Retail Inventory Management with Lost Sales

Retail Inventory Management with Lost Sales

This thesis is number D147 of the thesis series of the Beta Research School for Operations Management and Logistics. The Beta Research School is a joint effort of the departments of Industrial Engineering & Innovation Sciences, and Mathematics and Computer Science at Eindhoven University of Technology and the Centre for Production, 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, Eindhoven Cover designed by Paul Versparget

Retail Inventory Management with Lost Sales PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 23 januari 2012 om 16.00 uur door Alina Curşeu geboren te Alba-Iulia, Roemenië

Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. J.C. Fransoo en prof.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, have offered support over the past few years. First, I would like to thank my supervisors, prof. Jan Fransoo and prof. Tom van Woensel from TU/e, for giving me the opportunity to carry out this research. It certainly opened the door for many learning experiences. I thank prof. Fransoo for his 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 to my co-promotor Tom van Woensel. I also wish to acknowledge his feedback, valuable comments and view on retail problems, and his relentless involvement throughout the different phases of this project. Second, I owe special thanks to my second promoter prof. Nesim Erkip from Bilkent University who kindly accepted to join this project. This dissertation benefited a lot from his advice and constructive ideas over the years, his prompt and careful feedback on different parts of this dissertation. Thank you also for hosting my short visit to Bilkent University. For their willingness to serve on my doctoral committee, as well as for their time dedicated to evaluate this dissertation, I would like to thank prof. Stefan Minner from University of Vienna, prof. René 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 stimulating academic environment, to the fellow PhD students for providing peer support, and to the OPAC secretaries for their continuous assistance. In particular, I wish to thank Ola Jabali for many open and friendly discussions, and her lively support over the years. I am also thankful to my successive officemates Ingrid Vliegen and Youssef Boulaksil for their company, to Gergely Mincsovics for many fruitful discussions on MDPs and not only, and to Ingrid Reijnen for her support and a couple of extremely nice photos to treasure. On a personal note, I wish to thank all old and new friends over the years. I thank my old 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 the years. Life would have been very different without the Romanian community in the Netherlands. I thank you all for many familiar discussions, many birthday and children parties that energized the work-life balance. I am also fortunate to having made wonderful Dutch friendships. Thank you Sandra for your continuous support and for being part of my family s life. Thank you Apinya and Carel for our growing friendship over the years. Most of all, I am deeply thankful to my family, for their unconditional love and support 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 doubtful times. This dissertation would not have been possible without his boundless love and support. Finally, I am deeply grateful for the two wonderful children in my life, Dragoş and Antonia. Thank you both for changing my life so beautifully! Alina Curşeu November 2011, Eindhoven

Contents 1 Introduction 1 1.1 Motivation and objective.......................... 2 1.2 Scope of the dissertation.......................... 4 1.3 Research questions and contributions of the dissertation........ 8 1.3.1 Modeling handling operations in grocery retail stores...... 8 1.3.2 Lost-sales inventory models with batch ordering and handling costs................................. 9 1.3.3 Retail inventory control with shelf space and backroom consideration................................ 10 1.3.4 Efficient control of lost-sales inventory systems with batch ordering and setup costs...................... 11 1.4 Outline of the dissertation......................... 12 2 Modeling handling operations in grocery retail stores: an empirical analysis 15 2.1 Introduction and related literature.................... 15 2.2 Conceptual model and hypotheses development............. 18 2.3 Study design and data description.................... 21 2.4 Results.................................... 22 2.4.1 Sequential regression results.................... 24 2.4.2 Overall regression results..................... 28 2.4.3 Validation of the results...................... 29 2.5 Analytical insights and implications for retailers............. 32 2.5.1 Extending the EOQ-model with shelf stacking......... 32 2.5.2 Order of magnitude for efficiency gains in stacking....... 33 2.6 Conclusions................................. 36 Appendix A. Shelf stacking activities...................... 37 Appendix B. Descriptive statistics of the empirical datasets......... 37 Appendix C. Validation results for chain B.................. 37 3 Lost-sales inventory models with batch ordering and handling costs 45 3.1 Introduction................................. 45

3.2 Literature review.............................. 47 3.3 Mathematical model............................ 50 3.4 Old and new heuristics........................... 54 3.4.1 The (s, S, nq) and (s, Q, nq) policies............... 54 3.4.2 The (s, Q S, nq) policy....................... 54 3.5 Numerical study.............................. 55 3.5.1 On the structure of the optimal policy.............. 56 3.5.2 Sensitivity analysis......................... 58 3.5.3 Performance of the (s, Q S, nq) heuristic............. 62 3.6 Penalty for not taking handling into account.............. 67 3.7 Conclusions................................. 69 Appendix A. On unit vs. batch costs...................... 71 Appendix B. Computational issues....................... 72 4 Retail inventory control with shelf space and backroom consideration 75 4.1 Introduction................................. 75 4.2 System under study............................ 78 4.3 Model formulations............................. 80 4.3.1 Model with continuous backroom operations.......... 81 4.3.2 Model with fixed extra handling costs.............. 83 4.4 Numerical study: the model with continuous backroom operations.. 85 4.4.1 On the structure of the optimal policy.............. 85 4.4.2 Sensitivity analyses: the effect of V, K s and p.......... 86 4.4.3 Managerial insights......................... 92 4.4.4 Summary.............................. 95 4.5 Numerical study: the model with fixed extra handling costs...... 98 4.5.1 On the structure of the optimal policy.............. 99 4.5.2 Sensitivity analyses: the effect of V, K e and p.......... 101 4.5.3 Managerial insights......................... 106 4.5.4 Summary.............................. 109 4.6 Conclusion................................. 109 Appendix A. Transition probability matrix.................. 110 Appendix B. Additional numerical results................... 111 Appendix C. Related models.......................... 113 5 Efficient control of lost-sales inventory systems with batch ordering and setup costs 115 5.1 Introduction and related literature.................... 115 5.2 Model.................................... 120 5.3 On the structure of the optimal policy: K = 0 vs. K > 0....... 122 5.3.1 The case L = 0........................... 122 5.3.2 The case 0 < L < 1......................... 124 5.3.3 The case L = 1........................... 129

5.4 The effectiveness of the (s, Q S, nq) policy................ 132 5.4.1 The (s, Q S, nq) policy....................... 132 5.4.2 Effectiveness of the (s, Q S, nq) policy.............. 134 5.5 Performance of the (s, nq) policy..................... 139 5.5.1 Numerical results: L = 0..................... 140 5.5.2 Numerical results: L = 1..................... 141 5.5.3 Numerical results: 0 < L < 1................... 142 5.6 Conclusions................................. 143 Appendix A. Selected numerical results.................... 144 Appendix B. Approximate (s, Q S) policies.................. 148 6 Conclusions 153 6.1 Results.................................... 153 6.2 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 right place at the right time and at the right price (Fisher et al., 2000), many retailers concentrate on improving the efficiency of their operations. Efficient management of store operations is crucial to the retailer s own success (Pal and Byron, 2003) and often critical for the performance of the entire supply chain. Many believe that the last 100 feet of the supply chain from store receipt to the shelf represent both the highest supply chain cost and the biggest customer service risk (Supply Chain Effectiveness Survey, 2002). Typical tactical and operational decisions retailers face in managing their stores refer to product assortment and variety (which products to store), location and shelf-space allocation in the store (where and how much space should be assigned to each stock keeping unit (SKU)) and products replenishment (when and how much to reorder of each SKU). An essential objective for most retailers is to provide a high availability of their products at low operational costs. This ultimately challenges retailers to formulate 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 store operations: merchandise handling and inventory management. Decisions regarding the assortment of products, location and allocation of shelf space are outside our scope 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 presented further on.

2 Chapter 1. Introduction 1.1. Motivation and objective We place our research in the context of the grocery industry, a sector that contributes considerably to the total sales volume in retail (Guptill and Wilkens, 2002). European grocery retailing is an extremely competitive sector with high operating costs and low profit margins. One natural way to remain competitive is to reduce and/or manage costs in key areas. Proper control of store operating expenses typically requires balancing transportation, inventory, shelf space and handling costs. Currently, models that assess the overall operational costs in retail stores on multiple dimensions are not available. Much of the academic model-based research in retail operations has focused 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 considered explicitly. There is also a general lack of understanding of what drives handling costs in 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 better understanding of the main drivers of handling costs in retail stores. Second, our objective is to integrate inventory, handling and shelf space into a single model for analysis and optimization of replenishment decisions. Limited published evidence exists on store level processes (Falck, 2005, Småros et al., 2004) and even more limited research exists on models of in-store processes (Kotzab and Teller, 2005). For most retailers, in-store handling operations are not only labourintensive processes, but also very costly. An empirical study of Saghir and Jönson (2001) found that 75% of the total handling costs in a grocery retail chain occurs in the store. In another study, Broekmeulen et al. (2004) showed that the handling costs at the store level clearly dominate the other operational cost components in the retail supply chain consisting of the retail distribution center and the store (see Figure 1.1). Statistics from the Food Marketing Institute also suggest that labor costs make up more than 57% of store operating costs. However, academic approaches towards modeling of in-store handling operations and integration with operational decisions such as inventory management are rare. This dissertation explores these opportunities. Inventory management remains a key strategic weapon for many retailers. The academic research and studies on inventory management systems is rather abundant (see Silver, 2008 for a review). However, much of the research remains rather theoretical and there is still a gap between theory and practice. For example, many of the retail inventory management models and methods use assumptions that were developed for application areas other than retailing. For instance, it is often assumed that unmet demand is backordered (i.e., customers wait for the unavailable stock to be replenished), while in traditional retailing, unmet demand is typically lost. The consumer behavior studies reveal that when a product is out-of-stock, the customer typically buys a substitute product or visits another store. A study by Gruen et al.

1.1 Motivation and objective 3 Transportation cost (DC to store) 22% Inventory costs in store 7% Handling costs in store 38% Handling costs in DC 28% Inventory costs in DC 5% 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 when confronted with an out-of-stock situation, whereas the remaining 85% will either buy a 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 a direct result of inadequate store ordering and shelf restocking practices. Hence, many opportunities for improvement exist in these areas. Do not purchase item 9% Buy item at another store 31% Substitutedifferent brand 26% Substitutesame brand 19% Delay purchase 15% 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 a periodic 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. The orders placed in the morning are often received at the end of the day, or the beginning of the next day. In any case, the replenishment lead time is typically shorter than

4 Chapter 1. Introduction the length of the review period. Furthermore, the orders are usually constrained to batches of fixed sizes (the case packs), generally dictated by the manufacturer from the need to coordinate inventory and transportation of several products. Upon order receipt at the store, the replenishment stock needs to be stacked on the shelves, and this 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 characteristics are rarely taken into account in one comprehensive model, in the analysis of lost-sales inventory systems, as our literature review in the following chapters reveals. This dissertation also aims to contribute to the literature on single-location, single-item lost-sales inventory theory. More details about the scope of the dissertation, research questions 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 the grocery retail sector, in particular dry groceries. Our research mainly revolves around inventory management and shelf stacking practices at store level. The relationship between the store and the retailer s distribution center is not taken into account. The focus of the dissertation is on developing and solving single-location inventory control models 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 and sells them to consumers. In doing so, many logistics activities are carried out and need to be coordinated within a retail outlet from an incoming dock to the check out counters. A schematic representation of the general flow of goods in a retail store, 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 packs to end-consumer units, shelves (re)filling, or merchandise presentation. We leave outside our scope activities related to order receipt and check out. We also distinguish between the sales floor and the store backroom, and assume that the retailer uses the backroom to temporarily stock additional inventory that does not fit in the regular shelves upon delivery. However, questions on how should a retailer best operate his backroom are not addressed here. All store processes depend on stochastic end-customer demand, and their ultimate goal is efficiency, which means to satisfy the amount of items as requested by endcustomers at the lowest costs possible. Two types of costs are considered relevant in this dissertation: (i) inventory-related costs (for ordering, for holding products on stock, 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 Backroom Second replenishment process Incoming stock Receipt First replenishment process Shelves Check out Figure 1.3 Generic flow of goods in a retail store with stock from the backroom). As shown by Broekmeulen et al. (2004), handling costs (mostly shelf stacking) are much higher than inventory holding costs at store level. The novelty in this dissertation consists in the explicit consideration of handlingrelated costs in the optimization of inventory decisions. Since we target operational costs, assortment or shelf space related costs are not considered. Also, we focus on non-perishable products, thus obsolescence costs are not taken into account. To control inventories, we focus on the product level, and the main question under study is when and how much to order of a particular item to satisfy end-customer demand at minimum total costs. To address this question, we consider several inventory control problems, which retain a number of features that are observed in the replenishment practices of grocery retailers. Products are facing stochastic customer demand and excess demand is lost, rather than backordered. Two main assumptions about the consequences of stock outs prevail in the stochastic inventory control theory: backordering of unmet demand and lost sales, respectively. Models under the second assumption received far less attention in the literature, mainly because they are analytically less tractable than the backorder models. For the backorder models, simple classes of replenishment policies are proven to be optimal, but the results do not extend in general to lost-sales models. In this dissertation, we focus on stochastic lost-sales inventory control models in various settings. The replenishment decisions are taken under periodic review. Although continuous inventory monitoring (i.e. continuous review) could be justified by the emergence of technologies such as RFID (radio frequency identification) at item level (Metzger, 2008), retail stores usually prefer to inspect their inventories periodically at regular intervals. This offers them the opportunity to coordinate the replenishment and transportation of different products. Our research concentrates on periodic review inventory management systems. Many products are replenished by the store in case packs (batches), each pack

6 Chapter 1. Introduction containing several units of the product. The nature and size of the packs are usually determined by the supplier or manufacturer due to storage, transportation and handling rather then inventory considerations. Therefore from an inventory management perspective, the case pack size is a fixed rather then a decision parameter for the retailer. Depending on the store format and product type, products are displayed on the shelves in their original package, or case packs are first broken down into individual consumer units. The particular application of interest in this dissertation is where stock is supplied to the stores in pre-packed form, and the retailer displays individual units on the shelves in response to end-consumer demand. This distinction between the unit of demand and supply is rarely taken explicitly into account in the development of inventory control policies (Hill, 2006). The packaging design problem and its impact on the store s logistics processes is outside our scope (see, e.g., Hellström and Saghir, 2007, Van Stipdonk, 2007). In the literature, the problem with order quantities that are restricted to be integer multiples of the batch size is typically referred to as batch ordering. We consider the feature of full batch ordering (partial batches are not allowed, see for example Alp et al., 2009) and investigate the impact of different batch sizes on the performance of the inventory systems. Another characteristic of the inventory replenishment process in the grocery sector is that the time between placing an order and receiving it, called the replenishment lead time, is typically shorter than the review period length. This feature is also referred to as fractional lead time. The majority of inventory models in the literature assume that 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. In this 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 has to allocate limited shelf space among many different products, and the distribution of appropriate amounts, together with their location, is indicated in the so called planogram. The allocation of shelf space is typically decided at a tactical level, considering many marketing variables, and aims at stimulating customer purchases and maximizing profits (see e.g., Hübner and Kuhn, 2010 for a review on assortment and shelf space planning models). For inventory replenishment decisions, the shelf capacities at product level are usually predetermined. Due to insufficient shelf space, part of the retailer s assortment is split between the sales floor and the backroom. Additional handling operations can be expected in transferring stock from the backroom to the sales floor. Since the retailer costs are sensitive to in-store merchandise handling, we consider the effect of using the backroom on the combined cost 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 controlling a retailer inventory, which we do not address in this dissertation. Among others, the non-stationarity of the demand process (in particular cyclic demand patterns are common 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 and Smith, 1996, Raman and Zotteri, 2000), the efficient measuring of stockouts (Corsten and Gruen, 2004), the accuracy of inventory records (Raman and DeHoratius, 2001, Atali et al., 2009) and implications for inventory management (Kök and Shang, 2007, DeHoratius et al., 2008), the management of promotion items (Huchzermeier et al., 2002), or RFID applications (Lee and Özer, 2007). In short, in this dissertation we study stochastic single-item inventory control models under different valid assumptions inspired from the replenishment practice of grocery retailers. Optimal as well several alternative inventory control policies are considered throughout the dissertation in a multi-period inventory setting. The main definitions and notation are introduced next. Inventory control policies All inventory policies are studied under periodic-review, with R the fixed review period length. For notation expediency, we shall omit R in all policy notations. The well-known (s, Q) and (s, S) policies (Zipkin, 2000) with fixed (Q) or variable order size (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 at grocery retailers resembles an (s, nq) policy (Van Donselaar et al., 2009), which is also 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 period is less than or equal to s, order the largest integer multiple of q which results in an inventory position less than or equal to S. The (s, Q, nq) policy: The (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 that the order size Q is a nonnegative integer multiple of q. The (s, Q S, nq) policy: The newly proposed (s, Q S, nq) policy has three parameters s, S and Q with 0 max{s, Q} S s + Q. Under this policy, the order quantity in each period depends on the beginning inventory on hand x and is given by Q/q q if 0 x S Q/q q a(x) = (S x)/q q if S Q/q q < x s 0 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 minimum integer multiple of q such that, after ordering, the inventory will rise at or above s.

8 Chapter 1. Introduction The remainder of the chapter presents the main research questions and contributions of the dissertation, outlined by chapter. 1.3. Research questions and contributions of the dissertation Traditional store-based retailing heavily relies on two operations: inventory replenishment and merchandise handling. Therefore, many opportunities for improvement in these operations exist, once their most important characteristics and drivers are well understood. In this dissertation, we explore such opportunities in several studies. First, we focus on the shelf stacking process in retail stores, aiming for a better understanding of the specifics of this process. Then, we develop and solve several lostsales inventory control models, which take into account key characteristics of the retail environment: batch ordering, handling costs, shelf space and backroom operations. In the following, we discuss the specific research questions and methodologies for each individual 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 with products from new deliveries. Chapter 2 presents an empirical study of the shelf stacking process in grocery retail stores. We examine the complete process at the level of individual sub-activities and study the main factors that affect the execution time of this common operation. In Chapter 2 we address the following research question: 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 statistical analyses 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 waste disposal. Further, we find that the three most time-consuming sub-activities are stacking new inventory, grabbing/opening a case pack, and waste disposal. Based on the insights from the different sub-activities, a prediction model is developed that allows estimating the total stacking time per order line 1, solely on the basis of the number of case packs and consumer units. The model is tested and validated 1 An order line typically contains a number of consumer units from a specific article, or Stock Keeping 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 tool for evaluating the workload required for the usual shelf stacking operations. Finally, we illustrate the benefits of the model by analytically quantifying the potential time savings in the stacking process, and present a lot-sizing analysis to demonstrate the opportunities for extending inventory control rules with a handling component. These opportunities are further explored in Chapters 3 and 4. 1.3.2 Lost-sales inventory models with batch ordering and handling costs Empirical evidence shows that handling costs are relatively high in the grocery retail supply chain. Van Zelst et al. (2009) reported that the handling of goods in the store accounted for 75% of the total logistics store costs, while inventory accounted for the remaining 25% of total costs. However, handling-related costs are rarely acknowledged in inventory replenishment decisions. In this research, we aim to bridge this gap by explicitly recognizing the handling costs (for shelf stacking of replenishment orders) at the retailer as a critical cost component, and integrating inventory and handling into 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 in decision making? And what is the impact of adding this aspect on the overall system performance? In Chapter 2, we showed that we can reliably estimate the handling time per Stock Keeping Unit (SKU) required to execute the shelf stacking operation using an additive model (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, we assume a similar structure for the shelf stacking costs. This leads to a replenishment cost structure that allows economies of scale. For example, the retailer could decide to order less frequently but in a larger number of case packs in order to reduce the handling costs. However, less frequent deliveries lead to an increase in the average 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 restricted to integers multiple of a fixed batch size q, and the lead time is assumed to be less than the review period length. The objective of the system is to minimize the long-run average costs. We use stochastic dynamic programming to model and solve the inventory control problem. Since optimal policies have a rather complex structure, we propose a heuristic policy, referred to as the (s, Q S, nq) policy, which partially captures the

10 Chapter 1. Introduction structure of optimal policies and shows close-to-optimal performance in many settings. We also benchmark the performance of the heuristic against two reasonable alternative policies, the (s, S, nq) and (s, Q, nq) policy, and quantify the overall improvement. Furthermore, we present several insights from sensitivity analyses regarding the impact of problem parameters, in particular the batch size and material handling costs, on the system s performance. Finally, we show that material handling costs may substantially affect the overall system s performance, when ignored, especially for items with low profit margins. 1.3.3 Retail inventory control with shelf space and backroom consideration In Chapter 4, we extend the retail setting from Chapter 3 to include another realistic dimension of the retailer s inventory decision, namely the shelf space. As we mentioned earlier, the shelf space allocation for each product is typically dictated by marketing constraints (Dreze et al., 1981) and comes as a result of planograming decisions (Corsten and Doyle, 1981, Yang, 2001). Consequently, for operational decisions such as inventory replenishment, the shelf space is often an exogenous parameter. We also make this assumption here, and consider therefore that the available shelf space has been predetermined. We study the inventory management problem for a single product under similar assumptions to those formulated in Chapter 3. We extend the setting to include the following situation: upon arrival at the store, the replenishment stock has to be stacked on the shelves to serve end-customer demand. This operation is part of the shelf stacking process at the store (also referred to as the first replenishment process). Since shelves have limited capacity, store managers keep surplus stock that did not fit on the shelves temporarily in the store s backroom, which creates the need for a second restocking of the shelves (referred to as the second replenishment process). In turn, this translates into additional handling costs for the retailer. The following research questions are addressed in Chapter 4: How could inventory control models be adapted to account for shelf space limitations and use of the backroom? And what is the impact of including these features on the performance of the inventory control models? In Chapter 4, we adapt the lost-sales inventory model studied in Chapter 3 to include shelf space constraints and additional costs associated with the second replenishment process. Two models are developed, using stochastic dynamic programming: the first one assumes linear extra handling costs and continuous backroom operations, while the second one assumes that a fixed cost is charged to the system for exceeding the allocated shelf capacity. These costs are charged in addition to a fixed cost for placing an order, which further increases the system complexity. In a numerical study, we discuss 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 and the associated long-run average total costs. Furthermore, we measure the retailer s benefit of accounting for additional handling operations, as the marginal cost decrease the retailer may achieve relative to the case where neither the shelf space, nor the extra handling costs are included in the optimization of inventory decisions. Several interesting managerial insights into the tradeoff between the different cost components are also illustrated. 1.3.4 Efficient control of lost-sales inventory systems with batch ordering and setup costs In Chapter 5, we consider a variant of the classical periodic-review, lost-sales stochastic inventory problem, which has the following features: batch ordering and fractional lead time. We assume the standard cost structure, with a zero or fixed setup cost for ordering, and the objective is to minimize the long-run average cost of the system. Lost-sales inventory models are known to be analytically more challenging than their backorder counterparts (Hadley and Within, 1963), therefore various heuristics have been proposed in the literature to address this issue (Zipkin, 2008a, Nahmias, 1979). The features of batch ordering, fractional lead time and setup cost have been studied less frequently in the literature, especially in an all embracing model. Our research addresses this deficit and in Chapter 5, we aim to answer the following research questions: Can we derive an efficient heuristic to control the single-item lost-sales inventory problem 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 the optimal policies, in an extensive computational study. We provide numerical evidence to 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 optimal solution, 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 is adjusted to account for batch restrictions on the order quantity (when q > 1). In many instances, it even captures the true structure of the optimal policies. We also compare the performance of the heuristic with those commonly used, and demonstrate its superiority and effectiveness. In particular, we find that the best (s, S) policies are performing 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 with small penalty costs. Finally, we test the performance of the (s, nq) policy, often implemented in automated ordering systems at grocery retailers (Van Donselaar et al., 2009), and show that it may result in substantial cost increase when implemented

12 Chapter 1. Introduction in the presence of fixed setup costs, which is often the case in grocery stores due to the presence of fixed handling costs. Finally, in Chapter 6, we conclude the dissertation and discuss several future research directions. Concisely, in this dissertation, we develop adapted inventory control models, and design new solution approaches which take more effectively into account different retail characteristics. First, we build upon an empirical study and derive a formal model of handling costs, which includes fixed and variable components. Then, we study the performance of a lost-sales inventory system where we explicitly recognize the handling costs as a critical component, in addition to a standard cost structure. We show that optimal policies have quite complicated structures and propose a new heuristic policy, which is intuitive for practitioners, shows close to optimal performance, and proves superior to reasonable alternative policies. We also give several managerial insights from sensitivity analyses and quantify the added value of handling costs in the decision making. Next, we extend the model to account for limited shelf space and the cost of handling backroom stock, which leads to a more realistic system perspective for reordering decisions and enhanced cost control capabilities. Finally, we study a variant of the single-item lost-sales inventory model with standard cost structure to reinforce the proposed heuristic. While the inventory control models presented in this dissertation have been inspired by grocery retailing, the analysis, solution techniques and insights may be applicable to other settings, where the unsatisfied demand is lost, there is a non-unit size of stock transfer and there are economies of scale in the replenishment cost structure. 1.4. Outline of the dissertation Table 1.1 summarizes the different characteristics of the retail environment, as considered in each chapter of this dissertation. Each chapter of the dissertation is self contained and can be read independently. The research presented in Chapter 2 appeared also as Curşeu et al. (2009a) and the research described in Chapters 3, 4 and 5 is based upon Curşeu et al. (2009b) and Curşeu et al. (2010a,b), respectively.

1.4 Outline of the dissertation 13 Table 1.1 Dissertation outline: main features by chapter Chapter Comment Shelf stacking 2 Empirical study X 3 New heuristic testing Impact of handling 4 Numerical optimization Impact of shelf space 5 Numerical optimization Heuristics testing X Inventory control X Shelf space and backroom X X X X

15 Chapter 2 Modeling handling operations in grocery retail stores: an empirical analysis Abstract: Shelf stacking represents the daily process of manually refilling the shelves with products from new deliveries. For most retailers, handling operations are labourintensive and often very costly. This chapter presents an empirical study of the shelf stacking process in grocery retail stores. We examine the complete process at the level of individual sub-activities and study the main factors that affect the execution time of this common operation. Based on the insights from different sub-activities, a prediction model is developed that allows estimating the total stacking time per order line, solely on the basis of the number of case packs and consumer units. The model is tested and validated using real-life data from two European grocery retailers and serves as a useful tool for evaluating the workload required for the usual shelf stacking operations. Furthermore, we illustrate the benefits of the model by analytically quantifying the potential time savings in the stacking process, and present a lot-sizing analysis to demonstrate the opportunities for extending inventory control rules with a handling component. 2.1. Introduction and related literature In today s highly competitive market environment many retailers are concentrating on controlling costs, as a means of achieving operational excellence and their business success 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, retail operational costs include the cost for inventory, shelf space, transportation, and handling. Existing research in retail operations mainly concentrates on inventory, marketing, or planograming decisions separately (see e.g., Corstjens and Doyle, 1981, Drèze et al., 1994, Urban, 1998, Cachon, 2001, Hoare and Beasley, 2001). Store handling operations and associated costs are often not modeled explicitly in these studies (see e.g. Themido et al. 2000, where handling costs are treated in an aggregate way). This research focuses exclusively on the handling cost component of retail operations, 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 incurred in the part of the retail supply chain that includes the retailer s distribution center and the store (given assortment and shelf space allocation ). They identify inventory holding, transportation (from the distribution center to the store) and handling (order picking in the warehouse and shelf stacking in the retail store) as relevant cost components, and find that in-store handling costs represent the largest share of operational costs, accounting for more than one third of total operational costs. Another empirical study by Saghir and Jönson (2001) suggests that 75% of the total handling time in a grocery retail chain occurs in the store, and investigates how packaging evaluation methods may assist in reducing the total handling time. There is however, in general, a lack of understanding of what drives handling costs in 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 process in grocery retail stores and study the key factors that drive the execution time of this store operation. Shelf stacking represents the daily process of manually refilling the shelves in the store with products from new deliveries. As with most manual activities, such processes are often time consuming and costly. Furthermore, unless clear and reliable work standards are implemented, such activities may well suffer from 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 contributions of this chapter are threefold. First, we examine the shelf stacking process at the level of individual subtasks and analyze the impact of different drivers (e.g., number of case packs and consumer units, etc.) on the individual shelf stacking times, as well as the total stacking time. For the retail practice, we offer a better understanding of the distribution of workload in shelf stacking to the individual sub-activities, while we specifically recognize those sub-tasks that are mostly influenced by the key drivers identified, as compared to those for which the variation in workload is potentially affected by other factors. This has further implications for identifying inefficiencies in 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 multiple regressions, a prediction model is developed, which allows estimating the shelf stacking time to a large extent only on the basis of the number of case packs per order line and the number of consumer units (measures which are readily available). In contrast to common assumptions in the literature, we find that an additive, rather than a linear structure is appropriate for describing the specific relationship. Real-life data was used to test the model and to assure it has face validity, which is relevant for its general applicability to other settings. Thirdly, we investigate the potential of the prediction model for a better estimation and control of the overall logistical costs. Closed-form analytical expressions for expected efficiency gains are investigated to quantify the potential gains that could be achieved. Moreover, we present a lot-sizing analysis to illustrate the benefits of the model in extending currently available inventory control models with a handling component. This idea is further explored in Chapters 3 and 4. The remainder of the chapter is organized as follows: first, we give a brief overview of related literature; then in Section 4.3, we describe the shelf stacking process and derive a conceptual model for estimating the time required to fulfil this common store activity. Section 2.3 introduces the methodology we used to test the proposed model and describes the datasets supporting our analyses. Section 2.4 presents the results of our 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 for research in the field of store handling operations. An early study that considers both inventory and handling costs comes from 1960s (Chain Store Age, 1963). SLIM (Store Labor and Inventory Management), a system widely promoted in the mid-1960s, focused on minimizing store handling expense, by reducing backroom inventories and the double handling of goods (Chain Store Age, 1965). Two other studies carried out by the Swedish group DULOG in 1976 and 1997, measured package handling time in the store, in order to gather information about the impact of the type of package on handling efficiency in the grocery retail supply chain (DULOG, 1997). Time-study approaches are sometimes reported in the warehouse operations research for estimating order-picking times. Gray (1992) uses basic multiple regression to derive estimations of the necessary time to pick all items from a pick list for a customer order, and applies it for establishing labour productivity standards. Gray et al. (1992) consider the general problem of warehouse design and operation, and propose a model in which order-picking time includes three components: walking, stopping and grabbing. Varila et al. (2007) uses order-picking in a warehouse as a case activity to illustrate, using regression analysis, that a time-based accounting system is often suitable in tracing the cost behavior of an activity, especially when this

18 Chapter 2. Modeling handling operations in grocery stores is directly proportional to time. Their work is similar in objective to the time-driven ABC, a concept recently introduced by Kaplan and Anderson (2004) as a simpler and more accurate alternative to the traditional ABC systems. However, in retailing literature, time-studies are rarely reported. Recently, Van Zelst et al. (2009) showed that significant efficiency in terms of shelf stacking time could be gained once the impact of most important drivers is well understood. This chapter supports their main findings. Although both studies inherently start from the same underlying empirical dataset, a number of significant differences in this chapter are identified compared to Van Zelst et al. (2009). First, the current chapter explicitly focuses its analysis on the order line level 1, rather than the consumer unit level as in Van Zelst et al. (2009). Focusing on the consumer units involved 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 problems associated 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) towards the individual sub-activities and we support our results with extensive testing and validation. Finally, we follow an analytical approach to illustrate the benefits of our findings for the practice of retailers. This latter involves both analytically deriving gains in terms of handling and the consequence of incorporating the handling function into a lot sizing decision model. 2.2. Conceptual model and hypotheses development Generally, a store undertakes the following replenishment process: upon arrival of a new shipment, the truck is unloaded; next, the store clerks move the deliveries into the store and then restock the shelves with the newly arrived products. The shelf stacking process defined in this research starts after the incoming products are moved into the store and are taken to the shelves area (usually by rolling containers), in order to be stored on the shelves. Therefore, neither the walking with the rolling container in the store, nor the replenishment process from the backroom and the corresponding time delay are part of the defined shelf stacking process. Furthermore, we focus on products that are replenished in pre-packed form but presented to the final consumer in individual units. This situation is typical for a large part of the assortment of most retailers (we consider here dry groceries, and products which are comparable in terms of the stacking process and productivity). For each Stock Keeping Unit (SKU), the store clerks unpack the product and stock the consumer units on the shelves at the assigned location (as indicated in the planogram 2 ). An important sub-activity in this process is shelf maintenance: the store clerks need to check the best before date of the products on the shelf and 1 An order line typically contains a number of consumer units from a specific article or stock keeping unit (SKU) 2 The 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 facilitate First-In-First-Out retrieval or proper shelf display. For each SKU, the shelf stacking process ends with disposing the empty case packs. Although inherently not a complex task, the shelf stacking process is manually executed and thus may suffer from a lot of variation. If time drives costs, then it becomes valuable to understand what drives time. We are particularly interested in estimating the Total Stacking Time per order line (T ST ) (i.e. for each individual SKU), based on a set of underlying factors, given a specific inventory replenishment rule, assortment, shelf space and package. To better examine the causes and effects of time variation, we examine the total stacking process at the levels of individual subtasks. Breaking down the entire operation into small components allows, on the one hand, an assessment of the contribution of each individual sub-activity to the T ST, and on the other hand, a better indication of the potential variables affecting the T ST. 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 this research. Few remarks are worthwhile. Because grabbing or opening a case pack were difficult to separate, these activities were measured together. Also, the walking subactivity does not include walking with the rolling container to the right aisle within the store or between aisles, but occasionally does include moving the rolling container to the right shelf location in case of heavy products, for example. The difference between filling old versus new inventory is relevant as depending upon the inventory level just before filling, the activity filling old inventory will become important for higher inventory levels. The total stacking time per order line (T ST ) has been divided accordingly into seven time components and the key variables that could logically influence the execution time of each subtask are identified. It is expected that the time needed to stack new 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)

20 Chapter 2. Modeling handling operations in grocery stores grabbing and unpacking a case pack, traveling within the shelf aisle to and from the right location, or disposing the wasted case packs depend on the number of case packs being handled per order line. Lastly, searching for the right shelf location, preparing the shelf or restacking old inventory if necessary, are normally executed only once, for each SKU, independent of the number of case packs or consumer units. The set of potential 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 x 2 Searching (S) - - - 3 Walking (W) x - x 4 Preparing (P) - - x 5 Filling New Inventory (Fn) x x x 6 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 the shelf stacking time (such as SKU volume, weight or type of packaging, the distance traveled within the aisle, the old inventory position just before new replenishment, the labour, the environment, etc.; recently, Hellström and Saghir (2007) investigated the relationship between the packaging system and logistics processes in the retail supply chain). Herein, we concentrate only on order line-related (number of case packs (CP ) and number of consumer units (CU) per order line,) and product-related characteristics (product category) as the key drivers of time variation of the shelf stacking process. The product subgroup variable captures any time variation that could be attributed to differences in product-related characteristics not measured specifically 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 or consumer units) being handled, and is thus an appropriate cost driver, while the product information approximates the difficulty in handling products from different categories. These variables are selected as potential predictors in our subsequent analyses. The dependent variables are the individual times per sub-activities (T s, with s {G, S, W, P, F n, F o, D}) and the Total Stacking Time (T ST ), all expressed in seconds. The explanatory variables are hypothesized to have the following influence on the execution time of each sub-activity: Hypothesis 1 The number of case packs (CP ) has a positive effect on the individual times T G, T W, T F n, T D and T ST.

2.3 Study design and data description 21 Hypothesis 2 The number of consumer units to be stacked (CU) has a positive effect on sub-activities execution times T G, T F n and T ST. We expect that CP and CU have no significant effect on T S, T P, and T F o. Under these hypotheses, the Search, Prepare and Fill Old sub-activities could be regarded as fixed activities, while only the remaining activities are variable, depending on the set of hypothesized factors. 2.3. Study design and data description Two grocery retail chains (denoted here by A and B) agreed to participate in this study. Empirical data on the stacking process was collected using a motion and time study 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 for each supermarket chain), nine experienced employees, familiar with the operations, were videotaped during the shelf stacking process. The product subgroups (all dry groceries) 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 than one 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 (no seasonal changes or promotions) Finally, we note that the data collection period did not include days with peak or dropping 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 Stacking Time per order line (T ST ) was registered using a computerized time registration tool, and results were entered into a database. Additional information necessary to identify the stacking process for each order line was added as well, such as the SKU type, the number of case packs and case pack size per order line, and the product category each SKU belongs to.

22 Chapter 2. Modeling handling operations in grocery stores The final dataset contains 1048 observations, for chain A, across nine different product categories, and 563 observations, for chain B, across five different product categories. Tables B1 to B4 (Appendix B) contains descriptive statistics of the variables used in this study. The average total time to stack an order line into the shelves is 57.31 seconds, ranging from a low 10 seconds per order line (personal care category) to a high 334 seconds (coffee), with a standard deviation of 36.6 seconds. This reveals the degree of variation that exists in the T ST between different order lines and this study aims at gaining a better understanding of the factors causing this variation. We further note that some degree of variation exists also between the T ST corresponding to different product categories. The average T ST varies between 35.47 seconds (products of personal care) and 80.86 seconds (coffee milk). With reference to the explanatory variables of this study, we note that the average number of case packs per order line varies between 1 CP (all categories) to 9 CP (coffee), with an average of 1.3 CP and a standard deviation of 0.7 CP. The average number of consumer units per order line exhibits 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 Time and the relative contribution of each individual sub-activity to the T ST, as illustrated in Figure 5.1. We note that the most time consuming sub-activity in the shelf stacking process is the Stacking of new inventory (Fn) (about 48% of the T ST ), followed by the Grabbing and unpacking the case packs (G) (about 20% of T ST ) and Disposing the waste (D) (about 13% of T ST ), respectively. Together, they account for almost 81% of the T ST. Tables B2 and B4 (Appendix B) provides descriptive statistics of the dependent variables used in this study. The corresponding average times for execution of the three most time consuming sub-activities are 27.32 seconds (Stacking new inventory), 11.65 seconds (Grabbing/opening a case pack) and 7.28 seconds (Disposing waste), respectively. 2.4. Results In order to test our hypotheses, we performed several separate regression analyses with T s (s {G, S, W, P, F n, F o, D}) and T ST as dependent variables, and CP, CU and product category as the independent ones. We adopt two different strategies for estimating the Total Stacking Time per order line (T ST ), which we refer to as sequential regression and overall regression, respectively. Both approaches allow predicting the T ST as a function of the identified drivers using multiple linear regressions. However, the two approaches have different practical purposes. While the first approach reveals detailed information regarding the causes of variation in each subtask, as explained by the hypothesized variables, and provides more information regarding the contribution of each individual sub-activity to the variability of the entire process, the second approach is selected as a simple, less expensive alternative for practical forecasting of the T ST. The sequential regression starts from the

2.4 Results 23 Stack new inventory 48% Grab and unpack case 20% Sub-activity Dispose waste Walking Prepare the shelf 6% 8% 13% Search 4% Stack old inventory 1% 0% 10% 20% 30% 40% 50% 60% % of total shelf stacking time Figure 2.1 Distribution of the total shelf stacking time (Chain A) following functional form: T ST = s A T s, A = {G, S, W, P, F n, F o, D}, (2.1) where the duration of each individual sub-activity, T s per order line is estimated using the following general linear regression model: P C 1 T s = b s 0 + b s 1CP + b s 2CU + αpcd s pc + ε s, (2.2) for every sub-activity s A, and where P C represents the number of different product categories considered in the analysis. A set of dummy variables is used to account for differences between product categories {D pc } pc=1:p C. To avoid perfect multicollinearity, one category (from the group of product categories) will act as a reference for the others (Gujarati, 1995). The overall regression has the following functional form: pc=1 P C 1 T ST = b 0 + b 1 CP + b 2 CU + γ pc D pc + ε. Both approaches allow for an estimation of the expected T ST (in seconds). Sequential regression requires the T ST be estimated in two steps: first, an estimation of pc=1

24 Chapter 2. Modeling handling operations in grocery stores individual sub-activities times per order line is necessary (based on (2.2)), which then add up naturally into the Total Stacking Time according to (2.1). The overall regression on the other hand, allows one to predict the T ST directly on the basis of the 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 variable used in this study. We first estimate each model for the first dataset (chain A) and then validate the results on the second dataset (chain B). The tested models for each individual sub-activity are specified next. Similar models are used for the analysis of the overall total stacking time, too. P C 1 Model 1 : Ti s = b s 01 + αpc1d s pci + ε s1 i, pc=1 P C 1 Model 2 : Ti s = b s 02 + b s 12CP i + b s 22CU i + αpc2d s pci + ε s2 i, pc=1 Model 3 : T s i = b s 03 + b s 13CP i + b s 23CU i + ε s3 i, where s A, and ε s1 i, εs2 i, εs3 i are the error terms for each order line i = 1 : N. Model 1 is an ANOVA model with only the product category identifier as an explanatory variable, which is modeled here by the group of dummy variables {D pc } pc=1:p C 1. Therefore, this model estimates differences in execution time across products categories and is used as a reference in our analysis. Model 2 includes the main 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 variables from 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 significant effect on the execution times. 2.4.1 Sequential regression results For the derivation of the T ST, we carried out a sequential analysis. First, for each individual sub-activity, we tested regression Models 1 to 3 and derived estimates of the execution times T s (s A) for each sub-activity. Then, these estimates are used to predict the T ST, as indicated by equation (2.1). Separate analyses for each individual sub-activity correspond to our motivation of identifying which subactivities are mostly affected by the selected order line- and product-related factors. The final derivation of the T ST is in line with our purpose of deriving a predictive model for estimating the total time necessary to stack the products from an order line into the shelves.

2.4 Results 25 Models 1 and 2 are analyzed using hierarchical regression. The group of dummy variables representing the merchandising category was considered as a control variable, and it was introduced in the first step of hierarchical regressions. The reference category 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 category in this dataset is Personal care (see Table B2, Appendix B). In the second step of hierarchical regression, we added together the main effects CP and CU. The results of the ordinary least squares estimation for the first data set are presented in Table 5.3. Relevant collinearity diagnosis (such as coefficient of correlation, variance inflation 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 explains only a small proportion of the total variance in the execution times of corresponding sub-activity. The three largest adjusted R 2, obtained for Fill New, Prepare and Dispose in this sequence, vary from 10% to almost 17%. We also note that although some product categories dummies are not significant predictors, the group of dummies is overall significant (as confirmed by the overall F-statistics), and this holds true for every individual sub-activity. Results from the second regression step indicate that Model 2 explains a significantly higher proportion of the variance in sub-activities times. The adjusted R 2 ranges from.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 by the explanatory variables of Model 2 belong to Fill New (R 2 adj equals 67.9%), Grab and unpack (R 2 adj of 41.6%) and Dispose (R 2 adjof 31.6%) sub-activity, respectively. Recall from Figure 5.1 that these are also the three most influential sub-activities with respect to their relative contribution to the Total Stacking Time. The overall F-statistics indicate a significant joint contribution of the variables in predicting the execution times for all sub-activities (at p.05). However, we note that the explanatory variables CP and CU do not contribute significantly in explaining the time for searching, and have only a marginal contribution in explaining the time for preparing the shelves, filling old inventory and walking, respectively (R 2 change of 0.011 and 0.057). Further, we note that when the subgroup effect is removed from the analysis (Model 3), the adjusted R 2 for the Fn, G and D drops marginally from the previous model to 63.3%, 39.8% and 25.4%, respectively. The F-statistics show that the joint contribution of CP and CU is statistically significant for Fn, G and D and their standardized coefficients are both positive, thus showing support for our hypotheses for these sub-activities. Note that these results are also consistent between Models 2 and 3. While CU has a larger contribution for Fn, sub-activities G and D are mostly affected by CP, as indicated by the standardized coefficients. Comparing Models 2 and 3, we also find no support for S being affected by CP or CU. Although the results show a statistically significant effect of CP or CU for W, P and Fo, by inspecting the

26 Chapter 2. Modeling handling operations in grocery stores Table 2.2 Regression results for each individual sub-activity (standardized coefficients) (Chain A) Dependent Variables G S W P Fn Fo D Model 1 Baby food.033.039.052.075.065.028.065 Chocolate.205.085.111.207.238.083.284 Coffee.271.065.027.320.368.103.051 Coffee milk.106.043.062.165.283.083.181 Candy.076.007.201.042.247.004.165 Sugar.045.040.060.079.165.002.040 Canned meat.080.091.138.090.231.003.271 Canned fruits.071.011.032.040.161.003.080 R 2.072.017.071.109.172.018.122 R 2 adj.065.010.064.103.166.010.115 Mean SS Err. 111.167 13.520 13.855 49.434 405.135 12.581 39.847 Overall 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 2 Baby food.031.039.052.077.057.031.064 Chocolate.037.084.051.262.086.146.195 Coffee.124.062.084.338.147.128.044 Coffee milk.001.041.023.188.104.111.119 Candy.005.006.175.092.043.057.135 Sugar.032.037.092.072.083.005.019 Canned meat.055.087.085.077.052.008.177 Canned fruits.030.014.072.262.030.004.008 CP.422.023.187.338.184.144.401 CU.253.003.082.188.647.171.090 R 2.422.018.128.121.682.028.322 R 2 adj.416.008.120.112.679.019.316 R 2 change.350.000.057.011.510.011.200 F 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 30.816 Overall 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 3 CP.409.025.085.120.241.090.313 CU.271.030.183.007.606.072.232 R 2.399.003.063.013.634.004.256 R 2 adj.398.001.061.011.633.002.254 Mean SS Err. 71.616 13.643 13.894 54.455 178.035 12.681 33.568 Overall 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 R 2 we conclude that the impact of these variables on the execution times of the aforementioned sub-activities is weak. This result is consistent with our prediction that 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 time for Fill new, Grab/unpack and Dispose are mostly explained by CP and CU, while we found little evidence to show that these variables substantially affect the other sub-activities. Next, we obtain the T ST simply as the sum of the estimated execution times for each individual sub-activity, derived under Models 2 and 3. Thus the estimated T ST is derived as T ST = T G + T S + T W + T P + T F n + T F o + T D, where T s, s A, stands for the estimated execution time of the corresponding subactivity, as given by Model 2 or 3. To estimate the accuracy of this prediction we compare the estimated T ST with the actual T ST (obtained from empirical data) and the results are included in Table 5.4. Both variables have the same mean (57.31 seconds) as confirmed by a paired-samples t-test. The correlation coefficient between the predicted and the measured T ST is.819 (Model 2) and.798 (Model 3) and thus 67% (respectively 63.7%) of the variance in the measured T ST per order line is explained by the sum of time estimates for individual sub-activities. Thus, results show a slightly better performance of Model 2 as compared with Model 3 but the increase in adjusted R 2 is marginal. Therefore, given the simplicity of Model 3, we recommend choosing it for forecasting purposes. Table 2.3 Sequential regression: actual vs. predicted T ST (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.819.798 R 2.670.637 R 2 adj.670.636 Mean SS Err. 442.099 486.720 Overall F 2124.795 1834.109 df 1, 1046 1, 1046 Statistical significance at p.05, also p.01, p.001

28 Chapter 2. Modeling handling operations in grocery stores 2.4.2 Overall regression results Similarly, regression models 1 to 3 are analyzed for the dependent variable T ST. The results of ordinary least squares estimation for the first dataset (1048 observations) are presented in Table 2.4. Collinearity tests (correlation coefficients, variance inflation factors) indicated no significant problems with respect to multi-collinearity for the estimated models. In addition, upon preliminary inspection of the results, no significant outliers or influential points were detected, and thus the results included in Table 2.4 reflect the entire dataset. An alternative model formulation where interactions between the explanatory variable and the product category were included did 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 T ST (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 1.447 Baby food 14.978 6.298.069 13.898 3.995.064 Chocolate 30.872 3.239.310 5.896 2.375.059 Coffee 38.063 3.270.377 18.471 2.166.183 Coffee milk 45.383 4.868.279 21.527 3.227.132 Candy 19.968 2.892.232 7.932 2.007.092 Sugar 35.360 8.093.126 10.517 5.176.037 Canned meat 41.697 5.243.236 12.016 3.418.068 Canned fruits 29.401 6.209.138 2.922 3.998.014 CP 19.614 1.471.375 19.052 1.396.364 CU 1.180.081.442 1.327.071.496 R 2.178.670.637 R 2 adj.172.667.636 R 2 change.178.492.637 F change 28.130 773.434 916.178 Mean SS Err. 1108.989 445.936 487.185 Overall 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 T ST, while Model 2 yields a significantly larger adjusted R 2 (66.7%), with a significant 49.2% of the total variability in T ST accounted for by the group of variables CP and CU. Model 3 shows that the explanatory variables CP and CU together have a significantly high joint contribution in predicting the T ST, accounting for 63.7% of the variability in the T ST. The F-

2.4 Results 29 statistics for all three models are statistically significant (p.001). Thus, results provide evidence that the T ST is systematically explained in large measure by our model. Furthermore, comparing Models 2 and 3, we note that when the subgroupeffect is excluded from the model, the adjusted R 2 decreases only marginally and in both cases the hypotheses of our study are supported. The coefficient estimates presented in Table 2.4 show that T ST is positively correlated with CP and CU, thus providing support for our hypotheses at p.001. They also are fairly consistent between Models 2 and 3. The standardized values of the coefficients indicate that most of the explanatory power comes from the variables CP and CU, with a higher influence of CU. This could reflect the larger impact of the sub-activity Fn (mostly affected by CU) on the T ST. While all significant in Model 1, the coefficients of dummy variables for product category remain significant (at p.05), with one exception (Canned fruits), in Model 2. Compared with CP and CU however, they have small explanatory power. Also, recall that in our modeling we used Personal care as a reference category for the group of dummy variables (with the largest number of observations), and therefore the positive coefficients for the dummy variables confirm our expectations from previous descriptive statistics (see Table B1, Appendix B): according to this dataset, the personal care category is the fastest to handle on order line basis. Based on these results, we conclude that Models 2 and 3 already explain a large portion of the Total Stacking Time and the variables CP and CU are important predictors of T ST. Due to the simplicity of the model and its good accuracy, we recommend using Model 3 for forecasting T ST. 2.4.3 Validation of the results To validate the results from the previous section, we use the empirical data from chain B and replicate the analysis conducted for chain A. We do this in order to verify the reliability of the previously obtained results and the accuracy of the predictive models (Wang, 1994). Summary statistics for the variables in this study using the second dataset are included in Tables B3 and B4 from Appendix B. The average T ST across all five product categories is 49.29 seconds with a standard deviation of 27.06. The smallest average T ST is recorded for the products from the wine subgroup (39.69 seconds), while the most time consuming products in this set for handling are those from category cookies (mean T ST equals 60.62 seconds). The T ST shows significant variation between order lines, ranging from a minimum of 6 seconds (wine) to a maximum of 212 seconds per order line (canned vegetables). The variable CP ranges from 1 CP (all categories) to 8 CP (cookies) with an average of 1.22 CP across all categories, and a standard deviation of 0.6 CP. The variable CU has an overall mean of 15.5 consumer units (standard deviation equals 8.86), ranging from 6 to 80 CU per order line. To assure the general applicability of the approach proposed in this study, we are

30 Chapter 2. Modeling handling operations in grocery stores interested in how consistently the previous results replicate for the second data set. Regarding individual sub-activities, regression results (see Table C1 from Appendix C) for Models 1 to 3 confirm to a high extent previous findings: the CP, CU have a positive effect on execution times of sub-activities Fn, G and D and are the most important predictors of time variation for these sub-activities (adjusted R 2 is 52.6%, 48.3% and 20.4%, respectively for Model 2, and 50.9%, 47.6% and 19.6%, respectively for Model 3). When sequential regression is then used to estimate the T ST (see Table C2 from Appendix C), we found that the correlation coefficient between the predicted and the measured T ST is 0.724 (R 2 = 52.4%) using Model 2 and 0.684 (R 2 = 46.7%) for Model 3. The high values of these correlation coefficients indicate that the T ST for 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 3 to be used for deriving good estimations of the T ST. For forecasting purposes, the overall regression for estimating T ST provides a simple and less time-consuming procedure. Therefore, we tested the reliability of the results on the second data set as well. We found consistent support for our hypotheses regarding T ST (see Table 2.5). While the group of dummy variables related to product category have a significant, but small contribution in predicting T ST (adjusted R 2 about 10%), the most explanatory power comes again from the group of variables CP and CU, which affect significantly and positively the T ST. Compared to Model 2 (R 2 adj = 51.9%), CP and CU alone explain 46.5% of the variance in T ST, thus indicating only a marginal decrease in adjusted R 2. Their coefficients are both statistically significant at p.001 and consistent between the two models. Moreover, note that they have also comparable sizes with coefficients estimates for CP and CU derived for the first dataset (compare Tables 2.4 and 2.5). We can, therefore, be confident that the effects of both CP, and CU are needed to model the T ST to a large extent and that Model 3 represents a simple and reliable alternative for predicting T ST. We performed a final verification of the results, in which we used the data collected for chain B and the coefficients estimates for Model 3 derived for chain A (see again Table 2.4) to compute predicted values of T ST for each order line. We restate here for reference the model used for prediction: where a 0 = 10.240, a 1 = 19.052 and a 2 = 1.327. T ST = a 0 + a 1 CP + a 2 CU, (2.3) The high values of the correlation coefficient between the measured and predicted T ST (R = 0.683, R 2 = 46.6%) provide additional evidence that the T ST for chain B is also explained to a large degree by our model. Overall, the results of this section offer valuable support for the general applicability of our approach to similar settings, with important ramifications for retailers. Model 3 represents a simple and reliable method for predicting the T ST. The stacking-times for each order line can be estimated inexpensively in this way, and ultimately be used for management decisions. For

2.4 Results 31 Table 2.5 Overall regression: results for T ST (Chain B) 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) 60.620 1.923 19.981 2.361 9.960 1.971 Sandwich spread 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 R 2.102.524.467 R 2 adj.095.519.465 R 2 change.102.422.467 F change 15.820 246.753 245.558 Mean SS Err. 662.247 352.103 391.431 Overall F 15.820 102.087 245.558 df 4, 558 6, 556 2, 560 Statistical significance at p.05, p.01, p.001; Reference category: Cookies (N = 179)

32 Chapter 2. Modeling handling operations in grocery stores example, one can assess the amount of work necessary during a day to execute the restocking of the shelves (using readily available information about the number of units and case packs for each SKU), or can assess the individual labour performance of store employees. Also, by inspecting the individual sub-activities, one can get an indication 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 a better control of the overall logistical costs. In the literature on retail operations, the handling costs are rarely modeled or they are often assumed to be a linear function of the number of CU s. Our results indicate however that an additive cost structure is more appropriate. The T ST per SKU is linearly dependent on CP and CU, but not directly proportional with CP and CU (due to the constant parameter a 0 ) (see equation (2.3)). For each order line, a fixed setup time (a 0 ) is incurred, additionally to the positive time related to the number of units being handled. This cost structure of the T ST allows economies of scale, and thus can be further exploited in order replenishment decisions. For example, the retailer could decide to order less frequently but in a larger number of case packs in order to reduce the handling workload. On the other 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 handling costs, and in Section 2.5.2 we give an indication about the magnitude of potential efficiency 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 the store (C h ) and the ordering cost (C o ), but also a distinct shelf stacking cost (C s ), derived on the basis of the handling time (2.3). The objective is to determine an inventory control policies that minimizes the long run average total costs. We use the following 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 cost h = annual holding cost per consumer unit S = stacking cost per hour T C(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 in multiples 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 a function of the order size X yields the following average annual total cost: or equivalently, T C(X) = C o (X) + C h (X) + C s (X) = K λ X + h X λ + S T ST (X) 2 X = (K + S a 0 ) λ X + h X 2 + S (a 1 λ q + a 2λ) T C(m) = C o (mq) + C h (mq) + C s (mq) = λ (K + S a 0 ) mq + h mq 2 + S (a 1 λ q + a 2λ). Hence, when X (and m) is not restricted to be an integer value, and we assume fixed case pack sizes (q), the optimal ordering quantity (X ) is the value such that T C(X)/ X = 0, and is given by: X = m q = 2λ(K + Sa 0 )/h. (2.4) Because function T C(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, which give the lowest value of the total costs. The total annual costs corresponding to X is given by: T C(m ) = C o (m q) + C h (m q) + C s (m q) = 2hλ(K + Sa 0 ) + Sλ(a 1 /q + a 2 ). (2.5) Equation (2.4) resembles the classical EOQ formula, in which the ordering cost is replaced here by K + Sa 0, while in formula (2.5) we note that the first component represents the optimal total cost in the classical EOQ formula (but with the ordering cost modified as K +Sa 0 ) plus the extra term Sλ(a 1 /q +a 2 ). In this setting, the fixed setup time a 0 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 in the stacking process, obtained when it is possible to reduce the frequency of the replenishment, by ordering more products at once, rather than the same amount multiple times. We use the coefficient estimates in (2.3) as reliable indication of the size of the effects identified. Let n (n = 1, 2, 3,...) be the number of order lines for the same SKU in a replenishment order in the subsequent analysis. Two situations can then be considered: (1) increase the number of case packs per order line and (2) increase the case pack size q.

34 Chapter 2. Modeling handling operations in grocery stores 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) by ordering more CP per order line, while keeping the same case pack size (q) can be evaluated. We compare the time savings obtained if, instead of ordering n order lines with 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 for stacking n order lines with CP of size q per order line can be written as: T T (CP, q) = nt ST (CP, q) = na 0 + na 1 CP + a 2 ncp q. The total time needed for stacking the same amount at once is expressed as: T ST 1 (CP, q) = T ST (ncp, q) = a 0 + a 1 ncp + a 2 ncp q. Then the time savings can be derived as: T T T ST 1 = (n 1)a 0 > 0, for n > 1, which implies that we may save stacking time if we order, in each replenishment, more case packs at once, instead of ordering one case pack at a time, and this saving is due to the constant setup time a 0. The efficiency gain, compared with the case of multiple replenishments is then: S 1 (CP, q, n) := T T T ST 1 T T = (n 1)a 0 100%, for n > 1. (2.6) na 0 + na 1 CP + a 2 ncp q Case 2: Increase the case pack size q We evaluate the time savings obtained if, instead of ordering n order lines with 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 CP case packs, each of size nq. In this case, the shelf stacking time is derived as follows: T ST 2 (CP, q) = T ST (CP, nq) = a 0 + a 1 CP + a 2 CP nq. Then the time gains are now: T T T ST 2 = (n 1)a 0 + (n 1)a 1 CP > 0, for n > 1, (2.7) and the percentage of time saving is then: S 2 (CP, q, n) := T T T ST 2 T T = (n 1)a 0 + (n 1)a 1 CP 100%, for n > 1. (2.8) na 0 + na 1 CP + a 2 ncp q Again, in this case, reducing the frequency of the replenishments may result in time savings and efficiency gains as given by (2.7) and (2.8). Furthermore, by comparing

2.5 Analytical insights and implications for retailers 35 80 70 S1, CP = 1, q = 6 S1, CP = 4, q = 6 S2, CP = 1, q = 6 S2, CP = 4, q = 6 60 % Efficiency gains 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 Order lines (n) Figure 2.2 Effect of n on S 1 and S 2 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 second case, when the strategy is to increase the case pack size (q) instead of the number of case packs (CP ) per order line. The efficiency gains derived from (2.6) and (2.8) are illustrated in Figure 5.10, for two particular choices of CP and q. Typically, case pack sizes take values of 6, 12 or 24 consumer units. In Figure 5.10, the effect of n on S 1 and S 2 is illustrated for one and respectively four case packs per order line, each of size six. We note that the higher the reduction in the number of order lines, the higher the savings. Reducing one order line for the same SKU (n = 2 in Figure 5.10) as a consequence of ordering two times more case packs results in efficiency gains of 13% (if CP = 1) and 4% (if CP = 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 S 1 and S 2. Generally as n increases, S 1 and S 2 reach steady values, with a maximum around 30% (for S 1 ) and 80% (for S 2 ), respectively. However, as CP and q increases, the efficiency gains are decreasing. Notable is the behavior of S 1 with respect to n and CP, when the savings drop as CP increases, indicating that in formula (2.6), the estimated time due to CP and q, outweighs the fixed setup-time (due to a 0 ).

36 Chapter 2. Modeling handling operations in grocery stores Although in practice the case pack sizes are usually set by the manufacturers, it is still valuable to recognize the impact of reduced sizes on handling efficiency, perhaps especially for retailers that also carry their own private labels. These preliminary insights into the potential efficiency gains derived from the proposed model, offer interesting opportunities for developing adapted inventory control rules that take into account the handling component. Building new inventory replenishment policies that recognize the handling efficiency, should of course consider the possible tradeoffs (such as the shelf space availability and physical constraints of the shelves, the demand pattern, or restrictions with respect to possible case pack sizes). We consider such advancements in the following chapters. 2.6. Conclusions In this chapter, we focused on the shelf stacking process in grocery retail stores and studied the main factors that drive the shelf stacking time. This study has three main contributions: first, we provide insights into the specifics of the stacking process by analyzing the individual sub-activities and describe the interactions between key time-drivers (CP and CU) and logistics processes; secondly, we derive a reliable model for estimating the total stacking time per order line; finally, we use the empirical results to quantify potential time savings and demonstrate by a lot sizing analysis the opportunities for considering handling costs explicitly for inventory management decisions. We used real life empirical data from two European grocery retailers and adopted two strategies for estimating the T ST per order line and evaluating the relative impact of each factor identified (sequential vs. overall regression). The two approaches may serve two different practical purposes. On one hand, the sequential approach, allows one for a better insight into the details of the shelf stacking process, identifying those sub-activities that are mostly affected by the number of items being handled (CP and CU), and those for which the variation in workload is potentially affected by other factors. At the same time, the approach indicates which sub-activities contribute mostly to the total variation in the stacking time of a new order line. The three most relevant sub-activities are found to be: stacking new inventory, grabbing and opening of a case pack, and waste disposal, in this order. These sub-activities are also mostly affected by CP and CU, and we indicate the magnitude of their relative positive impact. We also find that the time for searching, preparing and filling old inventory can be regarded as fixed per order line. This information can further be used to identify inefficiencies in the stacking process. On the other hand, the overall regression strategy offers a simple, inexpensive tool for predicting the T ST per order line. In this study, we found enough support to conclude that a simple prediction model, depending only on the number of case packs and the number of consumer units, offers already a reliable estimate of the T ST.

2.6 Conclusions 37 Results from testing and validation show that the model is stable and it explains the T ST to a large extent. Although the influence of CP and CU on the shelf stacking process may be implicitly recognized, we demonstrate that both variables are relevant predictors for T ST and we also estimate the size of their effects. As compared to common assumptions in the literature, we find that the form of the relationship is additive, rather than purely linear. Using this structural insight, we investigate the magnitude of efficiency gains in the stacking process. Furthermore, we model the empirical findings into a lot-sizing analysis to demonstrate the opportunities for extending inventory control rules with a handling component. The empirical findings of this study offer the practitioners the opportunity for a better control of the overall logistical costs. We shall further investigate this opportunity in Chapter 3. While we illustrate the impact of some key drivers on time variation, we recognize there 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 assume that the type of package, the distance traveled within the aisle, the SKUs volume or weight, and the inventory level of the products on the shelves just before restocking might have an impact on the Total Stacking Time. These variables were however not available in the original dataset. Appendix A. Shelf stacking activities See Table A1. Appendix B. Descriptive statistics of the empirical datasets See Tables B1, B2, B3 and B4. Appendix C. Validation results for chain B See Tables C1 and C2.

38 Chapter 2. Modeling handling operations in grocery stores Sub-activity Table A1 Shelf stacking sub-activities Starting/Ending point of sub-activity Grab/open case pack (G)* Start The filler stands in front of the rolling container and reaches for a case pack. End The filler prepares to walk away from the rolling container and starts opening a case pack. or Start The filler has arrived at the shelf location and starts opening the case pack. End The filler is ready with opening the case pack and another sub-activity starts. Search (S) Start The filler starts with checking the product (from the rolling container) and he/she looks for the right shelf location. End The filler sees the right shelf location and prepares to approach it (walk). Walk (W)** Start The filler prepares to walk away from the rolling container or 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 End The filler reaches for the old inventory on the shelves and start to check the best before date (if needed). The filler is ready with preparing the shelves; old inventory is 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 filler starts putting it back on the shelves. End The filler is ready with putting old inventory back on the shelves and grabs the empty box or plastic. Waste disposal (D) Start The filler holds an empty box (or plastic) and starts to flatten it (sometimes the box is preserved to customers). End The moment the filler prepares to leave the waste disposal place (a trolley or a place near the rolling container). Extra (E) An activity not part of the first sub-activities, e.g., help a customer, customer is in the way, get or put away crate, process inventory remainder, organize labels, general cleaning, discuss with a colleague, take away waste, bring empty boxes for customers to check out area, get a new rolling container, take away misplaced products, repair a broken product, remove cord from rolling container, take a product to the kiosk, straighten separation plate. Notes: *Grabbing and opening the case pack are taken together, because the individual activities were difficult to separate. **Walking does not include walking with the rolling container from the storage area to the right aisle or walking with the rolling container between the aisles. But it does include (in exceptional cases) walking with the rolling container when the rolling container is moved to bring certain case packs to the right shelf locations (e.g. heavy products). It is possible that the filler performs multiple sub-activities at once, e.g. walking while opening the case pack, searching or disposing waste. When this happened, the following reasoning was used: if the walking time was significantly influenced by the attention focused on opening the case pack (or searching/waste disposal), the time for e.g. opening the case pack was measured as sub-activity G, and the remaining time as sub-activity W. If the walking time was not significantly influenced by one of these sub-activities, then the total time was measured as walking time (W).

2.6 Conclusions 39 Table B1 Descriptive statistics of explanatory variables (Chain A) Number Number Number of CP per Number of CU per TST per Category of SKUs of Order Order Line Order Line Order Line [sec.] Lines Avg. SD Min. Max. Avg. SD Min. Max. Avg. SD Min. Max. Baby food 26 31 1.13 0.34 1 2 8.90 4.03 4 16 50.45 17.47 20 88 Chocolate 91 168 1.36 0.72 1 4 25.26 16.58 6 80 66.35 41.22 15 294 Personal care 193 285 1.14 0.37 1 3 7.80 3.86 3 36 35.47 14.93 10 94 Coffee 91 163 1.47 1.08 1 9 18.88 18.74 6 135 73.54 48.62 20 334 Coffee milk 31 56 1.45 0.81 1 5 22.93 12.27 10 60 80.86 35.34 34 211 Candy 143 248 1.15 0.39 1 3 17.92 9.17 8 72 55.44 26.69 16 74 Sugar 11 18 1.83 0.99 1 4 17.33 9.43 8 40 70.83 32.25 22 151 Canned meat 40 47 1.77 0.96 1 5 22.55 15.20 6 72 77.17 51.23 12 245 Canned fruit 32 32 1.72 0.81 1 4 20.63 13.10 6 48 64.88 31.09 11 125 Aggregate 646 1048 1.30 0.70 1 9 16.78 13.69 3 135 57.31 36.59 10 334 statistics CP: case packs, CU : consumer units, TST : total stacking time, SD: standard deviation

40 Chapter 2. Modeling handling operations in grocery stores Table B2 Descriptive statistics of response variables (Chain A) N Mean Std. Std. Error Dev. Mean T ST 1048 57.31 36.59 1.13 Grab/Open 1048 11.65 10.91 0.34 Search 1048 2.31 3.70 0.11 Prepare 1048 3.54 7.42 0.23 Fill New 1048 27.32 22.04 0.68 Dispose 1048 7.28 6.71 0.21 Walking 1048 4.77 3.85 0.12 Fill 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. Error Dev. Mean T ST 563 49.29 27.06 1.14 Grab/open 563 7.49 6.92 0.29 Search 563 0.67 3.09 0.13 Prepare 563 5.87 7.51 0.32 FillNew 563 21.94 12.97 0.55 Dispose 563 4.56 4.57 0.19 Walking 563 7.26 5.78 0.24 FillOld 563 1.5 4.93 0.21 TST : total stacking time

2.6 Conclusions 41 Table B3 Descriptive statistics of explanatory variables (Chain B) Number Number Number of CP per Number of CU per TST per Category of SKUs of Order Order Line Order Line Order Line [sec.] Lines Avg. SD Min. Max. Avg. SD Min. Max. Avg. SD Min. Max. Sandwich spread 39 56 1.25 0.47 1 3 17.11 9.71 8 48 48.73 25.02 18 148 Canned vegetables 46 57 1.28 0.67 1 5 14.74 7.83 8 60 52.67 31.88 17 212 Cookies 125 179 1.22 0.68 1 8 18.41 9.60 8 80 60.62 27.34 19 194 Candy/Chocolate 103 142 1.13 0.35 1 3 18.05 7.22 6 50 42.59 20.04 13 132 Wine 84 129 1.29 0.69 1 5 8.29 4.36 6 30 39.69 26.29 6 168 Aggregate statistics 397 563 1.22 0.60 1 8 15.50 8.86 6 80 49.29 27.06 6 212 CP: case packs, CU : consumer units, TST : total stacking time, SD: standard deviation

42 Chapter 2. Modeling handling operations in grocery stores Table C1 Regression results for each individual sub-activity (standardized coefficients) (Chain B) Dependent Variables G S W P Fn Fo D Model 1 Sandwich spread.073.002.188.056.033.218.009 Canned vegetables.002.058.152.016.050.257.076 Candy/Chocolate.053.002.106.376.168.351.028 Wine.030.241.130.333.339.343.091 R 2.009.056.041.157.094.142.016 R 2 adj.002.049.034.151.088.136.009 Mean SS Err. 47.715 9.069 32.253 47.865 153.305 20.975 20.722 Overall F 1.298 8.279 6.015 25.955 14.546 23.039 2.34 df 4, 558 4, 558 4, 558 4, 558 4, 558 4, 558 4, 558 Model 2 Sandwich spread.075.006.194.053.014.223.015 Canned vegetables.001.049.165.005.008.271.064 Candy/Chocolate.011.007.088.372.143.345.001 Wine.076.212.159.288.099.389.106 CP.591.084.282.031.245.117.436 CU.155.051.032.097.523.083.014 R 2.489.060.109.169.531.148.213 R 2 adj.483.050.100.160.526.139.204 R 2 change.480.004.068.012.437.007.196 F 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 16.646 Overall 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 3 CP.634.193.220.004.238.008.388 CU.089*.211.054.180.546.091.087 R 2.478.033.066.032.510.009.199 R 2 adj.476.030.062.028.509.006.196 Mean SS Err. 25.053 9.256 31.326 54.780 82.599 24.128 16.825 Overall 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 T ST (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.724.684 R 2.524.467 R 2 adj.523.466 Mean SS Err. 348.965 390.733 Overall 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 models with batch ordering and handling costs Abstract: Consider a retailer who manages periodically the inventory of a singleitem facing stochastic demand. The retailer may only order in multiples of a fixed batch size q, the lead time is less than the review period length and all unmet demand is lost, which is a realistic situation for a large part of the assortment of grocery retailers. The replenishment cost includes fixed and variable components, dependent on the number of batches and units in the order. This structure captures the shelf stacking costs in retail stores. We investigate the optimal policy structure under the long-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-tooptimal 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 unitrelated costs, while (s, Q) policies may outperform (s, S) policies in environments with low service levels. The new heuristic performs, in both situations, consistently very well. Finally, we show that handling costs may substantially affect overall system s performance, when ignored, especially for items with low-profit margins. 3.1. Introduction As pointed out in earlier chapters, traditional store-based retailers face nowadays intense 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 and people (i.e. merchandize handling). While handling costs are relatively high in the grocery retail supply chain, they are rarely acknowledged in inventory replenishment decisions. 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 optimization of inventory decisions. In Chapter 2, we showed that we can reliably estimate the handling time per Stock Keeping Unit (SKU) required to execute the shelf stacking operation using an additive model (fixed plus linear terms), depending on the number of case packs (batches) and the number of consumer units stacked. This leads to a replenishment cost structure that allows economies of scale. For example, the retailer could decide to order less frequently but in a larger number of case packs in order to reduce the handling workload. However, less frequent deliveries lead to an increase in the average inventory level. In this chapter, we investigate inventory replenishment decisions that take into account a merchandize handling component due to shelf stacking. The basic setting is a retailer managing the inventory of a single item, facing stochastic demand. The stock is reviewed periodically (say daily or every 2nd or 3rd day) and new orders can be placed at the beginning of each review epoch. The ordered stock is typically received before the beginning of the next review epoch, which results in replenishment lead times shorter than the length of the review epoch. This situation is common in the European and Japanese grocery retail environment. For example, orders might be placed during the morning and delivered to the stores in the evening of the same day. Furthermore, the replenishment stock arrives in pre-packed form, while the consumer demand may be for individual units. The case packs have fixed, exogenously determined sizes, usually set by the manufacturer, due to limitations in packaging, transportation and coordination. Once arrived at the store, the packed deliveries are unwrapped and units are displayed onto the shelves in response to consumer s demand. 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 lost sale. We analyze such a system considering the following cost components: a cost for holding inventory in the store, a cost associated with the demand which is lost and a replenishment cost, which includes fixed ordering and shelf stacking costs. Based on the results obtained in Chapter 2, we assume that each replenishment order is associated with costs of the following structure: K + K 1 n + K 2 nq, where K and K 1, K 2 represent fixed and variable components, respectively, q is the case pack size, and orders are nonnegative integers (n) multiple of these q consumer units. The objective is to find an inventory policy which minimizes the long-run expected average costs of the system over an infinite horizon. We formulate the problem as a Markov decision problem and use it to explore the structure of the optimal policies. In particular, we numerically illustrate the impact of handling 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 as the (s, Q S, nq) policy, which combines the logic of the two policies, and accounts for quantized ordering 1. In an extensive numerical study, we show that the new heuristic has 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 that the best (s, S) policies are performing very well especially in settings where the penalty cost 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 service levels, they may eventually outperform the best (s, S) policy. The new heuristic performs, in both situations, consistently very well. Finally, we investigate the added value of including the handling costs into decision making. We find that, for items with low profit margins, ignoring handling related costs, may result in substantial cost penalty. The remainder of this chapter is organized as follows. The next section gives a brief review of related literature. Section 4.3 formally introduces the problem and formulates the model using a Markov decision process. In Section 3.4 we introduce the new heuristic. A numerical study is conducted in Section 3.5, in which we illustrate the complexity of the optimal policies ( 5.3), we conduct sensitivity analysis with respect to problem parameters ( 3.5.2), and we test the comparative performance of the newly proposed heuristic ( 3.5.3). In section 3.6 we investigate the added value of 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 sales inventory models with positive lead times and batch (quantized) ordering. The classical lost sales problem, originally formulated by Karlin and Scarf (1958), is known to be far less analytically tractable than the corresponding backorder problem. If the lead time is positive, the complete structure of the optimal ordering policy is unknown (Hadley and Within, 1963). Therefore, there are few papers that contain analytical results for periodic review problems with lost sales and positive lead times. Karlin and Scarf (1958) establish some basic properties of the optimal ordering policy for a model with a lead time of exactly one period. Later on, Morton (1969, 1971), extends these results to periodic review lost sales problems with fixed lead times, multiples of the review period length. More recently, Zipkin (2008b) provides a new derivation of this 1 Here and in the rest of the chapter, we study a periodic review system with fixed review period R, taken to be the time unit. Hence, for notation convenience, we shall not use R in denoting the different inventory policies, unlike in some existing literature.

48 Chapter 3. Inventory control with handling consideration result and extends it to more complex settings. Additionally, bounds on the optimal policy are also derived in these papers. Since the derivation of optimal inventory polices (via dynamic programming for example) becomes prohibitive as lead time increases, 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 policies are found to perform poorly (except for short lead times), but it is in general unclear which approach is best. Note that in all models mentioned so far, no fixed ordering cost is taken into consideration. Nahmias (1979) considers a more general periodic-review lost-sales problem with a fixed ordering cost and stochastic lead times. He proposes using an (s, S) policy (or order-up-to S policies for zero ordering cost) and derives the optimal parameters using simulation. Analysis of simple-structured policies are often preferred in a lost sales setting, due to intrinsic complexity of the original model, usually with the assumption of at most one order outstanding at any time. Order-up-to (or base stock) policies are studied mostly (see, e.g., Morse, 1959, Gaver, 1959 and Johansen, 2001), which are known to be optimal for the periodic-review backorder model, when there are no setup costs. For this class of policies, structural properties such as convexity of the underlying cost functions are often investigated (see, e.g., Downs et al., 2001 and Janakiraman and Roundy, 2004). In the presence of a positive setup cost, reorderpoint polices of (s, S) type (Wagner, 1962) or (s, Q) (Johansen and Hill, 2000) type are considered 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) type are known to be optimal under rather general conditions (Beyer and Sethi, 1999), the equivalent lost sales model is much more complex and (s, S) policies are not optimal for these systems. In almost all papers that deal with periodic review inventory problems, lead times are assumed to be integral multiples of the review period length. By contrast, the problem we consider assumes that the lead time is deterministic, and between zero and the length of a review period, as this accommodates situations when the inventory is reviewed weekly, for example, while the replenishment lead times are only one, or two days 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 an order-up-to policy the analytical difficulties inherited by lost sales models, and also Kapalka et al. (1999). The latter study considers the class of (s, S) policies and uses Markov chains to analyze and derive the best policy under the average cost criterion and service level constraints. They compute the optimal parameters using a search procedure based on an efficient updating scheme for the transition probability matrix, bounds on S and monotonicity assumptions on the cost and service constraint. In both studies, fixed costs are present.

3.2 Literature review 49 Janakiraman and Muckstadt (2004a) also consider a periodic review lost sales inventory 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 positive ordering cost. They investigate structural properties of the objective function as well as the optimal policy, and find similar results to those initially reported by Karlin and Scarf (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 function with respect to the order-up-to level. Chiang (2006, 2007a) recently proposed a dynamic programming model for periodic-review systems in which a replenishment cycle 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 small periods, rather than ending inventory of replenishment cycles. They analyze both backorder and lost sales inventory models (under the assumptions of lead times shorter than the replenishment cycle length), and for the latter provide some properties and computational results. Thus far, existing literature on lost sales inventory systems provides partial characterizations of the optimal ordering policy in the absence of fixed ordering costs. There exists limited analytical research for the case of a positive ordering cost and there are still many open questions when the restriction of a fixed batch size for ordering is assumed, as noted earlier by Veinott (1965). The incorporation of a fixed batch size is hardly taken into account in lost sales models. A notable exception is Hill and Johansen (2006). When excess demand is fully backordered, and there are no fixed ordering costs, then Veinott (1965) shows that the (s, nq) 2 policy is optimal for a single-location, single-item inventory model under both finite, and infinite period settings. He also points out that the (s, nq) policies are not optimal in general if a fixed cost is taken into consideration and he proposes a two parameter policy instead. Zheng and 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 the analysis and optimization of an (s, S, q) policy, where s and S are assumed to be multiples of q. The structure of the optimal policy for lost sales systems with batch ordering remains an open question (Veinott, 1965, Hill and Johansen, 2006). The research presented in this chapter makes the following contributions over existing literature. We study a periodic review, stochastic lost sales inventory system that combines attractive features for practice: fixed batch sizes and fixed plus variable (depending on the batch size) material handling costs. We investigate numerically the impact of material handling costs on the structure of the optimal policies and the corresponding 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 2 Veinott s original notation (k, Q), with Q a fixed positive constant. With the (k, Q) policy if the initial inventory on hand and on order in a period is less than k, an order is placed for the smallest multiple of Q that will bring the inventory on hand and on order to at least k; otherwise, no order is placed.

50 Chapter 3. Inventory control with handling consideration partially the structure of optimal policies and works as follows: when q is one, at every review moment, if the inventory position is not more than S Q, then place a replenishment order of size Q; if the inventory position is above S Q but not more 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. Figure 3.1 illustrates the logic of the (s, Q S, nq) policy by an example. In this chapter, we computationally investigate the performance of the heuristic against optimal policies as well as reasonable alternative policies. 30 25 Q Order quantity 20 15 10 q = 1 q = 4 5 S-Q s 0 0 4 8 12 16 20 24 28 32 36 40 Inventory on hand 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 inventory system under consideration. The main notation in this chapter is as follows. R Review period L Lead time (0 L R) t Period index, t = 0, 1, 2,... X t Inventory on hand at the beginning of period t = 0, 1, 2,... a t Quantity ordered in period t = 0, 1, 2,... D L Demand during lead time (i.e. the demand occurring between the start of a period and the time the order a t is received) Demand after the receipt of the orders (and until the end of the period) D R Total demand during a review period, D R = D L + D R L q Fixed (exogenously determined) batch size, q = 1, 2,... K Fixed cost per order K 1 Fixed cost per batch K 2 Variable unit cost h 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) D R L

3.3 Mathematical model 51 We assume the period demand to be a known, nonnegative discrete random variable and the demand process is i.i.d stationary. The lead time is fixed but less than the review period length as this accommodates situations when the stock is reviewed, for example, every three days and the lead time is say one or two days. We assume that the length of the review cycle R is exogenous to the model (for example determined by the need of coordinating replenishment of many different items) and we further assume it to be the time unit. The random variables D L and D R L are stochastically independent, and a t is restricted to be a nonnegative integer, multiple of the fixed batch size q, i.e. a t {0, q, 2q,... }. The sequence of events in each review period is as follows (see Figure 3.2). At the beginning 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 period arrives and afterwards stochastic demand continues to occur, up until the beginning of the next ordering moment. All demand occurring in this time period that cannot be directly satisfied is again assumed to be lost. Note that due to the assumption L R, lost sales might occur between the time an order is placed and received, as well as in the time following the order receipt up until the beginning of the next ordering moment. X t Place order a t X t+1 Place order a t+1 Order receipt Order receipt D L D R-L D L D R-L L R - L L R - L Figure 3.2 The sequence of events Dynamics of the system We formulate the problem as a Markov decision process, in which the decision epoch is the beginning of each review period and the inventory on hand at the beginning of a 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 is limited 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 deterministic and equal R. The evolution of on hand inventory from one decision epoch to the next is given by the following recursive relation: X t+1 = ((X t D L ) + + a t D R L ) +, t = 0, 1, 2,....

52 Chapter 3. Inventory control with handling consideration where (x) + = max{0, x} for any x R. Next, we define the transition probabilities and expected costs from one decision epoch to the next. Transition probabilities The probability p ij (a i ) of a transition from state i at one decision epoch to state j at the next epoch, given decision a i at the first decision epoch is defined as p ij (a i ) = P ( j = ((i D L ) + + a i D R L ) +), i, j = 0, 1,..., a i = 0, q, 2q,..., and is given by p i0 (0) = P(D R i), i = 0, 1,..., p ij (0) = P(D R = i j), i = 0, 1,..., j = 1, 2,..., i, p 00 (0) = 1, when there is no order, and when the order amounts to an integer a i = n i q > 0 is given by p i0 (a i ) = p ij (a i ) = p ij (a i ) = i 1 P(D L = k)p(d R L i + a i k) + P(D L i)p(d R L a i ), k=0 i = 1, 2,..., i 1 P(D L = k)p(d R L = i + a i j k) + P(D L i)p(d R L = a i j), k=0 i+a i j k=0 p 00 (a 0 ) = P(D R L a 0 ), i = 1, 2,..., j = 1, 2,..., a i, P(D L = k)p(d R L = i + a i j k), p 0j (a 0 ) = P(D R L = a 0 j), j = 1, 2,..., a 0, p ij (a i ) = 0, otherwise. Transition costs i = 1, 2,..., j = a i + 1,..., a i + i, The total expected cost from one decision epoch to the next (i.e., the one-period transition cost), given that the we are in state i and we order an integer amount a i = n i q 0, is defined as c i (a i ) = c r i (a i ) + c h i (a i ) + c p i (a i), a i A(i), i Ω, (3.1) where the expected replenishment cost c r i and the expected holding (ch i ) and penalty ) costs are given by (c p i c r i (0) = 0, c r i (a i ) = K + K 1 n i + K 2 n i q, a i = n i q > 0, (3.2)

3.3 Mathematical model 53 and c h i (0) + c p i (0) = he [ (i D R ) +] + pe [ (i D R ) ], c h i (a i ) + c p i (a i) = he [ ((i D L ) + + a i D R L ) +] = h +p { E [ (D L i) +] + E [ (D R L a i (i D L ) + ) +]} { i 1 P(D L = k)e [ (i k + a i D R L ) +] k=0 +P(D L i)e [ (a i D R L ) +] } { +p E [ (D L i) +] i 1 + P(D L = k)e [ (D R L a i i + k) +]} k=0 +p P(D L i)e [ (D R L a i ) +], for a i > 0, respectively, where (x) = max{0, x} = min{0, x} for any x R. From (3.1) and (3.2), we derive the total one-period transition cost as follows: c i (0) = he [ (i D R ) +] + pe [ (i D R ) ], c i (n i q) = K + K 1 n i + K 2 n i q +he [ ((i D L ) + + a i D R L ) +] +p { E [ (D L i) +] + E [ (D R L a i (i D L ) + ) +]}, n i > 0. We aim to find an inventory policy U, which solves the following optimization problem: g = min U C(U) = 1 R i Ω π i c i (a i ) R=1 = i Ω π i c i (a i ), where C(U) denotes the long-run expected average cost under policy U, and (π i ) 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 steadystate distribution may not be determined in closed form. For the exact conditions that 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, then there exist relative values (v i ) i Ω and the long-run expected average cost g, such that they satisfy the average-cost optimality equations: v i = min {c i(a) g + p ij (a)v j }, i Ω. (3.3) a A(i) j Ω Unique relative values (v i ) i Ω are obtained if we provide an initial condition, such as v 0 = 0. In order to solve the optimality equations and determine an optimal policy,

54 Chapter 3. Inventory control with handling consideration it is common to use an algorithm such as policy iteration, value iteration or linear programming (Puterman, 1994). 3.4. Old and new heuristics For the lost-sales inventory control problem introduced in Section 4.3, there is no apparent 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 is adjusted 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) constraint on the order size, and will be referred to as (s, S, nq) or (s, Q, nq), whenever q > 1. We mention that these policies are applied in a periodic-review setting, with the review period 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 review period is less than or equal to s, order the largest integer multiple of q which results in an inventory position less than or equal to S. For the corresponding (s, Q, nq) policy, an order is placed at a review moment whenever the inventory level is less than or equal to s, and the order size Q is constrained 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 max{s, Q} S s +Q. Under this policy, the order quantity in each period depends on the beginning inventory on hand x and is given by Q/q q if 0 x S Q/q q a(x) = (S x)/q q if S Q/q q < x s 0 if s < x, where x denotes the largest integer, smaller or equal to x (see Figure 3.1). particular, when q = 1 the order quantity equals In a(x) = { min{q, S x} if 0 x s 0 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, then order 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-to level S. Similarly, if S = Q, the (s, Q S) policy acts like an (s, Q) policy. Thus the new heuristic generalizes both (s, S) and (s, Q) policy. In Section 3.5.3 we conduct a computational investigation of the comparative performance of these policies under different problem parameters. 3.5. Numerical study The objective of this section is three-fold. First, to explore the effect of material handling costs, K 1 and K 2, on the structure of the optimal policy and the associated long-run average cost. Second, to determine the effectiveness of the newly proposed heuristic policy, referred to as the (s, Q S, nq) policy by testing its performance against the optimal policy determined via dynamic programming. Third, we benchmark the 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 of including the handling costs into decision making, which enable us to develop further managerial insights. We conduct several numerical studies involving the following parameters: the lead time L, the fixed ordering cost K and the material handling costs K 1 and K 2, as well as the fixed batch size q. Throughout all experiments, the holding cost h, the expected demand per period λ, and the penalty cost p are held constant. Furthermore, we assume that the random demands D L and D R L are stochastically independent and both follow a Poisson distribution. The main reason for this choice is that technically the mean period demand D R is than a readily available Poisson distributed random variable. The numerical examples are chosen as follows: h = 1, λ = 20, p = 50 remain unchanged, and L {0.25, 0.33, 0.50}, K {10, 50, 100}, K 1 {5, 10, 20, 40}, K 2 {0, 5, 10}, q {1, 2, 4, 6, 12, 20}. Altogether, there were a total of 648 instances in our computational study. The choice of handling cost values is related to real-life data. In Chapter 2 we reported approximate values of K = 10, K 1 = 20 and K 2 = 1.3, based on empirical data (see e.g. equation (2.3)). Also the case pack sizes are chosen as commonly reported in practice. The holding cost is set arbitrarily, while high penalty cost (relative to holding cost) in the retail setting are more likely to hold for nonperishable products

56 Chapter 3. Inventory control with handling consideration with long life cycles. To determine the optimal policy under the long-run average cost criterion, we used the standard value-iteration algorithm (Puterman, 1994, Bertsekas, 1995). The program was 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 optimal policies. We define the reorder level as rl = max{i Ω a i > 0}, i.e., the highest value of inventory on hand at which it is optimal to order a positive amount; the maximum stock level is defined to be the maximum optimal inventory position, after ordering, i.e., ms = max{i + a i 0 i rl}. The complexity of optimal policies for lost sales inventory models contrasts with that 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 sales formulation, the optimal control policy will, in general, be neither of the (s, S) nor of the (s, Q) type, but will depend in a much more complex way on the physical stock at the time of placing an order. The optimal policies for three scenarios (corresponding to different lead time L values) are depicted in Figure 3.3(a). Figure 3.3(b) exemplifies the effect of non-unit batch sizes on the structure of the optimal policy (here q = 4 vs. q = 1). In view of these results, a few observations are worthwhile mentioning. There is no apparent solution of simple form. That is, the optimal order quantity as a function of on-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 to place an order and beyond which it is never optimal to order. Thus, rl plays the role of a reorder level in a general inventory policy. This observation has been conjectured before (see Hill and Johansen (2006), in a continuous review setting), but no proofs exist 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, the advice is either to order a fixed amount Q (and thus act like an (s, Q) policy), for low levels of stock on hand, or order enough to reach a target stock level S (and thus act like an (s, S) policy), for high levels of inventory. Outside these regions however, the optimal policy structure remains unclear. This observation also suggests that a structured policy, which combines the logic of both policies might perform close to optimal. This is precisely the policy that has been introduced in Section 3.4 as the (s, Q S, nq) policy, and whose performance is investigated numerically in Section 3.5.3. Generally, when q > 1, the optimal order quantity is stepwise decreasing as a function of on-hand inventory (as illustrated by Figure 3.3(b)), and the step-down size equals q. We notice that the structure of the optimal solution differs from an (s, S, nq) policy,

3.5 Numerical study 57 K = 10, K1 = 10, K2= 5 35 30 Order quantity 25 20 15 10 L = 0.50 L = 0.33 L = 0.25 5 0 0 4 8 12 16 20 24 28 32 36 Inventory on hand (a) Structure of the optimal policy for q = 1 K = 10, K1 = 10, K2 = 5, L = 0.50 35 30 Order quantity 25 20 15 10 q = 1 q = 4 5 0 0 4 8 12 16 20 24 28 32 36 Inventory on hand (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 Poisson demand with λ = 20, h = 1, p = 50

58 Chapter 3. Inventory control with handling consideration as for low levels of stock on hand, the advice is to order the same quantity, multiple of the batch size q. 3.5.2 Sensitivity analysis In this section, we discuss the sensitivity of the optimal policy and the associated longrun 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 the optimal solution. Therefore, we use the reorder level (rl) and the maximum stock level (ms) as main operational indicators of change. (a) Impact of K 1, K 2 and q Table 3.1 provides a representative set of our results. The first part is meant to illustrate the impact of K 1 and K 2, while the other two parts are referred to illustrate the sensitivity of the results to K and L, respectively. In these cases, we use as a reference a base scenario with parameters: L = 0.5, K = 10 and K 1 = 10, and we vary only one parameter at a time, while keeping the others the same as in the base case. First, based on our results, we observed that for any given q value, the optimal policy for a problem with parameters (K, K 1, K 2, h, p, q) is the same as the optimal policy for a problem with parameters (K, 0, 0, h, p K 1 /q K 2, q), and on the longrun, the average cost difference between the former and the later model is given by (K 1 /q+k 2 ) λ, a term independent of the policy. This observation is in line with some earlier results. Janakiraman and Muckstadt (2004b) show that linear purchase costs (K 2 ) can be assumed to be zero without loss of generality, for general distribution and 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 lead times 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 cost and a positive unit purchasing cost can be transformed into a finite horizon problem with zero unit purchase cost. In view of this result, we may show that that for any fixed value of q, the optimal policy for a problem with parameters (K 1, K 2, h, p, q) and K = 0 is the same as the optimal policy for a problem with parameters (0, 0, h, p K 1 /q K 2, q), and on the long-run, the average cost difference between the former and the later model is given by (K 1 /q+ K 2 ) λ (see Proposition 3.1 in Appendix A). Therefore, when q is predetermined (i.e. determined outside of the system), the batch (K 1 ) and unit (K 2 ) handling costs can be assumed to be zero without loss of generality, for determining the optimal policy, provided that the penalty cost transformation p(q) := p K 1 /q K 2 is accounted for. However, since p(q) is nonlinearly dependent on q, the batch cost component K 1

3.5 Numerical study 59 Table 3.1 Sensitivity analysis with Poisson demands λ = 20, h = 1 and p = 50 (Selective results) K 2 = 0 K 2 = 5 K 2 = 10 L K K 1 q rl ms optcost rl ms optcost rl ms optcost 0.50 10 5 1 36 41 133.634 35 41 233.335 35 41 333.026 2 36 42 83.798 35 42 183.546 35 41 283.244 4 36 43 59.065 35 43 158.796 35 43 258.515 6 36 45 51.095 36 44 150.821 35 44 250.507 12 35 49 44.164 35 49 143.882 35 48 243.575 20 35 57 42.793 35 56 142.521 34 56 242.189 10 1 35 41 233.335 35 41 333.026 34 40 432.625 2 36 42 133.674 35 42 233.413 35 41 333.058 4 36 43 83.999 35 43 183.726 35 43 283.445 6 36 45 67.726 35 44 167.436 35 44 267.122 12 35 49 52.474 35 49 152.192 34 48 251.876 20 35 57 47.781 35 56 147.507 34 56 247.172 20 1 34 40 432.625 34 40 532.166 33 39 631.563 2 35 42 233.413 35 41 333.058 34 41 432.676 4 35 43 133.866 35 43 233.586 35 42 333.261 6 36 44 100.970 35 44 200.665 35 44 300.350 12 35 49 69.093 35 49 168.811 34 48 268.476 20 35 56 57.755 35 56 157.479 34 56 257.136 40 1 30 37 829.600 26 34 927.366-1 -1 1000.000 2 34 41 432.676 34 40 532.205 33 40 631.630 4 35 43 233.586 35 42 333.261 34 42 432.853 6 35 44 167.436 35 44 267.122 35 44 366.804 12 35 49 102.333 35 49 202.051 34 48 301.675 20 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 61 458.414 2 32 62 159.686 32 62 259.313 31 62 358.927 4 33 64 109.924 32 63 209.580 32 63 309.199 6 33 65 93.405 32 65 193.086 32 64 292.693 12 33 69 77.224 32 69 176.863 32 69 276.492 20 33 78 70.944 32 78 170.626 32 77 270.245 100 10 1 31 82 277.3099 30 81 376.8454 29 81 476.3208 2 31 83 177.6941 31 82 277.3151 30 82 376.8516 4 31 84 127.9013 31 84 227.5329 30 83 327.1256 6 31 85 111.3373 31 85 210.9684 30 84 310.5849 12 31 90 94.91544 31 89 194.5429 31 89 294.1674 20 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 36 428.6043 2 32 38 129.5804 31 38 229.3388 31 37 329.0111 4 32 39 79.90654 32 39 179.6496 31 39 279.3827 6 32 41 63.6399 32 40 163.3677 31 40 263.0756 12 31 45 48.46234 31 45 148.1952 31 45 247.9193 20 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 34 426.7099 2 30 36 127.6322 30 36 227.3894 29 35 327.1006 4 30 37 77.97284 30 37 177.709 29 37 277.445 6 30 39 61.68867 30 38 161.4476 29 38 261.1491 12 30 43 46.56806 29 43 146.2972 29 43 246.019 20 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 average cost equals p λ.

60 Chapter 3. Inventory control with handling consideration is particulary relevant for analyzing the impact of q on the system performance, and its 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 rl and ms decrease in general with K 1 and K 2. However, the change in policy is small for small values of K 1 (see Figure 3.4(a)). Moreover, unless K 1 is high, the impact of K 2 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 the batch 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 K 1 are similar. This observation is illustrated 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 (K 1 ) on the optimal solution, L = 1, h = 1, K 2 = 0, λ = 10, p = 10, K = 10 q K 1 optcost rl ms optpolicy 1 0 17.316 20 34 [24 9, 23 3, 22, 12 2, 18 : 13] 2 0 17.329 20 34 [24 11, 22 2, 12 2, 18 2, 16 2, 14 2 ] 4 0 17.373 20 35 [24 12, 12 3, 16 5, 12] 6 0 17.396 20 35 [24 12, 12 3, 18 3, 12 3 ] 12 0 17.475 20 35 [24 12, 12 9 ] 20 0 18.159 20 40 [20 21 ] 30 0 20.058 18 48 [30 19 ] 40 0 23.403 17 57 [40 18 ] 1 5 65.328 17 32 [22 7, 21 4, 20 2, 19, 17 : 14] 2 5 41.535 19 34 [24 6, 22 7, 20, 18 2, 16 2, 14 2 ] 4 5 29.532 19 34 [24 11, 12 3, 20, 16 4, 12] 6 5 25.52 20 35 [24 11, 12 3, 18 4, 12 3 ] 12 5 21.537 20 35 [24 12, 12 9 ] 20 5 20.607 20 40 [20 21 ] 30 5 21.677 18 48 [30 19 ] 40 5 24.618 17 57 [40 18 ] rl = optimal reorder level, ms = optimal maximum stock level; optpolicy U = [U 1, U 2,..., U rl ] means U = [U 1, U 2,..., U rl, 0, 0, 0,...]; Ui n denotes n times U i and U i : U j denotes unit decreasing values from U i to U j (U i > U j ). This observation echoes a point made earlier, namely that as q becomes much greater than K 1, the ratio K 1 /q goes to zero, and thus the optimal policy for the problem with parameters (K, K 1, K 2, h, p, q) (which is equivalent with the optimal policy for the problem with parameters (K, 0, 0, h, p K 1 /q K 2, q)), becomes insensitive to K 1. When the batch cost K 1 = 0, the optimal cost ( optcost in Table 3.2) increases

3.5 Numerical study 61 35 Effect of K1 on optimal policy for L = 0.50, K = 10, p= 50 30 25 Order quantity 20 15 10 K1 = 5 K1= 10 K1 = 20 K1= 40 5 0 0 4 8 12 16 20 24 28 32 36 Inventory on hand (a) Effect of K 1 on the optimal policy Effect of K2 on optimal policy for L = 0.50, K = 10, p= 50 Order quantity 35 30 25 20 15 10 K2 = 0, K1 = 0 K2 = 5, K1 = 0 K2 = 10, K1 = 0 K2 = 0, K1 = 40 K2 = 5, K1 = 40 K2 = 10, K1 = 40 5 0 0 4 8 12 16 20 24 28 32 36 Inventory on hand (b) Effect of K 2 on the optimal policy Figure 3.4 Sensitivity of the optimal policies to changes in K 1 and K 2 for Poisson demands with λ = 20, h = 1, p = 50, L = 0.50, K = 10 and q = 1

62 Chapter 3. Inventory control with handling consideration with q, reflecting having less flexibility in ordering, as q increases. However, when K 1 = 5, the optimal cost decreases with q, for smaller values, and then increases and it eventually converges to cost of the optimal policy without K 1. This is also illustrated in Figure 3.5. It is possible in this way to evaluate the added value of including K 1 into decision making. This idea is further pursued in Section 3.6. Regarding the optimal average cost, we notice from Table 3.1 that the average cost increases with K 1, K 2 and decreases with q, all other things being equal. 70 60 50 min Avg Cost 40 30 20 K1 = 5 K1 = 0 10 0 0 4 8 12 16 20 24 28 32 36 40 44 q Figure 3.5 The optimal average cost as a function of the batch size for K 1 = 0 vs. K 1 = 5, L = 1, h = 1, K 2 = 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 increases with L and q (and there is little interaction between L and q). Additionally, rl decreases with K, while ms increases with K and q. Furthermore, regarding the sensitivity of the average cost to changes in problem parameters, our numerical studies suggest that the optimal long-run average cost is increasing in L and K(all other things being equal). Note that similar monotonic results (w.r.t. L) of the average cost (as well as the infinite horizon discounted costs) are claimed by Zipkin (2008a) for the lost sales 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 the optimal policy determined via dynamic programming, as well as against each other.

3.5 Numerical study 63 First, we briefly describe our methodology. For any (s, Q S, nq) policy, we use a dynamic programming formulation similar to (3.3) in Section 4.3 in order to determine the long-run average cost of the policy. In this case, for any state i Ω, instead of minimizing over all possible order quantities as in the optimality equations (3.3), the order quantity is determined by the logic of the (s, Q S, nq) policy. We next solve the resulting system of equations to determine the long-run average cost C(s, Q, S). To determine the best (s, Q S, nq) policy, we used an exhaustive search over a sufficiently large feasible region to ensure we find a global optimum. We applied a similar methodology 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 the best (s, S, nq) and (s, Q, nq) policy are stated as percentage increase in average costs from the cost of the best (s, Q S, nq) policy. We report the minimum, maximum and average percentage errors over all scenarios ( Total ), and for each individual parameter. In computing the percentage errors, all average costs are first normalized by subtracting from the total costs the constant (K 1 /q + K 2 ) λ. Excluding unit handling-related costs preserves absolute deviations, but makes the percent error higher. The main observation made is that the best (s, Q S, nq) and (s, S, nq) policy are both performing close to optimal, in the range of parameters we considered, while the (s, Q, nq) heuristic is remarkably worse, on average. The average and maximum percentage 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 of 0.05%, deteriorates slightly (at maximum 0.83% in excess cost from the average cost of the best (s, Q S, nq) policy). The performance of the heuristics improves with K and deteriorates with K 1 and K 2, on average. Note that the performance of the (s, Q, nq) heuristic substantially improves for large K and also improves with L (unlike the other heuristics). In Table 3.4 we report similar information, for each value of the handling cost parameters K 1 and K 2. These results also indicate that the (s, Q S, nq) policy is consistently better than the (s, S, nq) heuristics; however, on average, both policies have close to optimal performance. Unlike the (s, Q, nq) heuristic, the performance deteriorates, in general, with increasing values of K 1 and K 2. In fact, as we later on illustrate by an example, when K 1 and K 2 are large, the best (s, S) policy may actually perform worse than the (s, Q) policy, while the best (s, Q S) policy is clearly superior. Notably, numerical results (not reported here) also show that the values of 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 the flatness of the cost curves C(s, Q, S) around the optimal Q value. To illustrate this idea, we plot the average cost of the (s, Q S) policy for one of our examples as a function of each individual parameter. In the example, the problem parameters are

64 Chapter 3. Inventory control with handling consideration Table 3.3 Comparative performance of the (s, Q S, nq) heuristic for 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) 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 16.321 0.33 0.008 0.000 0.109 0.056 0.000 0.843 4.667 0.000 14.961 0.50 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 16.321 50 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 K 1 5 0.009 0.000 0.158 0.042 0.000 0.135 4.794 0.447 16.321 10 0.009 0.000 0.164 0.045 0.000 0.140 4.759 0.435 16.246 20 0.010 0.000 0.176 0.053 0.000 0.211 4.669 0.377 15.958 40 0.011 0.000 0.130 0.083 0.000 0.863 4.062 0.000 15.460 K 2 0 0.008 0.000 0.127 0.049 0.000 0.440 4.726 0.173 16.321 5 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 16.321 2 0.010 0.000 0.066 0.075 0.017 0.222 5.369 0.370 16.310 4 0.006 0.000 0.051 0.063 0.014 0.161 5.200 0.479 15.419 6 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 8.330 20 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, K 1 = 10, K 2 = 5 and q = 1. Figure 3.6 illustrates the shape of the function around the optimal values (s = 34, S = 40, Q = 27). In each plot, we fixed two parameters to their optimal values and plot the cost as a function of the remaining parameter. We observe from the plot that the cost C(s, Q, S) is relatively flat around the optimal s and Q values, respectively and more sensitive to changes in S. Of course, we can t rule out the fact that the Poisson assumption might 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 the parameters are optimized taking the handling costs into account. Next, we report selected results with large material handling costs to clarify a point made earlier. Detailed results for K 1 = 40 and K 2 = 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

3.5 Numerical study 65 Table 3.4 Performance of the (s, Q S, nq) heuristic for different values of K 1 and K 2 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) K 1 K 2 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 16.321 5 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 15.923 Total 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 16.246 5 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 15.639 Total 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 15.958 5 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 15.460 Total 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 15.460 5 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 14.639 Total 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 16.321 5 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 for K = 10), the best (s, Q) policy outperforms the best (s, S) policy. This example demonstrates the following claim: the region where our heuristic differs from the (s, S, nq) policy is the set of inventory levels which are small. Hence, in this case, where the transformed penalty cost is small (in the example the transformed penalty cost p K 1 /q K 2 = 5) the chances of having lower inventory levels will be higher, and hence the percentage increase from optimality 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 policy for few scenarios. Clearly, the best (s, Q S) policy has a simpler structure, which partially captures those of the optimal policies. By restricting the (s, S) policy to a value 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

66 Chapter 3. Inventory control with handling consideration Table 3.5 Performance of the (s, Q S, nq) heuristic: selected results with λ = 20, h = 1 and q = 1 (Percent error) Optimal policy Best (s, Q S) policy Best (s, S) policy Best (s, Q) policy % Diff % Diff % Diff from from from Optimal optimal optimal optimal L p K K1 K2 rl ms cost s S Q Cost cost s S Cost cost r Q Cost cost 0.5 50 10 40 5 26 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 5 20 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 5 16 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 5 23 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 5 17 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 5 13 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 5 21 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 5 15 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 5 12 69 962.767 12 69 63 962.767 0.000 12 67 962.811 0.071 12 63 962.789 0.036 Note: Percent errors are computed based on the normalized costs, i.e. total average costs (K1/q + K2) λ

3.6 Penalty for not taking handling into account 67 660 Sensitivity of the average cost C(s,S*,Q) to changes in s S* = 40, Q* = 27 L=0.5, K = 10, K1 = 20, K2 = 5, q=1 900 Sensitivity of the average cost C(s*,Q, S*) to changes in Q s* = 34, S* = 40 L=0.5, K = 10, K1 = 20, K2 = 5, q=1 640 850 620 800 Avg. Cost (s,q S) 600 580 560 Avg. Cost C(s,Q S) 750 700 650 600 540 550 520 10 15 20 25 30 35 40 s 540 539 Sensitivity of the average cost C(s*,Q*,S) to changes in S s* = 34, Q* = 27 500 5 10 15 20 25 30 35 40 Q L=0.5, K = 10, K1 = 20, K2 = 5, q=1 538 Avg. Cost (s,q S) 537 536 535 534 533 532 30 35 40 45 50 55 60 65 S Figure 3.6 Sensitivity of the average cost C(s, Q, S) around optimal values Example with Poisson demands λ = 20, h = 1, p = 50, L = 0.5, K = 10, K 1 = 10, K 2 = 5, q = 1 performing better than the (s, S, nq) policy (as statistically confirmed by a t-test on total average costs), from the average cost perspective, both heuristics appear to perform very well, especially when the penalty cost in high and the batch and unit handling costs are low. In the reverse situation, the best (s, Q) may actually outperform the best (s, S), while the performance of the new heuristic remains consistently better. Unlike the other two heuristics, the (s, Q S, nq) heuristic better approximates, 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 in practice, are not taken explicitly into account when making inventory replenishment decisions. Hence, it is interesting to study the cost penalty of using a suboptimal policy (obtained by ignoring the handling costs) for a situation when the handling

68 Chapter 3. Inventory control with handling consideration L = 0.50, K1 = 10, K2 = 5, q = 1, md = 20, p= 50 Order quantity 70 60 50 40 30 20 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 10 0 0 4 8 12 16 20 24 28 32 36 Inventory on hand 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 values remain the same. In doing so, we take the following steps: 1. First, we compute the optimal policy and the optimal average cost under the assumption of no handling costs. 2. Then, we plug in this policy into the lost sales model with handling costs to evaluate the corresponding average cost. 3. Finally, we compare the result against the true optimal cost, associated to the optimal 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 K 1 2. We acknowledge both K and K 1 as relevant handling components. In both situations, we assume the unit variable cost K 2 to be zero. Therefore, we measure the penalty for not taking handling into account in two situations, as follows: Gap1 = 100 C (K,K 1)(U (K,0) ) C (K,K 1) C (K,K 1 ) (K 1/q + K 2 ) λ (ignoring only K 1), Gap2 = 100 C (K,K 1 )(U (0,0) ) C (K,K 1 ) C (K,K 1 ) (K 1/q + K 2 ) λ (ignoring K and K 1), where U (K,0) denotes the optimal policy determined under the assumption K 1 = 0, U (0,0) denotes the optimal policy obtained under the assumption K = K 1 = 0,

3.7 Conclusions 69 C (K,K1 )(U) is the average cost of the original model with cost components K and K 1 corresponding to policy U, and C(K,K 1) 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 of the ordering cost (K), the batch cost (K 1 ) and the unit penalty cost (p) as indicated in Tables 3.6 and 3.7. The results reported Table 3.6 consider smaller values of the penalty cost (p = {5, 10}), while those reported in Table 3.7 correspond to higher penalty cost values (p = 50). In choosing the parameter values, we considered the fact that as p K 1 approaches zero, the optimal ordering policy is never order. These results indicate that the cost impact of ignoring the handling costs is higher when K and K 1 are large (in absolute and percentage deviations). Intuitively, the fixed ordering cost, K, affects quite substantially the total costs, when ignored. The penalty for ignoring the unit batch size K 1 increases with K 1, but the percentages are much smaller than those corresponding to K. Finally, we observe that the penalty resulting from ignoring the variable handling costs (K 1 ) is more significant for smaller p values. Low unit penalty costs corresponds to items with lower profit margin, and it represents a large assortment of products in grocery retailing. Thus, for these items, ignoring the handling costs in the decision making, may result in substantial cost penalty. We explain this finding in view of the fact that when the penalty cost is quite high (e.g. p = 50 in Table 3.7) compared to the holding costs, it results in high service levels. With a very high service level, shortages are rare events, and the lost sales model could be approximated by a backorder one. Thus the choice of K 1 (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 lostsales inventory control problem with the following features: there are stochastic customer 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 integer multiples of the batch size. Furthermore, we assume a replenishment cost structure that includes a fixed cost, as well as linear components depending on the number of batches, and the number of units in a replenishment order. Our framework has been inspired from the retail environment, but the analysis is appropriate for systems in which there is a fixed unit-size of stock transfer and there are economies of scale in the replenishment component. Using Markov decision processes, we explored numerically the structure of the optimal policies and investigated, in particular, the impact of material handling costs on the optimal policy and the long-run average cost. Optimal policies have rather complicated structures, which make them difficult to implement in practice, so heuristics are used. In this chapter we present a new

70 Chapter 3. Inventory control with handling consideration Table 3.6 Cost penalty for ignoring K and/or K 1, L = 0.5, λ = 20, h = 1, K 2 = 0, q = 1 p K K 1 OptCost AvgCost1 AvgCost2 Gap1(%) Gap2(%) 10 0 0 19.601 - - - - 10 50 0 54.397-69.601-27.952 5 151.097 152.199 168.180 2.156 33.431 6 169.797 171.759 187.895 3.942 36.346 7 187.626 191.320 207.611 7.756 41.963 8 200 210.880 227.327 27.201 68.317 5 0 0 173.674 - - - - 5 50 0 51.097-67.367-31.842 1 69.797 70.063 86.672 0.535 33.888 2 87.626 89.030 105.976 2.948 38.529 2.5 95.983 98.513 115.628 5.501 42.721 2.6 97.586 100.409 117.559 6.192 43.812 3 100 107.996 125.280 19.989 63.201 OptCost = C (K,K 1 ); AvgCost1 = C (K,K1 )(U (K,0) ); C (K,K1 )(U (0,0) ); Optimal policy is never order AvgCost2 = 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 closeto-optimal performance in both policy and cost space and is consistently performing very well in many settings. We further compare the performance of the heuristic against reasonable alternative policies, the (s, S, nq) and (s, Q, nq) policy. We find that the best (s, S) policies are performing very well especially in settings where the 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, in environments 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 handling costs from decision making. Our numerical studies show that, in general, ignoring the fixed ordering cost (K) may result in substantial cost penalty, while the effect of the fixed batch cost (K 1 ) is better noticed when high, and the unit penalty cost is low. We provide additional insights with respect to the effect of the problem parameters on the system s performance. Our conclusions clearly show that it is worthwhile to explicitly take handling costs into account when making inventory decisions. We have used parameter values that are typical of grocery retail environments, where decision models typically abstain from including these costs. While our heuristic has an excellent performance and clearly improves 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 K 1, L = 0.5, λ = 20, h = 1, p = 50, K 2 = 0, q = 1 K K 1 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,K 1 ); 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 performance of our heuristic for the standard lost-sales inventory control problem, with or without setup 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 cost parameters (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 (x n, q n ) = f n (0;h;p) (x n, q n ) + k, where f n (c;h;p) (x n, q n ) 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 n with x n units of inventory on hand and we order q n units, and k is a term independent of the policy. For the proof, we refer the reader to Janakiraman and Muckstadt (2001). In view of this result, we derive the following result for the infinite horizon, average cost model with batch ordering and no setup cost.

72 Chapter 3. Inventory control with handling consideration Proposition 3.1 Consider the inventory system introduced in Section 4.3 and assume there is no setup cost. For any given batch size q, and all sets of cost parameters (K 1, K 2, h, p) such that p h K 1 /q +K 2, the parameter transformation (K 1, K 2, h, p, q) (0, 0, h, p K 1 /q K 2, q) leads to the following cost transformation: C (K 1,K 2,h,p,q) = C (0,0,h,p K 1 /q K 2,q) + (K 1/q + K 2 ) λ, where C(K 1,K 2,h,p,q) denotes the minimum long-run average cost corresponding to parameters (K 1, K 2, 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 the replenishment cost c r (defined in equation (3.2)) as follows c r (nq) = δ(nq)k + K 1 n + K 2 nq = δ(nq)k + (K 1 /q + K 2 )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) = K 1 /q + K 2 is the per unit purchasing cost (given q). Then, since K = 0, we apply Lemma 3.1 and a limiting argument and deduce that C(K = 1,K 2,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 policy corresponding to (0, 0, h, p c(q), q) is to never order. In this case the minimum long-run average cost equals C(0,0,h,p c(q),q) = (p c(q)) λ = (p K 1/q K 2 ) λ, since all demand is lost. It follows that the optimal policy for the problem with parameters (K 1, K 2, h, p, q) is also the policy of never ordering and the corresponding minimum long-run average cost equals C(K 1,K 2,h,p,q) = p λ. Replacing the average costs in (3.4) it follows that p λ = (p c(q)) λ + k, and thus k = c(q) λ = (K 1 /q + K 2 ) λ. Appendix B. Computational issues A few additional comments regarding the computational study conducted in this chapter are worthwhile mentioning.

3.7 Conclusions 73 Finding the optimal policy: The average cost optimal policies can be obtained either by solving linear programs, or by solving the average cost optimality equations using policy or value iteration algorithms (Puterman, 1994). In this chapter, we solved numerically the average cost optimality equations (3.3) using the 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 was truncated to a size sufficiently large to ensure we find a global optimum. The state space size increases with the mean demand per period. The truncated state space is determined by testing larger and larger sizes until the results are insensitive to the increments. Speed of execution: We observed that the computational time increases with Ω, which indicates the size of the system of equations to be solved in the policy evaluation step. The most time consuming part is the generation of transition probabilities at each step. Policy evaluation: For any given policy, we evaluate numerically the long run average cost by solving the average-cost optimality equations (3.3). The best policy within a given policy class is determined by exhaustive search over its defining parameters.

75 Chapter 4 Retail inventory control with shelf space and backroom consideration Abstract: In an infinite-horizon, periodic-review, single-item inventory system with random demands and lost sales, we study the impact of shelf space constraints on the system s performance. Unlike traditional approaches, we assume that inventory levels may exceed the allocated shelf capacity. This situation captures many retail settings in which the retailer stores surplus stock, which does not fit on the shelves, in the store s backroom. Consequently, each period, the stock is transferred from the backroom to the sales floor to serve end-customer demand, and there is an associated extra handling cost. Two models are considered that include (i) a linear or (ii) a fixed cost component for exceeding the allocated shelf capacity, additionally to a fixed cost for placing an order. In a numerical study, we discuss several qualitative properties of the optimal solutions, conduct sensitivity analyses and quantify the impact of accounting for shelf space constraints explicitly in inventory decisions. 4.1. Introduction In the previous chapter, we considered a single-item inventory replenishment decision faced by a retailer under several realistic constraints, in particular non-negligible shelf stacking costs. We investigated (sub)optimal ordering decisions and illustrated the impact of shelf stacking costs on the system s performance. In this chapter, we extend the retail setting in Chapter 3 to incorporate another real dimension of the retailer s inventory 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 shelf space that accommodates consumer units set to serve end-customer demand. Many products are typically competing for limited shelf space, and decisions on which products to store and where, and how much space should be devoted to each product are usually taken at a tactical level. The ultimate goal is to stimulate demand and maximize sales under various constraints such as limited budget for purchase of products or limited store space for displaying products. We refer to Agrawal and Smith (2009) for a recent review on assortment and shelf space allocation models. Therefore, for operational decisions such as inventory replenishment, the allocated shelf space is typically an exogenous variable, dictated by the planogram 1, and not a decision parameter (Broekmeulen et al., 2004). In this chapter, we make the same assumption and consider an inventory problem of a single product given its allocated shelf space. Due to limited store space, the shelf space allocations at product level are often insufficient to accommodate the demand. Therefore, store managers typically keep surplus stock that does not fit on the shelves in the store backroom. This is particulary true in the grocery sector. Wong and McFarlane (2003) discuss several reasons why retailers keep backroom inventory (including buffering against imperfect deliveries, or insufficient shelf space for some bulky or fast-moving products), while investigating the impact of new technologies such as RFID on shelf replenishment. As a result, the stock temporarily stored at the backroom needs to be transferred from the backroom to the sales floor, to satisfy customer demand. This situation leads to additional store handling operations. The objective of this chapter is to extend the periodic-review single-item lostsales inventory control model studied in Chapter 3 to the case in which there are physical storage constraints at the retailer, and the retailer may use the backroom to temporarily store surplus stock. We are interested in the impact of including these features on the performance of the inventory control models, where performance is measured with respect to the optimal ordering decision and associated long-run average cost. Since the retailer s operational costs are greatly influenced by the material handling operations, aside from the regular inventory-related and shelf stacking costs, an additional cost is charged to the system for exceeding the shelf capacity, to justify the need for additional handling operations. Our chapter is related to the literature on inventory control in single-item periodicreview stochastic systems with lost sales. Consistent with the earlier chapter, we assume here fractional lead times (i.e. lead times shorter in length than the review period) and batch ordering (see e.g. Janakiraman and Muckstadt, 2004a, Hill and Johansen, 2006). Available results on optimality and various heuristics for such systems are reviewed in the previous chapter. Our chapter is also related to the 1 The planogram is a diagram of fixtures and products that illustrates where and how every stock keeping unit should be displayed on the shelf in order to increase customer purchase (Levy and Weitz, 2001)

4.1 Introduction 77 literature on capacitated inventory systems. Traditional capacitated productioninventory models (e.g. Federgruen and Zipkin, 1986a and 1986b) typically impose that the ordering quantity in each period may not exceed a given capacity, and show that when fixed costs are present, the optimal decision is rather complex. Motivated by supply contracts, the inventory literature contains models that allow adjustments on the contracted order quantity. For example, Henig et al. (1997) explore the optimal inventory policies when both upward and downward adjustments on the order quantity are allowed, and the ordering costs are piecewise linear convex. Chao and Zipkin (2008) extend their model to include fixed costs for periodically adjusting the order quantity, and partially characterize the optimal policy. Also, Chiang (2007b) considers storage constraints in a standing order inventory system with both backorder and 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, and solutions using Lagrangian multipliers are classically presented (see e.g. Hadley and Whitin, 1963). Downs et al. (2001) study base stock policies in a multi-period version of this problem with lost sales. They used a linear-programming-based policy to determine optimal order up to levels for multiple products in the presence of resources constraints. Cachon (2001) optimizes the shelf space allocation, in a warehouseretailer setting where the objective is to select a truck dispatching policy, shelf space allocation and an inventory policy that minimize the sum of retailer s transportation, shelf space and inventory costs. In these models, handling costs are typically not included. Our approach differs from many in the literature, in the sense that we do not impose limitations on the order quantity, or maximum inventory levels, but rather acknowledge the presence of storage capacities, allow the maximum stock levels to exceed the available capacity, but then we charge additional costs to the system (see e.g. Kotzab and Teller, 2005 for an empirical study on the additional handling effort needed 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, and investigate the effect of using the backroom on the system s performance. The main contributions of this chapter are as follows. We develop single-item lost-sales inventory control models to explicitly take into account the features of batch ordering, shelf space limitations and backroom operations. We propose two models: (1) the first one assumes continuous replenishment from the backroom and extra handling costs that are proportional to the expected number of units at the backroom at the moment the order arrives; (2) the second model assumes that an additional fixed cost is charged if the inventory position (on-hand plus on order) at each review moment exceeds the 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

78 Chapter 4. Inventory control with shelf space consideration the cost penalty for ignoring the additional handling costs in inventory related decisions. For the second model, we illustrate the additional complexity of the optimal policies, in relation with the first model, as well as with the model that assumes ample shelf capacity. In a numerical study, we illustrate the impact of shelf space (and other parameters) on the system s performance, quantify the effect of additional handling operations, and provide several interesting managerial insights. The remainder of the chapter is organized as follows. Section 4.2 describes the retailer s system that we consider in more detail. Then, in Section 4.3, we introduce two inventory control models for the considered system, to account for limited shelf space and backroom operations. In Section 4.4, we conduct a numerical study on the model with linear extra handling costs, and in Section 4.5, we study the model with fixed 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 stochastic demands. The replenishment lead time is fixed but less than the review period length, the ordering is quantized, i.e., the order quantities are restricted to nonnegative integers, multiple of a fixed batch size q, and any unfilled demand during a review period is assumed to be lost. These features are commonly encountered in grocery retailing, as we already mentioned in earlier chapters. We further consider the following situation. Upon order receipt at the store, the pre-packed deliveries are unwrapped and units are displayed onto the shelves to serve consumer demand. This store handling operation was referred to as shelf stacking in earlier chapters, and will also be referred to as the first replenishment process in this chapter. We assume that shelves have limited storage capacity, and store managers keep surplus stock that did not fit on the shelves in the store s backroom, which creates the need for a second restocking of the shelves. The manual process of shelf restocking with products which are located in the backroom will be referred to in this chapter as the second replenishment process, or simply extra handling. Figure 4.1 illustrates the generic material flow in a retail system with backroom operations. We analyze the system from a total expected cost point of view considering the following 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 Backroom Second replenishment process Incoming stock Receipt First replenishment process Shelves Check out 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 costs of the system in an infinite horizon. The system analyzed in this chapter extends the one considered in Chapter 3. In this chapter, we impose extra handling operations be taken into account in inventory replenishment decisions. Consequently, the cost structure is adjusted to take explicitly into 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 shelves during shelves stocking with new deliveries is temporarily stored at the backroom), the store manager needs to decide not only on the replenishment of inventory, but also on when and how often during the review period should the shelf replenishment from the backroom be executed. In this chapter, we do not address the question of what is the right moment and/or frequency to conduct the second replenishment process. While we acknowledge here several possibilities for shelves restocking with backroom stock (e.g. at the beginning of the review period only, only at the moment the order arrives, at the end of the period only, at any predetermined moment during the 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 second replenishment process, also referred to as extra handling costs, or second replenishment costs 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.

80 Chapter 4. Inventory control with shelf space consideration 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 and 4.5, respectively, we study the models performance, and investigate the impact of including the new retail features (i.e. limited shelf space and extra handling costs) on the performance of the inventory systems. 4.3. Model formulations In this section, we present our basic assumptions on the system under study described in Section 4.2, and introduce two alternative models for capturing the retailer s backroom operations and the associated extra handling costs. In Section, 4.3.1 we assume continuous replenishment from the backroom, at the expense of additional per unit handling costs, while in Section 4.3.2 we assume a one time opportunity for restocking the shelves with backroom stock, at the expense of a fixed cost. The models presented in this chapter are extensions of the inventory control model studied in Chapter 3, which will be referred to in this chapter as the basic lost-sales inventory control model. Therefore, the basic notation in this chapter is consistent with the one in Chapter 3, and is summarized below. R Review period length L Lead time length (0 L R) V Allocated shelf space capacity D L Random demand during lead time Random demand during period R L D R Total demand during a review period, D R = D L + D R L q Fixed (exogenously determined) batch size, q = 1, 2,... K Fixed cost per order K 1 Fixed cost per batch K 2 Variable unit cost K s Unit cost of handling excess inventory 2 K e Fixed cost for extra handling h 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) D R L The main assumptions formulated in Chapter 3 hold in this chapter as well: period demand is stochastic, stationary, independent and identically distributed over time; the random variables D L and D R L are stochastically independent, the order quantities are restricted to the set {0, q, 2q,... }, and an order is delivered within the same review period it has been placed (i.e. 0 L R with L and R constant parameters). 2 Excess inventory is defined as the amount of stock required by the inventory replenishment operations, which exceeds the allocated shelf space.

4.3 Model formulations 81 Additionally, we assume throughout the analysis that the allocated shelf space V 0 is a parameter exogenous to the system, and thus is a fixed parameter. Unlike most approaches in the literature (see e.g., Cachon 2001), we further assume that inventory levels may exceed the capacity V, in which case the system will incur an additional cost, to justify the need for additional handling operations. Obviously, if V is large enough, the backroom usage will be avoided at all times. In this case, the shelf space is a non-binding parameter in the inventory system, which simplifies to the one studied in Chapter 3 (herein called the basic system). The system incurs a number of costs, assumed time invariant. We assume per unit holding and lost-sales penalty costs, with positive rates h and p, respectively. Two types 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, including fixed (K) and variable components (with nonnegative rates K 1 and K 2 ). The novelty in this chapter is the explicit incorporation of the second replenishment costs into the modeling and analysis of inventory decisions. We present in the following subsections two model formulations: one assuming extra handling costs that are proportional to the average excess inventory, and another one assuming a fixed cost for exceeding the shelf capacity. 4.3.1 Model with continuous backroom operations We assume continuous restocking from the backroom, meaning that whenever an item is demanded and is not directly available on the shelf, but it is available in the backroom, it will be used to satisfy the demand. Only in the event that there is a demand 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 possible service, 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 the period, inventory on hand X t is observed; (ii) an order a t is placed (in batches of q units), which will arrive L time units later, but within the same review period, due to our assumption L R; (iii) the demand D L during the leadtime is realized and satisfied with on-hand inventory; unsatisfied demand in lost; (iv) the order placed at the beginning of the period arrives; (iv) the demand D R L in the period R L continues to occur up until the end of the period; any demand that cannot be directly satisfied 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 (K s ) for each unit of stock that exceeds, on average, the

82 Chapter 4. Inventory control with shelf space consideration available shelf capacity V, to reflect additional handling activities such as going back and forth to the backroom. The objective is to minimize the long-run expected average cost of the system. Similar to the earlier chapter, we use Markov decision processes (MDPs) to formulate the mathematical model. Every review moment is a decision epoch, the state of the system is the inventory on hand, X t, at the beginning of the review period, with state space Ω = {0, 1, 2,...}, and the ordering decisions, a t, are limited to the set A = {0, q, 2q,...}. The stock levels of two consecutive review points are related by the balance equation: X t+1 = ((X t D L ) + + a t D R L ) +, t = 0, 1, 2,.... (4.1) where (x) + = max{0, x} for any x R. The probability p xy (a x ) of a transition from state x at one decision epoch to state y at the next decision epoch, given decision a x is defined as p xy (a x ) = P (y = ((x D L ) + + a x D R L ) + ), x, y = 0, 1,..., a x = 0, q, 2q,..., and is detailed in Appendix A. The expected transition costs from one period to the next, given initial inventory x t = x, and decision a t = a is denoted by C t (x, a), and includes the following components: (i) ordering and shelf stacking costs C r t (x, a) = δ(a)k + K 1 a/q + K 2 a, (4.2) where δ(a) = 0, if a = 0 and δ(a) = 1, otherwise, for all a 0, and y denotes the largest integer, smaller or equal than y 0, (ii) costs for holding inventory C h t (x, a) = h E DL,D R L [ ((x DL ) + + a D R L ) +], (4.3) (iii) lost-sales penalty costs and additionally, (iv) extra handling costs C p t (x, a) = p E DL [ (DL x) +] + p E DL,D R L [ (DR L a (x D L ) + ) +], (4.4) C extra t (x, a) = K s E DL [((x D L ) + + a V ) + ]. (4.5) Therefore, the total expected transition cost, defined as C t (x, a) = C r t (x, a) + C h t (x, a) + C p t (x, a) + C extra t (x, a)

4.3 Model formulations 83 is given by C t (x, a) = δ(a)k + K 1 a/q + K 2 a + h E DL,D R L [ ((x DL ) + + a D R L ) +] + p E DL [ (DL x) +] + p E DL,D R L [ (DR L a (x D L ) + ) +]} + K s E DL [((x D L ) + + a V ) + ]. Note that equations (4.1) to (4.4) characterize the basic lost-sales inventory model presented in Chapter 3. We extend this model with an additional cost component, equation (4.5), in which a positive cost K s is charged for the average number of units in excess of V, at the moment the order arrives. Observe that if the shelf space V is large enough to accommodate the incoming stock, then there will be no additional handling costs and the model simplifies to the basic one. Otherwise, we interpret equation (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 a more complete representation of handling-related costs, in the presence of shelf space limitations. As with many lost-sales systems (Hadley and Whitin, 1963), the optimal solution is likely to be quite complex in general. In Section 4.4, we present a numerical analysis of the problem, derive qualitative solution properties and several managerial insights 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 notation and assumptions have already been set earlier in this section. Alternatively to the model presented in Section 4.3.1, we introduce a different costing scheme for the second replenishment process. A store backroom is often a small space, poorly organized. Therefore shelves restocking with backroom stock often requires activities such as finding the items in the backroom, extra processing or administrative costs. These operations are typically independent of the volume of back stock, and the associated handling costs are fixed. Hence, in this section we assume a fixed cost is incurred by the system in the event of using the backroom. As before, our focus is to decide on the inventory control policy which minimizes the long-run average expected cost of the system. Define C extra (x, a) = K e δ ( (x + a V ) +), (4.6) where K e 0. If x + a V, then the above expression is zero. Otherwise, we interpret C extra (x, a) as the additional handling cost from using the backroom for

84 Chapter 4. Inventory control with shelf space consideration 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 the inventory position at the beginning of the period to reflect the worst case scenario in which lead time demand could be zero. Furthermore, we assume the same ordering (plus shelf stacking) and holding costs per period as in the basic model: (i) ordering and shelf stacking costs (ii) costs for holding inventory C r t (x, a) = δ(a)k + K 1 a/q + K 2 a, (4.7) C h t (x, a) = h E DL,D R L [ ((x DL ) + + a D R L ) +]. (4.8) However, with respect to the lost-sales penalty cost in period t, we make an additional assumption. Namely, we calculate the lost-sales cost depending on the amount of stock visible on the shelf at the beginning of the period, i.e. min{x t, V } instead of the inventory on hand x t 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 on order, is given by (iii) lost-sales penalty costs C p t (x, a) = p E DL [ (DL min{x, V }) +] + p E DL,D R L [ (DR L a (min{x, V } D L ) + ) +]. (4.9) Assuming that the extra handling costs are calculated using relation (4.6), i.e. (iv) extra handling costs C extra t (x, a) = K e δ ( (x + a V ) +), (4.10) the total expected cost in period t, defined as the sum of all cost components, is given by C t (x, a) = Ct r (x, a) + Ct h (x, a) + C p t (x, a) + Ct extra (x, a). C t (x, a) = δ(a)k + K 1 a/q + K 2 a + he DL,D R L [ ((x DL ) + + a D R L ) +] + p E DL [ (DL min{x, V }) +] + p E [ (D R L a (min{x, V } D L ) + ) +] + K e δ ( (x + a V ) +). (4.11) The performance of this model, under the long-run average expected cost criterion, is investigated numerically in Section 4.5. Note that the basic model (studied in Chapter 3) 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 continuous backroom operations We focus on the model with continuous backroom operations described in Section 4.3.1. The purpose of this section is threefold: (1) to provide insights into the structure of optimal policies (Section 4.4.1), (2) to conduct sensitivity analyses with respect to the problem parameters (Section 4.4.2), and (3) to quantify the cost penalty for excluding the second replenishment process from the decision making (Section 4.4.3). This problem is difficult to tackle analytically, thus we rely on numerical studies to address it. We have computed optimal policies and the associated long-run average cost for a variety of instances, all with a Poisson demand process with mean λ, constant leadtime L, linear holding, penalty and extra handling costs with rates h, p and K s, respectively, and fixed ordering cost K. Unless otherwise specified, all problem instances in this section have R = 1, L = 0.50, λ = 20, h = 1, K 1 = K 2 = 0 and q = 1. The remaining problem parameters are set as follows: p = {2, 5, 10, 20, 40, 100, 200} K = {5, 10} K s = {5, 10, 20, 40, 100, 200} V = {10, 15, 20, 25, 30, 40, 50} The combination of all parameters resulted in 588 problem instances in our numerical study. The values for the penalty cost p are chosen to reflect products with low and hight profit margins, while the values for the shelf space V vary around the mean demand. When V is much larger than the mean demand, it will become ineffective in the model, which simplifies to the basic lost-sales inventory model. We assume the extra handling costs to be generally higher than the shelf stacking costs, and several values were selected to facilitate sensitivity analyses. To determine an optimal policy and the corresponding long-run average cost for an inventory system with a cost objective as described in Section 4.3.1, we used a similar methodology to the one reported in Chapter 3. We applied the value iteration algorithm to solve the average-cost optimality equations. 4.4.1 On the structure of the optimal policy In this section, we investigate numerically the structure of optimal policies for the inventory model presented in Section 4.3.1 The extra handling cost structure, given by 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 optimal policies for the basic model in Chapter 3.

86 Chapter 4. Inventory control with shelf space consideration For a given problem instance, we denote by rl and ms the reorder and maximum stock levels, respectively, associated with an optimal policy of the basic model (as discussed in Chapter 3, these values exist). An optimal policy defines a rule that indicates, depending on the initial stock on hand x, the amount a(x) to be ordered at each review moment. Therefore, similarly to Chapter 3, we define rl = max{x Ω : a(x) > 0} (i.e. the highest value of inventory on hand at which it is optimal to order a positive amount) and ms = max{x + a(x) : 0 x rl }. Given the shelf space V, the optimal policies for the extended inventory model are depicted in Figure 4.2 for few scenarios with K = 10, p = 10 and K s = 20. Similarly to the solutions of the basic model (see Chapter 3), these results reveal that there seems to exist a non-negative reorder level, denoted by rl V, and for inventory levels higher than rl V, it is not optimal to place an order, i.e. rl V = max{x Ω a V (x) > 0}. Furthermore, the optimal order quantity a V (x), as a function of on hand inventory, has no simple closed-form expression, but for a given V, it may be approximated by an (s, Q S) policy (see again Chapter 3). Figure 4.2 also illustrates the effect of V on optimal solutions. As V increases, the reorder level rl V and order quantities a V (x) increase, resulting in higher maximum stock levels ms V (defined as ms V = max{x + a(x) : 0 x rl V }), in general. This suggests higher average stock on hand, but on average, less sales lost and less stock in excess 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 accommodate the incoming stock, thus resulting in a need for a second replenishment process, with additional handling costs. 4.4.2 Sensitivity analyses: the effect of V, K s and p In this section, we discuss the sensitivity of the optimal solution and the associated long-run average cost to the system parameters. In particular, we investigate the effect of V, Ks and p on the optimal solution and minimum average cost. Since the optimal policy is not simple-structured, we use the reorder level (rl V ) and the maximum stock level (ms V ) 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 report the reorder level (rl V ), the maximum stock level (ms V ) associated to the optimal policy, 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 rl V and ms V are monotone increasing (i) or decreasing (d) in p, K s and V as follows:

4.4 Numerical study: the model with continuous backroom operations87 30 25 Order quantity 20 15 10 5 V = 10 V = 15 V = 20 V = 25 V = 30 V = 40 0 0 5 10 15 20 25 30 Inventory on hand Figure 4.2 Optimal order quantity as a function of on hand inventory: λ = 20, L = 0.50, h = 1, p = 10, K = 10, K 1 = K 2 = 0, K s = 20, q = 1. Note. (rl V =10, ms V =10 ) = (16, 19), (rl V =15, ms V =15 ) = (20, 24), (rl V =20, ms V =20 ) = (24, 27), (rl V =25, ms V =25 ) = (27, 31), (rl V =30, ms V =30 ) = (29, 35), (rl V =40, ms V =40 ) = (rl, ms ) = (30, 37)

88 Chapter 4. Inventory control with shelf space consideration Table 4.1 Sensitivity analysis with Poisson demands with mean λ = 20, h = 1, L = 0.50 and q = 1 K s = 5 K s = 10 K s = 20 K s = 40 K s = 100 K s = 200 p K V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V 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 57.152 15 22 25 36.999 20 23 37.985 19 22 39.095 18 21 40.184 17 20 41.532 17 19 42.417 20 24 28 27.472 24 27 28.619 23 26 29.706 22 25 30.727 22 24 31.954 21 24 32.819 25 27 32 23.572 27 31 23.984 26 30 24.436 26 30 24.973 25 29 25.587 25 28 26.052 30 28 34 22.412 28 34 22.451 28 33 22.496 28 33 22.546 28 33 22.677 28 32 22.776 40 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 28 34 22.367 50 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 62.152 15 20 25 41.999 19 23 42.985 18 22 44.095 17 21 45.184 16 20 46.532 16 19 47.417 20 23 28 32.472 22 27 33.619 21 26 34.706 21 25 35.727 20 24 36.954 20 24 37.819 25 25 32 28.572 25 31 28.984 24 30 29.436 24 30 29.973 24 29 30.587 23 28 31.052 30 26 34 27.411 26 34 27.450 26 33 27.495 26 33 27.545 26 33 27.677 26 32 27.776 40 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 26 34 27.366 50 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 15 107.472 15 25 29 58.190 23 26 63.664 21 24 65.852 20 22 68.279 19 21 71.540 18 20 73.781 20 27 30 37.317 26 29 40.514 25 27 43.511 24 26 46.223 23 25 49.403 22 25 51.704 25 30 33 28.492 29 32 29.827 28 31 31.259 28 31 32.784 27 30 34.673 27 29 35.876 30 31 36 25.102 31 35 25.326 31 35 25.670 31 34 26.021 30 34 26.659 30 33 27.036 40 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 31 37 24.601 50 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 15 112.472 15 24 29 63.190 22 26 68.664 20 24 70.852 19 22 73.279 18 21 76.540 17 20 78.781 20 26 30 42.317 25 29 45.514 24 27 48.511 23 26 51.223 22 25 54.403 22 25 56.704 25 28 33 33.492 28 32 34.827 27 31 36.259 27 31 37.784 26 30 39.673 26 29 40.876 30 30 36 30.102 29 35 30.326 29 35 30.670 29 34 31.021 29 34 31.659 29 33 32.036 40 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 30 37 29.600 50 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 16 207.742 15 29 32 75.466 27 29 102.725 24 26 116.955 22 24 121.542 20 22 128.432 19 21 133.533 20 30 33 51.165 28 31 59.033 27 29 66.319 26 28 73.156 25 26 80.857 24 25 86.115 25 32 35 35.322 31 33 38.574 30 32 42.019 29 31 45.597 28 30 50.091 28 30 53.440 30 33 37 28.331 33 36 29.188 33 36 30.159 32 35 31.200 32 34 32.903 32 34 34.106 40 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 50 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 16 212.742 15 28 32 80.466 26 29 107.725 23 26 121.955 21 24 126.542 20 22 133.432 19 21 138.533 20 29 33 56.165 28 31 64.033 26 29 71.319 25 28 78.156 24 26 85.857 23 25 91.115 25 31 35 40.322 30 33 43.574 29 32 47.019 29 31 50.597 28 30 55.091 27 30 58.440 30 32 37 33.331 32 36 34.188 32 36 35.159 31 35 36.200 31 34 37.903 31 34 39.106 40 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 50 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563

4.4 Numerical study: the model with continuous backroom operations89 p K s V rl V i d i ms V i d i Monotonicity with respect to p aims to reduce shortages. Monotonicity with respect to K s and V aims to reduce the expected stock above V. As V increases, so do rl V and ms V (see Figure 4.3 for an example). However, the higher the storage capacity V, the lower the effect of K s on the optimal solution. Furthermore, the optimal policy is more sensitive to changes in the values of V for higher values of K s, rather than for lower values. Intuitively, as K s converges to zero, the optimal solution converges to the one of the basic model, and thus becomes insensitive to V. Numerical results show that the long-run average cost is decreasing in V, and increasing in K s and p, all other parameters being equal. Figure 4.4 illustrates this behavior (see Appendix B for numerical details). p K s V C V i i d Monotonicity of the minimum average cost with respect to V confirms an earlier observation 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 represents a lower bound on the optimal cost of the extended model. All other parameters being fixed, the minimum cost appears convex in V, as Figure 4.5 illustrates for few scenarios. The cost increase with K s is also rather intuitive in that if the unit cost for extra handling is higher, the minimum long-run average cost is higher. However, as can be seen from Figure 4.5, the higher V, the smaller the effect of an increase in K s on the optimal cost. Figure 4.4 also indicates that the relative impact of K s on the minimum cost does not only decrease with V, but also increases with the unit penalty cost p. The interaction effect of p and K s on the minimum average cost at V = 20 is illustrated in Figure 4.6. The minimum average cost increases with both p and K s, but for smaller values of p, the relative effect of K s on cost is smaller. Finally, we observe that the size of the positive effect of p on the minimum average cost decreases with V (see Figure 4.4). Note that even when V is high enough to become non-binding in the extended model, p still has a positive impact on the minimum cost (unlike K s, which becomes irrelevant in the model).

90 Chapter 4. Inventory control with shelf space consideration 40 35 Reorder level (rl V ) 30 25 20 15 10 Ks = 5 Ks = 10 Ks = 20 Ks = 40 Ks =100 Ks = 200 5 5 10 15 20 25 30 35 40 Shelf space (V) (a) Sensitivity of rl V to changes in V and K s Max. stock level (ms V ) 40 35 30 25 20 15 10 Ks = 5 Ks = 10 Ks = 20 Ks = 40 Ks =100 Ks = 200 5 5 10 15 20 25 30 35 40 Shelf space (V) (b) Sensitivity of ms V to changes in V and K s Figure 4.3 Sensitivity of rl V and ms V to V and K s. Fixed parameters λ = 20, h = 1, L = 0.50, p = K = 10, q = 1.

4.4 Numerical study: the model with continuous backroom operations91 450 400 350 300 Min. Avg. Cost 250 200 150 100 Ks = 5 Ks = 10 Ks = 20 Ks = 40 Ks = 200 50 0 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 V Figure 4.4 Effect of V, K s and p on the minimum long-run average expected cost: λ = 20, L = 0.50, K = 10, q = 1 250 200 Min. Avg. Cost 150 100 50 Ks = 5 Ks = 10 Ks = 20 Ks = 40 Ks = 100 Ks = 200 0 0 0.5 1 1.5 2 2.5 3 Shelf space vs. mean demand (V/λ) Figure 4.5 Effect of V, K s on the minimum long-run average expected cost: λ = 20, L = 0.50, K = 10, q = 1, p = 20

92 Chapter 4. Inventory control with shelf space consideration 450 400 350 300 Min. Avg. Cost 250 200 150 100 Ks = 5 Ks = 10 Ks = 20 Ks = 40 Ks = 100 50 0 0 20 40 60 80 100 120 140 160 180 200 Penalty cost (p) Figure 4.6 Effect of p and K s 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 the shelf space capacity (V ) and all relevant costs in inventory related decisions. We focus on: (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 additional handling costs from the optimization of inventory decisions. Cost decomposition First, we illustrate the effect of V on the different cost components by an example. The following parameter values are chosen in our numerical example: λ = 20, L = 0.50, q = 1, K = 10, K 1 = K 2 = 0, h = 1, p = 10 are fixed parameters, and we vary the shelf space V {10, 15, 20, 25, 30, 40, 50} and the unit extra handling cost K s {5, 20}. The retailer s long-run average expected cost (herein simply referred to as 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

4.4 Numerical study: the model with continuous backroom operations93 (referred here as Optimal cost ) is computed numerically by value iteration. The contribution of each cost component to the optimal cost is determined as follows. When K > 0 and h = p = K s = 0 we obtain the average order cost, when h > 0 and K = p = K s = 0 we obtain the average inventory-holding cost, when p > 0 and K = h = K s = 0 we obtain the average lost-sales penalty cost, and similarly, when K s > 0 and K = h = p = 0 we obtain the average extra handling costs. For each scenario, we determined numerically the optimal inventory control policy, the associated long-run average cost and the individual cost components. We summarize our 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 penalty costs 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 = 20 and L = 0.50, h = 1, p = 10, K = 10, q = 1, K s = 5 Optimal Cost Lost-sales Extra V K s Ordering Holding penalty handling Total 10 5 10.000 8.360 28.073 41.714 88.147 15 10.000 8.334 28.248 16.607 63.190 20 10.000 10.471 17.333 4.513 42.317 25 10.000 13.208 8.663 1.829 33.492 30 10.000 15.799 3.897 0.406 30.102 40 9.998 16.757 2.845 0.000 29.600 50 9.998 16.757 2.845 0.000 29.600 10 20 10.000 1.256 100.001 0.167 111.424 15 10.000 5.144 52.416 3.292 70.852 20 10.000 8.758 26.220 3.533 48.511 25 10.000 11.824 12.666 1.769 36.259 30 9.999 14.957 5.086 0.627 30.670 40 9.998 16.757 2.845 0.000 29.600 50 9.998 16.757 2.845 0.000 29.600 We also notice in Table 4.2 that when the unit extra handling cost K s increases from 5 to 20, for given V, the expected holding costs decrease, the expected lost-sales penalty costs increase, the expected extra handling costs decrease, while the total average 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 face by ignoring the second replenishment costs in inventory replenishment decisions. As mentioned in earlier chapters, handling-related costs (though generally much larger

94 Chapter 4. Inventory control with shelf space consideration Cost decomposition 45 Ordering Holding Lost-sales penalty Extra handling 40 35 30 Cost 25 20 15 10 5 0 0 0.5 1 1.5 2 2.5 Shelf space vs. mean demand (V/λ) Figure 4.7 Individual cost components as a function of V for Poisson demand with λ = 20 and L = 0.5, h = 1, p = 10, K = 10, q = 1, K s = 5 than inventory-related costs at the store level) are usually not accounted for explicitly in inventory control models. Thus, we quantify the cost penalties at the retailer by comparing the total average cost (which includes ordering, holding, lost-sales penalty, and extra handling costs) in two situations: (1) all cost components are included in the optimization of inventory decisions vs. (2) extra handling costs are not part of the optimization (but included in 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 (K s ); see Section 4.3.1), denoted by U V, and the optimal policy of the basic model (without the shelf space and extra handling costs), denoted by U. The long-run average cost corresponding to each policy is denoted by CV = C(U V, V ) and C V = C(U, V ), respectively, where C(U, V ) denotes the total average cost of the extended model under any inventory policy U. Note that since U is generally suboptimal for the extended model, it holds that C V CV. We quantify the impact of the second replenishment process as the percentage

4.4 Numerical study: the model with continuous backroom operations95 difference (denoted by % ) between C V and C V as follows: % = 100 C V C V C V, (4.12) and we interpret % as the percentage of cost penalty the retailer may face by ignoring the extra handling costs in the optimization of inventory decisions. We aim to obtain insight into the magnitude of expected cost penalties. We have computed C V, C V and % in a numerical experiment. To facilitate numerical comparison, we used the same parameter values as in the previous numerical example. Table 4.3 summarizes our results. In the table, we report not only on the total costs, but also on the individual cost components. For convenience, we repeat here the results already reported in Table 4.2. The results in Table 4.3 show that the cost penalty at the retailer may be substantial when the additional costs resulting from handling stock in excess of V are not explicitly included in the optimization of inventory decisions. The percentage of cost penalty (% ) increases with K s, and interestingly, it appears that if the allocated shelf capacity is close to the mean demand, we have the largest percentage of cost deviation (see also Figure 4.8(a)). Figure 4.8(b) illustrates the percentage deviation calculated for each individual cost 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 retailer from using the suboptimal policy U (instead of UV ) in the extended model, which appears to peak when V is around the mean demand. The trade-off, however, would be less sales lost, on average (i.e. negative % Lost-sales penalty ). Intuitively, at optimality, as the retailer tries to save on additional handling costs, she may order less, which in turn increases the average lost-sales. Finally, we note that when decisions are separated (i.e. the extra handling costs are not included in inventory optimization, yet are part of the total cost) substantial excess stock may result at the retailer, which decreases with V (as shown by Figure 4.9). Figure 4.9 also depicts the long-run average optimal (CV ) and expected cost (C V ), as a function of V. As expected, C V dominates CV by optimization, and the difference diminishes with higher values of V. We conclude based on these results that the retailer may indeed benefit from the integration of shelf space and resulting extra handling costs in the optimization of inventory decisions. 4.4.4 Summary In an infinite-horizon, periodic-review, single-item retail inventory system with random demand and lost-sales, we study the feature of limited shelf space and

96 Chapter 4. Inventory control with shelf space consideration Table 4.3 Percentage of cost penalties: an example with Poisson demand with mean λ = 20, L = 0.5, h = 1, p = 10, K = 10, q = 1 Optimal Cost (C V ) Cost (C V ) % Lost-sales Extra Lost-sales Extra V Ks Ordering Holding penalty handling Total Ordering Holding penalty handling Total Total 10 5 10.000 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 = optimal long-run average cost for the extended model CV = long-run average cost for the extended model under policy U

4.4 Numerical study: the model with continuous backroom operations97 300 250 Percentage cost penalty (% ) 200 150 100 50 Ks = 5 Ks = 20 0 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Shelf space vs. mean demand (V/λ) (a) Percentage of cost penalties in total average cost 710 649.7 610 Percentage cost penalty (% Δ) 510 410 310 210 110 10 100.4 100.9 101.1 254.0 60.0 26.9 507.5 6.1 136.6 %ΔOrdering %ΔHolding %ΔLost-sales penalty %ΔExtra handling -90-0.02-0.02-0.02-0.02-0.01-27.0-67.2-89.9-89.9-83.6 V/λ = 0.5 V/λ = 75 V/λ = 1 V/λ = 1.25 V/λ = 1.5 (b) Percentage of cost penalties in individual cost components for K s = 5 Figure 4.8 Percentage of cost penalties (% ) in total average cost (a) as well as individual components (b): an example with Poisson demand with mean λ = 20 and L = 0.50, K = 10, h = 1, p = 10, q = 1, V {10, 15, 20, 25, 30, 40, 50}

98 Chapter 4. Inventory control with shelf space consideration 120 100 80 Cost 60 40 Average cost Optimal cost Extra handling cost 20 0 0 0.5 1 1.5 2 2.5 3 Shelf space vs. mean demand (V/λ) Figure 4.9 Average costs for λ = 20, L = 0.5, K = 10, p = 10, q = 1, K s = 5 Note: Average cost = C V = C(U, V ), Optimal cost = CV = C(U V, V ) and Extra handling cost = K s EExcess(U, V ) backroom usage, and assume that additionally to a fixed cost per order, a linear cost is charged if the inventory position exceeds, on average, the available shelf space capacity. 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 give several managerial insights into the effect of problem parameters on the system s performance; additionally, we investigate the relevance of the second replenishment process for decision making. Our analysis demonstrates that the retailer s long-run average cost decreases with the shelf space V, and increases with the unit extra handling cost K s, and unit penalty cost p, and there is a combined nonlinear effect of the problem parameters on the minimum cost. Finally, we show that including the extra handling costs in the optimization of inventory decisions results not only in a better representation of system s costs, but also allows the retailer to achieve substantial cost savings from the optimization of all relevant costs. 4.5. Numerical study: the model with fixed extra handling costs In this section, we provide a numerical investigation on the inventory control model introduced in Section 4.3.2. This model extends the basic lost-sales inventory control model studied in Chapter 3 to the case in which a fixed cost K e 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 analysis complements the one conducted in Section 4.4, where we assumed that the extra handling cost is proportional to the average stock in excess of V. The purpose of this section is threefold. First, we provide insights into the structure of optimal policies (Section 5.3), then we conduct sensitivity analyses (Section 4.5.2) and provide additional managerial insights (Section 4.5.3). In our numerical experiments, we consider all possible combinations of the following problem parameters: Poisson demand distribution with mean λ = 10, R = 1, L = 1, h = 1, K 1 = K 2 = 0, q = 1 and p = {5, 10, 20} K = {5, 10}, K e = {0, 5, 10, 20, 40, 100, 200, 400} V = {10, 15, 20, 25, 30, 40} The combination of all parameters resulted in 288 problem instances. We vary mostly the shelf space V and the fixed extra handling cost K e, as we aim to conduct sensitivity analyses mostly around these two problem parameters. In practice, we expect the extra handling cost K e to be larger than the regular shelf stacking cost K. Intuitively, through very large values of K e, we attempt to capture situations in which 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 and the associated long-run average cost. 4.5.1 On the structure of the optimal policy In this section, we provide several qualitative insights into the structure of optimal policies. We start by observing that, when the shelf space V is sufficiently large, it becomes 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 the optimal policy of the basic model is known to posses no simple structure, it is expected that the structure of the optimal policy of the extended model is also intricate. Similar to Section 4.4.1, to an optimal policy of the basic model, we associate two operational parameters, denoted by rl and ms, which define the optimal reorder point and optimal maximum stock level, respectively. We computed the optimal policies for the extended model for each scenario in our numerical experiment. The optimal policies (expressed by the order quantity as a function of on hand inventory) are depicted in Figure 4.10 for selected scenarios. These examples illustrate that the optimal policies exhibit complex structures, in general. As anticipated, there is no apparent simple form solution, and the optimal order quantity as a function of on-hand inventory is usually not monotone and discontinuous. Nevertheless, results suggest that there exists an inventory level rl V

100 Chapter 4. Inventory control with shelf space consideration such that if the beginning inventory level is above rl V nothing should be ordered. Similar to Section 4.4, we define ms V = max{x + a V (x) : 0 x rl V }, where a V (x) denotes the order quantity as a function of on-hand inventory. K = 10, Ke = 20 K = 5, Ke = 20 30 16 Order quantity 25 20 15 10 5 V = 15 V = 40 Order quantity 14 12 10 8 6 4 2 V = 15 V = 40 0 0 5 10 15 20 25 0 0 5 10 15 20 25 Inventory on hand Inventory on hand (a) Case A and C K = 10, Ke = 20 K = 5, Ke = 20 30 16 Order quantity 25 20 15 10 5 V = 20 V = 40 Order quantity 14 12 10 8 6 4 2 V = 20 V = 40 0 0 5 10 15 20 25 0 0 5 10 15 20 25 Inventory on hand Inventory on hand (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, K 1 = K 2 = 0, q = 1, K e = 20, K = {5, 10} Note: Case (A): (rl V, ms V ) = (19, 33) for K = 10, (rl V, ms V ) = (21, 25) for K = 5 Case (B): (rl V, ms V ) = (17, 20) for K = 10, (rl V, ms V ) = (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 optimal policies are worthwhile mentioning. First, it appears that there exist two threshold values (possibly rl and ms ) such that, depending on the value of V, the optimal ordering 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

4.5 Numerical study: the model with fixed extra handling costs 101 in situations (A) and (C) is depicted in Figure 4.10(a) (for two scenarios with K = 5 and 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 space V is sufficiently large such that the extended and the basic model have identical solutions; 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 numerical results that the optimal decision is to order such that the maximum inventory level ms V 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 the inventory position after ordering does not exceed V, i.e. a V (x) V x; however for higher inventory levels, although the storage capacity V after ordering is exceeded, it appears that there exists an order-up-to level S such that V x < a V (x) S x, with V S. The intuition behind this observation could be as follows: if there are few items on hand at a review moment, then we may take the opportunity of ordering in such a way that we avoid paying the additional fixed cost K e, and thus we order up to V (on hand plus on order V ); on the other hand, if the initial stock on hand is higher than V, then whatever we order will result in an inventory position above V and the cost K e is unavoidable. In such a case, the optimal solution is less sensitive to V, as if we are solving a basic lost-sales inventory model. These examples also suggest that many parameters would be required to fully characterize the structure of the optimal policy, which is more complicate than the (s, Q S) heuristic introduced in Chapter 3. Furthermore, by comparison with the model discussed in Section 4.4, the model with a fixed cost for extra handling generates additional complexities in the optimal decision pattern. 4.5.2 Sensitivity analyses: the effect of V, K e and p In this section, we discuss the sensitivity of the solution and the associated long-run average cost to changes in V, K e and p. Therefore, we assume for simplicity that the batch and unit handling costs are fixed to zero. Table 4.4 presents a summary of our results, for different values of shelf space V, extra handling cost K e, and unit lost-sales penalty cost p. For each scenario, we report the reorder level (rl V ), the maximum stock level (ms V ), 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 of the optimal ordering pattern, we mainly study the variation of these indicators with changes in the model parameters. Based on these results we make the following observations.

102 Chapter 4. Inventory control with shelf space consideration Table 4.4 Sensitivity analysis with Poisson demands λ = 10, L = 1, h = 1, K1 = K2 = 0, q = 1 Ke = 0 Ke = 5 Ke = 10 Ke = 20 Ke = 40 Ke = 100 Ke = 200 Ke = 400 p K 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 rlv msv C V 5 5 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 6 10 30.182 6 10 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 5 10 10 15 22 18.986 15 22 23.970 15 23 28.933 13 20 32.729 11 17 32.729 4 10 32.729 4 10 32.729 4 10 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 5 10 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 5 10 60.086 5 10 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 5 10 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

4.5 Numerical study: the model with fixed extra handling costs 103 First, we note that when K e = 0, the shelf space V may still affect the system s performance (unless V is sufficiently high to become non-binding in the extended model). This observation stems from the fact that the cost structure of the extended model differs from the one of the basic inventory model in two ways: the way we charge the lost-sales penalty cost, and the way we charge the extra handling costs (see Section 4.3.2 and Appendix C for alternative model formulations). When V is sufficiently high, the extended model reduces to the basic inventory control model (studied in Chapter 3), and the system becomes insensitive to K e. On the other hand, when the fixed cost for extra handling K e is very high, the optimal decision aims to avoid stock in excess of V, which results in maximum stock levels below the shelf capacity (i.e. ms V V ). Figure 4.11 illustrates the effect of V and K e on the structure of the optimal policy for few scenarios. In our numerical settings, the values of rl V and ms V are monotone increasing (i), decreasing (d), or non-monotone (x) in p, K e and V as follows: p K e V rl V i d x ms V i d x Monotonicity with respect to p aims at reducing the expected lost sales, while monotonicity with respect to K e aims at cutting deliveries above V. The complex pattern in optimal decisions, as discussed in the previous section, is also reflected in a non-monotone behavior of rl V and ms V with respect to V. On the cost side, numerical results show that the optimal long-run average cost decreases with V, and increases with K e and p, all other parameters being equal. p K e V C V i i d Figure 4.12 illustrates this behavior by some examples. As shown in Figure 4.12(a), the positive effect of K e on the optimal tcost decreases with increasing values of V, and for large V, the solution becomes insensitive to K e. For a given V, the optimal cost increases with K e up to a maximum value (see also Figure 4.12(b)). By inspection of results in Table 4.4, we note that when the upper limit is reached, the model s optimal decision is usually to order at most to the capacity V. Figure 4.12(b) illustrates not only the main effects of p and K e on the optimal cost, but also the interaction effects. The rate of cost increase with p is accentuated by K e.

104 Chapter 4. Inventory control with shelf space consideration 25 Ke = 20 20 Order quantity 15 10 5 V = 10 V = 15 V = 25 V = 40 0 0 5 10 15 20 25 Inventory on hand (a) λ = 10, K = 10, p = 10, K 1 = 0, q = 1, K e = 20 25 V = 15 20 Order quantity 15 10 Ke = 5 Ke = 10 Ke = 20 5 0 0 5 10 15 20 25 Inventory on hand (b) λ = 10, K = 10, p = 10, K 1 = 0, q = 1, V = 15 Figure 4.11 Effect of V and K e on the optimal policy

4.5 Numerical study: the model with fixed extra handling costs 105 70 60 Min. avg. cost 50 40 30 20 Ke = 0 K3 = 5 Ke = 10 Ke = 20 Ke = 40 Ke = 100 10 0 10 20 30 40 50 60 Shelf space (V) 70 (a) Effect of V and K e on the average cost for p = 10 60 50 Min.avg.cost 40 30 20 p = 5, V = 15 p = 10, V = 15 p = 20, V = 15 10 0 0 5 10 20 40 100 200 400 Fixed extra handling cost (Ke) (b) Effect of p and K e on the average cost for V = 15 Figure 4.12 Effect of V, K e and p on the minimum long-run average expected cost: λ = 10, L = 1, h = 1, q = 1, K = 10

106 Chapter 4. Inventory control with shelf space consideration Finally, based on these results we infer that, given V, the following bounds on the optimal cost (C V ) of the extended model can be identified: when K e = 0, the optimal cost of the resulting reduced model (see Appendix C for the cost structure) is a lower bound on C V ; alternatively, when K e is large enough, the shelf space V will act like a hard constraint in the extended model, and the corresponding 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 4.4.3 in order to gain additional insights on the impact of V and K e on the system s performance. Therefore, we focus on two aspects: (i) first, we illustrate the impact of V on the different cost components of the long-run average cost and (ii) second, we quantify the cost penalty for excluding the extra handling costs from the optimization of inventory decisions. We accomplish these goals by means of a numerical example. The following parameters are chosen in our numerical experiment: λ = 10, L = 1, q = 1, K = 10, K 1 = K 2 = 0, h = 1, p = 10, K e = 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 expected ordering, holding, lost-sales penalty, and extra handling cost for exceeding the shelf capacity V as follows: Cost = K OF + h EOH + p ELS(V ) + K e 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 of ordering above V on the long run. We used a similar methodology to the one described in Section 4.4.3 to compute, for each scenario, the optimal solution and the associated long-run average cost (CV ), as well as the individual cost components. We present our findings in Table 4.5 (columns one to six). We remark, based on this example, that the optimal cost components do not exhibit a 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 (possibly v = rl = 20), such that when V v there is no extra handling cost (suggesting that there are no orders exceeding the shelf capacity), while the ordering and lost-sales penalty cost decrease, and the holding cost increases with V, respectively. Thus, only when V < v, the system incurs extra handling costs. When V becomes sufficiently high, it has no effect on system s performance. These observations are consistent with earlier findings and reflect the complexity of the ordering pattern.

4.5 Numerical study: the model with fixed extra handling costs 107 Table 4.5 Impact of shelf space V on individual cost components for Poisson demand with λ = 10 and L = 1, h = 1, p = 10, K = 10, q = 1, Ke = 20 Optimal Cost (C V ) Cost (C V ) % Lost-sales Extra Lost-sales Extra V Ordering Holding penalty handling Total Ordering Holding penalty 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 = optimal long-run average cost for the extended model CV = long-run average cost for the extended model under policy U When V large enough, (rl, ms ) = (20, 34)

108 Chapter 4. Inventory control with shelf space consideration Quantifying the effect of second replenishment Next, we investigate the impact of the second replenishment process on the system s performance. More precisely, we quantify the cost penalties at the retailer by comparing the total cost (including extra handling costs) of two policies: (1) optimal policy of the extended model (UV ) 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 are reported 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 C V decreases with V. The ordering 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 case in 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 costs are unavoidable (in both optimal and suboptimal case) and smaller penalties result in total cost. Finally, we note that, by using an integrated optimization of all relevant costs, always saves the costs for extra handling. However, this gain trades-off with other cost components, especially the cost of ordering and the penalty cost for expected demand lost (see Figure 4.13). Ordering Holding Lost-sales penalty Extra handling Total 20 15 Subopt -Opt Cost 10 5 0-5 -10 10 15 20 25 30 Shelf space (V) Figure 4.13 Cost difference C V CV 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, K e = 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 with random demand and lost-sales, we study the feature of limited shelf space, and assume that in addition to a fixed cost per order, a fixed cost is charged in the event of inventory position exceeding the available shelf space. Furthermore, we distinguish between sales floor and backroom stock, and assume that the average demand lost is measured against the amount of stock visible on the shelf. In Section 4.3.2, we introduce an extended model that particularly takes these features into account. In this section, via a numerical study, we investigate qualitative properties of the optimal solutions, draw managerial insights regarding the effect of problem parameters on the system s performance, and quantify the effect of the second replenishment process for decision making. Our analysis demonstrates that the presence of fixed extra handling costs into the model highly complicates the structure of the optimal policies, and we provide several qualitative structural insights. We show that the retailer s long-run average cost decreases with the shelf space V, and increases with the extra handling cost K e and the unit lost-sales penalty cost p. As V increases (i.e. more shelf space), the solution of the extended model converges to the one of the basic lost-sales inventory model, while as K e increases (i.e. more expensive backroom operations), an incentive is created of ordering up to the available shelf capacity V. Finally, we provide 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 item facing stochastic demand and periodically reviews its stock for replenishment. The replenishment lead time is less than the review period length and excess demand that cannot be satisfied from stock on hand is lost. We allow batch ordering and assume that the batch size has been predetermined. Furthermore, due to insufficient shelf space, the retailer may temporarily keep surplus stock, which does not fit onto the shelves upon delivery, in the store s backroom. Hence a distinction is made between the sales floor and back stock. Each period, the surplus stock is transferred from the backroom to the regular shelves to serve end-customer demand (referred to as the second replenishment process), and there is an associated extra cost for handling backroom stock. Since the retailer s total costs are greatly influenced by the material handling operations, we investigate the effect of using the backroom on the combined cost of ordering, holding, lost sales and merchandise handling, in an infinite-horizon inventory system. Two models are proposed to control the retailer s inventory, given a limited allocated shelf space and backroom usage. The first one assumes continuous backroom operations and extra handling costs that are proportional to the average back stock.

110 Chapter 4. Inventory control with shelf space consideration Alternatively, the second model assumes that the extra handling costs are independent of the number of units at the backroom. Instead, a fixed cost is incurred by the system when the maximum inventory levels exceed the allocated shelf capacity. We investigated the performance of the two models with respect to the optimal ordering policies and associated long-run average cost. Furthermore, we investigated the effect of including the shelf space and the additional handling costs on the inventory replenishment decisions. Similar to lost-sales inventory control models, the proposed variants are analytically involved. Hence, we solved each model numerically in an extensive computational study, and provided qualitative insights into the structure of the optimal policies. We illustrated the additional structural complexity inherited by the solutions of the model with fixed extra handling costs in comparison with the solutions of the model with linear costs, as well as the solutions of the basic lost-sales model (i.e. without shelf space consideration). Furthermore, we provided several managerial insights into the effect of problem parameters (in particular the shelf space) on the optimal solution and associated long-run average cost. Finally, we showed that the retailer may greatly benefit from the explicit consideration of the second replenishment process, by quantifying the cost penalty the retailer may face by excluding the extra handling costs from the optimization of inventory decisions. The results and insights from this study could serve in addressing more general questions regarding the shelf space optimization, or the effective management of the backroom, for example. Furthermore, extensions into the direction of managing the inventory 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 the models introduced in Section 4.3. The probability p ij (a i ) of a transition from state i at one decision epoch to state j at the next epoch, given decision a i at the first decision epoch is defined as p ij (a i ) = P ( j = ((i D L ) + + a i D R L ) +), i, j = 0, 1,..., a i = 0, q, 2q,..., and is given by p i0 (0) = P(D R i), i = 0, 1,..., p ij (0) = P(D R = i j), i = 0, 1,..., j = 1, 2,..., i, p 00 (0) = 1,

4.6 Conclusion 111 when there is no order, and when the order amounts to a i = n i q > 0 by p i0 (a i ) = p ij (a i ) = p ij (a i ) = i 1 P(D L = k)p(d R L i + a i k) + P(D L i)p(d R L a i ), i = 1, 2,..., k=1 i 1 P(D L = k)p(d R L = i + a i j k) + P(D L i)p(d R L = a i j), k=1 i+a i j k=1 p 00 (a 0 ) = P(D R L a 0 ), i = 1, 2,..., j = 1, 2,..., a i, P(D L = k)p(d R L = i + a i j k), p 0j (a 0 ) = P(D R L = a 0 j), j = 1, 2,..., a i, p ij (a i ) = 0, otherwise. i = 1, 2,..., j = a i + 1,..., a i + i, Appendix B. Additional numerical results: with continuous backroom operations model Table 4.6 Sensitivity analysis with Poisson demand with mean λ = 20, h = 1, L = 0.5 and q = 1 K e = 5 K e = 10 K e = 20 K e = 40 K e = 100 K e = 200 p K V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C 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 26.759 15 17 22 21.307 16 21 21.588 16 20 21.861 15 20 22.134 15 19 22.463 14 18 22.677 20 20 26 19.475 19 25 19.642 19 25 19.822 19 24 19.979 18 23 20.189 18 23 20.354 25 21 29 18.883 21 29 18.905 21 28 18.925 21 28 18.957 21 28 19.018 21 27 19.048 30 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 40 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 21 30 18.855 50 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 16 207.742 15 29 32 75.466 27 29 102.725 24 26 116.955 22 24 121.542 20 22 128.432 19 21 133.533 20 30 33 51.165 28 31 59.033 27 29 66.319 26 28 73.156 25 26 80.857 24 25 86.115 25 32 35 35.322 31 33 38.574 30 32 42.019 29 31 45.597 28 30 50.091 28 30 53.440 30 33 37 28.331 33 36 29.188 33 36 30.159 32 35 31.200 32 34 32.903 32 34 34.106 40 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 34 39 26.563 50 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 17 407.891 15 33 35 89.991 30 33 133.614 27 30 191.346 24 26 223.538 22 23 236.105 21 22 247.017 20 33 35 65.178 31 33 84.703 29 31 102.214 27 29 117.830 26 27 136.306 25 26 148.512 25 34 36 43.949 33 35 50.701 32 34 58.340 31 33 66.230 30 31 76.155 29 30 83.939 30 35 38 32.561 35 37 34.581 34 37 36.782 34 36 39.221 33 35 42.894 33 35 46.324 40 36 41 28.335 36 41 28.335 36 41 28.336 36 41 28.336 36 41 28.339 36 41 28.343 50 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

112 Chapter 4. Inventory control with shelf space consideration Table 4.6 (continued) K e = 5 K e = 10 K e = 20 K e = 40 K e = 100 K e = 200 p K V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V rl V ms V C V 15 36 38 106.542 34 36 167.519 32 34 266.196 29 31 407.168 25 26 543.286 23 24 566.885 20 36 39 81.561 34 37 117.678 32 34 167.448 30 32 218.073 28 29 272.331 27 28 309.609 25 36 39 57.396 35 37 72.243 34 36 90.616 33 34 110.386 31 33 138.410 30 32 159.042 30 37 40 39.850 37 39 44.456 36 38 50.097 35 37 56.663 35 36 66.263 34 35 74.725 40 39 43 30.485 39 43 30.501 39 43 30.533 39 43 30.597 39 43 30.784 39 42 30.951 50 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 2007.931 15 38 41 117.495 37 39 189.462 35 37 313.699 33 34 513.564 29 30 899.410 25 26 1076.200 20 38 41 92.497 37 39 139.486 35 37 213.890 33 34 316.132 30 31 447.316 28 29 529.831 25 38 41 67.743 37 39 90.977 36 37 122.551 34 36 160.464 33 34 215.270 32 33 258.668 30 39 41 46.749 38 40 54.717 37 39 64.898 37 38 77.131 36 37 95.711 35 36 111.299 40 41 45 32.129 41 44 32.253 41 44 32.388 41 44 32.655 40 43 33.271 40 43 33.598 50 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 31.756 15 14 22 26.307 14 21 26.587 13 20 26.861 13 20 27.133 12 19 27.461 12 18 27.675 20 17 26 24.473 16 25 24.640 16 25 24.821 16 24 24.977 15 23 25.188 15 23 25.353 25 18 29 23.872 18 29 23.895 17 28 23.918 17 28 23.950 17 28 24.013 17 27 24.043 30 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 40 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 18 30 23.840 50 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 16 212.742 15 28 32 80.466 26 29 107.725 23 26 121.955 21 24 126.542 20 22 133.432 19 21 138.533 20 29 33 56.165 28 31 64.033 26 29 71.319 25 28 78.156 24 26 85.857 23 25 91.115 25 31 35 40.322 30 33 43.574 29 32 47.019 29 31 50.597 28 30 55.091 27 30 58.440 30 32 37 33.331 32 36 34.188 32 36 35.159 31 35 36.200 31 34 37.903 31 34 39.106 40 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 33 39 31.563 50 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 17 412.891 15 32 35 94.991 30 33 138.614 27 30 196.346 24 26 228.538 22 23 241.105 20 22 252.017 20 32 35 70.178 30 33 89.703 29 31 107.214 27 29 122.830 25 27 141.306 25 26 153.512 25 33 36 48.949 32 35 55.701 31 34 63.340 30 33 71.230 29 31 81.155 29 30 88.939 30 34 38 37.561 34 37 39.580 33 37 41.782 33 36 44.221 33 35 47.894 32 35 51.324 40 35 41 33.335 35 41 33.335 35 41 33.335 35 41 33.336 35 41 33.339 35 41 33.343 50 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 1012.930 15 35 38 111.542 34 36 172.519 32 34 271.196 29 31 412.168 25 26 548.286 23 24 571.885 20 35 39 86.561 34 37 122.678 32 34 172.448 30 32 223.073 28 29 277.331 26 28 314.609 25 36 39 62.396 35 37 77.243 33 36 95.616 32 34 115.386 31 33 143.410 30 32 164.042 30 37 40 44.850 36 39 49.456 36 38 55.097 35 37 61.663 34 36 71.263 34 35 79.725 40 38 43 35.485 38 43 35.501 38 43 35.533 38 43 35.597 38 43 35.784 38 42 35.951 50 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 2012.931 15 37 41 122.495 36 39 194.462 34 37 318.699 32 34 518.564 28 30 904.410 25 26 1081.200 20 37 41 97.497 36 39 144.486 35 37 218.890 33 34 321.132 30 31 452.316 28 29 534.831 25 38 41 72.743 37 39 95.977 35 37 127.551 34 36 165.464 32 34 220.270 31 33 263.668 30 38 41 51.749 38 40 59.717 37 39 69.898 36 38 82.131 35 37 100.711 35 36 116.299 40 40 45 37.129 40 44 37.253 40 44 37.388 40 44 37.655 40 43 38.271 39 43 38.598 50 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 min{x, V } No extra cost basic reduced Extra cost - extended We detail here, for ease of reference, the underlying cost structure for each of the models. Given x units on hand at the beginning of a review period and a units ordered, the expected transition cost C(x, a) is defined as follows. Basic model: no extra cost, no V C(x, a) = δ(a)k + K 1 a/q + K 2 a + h E DL,D R L [ ((x DL ) + + a D R L ) +] + p E DL [ (DL x) +] + p E DL,D R L [ (DR L a (x D L ) + ) +] Reduced model: no extra cost, V included C(x, a) = δ(a)k + K 1 a/q + K 2 a + h E DL,D R L [ ((x DL ) + + a D R L ) +] + p E DL [ (DL min{x, V }) +] + p E DL,D R L [ (DR L a (min{x, V } D L ) + ) +] Extended model: extra cost, V included C t (x, a) = δ(a)k + K 1 a/q + K 2 a + h E DL,D R L [ ((x DL ) + + a D R L ) +] + p { E DL [ (DL min{x, V }) +] + E DL,D R L [ (DR L a (min{x, V } D L ) + ) +]} + K e δ ( (x + a V ) +)

115 Chapter 5 Efficient control of lost-sales inventory systems with batch ordering 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 case have 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 a numerical study, we compare the performance of the heuristic with those commonly used, as well as optimal solutions, under a variety of conditions and demonstrate its effectiveness. Furthermore, we conduct sensitivity analysis with respect to the problem parameters and provide managerial insights. Additionally, we compare the performance of the optimal policy against the most common heuristic used in the retail practice, the (s, nq) policy 1. 5.1. Introduction and related literature In the previous chapters we dealt with a single-item inventory replenishment decision commonly faced by grocery retailers. In Chapter 3, we incorporated a special material 1 Also denoted in the literature by (k, Q) (Veinott, 1965) with k the reorder level and Q the batch size (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 batch size (assumed a decision variable). Similar to earlier chapters, we assume that Q is an exogenous parameter, hence denoted by q, and we exclude from our policy notations the review period length R (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 Chapter 4, 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 a variant of the fundamental lost-sales inventory control problem, i.e., the one-location, one-product, periodic-review inventory replenishment system, which includes the following features: batch ordering and fractional lead times. The general description of the system is that the item s inventory is managed periodically and a decision on how much to order has to be made; a replenishment order, when placed, is delivered before the beginning of the next review moment, which results in lead times (L) shorter than the length of the review period (i.e. fractional lead times); multiple units of items are 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 is lost, rather than backlogged. Three types of costs are involved: for ordering, for holding inventory and for losing demand, and we assume that the cost of ordering is non-negative and fixed (i.e., setup cost K 0). The objective is to determine an inventory policy that minimizes the long-run average cost of the system. Compared to earlier chapters, we therefore assume in this chapter that the ordering cost structure is independent of the number of batches and units in the order, and there is ample storage capacity at the retailer. We complement the analysis in the previous 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 alternative assumption 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 policy parameters (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, though a common assumption in practice. If K = 0 and q > 1 then the optimal policy is of (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, nothing should be ordered (see, e.g., Veinott, 1965, Iwaniec, 1979). This policy is a natural extension 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 optimal solution for this problem has yet not been fully demonstrated for the backordering (or lost-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 2 In the literature, the terminology batch ordering typically refers to the problem with the restriction 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). 3 Veinott s original notation (k, Q), with Q a fixed positive constant.

5.1 Introduction and related literature 117 q when the inventory position is at or below the threshold, else do nothing) for the backorder model. A reasonable extension of the well known (s, S) 4 policy to the batch ordering case would result in an (s, S, nq) policy, as already described in Chapters 1 and 3, where an order is placed if the inventory position falls to or below s, and the order size is the largest integer multiple of q which results in the inventory position not 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 batch ordering 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 Base stock or (S-1,S) policy (s,s) policy Order quantity Order quantity S-s q = 1 Inventory on hand S s Inventory on hand S (s,nq) or (S-q,S,nq) policy (s,s, nq) policy Order quantity Order quantity [(S-s)/q]q q q > 1 Inventory on hand s s Inventory on hand S Figure 5.1 A literature review for the backorder case The inventory control problem addressed in this chapter assumes lost-sales instead of backordering of unmet demand. The lost-sales model has been recognized to be more complex than the equivalent backorder model, and for positive leas times, the complete 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 for the lost-sales case, unless restrictive conditions apply (see e.g., Huh et al., 2009). The approximation of the lost-sales model by its backorder counterpart performs badly, in 4 In an (s, S) policy a replenishment order is placed if, at a review moment, the inventory position drops to or below s, and the order size is the amount required to raise the inventory position to S.

118 Chapter 5. Efficient control of lost-sales inventory systems general (see e.g., Zipkin, 2008b). In Chapter 3, we presented a more extensive review of the related literature. Table 5.1 summarizes the findings for easy reference. We conclude based on our overview that mostly studied are inventory models with no setup 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 Lead Assump. Batch Order Obj Policy time cost Karlin & Scarf 1958 G D L = R no zero C optimal Gaver 1959 G D L = R no zero C base stock Morse 1959 G D L = R no zero C base stock Morton 1969 G D L = nr no zero C optimal Morton 1971 G D L = nr no zero C myopic, approximate Nahmias 1979 G G L = nr no zero C optimal, base stock, approximate Downs et al. 2001 G D L = nr no zero C base stock Johansen 2001 P D L = nr no zero C optimal, base stock Janakiraman& 2004 G G L = nr no zero C base stock Roundy Janakiraman& Muckstadt 2004a G G L R no zero C optimal, myopic, base stock Reiman 2004 G D L = nr no zero C constant order Chiang 2006 G D L R no zero C optimal, base stock Levi et al. 2008 G D L = nr no zero C dual-balancing Zipkin 2008a G D L = nr no zero C several, compared numerically Zipkin 2008b G D L = nr no zero C optimal Huh et al. 2009 G D L = nr no zero C base stock Nahmias 1979 G G L = nr no pos. C optimal, (s, S), approximate 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 optimal Chiang 2007a G G L R no pos. C optimal Demand distribution P = Poisson, N = Normal, G = general L = lead time (D = deterministic, G = stochastic), R = review period length L = 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 batch ordering constraint into account, assume fractional lead times, and allow zero or fixed ordering costs. We show numerically that the optimal policy structure of the most generic model is, in general, more complex than the structure of the optimal solutions of the model with fixed ordering cost, or batch ordering only. In particular, we provide numerical examples to show that the optimal policy in case of positive lead times and zero, or fixed setup costs is not of (s, nq), or

5.1 Introduction and related literature 119 (s, S, nq) type, respectively. 2. Since optimal policies have rather complicated structures, we propose as an alternative heuristic the (s, Q S, nq) policy, and we analyze it numerically for zero and positive setup costs, unit and non-unit batch sizes, as illustrated in Figure 5.2. In a numerical study, we compare the performance of the heuristic with those commonly used as well as optimal solutions, and demonstrate its superiority and effectiveness. Hence, our analysis complements the one in Chapter 3, and we provide additional insights. 3. We discuss several managerial insights regarding the impact of problem parameters, in particular the batch size, on the optimal policies and the associated long-run average cost. 4. Finally, we investigate numerically the performance of the (s, nq) policy, a heuristic commonly implemented in automatic ordering systems at grocery retailers (Van Donselaar et al., 2009). K = 0 K > 0 (S-1,Q S) policy (s,q S) policy Q Q Order quantity Order quantity S-s q = 1 Inventory on hand S s Inventory on hand S (S-q,Q S,nq) policy (s,q S, nq) policy [(Q)/q]q [(Q)/q]q Order quantity Order quantity [(S-s)/q]q q q > 1 Inventory on hand s Inventory on hand s S 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 numerical study, we present in Section 5.3 a qualitative investigation of the optimal policies, as well as sensitivity analyses with respect to the problem parameters. Then, we test

120 Chapter 5. Efficient control of lost-sales inventory systems the performance of the (s, Q S, nq) heuristic and demonstrate its effectiveness against common alternatives, the (s, S, nq) and (s, Q, nq) policies (Section 5.4). In Section 5.5, we study the performance of the (s, nq) against the optimal policy. Finally, we conclude the study. 5.2. Model We consider a periodic-review single-item stochastic inventory system in which any unfilled demand is assumed to be completely lost, the replenishment orders are restricted to non-negative integers multiple of a fixed batch size, and the replenishment lead time is fixed but less than the review period length. The notation in this chapter follows that of earlier chapters. R Review period length L Lead time length (0 L R) t Period index (t = 0, 1, 2,... ) X t On-hand inventory at the beginning of period t a t Quantity ordered in period t Random demand during lead time 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) D L D R L 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 levels X t are observed and the current period s ordering decision (a t ) is made; the orders are restricted to integral multiples of q; (ii) next, demand D L occurs during the lead time; (iii) then, orders arrive following their respective lead time L; (iii) then, demand D R L continues to occur up until of the end of the current period. Any unsatisfied demand during the period is lost. Also, any positive order is charged a fixed cost K. The holding and penalty costs are charged at the end of the period. We assume that D R is stochastic, independent and identically distributed across periods and follows a non-negative known discrete distribution. Furthermore, similar to earlier chapters, we assume that the length of the review cycle R is exogenous to the model (for example determined by the need of coordinating replenishment of many different items) and we further assume it to be the time unit. Next, we use Markov Decision Processes (MDPs) to formulate the problem. The decision epochs are the beginning of review periods, the state of the system is X t, defined on Ω = {0, 1, 2,...}, and the action is a t, with action space A = {0, q, 2q...}.

5.2 Model 121 The system dynamics are as follows X t+1 = ((X t D L ) + + a t D R L ) +, t = 0, 1, 2,..., where (x) + = max{0, x} for any x R. We consider the infinite horizon averagecost model with the objective to determine an inventory policy that minimizes the long-run average cost of the system. The one-period transition probabilities and expected costs are given next. The probability p xy (a) of a transition from state x at one decision epoch to state y at the next epoch, given action a at the first decision epoch is defined as p xy (a) = P ( y = ((x D L ) + + a D R 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 D L and D R L are stochastically independent. The total expected cost from one decision epoch to the next, given initial starting inventory x and the amount ordered a = nq (n Z + ), is denoted by C(x, a) and is given by C(x, a) = δ(a)k + h E DL,D R L [ ((x DL ) + + a D R L ) +] + p E DL [ (DL x) +] + p E DL,D R L [ (DR L a (x D L ) + ) +], where δ(a) = 0 if a = 0 and δ(a) = 1, otherwise, for all a 0. Note that the difference between this formulation and the one in Chapter 3 is the ordering cost, assumed here to be independent of the number of batches and units in the order. Commonly, an expected average cost optimal stationary policy is determined by solving the following average cost optimality equations: v x = min {C(x, a) g + p xy (a)v y }, x Ω, (5.1) a A y Ω where (v x ) x Ω are called the relative values (proven unique if an initial condition such as v 0 = 0 is imposed), and g is the long-run average cost of the system. Similar to earlier chapters, we solve the optimality equations via the value iteration algorithm. In the remainder of the chapter, we conduct a numerical study whose objective is manifold: first, to explore the structural form of the optimal policy for the inventory system described in Section 5.2, and to conduct sensitivity analyses with respect to the problem parameters, in particular the fixed batch size q (Section 5.3), second to determine the effectiveness of the (s, Q S, nq) policy (Section 5.4), and finally, to investigate the performance of the (s, nq) policy, a commonly used heuristic in the retail environment (Section 5.5).

122 Chapter 5. Efficient control of lost-sales inventory systems 5.3. On the structure of the optimal policy: K = 0 vs. K > 0 As our literature review reveals, it is difficult to analytically derive properties of the optimal ordering solutions for the lost-sales case. The few approaches that exist are usually limited to the case of no setup cost and no batch ordering. Hence, in this section, we numerically explore the structure of the optimal policies for a zero, as well as a positive ordering cost, assuming unconstrained orders (i.e. q = 1), as well as batch ordering (i.e. q > 1). The lead time is either zero (Section 5.3.1), a fraction of the review period length (Section 5.3.2), or equal to the review period length (Section 5.3.3). Additionally, we present several insights from sensitivity analyses with respect to the impact of problem parameters on the optimal solutions and the corresponding long-run average expected costs. We conduct several numerical studies involving the following parameters, the lead time L, the penalty cost ratio p/h, the ordering cost parameter K, the fixed batch size q and the mean demand per review period λ. Throughout, we assume that the random demands D L and D R L are stochastically independent and both follow a Poisson 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} {1,2,4,6,12,20} 1 {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. Our selection of parameter values are reasonable choices in a retail environment (see Chapter 2). To determine an optimal policy and the associated long-run average cost we applied the value-iteration algorithm (to solve equations (5.1)), for 1000 iterations or until the change between two iterations was less than 10 12 (for more details see e.g. Puterman, 1994, and Bertsekas, 1995). 5.3.1 The case L = 0 The special case of zero lead time approximates environments where products have a very short replenishment lead times, compared to their review period length. In grocery retailing, for example, many products ordered in the morning are received by the stores at the end of the day, or the beginning of the next day at the latest. When the 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 aware of 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; if K = 10, (s, S ) = (20, 33) and C(s, S ) = 17.443). In fact, for zero lead time and no 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 35 30 Order quantity 25 20 15 10 K=0, q=1 K=0, q=4 K=10, q=1 K=10, q=4 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Inventory on hand (a) Optimal order quantity 40 L = 0, md=10, h=1, p=10 35 Order up to levels 30 25 20 15 10 K=0, q=1 K=0, q=4 K=10, q=1 K=10, q=4 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 Inventory on hand (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 a reorder point such that if inventory on hand is above the reorder point no orders should be placed, otherwise, the optimal order quantity is a stepwise decreasing function of 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

124 Chapter 5. Efficient control of lost-sales inventory systems illustrative example when q = 4. In this example, if K = 0, the optimal policy advises to order 28, if there is nothing on stock, and zero if stock on hand is higher than or equal to 25. Otherwise, the order is an integer multiple of 4, identical for 4 consecutive values, and decreasing in steps of 4 units. The corresponding minimum average cost is C(U ) = 8.641. When there is a fixed order cost of K = 10, it is optimal to order if there 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 minimum average cost in this example is C(U ) = 17.482. As a result, the optimal order-up-to level is a stepwise function of on-hand inventory, which takes values in an interval of length q. In other words, it is optimal to order-up-to an interval of length q > 1 (see again 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 the class of (s, S, nq) policies is a straighforward candidate to optimality (see also Section 5.5.1). Unlike some assumptions in the literature (Hill, 2006), our numerical results show that, in general, the best s and S parameters are not multiples of q, and the optimal 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 examine the effect of changes in the values of different parameters on the system performance. Detailed numerical results for selected parameter values are included in Appendix A. Here and in the rest of the chapter, we define the optimal reorder level as rl = max{i Ω a i > 0}, i.e. the highest value of inventory on hand at which it is optimal to order a positive amount; the optimal maximum stock level is defined to be the maximum inventory position, after ordering, i.e. ms = max{i + a i 0 i rl}. (a) The case K = 0 First, let us focus on the inventory system with no setup cost. The optimal policies for 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 are worthwhile 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 a simple closed-form. It is observed however, in all scenarios, that there exists an optimal reorder point (rl) and for stock levels x rl, the optimal order function is monotonically decreasing, and the rate of decrease is less than or equal to q. Our findings are consistent with earlier studies of Janakiraman and Muckstadt (2004a) and Chiang (2006), who extend previously known structural properties of Karlin and Scarf (1958) and Morton (1969) from the case of integer lead times to the case of fractional lead times. However, none of these studies consider batch ordering and thus our observations are more general.

5.3 On the structure of the optimal policy: K = 0 vs. K > 0 125 p=10, md=20,k=0,q=1 30 25 Order quantity 20 15 10 L=1/2 L=1/3 L=1/4 5 0 0 3 6 9 12 15 18 21 24 27 30 33 36 Inventory on hand (a) Optimal order quantity for K = 0, q = 1 L = 1/2, md = 20, K = 0 30 25 Order quantity 20 15 10 p=10, q=1 p=10, q=4 5 0 0 3 6 9 12 15 18 21 24 27 30 33 36 39 Inventory on hand (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.

126 Chapter 5. Efficient control of lost-sales inventory systems 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 Table 5.9 in Appendix A), there seems to exist a region of higher inventory levels in which the optimal policy specifies the order quantity according to an (s, nq) policy. Outside this region however, the optimal policy structure is more involved. Sensitivity analysis In the absence of simple-structured policies, it is difficult to evaluate the impact of changes in the problem parameters on the optimal solution. Therefore, similar to earlier chapters, we use the reorder level (rl) and the maximum stock level (ms) as main operational indicators of change. Numerical results suggest that rl is 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 in problem parameters, our numerical studies suggest that the optimal long-run average cost 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 are demonstrated by Zipkin (2008a) for the lost-sales model with lead times that are integers multiple of the review period length. Impact of q > 1 When q > 1, the optimal order quantity is a stepwise decreasing function of onhand inventory; see Figure 5.4 (b) and also Appendix A for more results. In our numerical studies we observed that the step-down size equals q and thus relation qn x+1 qn x qn x+1 + q, x = 0, 1,... holds true, which is a generalization of previously 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 results show that rl + q = ms, when K = 0. Furthermore, sensitivity studies reveal that the optimal long-run average cost is non-decreasing in q (all else being equal), indicating that a higher batch size leads to a higher average cost. Intuitively, there is less flexibility in ordering with higher batch sizes, hence increased costs. Moreover, we observe 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 parameter q = mq (m {1, 2, 3... }) is also a feasible solution for the problem with parameter q (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, then Cq C q, where Cq (and C q, respectively) denotes the optimal expected average

5.3 On the structure of the optimal policy: K = 0 vs. K > 0 127 cost corresponding to the batch size q (and q, respectively). As a consequence, we derive the following relation between the optimal average costs for models with different batch sizes: C q=1 C q C 2q C(s, n(2q)) (the last inequality results from the suboptimality of the (s, n(2q)) policy). This gives lower and 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. The optimal policies for few problem instances are illustrated in Figure 5.5. The fixed ordering 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 effect of non-unit batch sizes on the optimal policy structure (here q = 4 vs. q = 1). We also refer to Appendix A (Table 5.10) for further numerical results. The main insights from our numerical study are as follows. The optimal policy has no clear or simple structure (see again Figure 5.5). In fact, while the optimal order quantity is monotonically decreasing for K = 0, in the presence of a positive order cost (K > 0), there are many scenarios where the optimal order quantity is nonmonotone in on-hand inventory (as Figure 5.5 (c)-(d) illustrates). However, as noted before for the case K = 0, there seems to exist an optimal reorder level (rl), below which it is always optimal to place an order and beyond which it is never optimal to order. Thus, rl plays the role of a reorder level in a general inventory policy. This observation has been conjectured before for continuous review systems (see Hill and Johansen, 2006), but to our knowledge no proofs exist so far in the literature, in a continuous 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 1 when initial stock on hand is not too low. Hence, the optimal maximum stock level is 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, while for K = 0 the optimal order quantity satisfies rl = ms 1, when K > 0, an order size 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 the structure of the optimal policy. An inventory policy with such complex structure, which exhibits jumps and nonmonotonicities (and it may be possibly described by many parameters) could be difficult to explain to practitioners, thus hindering the possibility of implementation. Therefore, it is desirable that in many environments facing lost sales reasonable heuristic policies are being applied. In Section 5.4, we study the effectiveness of the new (s, Q S, nq) heuristic policy, as compared to that of two reasonable alternative heuristics, the (s, Q, nq) and (s, S, nq) policy. Finally, in Section 5.5 we investigate the performance of the (s, nq) policy commonly applied in a retail setting.

128 Chapter 5. Efficient control of lost-sales inventory systems p = 10, md=20, K = 10, q=1 p=10, md = 20, K=10 30 30 25 25 Order quantity 20 15 10 L =1/2 L =1/3 L =1/4 Order quantity 20 15 10 L=1/2, q=1 L=1/2, q=4 5 5 0 0 3 6 9 12 15 18 21 24 27 30 33 0 0 3 6 9 12 15 18 21 24 27 30 33 Inventory on hand Inventory on hand (a) (b) p = 10, md=20, K = 20, q=1 p=10, md= 20, K=20; q = 4 45 50 Order quantity 40 35 30 25 20 15 10 5 L = 1/2 L = 1/3 L = 1/4 Order quantity 45 40 35 30 25 20 15 10 5 L = 1/2 L = 1/3 L = 1/4 0 0 3 6 9 12 15 18 21 24 27 30 0 0 3 6 9 12 15 18 21 24 27 30 Inventory on hand Inventory on hand (c) (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))

5.3 On the structure of the optimal policy: K = 0 vs. K > 0 129 Sensitivity analysis Regarding the impact of problem parameters on the optimal policy, we make the following observations: as K increases, the optimal reorder point (rl) decreases, while the maximum stock on hand (ms) increases. This observation is quite intuitive. As the fixed ordering cost increases, the retailer would like to order as few times as possible to avoid high ordering costs. Furthermore, our numerical results show that rl 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 nondecreasing 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) on the optimal policy. The optimal ordering function is generally stepwise decreasing in inventory 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 form of the policy. Hence, the relation qn x+1 qn x qn x+1 + q, x = 0, 1,... (i.e. the optimal order quantity non-increasing in x, and the rate of decrease less than or equal to q) doesn t always hold for periodic review models and Poisson demand. When K > 0, Hill and Johansen (2006)) remarked that the above relation has been proven by Johansen and Thorstenson (1993) for a continuous review lost-sales model under Poisson demand and q = 1. The authors also point out that the relation doesn t hold for compound Poisson demands. Seemingly, it doesn t hold, in general, in a periodic review setting as well. 5.3.3 The case L = 1 In this section, we assume that the lead time equals the review period length, and provide additional insights on the structure of the optimal policy. Results from our numerical investigation for selected parameter values are included in Table 5.11 from Appendix A. As in the previous cases, numerical results indicate a complex structure of the optimal ordering 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 observation that the relative value function (v x ) x Ω (given by (5.1)) may exhibit in such cases multiple local minima (see Figure 5.7). As a result, the cost objective may have multiple local minima, which prohibits a simple optimal policy structure. Moreover, our numerical results confirm that neither an (s, S), nor an (s, Q) policy

130 Chapter 5. Efficient control of lost-sales inventory systems 30 L=1, md=10, p=10, q=1 25 Order quantity 20 15 10 K=5 K=10 5 0 0 5 10 15 20 Inventory on hand Figure 5.6 Optimal order quantity for L = 1, h = 1, K = {5, 10} L=1, md=10, K=10, p=10, q=1-300 -200-220 v(i) -301-302 -303-304 -305 v(i) -240-260 -306-307 -308 0 10 20 30 40 50 60 i -280-300 -320 0 10 20 30 40 50 60 i Figure 5.7 The relative value function showing two local minima

5.3 On the structure of the optimal policy: K = 0 vs. K > 0 131 is average cost optimal. The optimal policy behaves like (s, Q) for low levels of stock on hand and like (s, S) for higher inventory levels (see example illustrated in Figure 5.8). Intuitively, when the stock level is low, one expects it to be used up during the review period, and thus order enough to meet the expected demand during the review period. When the stock level is high, there is a higher chance for surplus inventory and therefore one aims to have enough inventory on hand plus on order to 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 that the corresponding cost functions may have more than one local minimum. In such cases, 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 the optimal policy. Hence we propose the class of (s, Q S, nq) policies as an alternative heuristic. We already introduced this policy class in Chapter 3. In the next section we investigate its effectiveness for the cases of zero, and positive setup costs. L=1, md=10, p=20, K=10, q=1 21 18 Order quantity 15 12 9 6 18 17 16 15 15 15 15 15 15 15 15 15 15 15 15 14 14 14 13 13 13 12 12 3 0 0 3 6 9 12 15 18 21 Inventory on hand 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 note the 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 an interval of values for the setup cost, l K u, in which the optimal policy exhibits discontinuities; outside this parameter region the optimal order quantity appears monotone in on-hand inventory (see Figure 5.9). Regarding the impact of mean 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 as well, suggesting larger orders for higher demand rates. Finally, as the penalty cost p increases, both rl and ms increase. Intuitively, we tend to order more to avoid higher costs of losing sales.

132 Chapter 5. Efficient control of lost-sales inventory systems L=1, md=10, p=20, q=1 40 35 30 Order quantity 25 20 15 10 5 0 0 5 10 15 20 25 Inventory on hand 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 structural form of the optimal policies, we propose next the class of (s, Q S, nq) policies, and investigate its performance in a numerical study. The performance is measured against the optimal policy determined via dynamic programming, as well as against two reasonable 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. We repeat 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 such as 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, S and Q, where 0 max{s, 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. Given an initial inventory on hand x, the policy defines the order amount a(x) as follows Q/q q if 0 x S Q/q q a(x) = (S x)/q q if S Q/q q < x s (5.2) 0 if s < x, 5 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 period is less than or equal to s, order the largest integer multiple of q which results in an inventory position less than or equal to S. 6 The (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 that the 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 = 1 the order quantity equals { min{q, S x} if 0 x s a(x) = 0 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 to S 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 to this policy, the order-up-to level is not constant, but a two-piecewise linear function of on-hand inventory. We refer to Figure 5.2 for an illustration of the logic of the heuristic in four particular cases. The well known (s, S) and (s, Q) policies (see e.g. Zipkin, 2000) are special cases of the (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 setup cost), then s = S, and the policy has only two parameters S and Q. In Section 5.4.2, we use these widely known policies to benchmark the performance of the new heuristic. When q > 1, the (s, S) and (s, Q) policies are modified such as to account for the restriction on the order quantity being an integer multiple of the fixed batch size q. We refer to these policies as the (s, S, nq) and (s, Q, nq) policy, when K > 0, and the (s, nq) policy 7 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 assumptions remain the same. As before, the aim is to find the best (s, Q S, nq) policy that minimizes the long-run average cost. Let C(s, S, Q; q) denote the expected long-run average cost of the system controlled by the (s, Q S, nq) policy, i.e., C(s, S, Q; q) = lim T T 1 1 T t=0 C t (s, S, Q), where C t (s, S, Q) is the expected cost incurred by the system in period t, i.e., [ ] + C t (s, S, Q; q) = δ(a t ) K + h E DL,D R L (X (s,s,q) t D L ) + + a (s,s,q) t D R L [ + p E DL,D R L D R L a (s,s,q) t [ ] + + p E DL D L X (s,s,q), t (X (s,s,q) t D L ) +] + 7 The (s, nq) policy has one parameter s 0 and may be described as follows. Whenever the inventory 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.

134 Chapter 5. Efficient control of lost-sales inventory systems 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 lostsales system, controlled by a base stock policy, it does not seem possible to determine explicitly the long-run average cost. Thus, analytically determining the optimal policy within the class of (s, Q S, nq) policies may not be possible. Therefore, the optimal solutions will be determined numerically, according to following methodology. For any parameters s, Q, and S, a dynamic programming formulation similar to (5.1) is used to determine the long-run average cost of the (s, Q S, nq) policy. In this case however, for any x Z +, instead of minimizing over all possible ordering quantities as 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 algorithm to determine the long-run average cost of the (s, Q S, nq) policy. To determine the best s, Q and S parameters, we used an exhaustive search over a sufficiently large feasible region to ensure we have found a global optimum. We also applied a similar methodology 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, we evaluate 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 conducted in Chapter 3, here we do not consider the per-batch and per-unit costs, but only the setup costs. We extend the analysis to include more parameter settings, in particular by considering the case of no setup cost. We evaluate the effectiveness of the new heuristic 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 alternative policies, 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 the optimal policy by observing the percentage difference between the average costs of these 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 of commonly known heuristics: the (s, nq) policy (if K = 0) and the (s, S, nq) as well as the (s, Q, nq) policy (if K > 0), respectively. The percentage difference between the average costs of these policies is denoted by Gap 2 and is computed using the

5.4 The effectiveness of the (s, Q S, nq) policy 135 following formula Gap 2 = 100 Best(s, S, nq) average costs Best(s, Q S, nq) average costs, Best(s, Q S, nq) average costs where the (s, S, nq) policy is replaced, alternatively, with the (s, Q, nq) policy. When K = 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 optimal solution 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 original experiment, 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 the optimal policy, as well as the alternative heuristics. The reported results are averages over all other parameters values. Later on, we report detailed results for selected parameter 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. (s, Q S) vs. Optimum q = 1 q > 1 (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 0.000 K > 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) policies performs very well. When K = 0, the maximum percentage cost increase from optimality 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 all scenarios. In Figure 5.10 we illustrate the performance of the (s, Q S, nq) heuristic against optimal policies in Policy space, for selected parameter values. We notice that

136 Chapter 5. Efficient control of lost-sales inventory systems the (s, Q S, nq) captures partially the structure of the optimal policy. As depicted in Figure 5.10(b), it may even represent the true optimal policy. In fact, we notice that in 92 out of the 216 scenarios (i.e. about 43%), the best (s, Q S, nq) policy captures exactly the optimal policy (and then q 4). Furthermore, our results show that the best (s, Q S, nq) policy is within 0.24% of the optimal cost in about 96% of the 216 scenarios. However, the optimal ordering policy does not posses the same structure 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. Observe that the optimal policy may have one (or more) peaks or dips, while the heuristic is much 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% K=10, L=1/2, p=10, md=20, q=4, Diff=0.000% 30 30 25 25 Order quantity 20 15 10 optpolicy best(s,q S) Order quantity 20 15 10 optpolicy bes(s,q S,nq) 5 5 0 0 3 6 9 12 15 18 21 24 27 30 0 0 3 6 9 12 15 18 21 24 27 30 Inventory on hand Inventory on hand (a) (b) K=20, L=1/2, p=10, md=20, q=4, Diff=0.056% K=20, L=1/2, p=10, md=20, q=4, Diff=0.056% Order quantity 50 45 40 35 30 25 20 15 10 5 0 0 3 6 9 12 15 18 21 24 27 30 Inventory on hand optpolicy best(s,q S) Order quantity 50 45 40 35 30 25 20 15 10 5 0 0 3 6 9 12 15 18 21 24 27 30 Inventory on hand optpolicy best(s,q S,nq) (c) (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. In Table 5.2 we reported the overall percentage of cost increase of these policies over the 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 by 0.8% (q = 1) and 1% (q > 1), respectively. When K > 0, the best (s, Q S, nq) policy

5.4 The effectiveness of the (s, Q S, nq) policy 137 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 that the former is performing, on average about 5.5% (q = 1) to 3.6% (q > 1) worse than the (s, Q S, nq) heuristic, with maximum percentage differences of 14.4% (q = 1), and 14.6% (q > 1), respectively. Unit batch size Tables 5.3 and 5.4 summarize the results for unit batch size (i.e. q = 1), in the case of zero or fixed setup cost, respectively. In the tables, for each combination of the parameters, we report the cost of the optimal policy, the best (s, Q S, nq) policy parameters and the associated long-run average cost C(s, S, Q ), as well as the best policy 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 %Diff Optimal cost from optimal L p q rl ms s S Q Cost cost S Cost %Diff from cost of best (s, Q S) 0.50 5 1 33 34 17.367 33 34 23 17.386 0.108 34 17.617 1.328 10 36 37 19.601 36 37 24 19.628 0.136 37 19.757 0.658 20 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 1.463 10 32 33 15.774 32 33 25 15.780 0.036 33 15.893 0.718 20 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 1.587 10 30 31 13.980 30 31 25 13.983 0.017 31 14.077 0.674 20 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 values of the parameters. We also note that, independent of the ordering cost K, the performance of the heuristics improves as the penalty cost p increases, except for the (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 no setup costs, the order-up-to policies are asymptotically optimal as the penalty cost becomes large compared to the holding cost and the lead time is an integral multiple of 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. Although the (s, Q) policies generally perform worse than the (s, S) policies, there are a couple of instances when they may actually outperform the (s, S) policies (as indicated in bold in Table 5.4). We note that these are scenarios where p is relatively small. Also in these settings, (s, Q S) heuristic has improved performance. Finally, these results

138 Chapter 5. Efficient control of lost-sales inventory systems Table 5.4 Performance of the (s, Q S, nq), (s, S, nq) and (s, Q, nq) policy for K > 0 and q = 1 Optimal policy Best (s, Q S) Best (s, S) Best (s, Q) Optimal cost %Diff from optimal cost %Diff from cost of best (s, Q S) K L p rl ms s S Q Cost s S Cost s Q Cost %Diff from cost of best (s, Q S) 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 4.554 10 30 37 29.600 30 37 24 29.627 0.090 30 37 29.756 0.436 28 21 32.019 8.075 20 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 6.133 10 26 33 25.774 26 33 25 25.779 0.022 27 33 25.893 0.440 24 22 28.466 10.423 20 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 7.356 10 25 31 23.980 25 31 25 23.982 0.010 25 31 24.077 0.392 23 22 26.786 11.688 20 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 1.588 10 28 55 39.421 28 55 44 39.443 0.056 28 54 39.581 0.350 28 30 40.634 3.019 20 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 2.041 10 24 51 35.697 24 51 44 35.738 0.114 24 51 35.875 0.383 24 30 36.926 3.325 20 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 2.338 10 23 49 33.968 23 31 25 33.978 0.030 23 31 34.070 0.272 23 30 35.199 3.595 20 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 0.151 10 25 56 54.397 25 56 45 54.399 0.005 25 56 54.568 0.310 20 40 57.097 4.959 20 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 0.263 10 22 53 50.756 22 53 45 50.759 0.007 22 52 50.891 0.260 19 41 52.053 2.549 20 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 0.319 10 20 51 49.055 20 51 45 49.057 0.005 20 51 49.169 0.228 19 41 49.983 1.888 20 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 else being equal. Regarding the solution space, it is striking to observe that, in nearly all cases, the order-up-to levels (ms or S) and the reorder points (rl or s) are nearly identical across the three types of policies (optimal, best (s, Q S) and best (s, S) policy); the occasional differences are small. Although, as Figure 5.10 illustrates, the optimal solutions are structurally different than the (s, S) and (s, Q) policies. Apparently, the average cost C(s, Q, S) is less sensitive to changes in the values of parameter Q. These insights could be exploited in the derivation of easily computable policy parameters. In summary, the numerical study conducted in this section demonstrates that the best (s, Q S, nq) heuristic performs consistently well in all instances, while it captures partially 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 the optimal policy, in the general setting of zero of positive setup costs. Our motivation stems from the fact that many grocery retailers apply this heuristic for making inventory replenishment decisions (Broekmeulen et al., 2004). Although the retailer s replenishment cost structure includes a fixed component, due to the presence of material handling costs, as we already demonstrated in Chapter 2, heuristics with a similar logic to the (s, nq) policy are often implemented in Automated Ordering Systems at grocery retailers (Van Donselaar et al., 2009). Recall that the (s, nq) heuristic operates as follows. At the beginning of each review period, if the inventory on hand is less that s, an order is placed for the smalest integral multiple of q such that, after ordering, the inventory level will rise at or above s. Namely, given a non-negative integer s, the amount ordered in any period that 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 to the case of batch ordering (see Figure 5.1 for an illustration of the ordering function under this policy). The experimental set up is identical to the one described in Section 5.3, and hence includes 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.

140 Chapter 5. Efficient control of lost-sales inventory systems The best policy parameter (denote by s ) was determined using enumeration over a sufficiently large feasible region. Next, we summarize our results separately for the case 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 and average E% over all 72 instances is essentially zero, which suggests that for the case of zero lead time, the optimal policy is of (s, nq) type. However, when there is a positive order 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 indicates that the (s, nq) policy is suboptimal in many cases. Table 5.5 summarizes the minimum, maximum and average percentage gap between the optimal and the best (s, nq) policies, over all instances with K > 0. The information is presented for different values of the parameters, averaged over all other parameter values. The mean E% is significant for most parameters; it decreases with mean 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 28.876 10.000 78.906 20.659 20.000 41.481 9.196 p 5.000 72.573 17.986 10.000 116.198 21.758 20.000 130.597 20.130 40.000 122.763 18.433 K 5.000 13.190 1.688 10.000 35.224 6.423 20.000 72.573 19.286 50.000 130.597 50.910 q 1.000 130.597 27.742 2.000 127.200 27.389 4.000 115.904 25.493 6.002 91.209 21.597 12.000 57.956 11.074 20.000 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 error becomes. However, the order of magnitude of E% decreases as q increases. This can be intuitively explained by having less flexibility in ordering, as q increases. The mean E% indicates that the (s, nq) performs better for small values of K and large values of q. The large percentage gaps in most cases clearly indicate that the optimal policies do not have the same structure as the (s, nq) policy. This finding is not surprising; we know that if demand is fully backlogged and q = 1, an (s, S) policy is optimal (and the 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) is suboptimal 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 maximum E% equals 129%, while the average E% is 18.93%, indicating that on average, the cost 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).

142 Chapter 5. Efficient control of lost-sales inventory systems Table 5.6 The performance of the (s, nq) policy (K > 0, L = 1) Parameter Value Min.E(%) Max.E(%) Avg.E(%) λ 5.000 129.532 27.673 10.000 77.909 19.887 20.000 40.774 9.238 p 5.000 71.004 17.022 10.000 116.737 21.063 20.000 129.532 19.584 40.000 122.077 18.060 K 5.000 9.789 1.684 10.000 33.197 5.727 20.000 71.004 18.212 50.000 129.532 50.107 q 1.203 129.532 27.236 2.212 126.426 26.864 4.232 116.501 24.863 6.211 86.198 20.645 12.000 49.115 10.333 20.000 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 average relative 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 lead times are integers (see e.g., Zipkin, 2008a). Our numerical results indicate that, when K = 0, the best (s, nq) policy has an average difference of 1.15% and a maximum difference of 2.89% from the optimal average cost. We observe that the performance of the (s, nq) heuristic improves, on average, as p increases. We do not observe any other strong trend in the performance of the (s, nq) heuristic with respect to the other problem 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, the maximum and average relative difference between the cost of the best (s, nq) policy and the optimal average cost is 36.06% and 10.87%, respectively. The performance of the (s, nq) heuristic decreases with K and increases with p. Beyond theses trends, it is difficult 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) Parameter Value Min.E(%) Max.E(%) Avg.E(%) L 0.25.000 2.000 0.860 0.33.335 2.877 1.515 0.50.197 2.887 1.070 p 5.000 2.529 1.605 10.654 2.887 1.341 20.000 1.184 0.500 q 1.320 2.166 1.049 2.367 2.078 1.020 4.291 2.140 1.030 6.190 2.183 1.131 12.000 2.469 1.280 20.000 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) heuristic performs poorly for all batch sizes (especially when K > 0). Furthermore, except for the largest batch size, the average performance deteriorates as q increases (when q = 20 the mean E% drops however from 12.29% to 9.56%). In summary, the numerical results indicate that (s, nq) policies are, in general inefficient 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 result in large cost penalties. 5.6. Conclusions In this chapter, we studied a single-location, single-item periodic-review lost-sales inventory control problem with the following features: there are stochastic customer demands, 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 multiples of q. We analyzed the inventory system from a cost perspective considering holding and lost-sales penalty costs, as well as non-negative ordering costs. We distinguished between lost-sales systems with no ordering costs and lost-sales systems with fixed setup costs. Using Markov decision processes, we explored numerically the structure of the optimal policies and investigated, in particular, the impact of q on the optimal solution and the long-run average cost of the system. We showed that optimal policies have rather complicated structures, which hinders their practical applicability. In

144 Chapter 5. Efficient control of lost-sales inventory systems Table 5.8 The performance of the (s, nq) policy (K > 0, 0 < L < 1) Parameter Value Min.E(%) Max.E(%) Avg.E(%) K 10 0.000 1.489 0.633 20 0.115 5.817 2.165 50 23.010 36.061 29.800 L 0.25 0.000 36.061 11.098 0.33 0.312 35.145 11.114 0.50 0.023 33.302 10.387 p 5 0.000 36.061 12.619 10 0.023 32.297 10.527 20 0.115 29.875 9.452 q 1 0.146 34.872 10.693 2 0.172 34.889 10.715 4 0.146 35.107 10.851 6 0.115 35.658 11.080 12 0.327 36.061 12.295 20 0.000 29.114 9.561 Total.000 36.061 10.866 particular, we compared the optimal ordering patterns for systems with fixed ordering cost 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 and shows 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 than 0.2% (when K = 0) and 1.7% (when K > 0), respectively. We also benchmarked the performance of the heuristic against reasonable alternative policies, the (s, S, nq) and (s, Q, nq) policies, and quantified the overall improvement. In particular, our numerical results indicate that the best (s, S) policies are performing increasingly better and close to optimality as the penalty cost increases. On the other hand, the widely applied (s, nq) policy in retailing may result in substantial cost penalties when implemented in the presence of fixed costs. A relevant area for future research is the derivation of easily computable heuristic (s, Q S, nq) policies that are also Appendix A. Selected numerical results We denote by U = [U 1, U 2,..., U rl+1 ] the optimal order quantity as a function of on-hand inventory {0, 1, 2,...}, with rl the optimal reorder level (i.e. U = [U 1, U 2,..., U rl+1, 0, 0,...]). By convention, Ui n denotes n times the value U i and

Appendix A. Selected numerical results 145 U i : U j denotes unit decreasing values from U i to U j (U j < U i ). Furthermore, rl = 1 is used to denote the policy of never ordering. Note that, under such a policy, all expected demand is lost on the long run, and consequently the system incurs only the penalty cost of loosing demand, i.e. the optimal long-run average cost equals 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 [24 3, 23 3, 22 2, 21 : 1] 11.993 28 29 28 12.189 10 [26 4, 25 3, 24 : 1] 13.980 30 31 30 14.077 20 [28 2, 27 4, 26 2, 25 : 1] 15.746 32 33 32 15.797 0.33 5 [23 7, 22 2, 21, 20 2, 19 : 1] 13.716 30 31 29 14.013 10 [26 5, 25 3, 24 2, 23 : 1] 15.774 32 33 32 15.968 20 [27 8, 26, 25 2, 24 : 1] 17.609 34 35 34 17.752 0.50 5 [23 10, 22 2, 21 2, 20 : 1] 17.367 33 34 33 17.617 10 [26 4, 25 7, 24 2, 23, 22 2, 21 : 1] 19.601 36 37 36 19.757 20 [27 11, 26 2, 25, 24 2, 23 : 1] 21.563 38 39 38 21.642 0.25 5 4 [24 7, 20 5, 16 4, 12 4, 8 4, 4 4 ] 12.170 27 31 26 12.364 10 [24 10, 20 4, 16 4, 12 4, 8 4, 4 4 ] 14.177 29 33 29 14.284 20 [28 7, 24 5, 20 4, 16 4, 12 4, 8 4, 4 4 ] 15.964 31 35 31 16.010 0.33 5 [24 9, 20 4, 16 4, 12 4, 8 4, 4 4 ] 13.887 28 32 28 14.185 10 [24 12, 20 4, 16 4, 12 4, 8 4, 4 4 ] 15.983 31 35 31 16.183 20 [28 9, 24 5, 20 4, 16 4, 12 4, 8 4, 4 4 ] 17.837 33 37 33 17.979 0.50 5 [24 12, 20 5, 16 4, 12 4, 8 4, 4 4 ] 17.555 32 36 31 17.790 10 [24 15, 20 4, 16 4, 12 4, 8 4, 4 4 ] 19.795 34 38 34 19.936 20 [28 12, 24 6, 20 4, 16 4, 12 4, 8 4, 4 4 ] 21.801 37 41 37 21.888 0.25 5 12 [24 12, 12 12 ] 13.400 23 35 23 13.400 10 [24 15, 12 12 ] 15.504 26 38 26 15.768 20 [24 17, 12 12 ] 17.602 28 40 28 17.731 0.33 5 [24 14, 12 12 ] 15.099 25 37 25 15.472 10 [24 17, 12 12 ] 17.287 28 40 28 17.621 20 [24 19, 12 12 ] 19.393 30 42 30 19.623 0.50 5 [24 17, 12 12 ] 18.807 28 40 28 19.099 10 [24 20, 12 12 ] 21.073 31 43 31 21.318 20 [24 23, 12 12 ] 23.239 34 46 34 23.422 0.25 5 20 [20 21 ] 14.100 20 40 19 14.112 10 [20 25 ] 17.956 24 44 24 18.315 20 [40 8, 20 20 ] 20.534 27 47 27 20.534 0.33 5 [20 22 ] 15.672 21 41 19 15.906 10 [20 27 ] 19.552 26 46 26 20.115 20 [40 9, 20 21 ] 22.363 29 49 29 22.438 0.50 5 [20 25 ] 19.125 24 44 23 19.609 10 [20 30 ] 23.027 29 49 29 23.692 20 [40 13, 20 21 ] 26.206 33 53 32 26.258

146 Chapter 5. Efficient control of lost-sales inventory systems Table 5.10 Optimal policies, minimum costs and comparison with the (s, nq) policy, K > 0, λ = 20, K 1 = 0, h = 1 (Selected cases) Optimal Policy Best (s, nq) K L p q U C(U ) rl ms s 1 C(s, nq) 10 0.25 5 1 [24 3, 23 3, 22 2, 21 : 8] 21.992 21 29 28 22.189 10 [26 4, 25 3, 24 : 6] 23.98 25 31 30 24.077 20 [28 2, 27 4, 26 2, 25 : 6] 25.746 27 33 32 25.797 0.33 5 [23 7, 22 2, 21, 20 2, 19 : 8] 23.715 23 31 29 24.013 10 [26 5, 25 3, 24 2, 23 : 7] 25.774 26 33 32 25.968 20 [27 8, 26, 25 2, 24 : 6] 27.609 29 35 34 27.752 0.50 5 [23 10, 22 2, 21 2, 20 : 8] 27.366 26 34 33 27.617 10 [26 4, 25 7, 24 2, 23, 22 2, 21 : 7] 29.600 30 37 36 29.757 20 [27 11, 26 2, 25, 24 2, 23 : 6] 31.563 33 39 38 31.642 10 0.25 5 4 [24 7, 20 5, 16 4, 12 4, 8 2 ] 22.167 21 31 26 22.364 10 [24 10, 20 4, 16 4, 12 4 ] 24.176 25 33 29 24.284 20 [28 7, 24 5, 20 4, 16 4, 12 4, 8 4 ] 25.963 27 35 31 26.010 0.33 5 [24 9, 20 4, 16 4, 12 4, 8 3 ] 23.885 23 32 28 24.184 10 [24 12, 20 4, 16 4, 12 4, 8 3 ] 25.982 26 35 31 26.183 20 [28 9, 24 5, 20 4, 16 4, 12 4, 8 4 ] 27.837 29 37 33 27.979 0.50 5 [24 12, 20 5, 16 4, 12 4, 8 2 ] 27.551 26 36 31 27.790 10 [24 15, 20 4, 16 4, 12 4, 8 4 ] 29.795 30 38 34 29.936 20 [28 12, 24 6, 20 4, 16 4, 12 4, 8 4 ] 31.800 33 41 37 31.888 10 0.25 5 20 [20 20 ] 23.615 19 39 19 23.615 10 [40 6, 20 19 ] 27.192 24 45 24 27.201 20 [40 9, 20 18 ] 29.462 26 48 27 29.549 0.33 5 [20 21 ] 25.190 20 40 19 25.327 10 [40 7, 20 19 ] 28.903 25 46 25 28.994 20 [40 11, 20 18 ] 31.293 28 50 29 31.459 0.50 5 [20 25 ] 28.632 24 44 23 28.793 10 [40 9, 20 21 ] 32.571 29 49 29 32.578 20 [40 14, 20 19 ] 35.189 32 53 32 35.253 20 0.25 5 1 [41 5, 40 2, 39 : 27] 31.326 19 46 28 32.189 10 [26 4, 25 3, 24 : 20, 37 : 26] 33.968 23 49 30 34.077 20 [28 2, 27 4, 26 2, 25 : 7] 35.745 26 33 32 35.797 0.33 5 [41 7, 40, 39 2, 38 : 28] 32.921 20 48 29 34.013 10 [26 5, 25 3, 24 2, 23, 22, 39 : 27] 35.697 24 51 32 35.968 20 [27 8, 26 2, 25 : 8] 37.607 27 35 34 37.752 0.50 5 [41 9, 40 2, 39 2, 38 : 28] 36.391 23 51 33 37.617 10 [26 7, 25 4, 24 2, 23, 41 : 27] 39.421 28 55 36 39.757 20 [27 11, 26 2, 25, 24 2, 23 : 8] 41.559 31 39 38 41.642 20 0.25 5 4 [40 9, 36 4, 32 4, 28 3 ] 31.392 19 48 26 32.364 10 [44 5, 24 4, 40 2, 20 2, 36 2, 16 2, 32 2, 12 2, 28 2, 8] 34.017 23 50 29 34.284 20 [28 8, 24 4, 20 4, 16 4, 12 4, 8 3 ] 35.958 26 35 31 36.010 0.33 5 [40 10, 36 4, 32, 28 3 ] 32.979 20 49 28 34.184 10 [44 7, 24 3, 40 3, 20, 36 3, 16, 32 3, 12, 28] 35.759 24 52 31 36.183 20 [28 9, 24 5, 20 4, 16 4, 12 4, 8 2 ] 37.829 27 37 33 37.979 0.50 5 [40 13, 36 4, 32 4, 28 3 ] 36.45 23 52 31 37.790 10 [44 11, 24 3, 40 3, 20, 36 3, 32 4, 28 4 ] 39.475 28 56 34 39.936 Continued on next page

Appendix A. Selected numerical results 147 Table 5.10 (continued) Optimal Policy Best (s, nq) K L p q U C(U ) rl ms s 1 C(s, nq) 20 [28 12, 24 5, 40, 20 3, 36, 16 3, 32, 12 3, 28, 8 2 ] 41.757 31 57 37 41.888 20 0.25 5 20 [40 11, 20 7 ] 31.984 17 50 20 32.952 10 [40 13, 20 10 ] 35.192 22 52 24 36.087 20 [40 13, 20 13 ] 37.815 25 52 27 38.563 0.33 5 [40 13, 20 7 ] 33.538 19 52 20 34.572 10 [40 15, 20 10 ] 36.928 24 54 25 37.871 20 [40 16, 20 12 ] 39.632 27 55 29 40.480 0.50 5 [40 17, 20 6 ] 36.953 22 56 23 37.978 10 [40 19, 20 9 ] 40.599 27 58 29 41.465 20 [40 20, 20 12 ] 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) Optimal Policy Best (s, nq) λ K p U C(U ) rl ms s 1 C(s, nq) 5 10 5 [12 6, 11 2, 10] 10.853 8 18 11 14.456 10 [14 4, 13 3, 12 2, 11, 9] 12.288 10 20 13 15.632 50 5 [23 5 ] 21.184 4 27-1 25 10 [25 5, 24 4, 23, 22] 23.069 8 30-1 50 10 5 20 [15 10, 14 3, 13 2, 12 : 4] 14.393 23 27 25 14.714 10 5 [22 7, 21 4, 20 2, 19, 17 : 14] 15.328 17 32 22 16.362 10 [24 9, 23 3, 22, 12 2, 18 : 13] 17.316 20 34 24 17.952 20 [15 11, 14 2, 13 2, 12 2, 18 : 13] 19.040 22 35 26 19.476 20 20 [26 10, 25 3, 24 2, 23 2, 21 : 17] 23.986 21 39 26 29.475 50 5 [34 10, 33 2, 32, 31] 30.180 13 44-1 50 10 [36 9, 35 3, 34, 33 2, 32 : 30] 32.601 17 47 24 57.95 20 [38 8, 37 4, 36, 35 2, 34 : 29] 34.679 20 49 26 59.474 20 10 5 [23 20, 22 3, 21, 20 2, 19 : 8] 18.750 37 45 43 18.978 10 [25 20, 24 3, 23 2, 22, 21 2, 20 : 7] 20.980 41 48 46 21.154 50 5 [42 20, 41 3, 40 2, 39, 38 2, 37 : 34] 42.432 31 65 44 58.974 10 [45 20, 44 3, 43, 42 2, 41, 40 2, 39 : 37, 35 : 31] 45.500 36 68 46 61.154

148 Chapter 5. Efficient control of lost-sales inventory systems Appendix B. Approximate (s, Q S) policies In this chapter, we studied a variant of the traditional periodic-review lost-sales inventory control problem, with batch ordering and non-negative setup costs under the 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 policies in solving the problem, where the the best policy parameters were determined by enumeration. The best (s, Q S, nq) policy, although easy to explain in practice, could be computationally inefficient using an exhaustive search. In the following, we provide some computational details and few guidelines towards deriving easily computable approximate (s, Q S, nq) policies that are equally cost effective. More research is necessary to further explore such opportunities. Policy evaluation and optimization Analytically determining the (s, Q S, nq) policy, which minimizes the long-run average cost was not possible. In fact, we are not aware of analytical methods for obtaining explicit optimal solutions from a given class of policies for single-item lost-sales inventory problems with setup costs. Hence, exact optimal policies for the lostsales problem when an (s, Q S, nq) policy is in effect were obtained numerically by formulating a Markov decision process and solving it with the value iteration algorithm. More precisely, for any given (s, Q S, nq) policy (with s, Q and S nonnegative discrete parameters), we evaluated the long-run average cost of the policy for a given discrete demand distribution by formulating a dynamic program similar to (5.1). For a given starting 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 value iteration 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 integer values, 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 derive structural properties that could be exploited in the derivation of efficient optimization algorithms. 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) Z 3 0 max{s, Q} S s + Q, 0 S S max }, where S max was large enough to ensure we found a global optimum. Such an enumeration approach is however computationally inefficient, owing to the fact that one must search for a triplet of values (s, S, Q) over a response surface that is not necessarily convex in these variables. Hence, obtaining cost estimates for all possible values of (s, S, Q) in the grid search is computationally demanding. In Table 5.13, we give an example of average computation times resulted from applying a full grid 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 search technique, based on the assumption of unimodality of the cost function C(s, S, Q) in each of the three parameters, is in finding the best policy parameters. We summarize our 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 in Section 5.4. The exact parameter values are presented in Table 5.12, the combination of which resulted in 27 scenarios in the numerical study. Table 5.12 Parameter setting for numerical experiment Parameter Values λ 20 h 1 p 5, 10, 20 K 10,20, 50 L 0.25, 0.33, 0.50 q 1 Assuming unimodality of the objective function C(s, S, Q) in each parameter, we kept two of the parameters at their best values and we applied, sequentially, a simple one dimensional local search techniques to find a locally optimal solution. The aim was to test how much improvement, in terms of computation time, is to be expected from a simple one dimensional local search. Table 5.13 presents our findings. In the table, we reported the number of instances in which the local search found the best s, Q, or S value (as determined by full enumeration), which indicates the number of instances in which the local search failed to 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 time from the full enumeration approach. We observe that the local search technique managed to find the best values of the parameters s and Q in all 27 instances, while it failed to do so in 5 out of the 27 scenarios 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 by enumeration this is shown by underlining. Column 8 gives the long-run average cost

150 Chapter 5. Efficient control of lost-sales inventory systems Table 5.13 Results from numerical experiments on 1-dim unimodal search Algorithm Number of scenarios optimum found Time 1-dim search (sec) Total time (sec) avg max avg max Enumeration 27 292.3035 317.4102 Approx. s 27 0.017 0.062 70.181 72.170 Approx. Q 27 0.013 0.025 70.188 71.351 Approx. S 22 0.016 0.053 73.374 74.203 of the approximate (s, Q S) policy. Column 9 gives the percentage cost increase of Column 8 relative to the cost of the best (s, Q S, nq) policy, and thus provides a measure, in cost terms, of the error that results from applying the one dimensional local search technique. Table 5.14 also shows, for each scenarios, the computation time spent in generating the transition probabilities and cost functions for the dynamic program (Column 10), the computation time taken by the one dimensional local search (Column 11), as well as the total computation time (Column 12). We observe that the largest share of total computational time is related to the generation of transition probabilities and associated cost function. The one dimensional local search technique takes fraction of seconds. These findings suggest that further gains, in terms of computational time, could be obtained from a more efficient derivation of transition probability matrices and 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 probability matrices and associated costs. In an earlier study of a single-item lost-sales inventory control problem with service level constraints, restricted to the (s, S) policy class, Kapalka et al. (1999) also found that approximately 50% of the total computational time of the enumeration approach was spent in generating the transition probability matrices and cost functions. Therefore, to address the large computational times in generating the transition probabilities, Kapalka et al. (1999) developed a transition probability matrix updating procedure, which then combined with bounds on S and a monotone search algorithm reduced the time required for computing the transition probability matrices by 90%, and the total execution time by approximately 30%. Their approach is however not directly generalizable from the two dimensional to the three dimensional parameter space. It would be worthwhile to further investigate the consequence of applying a similar approach in our problem setting. A further way to decrease the computation time is to derive lower and upper bounds on 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. Q Approx. S / S cost % cost error Time to generate P, C (sec) Time local search (sec) Total run time (sec) 0.50 5 10 1 26 34 23 27.384 0.000 70.477 0.053 70.540 10 10 1 30 37 24 29.627 0.000 73.343 0.023 73.368 20 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 73.558 10 10 1 26 33 25 25.779 0.000 73.664 0.013 73.679 20 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 73.587 10 10 1 25 31 25 23.982 0.000 73.750 0.013 73.763 20 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 74.203 10 20 1 28 55 44 39.443 0.000 73.479 0.021 73.501 20 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 73.658 10 20 1 24 51 44 35.738 0.000 73.215 0.014 73.230 20 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 73.469 10 20 1 23 48/31 25 35.360 1.382 72.886 0.009 72.896 20 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 73.321 10 50 1 25 56 45 54.399 0.000 73.134 0.013 73.148 20 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 73.440 10 50 1 22 53 45 50.759 0.000 73.343 0.021 73.366 20 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 73.584 10 50 1 20 51 45 49.057 0.000 73.587 0.022 73.616 20 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

152 Chapter 5. Efficient control of lost-sales inventory systems number of polices that require evaluation during the grid search. This approach is common in the literature. For example, Zheng and Federgruen (1991) propose upper bounds on S for deriving efficient (s, S) policies for the backordering problem, but their approach does not appear as straight forward for our problem. Approximate bounds based on the EOQ model are proposed by Kapalka et al. (1999) and could also be investigated in our setting. Also, one might consider first to reduce the problem dimension by deriving approximations for one of the parameters. Another way for deriving efficient heuristics often relies on the derivation of structural properties of the cost function, such as monotonicity of the cost with respect to one or more of its parameters, or convexity results (see e.g. Huh et al., 2009, Janakiraman and Muckstadt, 2004a). It should be noted that it is not apparent that the longrun average cost C(s, Q S) is convex over the feasible policy space, therefore search algorithms based on convexity need not converge to an optimum. Further research to investigate if there is some structure in the cost function that could be exploited in the derivation of efficient heuristics might be valuable. Results regarding structural properties for the lost-sales problem with positive setup cost under a given policy class are still largely missing in the inventory literature. Finally, techniques to find approximate policies from a given policy class often propose transformations of the original model by relaxations, restrictions, or cost approximations (see e.g., Porteus, 1985). The effectiveness of such approaches for our problem setting could also be investigated.

153 Chapter 6 Conclusions In this dissertation, we studied inventory control systems that have been inspired from the practice of grocery retailers, where we focused on the incorporation of several features that we identified as relevant challenges: lost-sales, batch ordering, shelf space limitations, merchandise handling and backroom operations. We studied periodic review inventory control models under a cost perspective, where aside from the traditional inventory-related costs, we specifically included handling-related costs in the optimization of inventory decisions. First, a formal mathematical model of the handling costs is proposed based on insights from an empirical study. Then, several single-item lost-sales inventory models are developed, which are classified based on their main additional feature: (i) shelf stacking cost, (ii) shelf space capacity and backstock handling cost, and (iii) batch ordering and non-negative setup costs, respectively. In this concluding chapter, we briefly summarize the main results and insights from our research and discuss some future research directions. 6.1. Results In the introductory chapter, we raised a number of research questions related to the control of two essential store-based retail operations: merchandise handling and inventory replenishment. In this section, we provide answers to these questions by summarizing 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 and studied 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 found that stacking new inventory, grabbing and opening a case pack, and waste disposal are the three most time consuming sub-activities. They are mostly influenced by the number of case packs and consumer units in the replenishment order. Alternatively, activities such as searching for the right location on the shelf, preparing the shelves and filling old inventory require rather a fixed execution time. We demonstrated that a simple model, based on the number of case packs and consumer units, provides a reliable estimate of the total stacking time per order line, and the form of relationship is additive rather than purely linear. This structural insight was exploited into a lot sizing analysis to illustrate the opportunities for extending inventory control rules with a handling component. 2. How could the retail inventory control models be adapted to incorporate handling in decision making? And what is the impact of adding this aspect on the overall system performance? In Chapter 3, we recognized the shelf stacking cost at the retailer as a critical cost component for the analysis and optimization of replenishment decisions at item level. The handling cost structure assumes fixed and linear components, dependent on the number of batches and units in the replenishment order. This cost structure was embedded in a periodic-review lost-sales inventory model, along with the more standard inventory carrying and lost-sales costs. Assuming batch ordering and fractional lead times, the objective was to control the system in order to minimize the long-run average cost. Optimal replenishment policies for lost-sales inventory systems have been so far only partially characterized, mostly under simpler assumptions such as no order cost, and no batch ordering. Existing properties and numerical results for the optimal order quantities 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 captures the structure of optimal policies. Numerical experiments revealed that the best candidate within this class of heuristic policies comes close to being optimal in many settings. 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 illustrated the sensitivity of the solutions and associated long-run average cost to the handling cost parameters. In particular, we showed that if the handling costs are ignored in inventory replenishment decisions, the retailer s expected cost penalty may be substantial, especially for items with low profit margins. 3. How could inventory control models be adapted to account for shelf space limitations and use of the backroom? And what is the impact of including these features on the performance 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 space at the retailer and the fact that surplus stock is temporarily stored at the backroom, which generates additional handling costs. We proposed two inventory models to capture this situation. The first model assumes continuous replenishment from the backroom and extra handling costs that are proportional to the expected backstock. Alternatively, the second model assumes that an additional fixed cost is charged in the event of using the backroom. Both models generalize the situation when there is 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 longrun average costs. Furthermore, we quantified the retailer benefit from including the extra 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 the optimization of inventory decisions but nevertheless included in total cost calculation and (ii) a situation where the shelf space and extra handling costs were accounted for in the replenishment decision making, showed that the latter situation may lead to total cost savings of more than 50%. Several managerial insights are illustrated regarding the trade-off between the different cost components. 4. Can we derive an efficient heuristic to control the single-item lost-sales inventory problem 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 to optimal solutions. Four situations are considered, depending on (i) whether setup costs are present or not, and (ii) whether the fixed batch size is one or higher. Our numerical studies demonstrated that the cost increase from using the heuristic instead of 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 the behavior 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 showed that 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 with small 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 costs are 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 worth further investigation. Our models have been inspired from the practice of grocery retailers, 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, while in the retail environment, especially grocery retailing, the demand is non-stationary, following a cyclic (typically weekly) pattern. Also it is worth exploring the robustness of the results reported in this dissertation to different demand distributions, or even consider more challenging extensions to a setting with unknown demand, unobserved lost sales, or inaccurate inventory records. One of the characteristics included in our research was batch ordering, i.e., the replenishment order was restricted to multiples of a fixed batch size. However, in some settings, such as retailers carrying their own brands, the retailer may have the flexibility of replenishing not only in batches, but also with loose items. This situation brings up interesting challenges for inventory managers, if the replenishment cost structure assumes for example paying only for the loose items, or paying a fixed cost for each batch ordered (even if it is incomplete), and has been rarely addressed in the literature. 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 to optimize also the shelf space capacity, and our approach could serve as a subproblem in addressing this issue. Further research could be also dedicated to designing efficient backroom operations, and our research showed that handling costs are a relevant component. An interesting extension towards the real life retail setting would be to consider 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 problem with lost sales, and the practice could definitely benefit from the development of easily implementable, accurate approximations for the policy parameters, since this may well serve in addressing the more general constraint, multi-item stochastic inventory problem.

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Summary Retail Inventory Management with Lost Sales The inventory control problem of traditional store-based grocery retailers has several challenging features. Demand for products is stochastic, and is typically lost when no inventory is available on the shelves. As the consumer behavior studies reveal, only a small percentage of customers are willing to wait when confronted with an out-of-stock situation, whereas the remaining majority will either buy a different product, visit another store, or entirely drop their demand. A store orders inventory on a periodic basis, and receives replenishment according to a fixed schedule. The ordered stock is typically delivered before the next ordering moment, which results in lead times shorter than the review period length. Order sizes are often constrained to integer multiples 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 serve customer demand. Shelf space allocation of many products is limited, dictated by marketing 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 designing solution 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, rather than backordered. Fractional lead time. Time between order placement and order delivery is shorter than the review period length. Batch ordering. Order sizes are constrained to integer multiples of a fixed batch size. Limited shelf space. Shelf space allocation is predetermined. The retailer s inventory is split between the sales floor and the backroom, which is used to temporarily 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. Two types of costs are primarily recognized in the inventory models developed in this dissertation: inventory related costs: for ordering, for holding products on stock, and penalty costs 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, remarkably little is reported in the academic literature on how to manage inventory in the presence of handling costs. A reason for this is that formal models of handling operations are still 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-item lost-sales periodic-review inventory control model with several realistic dimensions of the replenishment practices of grocery retailers: batch ordering, handling costs, shelf space and backroom operations. The models we consider are too complex to lend themselves to straightforward analytical tractability. As a result, numerical solution methods 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 in grocery retail stores, and the key factors that influence the execution time of this common store operation. Shelf stacking represents the regular store process of manually refilling the shelves with products from new deliveries, which is typically time consuming and costly. We focus on products that are replenished in pre-packed form but presented to the end-customer in individual units. A motion and time study is executed, and the complete shelf stacking process is broken down into several sub-activities. The main time drivers for each activity are identified, relationships are established, tested and validated using real-life data collected at two European grocery retailers. A simple prediction model of the total stacking time per order line is then inferred, in terms of the number of case packs and consumer units. The model can be applied to estimate the workload and potential time savings in the stacking process. Implications of our empirical findings for inventory replenishment decisions are 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 under periodic review, which is designed to handle fractional lead times, batch ordering and handling costs. We study the settings in which replenishment costs reflect shelf stacking costs and have an additive form with fixed and linear components, depending on the number of batches and units in the replenishment order. We explore the structure of optimal policies under the long-run average cost criterion and propose a new policy, referred to as the (s, Q S, nq) policy, which partially captures 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 the best (s, Q, nq) and (s, S, nq) policies, and demonstrate the relative improvements. Sensitivity analyses illustrate the impact of the different problem parameters, in particular the batch size and the handling cost parameters, on the optimal solutions and associated average costs. Managerial insights into the effect of ignoring handling costs in the optimization of replenishment decisions are also discussed. Chapter 4 extends the retail setting from Chapter 3 to situations in which there is a limited shelf space to display goods on the sales floor, and the retailer uses the store s backroom to temporarily store surplus stock. As a result, the back stock is regularly transferred from the backroom to the sales floor to satisfy end-customer demand, which results in additional handling costs for the retailer. We investigate the effect of using the backroom on the inventory system performance, where performance is measured with respect to the optimal ordering decisions, and the long-run average cost of ordering, holding, lost-sales and merchandise handling. Two extensions of the inventory 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 numerical study, we discuss several qualitative properties of the optimal solutions, illustrate the additional complexities of the second model, and compare the findings with those of the previous chapter. Furthermore, we build several managerial insights into the effect of problem parameters, in particular the shelf space capacity, on the system s performance. Finally, we quantify the expected cost penalty the retailer may face by ignoring the additional handling costs in the optimization of inventory decisions, and illustrate the trade-off between the different cost components. Chapter 5 studies a variant of the traditional infinite-horizon, periodic-review, singleitem inventory system with random demands and lost sales, where we assume fractional lead times and batch ordering, and allow for fixed non-negative ordering costs. 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 only partially known in general. We show in a numerical study that the optimal policy structure of the most general model is usually more complex than that of the models with positive setup cost, or batch ordering only. Based on the gained insights, we further test the performance of the near-optimal (s, Q S, nq) heuristic policy in the different cases, and demonstrate its effectiveness. Also, well-known inventory control policies of base-stock, or (s, S) type are extended to the case of batch ordering and studied in comparison with the new heuristic under several conditions.

About the author Alina Curşeu was born in Alba-Iulia, Romania (ROU), on July 31, 1975. After completing her pre-university education at the Horia, Cloşca and Crişan National College in Alba-Iulia, Romania, she studied Mathematics at the Babeş-Bolyai University of Cluj-Napoca, Romania. She graduated the top of her class in 1998, and one year later she received her Master of Science (M.Sc.) degree in Mathematics from the 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 and functional analysis. From 2002, she joined the postmaster program Mathematics for Industry of the Stan Ackermans Institute at Eindhoven University of Technology, The Netherlands (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 Eindhoven University of Technology (NL) under the supervision of Jan Fransoo, Nesim Erkip and Tom van Woensel, the results of which are presented in this dissertation. The cooperation with Nesim Erkip was initiated in 2007 during a short visit to Bilkent University, Ankara, Turkey. As of January 2011, Alina is working as a consultant at LIME BV in Eindhoven (NL).