribution for Fn, sub-activities G and D are mostlyaffected by CP , as indicated by the standardized coefficients. Comparing Models 2and 3, we also find no support for S being affected by CP or CU . Although the resultsshow a statistically significant effect of CP or CU for W, P and Fo, by inspecting the


Table 2.2 Regression results for each individual sub-activity (standardized coefficients)(Chain A)

Dependent Variables

G S W P Fn Fo D

Model 1Baby food .033 −.039 −.052 .075∗ .065∗ .028 .065∗

Chocolate .205∗∗∗ −.085∗ .111∗∗ .207∗∗∗ .238∗∗∗ .083∗ .284∗∗∗

Coffee .271∗∗∗ −.065 −.027 .320∗∗∗ .368∗∗∗ .103∗∗ .051Coffee milk .106∗∗ −.043 .062∗ .165∗∗∗ .283∗∗∗ .083∗ .181∗∗∗

Candy .076∗ .007 .201∗∗∗ .042 .247∗∗∗ .004 .165∗∗∗

Sugar .045 −.040 −.060∗ .079∗∗ .165∗∗∗ −.002 .040Canned meat .080∗ −.091∗∗ .138∗∗∗ .090∗∗ .231∗∗∗ −.003 .271∗∗∗

Canned fruits .071∗ .011 −.032 .040 .161∗∗∗ −.003 .080∗∗

R2 .072 .017 .071 .109 .172 .018 .122R2

adj .065 .010 .064 .103 .166 .010 .115Mean SS Err. 111.167 13.520 13.855 49.434 405.135 12.581 39.847Overall F 10.117∗∗∗ 2.284∗ 9.933∗∗∗ 15.949∗∗∗ 27.048∗∗∗ 2.329∗ 17.998∗∗∗

df 8, 1039 8, 1039 8, 1039 8, 1039 8, 1039 8, 1039 8, 1039

Model 2Baby food .031 −.039 −.052 .077∗ .057∗∗ .031 .064∗

Chocolate .037 −.084∗ .051 .262∗∗∗ −.086∗∗∗ .146∗∗∗ .195∗∗∗

Coffee .124∗∗∗ −.062 −.084∗ .338∗∗∗ .147∗∗∗ .128∗∗∗ −.044Coffee milk .001 −.041 .023 .188∗∗∗ .104∗∗∗ .111∗∗ .119∗∗∗

Candy −.005 .006 .175∗∗∗ .092∗ .043 .057 .135∗∗∗

Sugar −.032 −.037 −.092∗∗ .072∗ .083∗∗∗ −.005 −.019Canned meat −.055∗ −.087∗∗∗ .085∗∗ .077∗∗ .052∗∗ .008 .177∗∗∗

Canned fruits −.030 .014 −.072∗ .262 .030 .004 .008CP .422∗∗∗ −.023 .187∗∗∗ .338∗∗∗ .184∗∗∗ .144∗∗ .401∗∗∗

CU .253∗∗∗ .003 .082 .188∗∗∗ .647∗∗∗ −.171∗∗ .090∗

R2 .422 .018 .128 .121 .682 .028 .322R2

adj .416 .008 .120 .112 .679 .019 .316R2 change .350 .000 .057 .011 .510 .011 .200F change 313.815∗∗∗ .214 33.932∗∗∗ 6.710∗∗ 832.809∗∗∗ 5.678∗∗ 153.239∗∗∗

Mean SS Err. 69.387 13.540 13.029 48.896 155.751 12.468 30.816Overall F 75.730∗∗∗ 1.867∗ 15.237∗∗∗ 14.241∗∗ 222.847∗∗∗ 3.016∗∗∗ 49.266∗∗∗

df 10, 1037 10, 1037 10, 1037 10, 1037 10, 1037 10, 1037 10, 1037

Model 3CP .409∗∗∗ −.025 .085∗ .120∗∗ .241∗∗∗ .090∗ .313∗∗∗

CU .271∗∗∗ −.030 .183∗∗∗ −.007 .606∗∗∗ −.072 .232∗∗∗

R2 .399 .003 .063 .013 .634 .004 .256R2

adj .398 .001 .061 .011 .633 .002 .254Mean SS Err. 71.616 13.643 13.894 54.455 178.035 12.681 33.568Overall F 346.724∗∗∗ 1.361 35.157∗∗∗ 7.007∗∗∗ 905.874∗∗∗ 2.118 179.629∗∗∗

df 2, 1045 2, 1045 2, 1045 2, 1045 2, 1045 2, 1045 2, 1045

Statistical significance at ∗p ≤ .05, also ∗∗p ≤ .01, ∗∗∗p ≤ .001;Reference category: Personal care (N = 285)

2.4 Results 27

adjusted R2 we conclude that the impact of these variables on the execution times ofthe aforementioned sub-activities is weak. This result is consistent with our predictionthat CP and CU do not affect the Search, Prepare and Fill old sub-activities.

In summary, we conclude that the results provide evidence that the execution timefor Fill new, Grab/unpack and Dispose are mostly explained by CP and CU , whilewe found little evidence to show that these variables substantially affect the othersub-activities.

Next, we obtain the TST simply as the sum of the estimated execution times for eachindividual sub-activity, derived under Models 2 and 3. Thus the estimated TST isderived as

T ST = TG + TS + TW + TP + TFn + TFo + TD,

where T s, s ∈ A, stands for the estimated execution time of the corresponding sub-activity, as given by Model 2 or 3. To estimate the accuracy of this prediction wecompare the estimated TST with the actual TST (obtained from empirical data)and the results are included in Table 5.4. Both variables have the same mean (57.31seconds) as confirmed by a paired-samples t-test. The correlation coefficient betweenthe predicted and the measured TST is .819 (Model 2) and .798 (Model 3) andthus 67% (respectively 63.7%) of the variance in the measured TST per order lineis explained by the sum of time estimates for individual sub-activities. Thus, resultsshow a slightly better performance of Model 2 as compared with Model 3 but theincrease in adjusted R2 is marginal. Therefore, given the simplicity of Model 3, werecommend choosing it for forecasting purposes.

Table 2.3 Sequential regression: actual vs. predicted TST (Chain A)

Model 2 Model 3

Variables Unstd. Std. Std. Unstd. Std. Std.Coeff. Err. Coeff. Coeff Err. Coeff.

(Constant) .000 1.403 .000 1.502

T ST 1.000∗∗∗ .022 .819 1.000∗∗∗ .023 .798

R .819 .798R2 .670 .637R2

adj .670 .636Mean SS Err. 442.099 486.720Overall F 2124.795 1834.109∗∗∗

df 1, 1046 1, 1046

Statistical significance at ∗p ≤ .05, also ∗∗p ≤ .01, ∗∗∗p ≤ .001


2.4.2 Overall regression results

Similarly, regression models 1 to 3 are analyzed for the dependent variable TST . Theresults of ordinary least squares estimation for the first dataset (1048 observations)are presented in Table 2.4. Collinearity tests (correlation coefficients, varianceinflation factors) indicated no significant problems with respect to multi-collinearityfor the estimated models. In addition, upon preliminary inspection of the results, nosignificant outliers or influential points were detected, and thus the results includedin Table 2.4 reflect the entire dataset. An alternative model formulation whereinteractions between the explanatory variable and the product category were includeddid not improve the model specification and were not significant (Aiken and West,1991). Results for Model 1 indicate that the product category variable alone explains

Table 2.4 Overall regression: results for TST (Chain A)

Model 1 Model 2 Model 3

Variables Unstd. Std. Std. Unstd. Std. Std. Unstd. Std. Std.Coeff. Err. Coeff. Coeff. Err. Coeff. Coeff. Err. Coeff.

(Constant) 35.474∗∗∗ 1.973 3.902∗ 1.785 10.240∗∗∗ 1.447Baby food 14.978∗ 6.298 .069 13.898∗∗∗ 3.995 .064Chocolate 30.872∗∗∗ 3.239 .310 5.896∗ 2.375 .059Coffee 38.063∗∗∗ 3.270 .377 18.471∗∗∗ 2.166 .183Coffee milk 45.383∗∗∗ 4.868 .279 21.527∗∗∗ 3.227 .132Candy 19.968∗∗∗ 2.892 .232 7.932∗∗∗ 2.007 .092Sugar 35.360∗∗∗ 8.093 .126 10.517∗ 5.176 .037Canned meat 41.697∗∗∗ 5.243 .236 12.016∗∗∗ 3.418 .068Canned fruits 29.401∗∗∗ 6.209 .138 2.922 3.998 .014CP 19.614∗∗∗ 1.471 .375 19.052∗∗∗ 1.396 .364CU 1.180∗∗∗ .081 .442 1.327∗∗∗ .071 .496

R2 .178 .670 .637R2

adj .172 .667 .636R2 change .178 .492 .637F change 28.130∗∗∗ 773.434∗∗∗ 916.178∗∗∗

Mean SS Err. 1108.989 445.936 487.185Overall F 28.130∗∗∗ 210.651∗∗∗ 916.178∗∗∗

df 8, 1039 10, 1037 2, 1045

Statistical significance at ∗p ≤ .05, ∗∗p ≤ .01, ∗∗∗p ≤ .001;

Reference category: Personal care (N = 285)

in proportion of 17.2% the variance in TST , while Model 2 yields a significantlylarger adjusted R2 (66.7%), with a significant 49.2% of the total variability in TSTaccounted for by the group of variables CP and CU . Model 3 shows that theexplanatory variables CP and CU together have a significantly high joint contributionin predicting the TST , accounting for 63.7% of the variability in the TST . The F-

2.4 Results 29

statistics for all three models are statistically significant (p ≤ .001). Thus, resultsprovide evidence that the TST is systematically explained in large measure by ourmodel. Furthermore, comparing Models 2 and 3, we note that when the subgroup-effect is excluded from the model, the adjusted R2 decreases only marginally andin both cases the hypotheses of our study are supported. The coefficient estimatespresented in Table 2.4 show that TST is positively correlated with CP and CU , thusproviding support for our hypotheses at p ≤ .001. They also are fairly consistentbetween Models 2 and 3. The standardized values of the coefficients indicate thatmost of the explanatory power comes from the variables CP and CU , with a higherinfluence of CU . This could reflect the larger impact of the sub-activity Fn (mostlyaffected by CU) on the TST . While all significant in Model 1, the coefficients ofdummy variables for product category remain significant (at p ≤ .05), with oneexception (Canned fruits), in Model 2. Compared with CP and CU however, theyhave small explanatory power. Also, recall that in our modeling we used Personal careas a reference category for the group of dummy variables (with the largest number ofobservations), and therefore the positive coefficients for the dummy variables confirmour expectations from previous descriptive statistics (see Table B1, Appendix B):according to this dataset, the personal care category is the fastest to handle on orderline basis.

Based on these results, we conclude that Models 2 and 3 already explain a largeportion of the Total Stacking Time and the variables CP and CU are importantpredictors of TST . Due to the simplicity of the model and its good accuracy, werecommend using Model 3 for forecasting TST .

2.4.3 Validation of the results

To validate the results from the previous section, we use the empirical data from chainB and replicate the analysis conducted for chain A. We do this in order to verify thereliability of the previously obtained results and the accuracy of the predictive models(Wang, 1994). Summary statistics for the variables in this study using the seconddataset are included in Tables B3 and B4 from Appendix B. The average TST acrossall five product categories is 49.29 seconds with a standard deviation of 27.06. Thesmallest average TST is recorded for the products from the wine subgroup (39.69seconds), while the most time consuming products in this set for handling are thosefrom category cookies (mean TST equals 60.62 seconds). The TST shows significantvariation between order lines, ranging from a minimum of 6 seconds (wine) to amaximum of 212 seconds per order line (canned vegetables). The variable CP rangesfrom 1 CP (all categories) to 8 CP (cookies) with an average of 1.22 CP across allcategories, and a standard deviation of 0.6 CP . The variable CU has an overall meanof 15.5 consumer units (standard deviation equals 8.86), ranging from 6 to 80 CU perorder line.

To assure the general applicability of the approach proposed in this study, we are


interested in how consistently the previous results replicate for the second data set.Regarding individual sub-activities, regression results (see Table C1 from AppendixC) for Models 1 to 3 confirm to a high extent previous findings: the CP , CU havea positive effect on execution times of sub-activities Fn, G and D and are the mostimportant predictors of time variation for these sub-activities (adjusted R2 is 52.6%,48.3% and 20.4%, respectively for Model 2, and 50.9%, 47.6% and 19.6%, respectivelyfor Model 3). When sequential regression is then used to estimate the TST (see TableC2 from Appendix C), we found that the correlation coefficient between the predictedand the measured TST is 0.724 (R2 = 52.4%) using Model 2 and 0.684 (R2 = 46.7%)for Model 3. The high values of these correlation coefficients indicate that the TSTfor chain B is also explained to a large degree by the chosen models. Furthermore,comparing results for Models 2 and 3, we are again in favor of the simplest Model 3to be used for deriving good estimations of the TST .

For forecasting purposes, the overall regression for estimating TST provides a simpleand less time-consuming procedure. Therefore, we tested the reliability of the resultson the second data set as well. We found consistent support for our hypothesesregarding TST (see Table 2.5). While the group of dummy variables relatedto product category have a significant, but small contribution in predicting TST(adjusted R2 about 10%), the most explanatory power comes again from the group ofvariables CP and CU , which affect significantly and positively the TST . Comparedto Model 2 (R2

adj = 51.9%), CP and CU alone explain 46.5% of the variance in TST ,

thus indicating only a marginal decrease in adjusted R2. Their coefficients are bothstatistically significant at p ≤ .001 and consistent between the two models. Moreover,note that they have also comparable sizes with coefficients’ estimates for CP andCU derived for the first dataset (compare Tables 2.4 and 2.5). We can, therefore, beconfident that the effects of both CP , and CU are needed to model the TST to a largeextent and that Model 3 represents a simple and reliable alternative for predictingTST .

We performed a final verification of the results, in which we used the data collectedfor chain B and the coefficients estimates for Model 3 derived for chain A (see againTable 2.4) to compute predicted values of TST for each order line. We restate herefor reference the model used for prediction:

TST = a0 + a1 CP + a2 CU, (2.3)

where a0 = 10.240, a1 = 19.052 and a2 = 1.327.

The high values of the correlation coefficient between the measured and predicted TST(R = 0.683, R2 = 46.6%) provide additional evidence that the TST for chain B isalso explained to a large degree by our model. Overall, the results of this section offervaluable support for the general applicability of our approach to similar settings, withimportant ramifications for retailers. Model 3 represents a simple and reliable methodfor predicting the TST . The stacking-times for each order line can be estimatedinexpensively in this way, and ultimately be used for management decisions. For

2.4 Results 31

Table

2.5

Overallregression:resultsforTST

(Chain

B)

Model1

Model2

Model3

Variables

Unstd.

Std.

Std.

Unstd.

Std.

Std.

Unstd.

Std.

Std.

Coeff.

Err.

Coeff.

Coeff.

Err.

Coeff.

Coeff.

Err.

Coeff.

(Constant)

60.620∗∗

∗1.923

19.981∗∗

∗2.361

9.960∗∗

∗1.971

Sandwichspread

−11.888∗∗

3.940

−.132

−11.367∗∗

∗2.883

−.126

Canned

vegetables

−7.953∗

3.914

−.089

−5.929∗

2.920

−.066

Candy/Chocolate

−18.029∗∗

∗2.892

−.290

−15.897∗∗

∗2.113

−.255

Wine

−20.930∗∗

∗2.972

−.325

−13.282∗∗

∗2.661

−.206

CP

19.890∗∗

∗1.887

.441

18.526∗∗

∗1.732

.411

CU

.892∗∗

∗.143

.292

1.079∗∗

∗.117

.353

R2

.102

.524

.467

R2 adj

.095

.519

.465

R2ch

ange

.102

.422

.467

Fch

ange

15.820∗∗

∗246.753∗∗

∗245.558∗∗

∗

MeanSSErr.

662.247

352.103

391.431

OverallF

15.820∗∗

∗102.087∗∗

∗245.558∗∗

∗

df

4,558

6,556

2,560

Statisticalsignifica

nce

at

∗p≤

.05,∗∗

p≤

.01,∗∗

∗p≤

.001;Referen

ceca

tegory:Cookies(N

=179)


example, one can assess the amount of work necessary during a day to execute therestocking of the shelves (using readily available information about the number ofunits and case packs for each SKU), or can assess the individual labour performanceof store employees. Also, by inspecting the individual sub-activities, one can get anindication about the possible inefficiencies in the whole process.

2.5. Analytical insights and implications for retailers

The empirical findings of this study offer the practitioners the opportunity for abetter control of the overall logistical costs. In the literature on retail operations, thehandling costs are rarely modeled or they are often assumed to be a linear functionof the number of CUs. Our results indicate however that an additive cost structureis more appropriate. The TST per SKU is linearly dependent on CP and CU , butnot directly proportional with CP and CU (due to the constant parameter a0) (seeequation (2.3)). For each order line, a fixed ’setup time’ (a0) is incurred, additionallyto the positive time related to the number of units being handled. This cost structureof the TST allows economies of scale, and thus can be further exploited in orderreplenishment decisions. For example, the retailer could decide to order less frequentlybut in a larger number of case packs in order to reduce the handling workload. On theother hand, less frequent deliveries lead to an increase in the average inventory level.In Section 2.5.1, we present a lot-sizing analysis which takes into account handlingcosts, and in Section 2.5.2 we give an indication about the magnitude of potentialefficiency gains in stacking.

2.5.1 Extending the EOQ-model with shelf stacking

Consider the classical single-item EOQ-model (Economic Order Quantity) (Zipkin,2000), where we recognize explicitly not only the cost for holding inventory in thestore (Ch) and the ordering cost (Co), but also a distinct shelf stacking cost (Cs),derived on the basis of the handling time (2.3). The objective is to determine aninventory control policies that minimizes the long run average total costs. We use thefollowing notation:

X = order quantity (in consumer units)q = fixed case pack size, q = 1, 2, 3, . . .λ = annual demand rate (assumed constant and known)K = fixed ordering costh = annual holding cost per consumer unitS = stacking cost per hour

TC(X) = average total annual costs, as a function of X

We further restrict our attention to the situation when we are only allowed to order inmultiples of a given case pack size q, i.e. X = mq, m = 0, 1, 2, . . .. Following a similar

2.5 Analytical insights and implications for retailers 33

reasoning as in the classical EQO analysis and expressing each cost component as afunction of the order size X yields the following average annual total cost:

TC(X) = Co(X) + Ch(X) + Cs(X)

= K · λ

X+ h · X

2+ S · TST (X) · λ

X

= (K + S · a0) ·λ

X+ h · X

2+ S · (a1 ·

λ

q+ a2λ)

or equivalently,

TC(m) = Co(mq) + Ch(mq) + Cs(mq)

= (K + S · a0) ·λ

mq+ h · mq

2+ S · (a1 ·

λ

q+ a2λ).

Hence, when X (and m) is not restricted to be an integer value, and we assumefixed case pack sizes (q), the optimal ordering quantity (X∗) is the value such that∂TC(X)/∂X = 0, and is given by:

X∗ = m∗q =√2λ(K + Sa0)/h. (2.4)

Because function TC(m) is convex in m for positive values and is minimized by m∗,the optimal integral order quantity is one of the two integers surrounding m∗, whichgive the lowest value of the total costs. The total annual costs corresponding to X∗

is given by:

TC(m∗) = Co(m∗q) + Ch(m

∗q) + Cs(m∗q)

=√

2hλ(K + Sa0) + Sλ(a1/q + a2). (2.5)

Equation (2.4) resembles the classical EOQ formula, in which the ordering cost isreplaced here by K + Sa0, while in formula (2.5) we note that the first componentrepresents the optimal total cost in the classical EOQ formula (but with the orderingcost modified as K+Sa0) plus the extra term Sλ(a1/q+a2). In this setting, the fixed’setup time’ a0 is relevant for the derivation of the optimal replenishment quantity.

2.5.2 Order of magnitude for efficiency gains in stacking

The model formulated in equation (2.3) allows quantifying the time savings inthe stacking process, obtained when it is possible to reduce the frequency of thereplenishment, by ordering more products at once, rather than the same amountmultiple times. We use the coefficient estimates in (2.3) as reliable indication of thesize of the effects identified. Let n (n = 1, 2, 3, . . .) be the number of order lines forthe same SKU in a replenishment order in the subsequent analysis. Two situationscan then be considered: (1) increase the number of case packs per order line and (2)increase the case pack size q.


Case 1: Increase the number of case packs per order line

The effect of reducing the replenishment size (i.e. the number of order lines) byordering more CP per order line, while keeping the same case pack size (q) can beevaluated. We compare the time savings obtained if, instead of ordering n order lineswith CP of size q per order line, it is possible to order the entire amount at once(i.e. in one order line), by ordering nCP , each of size q. The total time needed forstacking n order lines with CP of size q per order line can be written as:

TT (CP, q) = nTST (CP, q) = na0 + na1CP + a2nCP · q.

The total time needed for stacking the same amount at once is expressed as:

TST 1(CP, q) = TST (nCP, q) = a0 + a1nCP + a2nCP · q.

Then the time savings can be derived as:

TT − TST 1 = (n− 1)a0 > 0, for n > 1,

which implies that we may save stacking time if we order, in each replenishment, morecase packs at once, instead of ordering one case pack at a time, and this saving isdue to the constant ’setup time’ a0. The efficiency gain, compared with the case ofmultiple replenishments is then:

S1(CP, q, n) :=TT − TST 1

TT=

(n− 1)a0na0 + na1CP + a2nCP · q

· 100%, for n > 1. (2.6)

Case 2: Increase the case pack size q

We evaluate the time savings obtained if, instead of ordering n order lines with CPof size q per order line, it is possible to order the entire amount at once (i.e. in oneorder line), by ordering CP case packs, each of size nq. In this case, the shelf stackingtime is derived as follows:

TST 2(CP, q) = TST (CP, nq) = a0 + a1CP + a2CP · nq.

Then the time gains are now:

TT − TST 2 = (n− 1)a0 + (n− 1)a1CP > 0, for n > 1, (2.7)

and the percentage of time saving is then:

S2(CP, q, n) :=TT − TST 2

TT=

(n− 1)a0 + (n− 1)a1CP

na0 + na1CP + a2nCP · q· 100%, for n > 1. (2.8)

Again, in this case, reducing the frequency of the replenishments may result in timesavings and efficiency gains as given by (2.7) and (2.8). Furthermore, by comparing

2.5 Analytical insights and implications for retailers 35

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

Order lines (n)

% E

ffic

ien

cy g

ain

s

S1, CP = 1, q = 6

S1, CP = 4, q = 6

S2, CP = 1, q = 6

S2, CP = 4, q = 6

Figure 2.2 Effect of n on S1 and S2 for two particular choices of CP and q

equations (2.6) and (2.8), we notice that the time saving is always higher in the secondcase, when the strategy is to increase the case pack size (q) instead of the number ofcase packs (CP ) per order line.

The efficiency gains derived from (2.6) and (2.8) are illustrated in Figure 5.10, fortwo particular choices of CP and q. Typically, case pack sizes take values of 6, 12or 24 consumer units. In Figure 5.10, the effect of n on S1 and S2 is illustrated forone and respectively four case packs per order line, each of size six. We note that thehigher the reduction in the number of order lines, the higher the savings. Reducingone order line for the same SKU (n = 2 in Figure 5.10) as a consequence of orderingtwo times more case packs results in efficiency gains of 13% (if CP = 1) and 4% (ifCP = 4), respectively. Alternatively, we observe higher potential gains, up to 40%,when it is possible to place orders for higher case pack sizes.

Note again from (2.6) and (2.8) that n, CP and q have a combined effect on S1 andS2. Generally as n increases, S1 and S2 reach steady values, with a maximum around30% (for S1) and 80% (for S2), respectively. However, as CP and q increases, theefficiency gains are decreasing. Notable is the behavior of S1 with respect to n andCP , when the savings drop as CP increases, indicating that in formula (2.6), theestimated time due to CP and q, outweighs the fixed ’setup-time’ (due to a0).


Although in practice the case pack sizes are usually set by the manufacturers, it isstill valuable to recognize the impact of reduced sizes on handling efficiency, perhapsespecially for retailers that also carry their own private labels. These preliminaryinsights into the potential efficiency gains derived from the proposed model, offerinteresting opportunities for developing adapted inventory control rules that takeinto account the handling component. Building new inventory replenishment policiesthat recognize the handling efficiency, should of course consider the possible tradeoffs(such as the shelf space availability and physical constraints of the shelves, the demandpattern, or restrictions with respect to possible case pack sizes). We consider suchadvancements in the following chapters.

2.6. Conclusions

In this chapter, we focused on the shelf stacking process in grocery retail stores andstudied the main factors that drive the shelf stacking time. This study has threemain contributions: first, we provide insights into the specifics of the stacking processby analyzing the individual sub-activities and describe the interactions between keytime-drivers (CP and CU) and logistics processes; secondly, we derive a reliable modelfor estimating the total stacking time per order line; finally, we use the empiricalresults to quantify potential time savings and demonstrate by a lot sizing analysisthe opportunities for considering handling costs explicitly for inventory managementdecisions.

We used real life empirical data from two European grocery retailers and adopted twostrategies for estimating the TST per order line and evaluating the relative impactof each factor identified (sequential vs. overall regression). The two approaches mayserve two different practical purposes. On one hand, the sequential approach, allowsone for a better insight into the details of the shelf stacking process, identifying thosesub-activities that are mostly affected by the number of items being handled (CP andCU), and those for which the variation in workload is potentially affected by otherfactors. At the same time, the approach indicates which sub-activities contributemostly to the total variation in the stacking time of a new order line. The threemost relevant sub-activities are found to be: stacking new inventory, grabbing andopening of a case pack, and waste disposal, in this order. These sub-activities arealso mostly affected by CP and CU , and we indicate the magnitude of their relativepositive impact. We also find that the time for searching, preparing and filling oldinventory can be regarded as fixed per order line. This information can further beused to identify inefficiencies in the stacking process.

On the other hand, the overall regression strategy offers a simple, inexpensive toolfor predicting the TST per order line. In this study, we found enough support toconclude that a simple prediction model, depending only on the number of case packsand the number of consumer units, offers already a reliable estimate of the TST .

2.6 Conclusions 37

Results from testing and validation show that the model is stable and it explains theTST to a large extent. Although the influence of CP and CU on the shelf stackingprocess may be implicitly recognized, we demonstrate that both variables are relevantpredictors for TST and we also estimate the size of their effects. As compared tocommon assumptions in the literature, we find that the form of the relationshipis additive, rather than purely linear. Using this structural insight, we investigatethe magnitude of efficiency gains in the stacking process. Furthermore, we modelthe empirical findings into a lot-sizing analysis to demonstrate the opportunities forextending inventory control rules with a handling component. The empirical findingsof this study offer the practitioners the opportunity for a better control of the overalllogistical costs. We shall further investigate this opportunity in Chapter 3.

While we illustrate the impact of some key drivers on time variation, we recognizethere are more potential factors that may affect the execution time of a certain activity,which are worth further investigation. It seems reasonable for example to assumethat the type of package, the distance traveled within the aisle, the SKUs volume orweight, and the inventory level of the products on the shelves just before restockingmight have an impact on the Total Stacking Time. These variables were however notavailable in the original dataset.

Appendix A. Shelf stacking activities

See Table A1.

Appendix B. Descriptive statistics of the empiricaldatasets

See Tables B1, B2, B3 and B4.

Appendix C. Validation results for chain B

See Tables C1 and C2.


Table A1 Shelf stacking sub-activities

Sub-activity Starting/Ending point of sub-activity

Grab/open case pack (G)* Start The filler stands in front of the rolling container and reachesfor a case pack.

End The filler prepares to walk away from the rolling containerand starts opening a case pack.

or Start The filler has arrived at the shelf location and starts openingthe case pack.

End The filler is ready with opening the case pack and anothersub-activity starts.

Search (S) Start The filler starts with checking the product (from the rollingcontainer) and he/she looks for the right shelf location.

End The filler sees the right shelf location and prepares toapproach it (walk).

Walk (W)** Start The filler prepares to walk away from the rolling containeror walks after searching the right shelf location.

End The filler stands in front of the shelves.and Start The filler prepares to walk away from the shelf location or

waste disposal place, to the rolling container.End The filler stands in front of the rolling container and reaches

for a case pack.

Prepare the shelves/check’best before’ date (P)

Start The filler reaches for the old inventory on the shelves andstart to check the ’best before’ date (if needed).

End The filler is ready with preparing the shelves; old inventoryis straightened or is removed from the shelves.

Fill new inventory (Fn) Start The filler reaches for the new inventory in the case pack.End The filler reaches for the old inventory or grabs the empty

box or plastic.

Fill old inventory (Fo) Start In case old inventory was removed from the shelves, the fillerstarts putting it back on the shelves.

End The filler is ready with putting old inventory back on theshelves and grabs the empty box or plastic.

Waste disposal (D) Start The filler holds an empty box (or plastic) and starts toflatten it (sometimes the box is preserved to customers).

End The moment the filler prepares to leave the waste disposalplace (a trolley or a place near the rolling container).

Extra (E) An activity not part of the first sub-activities, e.g., help acustomer, customer is in the way, get or put away crate, processinventory remainder, organize labels, general cleaning, discuss witha colleague, take away waste, bring empty boxes for customers tocheck out area, get a new rolling container, take away misplacedproducts, repair a broken product, remove cord from rollingcontainer, take a product to the kiosk, straighten separation plate.

Notes:*Grabbing and opening the case pack are taken together, because the individual activities were difficultto separate.**Walking does not include walking with the rolling container from the storage area to the right aisle orwalking with the rolling container between the aisles. But it does include (in exceptional cases) walkingwith the rolling container when the rolling container is moved to bring certain case packs to the rightshelf locations (e.g. heavy products). It is possible that the filler performs multiple sub-activities atonce, e.g. walking while opening the case pack, searching or disposing waste. When this happened, thefollowing reasoning was used: if the walking time was significantly influenced by the attention focusedon opening the case pack (or searching/waste disposal), the time for e.g. opening the case pack wasmeasured as sub-activity ’G’, and the remaining time as sub-activity ’W’. If the walking time was notsignificantly influenced by one of these sub-activities, then the total time was measured as walking time(W).

2.6 Conclusions 39

Table

B1Descriptivestatisticsofexplanatory

variables(C

hain

A)

Number

ofSKUs

Number

ofOrder

Lines

NumberofCP

per

NumberofCU

per

TST

per

Cate

gory

Order

Line

Order

Line

Order

Line[sec.]

Avg.

SD

Min.

Max.

Avg.

SD

Min.

Max.

Avg.

SD

Min.

Max.

Babyfood

26

31

1.13

0.34

12

8.90

4.03

416

50.45

17.47

20

88

Chocolate

91

168

1.36

0.72

14

25.26

16.58

680

66.35

41.22

15

294

Personalcare

193

285

1.14

0.37

13

7.80

3.86

336

35.47

14.93

10

94

Coffee

91

163

1.47

1.08

19

18.88

18.74

6135

73.54

48.62

20

334

Coffee

milk

31

56

1.45

0.81

15

22.93

12.27

10

60

80.86

35.34

34

211

Candy

143

248

1.15

0.39

13

17.92

9.17

872

55.44

26.69

16

74

Sugar

11

18

1.83

0.99

14

17.33

9.43

840

70.83

32.25

22

151

Canned

meat

40

47

1.77

0.96

15

22.55

15.20

672

77.17

51.23

12

245

Canned

fruit

32

32

1.72

0.81

14

20.63

13.10

648

64.88

31.09

11

125

Aggregate

statistics

646

1048

1.30

0.70

19

16.78

13.69

3135

57.31

36.59

10

334

CP:ca

sepack

s,CU:co

nsu

mer

units,

TST:totalstack

ingtime,

SD:standard

dev

iation


Table B2 Descriptive statistics of response variables (Chain A)

N Mean Std. Std. ErrorDev. Mean

TST 1048 57.31 36.59 1.13Grab/Open 1048 11.65 10.91 0.34Search 1048 2.31 3.70 0.11Prepare 1048 3.54 7.42 0.23Fill New 1048 27.32 22.04 0.68Dispose 1048 7.28 6.71 0.21Walking 1048 4.77 3.85 0.12Fill Old 1048 0.44 3.57 0.11

TST : total stacking time

Table B4 Descriptive statistics of response variables (Chain B)

N Mean Std. Std. ErrorDev. Mean

TST 563 49.29 27.06 1.14Grab/open 563 7.49 6.92 0.29Search 563 0.67 3.09 0.13Prepare 563 5.87 7.51 0.32FillNew 563 21.94 12.97 0.55Dispose 563 4.56 4.57 0.19Walking 563 7.26 5.78 0.24FillOld 563 1.5 4.93 0.21

TST : total stacking time

2.6 Conclusions 41

Table

B3Descriptivestatisticsofexplanatory

variables(C

hain

B)

Number

ofSKUs

Number

ofOrder

Lines

NumberofCP

per

NumberofCU

per

TST

per

Cate

gory

Order

Line

Order

Line

Order

Line[sec.]

Avg.

SD

Min.

Max.

Avg.

SD

Min.

Max.

Avg.

SD

Min.

Max.

Sandwichspread

39

56

1.25

0.47

13

17.11

9.71

848

48.73

25.02

18

148

Canned

vegetables

46

57

1.28

0.67

15

14.74

7.83

860

52.67

31.88

17

212

Cookies

125

179

1.22

0.68

18

18.41

9.60

880

60.62

27.34

19

194

Candy/Chocolate

103

142

1.13

0.35

13

18.05

7.22

650

42.59

20.04

13

132

Wine

84

129

1.29

0.69

15

8.29

4.36

630

39.69

26.29

6168

Aggregate

statis-

tics

397

563

1.22

0.60

18

15.50

8.86

680

49.29

27.06

6212

CP:ca

sepack

s,CU:co

nsu

mer

units,

TST:totalstack

ingtime,

SD:standard

dev

iation


Table C1 Regression results for each individual sub-activity (standardized coefficients)(Chain B)

Dependent Variables

G S W P Fn Fo D

Model 1Sandwich spread −.073 −.002 −.188∗∗∗ −.056 −.033 −.218∗∗∗ −.009Canned vegetables −.002 .058 −.152∗∗∗ −.016 −.050 −.257∗∗∗ .076Candy/Chocolate −.053 .002 −.106∗ −.376∗∗∗ −.168∗∗∗ −.351∗∗∗ −.028Wine .030 .241∗∗∗ −.130∗∗∗ −.333∗∗∗ −.339∗∗∗ −.343∗∗∗ −.091

R2 .009 .056 .041 .157 .094 .142 .016R2

adj .002 .049 .034 .151 .088 .136 .009Mean SS Err. 47.715 9.069 32.253 47.865 153.305 20.975 20.722Overall F 1.298 8.279∗∗∗ 6.015∗∗∗25.955∗∗∗ 14.546∗∗∗23.039∗∗∗ 2.34df 4, 558 4, 558 4, 558 4, 558 4, 558 4, 558 4, 558

Model 2Sandwich spread −.075∗ −.006 −.194∗∗∗ −.053 −.014 −.223∗∗∗ −.015Canned vegetables −.001 .049 −.165∗∗∗ −.005 .008 −.271∗∗∗ .064Candy/Chocolate −.011 .007 −.088 −.372∗∗∗ −.143∗∗∗ −.345∗∗∗ .001Wine .076 .212∗∗∗ −.159∗∗ −.288∗∗∗ −.099∗ −.389∗ −.106∗

CP .591∗∗∗ .084 .282∗∗∗ .031 .245∗∗∗ .117 .436∗∗∗

CU .155∗∗ −.051 −.032 .097 .523∗∗∗ −.083 .014

R2 .489 .060 .109 .169 .531 .148 .213R2

adj .483 .050 .100 .160 .526 .139 .204R2 change .480 .004 .068 .012 .437 .007 .196F change 260.917∗∗∗ 1.075 21.264∗∗∗ 4.036∗ 258.759∗∗∗ 2.200 69.314∗∗∗

Mean SS Err. 24.702 9.067 30.069 47.35 79.686 20.885 16.646Overall F 88.644∗∗∗ 5.879∗∗∗11.389∗∗∗18.837∗∗∗104.909∗∗∗16.159∗∗∗25.046∗∗∗

df 4, 558 4, 558 4, 558 4, 558 4, 558 4, 558 4, 558

Model 3CP .634∗∗∗ .193∗∗∗ .220∗∗∗ −.004 .238∗∗∗ .008 .388∗∗∗

CU .089* −.211∗∗∗ .054 .180∗∗∗ .546∗∗∗ .091 .087R2 .478 .033 .066 .032 .510 .009 .199R2

adj .476 .030 .062 .028 .509 .006 .196Mean SS Err. 25.053 9.256 31.326 54.780 82.599 24.128 16.825Overall F 256.308∗∗∗ 9.594∗∗∗19.646∗∗∗ 9.137∗∗∗291.821∗∗∗ 2.588 69.384∗∗∗

df 2, 560 2, 560 2, 560 2, 560 2, 560 2, 560 2, 560

Statistical significance at ∗p ≤ .05, also ∗∗p ≤ .01, ∗∗∗p ≤ .001;Reference category: Cookies (N = 179)

2.6 Conclusions 43

Table C2 Sequential regression: actual vs. predicted TST (Chain B)

Model 2 Model 3

Unstd. Std. Std. Unstd. Std. Std.Coeff. Err. Coeff. Coeff. Err. Coeff.

(Constant) .000 2.133 .000 2.373

T ST 1.000∗∗∗ .040 1.000∗∗∗ .045 .684

R .724 .684R2 .524 .467R2

adj .523 .466Mean SS Err. 348.965 390.733Overall F 618.031 491.994∗∗∗

df 1, 561 1, 561

Statistical significance at ∗p ≤ .05, also ∗∗p ≤ .01, ∗∗∗p ≤ .001

45

Chapter 3

Lost-sales inventory modelswith batch ordering andhandling costs

Abstract: Consider a retailer who manages periodically the inventory of a single-item facing stochastic demand. The retailer may only order in multiples of a fixedbatch size q, the lead time is less than the review period length and all unmet demandis lost, which is a realistic situation for a large part of the assortment of groceryretailers. The replenishment cost includes fixed and variable components, dependenton the number of batches and units in the order. This structure captures the shelfstacking costs in retail stores. We investigate the optimal policy structure under thelong-run average cost criterion, and propose a new heuristic policy, the (s,Q|S, nq)policy, which captures most of the structure of optimal policies and shows close-to-optimal performance. We further compare its performance against the best (s, S, nq)and (s,Q, nq) policies and quantify the relative improvements. We find that (s, S)policies perform very well in settings with high penalty costs and low batch and unit-related costs, while (s,Q) policies may outperform (s, S) policies in environments withlow service levels. The new heuristic performs, in both situations, consistently verywell. Finally, we show that handling costs may substantially affect overall system’sperformance, when ignored, especially for items with low-profit margins.

3.1. Introduction

As pointed out in earlier chapters, traditional store-based retailers face nowadaysintense competition, which challenges them to reduce and/or manage costs in their

46 Chapter 3. Inventory control with handling consideration

key business areas. In a grocery retail store, these key areas are mainly inventory andpeople (i.e. merchandize handling). While handling costs are relatively high in thegrocery retail supply chain, they are rarely acknowledged in inventory replenishmentdecisions. In this chapter, we recognize the shelf stacking cost as a critical component,and integrate inventory and handling into a single model for analysis and optimizationof inventory decisions.

In Chapter 2, we showed that we can reliably estimate the handling time per StockKeeping Unit (SKU) required to execute the shelf stacking operation using an additivemodel (fixed plus linear terms), depending on the number of case packs (batches) andthe number of consumer units stacked. This leads to a replenishment cost structurethat allows economies of scale. For example, the retailer could decide to order lessfrequently but in a larger number of case packs in order to reduce the handlingworkload. However, less frequent deliveries lead to an increase in the average inventorylevel.

In this chapter, we investigate inventory replenishment decisions that take intoaccount a merchandize handling component due to shelf stacking. The basic settingis a retailer managing the inventory of a single item, facing stochastic demand. Thestock is reviewed periodically (say daily or every 2nd or 3rd day) and new orderscan be placed at the beginning of each review epoch. The ordered stock is typicallyreceived before the beginning of the next review epoch, which results in replenishmentlead times shorter than the length of the review epoch. This situation is common inthe European and Japanese grocery retail environment. For example, orders mightbe placed during the morning and delivered to the stores in the evening of the sameday. Furthermore, the replenishment stock arrives in pre-packed form, while theconsumer demand may be for individual units. The case packs have fixed, exogenouslydetermined sizes, usually set by the manufacturer, due to limitations in packaging,transportation and coordination. Once arrived at the store, the packed deliveriesare unwrapped and units are displayed onto the shelves in response to consumer’sdemand. This operation is part of the shelf stacking process at the store. Typically,if inventory is insufficient, resulting in an out of stock situation, this leads to a lostsale. We analyze such a system considering the following cost components: a cost forholding inventory in the store, a cost associated with the demand which is lost anda replenishment cost, which includes fixed ordering and shelf stacking costs. Basedon the results obtained in Chapter 2, we assume that each replenishment order isassociated with costs of the following structure: K +K1n+K2nq, where K and K1,K2 represent fixed and variable components, respectively, q is the case pack size, andorders are nonnegative integers (n) multiple of these q consumer units. The objectiveis to find an inventory policy which minimizes the long-run expected average costs ofthe system over an infinite horizon.

We formulate the problem as a Markov decision problem and use it to explore thestructure of the optimal policies. In particular, we numerically illustrate the impact ofhandling cost components on the optimal policy and the associated long-run average

3.2 Literature review 47

cost. In general, for a lost sales system, neither the best (s, S), nor the best (s,Q)policy is optimal. In this chapter, we propose a new heuristic policy, referred to asthe (s,Q|S, nq) policy, which combines the logic of the two policies, and accounts forquantized ordering1. In an extensive numerical study, we show that the new heuristichas close-to-optimal performance in many settings, with a cost that is within 0.01%of the optimal, on average.

We further benchmark its performance against reasonable alternative policies, the(s, S, nq) and (s,Q, nq) policy and quantify the relative improvements. We find thatthe best (s, S) policies are performing very well especially in settings where the penaltycost is high and the batch and unit-related handling costs are small. Alternatively, the(s,Q) policies have, on average, worst performance, although in case of lower servicelevels, they may eventually outperform the best (s, S) policy. The new heuristicperforms, in both situations, consistently very well. Finally, we investigate the addedvalue of including the handling costs into decision making. We find that, for itemswith low profit margins, ignoring handling related costs, may result in substantialcost penalty.

The remainder of this chapter is organized as follows. The next section gives abrief review of related literature. Section 4.3 formally introduces the problem andformulates the model using a Markov decision process. In Section 3.4 we introduce thenew heuristic. A numerical study is conducted in Section 3.5, in which we illustratethe complexity of the optimal policies (§5.3), we conduct sensitivity analysis withrespect to problem parameters (§3.5.2), and we test the comparative performance ofthe newly proposed heuristic (§3.5.3). In section 3.6 we investigate the added valueof handling costs. Finally, we present our conclusions.

3.2. Literature review

In this section, we give a brief review of the literature on periodic review lost salesinventory models with positive lead times and batch (quantized) ordering. Theclassical lost sales problem, originally formulated by Karlin and Scarf (1958), is knownto be far less analytically tractable than the corresponding backorder problem. If thelead time is positive, the complete structure of the optimal ordering policy is unknown(Hadley and Within, 1963). Therefore, there are few papers that contain analyticalresults for periodic review problems with lost sales and positive lead times. Karlin andScarf (1958) establish some basic properties of the optimal ordering policy for a modelwith a lead time of exactly one period. Later on, Morton (1969, 1971), extends theseresults to periodic review lost sales problems with fixed lead times, multiples of thereview period length. More recently, Zipkin (2008b) provides a new derivation of this

1Here and in the rest of the chapter, we study a periodic review system with fixed review periodR, taken to be the time unit. Hence, for notation convenience, we shall not use R in denoting thedifferent inventory policies, unlike in some existing literature.


result and extends it to more complex settings. Additionally, bounds on the optimalpolicy are also derived in these papers. Since the derivation of optimal inventorypolices (via dynamic programming for example) becomes prohibitive as lead timeincreases, various heuristics have been proposed. A recent paper by Zipkin (2008a)provides a numerical comparison of several inventory policies, such as a myopic policy(Morton, 1971), the base stock policy, the dual-balancing policy (Levi et al., 2008 ),the constant order policy (Reiman, 2004) and some variants. The base-stock policiesare found to perform poorly (except for short lead times), but it is in general unclearwhich approach is best. Note that in all models mentioned so far, no fixed orderingcost is taken into consideration.

Nahmias (1979) considers a more general periodic-review lost-sales problem with afixed ordering cost and stochastic lead times. He proposes using an (s, S) policy (ororder-up-to S policies for zero ordering cost) and derives the optimal parameters usingsimulation. Analysis of simple-structured policies are often preferred in a lost salessetting, due to intrinsic complexity of the original model, usually with the assumptionof at most one order outstanding at any time. Order-up-to (or base stock) policiesare studied mostly (see, e.g., Morse, 1959, Gaver, 1959 and Johansen, 2001), whichare known to be optimal for the periodic-review backorder model, when there areno setup costs. For this class of policies, structural properties such as convexity ofthe underlying cost functions are often investigated (see, e.g., Downs et al., 2001 andJanakiraman and Roundy, 2004). In the presence of a positive setup cost, reorder-point polices of (s, S) type (Wagner, 1962) or (s,Q) (Johansen and Hill, 2000) type areconsidered in a lost sales setting (see also Hill and Johansen (2006) for further review).In contrast to backorder inventory control systems, for which policies of (s, S) typeare known to be optimal under rather general conditions (Beyer and Sethi, 1999), theequivalent lost sales model is much more complex and (s, S) policies are not optimalfor these systems.

In almost all papers that deal with periodic review inventory problems, lead timesare assumed to be integral multiples of the review period length. By contrast, theproblem we consider assumes that the lead time is deterministic, and between zero andthe length of a review period, as this accommodates situations when the inventory isreviewed weekly, for example, while the replenishment lead times are only one, or twodays long. In this way, there can never be more than one order outstanding. However,the analysis remains complex due to the presence of lost sales during the lead time,as well as in the interval after the order receipt up until the end of the review period.Similar assumptions appear in Hadley and Within (1963, p.282), who illustrate for anorder-up-to policy the analytical difficulties inherited by lost sales models, and alsoKapalka et al. (1999). The latter study considers the class of (s, S) policies and usesMarkov chains to analyze and derive the best policy under the average cost criterionand service level constraints. They compute the optimal parameters using a searchprocedure based on an efficient updating scheme for the transition probability matrix,bounds on S and monotonicity assumptions on the cost and service constraint. Inboth studies, fixed costs are present.

3.2 Literature review 49

Janakiraman and Muckstadt (2004a) also consider a periodic review lost salesinventory model with fractional (i.e. less than the review period length) lead times,continuous demand, finite horizon and discounted costs, in the absence of a positiveordering cost. They investigate structural properties of the objective function as wellas the optimal policy, and find similar results to those initially reported by Karlin andScarf (1958) and Morton (1969). Using newly derived bounds on the optimal policy,they propose new heuristics and compare their performance against the optimal policy,as well as the order-up-to policy. They also prove the convexity of the cost functionwith respect to the order-up-to level. Chiang (2006, 2007a) recently proposed adynamic programming model for periodic-review systems in which a replenishmentcycle consists of a number of small periods (each of identical but arbitrary length)and holding and shortage costs are charged based on the ending inventory of smallperiods, rather than ending inventory of replenishment cycles. They analyze bothbackorder and lost sales inventory models (under the assumptions of lead times shorterthan the replenishment cycle length), and for the latter provide some properties andcomputational results.

Thus far, existing literature on lost sales inventory systems provides partial character-izations of the optimal ordering policy in the absence of fixed ordering costs. Thereexists limited analytical research for the case of a positive ordering cost and thereare still many open questions when the restriction of a fixed batch size for orderingis assumed, as noted earlier by Veinott (1965). The incorporation of a fixed batchsize is hardly taken into account in lost sales models. A notable exception is Hilland Johansen (2006). When excess demand is fully backordered, and there are nofixed ordering costs, then Veinott (1965) shows that the (s, nq)2 policy is optimal fora single-location, single-item inventory model under both finite, and infinite periodsettings. He also points out that the (s, nq) policies are not optimal in general if a fixedcost is taken into consideration and he proposes a two parameter policy instead. Zhengand Chen (1992) provide an efficient heuristic to compute the best s and q parameters(both s and q positive variables), and later on, Hill (2006) uses their heuristic in theanalysis and optimization of an (s, S, q) policy, where s and S are assumed to bemultiples of q. The structure of the optimal policy for lost sales systems with batchordering remains an open question (Veinott, 1965, Hill and Johansen, 2006).

The research presented in this chapter makes the following contributions over existingliterature. We study a periodic review, stochastic lost sales inventory system thatcombines attractive features for practice: fixed batch sizes and fixed plus variable(depending on the batch size) material handling costs. We investigate numericallythe impact of material handling costs on the structure of the optimal policies and thecorresponding long-run average cost. We propose and analyze a new heuristic policy,namely the (s,Q|S, nq) policy (with s, S and Q policy parameters), which captures

2Veinott’s original notation (k,Q), with Q a fixed positive constant. With the (k,Q) policy if theinitial inventory on hand and on order in a period is less than k, an order is placed for the smallestmultiple of Q that will bring the inventory on hand and on order to at least k; otherwise, no orderis placed.


partially the structure of optimal policies and works as follows: when q is one, atevery review moment, if the inventory position is not more than S − Q, then placea replenishment order of size Q; if the inventory position is above S − Q but notmore than s, then order up to S; otherwise, do not order; when q is greater than one,order quantities are adjusted to be integer multiples of the fixed batch size q. Figure3.1 illustrates the logic of the (s,Q|S, nq) policy by an example. In this chapter, wecomputationally investigate the performance of the heuristic against optimal policiesas well as reasonable alternative policies.

0

5

10

15

20

25

30

0 4 8 12 16 20 24 28 32 36 40

Order quantity

Inventory on hand

q = 1

q = 4

S-Q sS-Q

Q

Figure 3.1 The logic of the (s,Q|S, nq) policy; s, S and Q policy parameters

3.3. Mathematical model

In this section, we use Markov Decision Processes to model the dynamic inventorysystem under consideration. The main notation in this chapter is as follows.

R Review periodL Lead time (0 ≤ L ≤ R)t Period index, t = 0, 1, 2, . . .

Xt Inventory on hand at the beginning of period t = 0, 1, 2, . . .at Quantity ordered in period t = 0, 1, 2, . . .

DL Demand during lead time (i.e. the demand occurring between the start ofa period and the time the order at is received)

DR−L Demand after the receipt of the orders (and until the end of the period)DR Total demand during a review period, DR = DL +DR−L

q Fixed (exogenously determined) batch size, q = 1, 2, . . .K Fixed cost per orderK1 Fixed cost per batchK2 Variable unit costh Holding cost per unit of inventory (charged at the end of the period)p Penalty cost for each unit of sales lost during a period

(charged at the end of the period)

3.3 Mathematical model 51

We assume the period demand to be a known, nonnegative discrete random variableand the demand process is i.i.d stationary. The lead time is fixed but less than thereview period length as this accommodates situations when the stock is reviewed, forexample, every three days and the lead time is say one or two days. We assume thatthe length of the review cycle R is exogenous to the model (for example determinedby the need of coordinating replenishment of many different items) and we furtherassume it to be the time unit. The random variables DL and DR−L are stochasticallyindependent, and at is restricted to be a nonnegative integer, multiple of the fixedbatch size q, i.e. at ∈ 0, q, 2q, . . . .

The sequence of events in each review period is as follows (see Figure 3.2). At thebeginning of the period, the inventory on hand is observed and an order is placed,which will arrive L time units later (but within the same review cycle). Next,the stochastic demand is realized and satisfied with on-hand inventory (if possible);unsatisfied demand is lost. Then, the order placed at the beginning of the periodarrives and afterwards stochastic demand continues to occur, up until the beginningof the next ordering moment. All demand occurring in this time period that cannotbe directly satisfied is again assumed to be lost. Note that due to the assumptionL ≤ R, lost sales might occur between the time an order is placed and received, as wellas in the time following the order receipt up until the beginning of the next orderingmoment.

L R - L L R - L

Xt

Place order at

Xt+1

Place order at+1

DL DR-L DL DR-L

Order

receipt

Order

receipt

Figure 3.2 The sequence of events

Dynamics of the system

We formulate the problem as a Markov decision process, in which the decision epochis the beginning of each review period and the inventory on hand at the beginning ofa review moment characterizes the system state, with state space Ω = 0, 1, 2, . . ..At each review moment a decision is made regarding the ordering quantity, which islimited to the set A(i) = 0, q, 2q, . . ., for every i ∈ Ω. Due to assumption L ≤ R,the expected transition times from one decision epoch to the next are deterministicand equal R. The evolution of on hand inventory from one decision epoch to the nextis given by the following recursive relation:

Xt+1 = ((Xt −DL)+ + at −DR−L)

+, t = 0, 1, 2, . . . .


where (x)+ = max0, x for any x ∈ R. Next, we define the transition probabilitiesand expected costs from one decision epoch to the next.

Transition probabilities

The probability pij(ai) of a transition from state i at one decision epoch to state j atthe next epoch, given decision ai at the first decision epoch is defined as

pij(ai) = P(j = ((i−DL)

+ + ai −DR−L)+), i, j = 0, 1, . . . , ai = 0, q, 2q, . . . ,

and is given by pi0(0) = P(DR ≥ i), i = 0, 1, . . . ,

pij(0) = P(DR = i− j), i = 0, 1, . . . , j = 1, 2, . . . , i,

p00(0) = 1,

when there is no order, and when the order amounts to an integer ai = niq > 0 isgiven by

pi0(ai) =i−1∑k=0

P(DL = k)P(DR−L ≥ i+ ai − k) + P(DL ≥ i)P(DR−L ≥ ai),

i = 1, 2, . . . ,

pij(ai) =i−1∑k=0

P(DL = k)P(DR−L = i+ ai − j − k) + P(DL ≥ i)P(DR−L = ai − j),

i = 1, 2, . . . , j = 1, 2, . . . , ai,

pij(ai) =

i+ai−j∑k=0

P(DL = k)P(DR−L = i+ ai − j − k),

i = 1, 2, . . . , j = ai + 1, . . . , ai + i,

p00(a0) = P(DR−L ≥ a0),

p0j(a0) = P(DR−L = a0 − j), j = 1, 2, . . . , a0,

pij(ai) = 0, otherwise.

Transition costs

The total expected cost from one decision epoch to the next (i.e., the one-periodtransition cost), given that the we are in state i and we order an integer amountai = niq ≥ 0, is defined as

ci(ai) = cri (ai) + chi (ai) + cpi (ai), ai ∈ A(i), i ∈ Ω, (3.1)

where the expected replenishment cost cri and the expected holding (chi ) and penalty(cpi ) costs are given by

cri (0) = 0,

cri (ai) = K +K1ni +K2niq, ai = niq > 0, (3.2)

3.3 Mathematical model 53

and

chi (0) + cpi (0) = hE[(i−DR)

+]+ pE

[(i−DR)

−] ,chi (ai) + cpi (ai) = hE

[((i−DL)

+ + ai −DR−L)+]

+pE[(DL − i)+

]+ E

[(DR−L − ai − (i−DL)

+)+]

= h

i−1∑k=0

P(DL = k)E[(i− k + ai −DR−L)

+]

+P(DL ≥ i)E[(ai −DR−L)

+]

+p

E[(DL − i)+

]+

i−1∑k=0

P(DL = k)E[(DR−L − ai − i+ k)+

]+p P(DL ≥ i)E

[(DR−L − ai)

+],

for ai > 0, respectively, where (x)− = max0,−x = −min0, x for any x ∈ R.

From (3.1) and (3.2), we derive the total one-period transition cost as follows:

ci(0) = hE[(i−DR)

+]+ pE

[(i−DR)

−] ,ci(niq) = K +K1ni +K2niq

+hE[((i−DL)

+ + ai −DR−L)+]

+pE[(DL − i)+

]+ E

[(DR−L − ai − (i−DL)

+)+]

, ni > 0.

We aim to find an inventory policy U∗, which solves the following optimizationproblem:

g = minU

C(U) =1

R

∑i∈Ω

πici(ai)R=1=

∑i∈Ω

πici(ai),

where C(U) denotes the long-run expected average cost under policy U , and (πi)i∈Ω

represents the steady-state distribution of the inventory on hand (provided it exists).

Note that C(U) is generally a very complex function of U and in particular, the steady-state distribution may not be determined in closed form. For the exact conditionsthat guarantee the existence of an average-cost optimal policy see, e.g., Puterman(1994, Ch.8), or Cavazos-Catena and Senott (1992). If an optimal policy exists, thenthere exist relative values (vi)i∈Ω and the long-run expected average cost g, such thatthey satisfy the average-cost optimality equations:

vi = mina∈A(i)

ci(a)− g +∑j∈Ω

pij(a)vj, i ∈ Ω. (3.3)

Unique relative values (vi)i∈Ω are obtained if we provide an initial condition, such asv0 = 0. In order to solve the optimality equations and determine an optimal policy,


it is common to use an algorithm such as policy iteration, value iteration or linearprogramming (Puterman, 1994).

3.4. Old and new heuristics

For the lost-sales inventory control problem introduced in Section 4.3, there is noapparent analytic solution of simple form. Hence, we consider here three heuristics.Two heuristics, commonly known in the literature, the (s, S) and (s,Q) policy (Zipkin,2000), are compared with a newly proposed heuristic, referred to as the (s,Q|S, nq)policy. The new heuristic combines the logic of both (s,Q) and (s, S) policies, and isadjusted to take into account the batch constraint on the order size. The (s, S) and(s,Q) policies are also adjusted to accommodate the nq (n ≥ 0 an integer) constrainton the order size, and will be referred to as (s, S, nq) or (s,Q, nq), whenever q > 1. Wemention that these policies are applied in a periodic-review setting, with the reviewperiod as the time unit.

3.4.1 The (s, S, nq) and (s,Q, nq) policies

The (s, S, nq) policy’s advise is as follows: whenever the inventory level at a reviewperiod is less than or equal to s, order the largest integer multiple of q which resultsin an inventory position less than or equal to S.

For the corresponding (s,Q, nq) policy, an order is placed at a review momentwhenever the inventory level is less than or equal to s, and the order size Q isconstrained to be a nonnegative integer multiple of the batch size q.

3.4.2 The (s,Q|S, nq) policy

The newly proposed (s,Q|S, nq) policy has three parameters s, S and Q with 0 ≤maxs,Q ≤ S ≤ s+Q. Under this policy, the order quantity in each period dependson the beginning inventory on hand x and is given by

a(x) =

⌊Q/q⌋q if 0 ≤ x ≤ S − ⌊Q/q⌋q⌊(S − x)/q⌋q if S − ⌊Q/q⌋q < x ≤ s0 if s < x,

where ⌊x⌋ denotes the largest integer, smaller or equal to x (see Figure 3.1). Inparticular, when q = 1 the order quantity equals

a(x) =

minQ,S − x if 0 ≤ x ≤ s0 if s < x,

3.5 Numerical study 55

and we shall simply refer to this policy as the (s,Q|S) policy. That is, an (s,Q|S)policy works as follows. When the inventory on hand is smaller than or equal to S−Q,order exactly Q; when x is greater than S − Q but smaller than or equal to s, thenorder up to S; and when x is above s, do not order. Note that if s = S −Q, then the(s,Q|S) policy is the familiar (s, S) policy with a reorder level s and an order-up-tolevel S. Similarly, if S = Q, the (s,Q|S) policy acts like an (s,Q) policy. Thus thenew heuristic generalizes both (s, S) and (s,Q) policy. In Section 3.5.3 we conduct acomputational investigation of the comparative performance of these policies underdifferent problem parameters.

3.5. Numerical study

The objective of this section is three-fold. First, to explore the effect of materialhandling costs, K1 and K2, on the structure of the optimal policy and the associatedlong-run average cost. Second, to determine the effectiveness of the newly proposedheuristic policy, referred to as the (s,Q|S, nq) policy by testing its performance againstthe optimal policy determined via dynamic programming. Third, we benchmarkthe performance of the proposed policy against two reasonable alternative policies,the best (s, S, nq) and (s,Q, nq) policies. We further investigate the added value ofincluding the handling costs into decision making, which enable us to develop furthermanagerial insights.

We conduct several numerical studies involving the following parameters: the leadtime L, the fixed ordering cost K and the material handling costs K1 and K2, as wellas the fixed batch size q. Throughout all experiments, the holding cost h, the expecteddemand per period λ, and the penalty cost p are held constant. Furthermore, weassume that the random demands DL and DR−L are stochastically independent andboth follow a Poisson distribution. The main reason for this choice is that technicallythe mean period demand DR is than a readily available Poisson distributed randomvariable. The numerical examples are chosen as follows: h = 1, λ = 20, p = 50 remainunchanged, and

L ∈ 0.25, 0.33, 0.50,K ∈ 10, 50, 100,K1 ∈ 5, 10, 20, 40,K2 ∈ 0, 5, 10,q ∈ 1, 2, 4, 6, 12, 20.

Altogether, there were a total of 648 instances in our computational study. Thechoice of handling cost values is related to real-life data. In Chapter 2 we reportedapproximate values of K = 10, K1 = 20 and K2 = 1.3, based on empirical data(see e.g. equation (2.3)). Also the case pack sizes are chosen as commonly reportedin practice. The holding cost is set arbitrarily, while high penalty cost (relative toholding cost) in the retail setting are more likely to hold for nonperishable products


with long life cycles.

To determine the optimal policy under the long-run average cost criterion, we used thestandard value-iteration algorithm (Puterman, 1994, Bertsekas, 1995). The programwas written in Matlab and run on a standard computer.

3.5.1 On the structure of the optimal policy

We start with discussing the main insights regarding the structural form of the optimalpolicies. We define the reorder level as rl = maxi ∈ Ω|ai > 0, i.e., the highest valueof inventory on hand at which it is optimal to order a positive amount; the maximumstock level is defined to be the maximum optimal inventory position, after ordering,i.e., mS = maxi+ ai|0 ≤ i ≤ rl.

The complexity of optimal policies for lost sales inventory models contrasts withthat of classical backorder models, which are known to have solutions of the (s, S)type (Zipkin, 2000), when only fixed ordering cost are considered. For a lost salesformulation, the optimal control policy will, in general, be neither of the (s, S) nor ofthe (s,Q) type, but will depend in a much more complex way on the physical stock atthe time of placing an order. The optimal policies for three scenarios (correspondingto different lead time L values) are depicted in Figure 3.3(a). Figure 3.3(b) exemplifiesthe effect of non-unit batch sizes on the structure of the optimal policy (here q = 4vs. q = 1).

In view of these results, a few observations are worthwhile mentioning. There is noapparent solution of simple form. That is, the optimal order quantity as a function ofon-hand inventory exhibits no clear structure. It is observed however, in all scenarios,that there exists an optimal reorder level (rl) below which it is always optimal toplace an order and beyond which it is never optimal to order. Thus, rl plays the roleof a reorder level in a general inventory policy. This observation has been conjecturedbefore (see Hill and Johansen (2006), in a continuous review setting), but no proofsexist so far in the literature (in a continuous or periodic review setting). Moreover,as illustrated by the examples in Figure 3.3, whenever it is optimal to order, theadvice is either to order a fixed amount Q (and thus act like an (s,Q) policy), forlow levels of stock on hand, or order enough to reach a target stock level S (and thusact like an (s, S) policy), for high levels of inventory. Outside these regions however,the optimal policy structure remains unclear. This observation also suggests thata structured policy, which combines the logic of both policies might perform closeto optimal. This is precisely the policy that has been introduced in Section 3.4 asthe (s,Q|S, nq) policy, and whose performance is investigated numerically in Section3.5.3.

Generally, when q > 1, the optimal order quantity is stepwise decreasing as a functionof on-hand inventory (as illustrated by Figure 3.3(b)), and the step-down size equalsq. We notice that the structure of the optimal solution differs from an (s, S, nq) policy,


K = 10, K1 = 10, K2= 5

0

5

10

15

20

25

30

35

0 4 8 12 16 20 24 28 32 36

Inventory on hand

Ord

er q

uan

tity

L = 0.50

L = 0.33

L = 0.25

(a) Structure of the optimal policy for q = 1

K = 10, K1 = 10, K2 = 5, L = 0.50

0

5

10

15

20

25

30

35

0 4 8 12 16 20 24 28 32 36

Inventory on hand

Ord

er q

uan

tity

q = 1

q = 4

(b) The effect of q > 1 on the optimal policy structure

Figure 3.3 Optimal order quantity as a function of on hand inventory for Poissondemand with λ = 20, h = 1, p = 50


as for low levels of stock on hand, the advice is to order the same quantity, multipleof the batch size q.

3.5.2 Sensitivity analysis

In this section, we discuss the sensitivity of the optimal policy and the associated long-run average cost to system parameters. In the absence of simple-structured policies,it is difficult to evaluate the impact of changes in the problem parameters on theoptimal solution. Therefore, we use the reorder level (rl) and the maximum stocklevel (mS) as main operational indicators of change.

(a) Impact of K1, K2 and q

Table 3.1 provides a representative set of our results. The first part is meant toillustrate the impact of K1 and K2, while the other two parts are referred to illustratethe sensitivity of the results to K and L, respectively. In these cases, we use as areference a base scenario with parameters: L = 0.5, K = 10 and K1 = 10, and wevary only one parameter at a time, while keeping the others the same as in the basecase.

First, based on our results, we observed that for any given q value, the optimalpolicy for a problem with parameters (K,K1,K2, h, p, q) is the same as the optimalpolicy for a problem with parameters (K, 0, 0, h, p−K1/q −K2, q), and on the long-run, the average cost difference between the former and the later model is given by(K1/q+K2)·λ, a term independent of the policy. This observation is in line with someearlier results. Janakiraman and Muckstadt (2004b) show that linear purchase costs(K2) can be assumed to be zero without loss of generality, for general distributionand assembly systems with lost sales and/or backorders, when lead times are integers.Janakiraman and Muckstadt (2001) extend the result to a lost sales model with leadtimes which are a fraction of the review period length (see Lemma 3.1 in Appendix A).Their result states that the finite horizon, discounted cost problem with no setup costand a positive unit purchasing cost can be transformed into a finite horizon problemwith zero unit purchase cost.

In view of this result, we may show that that for any fixed value of q, the optimal policyfor a problem with parameters (K1,K2, h, p, q) and K = 0 is the same as the optimalpolicy for a problem with parameters (0, 0, h, p−K1/q−K2, q), and on the long-run,the average cost difference between the former and the later model is given by (K1/q+K2) · λ (see Proposition 3.1 in Appendix A). Therefore, when q is predetermined (i.e.determined outside of the system), the batch (K1) and unit (K2) handling costs canbe assumed to be zero without loss of generality, for determining the optimal policy,provided that the penalty cost transformation p(q) := p − K1/q − K2 is accountedfor. However, since p(q) is nonlinearly dependent on q, the batch cost component K1


Table 3.1 Sensitivity analysis with Poisson demands λ = 20, h = 1 and p = 50(Selective results)

K2 = 0 K2 = 5 K2 = 10

L K K1 q rl mS optCost rl mS optCost rl mS optCost

0.50 10 5 1 36 41 133.634 35 41 233.335 35 41 333.0262 36 42 83.798 35 42 183.546 35 41 283.2444 36 43 59.065 35 43 158.796 35 43 258.5156 36 45 51.095 36 44 150.821 35 44 250.507

12 35 49 44.164 35 49 143.882 35 48 243.57520 35 57 42.793 35 56 142.521 34 56 242.189

10 1 35 41 233.335 35 41 333.026 34 40 432.6252 36 42 133.674 35 42 233.413 35 41 333.0584 36 43 83.999 35 43 183.726 35 43 283.4456 36 45 67.726 35 44 167.436 35 44 267.122

12 35 49 52.474 35 49 152.192 34 48 251.87620 35 57 47.781 35 56 147.507 34 56 247.172

20 1 34 40 432.625 34 40 532.166 33 39 631.5632 35 42 233.413 35 41 333.058 34 41 432.6764 35 43 133.866 35 43 233.586 35 42 333.2616 36 44 100.970 35 44 200.665 35 44 300.350

12 35 49 69.093 35 49 168.811 34 48 268.47620 35 56 57.755 35 56 157.479 34 56 257.136

40 1 30 37 829.600 26 34 927.366 -1 -1 1000.0002 34 41 432.676 34 40 532.205 33 40 631.6304 35 43 233.586 35 42 333.261 34 42 432.8536 35 44 167.436 35 44 267.122 35 44 366.804

12 35 49 102.333 35 49 202.051 34 48 301.67520 35 56 77.700 34 56 177.419 34 56 277.066

0.50 50 10 1 32 61 259.307 31 61 358.899 31 61 458.4142 32 62 159.686 32 62 259.313 31 62 358.9274 33 64 109.924 32 63 209.580 32 63 309.1996 33 65 93.405 32 65 193.086 32 64 292.693

12 33 69 77.224 32 69 176.863 32 69 276.49220 33 78 70.944 32 78 170.626 32 77 270.245

100 10 1 31 82 277.3099 30 81 376.8454 29 81 476.32082 31 83 177.6941 31 82 277.3151 30 82 376.85164 31 84 127.9013 31 84 227.5329 30 83 327.12566 31 85 111.3373 31 85 210.9684 30 84 310.5849

12 31 90 94.91544 31 89 194.5429 31 89 294.167420 32 97 88.4552 31 96 188.084 31 96 287.7027

0.33 10 10 1 31 37 229.2587 31 37 328.9675 30 36 428.60432 32 38 129.5804 31 38 229.3388 31 37 329.01114 32 39 79.90654 32 39 179.6496 31 39 279.38276 32 41 63.6399 32 40 163.3677 31 40 263.0756

12 31 45 48.46234 31 45 148.1952 31 45 247.919320 31 53 43.75491 31 52 143.4969 30 52 243.1882

0.25 10 10 1 30 35 227.325 29 35 327.0298 29 34 426.70992 30 36 127.6322 30 36 227.3894 29 35 327.10064 30 37 77.97284 30 37 177.709 29 37 277.4456 30 39 61.68867 30 38 161.4476 29 38 261.1491

12 30 43 46.56806 29 43 146.2972 29 43 246.01920 29 51 41.84754 29 51 141.5945 29 50 241.3093

rl = mS = −1 is used to denote the policy of never ordering. In this case, the averagecost equals p · λ.


is particulary relevant for analyzing the impact of q on the system performance, andits role is more transparent in the original, rather than in the transformed model.Therefore, we considered the original model in our further investigations.

Next, we observe that the results are in agreement with expectations, in that both rland mS decrease in general with K1 and K2. However, the change in policy is smallfor small values of K1 (see Figure 3.4(a)). Moreover, unless K1 is high, the impact ofK2 on the optimal policy is also small (see Figure 3.4(b)).

Regarding the effect of q on the optimal policy parameters, we observe that thebatch size mostly affects the maximum stock level mS, which increases, in general,as q increases (other parameters being equal). Moreover, as the batch size increases,the optimal policies for the different values of K1 are similar. This observation isillustrated by an example in Table 3.2: as the batch size increases, the optimal policies(’optPolicy’ in the table) are identical in these examples.

Table 3.2 Effect of the batch cost (K1) on the optimal solution, L = 1, h = 1, K2 = 0,λ = 10, p = 10, K = 10

q K1 optCost rl† mS† optPolicy‡

1 0 17.316 20 34 [249, 233, 22, 122, 18 : 13]2 0 17.329 20 34 [2411, 222, 122, 182, 162, 142]4 0 17.373 20 35 [2412, 123, 165, 12]6 0 17.396 20 35 [2412, 123, 183, 123]

12 0 17.475 20 35 [2412, 129]20 0 18.159 20 40 [2021]30 0 20.058 18 48 [3019]40 0 23.403 17 57 [4018]

1 5 65.328 17 32 [227, 214, 202, 19, 17 : 14]2 5 41.535 19 34 [246, 227, 20, 182, 162, 142]4 5 29.532 19 34 [2411, 123, 20, 164, 12]6 5 25.52 20 35 [2411, 123, 184, 123]

12 5 21.537 20 35 [2412, 129]20 5 20.607 20 40 [2021]30 5 21.677 18 48 [3019]40 5 24.618 17 57 [4018]

† rl = optimal reorder level, mS = optimal maximum stock level; ‡ optPolicyU = [U1, U2, . . . , Url] means U = [U1, U2, . . . , Url, 0, 0, 0, . . .]; Un

i denotes ntimes Ui and Ui : Uj denotes unit decreasing values from Ui to Uj (Ui > Uj).

This observation echoes a point made earlier, namely that as q becomes much greaterthan K1, the ratio K1/q goes to zero, and thus the optimal policy for the problemwith parameters (K,K1,K2, h, p, q) (which is equivalent with the optimal policy forthe problem with parameters (K, 0, 0, h, p − K1/q − K2, q)), becomes insensitive toK1. When the batch cost K1 = 0, the optimal cost (’optCost’ in Table 3.2) increases


Effect of K1 on optimal policy for L = 0.50, K = 10, p= 50

0

5

10

15

20

25

30

35

0 4 8 12 16 20 24 28 32 36

Inventory on hand

Ord

er q

uan

tity

K1 = 5

K1= 10

K1 = 20

K1= 40

(a) Effect of K1 on the optimal policy

Effect of K2 on optimal policy for L = 0.50, K = 10, p= 50

0

5

10

15

20

25

30

35

0 4 8 12 16 20 24 28 32 36

Inventory on hand

Ord

er q

uan

tity

K2 = 0, K1 = 0

K2 = 5, K1 = 0

K2 = 10, K1 = 0

K2 = 0, K1 = 40

K2 = 5, K1 = 40

K2 = 10, K1 = 40

(b) Effect of K2 on the optimal policy

Figure 3.4 Sensitivity of the optimal policies to changes in K1 and K2 for Poissondemands with λ = 20, h = 1, p = 50, L = 0.50, K = 10 and q = 1


with q, reflecting having less flexibility in ordering, as q increases. However, whenK1 = 5, the optimal cost decreases with q, for smaller values, and then increases and iteventually converges to cost of the optimal policy without K1. This is also illustratedin Figure 3.5. It is possible in this way to evaluate the added value of including K1

into decision making. This idea is further pursued in Section 3.6. Regarding theoptimal average cost, we notice from Table 3.1 that the average cost increases withK1, K2 and decreases with q, all other things being equal.

0

10

20

30

40

50

60

70

0 4 8 12 16 20 24 28 32 36 40 44

q

min Avg Cost

K1 = 5

K1 = 0

Figure 3.5 The optimal average cost as a function of the batch size for K1 = 0 vs.K1 = 5, L = 1, h = 1, K2 = 0, λ = 10, p = 10, K = 10

(b) Impact of L and K

Numerical results suggest that, in general, rl increases with L, and mS also increaseswith L and q (and there is little interaction between L and q). Additionally, rldecreases with K, while mS increases with K and q. Furthermore, regarding thesensitivity of the average cost to changes in problem parameters, our numerical studiessuggest that the optimal long-run average cost is increasing in L andK(all other thingsbeing equal). Note that similar monotonic results (w.r.t. L) of the average cost (aswell as the infinite horizon discounted costs) are claimed by Zipkin (2008a) for the lostsales model with lead times which are integer multiples of the review period length.

3.5.3 Performance of the (s,Q|S, nq) heuristic

In this section, we report the computational results on the performance of the best(s,Q|S, nq), the best (s, S, nq) and the best (s,Q, nq) policies, compared against theoptimal policy determined via dynamic programming, as well as against each other.


First, we briefly describe our methodology. For any (s,Q|S, nq) policy, we use adynamic programming formulation similar to (3.3) in Section 4.3 in order to determinethe long-run average cost of the policy. In this case, for any state i ∈ Ω, instead ofminimizing over all possible order quantities as in the optimality equations (3.3), theorder quantity is determined by the logic of the (s,Q|S, nq) policy. We next solve theresulting system of equations to determine the long-run average cost C(s,Q, S). Todetermine the best (s,Q|S, nq) policy, we used an exhaustive search over a sufficientlylarge feasible region to ensure we find a global optimum. We applied a similarmethodology in determining the best (s, S, nq) and (s,Q, nq) policies.

Table 3.3 summarizes the results of our experiments. Results for the (s,Q|S, nq)heuristic are stated as percentage excess over the optimal cost, while results for thebest (s, S, nq) and (s,Q, nq) policy are stated as percentage increase in average costsfrom the cost of the best (s,Q|S, nq) policy. We report the minimum, maximumand average percentage errors over all scenarios (’Total’), and for each individualparameter. In computing the percentage errors, all average costs are first normalizedby subtracting from the total costs the constant (K1/q + K2) · λ. Excluding unithandling-related costs preserves absolute deviations, but makes the percent errorhigher.

The main observation made is that the best (s,Q|S, nq) and (s, S, nq) policy areboth performing close to optimal, in the range of parameters we considered, whilethe (s,Q, nq) heuristic is remarkably worse, on average. The average and maximumpercentage increase from optimality of the best (s,Q|S, nq) heuristic are very small(0.01% and 0.18%, respectively). The best (s, S, nq) policy, with an average error of0.05%, deteriorates slightly (at maximum 0.83% in excess cost from the average cost ofthe best (s,Q|S, nq) policy). The performance of the heuristics improves with K anddeteriorates with K1 and K2, on average. Note that the performance of the (s,Q, nq)heuristic substantially improves for large K and also improves with L (unlike theother heuristics).

In Table 3.4 we report similar information, for each value of the handling costparameters K1 and K2. These results also indicate that the (s,Q|S, nq) policy isconsistently better than the (s, S, nq) heuristics; however, on average, both policieshave close to optimal performance. Unlike the (s,Q, nq) heuristic, the performancedeteriorates, in general, with increasing values of K1 and K2. In fact, as we lateron illustrate by an example, when K1 and K2 are large, the best (s, S) policy mayactually perform worse than the (s,Q) policy, while the best (s,Q|S) policy is clearlysuperior. Notably, numerical results (not reported here) also show that the valuesof s and S parameters, are very close-to, if not identical, to the same values of the(s,Q|S, nq) policy, and are also close to the rl and mS optimal parameters.

A major reason for the optimality gap between these heuristics being so small is theflatness of the cost curves C(s,Q, S) around the optimal Q value. To illustrate thisidea, we plot the average cost of the (s,Q|S) policy for one of our examples as afunction of each individual parameter. In the example, the problem parameters are


Table 3.3 Comparative performance of the (s,Q|S, nq) heuristic for Poisson demandswith λ = 20, h = 1, p = 50 (Percent error)

Best (s,Q|S, nq) Best (s, S, nq) Best (s,Q, nq)vs. Optimum vs. Best (s,Q|S, nq) vs. Best (s,Q|S, nq)

Param. Value Mean Min. Max. Mean Min. Max. Mean Min. Max.

L 0.25 0.005 0.000 0.100 0.051 0.000 0.863 5.088 0.000 16.3210.33 0.008 0.000 0.109 0.056 0.000 0.843 4.667 0.000 14.9610.50 0.016 0.000 0.176 0.059 0.000 0.842 3.957 0.000 12.395

K 10 0.026 0.000 0.176 0.088 0.000 0.863 11.783 0.000 16.32150 0.003 0.000 0.015 0.057 0.000 0.382 1.372 0.000 1.952

100 0.001 0.000 0.006 0.021 0.000 0.132 0.558 0.000 0.843

K1 5 0.009 0.000 0.158 0.042 0.000 0.135 4.794 0.447 16.32110 0.009 0.000 0.164 0.045 0.000 0.140 4.759 0.435 16.24620 0.010 0.000 0.176 0.053 0.000 0.211 4.669 0.377 15.95840 0.011 0.000 0.130 0.083 0.000 0.863 4.062 0.000 15.460

K2 0 0.008 0.000 0.127 0.049 0.000 0.440 4.726 0.173 16.3215 0.011 0.000 0.176 0.063 0.000 0.863 4.582 0.014 16.239

10 0.011 0.000 0.135 0.053 0.000 0.222 4.405 0.000 15.923

q 1 0.010 0.000 0.090 0.118 0.000 0.863 4.513 0.000 16.3212 0.010 0.000 0.066 0.075 0.017 0.222 5.369 0.370 16.3104 0.006 0.000 0.051 0.063 0.014 0.161 5.200 0.479 15.4196 0.007 0.000 0.072 0.047 0.008 0.109 4.722 0.418 14.080

12 0.025 0.000 0.176 0.030 0.000 0.105 2.857 0.586 8.33020 0.000 0.000 0.000 0.000 0.000 0.000 4.765 0.537 13.301

Total 0.010 0.000 0.176 0.055 0.000 0.863 4.571 0.000 16.321

set as follows: L = 0.5, K = 10, K1 = 10, K2 = 5 and q = 1. Figure 3.6 illustrates theshape of the function around the optimal values (s∗ = 34, S∗ = 40, Q∗ = 27). In eachplot, we fixed two parameters to their optimal values and plot the cost as a functionof the remaining parameter. We observe from the plot that the cost C(s,Q, S) isrelatively flat around the optimal s and Q values, respectively and more sensitiveto changes in S. Of course, we can’t rule out the fact that the Poisson assumptionmight be causing this effect. Thus, if Q = S∗, the best (s∗, Q|S∗) policy becomes an(s, S) policy, but the costs are almost identical. Note, however, that in this case theparameters are optimized taking the handling costs into account.

Next, we report selected results with large material handling costs to clarify a pointmade earlier. Detailed results for K1 = 40 and K2 = 5 are included in Table 3.5.The entries in the table are best policy parameters as well as total average costs.Additionally, we report the percent increase from optimality for all heuristics. Note,as earlier, that the errors are reported relative to the normalized optimal cost. We


Table 3.4 Performance of the (s,Q|S, nq) heuristic for different values of K1 and K2

Poisson demands with λ = 20, h = 1, p = 50 (Percent error)

Best (s,Q|S, nq) Best (s, S, nq) Best (s,Q, nq)vs. Optimum vs. Best (s,Q|S, nq) vs. Best (s,Q|S, nq)

K1 K2 Mean Min Max. Mean Min. Max. Mean Min. Max.

5 0 0.007 0.000 0.086 0.035 0.000 0.103 4.865 0.491 16.3215 0.008 0.000 0.158 0.042 0.000 0.118 4.802 0.475 16.239

10 0.011 0.000 0.111 0.048 0.000 0.135 4.714 0.447 15.923Total 0.009 0.000 0.158 0.042 0.000 0.135 4.794 0.447 16.321

10 0 0.007 0.000 0.092 0.038 0.000 0.107 4.839 0.489 16.2465 0.011 0.000 0.164 0.044 0.000 0.133 4.762 0.472 15.958

10 0.010 0.000 0.119 0.051 0.000 0.140 4.676 0.435 15.639Total 0.009 0.000 0.164 0.045 0.000 0.140 4.759 0.435 16.246

20 0 0.007 0.000 0.104 0.044 0.000 0.136 4.763 0.477 15.9585 0.011 0.000 0.176 0.052 0.000 0.180 4.677 0.436 15.639

10 0.012 0.000 0.135 0.062 0.000 0.211 4.566 0.377 15.460Total 0.010 0.000 0.176 0.053 0.000 0.211 4.669 0.377 15.958

40 0 0.010 0.000 0.127 0.080 0.000 0.440 4.437 0.173 15.4605 0.013 0.000 0.130 0.114 0.000 0.863 4.087 0.014 14.963

10 0.010 0.000 0.115 0.054 0.000 0.222 3.662 0.000 14.639Total 0.011 0.000 0.130 0.083 0.000 0.863 4.062 0.000 15.460

Total 0 0.008 0.000 0.127 0.049 0.000 0.440 4.726 0.173 16.3215 0.011 0.000 0.176 0.063 0.000 0.863 4.582 0.014 16.239

10 0.011 0.000 0.135 0.053 0.000 0.222 4.405 0.000 15.923

Total 0.010 0.000 0.176 0.055 0.000 0.863 4.571 0.000 16.321

observe that, in most instances (except forK = 10), the best (s,Q) policy outperformsthe best (s, S) policy. This example demonstrates the following claim: the regionwhere our heuristic differs from the (s, S, nq) policy is the set of inventory levelswhich are small. Hence, in this case, where the transformed penalty cost is small(in the example the transformed penalty cost p − K1/q − K2 = 5) the chances ofhaving lower inventory levels will be higher, and hence the percentage increase fromoptimality will be more. In these instances, our heuristic, which combines (s, S, nq)and (s,Q, nq), is clearly superior.

Finally, we illustrate the performance of the (s,Q|S, nq) heuristic in ’policy’ space.In Figure 3.7, we plot the best (s,Q|S) and (s, S) policy against the optimal policyfor few scenarios. Clearly, the best (s,Q|S) policy has a simpler structure, whichpartially captures those of the optimal policies. By restricting the (s, S) policy to avalue below Q, the new heuristic better approximates the optimal policy structure.

In summary, although the results indicate that the (s,Q|S, nq) heuristic is, on average


Table

3.5

Perfo

rmance

ofthe(s,Q

|S,n

q)heu

ristic:selected

results

with

λ=

20,h=

1andq=

1(P

ercenterro

r)

Optim

alpolicy

Best

(s,Q|S)policy

Best

(s,S)policy

Best

(s,Q)policy

%Diff

%Diff

%Diff

from

from

from

Optim

al

optim

al

optim

al

optim

al

Lp

KK

1K

2rl

mS

cost

sS

QCost

cost

sS

Cost

cost

rQ

Cost

cost

0.5

50

10

40

526

34

927.366

26

34

23

927.384

0.067

27

34

927.615

0.910

24

20

928.632

4.625

0.5

50

50

40

520

54

951.097

20

53

42

951.103

0.010

20

53

951.298

0.393

19

41

951.180

0.162

0.5

50

100

40

516

75

967.541

16

74

63

967.541

0.001

16

71

967.616

0.112

16

62

967.551

0.014

0.33

50

10

40

523

31

923.715

23

31

23

923.734

0.083

23

30

923.935

0.927

20

20

925.190

6.220

0.33

50

50

40

517

50

947.679

17

50

42

947.683

0.008

17

49

947.850

0.359

16

42

947.808

0.271

0.33

50

100

40

513

71

964.269

13

70

63

964.270

0.001

13

68

964.330

0.094

13

62

964.285

0.025

0.25

50

10

40

521

29

921.992

21

29

23

921.997

0.022

22

29

922.187

0.886

19

20

923.615

7.380

0.25

50

50

40

515

49

946.109

15

48

43

946.112

0.005

15

47

946.253

0.312

15

42

946.259

0.324

0.25

50

100

40

512

69

962.767

12

69

63

962.767

0.000

12

67

962.811

0.071

12

63

962.789

0.036

Note:

Percen

terro

rsare

computed

based

onthenorm

alized

costs,

i.e.totalav

eragecosts−

(K1 /q+

K2 )·

λ

3.6 Penalty for not taking handling into account 67

10 15 20 25 30 35 40520

540

560

580

600

620

640

660

s

Avg

. Cos

t (s,

Q|S

)

Sensitivity of the average cost C(s ,S*,Q) to changes in sS* = 40, Q* = 27

L=0.5, K = 10, K1 = 20, K2 = 5, q=1

5 10 15 20 25 30 35 40500

550

600

650

700

750

800

850

900

Q

Avg

. Cos

t C(s

,Q|S

)

Sensitivity of the average cost C(s*,Q, S*) to changes in Qs* = 34, S* = 40

L=0.5, K = 10, K1 = 20, K2 = 5, q=1

30 35 40 45 50 55 60 65532

533

534

535

536

537

538

539

540

S

Avg

. Cos

t (s,

Q|S

)

Sensitivity of the average cost C(s*,Q*,S) to changes in Ss* = 34, Q* = 27

L=0.5, K = 10, K1 = 20, K2 = 5, q=1

Figure 3.6 Sensitivity of the average cost C(s,Q, S) around optimal valuesExample with Poisson demands λ = 20, h = 1, p = 50, L = 0.5, K = 10, K1 = 10,K2 = 5, q = 1

performing better than the (s, S, nq) policy (as statistically confirmed by a t-teston total average costs), from the average cost perspective, both heuristics appearto perform very well, especially when the penalty cost in high and the batch andunit handling costs are low. In the reverse situation, the best (s,Q) may actuallyoutperform the best (s, S), while the performance of the new heuristic remainsconsistently better. Unlike the other two heuristics, the (s,Q|S, nq) heuristic betterapproximates, consistently, the optimal policy in ’cost’ and ’policy’ space.

3.6. Penalty for not taking handling into account

As already mentioned in Section 4.1, retail handling costs, although acknowledged inpractice, are not taken explicitly into account when making inventory replenishmentdecisions. Hence, it is interesting to study the cost penalty of using a suboptimalpolicy (obtained by ignoring the handling costs) for a situation when the handling


L = 0.50, K1 = 10, K2 = 5, q = 1, mD = 20, p= 50

0

10

20

30

40

50

60

70

0 4 8 12 16 20 24 28 32 36

Inventory on hand

Ord

er q

uan

tity

Opt.policy, K = 10

Opt.policy, K = 50

Best (s,Q|S), K = 10

Best (s,Q|S), K = 50

Best (s,S), K = 10

Best (s,S), K = 50

Figure 3.7 Optimal policy, the best (s,Q|S) policy and the best (s, S) policy

costs are actually non-negligible, under the assumption that all other parameter valuesremain the same. In doing so, we take the following steps:

1. First, we compute the optimal policy and the optimal average cost under theassumption of no handling costs.

2. Then, we plug in this policy into the lost sales model with handling costs toevaluate the corresponding average cost.

3. Finally, we compare the result against the true optimal cost, associated to theoptimal policy determined while taking the handling costs into account.

We evaluate the added value of handling costs for decision making in two cases:

1. K is fixed, and we investigate only the added value of K1

2. We acknowledge both K and K1 as relevant handling components.

In both situations, we assume the unit variable cost K2 to be zero. Therefore, wemeasure the penalty for not taking handling into account in two situations, as follows:

Gap1 = 100×C(K,K1)(U(K,0))− C∗

(K,K1)

C∗(K,K1)

− (K1/q +K2) · λ(ignoring only K1),

Gap2 = 100×C(K,K1)(U(0,0))− C∗

(K,K1)

C∗(K,K1)

− (K1/q +K2) · λ(ignoring K and K1),

where U(K,0) denotes the optimal policy determined under the assumption K1 =0, U(0,0) denotes the optimal policy obtained under the assumption K = K1 = 0,

3.7 Conclusions 69

C(K,K1)(U) is the average cost of the original model with cost components K and K1

corresponding to policy U , and C∗(K,K1)

represents the true minimum average cost,which is rescaled to better reflect policy-related costs.

The different scenarios for this numerical study are generated by changing the value ofthe ordering cost (K), the batch cost (K1) and the unit penalty cost (p) as indicatedin Tables 3.6 and 3.7. The results reported Table 3.6 consider smaller values of thepenalty cost (p = 5, 10), while those reported in Table 3.7 correspond to higherpenalty cost values (p = 50). In choosing the parameter values, we considered thefact that as p−K1 approaches zero, the optimal ordering policy is never order.

These results indicate that the cost impact of ignoring the handling costs is higherwhen K and K1 are large (in absolute and percentage deviations). Intuitively, thefixed ordering cost, K, affects quite substantially the total costs, when ignored. Thepenalty for ignoring the unit batch size K1 increases with K1, but the percentagesare much smaller than those corresponding to K.

Finally, we observe that the penalty resulting from ignoring the variable handling costs(K1) is more significant for smaller p values. Low unit penalty costs corresponds toitems with lower profit margin, and it represents a large assortment of products ingrocery retailing. Thus, for these items, ignoring the handling costs in the decisionmaking, may result in substantial cost penalty. We explain this finding in view of thefact that when the penalty cost is quite high (e.g. p = 50 in Table 3.7) compared to theholding costs, it results in high service levels. With a very high service level, shortagesare rare events, and the lost sales model could be approximated by a backorder one.Thus the choice of K1 (unit cost) does not really matter in the decision making.

3.7. Conclusions

In this chapter, we studied a single-location, single-item periodic-review lost-sales inventory control problem with the following features: there are stochasticcustomer demands, lead time is less than the review period length, there is a fixed(predetermined) batch size (q) for ordering and orders are restricted to integermultiples of the batch size. Furthermore, we assume a replenishment cost structurethat includes a fixed cost, as well as linear components depending on the number ofbatches, and the number of units in a replenishment order. Our framework has beeninspired from the retail environment, but the analysis is appropriate for systems inwhich there is a fixed unit-size of stock transfer and there are economies of scale in thereplenishment component. Using Markov decision processes, we explored numericallythe structure of the optimal policies and investigated, in particular, the impact ofmaterial handling costs on the optimal policy and the long-run average cost.

Optimal policies have rather complicated structures, which make them difficult toimplement in practice, so heuristics are used. In this chapter we present a new


Table 3.6 Cost penalty for ignoring K and/or K1, L = 0.5, λ = 20, h = 1, K2 = 0,q = 1

p K K1 OptCost AvgCost1† AvgCost2‡ Gap1(%) Gap2(%)

10 0 0 19.601 - - - -

10 50 0 54.397 - 69.601 - 27.9525 151.097 152.199 168.180 2.156 33.4316 169.797 171.759 187.895 3.942 36.3467 187.626 191.320 207.611 7.756 41.9638∗ 200 210.880 227.327 27.201 68.317

5 0 0 173.674 - - - -

5 50 0 51.097 - 67.367 - 31.8421 69.797 70.063 86.672 0.535 33.8882 87.626 89.030 105.976 2.948 38.529

2.5 95.983 98.513 115.628 5.501 42.7212.6 97.586 100.409 117.559 6.192 43.8123∗ 100 107.996 125.280 19.989 63.201

OptCost = C∗(K,K1)

; †AvgCost1 = C(K,K1)(U(K,0));‡AvgCost2 =

C(K,K1)(U(0,0));∗Optimal policy is never order

heuristic, referred to as the (s,Q|S, nq) policy, which combines the logic of both(s, S) and (s,Q) policies. We demonstrate numerically that our heuristic has close-to-optimal performance in both policy and cost space and is consistently performingvery well in many settings. We further compare the performance of the heuristicagainst reasonable alternative policies, the (s, S, nq) and (s,Q, nq) policy. We findthat the best (s, S) policies are performing very well especially in settings wherethe unit penalty cost is high and the batch and unit-related handling costs are small.Alternatively, the (s,Q) policies, may eventually outperform the best (s, S) policies, inenvironments with low service levels. The new heuristic performs, in both situations,consistently very well.

We also quantified the impact of excluding the fixed and variable material handlingcosts from decision making. Our numerical studies show that, in general, ignoring thefixed ordering cost (K) may result in substantial cost penalty, while the effect of thefixed batch cost (K1) is better noticed when high, and the unit penalty cost is low.We provide additional insights with respect to the effect of the problem parameterson the system’s performance.

Our conclusions clearly show that it is worthwhile to explicitly take handling costs intoaccount when making inventory decisions. We have used parameter values that aretypical of grocery retail environments, where decision models typically abstain fromincluding these costs. While our heuristic has an excellent performance and clearlyimproves over traditionally used models, some more work is needed to heuristically

3.7 Conclusions 71

Table 3.7 Cost penalty for ignoring K and/or K1, L = 0.5, λ = 20, h = 1, p = 50,K2 = 0, q = 1

K K1 OptCost AvgCost1† AvgCost2‡ Gap1(%)(AbsDev1) Gap2(%)(AbsDev2)

0 0 23.897 - - - -

10 0 33.897 - 33.897 - 0.000 (0.000)10 233.335 233.544 233.481 0.629 (0.210) 0.439 (0.146)20 432.625 433.091 433.065 1.428 (0.466) 1.348 (0.440)40 829.600 832.231 832.232 8.889 (2.632) 8.892 (2.632)

50 0 59.964 - 73.897 - 23.238 (13.934)10 259.307 259.394 273.481 0.148 (0.008) 23.899 (14.174)20 458.414 458.826 473.065 0.706 (0.412) 25.081 (14.651)40 854.397 857.690 872.232 6.054 (3.293) 32.788 (17.836)

100 0 78.046 - 123.897 - 58.748 (45.851)10 277.311 277.473 344.603 0.211 (0.163) 87.043 (67.293)20 476.321 476.900 523.065 0.759 (0.579) 61.246 (46.744)40 871.705 875.754 922.232 5.647 (4.049) 70.467 (50.528)

OptCost = C∗(K,K1)

; †AvgCost1 = C(K,K1)(U(K,0));‡AvgCost2 = C(K,K1)(U(0,0))

determine the policy parameter values. In Chapter 5 we investigate the performanceof our heuristic for the standard lost-sales inventory control problem, with or withoutsetup cost, and with batch ordering.

Appendix A. On unit vs. batch costs

Lemma 3.1 (Janakiraman and Muckstadt, 2001, Lemma 1) For all valid sets of costparameters (c′;h′; p′) (i.e. (α(p′−h′) ≥ c′ )) there exists another set of cost parameters(0;h; p) with h = h′ + c′(1− α)/α and p = p′ − c such that

f (c′;h′;p′)n (xn, qn) = f (0;h;p)

n (xn, qn) + k,

where f(c;h;p)n (xn, qn) denotes the minimum expected sum of all discounted future

costs (with discount factor α and cost parameters c, h and p), if we start period nwith xn units of inventory on hand and we order qn units, and k is a term independentof the policy.

For the proof, we refer the reader to Janakiraman and Muckstadt (2001). In view ofthis result, we derive the following result for the infinite horizon, average cost modelwith batch ordering and no setup cost.


Proposition 3.1 Consider the inventory system introduced in Section 4.3 andassume there is no setup cost. For any given batch size q, and all sets of costparameters (K1,K2, h, p) such that p−h ≥ K1/q+K2, the parameter transformation

(K1,K2, h, p, q) 7→ (0, 0, h, p−K1/q −K2, q)

leads to the following cost transformation:

C∗(K1,K2,h,p,q)

= C∗(0,0,h,p−K1/q−K2,q)

+ (K1/q +K2) · λ,

where C∗(K1,K2,h,p,q)

denotes the minimum long-run average cost corresponding to

parameters (K1,K2, h, p, q) and λ denotes the average demand per review period.

Proof: For every fixed integer value of the batch size q ≥ 1, we can rewrite thereplenishment cost cr (defined in equation (3.2)) as follows

cr(nq) = δ(nq)K +K1n+K2nq = δ(nq)K + (K1/q +K2)nq

= δ(nq)K + c(q)nq, n = 0, 1, 2, . . . ,

where δ(a) = 0, if a = 0 and δ(a) = 1, otherwise and c(q) = K1/q+K2 is the per unitpurchasing cost (given q). Then, since K = 0, we apply Lemma 3.1 and a limitingargument and deduce that

C∗(K1,K2,h,p,q)

= C∗(0,0,h,p−c(q),q) + k. (3.4)

Next, assume that the problem parameters are such that the optimal policycorresponding to (0, 0, h, p − c(q), q) is to never order. In this case the minimumlong-run average cost equals C∗

(0,0,h,p−c(q),q) = (p − c(q)) · λ = (p − K1/q − K2) · λ,since all demand is lost. It follows that the optimal policy for the problem withparameters (K1,K2, h, p, q) is also the policy of never ordering and the correspondingminimum long-run average cost equals C∗

(K1,K2,h,p,q)= p · λ. Replacing the average

costs in (3.4) it follows that

p · λ = (p− c(q)) · λ+ k,

and thusk = c(q) · λ = (K1/q +K2) · λ.

Appendix B. Computational issues

A few additional comments regarding the computational study conducted in thischapter are worthwhile mentioning.

3.7 Conclusions 73

• Finding the optimal policy: The average cost optimal policies can be obtainedeither by solving linear programs, or by solving the average cost optimalityequations using policy or value iteration algorithms (Puterman, 1994). In thischapter, we solved numerically the average cost optimality equations (3.3) usingthe standard relative value iteration algorithm with epsilon = 10−12 (see, e.g.,Puterman 1994 or Bertsekas 1995).

• State space truncation: In our numerical computations, the state space Ω wastruncated to a size sufficiently large to ensure we find a global optimum. Thestate space size increases with the mean demand per period. The truncatedstate space is determined by testing larger and larger sizes until the results areinsensitive to the increments.

• Speed of execution: We observed that the computational time increases withΩ, which indicates the size of the system of equations to be solved in the policyevaluation step. The most time consuming part is the generation of transitionprobabilities at each step.

• Policy evaluation: For any given policy, we evaluate numerically the long runaverage cost by solving the average-cost optimality equations (3.3). The bestpolicy within a given policy class is determined by exhaustive search over itsdefining parameters.

75

Chapter 4

Retail inventory control withshelf space and backroomconsideration

Abstract: In an infinite-horizon, periodic-review, single-item inventory system withrandom demands and lost sales, we study the impact of shelf space constraints on thesystem’s performance. Unlike traditional approaches, we assume that inventory levelsmay exceed the allocated shelf capacity. This situation captures many retail settings inwhich the retailer stores surplus stock, which does not fit on the shelves, in the store’sbackroom. Consequently, each period, the stock is transferred from the backroom to thesales floor to serve end-customer demand, and there is an associated extra handlingcost. Two models are considered that include (i) a linear or (ii) a fixed cost componentfor exceeding the allocated shelf capacity, additionally to a fixed cost for placing anorder. In a numerical study, we discuss several qualitative properties of the optimalsolutions, conduct sensitivity analyses and quantify the impact of accounting for shelfspace constraints explicitly in inventory decisions.

4.1. Introduction

In the previous chapter, we considered a single-item inventory replenishment decisionfaced by a retailer under several realistic constraints, in particular non-negligible shelfstacking costs. We investigated (sub)optimal ordering decisions and illustrated theimpact of shelf stacking costs on the system’s performance. In this chapter, we extendthe retail setting in Chapter 3 to incorporate another real dimension of the retailer’sinventory decision, namely limited shelf space.

76 Chapter 4. Inventory control with shelf space consideration

In the traditional store-based retail environment, each product has an allocated shelfspace that accommodates consumer units set to serve end-customer demand. Manyproducts are typically competing for limited shelf space, and decisions on whichproducts to store and where, and how much space should be devoted to each productare usually taken at a tactical level. The ultimate goal is to stimulate demandand maximize sales under various constraints such as limited budget for purchaseof products or limited store space for displaying products. We refer to Agrawal andSmith (2009) for a recent review on assortment and shelf space allocation models.Therefore, for operational decisions such as inventory replenishment, the allocatedshelf space is typically an exogenous variable, dictated by the planogram1, and nota decision parameter (Broekmeulen et al., 2004). In this chapter, we make the sameassumption and consider an inventory problem of a single product given its allocatedshelf space.

Due to limited store space, the shelf space allocations at product level are ofteninsufficient to accommodate the demand. Therefore, store managers typically keepsurplus stock that does not fit on the shelves in the store backroom. This is particularytrue in the grocery sector. Wong and McFarlane (2003) discuss several reasons whyretailers keep backroom inventory (including buffering against imperfect deliveries, orinsufficient shelf space for some bulky or fast-moving products), while investigatingthe impact of new technologies such as RFID on shelf replenishment. As a result, thestock temporarily stored at the backroom needs to be transferred from the backroomto the sales floor, to satisfy customer demand. This situation leads to additional storehandling operations.

The objective of this chapter is to extend the periodic-review single-item lost-sales inventory control model studied in Chapter 3 to the case in which there arephysical storage constraints at the retailer, and the retailer may use the backroom totemporarily store surplus stock. We are interested in the impact of including thesefeatures on the performance of the inventory control models, where performanceis measured with respect to the optimal ordering decision and associated long-runaverage cost. Since the retailer’s operational costs are greatly influenced by thematerial handling operations, aside from the regular inventory-related and shelfstacking costs, an additional cost is charged to the system for exceeding the shelfcapacity, to justify the need for additional handling operations.

Our chapter is related to the literature on inventory control in single-item periodic-review stochastic systems with lost sales. Consistent with the earlier chapter, weassume here fractional lead times (i.e. lead times shorter in length than the reviewperiod) and batch ordering (see e.g. Janakiraman and Muckstadt, 2004a, Hill andJohansen, 2006). Available results on optimality and various heuristics for suchsystems are reviewed in the previous chapter. Our chapter is also related to the

1The planogram is a diagram of fixtures and products that illustrates where and how every stockkeeping unit should be displayed on the shelf in order to increase customer purchase (Levy andWeitz, 2001)

4.1 Introduction 77

literature on capacitated inventory systems. Traditional capacitated production-inventory models (e.g. Federgruen and Zipkin, 1986a and 1986b) typically imposethat the ordering quantity in each period may not exceed a given capacity, and showthat when fixed costs are present, the optimal decision is rather complex. Motivatedby supply contracts, the inventory literature contains models that allow adjustmentson the contracted order quantity. For example, Henig et al. (1997) explore theoptimal inventory policies when both upward and downward adjustments on the orderquantity are allowed, and the ordering costs are piecewise linear convex. Chao andZipkin (2008) extend their model to include fixed costs for periodically adjusting theorder quantity, and partially characterize the optimal policy. Also, Chiang (2007b)considers storage constraints in a standing order inventory system with both backorderand lost-sales, but the latter case is limited to the zero lead time situation.

A shelf space constraint is often assumed in multi-item inventory control models, andsolutions using Lagrangian multipliers are classically presented (see e.g. Hadley andWhitin, 1963). Downs et al. (2001) study base stock policies in a multi-period versionof this problem with lost sales. They used a linear-programming-based policy todetermine optimal order up to levels for multiple products in the presence of resourcesconstraints. Cachon (2001) optimizes the shelf space allocation, in a warehouse-retailer setting where the objective is to select a truck dispatching policy, shelf spaceallocation and an inventory policy that minimize the sum of retailer’s transportation,shelf space and inventory costs. In these models, handling costs are typically notincluded.

Our approach differs from many in the literature, in the sense that we do notimpose limitations on the order quantity, or maximum inventory levels, but ratheracknowledge the presence of storage capacities, allow the maximum stock levels toexceed the available capacity, but then we charge additional costs to the system (seee.g. Kotzab and Teller, 2005 for an empirical study on the additional handling effortneeded to replenish the shelves when two storage locations are available in the store).In doing so, we make a distinction between backroom and sales floor stock, andinvestigate the effect of using the backroom on the system’s performance. The maincontributions of this chapter are as follows.

• We develop single-item lost-sales inventory control models to explicitly take intoaccount the features of batch ordering, shelf space limitations and backroomoperations. We propose two models: (1) the first one assumes continuousreplenishment from the backroom and extra handling costs that are proportionalto the expected number of units at the backroom at the moment the orderarrives; (2) the second model assumes that an additional fixed cost is charged ifthe inventory position (on-hand plus on order) at each review moment exceedsthe storage capacity.

• For the first model, we provide insights into the structure of the optimal policies,via an extensive numerical study, conduct sensitivity analyses and quantify


the cost penalty for ignoring the additional handling costs in inventory relateddecisions.

• For the second model, we illustrate the additional complexity of the optimalpolicies, in relation with the first model, as well as with the model that assumesample shelf capacity. In a numerical study, we illustrate the impact of shelfspace (and other parameters) on the system’s performance, quantify the effectof additional handling operations, and provide several interesting managerialinsights.

The remainder of the chapter is organized as follows. Section 4.2 describes theretailer’s system that we consider in more detail. Then, in Section 4.3, we introducetwo inventory control models for the considered system, to account for limited shelfspace and backroom operations. In Section 4.4, we conduct a numerical study on themodel with linear extra handling costs, and in Section 4.5, we study the model withfixed extra handling costs. Finally, we draw some conclusions based on this study.

4.2. System under study

Consider a single-item periodic-review retail inventory system facing stochasticdemands. The replenishment lead time is fixed but less than the review periodlength, the ordering is quantized, i.e., the order quantities are restricted to non-negative integers, multiple of a fixed batch size q, and any unfilled demand duringa review period is assumed to be lost. These features are commonly encountered ingrocery retailing, as we already mentioned in earlier chapters.

We further consider the following situation. Upon order receipt at the store, thepre-packed deliveries are unwrapped and units are displayed onto the shelves to serveconsumer demand. This store handling operation was referred to as shelf stacking inearlier chapters, and will also be referred to as the first replenishment process in thischapter. We assume that shelves have limited storage capacity, and store managerskeep surplus stock that did not fit on the shelves in the store’s backroom, whichcreates the need for a second restocking of the shelves. The manual process of shelfrestocking with products which are located in the backroom will be referred to inthis chapter as the second replenishment process, or simply extra handling. Figure 4.1illustrates the generic material flow in a retail system with backroom operations.

We analyze the system from a total expected cost point of view considering thefollowing cost components:

• inventory-related costs (for ordering, holding and lost-sales penalty costs), and

• handling-related costs (for the first and second replenishment).

4.2 System under study 79

Check out

Shelves

Backroom

Incoming

stock

Second

replenishment

process

First

replenishment

process

Receipt

Figure 4.1 The flow of stock at the retailer using backroom operations

The objective is to find an inventory policy, which minimizes the total expected costsof the system in an infinite horizon.

The system analyzed in this chapter extends the one considered in Chapter 3. Inthis chapter, we impose extra handling operations be taken into account in inventoryreplenishment decisions. Consequently, the cost structure is adjusted to take explicitlyinto account the additional costs associated with backroom usage.

Second replenishment process and costs

In practice, when surplus stock exists (i.e. stock that does not fit on the regular shelvesduring shelves stocking with new deliveries is temporarily stored at the backroom),the store manager needs to decide not only on the replenishment of inventory, butalso on when and how often during the review period should the shelf replenishmentfrom the backroom be executed. In this chapter, we do not address the questionof what is the right moment and/or frequency to conduct the second replenishmentprocess. While we acknowledge here several possibilities for shelves restocking withbackroom stock (e.g. at the beginning of the review period only, only at the momentthe order arrives, at the end of the period only, at any predetermined moment duringthe review period), we focus in this chapter on two alternative situations:

1. continuous replenishment from the backroom, and

2. replenishment from the backroom only at the beginning of the review period.

Consequently, we assume two types of handling costs associated with the secondreplenishment process, also referred to as extra handling costs, or second replenishmentcosts interchangeably:

(i) a per unit cost of exceeding the shelf space capacity, and

(ii) a fixed cost per incidence of going above the shelf space capacity, respectively.


For each cost structure, we build an adapted single-item inventory control model.The details of these models are presented in Section 4.3. In Sections 4.4 and4.5, respectively, we study the models’ performance, and investigate the impact ofincluding the new retail features (i.e. limited shelf space and extra handling costs) onthe performance of the inventory systems.

4.3. Model formulations

In this section, we present our basic assumptions on the system under study describedin Section 4.2, and introduce two alternative models for capturing the retailer’sbackroom operations and the associated extra handling costs. In Section, 4.3.1 weassume continuous replenishment from the backroom, at the expense of additionalper unit handling costs, while in Section 4.3.2 we assume a one time opportunity forrestocking the shelves with backroom stock, at the expense of a fixed cost.

The models presented in this chapter are extensions of the inventory control modelstudied in Chapter 3, which will be referred to in this chapter as the basic lost-salesinventory control model. Therefore, the basic notation in this chapter is consistentwith the one in Chapter 3, and is summarized below.

R Review period lengthL Lead time length (0 ≤ L ≤ R)V Allocated shelf space capacity

DL Random demand during lead timeDR−L Random demand during period R− L

DR Total demand during a review period, DR = DL +DR−L

q Fixed (exogenously determined) batch size, q = 1, 2, . . .K Fixed cost per orderK1 Fixed cost per batchK2 Variable unit costKs Unit cost of handling excess inventory2

Ke Fixed cost for extra handlingh Holding cost per unit of inventory (charged at the end of the period)p Penalty cost for each unit of sales lost during a period

(charged at the end of the period)

The main assumptions formulated in Chapter 3 hold in this chapter as well: perioddemand is stochastic, stationary, independent and identically distributed over time;the random variables DL and DR−L are stochastically independent, the orderquantities are restricted to the set 0, q, 2q, . . . , and an order is delivered withinthe same review period it has been placed (i.e. 0 ≤ L ≤ R with L and R constantparameters).

2Excess inventory is defined as the amount of stock required by the inventory replenishmentoperations, which exceeds the allocated shelf space.

4.3 Model formulations 81

Additionally, we assume throughout the analysis that the allocated shelf space V ≥ 0is a parameter exogenous to the system, and thus is a fixed parameter. Unlike mostapproaches in the literature (see e.g., Cachon 2001), we further assume that inventorylevels may exceed the capacity V , in which case the system will incur an additionalcost, to justify the need for additional handling operations. Obviously, if V is largeenough, the backroom usage will be avoided at all times. In this case, the shelf space isa non-binding parameter in the inventory system, which simplifies to the one studiedin Chapter 3 (herein called the basic system).

The system incurs a number of costs, assumed time invariant. We assume per unitholding and lost-sales penalty costs, with positive rates h and p, respectively. Twotypes of replenishment costs are considered in this chapter:

1. for ordering and shelf stacking (first replenishment costs), and

2. for handling backroom stock (second replenishment costs).

The first replenishment costs are modeled consistently to earlier chapters, includingfixed (K) and variable components (with nonnegative rates K1 and K2). The noveltyin this chapter is the explicit incorporation of the second replenishment costs into themodeling and analysis of inventory decisions. We present in the following subsectionstwo model formulations: one assuming extra handling costs that are proportional tothe average excess inventory, and another one assuming a fixed cost for exceeding theshelf capacity.

4.3.1 Model with continuous backroom operations

We assume continuous restocking from the backroom, meaning that whenever anitem is demanded and is not directly available on the shelf, but it is available in thebackroom, it will be used to satisfy the demand. Only in the event that there is ademand for an item, which is not available directly on the shelf, or in the backroom,the demand is effectively lost. This situation corresponds to a maximum possibleservice, given the backroom. The trade-off, of course, is extra handling costs.

The sequence of events in each review period is as follows: (i) at the beginning of theperiod, inventory on hand Xt is observed; (ii) an order at is placed (in batches of qunits), which will arrive L time units later, but within the same review period, dueto our assumption L ≤ R; (iii) the demand DL during the leadtime is realized andsatisfied with on-hand inventory; unsatisfied demand in lost; (iv) the order placedat the beginning of the period arrives; (iv) the demand DR−L in the period R − Lcontinues to occur up until the end of the period; any demand that cannot be directlysatisfied from stock on hand in again assumed to be lost; (v) the (first and second)replenishment, holding and shortage costs are calculated.

We charge a positive cost (Ks) for each unit of stock that exceeds, on average, the


available shelf capacity V , to reflect additional handling activities such as going backand forth to the backroom. The objective is to minimize the long-run expected averagecost of the system.

Similar to the earlier chapter, we use Markov decision processes (MDPs) to formulatethe mathematical model. Every review moment is a decision epoch, the state of thesystem is the inventory on hand, Xt, at the beginning of the review period, withstate space Ω = 0, 1, 2, . . ., and the ordering decisions, at, are limited to the setA = 0, q, 2q, . . .. The stock levels of two consecutive review points are related bythe balance equation:

Xt+1 = ((Xt −DL)+ + at −DR−L)

+, t = 0, 1, 2, . . . . (4.1)

where (x)+ = max0, x for any x ∈ R.

The probability pxy(ax) of a transition from state x at one decision epoch tostate y at the next decision epoch, given decision ax is defined as pxy(ax) =P (y = ((x−DL)

+ + ax −DR−L)+) , x, y = 0, 1, . . . , ax = 0, q, 2q, . . . , and is detailed

in Appendix A.

The expected transition costs from one period to the next, given initial inventory xt =x, and decision at = a is denoted by Ct(x, a), and includes the following components:

(i) ordering and shelf stacking costs

Crt (x, a) = δ(a)K +K1⌊a/q⌋+K2a, (4.2)

where δ(a) = 0, if a = 0 and δ(a) = 1, otherwise, for all a ≥ 0, and ⌊y⌋ denotes thelargest integer, smaller or equal than y ≥ 0,

(ii) costs for holding inventory

Cht (x, a) = h EDL,DR−L

[((x−DL)

+ + a−DR−L)+], (4.3)

(iii) lost-sales penalty costs

Cpt (x, a) = p EDL

[(DL − x)+

]+ p EDL,DR−L

[(DR−L − a− (x−DL)

+)+], (4.4)

and additionally,

(iv) extra handling costs

Cextrat (x, a) = KsEDL [((x−DL)

+ + a− V )+]. (4.5)

Therefore, the total expected transition cost, defined as

Ct(x, a) = Crt (x, a) + Ch

t (x, a) + Cpt (x, a) + Cextra

t (x, a)

4.3 Model formulations 83

is given by

Ct(x, a) = δ(a)K +K1⌊a/q⌋+K2a

+ h EDL,DR−L

[((x−DL)

+ + a−DR−L)+]

+ p EDL

[(DL − x)+

]+ p EDL,DR−L

[(DR−L − a− (x−DL)

+)+]

+ KsEDL[((x−DL)

+ + a− V )+].

Note that equations (4.1) to (4.4) characterize the basic lost-sales inventory modelpresented in Chapter 3. We extend this model with an additional cost component,equation (4.5), in which a positive cost Ks is charged for the average number of unitsin excess of V , at the moment the order arrives. Observe that if the shelf space Vis large enough to accommodate the incoming stock, then there will be no additionalhandling costs and the model simplifies to the basic one. Otherwise, we interpretequation (4.5) as the expected cost for handling backroom stock.

In short, our model is novel compared to the basic one, in that we allow for amore complete representation of handling-related costs, in the presence of shelf spacelimitations. As with many lost-sales systems (Hadley and Whitin, 1963), the optimalsolution is likely to be quite complex in general. In Section 4.4, we present a numericalanalysis of the problem, derive qualitative solution properties and several managerialinsights into the trade-off between the different cost components.

4.3.2 Model with fixed extra handling costs

The general system under study was described in Section 4.2 and the main notationand assumptions have already been set earlier in this section. Alternatively to themodel presented in Section 4.3.1, we introduce a different costing scheme for thesecond replenishment process.

A store backroom is often a small space, poorly organized. Therefore shelvesrestocking with backroom stock often requires activities such as finding the itemsin the backroom, extra processing or administrative costs. These operations aretypically independent of the volume of back stock, and the associated handling costsare fixed. Hence, in this section we assume a fixed cost is incurred by the system inthe event of using the backroom. As before, our focus is to decide on the inventorycontrol policy which minimizes the long-run average expected cost of the system.

Define

Cextra(x, a) = Ke δ((x+ a− V )+

), (4.6)

where Ke ≥ 0. If x + a ≤ V , then the above expression is zero. Otherwise, weinterpret Cextra(x, a) as the additional handling cost from using the backroom for


storage of surplus stock, whenever the inventory position (on hand, x, plus on order,a) exceeds the shelf space V . Note that we assume the cost is charged based on theinventory position at the beginning of the period to reflect the worst case scenario inwhich lead time demand could be zero.

Furthermore, we assume the same ordering (plus shelf stacking) and holding costs perperiod as in the basic model:

(i) ordering and shelf stacking costs

Crt (x, a) = δ(a)K +K1⌊a/q⌋+K2a, (4.7)

(ii) costs for holding inventory

Cht (x, a) = h EDL,DR−L

[((x−DL)

+ + a−DR−L)+]. (4.8)

However, with respect to the lost-sales penalty cost in period t, we make an additionalassumption. Namely, we calculate the lost-sales cost depending on the amount ofstock visible on the shelf at the beginning of the period, i.e. minxt, V instead ofthe inventory on hand xt alone (compare equation (4.9) below with (4.4)). Hence,the expected lost-sales penalty cost in period t, given x units on hand and a units onorder, is given by

(iii) lost-sales penalty costs

Cpt (x, a) = p EDL

[(DL −minx, V )+

]+ p EDL,DR−L

[(DR−L − a− (minx, V −DL)

+)+]. (4.9)

Assuming that the extra handling costs are calculated using relation (4.6), i.e.

(iv) extra handling costs

Cextrat (x, a) = Ke δ

((x+ a− V )+

), (4.10)

the total expected cost in period t, defined as the sum of all cost components,

Ct(x, a) = Crt (x, a) + Ch

t (x, a) + Cpt (x, a) + Cextra

t (x, a).

is given by

Ct(x, a) = δ(a)K +K1⌊a/q⌋+K2a

+ hEDL,DR−L

[((x−DL)

+ + a−DR−L)+]

+ p EDL

[(DL −minx, V )+

]+ p E


+)+]

+ Keδ((x+ a− V )+

). (4.11)

The performance of this model, under the long-run average expected cost criterion, isinvestigated numerically in Section 4.5. Note that the basic model (studied in Chapter3) is a particular case, obtained when V is sufficiently large.

4.4 Numerical study: the model with continuous backroom operations85

4.4. Numerical study: the model with continuousbackroom operations

We focus on the model with continuous backroom operations described in Sec-tion 4.3.1. The purpose of this section is threefold: (1) to provide insights into thestructure of optimal policies (Section 4.4.1), (2) to conduct sensitivity analyses withrespect to the problem parameters (Section 4.4.2), and (3) to quantify the cost penaltyfor excluding the second replenishment process from the decision making (Section4.4.3). This problem is difficult to tackle analytically, thus we rely on numericalstudies to address it.

We have computed optimal policies and the associated long-run average cost fora variety of instances, all with a Poisson demand process with mean λ, constantleadtime L, linear holding, penalty and extra handling costs with rates h, p andKs, respectively, and fixed ordering cost K. Unless otherwise specified, all probleminstances in this section have R = 1, L = 0.50, λ = 20, h = 1, K1 = K2 = 0 andq = 1. The remaining problem parameters are set as follows:

p = 2, 5, 10, 20, 40, 100, 200K = 5, 10Ks = 5, 10, 20, 40, 100, 200V = 10, 15, 20, 25, 30, 40, 50

The combination of all parameters resulted in 588 problem instances in our numericalstudy. The values for the penalty cost p are chosen to reflect products with low andhight profit margins, while the values for the shelf space V vary around the meandemand. When V is much larger than the mean demand, it will become ineffectivein the model, which simplifies to the basic lost-sales inventory model. We assume theextra handling costs to be generally higher than the shelf stacking costs, and severalvalues were selected to facilitate sensitivity analyses.

To determine an optimal policy and the corresponding long-run average cost foran inventory system with a cost objective as described in Section 4.3.1, we used asimilar methodology to the one reported in Chapter 3. We applied the value iterationalgorithm to solve the average-cost optimality equations.


In this section, we investigate numerically the structure of optimal policies for theinventory model presented in Section 4.3.1 The extra handling cost structure, givenby relation (4.5), supports the observation made earlier that if V is sufficiently large,it becomes non-binding in the extended model. We examined the structure of optimalpolicies for the basic model in Chapter 3.


For a given problem instance, we denote by rl∞ and mS∞ the reorder and maximumstock levels, respectively, associated with an optimal policy of the basic model (asdiscussed in Chapter 3, these values exist). An optimal policy defines a rule thatindicates, depending on the initial stock on hand x, the amount a(x) to be orderedat each review moment. Therefore, similarly to Chapter 3, we define rl∞ = maxx ∈Ω : a(x) > 0 (i.e. the highest value of inventory on hand at which it is optimal toorder a positive amount) and mS∞ = maxx+ a(x) : 0 ≤ x ≤ rl∞.

Given the shelf space V , the optimal policies for the extended inventory model aredepicted in Figure 4.2 for few scenarios withK = 10, p = 10 andKs = 20. Similarly tothe solutions of the basic model (see Chapter 3), these results reveal that there seemsto exist a non-negative reorder level, denoted by rlV , and for inventory levels higherthan rlV , it is not optimal to place an order, i.e. rlV = maxx ∈ Ω | aV (x) > 0.Furthermore, the optimal order quantity aV (x), as a function of on hand inventory,has no simple closed-form expression, but for a given V , it may be approximated byan (s,Q|S) policy (see again Chapter 3).

Figure 4.2 also illustrates the effect of V on optimal solutions. As V increases, thereorder level rlV and order quantities aV (x) increase, resulting in higher maximumstock levels mSV (defined as mSV = maxx+ a(x) : 0 ≤ x ≤ rlV ), in general. Thissuggests higher average stock on hand, but on average, less sales lost and less stock inexcess of V . Further sensitivity analyses are presented in Section 4.4.2. As expected,the scenarios with V ≥ mS∞ yield identical optimal policies and average costs.However, when V < mS∞, the storage capacity may be insufficient to accommodatethe incoming stock, thus resulting in a need for a second replenishment process, withadditional handling costs.

4.4.2 Sensitivity analyses: the effect of V , Ks and p

In this section, we discuss the sensitivity of the optimal solution and the associatedlong-run average cost to the system parameters. In particular, we investigate theeffect of V , Ks and p on the optimal solution and minimum average cost. Sincethe optimal policy is not simple-structured, we use the reorder level (rlV ) and themaximum stock level (mSV ) as main operational indicators of change.

Table 4.1 summarizes the results of our numerical study for selected scenarios.Additional numerical results are included in Appendix B. For each scenario, we reportthe reorder level (rlV ), the maximum stock level (mSV ) associated to the optimalpolicy, as well as the corresponding minimum long-run average cost (denoted by C∗

V ).

Our conclusions from these numerical results are as follows.

• The values of rlV and mSV are monotone increasing (i) or decreasing (d) in p,Ks and V as follows:


15

20

25

30

Ord

er

qu

an

tity

V = 10

V = 15

V = 20

V = 25

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Ord

er

qu

an

tity

Inventory on hand

V = 10

V = 15

V = 20

V = 25

V = 30

V = 40

Figure 4.2 Optimal order quantity as a function of on hand inventory: λ = 20,L = 0.50, h = 1, p = 10, K = 10, K1 = K2 = 0, Ks = 20, q = 1.Note. (rlV =10,mSV =10) = (16, 19), (rlV =15,mSV =15) = (20, 24), (rlV =20,mSV =20) =

(24, 27), (rlV =25,mSV =25) = (27, 31), (rlV =30,mSV =30) = (29, 35), (rlV =40,mSV =40) =

(rl∞,mS∞) = (30, 37)


Table 4.1 Sensitivity analysis with Poisson demands with mean λ = 20, h = 1, L = 0.50and q = 1

Ks = 5 Ks = 10 Ks = 20 Ks = 40 Ks = 100 Ks = 200

p K V rlV mSV C∗V rlV mSV C∗

V rlV mSV C∗V rlV mSV C∗


V

5 5 10 18 22 56.298 16 19 56.340 15 18 56.422 14 16 56.562 13 15 56.855 12 14 57.15215 22 25 36.999 20 23 37.985 19 22 39.095 18 21 40.184 17 20 41.532 17 19 42.41720 24 28 27.472 24 27 28.619 23 26 29.706 22 25 30.727 22 24 31.954 21 24 32.81925 27 32 23.572 27 31 23.984 26 30 24.436 26 30 24.973 25 29 25.587 25 28 26.05230 28 34 22.412 28 34 22.451 28 33 22.496 28 33 22.546 28 33 22.677 28 32 22.77640 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.36750 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367

5 10 10 17 22 61.298 15 19 61.340 14 18 61.422 13 16 61.562 12 15 61.855 11 14 62.15215 20 25 41.999 19 23 42.985 18 22 44.095 17 21 45.184 16 20 46.532 16 19 47.41720 23 28 32.472 22 27 33.619 21 26 34.706 21 25 35.727 20 24 36.954 20 24 37.81925 25 32 28.572 25 31 28.984 24 30 29.436 24 30 29.973 24 29 30.587 23 28 31.05230 26 34 27.411 26 34 27.450 26 33 27.495 26 33 27.545 26 33 27.677 26 32 27.77640 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.36650 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366

10 5 10 24 28 83.147 20 23 106.340 17 19 106.424 16 18 106.588 14 16 106.996 13 15 107.47215 25 29 58.190 23 26 63.664 21 24 65.852 20 22 68.279 19 21 71.540 18 20 73.78120 27 30 37.317 26 29 40.514 25 27 43.511 24 26 46.223 23 25 49.403 22 25 51.70425 30 33 28.492 29 32 29.827 28 31 31.259 28 31 32.784 27 30 34.673 27 29 35.87630 31 36 25.102 31 35 25.326 31 35 25.670 31 34 26.021 30 34 26.659 30 33 27.03640 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.60150 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601

10 10 10 23 28 88.147 19 23 111.340 16 19 111.424 15 18 111.588 13 16 111.996 13 15 112.47215 24 29 63.190 22 26 68.664 20 24 70.852 19 22 73.279 18 21 76.540 17 20 78.78120 26 30 42.317 25 29 45.514 24 27 48.511 23 26 51.223 22 25 54.403 22 25 56.70425 28 33 33.492 28 32 34.827 27 31 36.259 27 31 37.784 26 30 39.673 26 29 40.87630 30 36 30.102 29 35 30.326 29 35 30.670 29 34 31.021 29 34 31.659 29 33 32.03640 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.60050 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600

20 5 10 29 32 100.457 26 29 152.658 21 23 206.424 18 20 206.591 16 17 207.073 15 16 207.74215 29 32 75.466 27 29 102.725 24 26 116.955 22 24 121.542 20 22 128.432 19 21 133.53320 30 33 51.165 28 31 59.033 27 29 66.319 26 28 73.156 25 26 80.857 24 25 86.11525 32 35 35.322 31 33 38.574 30 32 42.019 29 31 45.597 28 30 50.091 28 30 53.44030 33 37 28.331 33 36 29.188 33 36 30.159 32 35 31.200 32 34 32.903 32 34 34.10640 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.56350 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563

20 10 10 28 32 105.457 25 29 157.658 20 23 211.424 17 20 211.591 15 17 212.073 14 16 212.74215 28 32 80.466 26 29 107.725 23 26 121.955 21 24 126.542 20 22 133.432 19 21 138.53320 29 33 56.165 28 31 64.033 26 29 71.319 25 28 78.156 24 26 85.857 23 25 91.11525 31 35 40.322 30 33 43.574 29 32 47.019 29 31 50.597 28 30 55.091 27 30 58.44030 32 37 33.331 32 36 34.188 32 36 35.159 31 35 36.200 31 34 37.903 31 34 39.10640 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.56350 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563


p Ks V

rlV i d imSV i d i

Monotonicity with respect to p aims to reduce shortages. Monotonicity withrespect to Ks and V aims to reduce the expected stock above V . As V increases,so do rlV and mSV (see Figure 4.3 for an example). However, the higher thestorage capacity V , the lower the effect of Ks on the optimal solution.

• Furthermore, the optimal policy is more sensitive to changes in the values of Vfor higher values ofKs, rather than for lower values. Intuitively, asKs convergesto zero, the optimal solution converges to the one of the basic model, and thusbecomes insensitive to V .

• Numerical results show that the long-run average cost is decreasing in V , andincreasing in Ks and p, all other parameters being equal. Figure 4.4 illustratesthis behavior (see Appendix B for numerical details).

p Ks V

C∗V i i d

Monotonicity of the minimum average cost with respect to V confirms an earlierobservation that as V increases, it becomes non-binding in the extended model,and the minimum cost converges to the optimal cost of the basic model (i.e.without shelf space consideration). Thus, the cost of the basic model representsa lower bound on the optimal cost of the extended model. All other parametersbeing fixed, the minimum cost appears convex in V , as Figure 4.5 illustrates forfew scenarios.

• The cost increase with Ks is also rather intuitive in that if the unit cost for extrahandling is higher, the minimum long-run average cost is higher. However, ascan be seen from Figure 4.5, the higher V , the smaller the effect of an increasein Ks on the optimal cost.

• Figure 4.4 also indicates that the relative impact of Ks on the minimum costdoes not only decrease with V , but also increases with the unit penalty cost p.The interaction effect of p and Ks on the minimum average cost at V = 20 isillustrated in Figure 4.6. The minimum average cost increases with both p andKs, but for smaller values of p, the relative effect of Ks on cost is smaller.

• Finally, we observe that the size of the positive effect of p on the minimumaverage cost decreases with V (see Figure 4.4). Note that even when V ishigh enough to become non-binding in the extended model, p still has a positiveimpact on the minimum cost (unlikeKs, which becomes irrelevant in the model).


20

25

30

35

40

Re

ord

er

lev

el

(rl V

)

Ks = 5

Ks = 10

Ks = 20

Ks = 40

5

10

15

20

25

30

35

40

5 10 15 20 25 30 35 40

Re

ord

er

lev

el

(rl V

)

Shelf space (V)

Ks = 5

Ks = 10

Ks = 20

Ks = 40

Ks =100

Ks = 200

(a) Sensitivity of rlV to changes in V and Ks

20

25

30

35

40

Ma

x.

sto

ck l

ev

el

(mS

V)

Ks = 5

Ks = 10

Ks = 20

Ks = 40

5

10

15

20

25

30

35

40

5 10 15 20 25 30 35 40

Ma

x.

sto

ck l

ev

el

(mS

V)

Shelf space (V)

Ks = 5

Ks = 10

Ks = 20

Ks = 40

Ks =100

Ks = 200

(b) Sensitivity of mSV to changes in V and Ks

Figure 4.3 Sensitivity of rlV and mSV to V and Ks. Fixed parameters λ = 20, h = 1,L = 0.50, p = K = 10, q = 1.


150

200

250

300

350

400

450

Min

. A

vg

. C

ost

Ks = 5

Ks = 10

Ks = 20

Ks = 40

0

50

100

150

200

250

300

350

400

450

10 15 20 25 30 40 50 10 15 20 25 30 40 50 10 15 20 25 30 40 50

p=2 p=20 p=40

Min

. A

vg

. C

ost

Ks = 5

Ks = 10

Ks = 20

Ks = 40

Ks = 200

V

Figure 4.4 Effect of V , Ks and p on the minimum long-run average expected cost:λ = 20, L = 0.50, K = 10, q = 1

100

150

200

250

Min

. A

vg

. C

ost

Ks = 5

Ks = 10

Ks = 20

Ks = 40

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5 3

Min

. A

vg

. C

ost

Shelf space vs. mean demand (V/λλλλ)

Ks = 5

Ks = 10

Ks = 20

Ks = 40

Ks = 100

Ks = 200

Figure 4.5 Effect of V , Ks on the minimum long-run average expected cost: λ = 20,L = 0.50, K = 10, q = 1, p = 20


150

200

250

300

350

400

450

Min

. A

vg

. C

ost

Ks = 5

Ks = 10

Ks = 20

Ks = 40

0

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100 120 140 160 180 200

Min

. A

vg

. C

ost

Penalty cost (p)

Ks = 5

Ks = 10

Ks = 20

Ks = 40

Ks = 100

Figure 4.6 Effect of p and Ks on the minimum long-run average expected cost: λ = 20,L = 0.5, K = 10, q = 1, V = 20

4.4.3 Managerial insights

In this section, we build several managerial insights on the impact of including theshelf space capacity (V ) and all relevant costs in inventory related decisions. We focuson: (i) providing insights into the effect of V on the different cost components and(ii) quantifying the cost penalties at the retailer from excluding V and the additionalhandling costs from the optimization of inventory decisions.

Cost decomposition

First, we illustrate the effect of V on the different cost components by an example. Thefollowing parameter values are chosen in our numerical example: λ = 20, L = 0.50,q = 1, K = 10, K1 = K2 = 0, h = 1, p = 10 are fixed parameters, and wevary the shelf space V ∈ 10, 15, 20, 25, 30, 40, 50 and the unit extra handling costKs ∈ 5, 20. The retailer’s long-run average expected cost (herein simply referred toas ’Cost’) is calculated as the sum of expected ordering, holding, lost-sales penalty,and extra handling cost as follows:

Cost = K ·OF + h · EOH + p · ELS +Ks · EExcess,

where OF represents the order frequency, EOH denotes the expected stock on hand,ELS denotes the expected demand lost and EExcess denotes the expected excess(i.e. above V ) inventory on the long run.

As mentioned earlier in the chapter, the minimum long-run average expected cost


(referred here as ’Optimal cost’) is computed numerically by value iteration. Thecontribution of each cost component to the optimal cost is determined as follows.When K > 0 and h = p = Ks = 0 we obtain the average order cost, when h > 0and K = p = Ks = 0 we obtain the average inventory-holding cost, when p > 0 andK = h = Ks = 0 we obtain the average lost-sales penalty cost, and similarly, whenKs > 0 and K = h = p = 0 we obtain the average extra handling costs.

For each scenario, we determined numerically the optimal inventory control policy, theassociated long-run average cost and the individual cost components. We summarizeour findings in Table 4.2. These results indicate that as the shelf space V increases(other parameters being fixed), the retailer tends to hold on average more stock(holding costs increase), but will face less sales lost, in general (lost-sales penaltycosts decrease) and less surplus stock (extra handling costs for exceeding V decrease).Figure 4.7 graphs each cost component as a function of the shelf space V .

Table 4.2 Impact of V on individual cost components for Poisson demand with λ = 20and L = 0.50, h = 1, p = 10, K = 10, q = 1, Ks = 5

Optimal Cost

Lost-sales ExtraV Ks Ordering Holding penalty handling Total

10 5 10.000 8.360 28.073 41.714 88.14715 10.000 8.334 28.248 16.607 63.19020 10.000 10.471 17.333 4.513 42.31725 10.000 13.208 8.663 1.829 33.49230 10.000 15.799 3.897 0.406 30.10240 9.998 16.757 2.845 0.000 29.60050 9.998 16.757 2.845 0.000 29.600

10 20 10.000 1.256 100.001 0.167 111.42415 10.000 5.144 52.416 3.292 70.85220 10.000 8.758 26.220 3.533 48.51125 10.000 11.824 12.666 1.769 36.25930 9.999 14.957 5.086 0.627 30.67040 9.998 16.757 2.845 0.000 29.60050 9.998 16.757 2.845 0.000 29.600

We also notice in Table 4.2 that when the unit extra handling cost Ks increases from5 to 20, for given V , the expected holding costs decrease, the expected lost-salespenalty costs increase, the expected extra handling costs decrease, while the totalaverage expected costs increase, in general.

Quantifying the effect of second replenishment

Next, we aim to obtain additional insights into the cost penalties the retailer may faceby ignoring the second replenishment costs in inventory replenishment decisions. Asmentioned in earlier chapters, handling-related costs (though generally much larger


20

25

30

35

40

45

Co

st

Cost decomposition

Ordering Holding Lost-sales penalty Extra handling

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5

Co

st


Cost decomposition

Ordering Holding Lost-sales penalty Extra handling

Figure 4.7 Individual cost components as a function of V for Poisson demand withλ = 20 and L = 0.5, h = 1, p = 10, K = 10, q = 1, Ks = 5

than inventory-related costs at the store level) are usually not accounted for explicitlyin inventory control models.

Thus, we quantify the cost penalties at the retailer by comparing the total averagecost (which includes ordering, holding, lost-sales penalty, and extra handling costs) intwo situations: (1) all cost components are included in the optimization of inventorydecisions vs. (2) extra handling costs are not part of the optimization (but includedin the total cost pie).

In order to do so, we compare two inventory control policies:

• the optimal policy of the extended model (which includes the shelf space (V )and extra handling costs (Ks); see Section 4.3.1), denoted by U∗

V , and

• the optimal policy of the basic model (without the shelf space and extra handlingcosts), denoted by U∗

∞.

The long-run average cost corresponding to each policy is denoted by C∗V = C(U∗

V , V )and CV = C(U∗

∞, V ), respectively, where C(U, V ) denotes the total average cost ofthe extended model under any inventory policy U . Note that since U∗

∞ is generallysuboptimal for the extended model, it holds that CV ≥ C∗

V .

We quantify the impact of the second replenishment process as the percentage


difference (denoted by %∆) between CV and C∗V as follows:

%∆ = 100× CV − C∗V

C∗V

, (4.12)

and we interpret %∆ as the percentage of cost penalty the retailer may face byignoring the extra handling costs in the optimization of inventory decisions. We aimto obtain insight into the magnitude of expected cost penalties.

We have computed C∗V , CV and %∆ in a numerical experiment. To facilitate

numerical comparison, we used the same parameter values as in the previous numericalexample. Table 4.3 summarizes our results. In the table, we report not only on thetotal costs, but also on the individual cost components. For convenience, we repeathere the results already reported in Table 4.2.

The results in Table 4.3 show that the cost penalty at the retailer may be substantialwhen the additional costs resulting from handling stock in excess of V are notexplicitly included in the optimization of inventory decisions. The percentage of costpenalty (%∆) increases with Ks, and interestingly, it appears that if the allocatedshelf capacity is close to the mean demand, we have the largest percentage of costdeviation (see also Figure 4.8(a)).

Figure 4.8(b) illustrates the percentage deviation calculated for each individualcost component (according to formula (4.12)), and denoted here by %∆Ordering,%∆Holding, %∆Lost-sales penalty and %∆Extra handling, respectively. Componentwise,we observe that a substantial cost increase in ’Extra handling’ results for the retailerfrom using the suboptimal policy U∗

∞ (instead of U∗V ) in the extended model, which

appears to peak when V is around the mean demand. The trade-off, however, wouldbe less sales lost, on average (i.e. negative %∆Lost-sales penalty). Intuitively, atoptimality, as the retailer tries to save on additional handling costs, she may orderless, which in turn increases the average lost-sales.

Finally, we note that when decisions are separated (i.e. the extra handling costs arenot included in inventory optimization, yet are part of the total cost) substantialexcess stock may result at the retailer, which decreases with V (as shown byFigure 4.9). Figure 4.9 also depicts the long-run average optimal (C∗

V ) and expectedcost (CV ), as a function of V . As expected, CV dominates C∗

V by optimization, andthe difference diminishes with higher values of V .

We conclude based on these results that the retailer may indeed benefit from theintegration of shelf space and resulting extra handling costs in the optimization ofinventory decisions.

4.4.4 Summary

In an infinite-horizon, periodic-review, single-item retail inventory system withrandom demand and lost-sales, we study the feature of limited shelf space and


Table

4.3

Percen

tageofcost

pen

alties:

anexample

with

Poisso

ndem

andwith

meanλ=

20,L=

0.5,h=

1,p=

10,K

=10,

q=

1

Optim

alCost

(C∗V)

Cost

(CV)

%∆

Lost-sa

lesExtra

Lost-sa

lesExtra

VK

sOrd

ering

Holding

pen

alty

handlin

gTotal

Ord

ering

Holding

pen

alty

handlin

gTotal

Total

10

510.000

8.360

28.073

41.714

88.147

9.998

16.757

2.845

83.785

113.385

28.632

15

10.000

8.334

28.248

16.607

63.190

9.998

16.757

2.845

58.785

88.386

39.874

20

10.000

10.471

17.333

4.513

42.317

9.998

16.757

2.845

33.836

63.437

49.909

25

10.000

13.208

8.663

1.829

33.492

9.998

16.757

2.845

11.113

40.713

21.562

30

10.000

15.799

3.897

0.406

30.102

9.998

16.757

2.845

0.961

30.562

1.529

40

9.998

16.757

2.845

0.000

29.600

9.998

16.757

2.845

0.000

29.600

0.000

50

9.998

16.757

2.845

0.000

29.600

9.998

16.757

2.845

0.000

29.600

0.000

10

20

10.000

1.256

100.001

0.167

111.424

9.998

16.757

2.845

335.140

364.740

227.345

15

10.000

5.144

52.416

3.292

70.852

9.998

16.757

2.845

235.141

264.741

273.653

20

10.000

8.758

26.220

3.533

48.511

9.998

16.757

2.845

135.345

164.946

240.015

25

10.000

11.824

12.666

1.769

36.259

9.998

16.757

2.845

44.451

74.051

104.230

30

9.999

14.957

5.086

0.627

30.670

9.998

16.757

2.845

3.846

33.446

9.051

40

9.998

16.757

2.845

0.000

29.600

9.998

16.757

2.845

0.000

29.600

0.000

C∗V

=optim

allong-ru

navera

geco

stforth

eex

tended

model

CV

=long-ru

navera

geco

stforth

eex

tended

model

under

policy

U∗∞


100

150

200

250

300

Pe

rce

nta

ge

co

st p

en

alt

y (

% ∆∆ ∆∆)) ))

Ks = 5

Ks = 20

0

50

100

150

200

250

300

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

Pe

rce

nta

ge

co

st p

en

alt

y (

% ∆∆ ∆∆)) ))


Ks = 5

Ks = 20

(a) Percentage of cost penalties in total average cost

254.0

649.7

507.5

210

310

410

510

610

710

Pe

rce

nta

ge

co

st p

en

alt

y (

% Δ

)

%ΔOrdering

%ΔHolding

%ΔLost-sales penalty

100.9

254.0

649.7

507.5

136.6

10

0.4

10

1.1

60

.0

26

.9

6.1

-89.9 -89.9 -83.6-67.2

-27.0-0.02 -0.02 -0.02 -0.02 -0.01

-90

10

110

210

310

410

510

610

710

V/λ = 0.5 V/λ = 75 V/λ = 1 V/λ = 1.25 V/λ = 1.5

Pe

rce

nta

ge

co

st p

en

alt

y (

% Δ

)

%ΔOrdering

%ΔHolding

%ΔLost-sales penalty

%ΔExtra handling

(b) Percentage of cost penalties in individual cost components for Ks = 5

Figure 4.8 Percentage of cost penalties (%∆) in total average cost (a) as well asindividual components (b): an example with Poisson demand with mean λ = 20 andL = 0.50, K = 10, h = 1, p = 10, q = 1, V ∈ 10, 15, 20, 25, 30, 40, 50


40

60

80

100

120

Co

st

Average cost

Optimal cost

Extra handling cost

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3

Co

st


Average cost

Optimal cost

Extra handling cost

Figure 4.9 Average costs for λ = 20, L = 0.5, K = 10, p = 10, q = 1, Ks = 5Note: Average cost = CV = C(U∗

∞, V ), Optimal cost = C∗V = C(U∗

V , V ) and Extra handling

cost = Ks · EExcess(U∗∞, V )

backroom usage, and assume that additionally to a fixed cost per order, a linearcost is charged if the inventory position exceeds, on average, the available shelf spacecapacity. We build a model that particularly takes this cost structure into account,under the assumption of continuous backroom operations. In a numerical study,we discuss qualitative properties of the optimal solutions of this system, and giveseveral managerial insights into the effect of problem parameters on the system’sperformance; additionally, we investigate the relevance of the second replenishmentprocess for decision making. Our analysis demonstrates that the retailer’s long-runaverage cost decreases with the shelf space V , and increases with the unit extrahandling cost Ks, and unit penalty cost p, and there is a combined nonlinear effectof the problem parameters on the minimum cost. Finally, we show that includingthe extra handling costs in the optimization of inventory decisions results not onlyin a better representation of system’s costs, but also allows the retailer to achievesubstantial cost savings from the optimization of all relevant costs.

4.5. Numerical study: the model with fixed extrahandling costs

In this section, we provide a numerical investigation on the inventory control modelintroduced in Section 4.3.2. This model extends the basic lost-sales inventory controlmodel studied in Chapter 3 to the case in which a fixed cost Ke is charged to the

4.5 Numerical study: the model with fixed extra handling costs 99

system in the event that the shelf space V is exceeded. Our numerical analysiscomplements the one conducted in Section 4.4, where we assumed that the extrahandling cost is proportional to the average stock in excess of V . The purpose ofthis section is threefold. First, we provide insights into the structure of optimalpolicies (Section 5.3), then we conduct sensitivity analyses (Section 4.5.2) and provideadditional managerial insights (Section 4.5.3).

In our numerical experiments, we consider all possible combinations of the followingproblem parameters: Poisson demand distribution with mean λ = 10, R = 1, L = 1,h = 1, K1 = K2 = 0, q = 1 and

p = 5, 10, 20K = 5, 10,Ke = 0, 5, 10, 20, 40, 100, 200, 400V = 10, 15, 20, 25, 30, 40

The combination of all parameters resulted in 288 problem instances. We vary mostlythe shelf space V and the fixed extra handling cost Ke, as we aim to conductsensitivity analyses mostly around these two problem parameters. In practice, weexpect the extra handling cost Ke to be larger than the regular shelf stacking costK. Intuitively, through very large values of Ke, we attempt to capture situations inwhich the inventory position will not exceed the shelf space. Similarly to Section 4.4,we used the value iteration algorithm to determine numerically an optimal policy andthe associated long-run average cost.


In this section, we provide several qualitative insights into the structure of optimalpolicies. We start by observing that, when the shelf space V is sufficiently large, itbecomes non-binding in the model introduced in Section 4.3.2 (see Equation (4.11)),which reduces to the basic lost-sales inventory model, discussed in Chapter 3. Since theoptimal policy of the basic model is known to posses no simple structure, it is expectedthat the structure of the optimal policy of the extended model is also intricate. Similarto Section 4.4.1, to an optimal policy of the basic model, we associate two operationalparameters, denoted by rl∞ and mS∞, which define the optimal reorder point andoptimal maximum stock level, respectively.

We computed the optimal policies for the extended model for each scenario in ournumerical experiment. The optimal policies (expressed by the order quantity as afunction of on hand inventory) are depicted in Figure 4.10 for selected scenarios.These examples illustrate that the optimal policies exhibit complex structures, ingeneral. As anticipated, there is no apparent simple form solution, and the optimalorder quantity as a function of on-hand inventory is usually not monotone anddiscontinuous. Nevertheless, results suggest that there exists an inventory level rlV


such that if the beginning inventory level is above rlV nothing should be ordered.Similar to Section 4.4, we define mSV = maxx+ aV (x) : 0 ≤ x ≤ rlV , where aV (x)denotes the order quantity as a function of on-hand inventory.

10

15

20

25

30

Ord

er

qu

an

tity

K = 10, Ke = 20

V = 15

V = 40

0

5

10

15

20

25

30

0 5 10 15 20 25

Ord

er

qu

an

tity

Inventory on hand

K = 10, Ke = 20

V = 15

V = 40

4

6

8

10

12

14

16

Ord

er

qu

an

tity

K = 5, Ke = 20

V = 15

V = 40

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25

Ord

er

qu

an

tity

Inventory on hand

K = 5, Ke = 20

V = 15

V = 40

(a) Case A and C

10

15

20

25

30

Ord

er

qu

an

tity

K = 10, Ke = 20

V = 20

V = 40

0

5

10

15

20

25

30

0 5 10 15 20 25

Ord

er

qu

an

tity

Inventory on hand

K = 10, Ke = 20

V = 20

V = 40

4

6

8

10

12

14

16

Ord

er

qu

an

tity

K = 5, Ke = 20

V = 20

V = 40

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25

Ord

er

qu

an

tity

Inventory on hand

K = 5, Ke = 20

V = 20

V = 40

(b) Case B and C

Figure 4.10 Optimal order quantity as a function of on hand inventory: λ = 10,L = h = 1, p = 10, K1 = K2 = 0, q = 1, Ke = 20, K = 5, 10Note: Case (A): (rlV ,mSV ) = (19, 33) for K = 10, (rlV ,mSV ) = (21, 25) for K = 5

Case (B): (rlV ,mSV ) = (17, 20) for K = 10, (rlV ,mSV ) = (18, 20) for K = 5

(rl∞,mS∞) = (20, 34) for K = 10 and (rl∞,mS∞) = (21, 26) for K = 5

In view of our results, a few additional observations regarding the structure of optimalpolicies are worthwhile mentioning. First, it appears that there exist two thresholdvalues (possibly rl∞ and mS∞) such that, depending on the value of V , the optimalordering patterns differ in the following situations:

(A) 0 ≤ V ≤ rl∞,(B) rl∞ ≤ V ≤ mS∞,(C) mS∞ ≤ V .

Figure 4.10 exemplifies each situation by an example. The optimal policy structure


in situations (A) and (C) is depicted in Figure 4.10(a) (for two scenarios with K = 5and K = 10), while Figure 4.10(b) shows a general solution pattern in situations (B)and (C), respectively.

Next, we provide some intuition for each case. In the last case, the shelf spaceV is sufficiently large such that the extended and the basic model have identicalsolutions; hence V does not constitute an effective constraint in the extended model.We analyzed this situation in more details in Chapter 3.

In the second case, V becomes a binding constraint, and it appears from our numericalresults that the optimal decision is to order such that the maximum inventory levelmSV will not exceed V , i.e. to order at most to the capacity V .

Finally, in the first case, there seems to exist at least two ordering regions, in general:for lower initial inventory levels, the optimal decision is to order such that theinventory position after ordering does not exceed V , i.e. aV (x) ≤ V − x; however forhigher inventory levels, although the storage capacity V after ordering is exceeded, itappears that there exists an order-up-to level S such that V − x < aV (x) ≤ S − x,with V ≤ S. The intuition behind this observation could be as follows: if there arefew items on hand at a review moment, then we may take the opportunity of orderingin such a way that we avoid paying the additional fixed cost Ke, and thus we orderup to V (on hand plus on order ≤ V ); on the other hand, if the initial stock on handis higher than V , then whatever we order will result in an inventory position above Vand the cost Ke is unavoidable. In such a case, the optimal solution is less sensitiveto V , as if we are solving a basic lost-sales inventory model.

These examples also suggest that many parameters would be required to fullycharacterize the structure of the optimal policy, which is more complicate than the(s,Q|S) heuristic introduced in Chapter 3. Furthermore, by comparison with themodel discussed in Section 4.4, the model with a fixed cost for extra handling generatesadditional complexities in the optimal decision pattern.

4.5.2 Sensitivity analyses: the effect of V , Ke and p

In this section, we discuss the sensitivity of the solution and the associated long-runaverage cost to changes in V , Ke and p. Therefore, we assume for simplicity that thebatch and unit handling costs are fixed to zero. Table 4.4 presents a summary of ourresults, for different values of shelf space V , extra handling cost Ke, and unit lost-salespenalty cost p. For each scenario, we report the reorder level (rlV ), the maximumstock level (mSV ), as well as the optimal long-run average cost (denoted by C∗

V )associated to the optimal policy of the extended model. Due to the complexity ofthe optimal ordering pattern, we mainly study the variation of these indicators withchanges in the model parameters.

Based on these results we make the following observations.

102 Chapter 4. Inventory control with shelf space considerationTable

4.4

Sen

sitivity

analysis

with

Poisso

ndem

andsλ=

10,L=

1,h=

1,K

1=

K2=

0,q=

1

Ke=

0K

e=

5K

e=

10

Ke=

20

Ke=

40

Ke=

100

Ke=

200

Ke=

400

pK

VrlV

mSV

C∗V

rlV

mSV

C∗V

rlV

mSV

C∗V

rlV

mSV

C∗V

rlV

mSV

C∗V

rlV

mSV

C∗V

rlV

mSV

C∗V

rlV

mSV

C∗V

55

10

17

22

14.092

17

22

19.092

17

22

24.090

15

20

30.182

12

17

30.182

11

15

30.182

610

30.182

610

30.182

15

18

24

11.263

18

24

16.244

17

22

19.353

16

21

19.354

16

20

19.354

16

19

19.354

13

15

19.354

13

15

19.354

20

19

24

11.181

17

20

12.492

17

20

12.492

17

20

12.492

17

20

12.492

17

20

12.492

17

20

12.492

17

20

12.492

25

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

30

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

30

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

19

24

11.181

510

10

15

22

18.986

15

22

23.970

15

23

28.933

13

20

32.729

11

17

32.729

410

32.729

410

32.729

410

32.729

15

17

32

15.605

17

32

20.499

16

28

24.134

11

15

24.342

11

15

24.342

11

15

24.342

11

15

24.342

11

15

24.342

20

17

32

15.332

16

20

17.459

16

20

17.459

16

20

17.459

16

20

17.459

16

20

17.459

16

20

17.459

16

20

17.459

25

17

32

15.328

17

25

15.789

17

25

15.789

17

25

15.789

17

25

15.789

17

25

15.789

17

25

15.789

17

25

15.789

30

17

32

15.328

17

30

15.399

17

30

15.399

17

30

15.399

17

30

15.399

17

30

15.399

17

30

15.399

17

30

15.399

40

17

32

15.328

17

32

15.328

17

32

15.328

17

32

15.328

17

32

15.328

17

32

15.328

17

32

15.328

17

32

15.328

10

510

19

23

20.778

19

23

25.778

19

23

30.778

19

23

40.777

16

20

55.400

13

16

55.400

13

15

55.400

12

14

55.400

15

21

25

13.169

21

25

18.168

21

25

23.167

20

25

33.009

18

21

33.216

17

20

33.216

17

19

33.216

16

19

33.216

20

21

26

12.813

19

26

17.487

18

20

17.933

18

20

17.933

18

20

17.933

18

20

17.933

18

20

17.933

18

20

17.933

25

21

26

12.812

21

25

12.844

21

25

12.844

21

25

12.844

21

25

12.844

21

25

12.844

21

25

12.844

21

25

12.844

30

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

30

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

21

26

12.812

10

10

10

18

23

25.740

18

23

30.740

18

23

35.739

17

23

45.732

15

21

60.086

12

17

60.086

510

60.086

510

60.086

15

20

33

17.883

20

33

22.876

19

33

27.857

19

33

37.627

17

21

38.215

16

20

38.215

16

19

38.215

13

15

38.215

20

20

34

17.327

18

34

21.797

17

20

22.928

17

20

22.928

17

20

22.928

17

20

22.928

17

20

22.928

17

20

22.928

25

20

34

17.316

19

25

17.741

19

25

17.741

19

25

17.741

19

25

17.741

19

25

17.741

19

25

17.741

19

25

17.741

30

20

34

17.316

20

30

17.496

20

30

17.496

20

30

17.496

20

30

17.496

20

30

17.496

20

30

17.496

20

30

17.496

40

20

34

17.316

20

34

17.316

20

34

17.316

20

34

17.316

20

34

17.316

20

34

17.316

20

34

17.316

20

34

17.316

20

510

20

24

33.708

20

24

38.708

20

24

43.708

20

24

53.708

20

24

73.708

16

18

105.740

14

16

105.740

13

15

105.740

15

23

27

15.503

23

27

20.503

23

27

25.503

23

27

35.503

23

27

55.497

18

21

60.918

18

20

60.918

17

19

60.918

20

23

27

14.398

23

27

19.393

23

27

24.368

22

26

28.763

21

25

28.763

21

24

28.763

21

24

28.763

21

23

28.763

25

23

27

14.393

22

25

15.375

22

25

15.375

22

25

15.375

22

25

15.375

22

25

15.375

22

25

15.375

22

25

15.375

30

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

40

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

23

27

14.393

20

10

10

19

24

38.687

19

24

43.687

19

24

48.687

19

24

58.687

19

24

78.684

15

18

110.737

13

16

110.737

13

15

110.737

15

22

35

20.390

22

35

25.389

22

35

30.388

22

35

40.382

21

35

60.327

18

21

65.918

17

20

65.918

17

19

65.918

20

22

35

19.061

22

35

24.038

22

35

28.973

21

27

33.762

18

20

33.762

18

20

33.762

18

20

33.762

18

20

33.762

25

22

35

19.040

21

25

20.358

21

25

20.358

21

25

20.358

21

25

20.358

21

25

20.358

21

25

20.358

21

25

20.358

30

22

35

19.040

22

28

19.224

22

28

19.224

22

28

19.224

22

28

19.224

22

28

19.224

22

28

19.224

22

28

19.224

40

22

35

19.040

22

35

19.040

22

35

19.040

22

35

19.040

22

35

19.040

22

35

19.040

22

35

19.040

22

35

19.040


• First, we note that when Ke = 0, the shelf space V may still affect thesystem’s performance (unless V is sufficiently high to become non-binding in theextended model). This observation stems from the fact that the cost structureof the extended model differs from the one of the basic inventory model in twoways: the way we charge the lost-sales penalty cost, and the way we charge theextra handling costs (see Section 4.3.2 and Appendix C for alternative modelformulations).

• When V is sufficiently high, the extended model reduces to the basic inventorycontrol model (studied in Chapter 3), and the system becomes insensitive to Ke.On the other hand, when the fixed cost for extra handling Ke is very high, theoptimal decision aims to avoid stock in excess of V , which results in maximumstock levels below the shelf capacity (i.e. mSV ≤ V ). Figure 4.11 illustrates theeffect of V and Ke on the structure of the optimal policy for few scenarios.

• In our numerical settings, the values of rlV and mSV are monotone increasing(i), decreasing (d), or non-monotone (x) in p, Ke and V as follows:

p Ke V

rlV i d xmSV i d x

Monotonicity with respect to p aims at reducing the expected lost sales, whilemonotonicity with respect to Ke aims at cutting deliveries above V . Thecomplex pattern in optimal decisions, as discussed in the previous section, isalso reflected in a non-monotone behavior of rlV and mSV with respect to V .

• On the cost side, numerical results show that the optimal long-run average costdecreases with V , and increases withKe and p, all other parameters being equal.

p Ke V

C∗V i i d

Figure 4.12 illustrates this behavior by some examples. As shown in Figure4.12(a), the positive effect of Ke on the optimal tcost decreases with increasingvalues of V , and for large V , the solution becomes insensitive to Ke. For a givenV , the optimal cost increases with Ke up to a maximum value (see also Figure4.12(b)). By inspection of results in Table 4.4, we note that when the upperlimit is reached, the model’s optimal decision is usually to order at most to thecapacity V .

• Figure 4.12(b) illustrates not only the main effects of p and Ke on the optimalcost, but also the interaction effects. The rate of cost increase with p isaccentuated by Ke.


10

15

20

25

Ord

er

qu

an

tity

Ke = 20

V = 10

V = 15

V = 25

0

5

10

15

20

25

0 5 10 15 20 25

Ord

er

qu

an

tity

Inventory on hand

Ke = 20

V = 10

V = 15

V = 25

V = 40

(a) λ = 10, K = 10, p = 10, K1 = 0, q = 1, Ke = 20

10

15

20

25

Ord

er

qu

an

tity

V = 15

Ke = 5

Ke = 10

0

5

10

15

20

25

0 5 10 15 20 25

Ord

er

qu

an

tity

Inventory on hand

V = 15

Ke = 5

Ke = 10

Ke = 20

(b) λ = 10, K = 10, p = 10, K1 = 0, q = 1, V = 15

Figure 4.11 Effect of V and Ke on the optimal policy


30

40

50

60

70

Min

. a

vg

. co

st Ke = 0

K3 = 5

Ke = 10

Ke = 20

10

20

30

40

50

60

70

0 10 20 30 40 50 60

Min

. a

vg

. co

st

Shelf space (V)

Ke = 0

K3 = 5

Ke = 10

Ke = 20

Ke = 40

Ke = 100

(a) Effect of V and Ke on the average cost for p = 10

30

40

50

60

70

Min

.av

g.c

ost

p = 5, V = 15

p = 10, V = 15

p = 20, V = 15

0

10

20

30

40

50

60

70

0 5 10 20 40 100 200 400

Min

.av

g.c

ost

Fixed extra handling cost (Ke)

p = 5, V = 15

p = 10, V = 15

p = 20, V = 15

(b) Effect of p and Ke on the average cost for V = 15

Figure 4.12 Effect of V , Ke and p on the minimum long-run average expected cost:λ = 10, L = 1, h = 1, q = 1, K = 10


• Finally, based on these results we infer that, given V , the following bounds onthe optimal cost (C∗

V ) of the extended model can be identified: when Ke = 0,the optimal cost of the resulting reduced model (see Appendix C for the coststructure) is a lower bound on C∗

V ; alternatively, when Ke is large enough, theshelf space V will act like a hard constraint in the extended model, and thecorresponding optimal cost is an upper bound on C∗

V .

4.5.3 Managerial insights

In this section, we conduct a similar analysis to the one presented in Section 4.4.3in order to gain additional insights on the impact of V and Ke on the system’sperformance. Therefore, we focus on two aspects: (i) first, we illustrate the impact ofV on the different cost components of the long-run average cost and (ii) second, wequantify the cost penalty for excluding the extra handling costs from the optimizationof inventory decisions.

We accomplish these goals by means of a numerical example. The followingparameters are chosen in our numerical experiment: λ = 10, L = 1, q = 1, K = 10,K1 = K2 = 0, h = 1, p = 10, Ke = 20 and V ∈ 10, 15, 20, 25, 30, 40, 50.

Cost decomposition

The retailer’s total cost (referred to as ’Cost’) is calculated as the sum of expectedordering, holding, lost-sales penalty, and extra handling cost for exceeding the shelfcapacity V as follows:

Cost = K ·OF + h · EOH + p · ELS(V ) +Ke ·OExcess(V ),

where OF represents the order frequency, EOH denotes the expected stock on hand,ELS denotes the expected demand lost, and OExcess denotes the frequency ofordering above V on the long run.

We used a similar methodology to the one described in Section 4.4.3 to compute, foreach scenario, the optimal solution and the associated long-run average cost (C∗

V ), aswell as the individual cost components. We present our findings in Table 4.5 (columnsone to six).

We remark, based on this example, that the optimal cost components do not exhibita monotonic behavior with respect to V , although the minimum total cost (’Total’)decreases with V . In particular, there seems to exist a threshold value v (possiblyv = rl∞ = 20), such that when V ≥ v there is no extra handling cost (suggesting thatthere are no orders exceeding the shelf capacity), while the ordering and lost-salespenalty cost decrease, and the holding cost increases with V , respectively. Thus, onlywhen V < v, the system incurs extra handling costs. When V becomes sufficientlyhigh, it has no effect on system’s performance. These observations are consistent withearlier findings and reflect the complexity of the ordering pattern.


Table

4.5

Impact

ofshelfspace

Vonindividualcost

componen

tsforPoissondem

andwithλ=

10andL=

1,h=

1,p=

10,

K=

10,q=

1,K

e=

20

Optim

alCost

(C∗ V)

Cost

(CV)

%∆

Lost-sales

Extra

Lost-sales

Extra

VOrd

ering

Holding

pen

alty

handling

Total

Ord

ering

Holding

pen

alty

handling

Total

Total

10

9.761

2.705

13.297

19.968

45.732

5.735

8.943

13.307

20.000

47.985

4.928

15

7.415

6.191

7.188

16.832

37.627

5.735

8.943

3.279

20.000

37.957

0.877

20

9.973

2.097

10.858

0.000

22.928

5.735

8.943

2.649

20.000

37.327

62.801

25

9.458

5.067

3.216

0.000

17.741

5.735

8.943

2.638

12.566

29.882

68.435

30

7.885

7.082

2.529

0.000

17.496

5.735

8.943

2.638

9.135

26.451

51.180

40

5.735

8.943

2.638

0.000

17.316

5.735

8.943

2.638

0.000

17.316

0.000

50

5.735

8.943

2.638

0.000

17.316

5.735

8.943

2.638

0.000

17.316

0.000

C∗ V

=optimallong-runaverageco

stforth

eex

tended

model

CV

=long-runaverageco

stforth

eex

tended

model

under

policy

U∗ ∞

When

Vlargeen

ough,(rl ∞

,mS∞)=

(20,3

4)


Quantifying the effect of second replenishment

Next, we investigate the impact of the second replenishment process on the system’sperformance. More precisely, we quantify the cost penalties at the retailer bycomparing the total cost (including extra handling costs) of two policies: (1) optimalpolicy of the extended model (U∗

V ) vs. (2) optimal policy of the basic model (U∗∞).

As described in Section 4.4.3, we computed the percentage difference, denoted by%∆ (and given by Equation 4.12), between the cost of these policies. The results arereported in Table 4.5 (columns seven to twelve).

First, note that the application of the suboptimal policy U∗∞ in the extended model

results in higher than optimal total costs. The total cost CV decreases with V . Theordering and holding cost remain insensitive to changes in V , when U∗

∞ is applied.The system incurs extra handling costs for all V < mS∞, unlike in the optimal casein which, for V ≥ v, no extra handling costs are charged to the system.

Next, observe that for instances in which V ≥ v, the percentage increase in total cost(%∆) is rather substantial. On the other hand, when V < v, the extra handling costsare unavoidable (in both optimal and suboptimal case) and smaller penalties resultin total cost.

Finally, we note that, by using an integrated optimization of all relevant costs, alwayssaves the costs for extra handling. However, this gain trades-off with other costcomponents, especially the cost of ordering and the penalty cost for expected demandlost (see Figure 4.13).

5

10

15

20

Su

bo

pt

-O

pt

Co

st

Ordering Holding Lost-sales penalty Extra handling Total

-10

-5

0

5

10

15

20

10 15 20 25 30

Su

bo

pt

-O

pt

Co

st

Shelf space (V)

Ordering Holding Lost-sales penalty Extra handling Total

Figure 4.13 Cost difference CV − C∗V for the total as well as individual components:

an example with Poisson demand with mean λ = 10 and L = 1, K = 10, h = 1, p = 10,q = 1, Ke = 20, V ∈ 10, 15, 20, 25, 30, 40, 50

4.6 Conclusion 109

4.5.4 Summary

In an infinite-horizon, periodic-review, single-item retail inventory system withrandom demand and lost-sales, we study the feature of limited shelf space, and assumethat in addition to a fixed cost per order, a fixed cost is charged in the event ofinventory position exceeding the available shelf space. Furthermore, we distinguishbetween sales floor and backroom stock, and assume that the average demand lostis measured against the amount of stock visible on the shelf. In Section 4.3.2, weintroduce an extended model that particularly takes these features into account. Inthis section, via a numerical study, we investigate qualitative properties of the optimalsolutions, draw managerial insights regarding the effect of problem parameters on thesystem’s performance, and quantify the effect of the second replenishment process fordecision making. Our analysis demonstrates that the presence of fixed extra handlingcosts into the model highly complicates the structure of the optimal policies, and weprovide several qualitative structural insights. We show that the retailer’s long-runaverage cost decreases with the shelf space V , and increases with the extra handlingcost Ke and the unit lost-sales penalty cost p. As V increases (i.e. more shelfspace), the solution of the extended model converges to the one of the basic lost-salesinventory model, while as Ke increases (i.e. more expensive backroom operations),an incentive is created of ordering up to the available shelf capacity V . Finally, weprovide additional insights into the trade-off between the different cost components,and quantify the significance of the backroom usage on associated costs.

4.6. Conclusion

In this chapter, we considered a retailer who manages the inventory of a single itemfacing stochastic demand and periodically reviews its stock for replenishment. Thereplenishment lead time is less than the review period length and excess demand thatcannot be satisfied from stock on hand is lost. We allow batch ordering and assumethat the batch size has been predetermined. Furthermore, due to insufficient shelfspace, the retailer may temporarily keep surplus stock, which does not fit onto theshelves upon delivery, in the store’s backroom. Hence a distinction is made betweenthe sales floor and back stock. Each period, the surplus stock is transferred fromthe backroom to the regular shelves to serve end-customer demand (referred to asthe second replenishment process), and there is an associated extra cost for handlingbackroom stock. Since the retailer’s total costs are greatly influenced by the materialhandling operations, we investigate the effect of using the backroom on the combinedcost of ordering, holding, lost sales and merchandise handling, in an infinite-horizoninventory system.

Two models are proposed to control the retailer’s inventory, given a limited allocatedshelf space and backroom usage. The first one assumes continuous backroomoperations and extra handling costs that are proportional to the average back stock.


Alternatively, the second model assumes that the extra handling costs are independentof the number of units at the backroom. Instead, a fixed cost is incurred by thesystem when the maximum inventory levels exceed the allocated shelf capacity. Weinvestigated the performance of the two models with respect to the optimal orderingpolicies and associated long-run average cost. Furthermore, we investigated theeffect of including the shelf space and the additional handling costs on the inventoryreplenishment decisions.

Similar to lost-sales inventory control models, the proposed variants are analyticallyinvolved. Hence, we solved each model numerically in an extensive computationalstudy, and provided qualitative insights into the structure of the optimal policies.We illustrated the additional structural complexity inherited by the solutions of themodel with fixed extra handling costs in comparison with the solutions of the modelwith linear costs, as well as the solutions of the basic lost-sales model (i.e. withoutshelf space consideration). Furthermore, we provided several managerial insightsinto the effect of problem parameters (in particular the shelf space) on the optimalsolution and associated long-run average cost. Finally, we showed that the retailer maygreatly benefit from the explicit consideration of the second replenishment process,by quantifying the cost penalty the retailer may face by excluding the extra handlingcosts from the optimization of inventory decisions.

The results and insights from this study could serve in addressing more generalquestions regarding the shelf space optimization, or the effective management of thebackroom, for example. Furthermore, extensions into the direction of managing theinventory of multiple-items under storage constraints are also practically valuable.

Appendix A. The transition probability matrix

We define the transition probabilities from one decision epoch to the next for themodels introduced in Section 4.3. The probability pij(ai) of a transition from statei at one decision epoch to state j at the next epoch, given decision ai at the firstdecision epoch is defined as

pij(ai) = P(j = ((i−DL)

+ + ai −DR−L)+), i, j = 0, 1, . . . , ai = 0, q, 2q, . . . ,

and is given by

pi0(0) = P(DR ≥ i), i = 0, 1, . . . ,

pij(0) = P(DR = i− j), i = 0, 1, . . . , j = 1, 2, . . . , i,

p00(0) = 1,

4.6 Conclusion 111

when there is no order, and when the order amounts to ai = niq > 0 by

pi0(ai) =i−1∑k=1

P(DL = k)P(DR−L ≥ i+ ai − k) + P(DL ≥ i)P(DR−L ≥ ai), i = 1, 2, . . . ,

pij(ai) =i−1∑k=1

P(DL = k)P(DR−L = i+ ai − j − k) + P(DL ≥ i)P(DR−L = ai − j),

i = 1, 2, . . . , j = 1, 2, . . . , ai,

pij(ai) =

i+ai−j∑k=1

P(DL = k)P(DR−L = i+ ai − j − k),

i = 1, 2, . . . , j = ai + 1, . . . , ai + i,

p00(a0) = P(DR−L ≥ a0),

p0j(a0) = P(DR−L = a0 − j), j = 1, 2, . . . , ai,

pij(ai) = 0, otherwise.

Appendix B. Additional numerical results: modelwith continuous backroom operations

Table 4.6 Sensitivity analysis with Poisson demand with mean λ = 20, h = 1, L = 0.5and q = 1

Ke = 5 Ke = 10 Ke = 20 Ke = 40 Ke = 100 Ke = 200




V

2 5 10 13 18 26.298 13 17 26.337 12 16 26.399 11 15 26.485 10 14 26.630 10 13 26.75915 17 22 21.307 16 21 21.588 16 20 21.861 15 20 22.134 15 19 22.463 14 18 22.67720 20 26 19.475 19 25 19.642 19 25 19.822 19 24 19.979 18 23 20.189 18 23 20.35425 21 29 18.883 21 29 18.905 21 28 18.925 21 28 18.957 21 28 19.018 21 27 19.04830 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.85540 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.85550 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855

20 5 10 29 32 100.457 26 29 152.658 21 23 206.424 18 20 206.591 16 17 207.073 15 16 207.74215 29 32 75.466 27 29 102.725 24 26 116.955 22 24 121.542 20 22 128.432 19 21 133.53320 30 33 51.165 28 31 59.033 27 29 66.319 26 28 73.156 25 26 80.857 24 25 86.11525 32 35 35.322 31 33 38.574 30 32 42.019 29 31 45.597 28 30 50.091 28 30 53.44030 33 37 28.331 33 36 29.188 33 36 30.159 32 35 31.200 32 34 32.903 32 34 34.10640 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.56350 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563

40 5 10 33 35 114.990 30 33 183.607 27 29 291.207 21 23 406.591 18 19 407.093 16 17 407.89115 33 35 89.991 30 33 133.614 27 30 191.346 24 26 223.538 22 23 236.105 21 22 247.01720 33 35 65.178 31 33 84.703 29 31 102.214 27 29 117.830 26 27 136.306 25 26 148.51225 34 36 43.949 33 35 50.701 32 34 58.340 31 33 66.230 30 31 76.155 29 30 83.93930 35 38 32.561 35 37 34.581 34 37 36.782 34 36 39.221 33 35 42.894 33 35 46.32440 36 41 28.335 36 41 28.335 36 41 28.336 36 41 28.336 36 41 28.339 36 41 28.34350 36 41 28.335 36 41 28.335 36 41 28.335 36 41 28.335 36 41 28.335 36 41 28.335

100 5 10 36 38 131.542 34 36 217.518 32 34 366.189 29 31 607.082 22 24 1007.094 18 20 1007.930

Continued on next page


Table 4.6 (continued)

Ke = 5 Ke = 10 Ke = 20 Ke = 40 Ke = 100 Ke = 200




V

15 36 38 106.542 34 36 167.519 32 34 266.196 29 31 407.168 25 26 543.286 23 24 566.88520 36 39 81.561 34 37 117.678 32 34 167.448 30 32 218.073 28 29 272.331 27 28 309.60925 36 39 57.396 35 37 72.243 34 36 90.616 33 34 110.386 31 33 138.410 30 32 159.04230 37 40 39.850 37 39 44.456 36 38 50.097 35 37 56.663 35 36 66.263 34 35 74.72540 39 43 30.485 39 43 30.501 39 43 30.533 39 43 30.597 39 43 30.784 39 42 30.95150 39 43 30.469 39 43 30.469 39 43 30.469 39 43 30.469 39 43 30.469 39 43 30.469

200 5 10 38 41 142.495 37 39 239.462 35 37 413.698 33 34 713.548 28 29 1398.910 22 24 2007.93115 38 41 117.495 37 39 189.462 35 37 313.699 33 34 513.564 29 30 899.410 25 26 1076.20020 38 41 92.497 37 39 139.486 35 37 213.890 33 34 316.132 30 31 447.316 28 29 529.83125 38 41 67.743 37 39 90.977 36 37 122.551 34 36 160.464 33 34 215.270 32 33 258.66830 39 41 46.749 38 40 54.717 37 39 64.898 37 38 77.131 36 37 95.711 35 36 111.29940 41 45 32.129 41 44 32.253 41 44 32.388 41 44 32.655 40 43 33.271 40 43 33.59850 41 45 31.923 41 45 31.923 41 45 31.923 41 45 31.923 41 45 31.923 41 45 31.923

2 10 10 11 18 31.298 10 17 31.337 9 16 31.399 8 15 31.485 8 14 31.629 7 13 31.75615 14 22 26.307 14 21 26.587 13 20 26.861 13 20 27.133 12 19 27.461 12 18 27.67520 17 26 24.473 16 25 24.640 16 25 24.821 16 24 24.977 15 23 25.188 15 23 25.35325 18 29 23.872 18 29 23.895 17 28 23.918 17 28 23.950 17 28 24.013 17 27 24.04330 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.84040 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.84050 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840

20 10 10 28 32 105.457 25 29 157.658 20 23 211.424 17 20 211.591 15 17 212.073 14 16 212.74215 28 32 80.466 26 29 107.725 23 26 121.955 21 24 126.542 20 22 133.432 19 21 138.53320 29 33 56.165 28 31 64.033 26 29 71.319 25 28 78.156 24 26 85.857 23 25 91.11525 31 35 40.322 30 33 43.574 29 32 47.019 29 31 50.597 28 30 55.091 27 30 58.44030 32 37 33.331 32 36 34.188 32 36 35.159 31 35 36.200 31 34 37.903 31 34 39.10640 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.56350 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563

40 10 10 32 35 119.990 29 33 188.607 26 29 296.207 21 23 411.591 17 19 412.093 16 17 412.89115 32 35 94.991 30 33 138.614 27 30 196.346 24 26 228.538 22 23 241.105 20 22 252.01720 32 35 70.178 30 33 89.703 29 31 107.214 27 29 122.830 25 27 141.306 25 26 153.51225 33 36 48.949 32 35 55.701 31 34 63.340 30 33 71.230 29 31 81.155 29 30 88.93930 34 38 37.561 34 37 39.580 33 37 41.782 33 36 44.221 33 35 47.894 32 35 51.32440 35 41 33.335 35 41 33.335 35 41 33.335 35 41 33.336 35 41 33.339 35 41 33.34350 35 41 33.335 35 41 33.335 35 41 33.335 35 41 33.335 35 41 33.335 35 41 33.335

100 10 10 35 38 136.542 34 36 222.518 32 34 371.189 28 31 612.082 22 24 1012.094 18 20 1012.93015 35 38 111.542 34 36 172.519 32 34 271.196 29 31 412.168 25 26 548.286 23 24 571.88520 35 39 86.561 34 37 122.678 32 34 172.448 30 32 223.073 28 29 277.331 26 28 314.60925 36 39 62.396 35 37 77.243 33 36 95.616 32 34 115.386 31 33 143.410 30 32 164.04230 37 40 44.850 36 39 49.456 36 38 55.097 35 37 61.663 34 36 71.263 34 35 79.72540 38 43 35.485 38 43 35.501 38 43 35.533 38 43 35.597 38 43 35.784 38 42 35.95150 38 43 35.469 38 43 35.469 38 43 35.469 38 43 35.469 38 43 35.469 38 43 35.469

200 10 10 37 41 147.495 36 39 244.462 34 37 418.698 32 34 718.548 28 29 1403.910 22 24 2012.93115 37 41 122.495 36 39 194.462 34 37 318.699 32 34 518.564 28 30 904.410 25 26 1081.20020 37 41 97.497 36 39 144.486 35 37 218.890 33 34 321.132 30 31 452.316 28 29 534.83125 38 41 72.743 37 39 95.977 35 37 127.551 34 36 165.464 32 34 220.270 31 33 263.66830 38 41 51.749 38 40 59.717 37 39 69.898 36 38 82.131 35 37 100.711 35 36 116.29940 40 45 37.129 40 44 37.253 40 44 37.388 40 44 37.655 40 43 38.271 39 43 38.59850 40 45 36.923 40 45 36.923 40 45 36.923 40 45 36.923 40 45 36.923 40 45 36.923

End Table 4.6

4.6 Conclusion 113

Appendix C. Related models

Based on the transition cost structure introduced in Section 4.3.2 (see Equation(4.11)), we distinguished in Section 4.5 between the following models:

x minx, V No extra cost basic reducedExtra cost - extended

We detail here, for ease of reference, the underlying cost structure for each of themodels. Given x units on hand at the beginning of a review period and a unitsordered, the expected transition cost C(x, a) is defined as follows.

Basic model: no extra cost, no V

C(x, a) = δ(a)K +K1⌊a/q⌋+K2a+ h EDL,DR−L

[((x−DL)

+ + a−DR−L)+]

+ p EDL

[(DL − x)+

]+ p EDL,DR−L

[(DR−L − a− (x−DL)

+)+]

Reduced model: no extra cost, V included

C(x, a) = δ(a)K +K1⌊a/q⌋+K2a+ h EDL,DR−L

[((x−DL)

+ + a−DR−L)+]

+ p EDL

[(DL −minx, V )+

]+ p EDL,DR−L


+)+]

Extended model: extra cost, V included

Ct(x, a) = δ(a)K +K1⌊a/q⌋+K2a+ h EDL,DR−L

[((x−DL)

+ + a−DR−L)+]

+ pEDL

[(DL −minx, V )+

]+ EDL,DR−L


+)+]

+ Ke δ((x+ a− V )+

)

115

Chapter 5

Efficient control of lost-salesinventory systems with batchordering and setup costs

Abstract: We consider a single-item stochastic inventory system with lost sales,fixed batch ordering and non-negative ordering costs. Optimal policies for this casehave complicated structures. We analyze a heuristic policy, the (s,Q|S, nq) policy,which is intuitive and partially captures the structure of optimal policies. In anumerical study, we compare the performance of the heuristic with those commonlyused, as well as optimal solutions, under a variety of conditions and demonstrateits effectiveness. Furthermore, we conduct sensitivity analysis with respect to theproblem parameters and provide managerial insights. Additionally, we compare theperformance of the optimal policy against the most common heuristic used in the retailpractice, the (s, nq) policy1 .

5.1. Introduction and related literature

In the previous chapters we dealt with a single-item inventory replenishment decisioncommonly faced by grocery retailers. In Chapter 3, we incorporated a special material

1Also denoted in the literature by (k,Q) (Veinott, 1965) with k the reorder level and Q the batchsize (assumed exogenously determined), by (nQ, r) (Zheng and Chen, 1992) or (R,nQ) (Chen, 2000)with r (and R, respectively) the reorder level and Q the batch size (assumed a decision variable), by(R, s, nQ) (Silver et al., 1998) with R the review period length, s the reorder level, and Q the batchsize (assumed a decision variable). Similar to earlier chapters, we assume that Q is an exogenousparameter, hence denoted by q, and we exclude from our policy notations the review period lengthR (assumed given and fixed).

116 Chapter 5. Efficient control of lost-sales inventory systems

handling cost function into the optimization of inventory decisions, while in Chapter4, we extended the model to account for shelf space limitations at the retailer.

In this chapter, we relax some of the assumptions in earlier chapters, and consider avariant of the fundamental lost-sales inventory control problem, i.e., the one-location,one-product, periodic-review inventory replenishment system, which includes thefollowing features: batch ordering and fractional lead times. The general description ofthe system is that the item’s inventory is managed periodically and a decision on howmuch to order has to be made; a replenishment order, when placed, is delivered beforethe beginning of the next review moment, which results in lead times (L) shorter thanthe length of the review period (i.e. fractional lead times); multiple units of itemsare ordered and shipped together in batches of fixed size q ≥ 1 (i.e. batch ordering);the items face stochastic customer demands and any unfilled demand in a period islost, rather than backlogged. Three types of costs are involved: for ordering, forholding inventory and for losing demand, and we assume that the cost of orderingis non-negative and fixed (i.e., setup cost K ≥ 0). The objective is to determine aninventory policy that minimizes the long-run average cost of the system.

Compared to earlier chapters, we therefore assume in this chapter that the orderingcost structure is independent of the number of batches and units in the order, andthere is ample storage capacity at the retailer. We complement the analysis in theprevious chapters by explicitly considering the following cases: (a) K = 0 vs. K > 0,(b) q = 1 vs. q > 1 and (c) L = 0 vs. L > 0.

Numerous papers in inventory theory tackled this problem under the alternativeassumption of demand backordering. If there is no restriction on the order quantify(i.e. q = 1), then the optimal ordering policy is proven to be of base-stock (if K = 0)or (s, S) type (if K > 0) under a variety of conditions (see e.g., Bellman et al., 1955,Karlin and Scarf, 1958, Scarf, 1960, Iglehart, 1963, Veinott Jr, 1966, Chiang 2006,2007a). Exact and approximate algorithms have been proposed to compute the policyparameters (see e.g., Veinott Jr. and Wagner, 1965, Zheng and Federgruen, 1991).

The case of batch ordering 2 (i.e. q > 1) received considerably less attention, thougha common assumption in practice. If K = 0 and q > 1 then the optimal policy isof (s, nq)3 type: when the inventory level in a period falls below the critical level s,order the smallest multiple of q to bring total stock to or above s; otherwise, nothingshould be ordered (see, e.g., Veinott, 1965, Iwaniec, 1979). This policy is a naturalextension of the base-stock policy to the batch ordering restriction.

The last case is the one in which K > 0 and q > 1. To our knowledge, the optimalsolution for this problem has yet not been fully demonstrated for the backordering (orlost-sales) case. Under the restrictive condition of all-or-nothing (i.e 0 or q) ordering,Gallego and Toktay (2004) demonstrated the optimality of a threshold policy (order

2In the literature, the terminology ’batch ordering’ typically refers to the problem with therestriction imposed on the order quantity to be an integer multiple of a positive base quantity(batch); the problem with exogenously determined batch size was introduced by Veinott (1965).

3Veinott’s original notation (k,Q), with Q a fixed positive constant.


q when the inventory position is at or below the threshold, else do nothing) for thebackorder model. A reasonable extension of the well known (s, S)4 policy to the batchordering case would result in an (s, S, nq) policy, as already described in Chapters 1and 3, where an order is placed if the inventory position falls to or below s, and theorder size is the largest integer multiple of q which results in the inventory positionnot exceeding S (see also Hill, 2006). Similar policies have been proposed by Veinott(1965) and Zheng and Chen (1992), but whether or not they are optimal for the batchordering case with setup costs is yet unknown.

Figure 5.1 is a graphical summary of the four cases under the backorder assumption.

K = 0 K > 0

q = 1

Ord

er

qu

an

tity

Base stock or (S-1,S) policy

Ord

er

qu

an

tity

Inventory on hand

Base stock or (S-1,S) policy

S

Ord

er

qu

an

tity

(s,S) policy

S-sOrd

er

qu

an

tity

Inventory on hand

(s,S) policy

S-s

s S

q > 1

Ord

er

qu

an

tity

(s,nq) or (S-q,S,nq) policy

Ord

er

qu

an

tity

Inventory on hand

(s,nq) or (S-q,S,nq) policy

s

q

Ord

er

qu

an

tity

(s,S, nq) policy

[(S-s)/q]q

Ord

er

qu

an

tity

Inventory on hand

(s,S, nq) policy

s S

[(S-s)/q]q

Figure 5.1 A literature review for the backorder case

The inventory control problem addressed in this chapter assumes lost-sales insteadof backordering of unmet demand. The lost-sales model has been recognized to bemore complex than the equivalent backorder model, and for positive leas times, thecomplete structure of the optimal ordering policy is unknown (Hadley and Within,1963). The base stock (and respectively (s, S)) policy is not optimal in general forthe lost-sales case, unless restrictive conditions apply (see e.g., Huh et al., 2009). Theapproximation of the lost-sales model by its backorder counterpart performs badly, in

4In an (s, S) policy a replenishment order is placed if, at a review moment, the inventory positiondrops to or below s, and the order size is the amount required to raise the inventory position to S.


general (see e.g., Zipkin, 2008b). In Chapter 3, we presented a more extensive reviewof the related literature. Table 5.1 summarizes the findings for easy reference. Weconclude based on our overview that mostly studied are inventory models with nosetup cost and no batch ordering. The particular features considered in this chapter,batch ordering and fractional lead times, are rarely acknowledged (see Table 5.1).

Table 5.1 Brief summary of the periodic-review lost-sales inventory literature

Author Year Demand Leadtime

Assump. Batch Ordercost

Obj Policy

Karlin & Scarf 1958 G D L = R no zero C optimalGaver 1959 G D L = R no zero C base stockMorse 1959 G D L = R no zero C base stockMorton 1969 G D L = nR no zero C optimalMorton 1971 G D L = nR no zero C myopic, approximateNahmias 1979 G G L = nR no zero C optimal, base stock,

approximateDowns et al. 2001 G D L = nR no zero C base stockJohansen 2001 P D L = nR no zero C optimal, base stockJanakiraman&Roundy

2004 G G L = nR no zero C base stock

Janakiraman&Muckstadt

2004a G G L ≤ R no zero C optimal, myopic, basestock

Reiman 2004 G D L = nR no zero C constant orderChiang 2006 G D L ≤ R no zero C optimal, base stockLevi et al. 2008 G D L = nR no zero C dual-balancingZipkin 2008a G D L = nR no zero C several, compared nu-

mericallyZipkin 2008b G D L = nR no zero C optimalHuh et al. 2009 G D L = nR no zero C base stock

Nahmias 1979 G G L = nR no pos. C optimal, (s, S), ap-proximate

Kapalka et al. 1999 P G L ≤ R no pos. S (s, S)Johansen & Hill 2000 N D L = nR no pos. C (s,Q)Hill & Johansen 2006 G D - yes pos. C optimalChiang 2007a G G L ≤ R no pos. C optimalDemand distribution P = Poisson, N = Normal, G = generalL = lead time (D = deterministic, G = stochastic), R = review period lengthL = nR (integral multiple of R), L ≤ R (fractional)Obj = objective function (C = Cost, S = Service level)

The main contributions of this chapter can be summarized as follows.

1. We build stochastic lost-sales inventory control models, which take the full batchordering constraint into account, assume fractional lead times, and allow zeroor fixed ordering costs. We show numerically that the optimal policy structureof the most generic model is, in general, more complex than the structure of theoptimal solutions of the model with fixed ordering cost, or batch ordering only.In particular, we provide numerical examples to show that the optimal policyin case of positive lead times and zero, or fixed setup costs is not of (s, nq), or


(s, S, nq) type, respectively.

2. Since optimal policies have rather complicated structures, we propose as analternative heuristic the (s,Q|S, nq) policy, and we analyze it numerically forzero and positive setup costs, unit and non-unit batch sizes, as illustrated inFigure 5.2. In a numerical study, we compare the performance of the heuristicwith those commonly used as well as optimal solutions, and demonstrate itssuperiority and effectiveness. Hence, our analysis complements the one inChapter 3, and we provide additional insights.

3. We discuss several managerial insights regarding the impact of problemparameters, in particular the batch size, on the optimal policies and theassociated long-run average cost.

4. Finally, we investigate numerically the performance of the (s, nq) policy, aheuristic commonly implemented in automatic ordering systems at groceryretailers (Van Donselaar et al., 2009).

K = 0 K > 0

q = 1

Ord

er

qu

an

tity

(S-1,Q|S) policy

Q

Ord

er

qu

an

tity

Inventory on hand

(S-1,Q|S) policy

S

Q

Ord

er

qu

an

tity

(s,Q|S) policy

S-s

Q

Ord

er

qu

an

tity

Inventory on hand

(s,Q|S) policy

S-s

s S

Q

q > 1

Ord

er

qu

an

tity

(S-q,Q|S,nq) policy

[(Q)/q]q

Ord

er

qu

an

tity

Inventory on hand

(S-q,Q|S,nq) policy

s

q

[(Q)/q]q

Ord

er

qu

an

tity

(s,Q|S, nq) policy

[(S-s)/q]q

[(Q)/q]q

Ord

er

qu

an

tity

Inventory on hand

(s,Q|S, nq) policy

s S

[(S-s)/q]q

[(Q)/q]q

Figure 5.2 The logic of the (s,Q|S, nq) heuristic

The remainder of the chapter is organized as follows. Section 5.2 sets the notation,basic assumptions, and the Markov decision problem formulation. Via a numericalstudy, we present in Section 5.3 a qualitative investigation of the optimal policies, aswell as sensitivity analyses with respect to the problem parameters. Then, we test


the performance of the (s,Q|S, nq) heuristic and demonstrate its effectiveness againstcommon alternatives, the (s, S, nq) and (s,Q, nq) policies (Section 5.4). In Section5.5, we study the performance of the (s, nq) against the optimal policy. Finally, weconclude the study.

5.2. Model

We consider a periodic-review single-item stochastic inventory system in which anyunfilled demand is assumed to be completely lost, the replenishment orders arerestricted to non-negative integers multiple of a fixed batch size, and the replenishmentlead time is fixed but less than the review period length. The notation in this chapterfollows that of earlier chapters.

R Review period lengthL Lead time length (0 ≤ L ≤ R)t Period index (t = 0, 1, 2, . . . )

Xt On-hand inventory at the beginning of period tat Quantity ordered in period t

DL Random demand during lead time LDR−L Random demand during R− L

h Holding cost per unit of inventory per period,charged at the end of the period (h > 0)

p Penalty cost for every unit of sales lost during a period,charged at the end of the period (p > 0)

K Fixed cost per order (K ≥ 0)q Fixed batch size (q = 1, 2, 3, . . .)

In each period, the sequence of events is as follows: (i) inventory levelsXt are observedand the current period’s ordering decision (at) is made; the orders are restricted tointegral multiples of q; (ii) next, demand DL occurs during the lead time; (iii) then,orders arrive following their respective lead time L; (iii) then, demandDR−L continuesto occur up until of the end of the current period. Any unsatisfied demand during theperiod is lost. Also, any positive order is charged a fixed cost K. The holding andpenalty costs are charged at the end of the period. We assume that DR is stochastic,independent and identically distributed across periods and follows a non-negativeknown discrete distribution. Furthermore, similar to earlier chapters, we assume thatthe length of the review cycle R is exogenous to the model (for example determinedby the need of coordinating replenishment of many different items) and we furtherassume it to be the time unit.

Next, we use Markov Decision Processes (MDPs) to formulate the problem. Thedecision epochs are the beginning of review periods, the state of the system is Xt,defined on Ω = 0, 1, 2, . . ., and the action is at, with action space A = 0, q, 2q . . ..

5.2 Model 121

The system dynamics are as follows

Xt+1 = ((Xt −DL)+ + at −DR−L)

+, t = 0, 1, 2, . . . ,

where (x)+ = max0, x for any x ∈ R. We consider the infinite horizon average-cost model with the objective to determine an inventory policy that minimizes thelong-run average cost of the system.

The one-period transition probabilities and expected costs are given next. Theprobability pxy(a) of a transition from state x at one decision epoch to state y atthe next epoch, given action a at the first decision epoch is defined as

pxy(a) = P(y = ((x−DL)

+ + a−DR−L)+), x, y = 0, 1, . . . , a = 0, q, 2q, . . . ,

and has been detailed in Chapter 3 under the assumption (which we also make here)that the random variables DL and DR−L are stochastically independent.

The total expected cost from one decision epoch to the next, given initial startinginventory x and the amount ordered a = nq (n ∈ Z+), is denoted by C(x, a) and isgiven by

C(x, a) = δ(a)K + h · EDL,DR−L

[((x−DL)

+ + a−DR−L)+]

+ p · EDL

[(DL − x)+

]+ p · EDL,DR−L

[(DR−L − a− (x−DL)

+)+],

where δ(a) = 0 if a = 0 and δ(a) = 1, otherwise, for all a ≥ 0. Note that the differencebetween this formulation and the one in Chapter 3 is the ordering cost, assumed hereto be independent of the number of batches and units in the order.

Commonly, an expected average cost optimal stationary policy is determined bysolving the following average cost optimality equations:

vx = mina∈A

C(x, a)− g +∑y∈Ω

pxy(a)vy, x ∈ Ω, (5.1)

where (vx)x∈Ω are called the relative values (proven unique if an initial condition suchas v0 = 0 is imposed), and g is the long-run average cost of the system. Similar toearlier chapters, we solve the optimality equations via the value iteration algorithm.

In the remainder of the chapter, we conduct a numerical study whose objective ismanifold: first, to explore the structural form of the optimal policy for the inventorysystem described in Section 5.2, and to conduct sensitivity analyses with respect tothe problem parameters, in particular the fixed batch size q (Section 5.3), secondto determine the effectiveness of the (s,Q|S, nq) policy (Section 5.4), and finally, toinvestigate the performance of the (s, nq) policy, a commonly used heuristic in theretail environment (Section 5.5).


5.3. On the structure of the optimal policy: K = 0vs. K > 0

As our literature review reveals, it is difficult to analytically derive properties of theoptimal ordering solutions for the lost-sales case. The few approaches that exist areusually limited to the case of no setup cost and no batch ordering. Hence, in thissection, we numerically explore the structure of the optimal policies for a zero, as wellas a positive ordering cost, assuming unconstrained orders (i.e. q = 1), as well asbatch ordering (i.e. q > 1). The lead time is either zero (Section 5.3.1), a fraction ofthe review period length (Section 5.3.2), or equal to the review period length (Section5.3.3). Additionally, we present several insights from sensitivity analyses with respectto the impact of problem parameters on the optimal solutions and the correspondinglong-run average expected costs.

We conduct several numerical studies involving the following parameters, the leadtime L, the penalty cost ratio p/h, the ordering cost parameter K, the fixed batchsize q and the mean demand per review period λ. Throughout, we assume that therandom demands DL and DR−L are stochastically independent and both follow aPoisson distribution. The numerical examples are chosen as follows.

h L λ p K q

1 0, 1 5, 10, 20 5,10,20,40 0,5,10,20,50 1,2,4,6,12,201 0.25,0.33,0.50 20 5,10,20 0,10,20,50 1,2,4,6,12,20

Altogether, there are a total of 936 instances in our computational study. Ourselection of parameter values are reasonable choices in a retail environment (seeChapter 2). To determine an optimal policy and the associated long-run average costwe applied the value-iteration algorithm (to solve equations (5.1)), for 1000 iterationsor until the change between two iterations was less than 10−12 (for more details seee.g. Puterman, 1994, and Bertsekas, 1995).

5.3.1 The case L = 0

The special case of zero lead time approximates environments where products havea very short replenishment lead times, compared to their review period length. Ingrocery retailing, for example, many products ordered in the morning are received bythe stores at the end of the day, or the beginning of the next day at the latest. Whenthe order size is unconstrained (i.e. q = 1), the base stock (K = 0) and (s, S) policy(K > 0), respectively are optimal when the lead time is zero (see, e.g., Veinott, 1966,Shreve, 1976, Zipkin, 2000, and Cheng and Sethi, 1999). However, we are not awareof similar results for the case q > 1.

Our numerical studies confirm previous results for q = 1 (see Figure 5.3 for an

5.3 On the structure of the optimal policy: K = 0 vs. K > 0 123

illustrative example with L = 0, λ = 10, h = 1, p = 10 and q = 1; if K = 0,the base stock level S∗ = 26, and related minimum average cost C(S∗) = 8.405; ifK = 10, (s∗, S∗) = (20, 33) and C(s∗, S∗) = 17.443). In fact, for zero lead time andno constraints on the order quantity, the problem can be viewed as a backorder model,except that the unit penalty cost p should be higher in the lost-sales model.

L = 0, mD = 10, h = 1, p=10

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Inventory on hand

Order quantity

K=0, q=1

K=0, q=4

K=10, q=1

K=10, q=4

(a) Optimal order quantity

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20 22 24Inventory on hand

Order up to levels K=0, q=1

K=0, q=4

K=10, q=1

K=10, q=4

L = 0, mD=10, h=1, p=10

(b) Optimal order-up-to levels

Figure 5.3 Optimal order quantity and order-up-to levels for L = 0, q = 1 and q = 4

For non-unit batch sizes (i.e. q > 1), our numerical results indicate that there exists areorder point such that if inventory on hand is above the reorder point no orders shouldbe placed, otherwise, the optimal order quantity is a stepwise decreasing functionof on-hand inventory, with a ’step-down’ of size q and a ’step-length’ equal to q(except for very low/very high levels of stock on hand). We refer to Figure 5.3 for an


illustrative example when q = 4. In this example, if K = 0, the optimal policy advisesto order 28, if there is nothing on stock, and zero if stock on hand is higher than orequal to 25. Otherwise, the order is an integer multiple of 4, identical for 4 consecutivevalues, and decreasing in steps of 4 units. The corresponding minimum average costis C(U∗) = 8.641. When there is a fixed order cost of K = 10, it is optimal to order ifthere are 20 units or less on stock, the largest order size is 32 and smallest one is 12,and the order sizes are again decreasing in steps of 4 units. The associated minimumaverage cost in this example is C(U∗) = 17.482. As a result, the optimal order-up-tolevel is a stepwise function of on-hand inventory, which takes values in an interval oflength q. In other words, it is optimal to order-up-to an interval of length q > 1 (seeagain Figure 5.3).

Therefore, when K = 0 our numerical results suggest that the optimal policy is of(s, nq) type. When K > 0, the (s, nq) policy is suboptimal in general, and instead theclass of (s, S, nq) policies is a straighforward candidate to optimality (see also Section5.5.1). Unlike some assumptions in the literature (Hill, 2006), our numerical resultsshow that, in general, the best s and S parameters are not multiples of q, and theoptimal reorder point decreases with q, in general.

5.3.2 The case 0 < L < 1

Next, we provide a qualitative analysis of the optimal policy structure, and examinethe effect of changes in the values of different parameters on the system performance.Detailed numerical results for selected parameter values are included in AppendixA. Here and in the rest of the chapter, we define the optimal reorder level as rl =maxi ∈ Ω | ai > 0, i.e. the highest value of inventory on hand at which it is optimalto order a positive amount; the optimal maximum stock level is defined to be themaximum inventory position, after ordering, i.e. mS = maxi+ ai | 0 ≤ i ≤ rl.

(a) The case K = 0

First, let us focus on the inventory system with no setup cost. The optimal policiesfor three scenarios (corresponding to different L values) are depicted in Figure 5.4.For detailed results we refer to Table 5.9 in Appendix A. Several observations areworthwhile mentioning based on our numerical results. First, the optimal policies(defined as the order quantity as a function of on-hand inventory) do not have asimple closed-form. It is observed however, in all scenarios, that there exists anoptimal reorder point (rl) and for stock levels x ≤ rl, the optimal order function ismonotonically decreasing, and the rate of decrease is less than or equal to q. Ourfindings are consistent with earlier studies of Janakiraman and Muckstadt (2004a)and Chiang (2006), who extend previously known structural properties of Karlin andScarf (1958) and Morton (1969) from the case of integer lead times to the case offractional lead times. However, none of these studies consider batch ordering andthus our observations are more general.


p=10, mD=20,K=0,q=1

0

5

10

15

20

25

30

0 3 6 9 12 15 18 21 24 27 30 33 36

Inventory on hand

Order quantity

L=1/2

L=1/3

L=1/4

(a) Optimal order quantity for K = 0, q = 1

L = 1/2, mD = 20, K = 0

0

5

10

15

20

25

30

0 3 6 9 12 15 18 21 24 27 30 33 36 39

Inventory on hand

Order quantity

p=10, q=1

p=10, q=4

(b) Optimal order quantity for K = 0, q > 1

Figure 5.4 Optimal order quantity as a function of on hand inventory for K = 0. Fig.(a): q = 1, L ∈ 0.25, 0.33, 0.50 and Fig. (b): q = 4 vs. q = 1, L = 0.50.


Second, we point out that the optimal policy has a different structure than the (s, nq)policy, proven to be optimal for the backorder model with integer lead times (Veinott,1965). However, as illustrated by the examples in Figure 5.4 and many more (see Table5.9 in Appendix A), there seems to exist a region of higher inventory levels in whichthe optimal policy specifies the order quantity according to an (s, nq) policy. Outsidethis region however, the optimal policy structure is more involved.

Sensitivity analysis

In the absence of simple-structured policies, it is difficult to evaluate the impactof changes in the problem parameters on the optimal solution. Therefore, similarto earlier chapters, we use the reorder level (rl) and the maximum stock level(mS) as main operational indicators of change. Numerical results suggest that rlis non-decreasing in L and mS is non-decreasing in L or p (all else being equal).Furthermore, regarding the sensitivity of the long-run average cost to changes inproblem parameters, our numerical studies suggest that the optimal long-run averagecost is non-decreasing in L and p (all other things being equal) (see also Appendix A).Note that similar monotonic results (w.r.t. L) for the minimum average cost aredemonstrated by Zipkin (2008a) for the lost-sales model with lead times that areintegers multiple of the review period length.

Impact of q > 1

When q > 1, the optimal order quantity is a stepwise decreasing function of on-hand inventory; see Figure 5.4 (b) and also Appendix A for more results. In ournumerical studies we observed that the ’step-down’ size equals q and thus relationqnx+1 ≤ qnx ≤ qnx+1 + q, x = 0, 1, . . . holds true, which is a generalization ofpreviously known optimal structural results to the case of batch ordering.

With respect to the effect of q on the optimal solution, we observe that, on average,rl decreases as q increases and mS increases as q increases. In fact, numerical resultsshow that rl+ q = mS, when K = 0. Furthermore, sensitivity studies reveal that theoptimal long-run average cost is non-decreasing in q (all else being equal), indicatingthat a higher batch size leads to a higher average cost. Intuitively, there is lessflexibility in ordering with higher batch sizes, hence increased costs. Moreover, weobserve that the rate of cost increase is similar for different values of lead time L,indicating little interaction between q and L. Finally, we make the following remark,based on the observation that any feasible solution for the problem with parameterq′ = mq (m ∈ 1, 2, 3 . . . ) is also a feasible solution for the problem with parameterq (q ∈ 1, 2, 3, . . . ).

Remark 5.1 Consider the inventory system introduced in Section 5.2 with K = 0.If q′ = mq (with m a non-negative integer) and everything else is identical, thenC∗

q′ ≥ C∗q , where C∗

q′ (and C∗q , respectively) denotes the optimal expected average


cost corresponding to the batch size q′ (and q, respectively).

As a consequence, we derive the following relation between the optimal average costsfor models with different batch sizes: C∗

q=1 ≤ C∗q ≤ C∗

2q ≤ C(s∗, n(2q)) (the lastinequality results from the suboptimality of the (s, n(2q)) policy). This gives lowerand upper bounds on the optimal cost of the problem with parameter q > 1.

(b) The case K > 0

Next, we turn our attention to the inventory system with positive setup cost. Theoptimal policies for few problem instances are illustrated in Figure 5.5. The fixedordering cost changes from K = 10 (Figure 5.5(a)-(b)) to K = 20 (Figure 5.5(c)-(d))(all other parameters remain unchanged). Figure 5.5 (b) and (d) exemplifies the effectof non-unit batch sizes on the optimal policy structure (here q = 4 vs. q = 1). Wealso refer to Appendix A (Table 5.10) for further numerical results.

The main insights from our numerical study are as follows. The optimal policy hasno clear or simple structure (see again Figure 5.5). In fact, while the optimal orderquantity is monotonically decreasing for K = 0, in the presence of a positive ordercost (K > 0), there are many scenarios where the optimal order quantity is non-monotone in on-hand inventory (as Figure 5.5 (c)-(d) illustrates). However, as notedbefore for the case K = 0, there seems to exist an optimal reorder level (rl), belowwhich it is always optimal to place an order and beyond which it is never optimal toorder. Thus, rl plays the role of a reorder level in a general inventory policy. Thisobservation has been conjectured before for continuous review systems (see Hill andJohansen, 2006), but to our knowledge no proofs exist so far in the literature, in acontinuous or periodic review setting.

Moreover, when q = 1, numerical results suggest that the optimal ordering function,though generally intricate, simplifies to a linearly decreasing function with rate 1when initial stock on hand is not too low. Hence, the optimal maximum stock levelis constant in the region with higher values of stock on hand (see Figure 5.5 (a) and(c)). Comparative analysis under the same parameter settings also reveals that, whilefor K = 0 the optimal order quantity satisfies rl = mS − 1, when K > 0, an ordersize larger than one is placed at rl (compare for example Figures 5.4(a) and 5.5(a)).This indicates that when K > 0, an extra parameter will be necessary to describe thestructure of the optimal policy.

An inventory policy with such complex structure, which exhibits ’jumps’ and non-monotonicities (and it may be possibly described by many parameters) could bedifficult to explain to practitioners, thus hindering the possibility of implementation.Therefore, it is desirable that in many environments facing lost sales reasonableheuristic policies are being applied. In Section 5.4, we study the effectiveness of thenew (s,Q|S, nq) heuristic policy, as compared to that of two reasonable alternativeheuristics, the (s,Q, nq) and (s, S, nq) policy. Finally, in Section 5.5 we investigatethe performance of the (s, nq) policy commonly applied in a retail setting.


p = 10, mD=20, K = 10, q=1

0

5

10

15

20

25

30

0 3 6 9 12 15 18 21 24 27 30 33

Inventory on hand

Order quantity

L =1/2

L =1/3

L =1/4

(a)

p=10, mD = 20, K=10

0

5

10

15

20

25

30

0 3 6 9 12 15 18 21 24 27 30 33

Inventory on hand

Order quantity

L=1/2, q=1

L=1/2, q=4

(b)

p = 10, mD=20, K = 20, q=1

0

5

10

15

20

25

30

35

40

45

0 3 6 9 12 15 18 21 24 27 30

Inventory on hand

Order quantity

L = 1/2

L = 1/3

L = 1/4

(c)

p=10, mD= 20, K=20; q = 4

0

5

10

15

20

25

30

35

40

45

50

0 3 6 9 12 15 18 21 24 27 30

Inventory on hand

Order quantity

L = 1/2

L = 1/3

L = 1/4

(d)

Figure 5.5 Optimal order quantity as a function of on hand inventory for K = 10(subfigures (a),(b)) vs. K = 20 (subfigures (c),(d)), q = 1 (subfigures (a),(c)) vs. q > 1(subfigure (b),(d))


Sensitivity analysis

Regarding the impact of problem parameters on the optimal policy, we make thefollowing observations: as K increases, the optimal reorder point (rl) decreases, whilethe maximum stock on hand (mS) increases. This observation is quite intuitive. Asthe fixed ordering cost increases, the retailer would like to order as few times aspossible to avoid high ordering costs. Furthermore, our numerical results show thatrl increases with increasing values of L and p, and mS increases with L.

Our numerical experiments also indicate that the long-run average cost is non-decreasing in L, K and p (all else being identical)(see Appendix A for selected results).

Impact of q > 1

Figure 5.5 (b) and (d) depicts the impact of the batch constraint (here q = 4) onthe optimal policy. The optimal ordering function is generally stepwise decreasing ininventory on hand, but not always; it may exhibit ’up and down jumps’ when q > 1(as compared to the case q = 1) (Figure 5.5 (d)), which further complicates the formof the policy.

Hence, the relation qnx+1 ≤ qnx ≤ qnx+1 + q, x = 0, 1, . . . (i.e. the optimal orderquantity non-increasing in x, and the rate of decrease less than or equal to q) doesn’talways hold for periodic review models and Poisson demand. When K > 0, Hill andJohansen (2006)) remarked that the above relation has been proven by Johansen andThorstenson (1993) for a continuous review lost-sales model under Poisson demandand q = 1. The authors also point out that the relation doesn’t hold for compoundPoisson demands. Seemingly, it doesn’t hold, in general, in a periodic review settingas well.

5.3.3 The case L = 1

In this section, we assume that the lead time equals the review period length, andprovide additional insights on the structure of the optimal policy. Results from ournumerical investigation for selected parameter values are included in Table 5.11 fromAppendix A.

As in the previous cases, numerical results indicate a complex structure of the optimalordering policy, which may exhibit non-monotonicities, as well as ’up and down jumps’in the ordered quantity (see Figure 5.6). The reasoning is based on the observationthat the relative value function (vx)x∈Ω (given by (5.1)) may exhibit in such casesmultiple local minima (see Figure 5.7). As a result, the cost objective may havemultiple local minima, which prohibits a simple optimal policy structure.

Moreover, our numerical results confirm that neither an (s, S), nor an (s,Q) policy


L=1, mD=10, p=10, q=1

0

5

10

15

20

25

30

0 5 10 15 20

Inventory on hand

Order quantity

K=5

K=10

Figure 5.6 Optimal order quantity for L = 1, h = 1, K = 5, 10

0 10 20 30 40 50 60

-320

-300

-280

-260

-240

-220

-200

i

v(i)

L=1, mD=10, K=10, p=10, q=1

0 10 20 30 40 50 60-308

-307

-306

-305

-304

-303

-302

-301

-300

i

v(i)

Figure 5.7 The relative value function showing two local minima


is average cost optimal. The optimal policy behaves like (s,Q) for low levels of stockon hand and like (s, S) for higher inventory levels (see example illustrated in Figure5.8). Intuitively, when the stock level is low, one expects it to be used up duringthe review period, and thus order enough to meet the expected demand during thereview period. When the stock level is high, there is a higher chance for surplusinventory and therefore one aims to have enough inventory on hand plus on orderto meet the demand over the next two periods. The overlapping of the (s,Q) and(s, S) policies has also been observed by Hill and Johansen (2006), and suggests thatthe corresponding cost functions may have more than one local minimum. In suchcases, the optimal policy does not have a simple structure. In view of our results,we suggest that a three-parameter policy might describe better the behavior of theoptimal policy. Hence we propose the class of (s,Q|S, nq) policies as an alternativeheuristic. We already introduced this policy class in Chapter 3. In the next sectionwe investigate its effectiveness for the cases of zero, and positive setup costs.

L=1, mD=10, p=20, K=10, q=1

15 15 15 15 15 15 15 15 15 15 15

14 14

13 13

12 12

18

17

16

15

14

13

0

3

6

9

12

15

18

21

0 3 6 9 12 15 18 21

Inventory on hand

Order quantity

Figure 5.8 Optimal order quantity: a combination of (s,Q) and (s, S).

With respect to the impact of the problem parameters on the optimal policy, we notethe following rules of thumb. When the setup cost K increases, then rl decreases,and mS increases. Furthermore, all else being identical, there seems to exist aninterval of values for the setup cost, l ≤ K ≤ u, in which the optimal policyexhibits discontinuities; outside this parameter region the optimal order quantityappears monotone in on-hand inventory (see Figure 5.9). Regarding the impact ofmean demand on the optimal solution, we observe that when the mean demand (λ)increases, the optimal reorder level (rl) and maximum stock level (mS) increase aswell, suggesting larger orders for higher demand rates. Finally, as the penalty cost pincreases, both rl and mS increase. Intuitively, we tend to order more to avoid highercosts of losing sales.


L=1, mD=10, p=20, q=1

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25

Inventory on hand

Order quantity

K=5

K=10

K=20

K=50

Figure 5.9 Impact of K on the optimal policy.

5.4. The effectiveness of the (s,Q|S, nq) policy

In light of the numerical results obtained in Section 5.3 regarding the structuralform of the optimal policies, we propose next the class of (s,Q|S, nq) policies, andinvestigate its performance in a numerical study. The performance is measured againstthe optimal policy determined via dynamic programming, as well as against tworeasonable alternative heuristics, the (s, S, nq)5 and (s,Q, nq)6 policy.

5.4.1 The (s,Q|S, nq) policy

We discussed the (s,Q|S, nq) heuristic earlier in the dissertation, in Chapter 3. Werepeat here for convenience the main notation and definitions. The (s,Q|S, nq)heuristic has three non-negative parameters, s, Q and S, and is constructed suchas to combine the logic of two commonly known policies, the (s, S) and (s,Q) policy,respectively, and to account for the constraint on the order size. Given integers s, Sand Q, where 0 ≤ maxs,Q ≤ S ≤ s +Q, a fixed batch size q ≥ 1, the (s,Q|S, nq)policy defines a rule to decide on how much to order, at each review moment. Givenan initial inventory on hand x, the policy defines the order amount a(x) as follows

a(x) =

⌊Q/q⌋q if 0 ≤ x ≤ S − ⌊Q/q⌋q⌊(S − x)/q⌋q if S − ⌊Q/q⌋q < x ≤ s0 if s < x,

(5.2)

5The (s, S, nq) policy has two parameters, s and S (0 ≤ s ≤ S) and may be described as follows.Whenever the inventory level at a review period is less than or equal to s, order the largest integermultiple of q which results in an inventory position less than or equal to S.

6The (s,Q, nq) policy has two parameters s ≥ 0 and Q ≥ 0 and may be described as follows.Whenever the inventory level at a review period is less than or equal to s, order Q units such thatthe order size Q is a non-negative integer multiple of q.

5.4 The effectiveness of the (s,Q|S, nq) policy 133

where ⌊x⌋ denotes the largest integer, smaller or equal to x. In particular, when q = 1the order quantity equals

a(x) =

minQ,S − x if 0 ≤ x ≤ s0 if s < x,

and we shall simply refer to this policy as the (s,Q|S) policy. That is, an (s,Q|S)policy works as follows. When the inventory on hand is smaller than or equal toS −Q, order exactly Q; when x is greater than S −Q but smaller than or equal to s,then order up to S; and when x is above s, do not order. Notice that, according tothis policy, the order-up-to level is not constant, but a two-piecewise linear functionof on-hand inventory. We refer to Figure 5.2 for an illustration of the logic of theheuristic in four particular cases.

The well known (s, S) and (s,Q) policies (see e.g. Zipkin, 2000) are special cases ofthe (s,Q|S) policy, having S = Q and s = S − Q, respectively. Thus, the (s,Q|S)policy generalizes both (s,Q) and (s, S) policy. Additionally, when K = 0 (no setupcost), then s = S, and the policy has only two parameters S and Q. In Section5.4.2, we use these widely known policies to benchmark the performance of the newheuristic. When q > 1, the (s, S) and (s,Q) policies are modified such as to accountfor the restriction on the order quantity being an integer multiple of the fixed batchsize q. We refer to these policies as the (s, S, nq) and (s,Q, nq) policy, when K > 0,and the (s, nq) policy7 when K = 0, respectively.

Consequently, we restrict our attention to the class of stationary (s,Q|S, nq) policies,and study the original system under this restricted class. All other assumptionsremain the same. As before, the aim is to find the best (s,Q|S, nq) policy thatminimizes the long-run average cost.

Let C(s, S,Q; q) denote the expected long-run average cost of the system controlledby the (s,Q|S, nq) policy, i.e.,

C(s, S,Q; q) = limT→∞

1

T

T−1∑t=0

Ct(s, S,Q),

where Ct(s, S,Q) is the expected cost incurred by the system in period t, i.e.,

Ct(s, S,Q; q) = δ(at) K + h EDL,DR−L

[(X

(s,S,Q)t −DL)

+ + a(s,S,Q)t −DR−L

]++ p EDL,DR−L

[DR−L − a

(s,S,Q)t − (X

(s,S,Q)t −DL)

+]+

+ p EDL

[DL −X

(s,S,Q)t

]+,

7The (s, nq) policy has one parameter s ≥ 0 and may be described as follows. Whenever theinventory on hand is less than s, an order is placed for the minimum integer multiple of q such that,after ordering, the inventory will rise at or above s.


and X(s,S,Q)t is a nonnegative integer denoting the on-hand inventory at time t under

policy (s,Q|S, nq) and a(s,S,Q)t is the order quantity under policy (s,Q|S, nq) as given

by relation (5.2). Our objective is to find the optimal choice of parameters in the(s,Q|S, nq) class such that C(s, S,Q; q) is minimized.

As noted earlier by Hadley and Whitin (1963, Section 5.13), even for the simple lost-sales system, controlled by a base stock policy, it does not seem possible to determineexplicitly the long-run average cost. Thus, analytically determining the optimal policywithin the class of (s,Q|S, nq) policies may not be possible. Therefore, the optimalsolutions will be determined numerically, according to following methodology. Forany parameters s, Q, and S, a dynamic programming formulation similar to (5.1) isused to determine the long-run average cost of the (s,Q|S, nq) policy. In this casehowever, for any x ∈ Z+, instead of minimizing over all possible ordering quantitiesas in equation (5.1), the ordering quantity is given by the logic of the (s,Q|S, nq)policy, according to relation (5.2). We then apply the same value iteration algorithmto determine the long-run average cost of the (s,Q|S, nq) policy. To determine thebest s, Q and S parameters, we used an exhaustive search over a sufficiently largefeasible region to ensure we have found a global optimum. We also applied a similarmethodology for determining the best (s, nq), (s, S, nq) and (s,Q, nq) policies.

5.4.2 Effectiveness of the (s,Q|S, nq) policy

We evaluate the performance of the new policy by comparing the best (s,Q|S, nq)policies with optimal policies determined via dynamic programming. Additionally, weevaluate its performance against the performance of the best (s, S, nq) and (s,Q, nq)policies, and illustrate the overall improvement. Compared to the study conductedin Chapter 3, here we do not consider the per-batch and per-unit costs, but only thesetup costs. We extend the analysis to include more parameter settings, in particularby considering the case of no setup cost. We evaluate the effectiveness of the newheuristic in ’Policy’ as well as ’Cost’ space. In other words, we compare on one hand,the best (s,Q|S, nq) policy with the optimal solution, as well as the best alternativepolicies, and on the other hand we compare, the associated long-run average costs.

First, we measure the effectiveness of the (s,Q|S, nq) policy with respect to theoptimal policy by observing the percentage difference between the average costs ofthese two policies, denoted by Gap 1, using the formula

Gap 1 = 100× Best(s,Q|S, nq) average costs−Optimal average costs

Optimal average costs.

Second, we compare the average costs of the best (s,Q|S, nq) policy with those ofcommonly known heuristics: the (s, nq) policy (if K = 0) and the (s, S, nq) as wellas the (s,Q, nq) policy (if K > 0), respectively. The percentage difference betweenthe average costs of these policies is denoted by Gap 2 and is computed using the


following formula

Gap 2 = 100× Best(s, S, nq) average costs−Best(s,Q|S, nq) average costsBest(s,Q|S, nq) average costs

,

where the (s, S, nq) policy is replaced, alternatively, with the (s,Q, nq) policy. WhenK = 0, the average cost of the best (s, nq) policy is used instead. Note that, since the(s,Q|S, nq) policy generalizes the alternative policies, and might not be the optimalsolution in general, the above measures are well defined and positive.

We test the performance of the heuristics on a subset of 216 scenarios of the originalexperiment, with the following parameters.

L ∈ 0.25, 0.33, 0.50,K ∈ 0, 10, 20, 50,p ∈ 5, 10, 20,q ∈ 1, 2, 4, 6, 12, 20.

Table 5.2 shows the overall performance of the (s,Q|S, nq) heuristic against theoptimal policy, as well as the alternative heuristics. The reported results are averagesover all other parameters values. Later on, we report detailed results for selectedparameter values to gain further insights (see Tables 5.3 and 5.4).

Table 5.2 Overall performance of the (s,Q|S, nq) policy in ’Cost’ space.

q = 1 q > 1

(s,Q|S)vs.

Optimum

(s, S)vs.

(s,Q|S)

(s,Q)vs.

(s,Q|S)

(s,Q|S, nq)vs.

Optimum

(s, S, nq)vs.

(s,Q|S, nq)

(s,Q, nq)vs.

(s,Q|S, nq)Min 0.008 0.309 - Min. 0.000 0.000 -

K = 0 Avg. 0.062 0.819 - Avg. 0.047 0.973 -Max. 0.137 1.587 - Max. 0.197 2.887 -

Min. 0.000 0.101 0.151 Min. 0.000 0.000 0.000K > 0 Avg. 0.025 0.381 5.473 Avg. 0.124 0.279 3.593

Max. 0.114 0.932 14.444 Max. 1.654 1.089 14.639

Note: When K = 0, (s, S) means the best base stock policy, and (s, S, nq) means the best (s, nq)

Overall, the numerical experiments indicate that the class of (s,Q|S, nq) policiesperforms very well. When K = 0, the maximum percentage cost increase fromoptimality of the best (s,Q|S, nq) policy is below 0.2%, in all scenarios. When K > 0,the cost of using the heuristic is, on average, 0.025% (q = 1) and 0.12% (q > 1),respectively, higher than the true optimal cost, and at maximum 1.7% higher in allscenarios.

In Figure 5.10 we illustrate the performance of the (s,Q|S, nq) heuristic againstoptimal policies in ’Policy’ space, for selected parameter values. We notice that


the (s,Q|S, nq) captures partially the structure of the optimal policy. As depictedin Figure 5.10(b), it may even represent the true optimal policy. In fact, we noticethat in 92 out of the 216 scenarios (i.e. about 43%), the best (s,Q|S, nq) policycaptures exactly the optimal policy (and then q ≥ 4). Furthermore, our results showthat the best (s,Q|S, nq) policy is within 0.24% of the optimal cost in about 96% ofthe 216 scenarios. However, the optimal ordering policy does not posses the samestructure as the (s,Q|S, nq) heuristic. In fact, as visually illustrated in Figure 5.10(c)-(d), the optimal policies can be very complicated and difficult to identify. Observethat the optimal policy may have one (or more) ’peaks’ or ’dips’, while the heuristic ismuch simpler and it essentially eliminates the non-monotonicity in the optimal policy.However, in ’Cost’ space, the performance of the heuristic is fairly good.

K=10, L=1/2, p=10, mD=20, q=1, Diff=0.090%

0

5

10

15

20

25

30

0 3 6 9 12 15 18 21 24 27 30

Inventory on hand

Order quantity

optPolicy

best(s,Q|S)

(a)

K=10, L=1/2, p=10, mD=20, q=4, Diff=0.000%

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optPolicy

bes(s,Q|S,nq)

(b)

K=20, L=1/2, p=10, mD=20, q=4, Diff=0.056%

0

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25

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(c)

K=20, L=1/2, p=10, mD=20, q=4, Diff=0.056%

0

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Inventory on hand

Order quantity

optPolicy

best(s,Q|S,nq)

(d)

Figure 5.10 Optimal solution vs. best (s,Q|S, nq) policy in ’Policy’ space

Next, we compare the performance of the (s,Q|S, nq) heuristic against the best (s, nq)if K = 0, and the best (s, S, nq) and (s,Q, nq) policies, if K > 0, respectively. InTable 5.2 we reported the overall percentage of cost increase of these policies overthe cost of the best (s,Q|S, nq) policy. As expected from the definition, the best(s,Q|S, nq) policy outperforms the best (s, nq) (K = 0), and the best (s, S, nq) policy(K > 0), respectively. When K = 0, on average, it shows a better performance by0.8% (q = 1) and 1% (q > 1), respectively. When K > 0, the best (s,Q|S, nq) policy


outperforms the best (s, S, nq) policy on average by 0.4% (q = 1), and 0.3% (q > 1),respectively. Comparing the best (s,Q, nq) and (s,Q|S, nq) policies, we notice thatthe former is performing, on average about 5.5% (q = 1) to 3.6% (q > 1) worse thanthe (s,Q|S, nq) heuristic, with maximum percentage differences of 14.4% (q = 1), and14.6% (q > 1), respectively.

Unit batch size

Tables 5.3 and 5.4 summarize the results for unit batch size (i.e. q = 1), in thecase of zero or fixed setup cost, respectively. In the tables, for each combination ofthe parameters, we report the cost of the optimal policy, the best (s,Q|S, nq) policyparameters and the associated long-run average cost C(s∗, S∗, Q∗), as well as the bestpolicy parameters and the corresponding average costs for the other heuristics.

Table 5.3 Performance of the (S − 1, Q|S, nq) and (s, nq) policy for K = 0 and q = 1

Optimal policy Best (s,Q|S) Best base stock

Optimalcost

%Difffrom

optimalcost

%Difffrom costof best(s,Q|S)L p q rl mS s S Q Cost S Cost

0.50 5 1 33 34 17.367 33 34 23 17.386 0.108 34 17.617 1.32810 36 37 19.601 36 37 24 19.628 0.136 37 19.757 0.65820 38 39 21.563 38 39 26 21.575 0.056 39 21.642 0.309

0.33 5 30 31 13.716 29 30 23 13.735 0.137 30 13.936 1.46310 32 33 15.774 32 33 25 15.780 0.036 33 15.893 0.71820 34 35 17.609 34 35 27 17.613 0.022 35 17.670 0.322

0.25 5 28 29 11.993 28 29 23 11.998 0.042 29 12.189 1.58710 30 31 13.980 30 31 25 13.983 0.017 31 14.077 0.67420 32 33 15.746 32 33 27 15.748 0.008 33 15.797 0.313

Poisson demand with mean λ = 20, h = 1, K = 0, q = 1

Remarkably, the best (s,Q|S, nq) heuristic performs extremely well across all valuesof the parameters. We also note that, independent of the ordering cost K, theperformance of the heuristics improves as the penalty cost p increases, except forthe (s,Q, nq) heuristic, for which the performance actually deteriorates as p increases(see Table 5.4). A recent result of Huh et al. (2009) shows that, when there are nosetup costs, the order-up-to policies are asymptotically optimal as the penalty costbecomes large compared to the holding cost and the lead time is an integral multipleof the review period length. In view of our numerical results, we may expect that the(s, S) policies are also asymptotically optimal, as p gets large and K > 0. Althoughthe (s,Q) policies generally perform worse than the (s, S) policies, there are a coupleof instances when they may actually outperform the (s, S) policies (as indicated inbold in Table 5.4). We note that these are scenarios where p is relatively small. Alsoin these settings, (s,Q|S) heuristic has improved performance. Finally, these results


Table 5.4 Performance of the (s,Q|S, nq), (s, S, nq) and (s,Q, nq) policy for K > 0and q = 1

Optimal policy Best (s,Q|S) Best (s, S) Best (s,Q)

Opti-malcost

%Difffrom

optimalcost

%Difffrom costof best(s,Q|S)

%Difffrom costof best(s,Q|S)K L p rl mS s S Q Cost s S Cost s Q Cost

10 0.50 5 26 34 27.366 26 34 23 27.384 0.067 27 34 27.615 0.842 24 20 28.632 4.55410 30 37 29.600 30 37 24 29.627 0.090 30 37 29.756 0.436 28 21 32.019 8.07520 33 39 31.563 33 39 26 31.575 0.038 33 39 31.641 0.211 31 23 34.944 10.671

0.33 5 23 31 23.715 23 31 23 23.734 0.083 23 30 23.935 0.843 20 20 25.190 6.13310 26 33 25.774 26 33 25 25.779 0.022 27 33 25.893 0.440 24 22 28.466 10.42320 29 35 27.609 29 35 27 27.613 0.014 29 35 27.670 0.205 28 23 31.238 13.130

0.25 5 21 29 21.992 21 29 23 21.997 0.022 22 29 22.187 0.863 19 20 23.615 7.35610 25 31 23.980 25 31 25 23.982 0.010 25 31 24.077 0.392 23 22 26.786 11.68820 27 33 25.746 27 33 27 25.747 0.005 27 33 25.797 0.191 26 23 29.466 14.444

20 0.50 5 23 51 36.391 23 51 41 36.395 0.012 24 50 36.735 0.932 21 39 36.973 1.58810 28 55 39.421 28 55 44 39.443 0.056 28 54 39.581 0.350 28 30 40.634 3.01920 31 39 41.559 31 39 26 41.571 0.028 31 39 41.637 0.161 31 30 43.081 3.634

0.33 5 20 48 32.921 20 48 41 32.925 0.011 20 46 33.169 0.741 18 39 33.597 2.04110 24 51 35.697 24 51 44 35.738 0.114 24 51 35.875 0.383 24 30 36.926 3.32520 27 35 37.607 27 35 27 37.610 0.010 27 35 37.667 0.151 27 30 39.247 4.350

0.25 5 19 46 31.326 19 46 41 31.326 0.000 19 45 31.492 0.529 17 39 32.058 2.33810 23 49 33.968 23 31 25 33.978 0.030 23 31 34.070 0.272 23 30 35.199 3.59520 26 33 35.745 26 33 27 35.746 0.003 26 33 35.795 0.137 25 30 37.464 4.807

50 0.50 5 20 54 51.097 20 53 42 51.103 0.010 20 53 51.298 0.382 19 41 51.180 0.15110 25 56 54.397 25 56 45 54.399 0.005 25 56 54.568 0.310 20 40 57.097 4.95920 29 59 57.043 29 59 48 57.048 0.009 29 59 57.130 0.144 28 32 62.554 9.653

0.33 5 17 50 47.679 17 50 42 47.683 0.008 17 49 47.850 0.351 16 42 47.808 0.26310 22 53 50.756 22 53 45 50.759 0.007 22 52 50.891 0.260 19 41 52.053 2.54920 25 55 53.268 25 55 48 53.270 0.003 26 55 53.334 0.121 25 35 56.961 6.930

0.25 5 15 49 46.109 15 48 43 46.112 0.005 15 47 46.253 0.307 15 42 46.259 0.31910 20 51 49.055 20 51 45 49.057 0.005 20 51 49.169 0.228 19 41 49.983 1.88820 24 54 51.475 24 54 48 51.480 0.009 24 53 51.532 0.101 24 36 54.513 5.891

Poisson demand with mean λ = 20, h = 1, K > 0, q = 1

5.5 Performance of the (s, nq) policy 139

also indicate that the performance of the heuristic improves as K increases, all elsebeing equal.

Regarding the solution space, it is striking to observe that, in nearly all cases, theorder-up-to levels (mS or S) and the reorder points (rl or s) are nearly identicalacross the three types of policies (optimal, best (s,Q|S) and best (s, S) policy); theoccasional differences are small. Although, as Figure 5.10 illustrates, the optimalsolutions are structurally different than the (s, S) and (s,Q) policies. Apparently, theaverage cost C(s,Q, S) is less sensitive to changes in the values of parameter Q. Theseinsights could be exploited in the derivation of easily computable policy parameters.

In summary, the numerical study conducted in this section demonstrates that thebest (s,Q|S, nq) heuristic performs consistently well in all instances, while it capturespartially the structure of optimal policies.

5.5. Performance of the (s, nq) policy

In this section, we investigate the performance of the (s, nq) policy against theoptimal policy, in the general setting of zero of positive setup costs. Our motivationstems from the fact that many grocery retailers apply this heuristic for makinginventory replenishment decisions (Broekmeulen et al., 2004). Although the retailer’sreplenishment cost structure includes a fixed component, due to the presence ofmaterial handling costs, as we already demonstrated in Chapter 2, heuristics witha similar logic to the (s, nq) policy are often implemented in Automated OrderingSystems at grocery retailers (Van Donselaar et al., 2009).

Recall that the (s, nq) heuristic operates as follows. At the beginning of each reviewperiod, if the inventory on hand is less that s, an order is placed for the smalestintegral multiple of q such that, after ordering, the inventory level will rise at orabove s. Namely, given a non-negative integer s, the amount ordered in any periodthat starts with x items of inventory on hand equals

a(x) = q⌈(s− x)/q⌉, if x < s,

and 0 otherwise, where ⌈x⌉ denotes the smallest integer larger than or equal to x.Hence, the (s, nq) heuristic is a generalization of the well known base-stock policy tothe case of batch ordering (see Figure 5.1 for an illustration of the ordering functionunder this policy).

The experimental set up is identical to the one described in Section 5.3, and henceincludes 936 scenarios. For each scenario, we measure the effectiveness of the (s, nq)policy by the % of relative error between the average cost of the best (s, nq) policy(UB) and the true optimal cost (C∗), as follows:

E% = 100× (UB − C∗)/C∗.


The best policy parameter (denote by s∗) was determined using enumeration over asufficiently large feasible region. Next, we summarize our results separately for thecase of zero lead time (Section 5.5.1), lead time equal to the review period length(Section 5.5.2) and fractional lead time (Section 5.5.3).

5.5.1 Numerical results: L = 0

We find that when there is no setup cost (K = 0), the minimum, maximum andaverage E% over all 72 instances is essentially zero, which suggests that for the case ofzero lead time, the optimal policy is of (s, nq) type. However, when there is a positiveorder cost (K > 0), the maximum and the average relative error are substantial(average E% = 19.58% and maximum E% = 130.60% over 288 cases), which indicatesthat the (s, nq) policy is suboptimal in many cases.

Table 5.5 summarizes the minimum, maximum and average percentage gap betweenthe optimal and the best (s, nq) policies, over all instances with K > 0. Theinformation is presented for different values of the parameters, averaged over all otherparameter values. The mean E% is significant for most parameters; it decreases withmean review demand and batch size q, and increases with K.

Table 5.5 The performance of the (s, nq) policy (K > 0, L = 0)

Parameter Value Min.E(%) Max.E(%) Avg.E(%)

λ 5 .000 130.597 28.87610 .000 78.906 20.65920 .000 41.481 9.196

p 5 .000 72.573 17.98610 .000 116.198 21.75820 .000 130.597 20.13040 .000 122.763 18.433

K 5 .000 13.190 1.68810 .000 35.224 6.42320 .000 72.573 19.28650 .000 130.597 50.910

q 1 .000 130.597 27.7422 .000 127.200 27.3894 .000 115.904 25.4936 .002 91.209 21.597

12 .000 57.956 11.07420 .000 35.901 4.166

Total .000 130.597 19.577

Figure 5.11 illustrates the impact of the setup cost parameter (K) on the mean

5.5 Performance of the (s, nq) policy 141

E%, for different values of q: the higher K is, the higher the mean relative errorbecomes. However, the order of magnitude of E% decreases as q increases. This canbe intuitively explained by having less flexibility in ordering, as q increases. The meanE% indicates that the (s, nq) performs better for small values of K and large values ofq. The large percentage gaps in most cases clearly indicate that the optimal policiesdo not have the same structure as the (s, nq) policy. This finding is not surprising; weknow that if demand is fully backlogged and q = 1, an (s, S) policy is optimal (andthe order-up-to policy is generally suboptimal) in the presence of a fixed setup cost.

Figure 5.11 The impact of K on the mean E%, for different values of q (L = 0)

To conclude, when the lead time is zero and there is no setup cost (K = 0), the (s, nq)policy turns out to be the optimal policy in our numerical examples. The (s, nq) issuboptimal in most cases when K > 0.

5.5.2 Numerical results: L = 1

Table 5.6 reports the minimum, maximum and average E%, for each parameter(reported values are averages over all other parameters). We find that the (s, nq)policy performs poorly, in general, when K > 0 (over all 288 cases). The maximumE% equals 129%, while the average E% is 18.93%, indicating that on average, thecost of the best (s, nq) policy deviates substantially from the optimal average cost.

Our numerical results further suggest that the optimal policy does not have, in general,the same structure as the (s, nq) policy. The mean E% is significant for all parameters;it decreases with mean demand λ and batch size q, and increases with setup cost K(see Table 5.6).


Table 5.6 The performance of the (s, nq) policy (K > 0, L = 1)


λ 5 .000 129.532 27.67310 .000 77.909 19.88720 .000 40.774 9.238

p 5 .000 71.004 17.02210 .000 116.737 21.06320 .000 129.532 19.58440 .000 122.077 18.060

K 5 .000 9.789 1.68410 .000 33.197 5.72720 .000 71.004 18.21250 .000 129.532 50.107

q 1 .203 129.532 27.2362 .212 126.426 26.8644 .232 116.501 24.8636 .211 86.198 20.645

12 .000 49.115 10.33320 .000 34.002 3.654

Total .000 129.532 18.932

5.5.3 Numerical results: 0 < L < 1

Tables 5.7 (K = 0) and 5.8 (K > 0) summarize the minimum, maximum and averagerelative error (E%) between the optimal cost and the average cost of the best (s, nq)policy, for each individual parameter, averaged over all other parameters values.

When K = 0, it has been shown that base stock policies are suboptimal when leadtimes are integers (see e.g., Zipkin, 2008a). Our numerical results indicate that, whenK = 0, the best (s, nq) policy has an average difference of 1.15% and a maximumdifference of 2.89% from the optimal average cost. We observe that the performanceof the (s, nq) heuristic improves, on average, as p increases. We do not observe anyother strong trend in the performance of the (s, nq) heuristic with respect to the otherproblem parameters. Finally, we also note that the reorder point for the best (s, nq)policy is often equal or very close to rl for the optimal policy.

Numerical results in this case confirm previous expectations. When K > 0, the (s, nq)policies are suboptimal under most parameter settings. As indicated in Table 5.8, themaximum and average relative difference between the cost of the best (s, nq) policyand the optimal average cost is 36.06% and 10.87%, respectively. The performance ofthe (s, nq) heuristic decreases with K and increases with p. Beyond theses trends, it isdifficult to draw clear conclusions with respect to the impact of problem parameters.

5.6 Conclusions 143

Table 5.7 The performance of the (s, nq) policy (K = 0, 0 < L < 1)


L 0.25 .000 2.000 0.8600.33 .335 2.877 1.5150.50 .197 2.887 1.070

p 5 .000 2.529 1.60510 .654 2.887 1.34120 .000 1.184 0.500

q 1 .320 2.166 1.0492 .367 2.078 1.0204 .291 2.140 1.0306 .190 2.183 1.131

12 .000 2.469 1.28020 .000 2.887 1.379

Total .000 2.887 1.148

There is no clear trend w.r.t the batch size q. We do observe that the (s, nq) heuristicperforms poorly for all batch sizes (especially when K > 0). Furthermore, exceptfor the largest batch size, the average performance deteriorates as q increases (whenq = 20 the mean E% drops however from 12.29% to 9.56%).

In summary, the numerical results indicate that (s, nq) policies are, in generalinefficient for the cases of positive lead times and positive order cost. Therefore,using an (s, nq) policy in a setting where the replenishment cost is non negligible(such as grocery retailing, due to the presence of material handling costs) may resultin large cost penalties.

5.6. Conclusions

In this chapter, we studied a single-location, single-item periodic-review lost-salesinventory control problem with the following features: there are stochastic customerdemands, the lead time is a fraction of the review period length, there is a fixed(predetermined) batch size q for ordering, and orders are restricted to integer multiplesof q. We analyzed the inventory system from a cost perspective considering holdingand lost-sales penalty costs, as well as non-negative ordering costs. We distinguishedbetween lost-sales systems with no ordering costs and lost-sales systems with fixedsetup costs. Using Markov decision processes, we explored numerically the structureof the optimal policies and investigated, in particular, the impact of q on the optimalsolution and the long-run average cost of the system. We showed that optimal policieshave rather complicated structures, which hinders their practical applicability. In


Table 5.8 The performance of the (s, nq) policy (K > 0, 0 < L < 1)


K 10 0.000 1.489 0.63320 0.115 5.817 2.16550 23.010 36.061 29.800

L 0.25 0.000 36.061 11.0980.33 0.312 35.145 11.1140.50 0.023 33.302 10.387

p 5 0.000 36.061 12.61910 0.023 32.297 10.52720 0.115 29.875 9.452

q 1 0.146 34.872 10.6932 0.172 34.889 10.7154 0.146 35.107 10.8516 0.115 35.658 11.080

12 0.327 36.061 12.29520 0.000 29.114 9.561

Total .000 36.061 10.866

particular, we compared the optimal ordering patterns for systems with fixed orderingcost to those of the systems with no setup cost, or batch restrictions on the order size.

Based on the insights from the numerical study, we further studied the class of(s,Q|S, nq) policies, which partially captures the structure of optimal policies andshows very good performance in a variety of settings. In our computational study,we observed that the percentage of cost increase over the optimal cost is smaller than0.2% (when K = 0) and 1.7% (when K > 0), respectively. We also benchmarkedthe performance of the heuristic against reasonable alternative policies, the (s, S, nq)and (s,Q, nq) policies, and quantified the overall improvement. In particular, ournumerical results indicate that the best (s, S) policies are performing increasinglybetter and close to optimality as the penalty cost increases. On the other hand, thewidely applied (s, nq) policy in retailing may result in substantial cost penalties whenimplemented in the presence of fixed costs. A relevant area for future research is thederivation of easily computable heuristic (s,Q|S, nq) policies that are also

Appendix A. Selected numerical results

We denote by U∗ = [U1, U2, . . . , Url+1] the optimal order quantity as a functionof on-hand inventory 0, 1, 2, . . ., with rl the optimal reorder level (i.e. U∗ =[U1, U2, . . . , Url+1, 0, 0, . . .]). By convention, Un

i denotes n times the value Ui and

Appendix A. Selected numerical results 145

Ui : Uj denotes unit decreasing values from Ui to Uj (Uj < Ui ). Furthermore,rl = −1 is used to denote the policy of never ordering. Note that, under such apolicy, all expected demand is lost on the long run, and consequently the systemincurs only the penalty cost of loosing demand, i.e. the optimal long-run average costequals p · λ, with p the unit penalty cost and λ the mean period demand.

Table 5.9 Optimal policies, minimum costs and comparison with the (s, nq) policy,K = 0, λ = 20, h = 1 (Selected cases)

Optimal Policy Best (s, nq)

L p q U∗ C(U∗) rl mS s∗ − 1 C(s∗, nq)

0.25 5 1 [243, 233, 222, 21 : 1] 11.993 28 29 28 12.18910 [264, 253, 24 : 1] 13.980 30 31 30 14.07720 [282, 274, 262, 25 : 1] 15.746 32 33 32 15.797

0.33 5 [237, 222, 21, 202, 19 : 1] 13.716 30 31 29 14.01310 [265, 253, 242, 23 : 1] 15.774 32 33 32 15.96820 [278, 26, 252, 24 : 1] 17.609 34 35 34 17.752

0.50 5 [2310, 222, 212, 20 : 1] 17.367 33 34 33 17.61710 [264, 257, 242, 23, 222, 21 : 1] 19.601 36 37 36 19.75720 [2711, 262, 25, 242, 23 : 1] 21.563 38 39 38 21.642

0.25 5 4 [247, 205, 164, 124, 84, 44] 12.170 27 31 26 12.36410 [2410, 204, 164, 124, 84, 44] 14.177 29 33 29 14.28420 [287, 245, 204, 164, 124, 84, 44] 15.964 31 35 31 16.010

0.33 5 [249, 204, 164, 124, 84, 44] 13.887 28 32 28 14.18510 [2412, 204, 164, 124, 84, 44] 15.983 31 35 31 16.18320 [289, 245, 204, 164, 124, 84, 44] 17.837 33 37 33 17.979

0.50 5 [2412, 205, 164, 124, 84, 44] 17.555 32 36 31 17.79010 [2415, 204, 164, 124, 84, 44] 19.795 34 38 34 19.93620 [2812, 246, 204, 164, 124, 84, 44] 21.801 37 41 37 21.888

0.25 5 12 [2412, 1212] 13.400 23 35 23 13.40010 [2415, 1212] 15.504 26 38 26 15.76820 [2417, 1212] 17.602 28 40 28 17.731

0.33 5 [2414, 1212] 15.099 25 37 25 15.47210 [2417, 1212] 17.287 28 40 28 17.62120 [2419, 1212] 19.393 30 42 30 19.623

0.50 5 [2417, 1212] 18.807 28 40 28 19.09910 [2420, 1212] 21.073 31 43 31 21.31820 [2423, 1212] 23.239 34 46 34 23.422

0.25 5 20 [2021] 14.100 20 40 19 14.11210 [2025] 17.956 24 44 24 18.31520 [408, 2020] 20.534 27 47 27 20.534

0.33 5 [2022] 15.672 21 41 19 15.90610 [2027] 19.552 26 46 26 20.11520 [409, 2021] 22.363 29 49 29 22.438

0.50 5 [2025] 19.125 24 44 23 19.60910 [2030] 23.027 29 49 29 23.69220 [4013, 2021] 26.206 33 53 32 26.258


Table 5.10 Optimal policies, minimum costs and comparison with the (s, nq) policy,K > 0, λ = 20, K1 = 0, h = 1 (Selected cases)


K L p q U∗ C(U∗) rl mS s∗ − 1 C(s∗, nq)

10 0.25 5 1 [243, 233, 222, 21 : 8] 21.992 21 29 28 22.18910 [264, 253, 24 : 6] 23.98 25 31 30 24.07720 [282, 274, 262, 25 : 6] 25.746 27 33 32 25.797

0.33 5 [237, 222, 21, 202, 19 : 8] 23.715 23 31 29 24.01310 [265, 253, 242, 23 : 7] 25.774 26 33 32 25.96820 [278, 26, 252, 24 : 6] 27.609 29 35 34 27.752

0.50 5 [2310, 222, 212, 20 : 8] 27.366 26 34 33 27.61710 [264, 257, 242, 23, 222, 21 : 7] 29.600 30 37 36 29.75720 [2711, 262, 25, 242, 23 : 6] 31.563 33 39 38 31.642

10 0.25 5 4 [247, 205, 164, 124, 82] 22.167 21 31 26 22.36410 [2410, 204, 164, 124] 24.176 25 33 29 24.28420 [287, 245, 204, 164, 124, 84] 25.963 27 35 31 26.010

0.33 5 [249, 204, 164, 124, 83] 23.885 23 32 28 24.18410 [2412, 204, 164, 124, 83] 25.982 26 35 31 26.18320 [289, 245, 204, 164, 124, 84] 27.837 29 37 33 27.979

0.50 5 [2412, 205, 164, 124, 82] 27.551 26 36 31 27.79010 [2415, 204, 164, 124, 84] 29.795 30 38 34 29.93620 [2812, 246, 204, 164, 124, 84] 31.800 33 41 37 31.888

10 0.25 5 20 [2020] 23.615 19 39 19 23.61510 [406, 2019] 27.192 24 45 24 27.20120 [409, 2018] 29.462 26 48 27 29.549

0.33 5 [2021] 25.190 20 40 19 25.32710 [407, 2019] 28.903 25 46 25 28.99420 [4011, 2018] 31.293 28 50 29 31.459

0.50 5 [2025] 28.632 24 44 23 28.79310 [409, 2021] 32.571 29 49 29 32.57820 [4014, 2019] 35.189 32 53 32 35.253

20 0.25 5 1 [415, 402, 39 : 27] 31.326 19 46 28 32.18910 [264, 253, 24 : 20, 37 : 26] 33.968 23 49 30 34.07720 [282, 274, 262, 25 : 7] 35.745 26 33 32 35.797

0.33 5 [417, 40, 392, 38 : 28] 32.921 20 48 29 34.01310 [265, 253, 242, 23, 22, 39 : 27] 35.697 24 51 32 35.96820 [278, 262, 25 : 8] 37.607 27 35 34 37.752

0.50 5 [419, 402, 392, 38 : 28] 36.391 23 51 33 37.61710 [267, 254, 242, 23, 41 : 27] 39.421 28 55 36 39.75720 [2711, 262, 25, 242, 23 : 8] 41.559 31 39 38 41.642

20 0.25 5 4 [409, 364, 324, 283] 31.392 19 48 26 32.36410 [445, 244, 402, 202, 362, 162, 322, 122, 282, 8] 34.017 23 50 29 34.28420 [288, 244, 204, 164, 124, 83] 35.958 26 35 31 36.010

0.33 5 [4010, 364, 32, 283] 32.979 20 49 28 34.18410 [447, 243, 403, 20, 363, 16, 323, 12, 28] 35.759 24 52 31 36.18320 [289, 245, 204, 164, 124, 82] 37.829 27 37 33 37.979

0.50 5 [4013, 364, 324, 283] 36.45 23 52 31 37.79010 [4411, 243, 403, 20, 363, 324, 284] 39.475 28 56 34 39.936

Continued on next page

Appendix A. Selected numerical results 147

Table 5.10 (continued)


K L p q U∗ C(U∗) rl mS s∗ − 1 C(s∗, nq)

20 [2812, 245, 40, 203, 36, 163, 32, 123, 28, 82] 41.757 31 57 37 41.888

20 0.25 5 20 [4011, 207] 31.984 17 50 20 32.95210 [4013, 2010] 35.192 22 52 24 36.08720 [4013, 2013] 37.815 25 52 27 38.563

0.33 5 [4013, 207] 33.538 19 52 20 34.57210 [4015, 2010] 36.928 24 54 25 37.87120 [4016, 2012] 39.632 27 55 29 40.480

0.50 5 [4017, 206] 36.953 22 56 23 37.97810 [4019, 209] 40.599 27 58 29 41.46520 [4020, 2012] 43.526 31 59 32 44.249

End Table 5.10

Table 5.11 Optimal policies, minimum costs and comparison with the (s, nq) policy,L = 1, q = 1 (Selected cases)


λ K p U∗ C(U∗) rl mS s∗ − 1 C(s∗, nq)

5 10 5 [126, 112, 10] 10.853 8 18 11 14.45610 [144, 133, 122, 11, 9] 12.288 10 20 13 15.632

50 5 [235] 21.184 4 27 -1 2510 [255, 244, 23, 22] 23.069 8 30 -1 50

10 5 20 [1510, 143, 132, 12 : 4] 14.393 23 27 25 14.714

10 5 [227, 214, 202, 19, 17 : 14] 15.328 17 32 22 16.36210 [249, 233, 22, 122, 18 : 13] 17.316 20 34 24 17.95220 [1511, 142, 132, 122, 18 : 13] 19.040 22 35 26 19.476

20 20 [2610, 253, 242, 232, 21 : 17] 23.986 21 39 26 29.475

50 5 [3410, 332, 32, 31] 30.180 13 44 -1 5010 [369, 353, 34, 332, 32 : 30] 32.601 17 47 24 57.9520 [388, 374, 36, 352, 34 : 29] 34.679 20 49 26 59.474

20 10 5 [2320, 223, 21, 202, 19 : 8] 18.750 37 45 43 18.97810 [2520, 243, 232, 22, 212, 20 : 7] 20.980 41 48 46 21.154

50 5 [4220, 413, 402, 39, 382, 37 : 34] 42.432 31 65 44 58.97410 [4520, 443, 43, 422, 41, 402, 39 : 37, 35 : 31] 45.500 36 68 46 61.154


Appendix B. Approximate (s,Q|S) policies

In this chapter, we studied a variant of the traditional periodic-review lost-salesinventory control problem, with batch ordering and non-negative setup costs underthe average cost criterion, with an emphasis on the class of (s,Q|S, nq) policies.We demonstrated the efficiency of the best candidate within this class of policiesin solving the problem, where the the best policy parameters were determined byenumeration. The best (s,Q|S, nq) policy, although easy to explain in practice, couldbe computationally inefficient using an exhaustive search. In the following, we providesome computational details and few guidelines towards deriving easily computableapproximate (s,Q|S, nq) policies that are equally cost effective. More research isnecessary to further explore such opportunities.

Policy evaluation and optimization

Analytically determining the (s,Q|S, nq) policy, which minimizes the long-run averagecost was not possible. In fact, we are not aware of analytical methods for obtainingexplicit optimal solutions from a given class of policies for single-item lost-salesinventory problems with setup costs. Hence, exact optimal policies for the lost-sales problem when an (s,Q|S, nq) policy is in effect were obtained numericallyby formulating a Markov decision process and solving it with the value iterationalgorithm.

More precisely, for any given (s,Q|S, nq) policy (with s, Q and S nonnegative discreteparameters), we evaluated the long-run average cost of the policy for a given discretedemand distribution by formulating a dynamic program similar to (5.1). For a givenstarting inventory on hand, the ordering quantity is dictated by the logic of the(s,Q|S, nq) policy as expressed in relation (5.2). We then applied the relative valueiteration method to compute the long-run average cost C(s, S,Q).

Optimization concerns the process of finding the best nonnegative s, S and Q integervalues, which minimize the long-run average cost C(s, S,Q). The cost C(s, S,Q)is an involved function of its defining parameters, and we were not able to derivestructural properties that could be exploited in the derivation of efficient optimizationalgorithms. We recognize this as an interesting line for further research. Therefore,we evaluated the cost C(s, S,Q) over the grid (s, S,Q) ∈ Z3 | 0 ≤ maxs,Q ≤ S ≤s + Q, 0 ≤ S ≤ Smax, where Smax was large enough to ensure we found a globaloptimum.

Such an enumeration approach is however computationally inefficient, owing to thefact that one must search for a triplet of values (s, S,Q) over a response surface thatis not necessarily convex in these variables. Hence, obtaining cost estimates for allpossible values of (s, S,Q) in the grid search is computationally demanding. In Table5.13, we give an example of average computation times resulted from applying a fullgrid search in a short numerical study, in which the chosen problem parameters are

Appendix B. Approximate (s,Q|S) policies 149

selected as presented in Table 5.12.

As an alternative to enumeration, we explored how effective a simple local searchtechnique, based on the assumption of unimodality of the cost function C(s, S,Q) ineach of the three parameters, is in finding the best policy parameters. We summarizeour findings in the following.

Simple local search

For simplicity, we restricted ourselves to the case q = 1, hence to the policy class(s,Q|S), and K > 0. The parameters and model assumptions are the same as those inSection 5.4. The exact parameter values are presented in Table 5.12, the combinationof which resulted in 27 scenarios in the numerical study.

Table 5.12 Parameter setting for numerical experiment

Parameter Values

λ 20h 1p 5, 10, 20K 10,20, 50L 0.25, 0.33, 0.50q 1

Assuming unimodality of the objective function C(s, S,Q) in each parameter, we kepttwo of the parameters at their best values and we applied, sequentially, a simple onedimensional local search techniques to find a locally optimal solution. The aim wasto test how much improvement, in terms of computation time, is to be expected froma simple one dimensional local search.

Table 5.13 presents our findings. In the table, we reported the number of instancesin which the local search found the best s, Q, or S value (as determined by fullenumeration), which indicates the number of instances in which the local search failedto do so. We also reported the average and maximum computation time (in seconds)from using the one dimensional local search, as well as the total computation timefrom the full enumeration approach.

We observe that the local search technique managed to find the best values of theparameters s and Q in all 27 instances, while it failed to do so in 5 out of the 27scenarios for parameter S. These results suggest that the cost function C(s, S,Q)might be unimodal in s and Q, but not necessarily in S.

In Table 5.14, we present the detailed findings from applying local search along S.Where the corresponding approximate S value differ from the best value found byenumeration this is shown by underlining. Column 8 gives the long-run average cost


Table 5.13 Results from numerical experiments on 1-dim unimodal search

Number ofscenariosoptimum found

Time 1-dim search (sec) Total time (sec)

Algorithmavg max avg max

Enumeration 27 292.3035 317.4102Approx. s 27 0.017 0.062 70.181 72.170Approx. Q 27 0.013 0.025 70.188 71.351Approx. S 22 0.016 0.053 73.374 74.203

of the approximate (s,Q|S) policy. Column 9 gives the percentage cost increase ofColumn 8 relative to the cost of the best (s,Q|S, nq) policy, and thus provides ameasure, in cost terms, of the error that results from applying the one dimensionallocal search technique.

Table 5.14 also shows, for each scenarios, the computation time spent in generatingthe transition probabilities and cost functions for the dynamic program (Column 10),the computation time taken by the one dimensional local search (Column 11), as wellas the total computation time (Column 12). We observe that the largest share oftotal computational time is related to the generation of transition probabilities andassociated cost function. The one dimensional local search technique takes fraction ofseconds. These findings suggest that further gains, in terms of computational time,could be obtained from a more efficient derivation of transition probability matricesand cost function. More research is necessary to exploit this opportunity.

Further enhancements

Next, we discuss some enhancements to enumeration as directions for future research.First, one could benefit from faster evaluation methods of the transition probabilitymatrices and associated costs. In an earlier study of a single-item lost-sales inventorycontrol problem with service level constraints, restricted to the (s, S) policy class,Kapalka et al. (1999) also found that approximately 50% of the total computationaltime of the enumeration approach was spent in generating the transition probabilitymatrices and cost functions. Therefore, to address the large computational times ingenerating the transition probabilities, Kapalka et al. (1999) developed a transitionprobability matrix updating procedure, which then combined with bounds on S anda monotone search algorithm reduced the time required for computing the transitionprobability matrices by 90%, and the total execution time by approximately 30%.Their approach is however not directly generalizable from the two dimensional to thethree dimensional parameter space. It would be worthwhile to further investigate theconsequence of applying a similar approach in our problem setting.

A further way to decrease the computation time is to derive lower and upper boundson the policy parameters, in particular s and S (Q ≤ S ≤ s+Q). This will reduce the

Appendix B. Approximate (s,Q|S) policies 151

Table 5.14 Results from local search along S

L p K q s∗ Approx.S / S∗

Q∗ Approx.cost

%costerror

Time togenerateP , C(sec)

Timelocalsearch(sec)

Totalruntime(sec)

0.50 5 10 1 26 34 23 27.384 0.000 70.477 0.053 70.54010 10 1 30 37 24 29.627 0.000 73.343 0.023 73.36820 10 1 33 39 26 31.575 0.000 73.291 0.015 73.312

0.33 5 10 1 23 31 23 23.734 0.000 73.540 0.010 73.55810 10 1 26 33 25 25.779 0.000 73.664 0.013 73.67920 10 1 29 35 27 27.613 0.000 73.851 0.014 73.865

0.25 5 10 1 21 29 23 21.997 0.000 73.573 0.013 73.58710 10 1 25 31 25 23.982 0.000 73.750 0.013 73.76320 10 1 27 33 27 25.747 0.000 73.878 0.019 73.898

0.50 5 20 1 23 51 41 36.395 0.000 74.186 0.015 74.20310 20 1 28 55 44 39.443 0.000 73.479 0.021 73.50120 20 1 31 57/39 26 43.167 1.597 73.679 0.009 73.689

0.33 5 20 1 20 48 41 32.925 0.000 73.646 0.011 73.65810 20 1 24 51 44 35.738 0.000 73.215 0.014 73.23020 20 1 27 54/35 27 39.311 1.701 73.355 0.011 73.367

0.25 5 20 1 19 46 41 31.326 0.000 73.455 0.013 73.46910 20 1 23 48/31 25 35.360 1.382 72.886 0.009 72.89620 20 1 26 52/33 27 37.516 1.771 73.092 0.009 73.103

0.50 5 50 1 20 53 42 51.103 0.000 73.308 0.013 73.32110 50 1 25 56 45 54.399 0.000 73.134 0.013 73.14820 50 1 29 59 48 57.048 0.000 73.428 0.015 73.444

0.33 5 50 1 17 50 42 47.683 0.000 73.422 0.014 73.44010 50 1 22 53 45 50.759 0.000 73.343 0.021 73.36620 50 1 25 55 48 53.270 0.000 72.819 0.030 72.850

0.25 5 50 1 15 48 43 46.112 0.000 73.564 0.019 73.58410 50 1 20 51 45 49.057 0.000 73.587 0.022 73.61620 50 1 24 69/54 48 52.329 0.849 73.641 0.014 73.655

P : the transition probability matrix, C: the cost function


number of polices that require evaluation during the grid search. This approach iscommon in the literature. For example, Zheng and Federgruen (1991) propose upperbounds on S for deriving efficient (s, S) policies for the backordering problem, buttheir approach does not appear as straight forward for our problem. Approximatebounds based on the EOQ model are proposed by Kapalka et al. (1999) and could alsobe investigated in our setting. Also, one might consider first to reduce the problemdimension by deriving approximations for one of the parameters.

Another way for deriving efficient heuristics often relies on the derivation of structuralproperties of the cost function, such as monotonicity of the cost with respect to one ormore of its parameters, or convexity results (see e.g. Huh et al., 2009, Janakiramanand Muckstadt, 2004a). It should be noted that it is not apparent that the long-run average cost C(s,Q|S) is convex over the feasible policy space, therefore searchalgorithms based on convexity need not converge to an optimum. Further research toinvestigate if there is some structure in the cost function that could be exploited inthe derivation of efficient heuristics might be valuable. Results regarding structuralproperties for the lost-sales problem with positive setup cost under a given policyclass are still largely missing in the inventory literature.

Finally, techniques to find approximate policies from a given policy class oftenpropose transformations of the original model by relaxations, restrictions, or costapproximations (see e.g., Porteus, 1985). The effectiveness of such approaches for ourproblem setting could also be investigated.

153

Chapter 6

Conclusions

In this dissertation, we studied inventory control systems that have been inspiredfrom the practice of grocery retailers, where we focused on the incorporation ofseveral features that we identified as relevant challenges: lost-sales, batch ordering,shelf space limitations, merchandise handling and backroom operations. We studiedperiodic review inventory control models under a cost perspective, where aside fromthe traditional inventory-related costs, we specifically included handling-related costsin the optimization of inventory decisions. First, a formal mathematical model ofthe handling costs is proposed based on insights from an empirical study. Then,several single-item lost-sales inventory models are developed, which are classifiedbased on their main additional feature: (i) shelf stacking cost, (ii) shelf space capacityand backstock handling cost, and (iii) batch ordering and non-negative setup costs,respectively. In this concluding chapter, we briefly summarize the main results andinsights from our research and discuss some future research directions.

6.1. Results

In the introductory chapter, we raised a number of research questions related tothe control of two essential store-based retail operations: merchandise handling andinventory replenishment. In this section, we provide answers to these questions bysummarizing the main findings and insights from our studies.

1. What are the key factors that drive the shelf stacking time in retail stores?

In Chapter 2, we considered the shelf stacking process in grocery retail stores andstudied the main factors that have an influence on the execution time of this operation.

154 Chapter 6. Conclusions

We investigated the entire process at the level of individual sub-activities, and foundthat stacking new inventory, grabbing and opening a case pack, and waste disposalare the three most time consuming sub-activities. They are mostly influenced by thenumber of case packs and consumer units in the replenishment order. Alternatively,activities such as searching for the right location on the shelf, preparing the shelvesand filling old inventory require rather a fixed execution time. We demonstrated thata simple model, based on the number of case packs and consumer units, provides areliable estimate of the total stacking time per order line, and the form of relationshipis additive rather than purely linear. This structural insight was exploited into a lotsizing analysis to illustrate the opportunities for extending inventory control ruleswith a handling component.

2. How could the retail inventory control models be adapted to incorporate handling indecision making? And what is the impact of adding this aspect on the overall systemperformance?

In Chapter 3, we recognized the shelf stacking cost at the retailer as a critical costcomponent for the analysis and optimization of replenishment decisions at item level.The handling cost structure assumes fixed and linear components, dependent onthe number of batches and units in the replenishment order. This cost structurewas embedded in a periodic-review lost-sales inventory model, along with the morestandard inventory carrying and lost-sales costs. Assuming batch ordering andfractional lead times, the objective was to control the system in order to minimizethe long-run average cost.

Optimal replenishment policies for lost-sales inventory systems have been so far onlypartially characterized, mostly under simpler assumptions such as no order cost, andno batch ordering. Existing properties and numerical results for the optimal orderquantities suggest no simple structure, which could be used in practical applications.We identified a simple class of so-called (s,Q|S, nq) policies, which partially capturesthe structure of optimal policies. Numerical experiments revealed that the bestcandidate within this class of heuristic policies comes close to being optimal in manysettings. Furthermore, we compared the performance of the heuristic against the best(s,Q, nq) and (s, S, nq) policies, and quantified its superiority. Finally, we illustratedthe sensitivity of the solutions and associated long-run average cost to the handlingcost parameters. In particular, we showed that if the handling costs are ignoredin inventory replenishment decisions, the retailer’s expected cost penalty may besubstantial, especially for items with low profit margins.

3. How could inventory control models be adapted to account for shelf space limitationsand use of the backroom? And what is the impact of including these features on theperformance of the inventory control models?

6.2 Future research directions 155

In Chapter 4, we extended the model in Chapter 3 to account for limited shelf spaceat the retailer and the fact that surplus stock is temporarily stored at the backroom,which generates additional handling costs. We proposed two inventory models tocapture this situation. The first model assumes continuous replenishment from thebackroom and extra handling costs that are proportional to the expected backstock.Alternatively, the second model assumes that an additional fixed cost is charged inthe event of using the backroom. Both models generalize the situation when thereis sufficient shelf space to avoid using the store’s backroom. In a numerical study,we illustrated how shelf space shortages affect optimal solutions and associated long-run average costs. Furthermore, we quantified the retailer benefit from including theextra handling costs in inventory optimization. A comparison of total costs between(i) a situation where the additional handling costs were not taken into account in theoptimization of inventory decisions but nevertheless included in total cost calculationand (ii) a situation where the shelf space and extra handling costs were accountedfor in the replenishment decision making, showed that the latter situation may leadto total cost savings of more than 50%. Several managerial insights are illustratedregarding the trade-off between the different cost components.

4. Can we derive an efficient heuristic to control the single-item lost-sales inventoryproblem with batch ordering and setup costs? And how efficient is the (s, nq) policy,a commonly applied heuristic in grocery retailing, in controlling the inventory system?

In Chapter 5, we investigated the class of (s,Q|S, nq) policies as an alternative tooptimal solutions. Four situations are considered, depending on (i) whether setupcosts are present or not, and (ii) whether the fixed batch size is one or higher. Ournumerical studies demonstrated that the cost increase from using the heuristic insteadof optimal solutions is at most 0.2% when there are no setup costs and at most 1.7%,when setup costs are present. The heuristic is intuitive and partially explains thebehavior of optimal policies. Compared to the more common (s,Q, nq) and (s, S, nq)policies, our heuristic is performing always better. In particular, our results showedthat the best (s, S) policy have improved performance as the penalty cost increases,while the best (s,Q) policy may outperform the best (s, S) policy in settings withsmall penalty costs. Our heuristic performs consistently very well in both settings.Finally, we showed that the (s, nq) policies may perform quite poorly when setup costsare present, always the case in a retail setting where handling costs are prevalent.

6.2. Future research directions

Based on the research conducted in this dissertation, a few additional issues are worthfurther investigation. Our models have been inspired from the practice of groceryretailers, yet some of our assumptions might be limiting and could be relaxed to better

156 Chapter 6. Conclusions

reflect current practices. For example, we assumed a stationary demand process, whilein the retail environment, especially grocery retailing, the demand is non-stationary,following a cyclic (typically weekly) pattern. Also it is worth exploring the robustnessof the results reported in this dissertation to different demand distributions, or evenconsider more challenging extensions to a setting with unknown demand, unobservedlost sales, or inaccurate inventory records.

One of the characteristics included in our research was batch ordering, i.e., thereplenishment order was restricted to multiples of a fixed batch size. However, insome settings, such as retailers carrying their own brands, the retailer may have theflexibility of replenishing not only in batches, but also with loose items. This situationbrings up interesting challenges for inventory managers, if the replenishment coststructure assumes for example paying only for the loose items, or paying a fixed costfor each batch ordered (even if it is incomplete), and has been rarely addressed in theliterature.

Another practical constraint we tackled in our research was the shelf space limitation.In our models, the shelf space was an input parameter. A logical extension would be tooptimize also the shelf space capacity, and our approach could serve as a subproblem inaddressing this issue. Further research could be also dedicated to designing efficientbackroom operations, and our research showed that handling costs are a relevantcomponent. An interesting extension towards the real life retail setting would be toconsider not only a maximum constraint on inventory but also a minimum stock level,which is often imposed by marketing considerations to attract customers in the store.

We proposed an intuitive heuristic approach for the retail inventory control problemwith lost sales, and the practice could definitely benefit from the development of easilyimplementable, accurate approximations for the policy parameters, since this maywell serve in addressing the more general constraint, multi-item stochastic inventoryproblem.

References 157

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Summary


The inventory control problem of traditional store-based grocery retailers has severalchallenging features. Demand for products is stochastic, and is typically lost when noinventory is available on the shelves. As the consumer behavior studies reveal, only asmall percentage of customers are willing to wait when confronted with an out-of-stocksituation, whereas the remaining majority will either buy a different product, visitanother store, or entirely drop their demand. A store orders inventory on a periodicbasis, and receives replenishment according to a fixed schedule. The ordered stockis typically delivered before the next ordering moment, which results in lead timesshorter than the review period length. Order sizes are often constrained to integermultiples of a fixed batch size, the case packs, generally dictated by the manufacturer.Upon order receipt at the store, the stock is manually stacked on the shelves, to servecustomer demand. Shelf space allocation of many products is limited, dictated bymarketing constraints. Hence, surplus stock, which does not fit on the regular shelf,is temporarily stored in the store’s backroom, often a small place, poorly organized.

The focus of this dissertation is on developing quantitative models and designingsolution approaches for managing the inventory of a single item, under periodic review,when some or all of the following characteristics are taken into account:

• Lost sales. Demand that occurs when no inventory is available is lost, ratherthan backordered.

• Fractional lead time. Time between order placement and order delivery isshorter than the review period length.

• Batch ordering. Order sizes are constrained to integer multiples of a fixed batchsize.

• Limited shelf space. Shelf space allocation is predetermined. The retailer’sinventory is split between the sales floor and the backroom, which is used totemporarily store surplus inventory not accommodated by the regular shelves.

166 Summary

We consider optimal, as well as easy-to-understand inventory replenishment policies,where the objective is to minimize the long-run average cost of the system. Twotypes of costs are primarily recognized in the inventory models developed in thisdissertation:

• inventory related costs: for ordering, for holding products on stock, and penaltycosts for not being able to satisfy end-customer demand, and

• handling related costs: for shelf stacking, and for handling backroom stock.

Despite empirical evidence on the dominance of handling costs in the store, remarkablylittle is reported in the academic literature on how to manage inventory in the presenceof handling costs. A reason for this is that formal models of handling operations arestill scarce. In this dissertation, we first formalize a model of shelf stacking costs,using insights from an empirical study. Then, we extend the traditional single-itemlost-sales periodic-review inventory control model with several realistic dimensions ofthe replenishment practices of grocery retailers: batch ordering, handling costs, shelfspace and backroom operations. The models we consider are too complex to lendthemselves to straightforward analytical tractability. As a result, numerical solutionmethods based on stochastic dynamic programming are proposed in this dissertation,and near-optimal alternative replenishment policies are investigated.

Chapter 2 addresses operational concerns regarding the shelf stacking process ingrocery retail stores, and the key factors that influence the execution time of thiscommon store operation. Shelf stacking represents the regular store process ofmanually refilling the shelves with products from new deliveries, which is typicallytime consuming and costly. We focus on products that are replenished in pre-packedform but presented to the end-customer in individual units. A motion and timestudy is executed, and the complete shelf stacking process is broken down into severalsub-activities. The main time drivers for each activity are identified, relationshipsare established, tested and validated using real-life data collected at two Europeangrocery retailers. A simple prediction model of the total stacking time per order lineis then inferred, in terms of the number of case packs and consumer units. The modelcan be applied to estimate the workload and potential time savings in the stackingprocess. Implications of our empirical findings for inventory replenishment decisionsare illustrated by a lot-sizing analysis in Chapter 2, and further explored in Chapter 3.

Chapter 3 defines a single item stochastic lost sales inventory control model underperiodic review, which is designed to handle fractional lead times, batch orderingand handling costs. We study the settings in which replenishment costs reflectshelf stacking costs and have an additive form with fixed and linear components,depending on the number of batches and units in the replenishment order. Weexplore the structure of optimal policies under the long-run average cost criterionand propose a new policy, referred to as the (s,Q|S, nq) policy, which partiallycaptures the optimal policy structure and shows close-to-optimal performance in many

Summary 167

settings. In a numerical study, we compare the performance of the policy against thebest (s,Q, nq) and (s, S, nq) policies, and demonstrate the relative improvements.Sensitivity analyses illustrate the impact of the different problem parameters, inparticular the batch size and the handling cost parameters, on the optimal solutionsand associated average costs. Managerial insights into the effect of ignoring handlingcosts in the optimization of replenishment decisions are also discussed.

Chapter 4 extends the retail setting from Chapter 3 to situations in which there is alimited shelf space to display goods on the sales floor, and the retailer uses the store’sbackroom to temporarily store surplus stock. As a result, the back stock is regularlytransferred from the backroom to the sales floor to satisfy end-customer demand,which results in additional handling costs for the retailer. We investigate the effectof using the backroom on the inventory system performance, where performance ismeasured with respect to the optimal ordering decisions, and the long-run averagecost of ordering, holding, lost-sales and merchandise handling. Two extensions of theinventory model with ample shelf space are proposed in Chapter 4, which include a(i) linear or (ii) fixed cost structure for additional handling operations. In a numericalstudy, we discuss several qualitative properties of the optimal solutions, illustrate theadditional complexities of the second model, and compare the findings with thoseof the previous chapter. Furthermore, we build several managerial insights into theeffect of problem parameters, in particular the shelf space capacity, on the system’sperformance. Finally, we quantify the expected cost penalty the retailer may face byignoring the additional handling costs in the optimization of inventory decisions, andillustrate the trade-off between the different cost components.

Chapter 5 studies a variant of the traditional infinite-horizon, periodic-review, single-item inventory system with random demands and lost sales, where we assumefractional lead times and batch ordering, and allow for fixed non-negative orderingcosts. We present a comparison of four situations: zero vs. positive setup costs,and unit vs. non-unit batch sizes. For all cases, the optimal policy structure is onlypartially known in general. We show in a numerical study that the optimal policystructure of the most general model is usually more complex than that of the modelswith positive setup cost, or batch ordering only. Based on the gained insights, wefurther test the performance of the near-optimal (s,Q|S, nq) heuristic policy in thedifferent cases, and demonstrate its effectiveness. Also, well-known inventory controlpolicies of base-stock, or (s, S) type are extended to the case of batch ordering andstudied in comparison with the new heuristic under several conditions.

About the author

Alina Curseu was born in Alba-Iulia, Romania (ROU), on July 31, 1975. Aftercompleting her pre-university education at the ”Horia, Closca and Crisan” NationalCollege in Alba-Iulia, Romania, she studied Mathematics at the ”Babes-Bolyai”University of Cluj-Napoca, Romania. She graduated the top of her class in 1998, andone year later she received her Master of Science (M.Sc.) degree in Mathematics fromthe same university, with a specialization in Convex Analysis and Approximation.Between 1999 and 2002 she worked as a junior researcher at the ”T. Popoviciu”Institute of Numerical Analysis of Romanian Academy, in Cluj-Napoca (ROU).She carried out research on multicriteria, continuous optimization, numerical andfunctional analysis. From 2002, she joined the postmaster program Mathematics forIndustry of the Stan Ackermans Institute at Eindhoven University of Technology, TheNetherlands (NL). Within this program, she carried out several industrial projects,and in 2004 she received the Professional Doctorate in Engineering (PDEng) degree.In 2005, she started a PhD project on retail inventory management at the EindhovenUniversity of Technology (NL) under the supervision of Jan Fransoo, Nesim Erkipand Tom van Woensel, the results of which are presented in this dissertation. Thecooperation with Nesim Erkip was initiated in 2007 during a short visit to BilkentUniversity, Ankara, Turkey. As of January 2011, Alina is working as a consultant atLIME BV in Eindhoven (NL).

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