Introduction. Chapter 1

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1 Chapter 1 Introduction The success of Japanese companies in the second half of the 20th century has lead to an increased interest in inventory management. Typically, these companies operated with far less inventory than their Western counterparts. While initially the attention focused on the inventory within the production process, it has meanwhile also turned towards raw materials and finished goods. With 37.5% the latter presents the major part of the total inventory kept by US manufacturing companies (Survey of Current Business, April 2006). Lower inventory leads to less fixed capital and allows companies to react more flexible to market changes. Thus well-managed inventory might lead to a competitive advantage. Chen et al. (2005) provide empirical evidence that poor inventory management can harm a company s shareholder value. The last decades have additionally seen an increasing diversity of customer expectations and growing competitive pressure for a wide variety of industries. To cope with those customer demands while maintaining a competitive offer, many companies have grouped their customers. This might be an internal process where customers are assigned a certain priority, e.g. based on annual sales volume, or a result of customers having signed up for special services. The segmented customer basis allows implementing an inventory management approach that resembles the yield management practised in the airline or hotel industries: Demand fulfillment for low priority customers 1

2 2 CHAPTER 1. INTRODUCTION might be refused or delayed in order to reserve stock for more important clients. 1.1 Motivation Our research was originally motivated by an application at a European mobile communications provider. Some service parts are used at different levels of the telecommunication network, for instance in antennas and network computers. If a part fails in an antenna, the antenna goes down. If the same part fails in the network computer, the network computer goes down and about 30 antennas become unavailable. Thus the failure of this part in the network computer causes at least 30 times higher penalty cost than in the antenna. In such situations, the approach outlined above helps to reduce costs because we reduce the number of expensive shortages. However, to apply this strategy two core questions have to be answered: Firstly, what is the optimal strategy? The formulation before suggests that one demand occurs after the other. In a lot of practical situations, this is not the case. Demands are not unit-sized and are collected over a certain time interval. In these situations, it is not immediately obvious in which way the available inventory should be divided amongst the different customer classes. For instance, demand from the highest customer class could always be filled if there is enough inventory or each customer class obtains a share that corresponds to its overall importance. Additionally, there is not only the delivery, but also the replenishment that has to be controlled. Secondly, the characterization of a strategy as such does not necessarily include a way to derive the optimal parameters. Thus optimization is the other question that requires an answer. Intuitively, a lot of aspects of such a strategy are clear: For instance, it seems reasonable that the point after which we do not deliver anymore to low priority customers is lower if we expect a significant replenishment in the near future. Thus the orders in the pipeline have to be tracked. Unfortunately, this is exactly the point that most existing research tries to avoid. Only for unit-sized demands and assuming that both, demands of different classes and replenishments, cannot occur simultaneously, the optimal strategy has

3 1.2. RESEARCH OBJECTIVES 3 been completely characterized. 1.2 Research Objectives Our main objective is to provide and optimize a mathematical model for more practical situations. Orders are not placed whenever a demand occurs but at certain discrete time instances. Demands are not necessarily unitsized and demands of different priority (including waiting customers) may be observed at once. Furthermore, the time between placing an order and order arrival is greater than zero. Building a mathematical model, it is temptingtoincludeamaximumofreallife aspects. But the main contribution of a model lies in its solution. Thus simplifications and abstractions cannot be avoided completely. We restrict our attention to two customer classes and assume that a certain control policy is in place, the so-called critical level policy. Under this policy, if the inventory hits or drops below this rationing level, low priority customers are no longer served. In our context of larger incoming orders and demands, the order of events is decisive in which this policy is enforced. Cost-optimally, upon arrival of stock, all high priority customers should be considered before low priority customers, independent of the real sequence in which the demand occurred. This includes also all waiting customers. Although this is easy to see and to state verbally,this affects the mathematical tractability. For constant critical levels, we present a new modeling approach based on a Markov chain with multi-dimensional states. This approach allows us to optimize the policy parameters in the presence of lead time and obeying by the cost-optimal sequence of events. We apply this approach to two problems. In one case, the two customer classes are prioritized based on the cost associated with each period that the customer has to wait. The second problem addresses the case in which the two customer classes require adifferent level of service. As a second objective, we want to get some insight into the advantages of state-dependent critical levels as opposed to constant critical levels. We therefore introduce two state-dependent rules of setting the critical levels

4 4 CHAPTER 1. INTRODUCTION that allow us to maintain the advantage of a simple-to-implement strategy and evaluate the benefits. 1.3 Outline To solve a mathematical model, the processes and characteristics that are supposed to be modeled have to be identified. Additionally, a certain amount of mathematics is required. In line with this observation, this thesis is divided into three parts: One part introducing the subject of inventory control, one part laying the mathematical foundations and finally the part introducing our model. Figure 1.1 gives an overview of this thesis. The first part (Chapters 2 and 3) introduces stochastic inventory control. Chapter 2 focuses on general aspects of inventory management. While inventory control is concerned with the steering of stock in a specific environment, inventory management additionally includes all efforts influencing the environment e.g. by reducing lead times or quality defects (Zipkin 2000). We give evidence for the importance of inventory management, provide reasons for holding inventories, classify inventories into different types and look into the costs and service requirements associated with holding inventory. In Chapter 3, we concentrate on inventory control and identify decisive characteristics of the environment that have to be considered in the control of inventory. We provide an overview of the most common control policies and optimize a periodic review control policy for a homogeneous customer basis that we will extend to a setting with more than one customer class lateron. In the second part (Chapters 4-6), we introducestochasticprocessesas far as required for solving our model. We focus on Markov chains and here in particular on those with infinite state space. Ross (1983) and Wolff (1989) provide a more extensive account of Markov chain theory additionally covering reversible Markov chains. We first discuss the theory of Markov chains (Chapter 4) before we focus on their numerical solution (Chapter 5). In Chapter 6, we discuss stochastic ordering. In the last part (Chapters 7-11), we present our results. In Chapter 7, we provide examples of situations with segmented customer basis. We introduce

5 1.3. OUTLINE 5 Chapter 1: Introduction PART I: Foundations of Stochastic Inventory Control Chapter 2: Basic Concepts of Inventory Management Chapter 3: Stochastic Inventory Control PART II: Essential Stochastic Processes Chapter 4: Markov Chains Chapter 5: Numerical Solution of Infinite Markov Chains Chapter 6: Comparing Stochastic Processes PART III: Stochastic Inventory Control with Customer Segmentation Chapter 7: Introduction to Inventory Rationing Chapter 8: A Markov Chain Based Modeling Approach Chapter 9: Prioritization by Penalty Costs Chapter 10: Prioritization by Service Levels Chapter 11: Dynamic Rationing Policies Chapter 12: Conclusion and Critical Review Figure 1.1: Structure of research

6 6 CHAPTER 1. INTRODUCTION critical level rationing as one rule that has been studied to split inventory between different demand classes. We review the relevant literature and explain how our work fits in. Chapter 8 introduces our model and presents some basic results. In Chapter 9, this model is analyzed and optimized for the case of two customer classes prioritized based on the penalty costs caused by shortages. Chapter 10 treats the case of service levels instead of penalty costs. Up to this point, we have assumed that all parameters determining our control policy are constant over time. In Chapter 11, we develop some dynamic rationing policies and apply simulation to study their performance. Chapter 12 concludes this thesis. We summarize the results and point out the major contributions of this research. In addition, we critically review the drawbacks and end with suggestions for future research.

7 Quelle: Karin Möllering: Inventory rationing A new modeling approach using Markov Chain Theory, Kölner Wissenschaftsverlag, Köln, Kölner Wissenschaftsverlag und Karin Möllering

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