Spare Parts Inventory Control at Central Warehouses

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Spare Parts Inventory Control at Central Warehouses

E.J. Hoving

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Master thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Prof. Sc.D. B. Goldengorin

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Master thesis Econometrics, Operations Research and Actuarial Studies Title: Spare Parts Inventory Control at Central Warehouses Author: Erik Hoving BSc

1385844

Student Econometrics, Operations Research and Actuarial Studies (MSc) Specialization Operations Research

University of Groningen

Corresponding e-mail address erikhoving@gmail.com Date: January - July 2008

To cite: Hoving, E.J. (2008). Spare Parts Inventory Control at Central Warehouses. MSc Thesis, University of Groningen.

Supervision: Prof. Sc.D. B. Goldengorin (University of Groningen) Dr.ir. A.A. Kranenburg MTD (CQM)

Dr. B. Sel¸cuk

Abstract: In this thesis we focus on a central warehouse in a multi-echelon service supply chain for spare parts inventory control. A multi-item service opti-mization problem is formulated based on a practical situation at a company in the semiconductor industry. The objective is to minimize inventory costs under the condition that overall and individual constraints on the customer service degree are met. Critical level policies are used to differentiate be-tween the demand streams. For given critical level policies we explain meth-ods for evaluating performance measures like the customer service degree and the expected number of backorders.

A Dantzig-Wolfe decomposition is used to find a lower bound on the inven-tory costs in the multi-item spare parts problems. Several greedy heuristics are proposed. The performance of the greedy heuristics is compared with the Dantzig-Wolfe lower bound solution in a computational experiment. For both the computational experiment as well as a case study on real life com-pany data it is shown that substantial cost reductions can be obtained by using the new developed greedy heuristics compared to more naive policies. Keywords: Spare parts inventory control, service logistics, queuing systems,

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Preface

This master thesis is a result of an internship at consultancy company CQM (Centre for Quantitative Methods in Eindhoven) in collaboration with a company in the semiconductor business. It is the final project of five years of enrollment in the study program of Economet-rics, Operations Research and Actuarial Studies at the University of Groningen. During these past five years I have been taught several quantitative methods and, equally important, how to tackle problems. In my final project I used this knowledge to develop solution techniques for a challenging project in spare parts inventory control.

I am grateful for the help and support I received during my project. I would like to highlight my thanks for my supervisors Boris Goldengorin, from the University of Groningen, Bram Kranenburg, on behalf of CQM, and Baris Sel¸cuk, who guided me at the semiconductor company. They all gave me good suggestions and comments at different stages in my project. At this moment I also look back to the pleasant collaboration with all my fellow-students over the last five years. In particular, I would like to thank Johan Bos. We know each other from secondary school and completed several assignments together in an inspiring, comfortable working atmosphere. I appreciate his willingness to correct my thesis. He provided me with useful suggestions. Furthermore, I would like to thank all other people who read my thesis. They gave me several useful suggestions for improvements in the lay-out, the content and the English usage. Thanks to all.

Besides all the professional support I have got from my supervisors and numerous colleagues of them, I would like to thank my family and friends. They created the preconditions under which I could easily concentrate on my thesis project and my entire college life. I would like to mention the names of my parents Klaas and Lies, my brothers Peter and Marcel, my sister Annet, and my girlfriend Tryntsje. Without their support I would not have been where I am right now.

Depending on the reader’s knowledge in operations research and spare parts inventory control, several sections might not be relevant for the reader’s interest. Suggestions on how to read the thesis will be given in Section 1.1. Finally, I would like to wish the reader fun reading this thesis.

Erik Hoving

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6.4.2 Savings . . . 76 6.5 Sensitivity analysis . . . 77 6.6 Case study . . . 83 7 Conclusion 85 7.1 Contributions . . . 85 7.2 Overview of findings . . . 86 7.3 Recommendations . . . 88 7.3.1 Practical recommendations . . . 88 7.3.2 Future research . . . 90 Bibliography 92 List of figures 93 List of tables 95 Glossary 95 List of symbols 100 A Figures and tables 101 B Manufacturing of semiconductors 117 C Example of spread measures 119 C.1 Introduction. . . 119

C.2 Results. . . 120

C.2.1 Single scenario . . . 120

C.2.2 Sensitivity analysis . . . 123

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D Knapsack problem 129

E Evaluation method for a critical level policy 130

E.1 Steady state probabilities . . . 130 E.2 Bounds on the performance measures. . . 132

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Chapter 1

Introduction

In this chapter the topics considered in the thesis will be introduced. First, in Section1.1the structure of the report will be described. Section1.2 introduces the semiconductor company and the market it operates in. Special attention will be paid to the tasks and purposes of the Global Logistics Services (GLS) department. In Section1.3attention will be paid to past research in the spare parts inventory control problem. Finally, in Section1.4an introduction will be given to the spare parts inventory control problem at the company. The structure of the supply chain will be explained, as well as common performance indicators. Besides, some important terminology will be introduced in Section1.4.

1.1 Outline of the thesis

The general spare parts inventory control problem will be introduced in this chapter. Amongst other things, the starting literature on spare parts inventory control problems will be men-tioned in this chapter. The situation at the (semiconductor) company will be introduced in this chapter and will be discussed in more detail in Chapter 2. In Chapter 2 the research question will be stated and several assumptions will be explained.

Chapter3 will treat with the mathematical modeling of the situation introduced in the first chapters. First, simple cases will be discussed and extended towards the central multi-item service optimization problem, which will be presented in Section 3.5. The evaluation re-quired for the solution techniques will be discussed in Chapter4. The mathematical solution techniques for the central problem will be developed in Chapter5.

In Chapter6 we will present the results of the solution techniques based on a computational experiment. Sensitivity analysis will be carried out on selected parameters in the model. Based on the results in Chapter6a conclusion will be presented in Chapter7. The thesis will be discussed in this chapter and further research directions will be elaborated in Section7.3.2. Background information, advanced mathematical reasoning, figures and tables not directly relevant for the main text can be found in AppendicesA toF.

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of readers. For academic purposes all chapters, including appendices, are relevant material. Practitioners in the field of spare parts inventory control may find the regular chapters most relevant. AppendicesC.3,D,EandFmight be skipped by this group of readers. Chapters1, 2,6and 7 will be recommended to readers without any background in mathematics.

General abbreviations can be found in the glossary on page 95. Abbreviations from pro-fessional jargon and other relevant abbreviations will be explained in the text as well. An alphabetical list of symbols used in the main text of this thesis can be found on page96.

1.2 The company

1.2.1 Overall company description

The company we use for our study is a worldwide company founded in 1984, with its head-quarters in Veldhoven, the Netherlands. The company manufactures advanced lithography systems for the semiconductor industry. The company designs, develops, integrates, markets and services these systems. Lithography systems are used to make integrated circuits (ICs or chips) for chip producers. Background information about lithography technology and the production of semiconductors can be found in AppendixB.

Besides selling lithography systems to their customers, the company provides and sells services to their customers. These services comprise maintenance of machines and quick repair in case of breakdowns. The company has over 50 facilities in 16 countries providing service to their customers, see FigureA.1in AppendixA. The company possesses R&D facilities in Veldhoven and Wilton (CT, USA). Central warehouses are located in Veldhoven, Tempe (AZ, USA) and Singapore. Local warehouses are in close distance to the customer sites at various locations in Asia, Europe and Northern America.

The objective of the company is to pursue market leadership through customer satisfaction and technological expertise. Other business strategies imply high value drivers for customers, operational excellence and top financial performance. For this thesis the objectives of cus-tomer satisfaction and operational excellence are most relevant.

1.2.2 Market environment

In Section1.2.1it was mentioned that the core business of the company is the manufacturing of lithography systems. Customers include manufacturers of chips, R&D centers and some research laboratories. At the end of 2007, the company employed about 6,500 people generat-ing revenues of 3,809 million euro. Since 2003, the revenues increased with 150%, particularly due to the growing market in Asia.

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also commonality between systems will increase. Commonality means that different machine types have parts in common. Notice that good control of this knowledge can result in lower inventory levels than if one would keep inventory for spare parts of the different machine types apart. Especially if costs of spare parts are high and demand is low. It is generally preferable to have one SKU instead of more SKUs per spare part. Notice that demand for spare parts is the same as failure of those parts in the context of spare parts inventory control. Demand and failure are used as synonyms throughout this thesis.

Definition 1.1 A stock keeping unit (SKU) is an inventory item at a particular geographic location. See Sch¨onsleben (2007).

The machines are divided into four classes: non-AT, AT, XT and EUV, in chronological order of market introduction. The most recent machine type EUV is introduced in the market in 2007. The difference between machine types is the output they produce. They all produce chips, but the newer machines produce smaller, faster and smarter chips with more power on each chip. The machines are able to produce more chips on each wafer. The XT machines already produce with an accuracy of 40 nanometers and below. Over 10,000 spare parts and tools are defined for these four system classes.

1.2.3 Global Logistics Services department

The GLS department at the company is amongst other things responsible for the worldwide spare parts logistics. In this section the structure and objectives of GLS will be explained.

Structure

One can find the organizational chart of the company in Figure1.1. It is indicated how GLS is situated within the company. Worldwide GLS is divided into several functional units. For this thesis the most important units are the central planning, regional planning and part definition and quality group for the central warehouse in Veldhoven. The central planning group is responsible for the planning of SKUs at central warehouses. Local warehouses are planned by the regional group. Maintenance of BOM structures and failure rate reduction are tasks of the part definition and quality group.

Inventory policies and transactions are currently controlled using an enterprise resource plan-ning (ERP) system. The software package used at the company is SAP. All central and regional planners have online access to this information system. At any moment in time the planners can view the worldwide inventory levels of all SKUs, including all outstanding orders from customers and to suppliers. The data used in this thesis is collected from SAP.

Business objective

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Figure 1.1: Organizational chart, with customer support department folded out.

process, i.e. the required capacity for lithography systems is greater than the available ca-pacity. This results in extraordinary high downtime costs for the company its customers if a lithography system fails. For that reason the purpose of GLS is to maximize the equipment availability of their customers. Of course, maximizing equipment availability would yield un-economically high costs. Therefore GLS agrees on service level agreements (SLAs) with their customers. Agreements are made on fill rate or on waiting time for parts. Explanations of these performance criteria can be found in Section 1.4.3. On the basis of these SLAs the purpose of GLS shifts to meeting the uptime requirements for all customers while minimizing their own costs. The GLS objective will form the basis of the modeling concepts in this thesis.

1.3 Literature review

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some starting literature on inventory management will be mentioned. These books also treat general stock setting policies, which is the topic of Section1.3.3. This section will be concluded with a discussion on the relationship with other theories in Section1.3.4.

1.3.1 Characteristics of spare parts inventory control problems

Features of spare parts inventory control systems include the number of SKUs involved, SKU characteristics (i.e. cost, lead time, demand rate and physical characteristics), service differentiation, obsolescence, single-echelon versus multi-echelon, lateral transshipments, sub-stitutability, commonality, static versus dynamic environment, transportation modes, back-ordering versus lost sales and reparability. The methods used for inventory control depend strongly on the status of all these characteristics. In this section the meaning of these features will be explained.

• Number of SKUs. More SKUs will mean more differences between the SKUs. These differences express themselves in the SKU characteristics, see below. Classifications can be used to divide the SKUs. Then different methods for inventory control can be used for the classes.

• SKU characteristics. The standard cost price (SCP) of a SKU is relevant for inventory management. High cost SKUs have larger impact on total investment costs. Other SKU characteristics which might be relevant for modeling an inventory system are lead time, demand rate, shortage cost, holding cost, transportation cost, volume, weight, shape and storage requirements.

• Service differentiation. One speaks of service or customer differentiation if different customers or request types require different service level constraints.

• Obsolescence. After some years, SKUs and/or machines might become out of use be-cause of several reasons, e.g. introduction of substitutable, better machines. This aging problem is called obsolescence.

• Redundancy. In some production processes one might have two parallel subprocesses. If an item in one subprocess fails the machine still works, because it switches to the other

-Warehouse - Demand -Central J J J ^ Regionals @ @ R @ @ R -Locals Demand

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subprocess. With respect to inventory management the items in such parallel processes are called redundant items.

• Criticality. If a failure of a SKU causes failure of the machine, the SKU is critical. Some SKUs are not critical, e.g. they appear in the machines for lay-out reasons or failure of the SKU causes noise but no breakdown of the machine.

• Single-echelon versus multi-echelon. In Figure 1.2 one can see the difference between single-echelon and multi-echelon models. On the left side one warehouse is in the system. Consequently, stock levels only have to be set for this warehouse. On the right side of Figure 1.2 one can see a multi-echelon inventory system, in this case a three-echelon system. In this situation central warehouse replenishes stock in regional warehouses who replenish stock at the local storages. It is self-evident that increasing stock at one storage location affects the other sites. This indicates the necessity for integrated decisions on the stock levels at all warehouses in such a multi-echelon supply chain. • Multi-indenture. One speaks of a multi-indenture problem if one takes into account the

BOM position of the SKUs in inventory planning and control.

• Lateral transshipments. The provisioning of an item by a local warehouse to a customer of another local warehouse that is out of stock is called lateral transshipment. One may remark that lateral transshipments can only take place in a problem with multiple locations.

• Substitutability. In some situations one can replace a SKU with stockout by a replaceable SKU with similar technical characteristics. This SKU is supposed to have higher usage cost (SCP plus change cost) and it might be a SKU with higher position in the BOM, i.e. an assembly of the original SKU and other materials.

• Commonality. This is the phenomenon that different types of equipment have spare parts in common.

• Static versus dynamic environment. If a system has to be evaluated at one moment in time, one considers a static environment where variables might be stochastic but not time-varying. A dynamic environment takes into account changes over time, i.e. changes in demand pattern. One might want to determine the optimal values for stocks for different moments in time, where the total objective over time has to be optimized. • Transportation modes. Transportation between warehouses or from warehouses to cus-tomers can be done in a regular way. It might also be possible to do it in a quicker and possibly more expensive way. This is called an emergency mode in the literature. • Backordering versus lost sales. If an item is out of stock there are two possibilities. The

demand is lost, i.e. the demand is fulfilled from outside the system, or the demand is backordered, i.e. the item is provided to the customer as soon the item is replenished. • Reparability. Some SKUs might be considered reparable if reparation of these spare

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1.3.2 Starting literature

The books of Silver et al. (1998) and Axs¨ater (2006) present some nice overviews of avail-able methods in inventory management. General stock setting policies as in Section 1.3.3 are explained and the concept of material requirements planning (MRP) is considered. This method assumes a predictable demand which is generally not the case in spare parts man-agement. Silver et al. (1998) do not explicitly address the spare parts case. Axs¨ater (2006) also discusses cases with stochastic, discrete demand in several inventory models. In general, spare parts inventory control problems also have stochastic, discrete demand. Specific for spare parts inventory control problems is the low and unpredictable demand. Other books on spare parts inventory management are those ofMuckstadt(2005) andSherbrooke(2004).

Muckstadt(2005) andSherbrooke(2004) describe methods for spare parts inventory manage-ment in the application area of military systems. This area shows similar characteristics as for advanced equipment manufacturers like our semiconductor company. High cost spares, low and unpredictable demand and multiple storage locations. They explain a multi-echelon tech-nique for recoverable item control (METRIC) as was first introduced inSherbrooke (1968). Local warehouse stock levels at the company are currently set using a METRIC based pro-cedure which is mainly based onAxs¨ater (1990),Kranenburg(2006) and Sherbrooke(1968). For a clear distinction with the METRIC of Sherbrooke (1968), the procedure at the com-pany is called comcom-pany-METRIC from now on. Comcom-pany-METRIC spare parts planning is based on the BOM, failure rates, prices, replenishment lead times and emergency lead times. The inventory budgets are minimized subject to a SLA. Most important deviation from the original METRIC of Sherbrooke (1968) is that the company does not use their method for multi-echelon planning. They just use the model for planning the local inventories.

In inventory control, SKUs are often classified using an ABC classification. The SKUs are then divided into three classes called A, B and C. A items (SKUs) are a small portion of all SKUs constituting a high percentage of total inventory investments. C items are a large number of SKUs which account for only a small percentage of total inventory investments.

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The remaining SKUs are called B items. In Figure 1.3 a graphical representation is given. The percentages are indications taken from Sch¨onsleben (2007). In real world cases these values will differ from case to case.

1.3.3 Stock setting policies

In this section, stocking policies in general inventory control literature will be described. We start with the difference of continuous and periodic review policies. Backordering versus lost sales is the second topic in this section. We end with the addition of critical levels to these stocking policies.

Continuous review policies assume that the system is continuously updated with information regarding inventory transactions. If the actual stock level drops below the reorder point a production or procurement order is generated. Periodic review policies evaluate the actual stock level after a certain time period. If at that moment stock is below the reorder point, a purchase order is generated. From now on we assume continuous review policies, see Sec-tion 2.6 for arguments validating this assumption. The following continuous review stock policies are often used.

(s, Q) Stock policy with reorder point s and reorder quantity Q. (s, S) Stock policy with re-order point s and order-up-to-level S.

(S − 1, S) Base stock policy with re-order point S − 1 and order-up-to-level S.

Notice that (S − 1, S) is a special case of (s, Q) and (s, S), with s = S − 1 and Q = 1. A detailed description of this kind of policies, also in case of periodic review can be found in

Silver et al.(1998). The policies can be extended by adding reserved stock. Reserved stock is implemented to differentiate between two (or more) particular classes, e.g. customer groups. A so-called critical level c is introduced for the height of the reserved stock. In Figure 1.4 an example is given where emergency demand is the most important class. Reserved stock is created for this class. If the stock level drops to S = 2 an incoming replenishment demand is backordered or lost, while an emergency order is still allocated at once. Whether a demand is backordered or lost is case specific. It is one of the features we explained in Section1.3.1.

Base stock level, e.g. S = 5

Critical level, e.g. c = 2

Stockout

Physical stock available for all demand

(Reserved) physical stock only available for emergency demand

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Some interesting remarks can be cited from literature. InMuckstadt(2005) it is claimed and proven that the base stock policy is optimal in many cases, especially if demand is low and SKU costs are high. Silver et al. (1998) on their turn combine general stock setting policies with the ABC classification. They state that an (s, S) policy is best for A items, while an (s, Q) policy is best for B items. Possible savings on inventory decisions on C items are quite small, so Silver et al.(1998) recommend simple rules of thumb for controlling C items where the objective is to ensure high enough inventory at all time.

1.3.4 Relationship to other theories

In almost all stochastic inventory models queuing theory is in the background. An intro-duction to the most important aspects from queuing theory is given in Gross and Harris

(1985).

Another theory which might be used in inventory control is dynamic programming. In the literature, dynamic programming is used if demand is time-varying. A classical example discussed by Axs¨ater (2006) is the decision on batch sizes if demand for a SKU is time-varying (not stochastic). One can use the Wagner-Whitin algorithm to solve this problem. This algorithm is explained by Axs¨ater (2006) and first published in Wagner and Whitin

(1958). The reader is referred to these references for more information.

Furthermore, the inventory control problem shows similarities with other problems in opera-tions research: the knapsack problem and the assortment problem. The spare parts inventory control problem is in fact a complex knapsack problem. This relationship will be explained with a small description of the knapsack problems for the reader unfamiliar with this problem. The name corresponds to the problem of packing valuable objects into a knapsack. The objective of the knapsack owner is to maximize the value of items in the knapsack with the volume constraint limiting the number of possible outcomes. This problem is an example of an integer linear programming (ILP) model. The model is mathematically stated in AppendixD. Books with more information on knapsack problems are written byKellerer et al.(2004) and

Martello and Toth(1990).

The accordance with the bounded knapsack problem is that an integer number of items have to be stocked. The maximum number of items to be stocked is not specified. The objective function is to minimize costs subject to a minimal service level. In the knapsack problem the constraint is maximal storage space. With a simple transformation one can reformulate greater and equal constraints in less and equal constraints and vice versa. In Kellerer et al.

(2004) a proof is given of the computational complexity of the knapsack problem, i.e. even the simplest knapsack problem is non-deterministic polynomial time hard (N P -hard). Besides,

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The spare parts inventory control problem is also related to the assortment problem. Pentico

(2007) gives a description and a review of assortment problems in practice. The assortment (or catalog) problem can be used do determine which possible set of SKUs to stock to minimize total costs. This set is traditionally a subset of sizes of a product. Demand for sizes not in stock can be replaced by a SKU with larger stock size against some additional substitution cost. The assortment problem originally uses product sizes, but equivalently one can think of other hierarchies like quality of SKUs as well. The substitutability is one-way, i.e. demand can only be filled with larger size or higher quality SKUs.

The connection with the spare parts inventory control problem is limited to the substitutabil-ity characteristic. The assortment problem does not take into account service level constraints. These constraints are in most cases of big importance in the spare parts inventory control problem. One can handle this by adding substitution costs, as kind of penalty costs, to the objective function. In practical applications this might cause problems, because data is not available on substitution costs. Another difficulty in assortment problems stated byPentico

(2007) is that stochastic models are hard to handle. Stochastic demand is an important as-sumption in the case of the spare parts inventory control problem where demand is irregular and unpredictable. Rao et al. (2004) treat a stochastic programming model to deal with stochastic demand. They construct heuristics to solve the problem, where they even allow for two-way substitutability.

To summarize, this section explained the starting literature, the characteristics of a spare parts inventory control problem and the relationship with other theories. It is in no way an all-embracing treatment of all spare parts inventory control literature. Relevant literature for other chapters in this thesis will be mentioned throughout this thesis as well.

1.4 Spare parts logistics

In this section the following topics will be discussed. In Section1.4.1an introduction will be given to spare parts logistics in relation to logistics. In Section1.4.2an overview will be given of the service supply chain for spare parts at the semiconductor company which we study in this thesis. Key performance indicators (KPIs) used at the company will be explained in Section1.4.3.

1.4.1 Relationship with logistics

Spare parts logistics is a specific application of the field known as service logistics. Service logistics encompasses all logistics in the use phase of a product life cycle. According to

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@ @ @ Logistics @ @ @ @

Service logistics @@ Spare parts logistics @@

Figure 1.5: Spare parts logistics as part of logistics.

Figure 1.6: Service supply chain structure.

1.4.2 Structure of the service supply chain

Figure1.6 gives a simplified graphical representation of the service supply chain of the com-pany. Figure 1.6 will be used to explain how demand for spare parts is managed at the company. Subsequently, the flows in Figure1.6will be interpreted.

Demand allocation procedure

For the description of the demand allocation procedure it is first important to notice that we only treat customers with SLA. The company should meet service constraints for this class of customers. Customers without SLA do not have service constraints and are therefore less important to spare parts management. Demand for spare parts from customers without SLA is very unpredictable and specific. Moreover, the number of customers without SLA show a decreasing trend. Therefore we only look at customers with SLA in this project.

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Step 1: Local supply. The part is delivered from the standard local (warehouse) belonging to that particular customer site.

Step 2: Lateral supply. If the standard local has not enough stock, the part is delivered from one of the local warehouses in the neighborhood. Not all local warehouses provide lateral supply. Local warehouses close to airports and/or shipping routes are preferred for lateral supply. The order in which lateral supply at local ware-houses in the neighborhood takes place is predefined.

Step 3: Emergency supply. If the first two options fail, the part is shipped from the central warehouse.

Step 4: Rest of world (ROW) supply. If even the central warehouse is out of stock the demand is fulfilled using items from test machines (cannibalism) or from new machines. The demand is considered as lost demand. It is self-evident that this stream should be avoided anyhow.

Flows

In the demand allocation procedure some flows in Figure 1.6 are already mentioned. The normal local deliveries are indicated with L and the lateral transshipments are denoted by T . The other flows are all related to the central warehouse. With respect to the central ware-houses two main lead times can be identified. Definitions1.2 and 1.3are stated here to give a clear distinction between these lead times. Both production and delivery lead time consist of different parts.

Definition 1.2 The production (lead) time is the lead time from supply to central warehouse. It consists of manufacturing lead time P 1 and repair lead time P 2.

Definition 1.3 The delivery (lead) time is the lead time from central warehouse to local warehouses. It consists of emergency lead time C1 and replenishment lead time C2.

The emergency flow C1 and the replenishment flow C2 are two flows leaving the central ware-house towards the local wareware-houses. The difference between emergency and replenishment towards the local warehouse is new. Emergencies are explained above. If demand cannot be allocated from local warehouses it should be shipped from the central warehouse. However, in most cases demand will be allocated from local warehouses (step 1 and 2). The stock in the local warehouse should then be replenished. A replenishment order is placed at the central warehouse. In fact, a third flow could be mentioned as well. The stocking of new defined SKUs and of SKUs of which the reorder point recently has been increased is called initial stocking. This flow will not be treated explicitly in this thesis, because it has an incidental character and therefore needs specific techniques.

ROW supply is indicated with dashed lines in Figure1.6. This flow has no lead time, because it is defined as lost demand with respect to the central warehouse.

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Flow Lead time Local delivery L 0-4 hours Lateral transshipments T 6-8 hours

Emergency C1 12-36 hours

Replenishment C2 1-14 days Manufacturing P 1 3-6 months

Repair P 2 1-5 months

Table 1.1: Flows with their lead time.

in case of reparable SKUs. Then the item is replaced using the normal sequence but no production order is placed at supply. Instead the FSD will enter the repair network. At the central warehouse the FSD will be looked at and send to factory or original external supplier. Parameters related to the repair flow are the repair lead time P 2 and the scrap rate, see Definition1.4.

Definition 1.4 The scrap rate is the percentage of items that cannot be repaired.

About 56% of all SKUs of the company are considered non-reparable because of economical or technological reasons. For these SKUs the scrap rate is 100% and failures are replenished by means of manufacturing immediately. The corresponding lead time is called manufacturing lead time P 1. Under ideal circumstances for reparable SKUs, i.e. stock levels high enough and scrap rate equal to 0%, one can observe that replenishment can be completely fulfilled from repair and no items have to be manufactured from scratch.

In Table1.1 an indication of the lead time for the different flows is given. In about 10% of the cases the lead time is outside the interval in Table1.1. It will immediately be clear that production lead time is substantially longer than the delivery lead time. This is one reason why inventory control for spare parts is important for the company.

1.4.3 Performance indicators

In Section1.2.3the fill rate and waiting time for parts were mentioned as performance mea-sures for GLS. Definitions of these two KPIs are given in Definitions1.5 and1.6.

Definition 1.5 The customer service degree (CSD), also fill rate, is the percent-age of items that can be delivered immediately upon demand.

Definition 1.6 The down time waiting parts (DTWP) is the percentage of time equipment is waiting for a service part.

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Figure 1.7: Down time waiting parts as part of equipment availability.

Currently, both CSD and DTWP are used by the company. CSD is measured for local supply. CSD can be used both on item and on system (multi-item) level to measure the performance.

Thonemann et al. (2002) argue that the system approach results in lower costs, but the drawbacks are the needs for higher skilled personnel and the implementation is often quite complex. It is appropriate to use DTWP as system performance indicator, because it is only relevant whether equipment is down and not why, i.e. by which part, it is down. Typical targets for the company are 95% for system CSD and 0.5-0.7% for DTWP.

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Chapter 2

Problem formulation

The spare parts inventory control problem was introduced in Chapter1. In this chapter the motives for initializing this thesis project will be described in Sections 2.1 and 2.2. First, the relation with the overall spare parts inventory control problem will be discussed. In Section2.2 the focus will be on planning at central warehouses, the main topic of this thesis. In Section 2.3, the current situation for central planning will be explained. The resulting research project is explained in Section 2.4. Sections 2.5 and 2.6 will explain the scope of the project. The nature of the variables will be described in Section2.5. The features of the spare parts inventory control problem mentioned in Section1.3.1will be treated again for the real life company case in Section2.6.

2.1 Spare parts inventory control problem

The company currently plans central and local warehouses separately, as described in Sec-tion 1.2.3. No integration is made on the decisions between these two planning levels. The local warehouses are planned using company-METRIC, see Section1.3.2. The central ware-houses are planned based on intuition and some guidelines. The overall objective is to mini-mize holding costs plus transportation costs while meeting the service level constraints. The overall spare parts inventory control problem can be divided into two subproblems. The central (warehouse) planning problem and the local (warehouse) planning problem. Central planning refers to the planning of inventories at central warehouses. Similarly, local planning is the planning of local warehouse inventory levels.

The connection between the central planning and local planning problem is the delivery lead time, see Figure 1.6. The objective of the GLS department can be explained on the basis of Figure 2.1. Here the costs of stock are split into the two problems, central and local planning. Changes in costs at the central and local warehouse are evaluated against changes in the delivery lead time. The parameters LT1 and LT2 are defined the delivery lead times

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Costs LT1 Central planning Costs LT2 Local planning

Figure 2.1: Costs related to stock in central warehouse (left) and local warehouses (right) if delivery lead time (LT ) changes.

The ultimate goal of the company is to minimize the sum of costs involved in central and local planning. If the shape of the graphs in Figure2.1is known, the company wants to minimize the sum of these two graphs under the condition that LT1 = LT2. Obviously, if the delivery

lead time is zero, no stock is required at the local warehouse and all costs are made at the central warehouse. Notice that these costs are finite and computable. The contrary is also true. For high enough delivery lead time the costs at central planning are zero and all costs are made in local planning. One might remark that the costs at central planning (if delivery lead time is zero) and local planning (if delivery lead time is high) do not necessarily be the same.

The delivery lead time consists of two parts as explained in Section1.4.2. For every flow a different delivery lead time guarantee is appropriate. This makes the problem more complex. However, the importance of the simple representation of Figure2.1 is the interest of central and local planning. For central planning the objective is to guarantee a specific delivery lead time to local planning. Local planning should meet the SLAs. To be able to meet the SLAs local planning should have information about the delivery lead time from central warehouse. Logically one should start solving central planning. Management may decide on the best spot on the curve in the left graph in Figure2.1. The corresponding delivery lead time is used as input in local planning. It is possible to take a particular configuration for one of the two problems as given and solve the other problem using this input.

An integrated solution based on the previous reasoning is proposed byEnders(2004). He uses an iterative heuristic procedure based on two building blocks. One block is called the lateral transshipment model and the second block is the central warehouse model. The iterative heuristic procedure adjusts the delivery lead time in every iteration step.

A drawback of the proposal ofEnders(2004) is that spread in delivery lead time is not dealt with. Definition 2.1 explains what is meant with spread in this thesis. In Section 2.2.2 the objectives for central planning will be discussed. Objectives for spread will also be discussed.

Definition 2.1 Spread is the variation between the delivery lead times for differ-ent SKUs.

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recommends an integrated approach between local and central planning instead of an iterative procedure to reduce computation time.

2.2 Central planning

Recall from Section2.1that the central planning problem is a subproblem of the spare parts inventory control problem at the company. In this section it will be discussed why it does not hurt to focus on the central planning subproblem in first instance. Subsequently, the objective for central planning will be discussed. The current situation regarding central planning is explained in Section2.3.

2.2.1 Subproblem

Local warehouses are currently planned using company-METRIC. The SLAs, CSD or DTWP, are entered into this system. The delivery lead time for emergencies and replenishments are also entered into the system. The resulting stock levels at the local warehouses are such that the SLAs are satisfied. For central planning the challenge is to satisfy the assumed delivery lead times for the local planning subproblem.

Recall from Section2.1thatEnders(2004) treats central planning, namely the second building block of his iterative heuristic procedure. Because it is a part of his overall approach some simplifying assumptions are made. In this thesis project we will incorporate the repair flow into the model. Two production lead times are considered, namely manufacturing and repair lead time. Moreover, as desired by the company, it will be investigated whether the spread in delivery lead time can be considered as a service measure as well.

If spread is not dealt with, one can consider the item and system approach. These two inventory control approaches will be used a number of times throughout this thesis. Therefore we will first give definitions of these approaches in Definitions 2.2and2.3before we continue with the discussion about spread.

Definition 2.2 The item approach minimizes holding costs subject to a perfor-mance constraint per SKU. The perforperfor-mance constraint per SKU is equal to the overall performance constraint for each SKU.

Definition 2.3 The system approach minimizes holding costs subject to an over-all performance constraint.

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Costs

System

Spread Item

Figure 2.2: Total costs of item, system and spread approach.

It does not hurt to focus on the central planning problem in first instance because the prob-lematic nature is different from local planning. Local planning is necessary for fast delivery with an inventory buffer available in case of stockout. This inventory buffer, the central warehouse, is used to protect the fast delivery of items against negative effects caused by statistical fluctuations, i.e. unexpected high demand in a small time period. No inventory buffer is available in the central planning problem.

Moreover, local planning deals with one demand stream while central planning has two in-coming demand flows. Ultimately, it would be interesting to find an integral solution to the overall spare parts inventory control problem. The overall multi-echelon problem is neverthe-less much more complex. A thorough understanding of the subproblems is a good starting point for an integral solution method. It was mentioned before in Section 1.3.2 that local planning has been treated in the literature already, but central planning did not yet received similar attention. This is more remarkable if one observes that 33%1 of the worldwide stock value at the company is at the central warehouse in Veldhoven. The lack of research and the high impact in worldwide stock value are two reasons for initializing this thesis project. The contribution of this thesis to the available literature on central planning is the develop-ment of methods with two production flows (manufacturing and repair) and spread constraints in the model. Adding spread to the model means adding flexibility. The support function of the central warehouse will be emphasized with the inclusion of spread. Moreover, methods will be developed where delivery lead time is considered at central planning instead of local planning which is better for controlling the overall spare parts inventory problem.

2.2.2 Objective

The objective in the central planning problem is to minimize expected costs subject to partic-ular service constraints. The service constraints for central planning will be explained here. In Section1.4.2it is explained that two flows can be considered from central warehouse to local warehouse. The demand for each flow is different. It is generally assumed that demand for items occur according to a Poisson process. Because local warehouses trigger a replenishment as soon as they fulfill an order one can assume a Poisson process for replenishment towards the local warehouse as well. This is different for emergency orders. They are only issued if local warehouses cannot provide an item. So the demand for emergencies is much more spiky.

1

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We state the demand difference here to address the difficulty of the different flows. We know that emergency demand is of higher importance for central planning than replenishment orders. If a replenishment order is not issued immediately it does not affect the down time at the customer sites. If on the other hand, an emergency order cannot be allocated from central warehouse, the DTWP at the customer site will increase. Therefore, two different service constraints should be set in the central planning problem. A service constraint for replenishment orders and a more stringent constraint for emergencies. The constraint for emergencies should keep the number of long downs to a minimum. A machine is considered long down if the DTWP is greater than 12 hours.

The actual objective of central planning is to guarantee a specific delivery lead time to the local warehouse. Of course, divided into a guarantee on replenishments and on emergencies. Local planning assumes a replenishment lead time of 14 days over all SKUs. Emergency orders should be replenished at the latest in 2 days. These assumptions for local planning are the objectives for central planning. Not only the average lead time towards the local warehouse is important, also the spread is important. The company does not want general good performance and some extreme outliers. They want a reliable guarantee to the local warehouse.

Notice that the actual lead time that can be promised to local planning is output based on the reorder points and critical levels at the central warehouse. Secondly, note that the output lead time for some SKUs will be higher than 14 days because 14 days is the average over all SKUs. Sometimes the actual promised lead time for a SKU is two or three times the 14 days requirement, while for other items the output lead time is 1 day on average.

This spread is currently not considered. Several measures can be included to deal with spread. One way is to include additional hard constraints on the maximum average lead time per SKU to the model. A second option is to include penalty costs for the aggregate overall deviation from the average. In Appendix C, an example will be given to demonstrate the spread measures.

2.3 Current situation

The performance of central planning with respect to the objectives of 14 days replenishment lead times and 2 days emergency lead times is not measured exactly. The output for average replenishment lead time is not known. Local planning is responsible for satisfying the SLAs. As stated in the previous paragraph, local planning is executed with an input of 14 days average replenishment lead time. It was mentioned that the actual replenishment lead time is output of the central planning reorder points and critical levels. So the actual fulfillment of SLAs is dependent on central planning.

The current inventory policy for most SKUs can best be described as an (S − 1, S) policy, as described in Section1.3.3. For some (low cost) SKUs an (s, Q) policy is used.

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of one planner consists of both reparable and non-reparable items. The planners do not have any reliable information about the repair parameters, i.e. scrap rate and repair lead time. Central planning is currently not based on an item nor a system approach. Inventory of all SKUs is controlled and planned separately aiming at the overall company objective, see Sections 1.2.3 and 2.2.2. It is not based on quantitative techniques. Intuition and some practical guidelines are used for central inventory planning. An example of such a guideline is that the reorder point is currently set at the expected usage during the replenishment lead time of a SKU plus an extra 25% safety margin.

2.4 Research question

Based on the situation described in the previous sections of this chapter, the following research question will be stated:

Develop a technique for the setting of base stock levels and critical levels at the central warehouse. Take into account two production flows to the central ware-house and two delivery flows from the central wareware-house. Specify the model char-acteristics of the service supply chain of the semiconductor company. Compare heuristics with respect to optimality and required calculation times.

In continuation on this research question several other topics can be identified to investigate. In this thesis the following aspects are discussed:

• Item, system and spread approach. One of the discussions will involve the choice for an item or a system approach in the stock setting policy. The discussion will involve consideration of the application of one of the approaches in practical situations as well as a computable comparison. The comparison should indicate the performance of the (new) spread approaches developed in this project, see Appendix C.

• CSD versus delay. For the central planning problem service constraints need to be set. One can choose for different constraints than in the SLAs with customers and/or in the local planning problem. For emergencies one would like a high fill rate, so CSD looks appropriate. But dealing with spread may influence the choice of constraints. We will compare the pros and cons of CSD and delay as service constraints in the central planning problem. One question to investigate is whether delay can be transformed into CSD and vice versa.

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2.5 Variables

In this section the variables and parameters that will be used in the model will be introduced. In Chapter3, symbols will be introduced to represent the variables.

• SKUs. The number of SKUs is a parameter in the model.

• Demand. The demand is different per SKU and it is assumed that failures in the field occur one by one. Occurrence of failures over time do not depend on each other. Then the counting process called Poisson process is often used. The demand is assumed to follow a Poisson process.

• Scrap factor. The scrap factor is defined per SKU. No precise data is available. The scrap factor is assumed to be deterministic.

• Lead time. The production lead time is dependent on the reparability of the SKU. Therefore one can distinguish two production lead times. The production lead times are stochastic and defined per SKU.

– Manufacturing lead time. If the item cannot be repaired it will take a value from the manufacturing lead time distribution.

– Repair lead time. If the item can be repaired it will take a value from the repair lead time distribution.

The delivery lead time is divided into replenishment and emergency lead time. Recall that replenishment orders are backordered in case of stockout. Emergency orders are lost. That means that the stock levels determine how long average replenishment lead time takes, i.e. higher stock results in less stockout and therefore less delay. For emergencies the stock levels determine the number of ROW supplies. The delivery lead time is different per local warehouse, not per SKU.

• Costs. The costs can be split into transportation and holding costs. Other costs like ordering and setup costs are negligible and therefore neglected, see Section2.6 for vali-dating arguments. Stockout costs are not directly available at the company. Van Hou-tum and Zijm(2000) show a general relation between (penalty) cost models and service models in inventory management. They conclude with a remark that the one-to-one relation between the two different inventory models does not necessarily exist for spare parts management problems. Because the company has SLAs with their customers, service models are preferable and will be used in this thesis. Costs are assumed to be deterministic and given.

– Transportation costs. These are the costs of transportation of an item to a par-ticular local warehouse. Transportation costs depend on the stocks at the local warehouses. The local warehouse stock levels will be fixed in our optimization model, so the transportation costs are not of particular interest from an optimiza-tion perspective.

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• Decision variables. The decision one wants to make is the level of stock one has to keep to meet the service level constraints under minimal costs. For every SKU one should decide on the base stock level and the critical level.

2.6 Assumptions

In this section, the assumptions related to the practical situation at the semiconductor com-pany will be enumerated and explained. Some of the assumptions were already mentioned in Section2.5. In Section 2.6.1 the assumptions will be listed and validating arguments will be given. Section2.6.2continues the discussion on spare parts features in Section1.3.1. The characteristics for the central planning problem at the company will be listed here.

2.6.1 Validation of assumptions

A list of assumptions for the central planning problem at the company will be given below. • Continuous review policy. In reality, stock levels are not reviewed continuously, but

decision makers at the company give notice of the fact that base stock levels (or reorder points) may be changed any moment in time. Moreover, the company uses an advanced SAP system enabling the planners to view all stock levels and inventory transactions worldwide at any moment in time.

• No setup and ordering costs. If the SCP, holding and transportation costs of an item are high then the setup and ordering costs are assumed to be negligible. For most spare parts at the semiconductor company this is the case. For smaller cost SKUs or SKUs with high transportation costs, the setup and ordering cost are underestimated. Despite this inconvenience it is a practical assumption simplifying the modeling of the situation. The company agreed upon this assumption.

• Single size batch size. Batch sizes of one item are equivalent to a base stock policy. According to Feeney and Sherbrooke(1966) this policy is optimal if the SCP of a SKU is high (hundreds of dollars) and demand is low (demanded only few times a year). This assumption is related to the previous assumption. The inverse is also true. If every demand causes a single size order the total ordering costs are the straightforward consequence of total demand and do not play a role in optimization.

If setup and ordering costs are not negligible these costs will make it more attractive to order more items at once. This result follows from the economic order quantity (EOQ) model. One can find more information on the EOQ model and EOQ formula in Silver et al. (1998) or Sch¨onsleben (2007).

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• No obsolescence. In real world one often encounters the problem of obsolescence. In the spare parts situation at the semiconductor company there are two sources of ob-solescence. First of all, the obsolescence of spare parts themselves and secondly the obsolescence of the machines they are part of. The first is not a real problem for the company because technical material is not liable to perishableness like food industry, e.g. dairy products. Spare parts are quite stable products.

The obsolescence of machines is more particularly a problem for the company. The product life cycle of machines is quite short. Technological development is the main cause for this trend. This means that if machines are not sold anymore or if they are completely out of the market, the spare parts used for that machine might become redundant. However, in real world it is often the case that these parts can be used for other (newer) machine types. Sometimes the parts can be reconstructed relatively easy to another spare part. It does not occur frequently that spare parts have to be disposed of. Because of this, it is reasonable to assume no obsolescence in the model.

• Demand static over time. The number of failures in time periods of equal lengths may differ. In Section 2.5 it was mentioned that demand is supposed to be stochastic. It is supposed that variation of demand is completely due to the stochasticity. It is not dependent on trends. So demand is stochastic and static.

Time-varying elements are important for planning of spare parts inventory. Unfortu-nately, time-varying objects like macro economic variables and demand are unclear and unpredictable. Assumptions about the future are susceptible to mistakes. This is an important reason to use a static model. Another reason is that demand is only dynamic for new spare parts and/or modified parts. Here we talk more about initial stocking which is not in the scope of this thesis.

This assumption is important to realize in the evaluation of the model. Decision makers should take care of changes in the environment. This should always be in the back of mind. Notice that a solution of a static model is never optimal. An (S − 1, S) policy will generally not exactly satisfy the service level constraints. So over time one could vary the base stock and critical level(s) to approach the service level constraints. One can periodically adjust the base stock and critical levels as new information about the future becomes available.

Sensitivity analysis can be used to evaluate the effects of some assumptions. In the discussion of demand one could apply sensitivity analysis by increasing/decreasing the Poisson parameters. The effect of over/underestimating the demand can then be eval-uated.

• Independent Poisson demand. This assumption was already mentioned and explained in Section 2.5. We assume that failures occur one by one and failures do not depend on each other. The demand is assumed to follow a SKU independent Poisson process with different rates for all SKUs.

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• All items critical. If an item fails the machine will also fail. In practice this is the case for about 95% of the situations. So it is reasonable to model it this way.

• Stochastic production lead times. In Section 2.5 the stochasticity of production lead times was already mentioned. Palm’s theorem (Palm(1938)) states that if demand for a SKU is given by a Poisson process with rate m and the repair time for each failed unit is independently and identically distributed according to any distribution with mean production time _µ1, then the steady state probability distribution for the number of units in repair has a Poisson distribution with mean m_µ. A statement and proof of this theorem can be found inSherbrooke(2004). Using Palm’s theorem enables us to choose an arbitrary distributed production lead time. As agreed with the company, the SKUs are assumed to have independent identically distributed exponential production lead times.

• Backordering for replenishments and lost sales for emergencies. Reflecting the real world flows, it should be assumed that replenishment orders are backordered if central warehouse is out of stock while emergency orders are lost. An order is called lost if the demand has to be satisfied in any other way, e.g. using cannibalism or from ROW supply.

• Critical level policy. A critical level policy is used to distinguish emergency orders from backorders. In the previous assumption this difference was also mentioned. Reserved stock is used to prioritize emergency over replenishment orders.

• Unlimited storage capacity. There are no limitations on the number of items that can be stored. Up to now, the central warehouse did always find a way to stock items required by the company.

• Given stock levels at local warehouses. Stock levels at local warehouses influence the number of replenishment and emergency orders. Higher levels mean more replenishment orders and less emergencies. The stock levels should be deterministic input parameters for the model.

• Transportation times fixed. The time it takes to ship an item from the central warehouse to a local warehouse is fixed. This assumption is convenient because the objective of promising an average delivery lead time to the local warehouse is equivalent to promising an average delay to the local warehouse. The assumption is validated because the shipment has to be done anyway. The only relevant shipment time from central to local warehouse is the average shipment time.

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• Only customers with SLA. Demand for spare parts from customers without SLA is very unpredictable and specific. Moreover, the number of customers without SLA shows a decreasing trend. For these reasons the company indicated that this class of customers can be neglected in this project.

• One central warehouse. Despite the fact that the company possesses three central ware-houses we only look at one central warehouse, the one in Veldhoven. This is permittable because central warehouses do have disjoint sets of SKUs.

• Spare parts only. Within the company the SKUs are divided into service (spare) parts and service tools. Service tools are used, not consumed. This results in a totally different problem which is not in the scope of this thesis project.

2.6.2 Spare parts features for the practical case study

The characteristics of a spare parts inventory control problem were treated in Section 1.3.1. One can find an overview of the characteristics for the central planning problem at the com-pany in Table2.1. For explanation of the features one is referred to Section 1.3.1.

The values in the right column of Table2.1 are based on the central planning problem. For example, for transportation the volume, weight and height of SKUs might be relevant to know whether they can be shipped together or apart, resulting in different costs. Besides, the table focuses on the central planning problem instead of on the overall spare parts inventory control problem. For that reason we state no for lateral transshipments, because we look at one central warehouse. Substitutability and commonality are not considered in central planning as well.

For local planning these three features are important. The placement of machines in the field provide knowledge of commonality and substitutability in regions. With the knowledge that lateral transshipments are allowed, inventories can be planned such that two neighbor local warehouses serving the same spare parts can combine their inventories. This will result in less holding costs.

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Feature Central planning problem

Number of SKUs ± 6,000

SKU characteristics

- SCP 0 - 5,000 euro

- Production lead time

o Manufacturing 5 - 145 working days

o Repair 15 - 65 working days

- Total demand rate 0 - 6 items a year - Transportation costs Deterministic, given - Holding costs Deterministic, given

- Volume Not important

- Weight Not important

- Height Not important

- Shape Not important

- Storage requirements Not important

- Shortage costs Not necessary

Service differentiation Yes

Obsolescence No

Redundancy No

Criticality Yes, all items

Single-echelon versus multi-echelon Single-echelon

Multi-indenture No

Lateral transshipments No

Substitutability No

Commonality No

Static versus dynamic environment Static

Transportation modes Two

Backordering versus lost sales Combination

Reparability Yes

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Chapter 3

Mathematical model

In this chapter the questions and problem stated in Chapter2 will be captured in a mathe-matical model. We will start with simplified models to introduce the steady state behavior of queuing systems. First, cases of one SKU with either one lost sales class or one backorder demand class will be discussed in Sections3.1and 3.2.

The single item models from Sections 3.1 and 3.2 will be combined in Section 3.3. Still for a single item, a mixed two demand class system will be introduced according to the research by Enders et al. (2008). This work presents an evaluation method for critical level policies in the situation with two different demand classes. This method can be used for multi-item situations as well, since we assumed that failure of items occur independent of each other, i.e. steady state behavior of an item is independent of other items. The evaluation method ofEnders et al. (2008) will be described in AppendixE and briefly in Section 4.1.

In Section 3.4 the choice for CSD or delay will be discussed. Using the knowledge from Section 3.3 for a single item situation and the discussion on the performance measures in Section 3.4, a multi-item service model with two demand classes will be formulated in Sec-tion3.5. This model will be the starting point for the evaluation and optimization procedures that will be developed in Chapters4 and 5.

3.1 Lost sales

The inventory position (or net inventory), given in Equation3.1, is the stock on hand (physical stock, OH) plus the number of items in the pipeline (due-in, DI) minus the number of items in the backorder queue (BO):

S = OH + DI − BO. (3.1)

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State Stock level 0 S - m µ 1 S − 1 - m 2µ 2 S − 2 -

· · ·

- S − 1 1 - m Sµ S 0

Figure 3.1: Markov diagram for the lost sales model.

by one of DI. The contrary holds in case a replenishment order arrives. Then DI will decrease by one with as neutralizing transactions an increase of one in OH if BO = 0 or a decrease of one in BO if there are outstanding backorders.

The inventory position can be evaluated at any moment in time. For this thesis the expected values are more important than the exact values at a particular time. Poisson arrivals are assumed throughout the thesis. Because Poisson arrivals see time averages (PASTA pro-perty) the time is not important for steady state behavior of the system. The time index in Equation 3.1 is omitted.

Figure3.1shows the Markov diagram with transition probabilities between the states for the lost sales model. A queuing model like the one depicted in Figure 3.1 can be written as a M/M/S/S system using Kendall’s notation1. The first M indicates that arrivals follow a Poisson process with rate m. The second M is the exponential distribution with mean 1/µ indicating the time spend in a state. The first S is the number of available servers in the queuing model and the second S indicates the number of states in the system. So if all servers are busy, no entry is possible for new arrivals. In our inventory context, the number of servers can be interpreted as the maximum number of items that can physically be stored.

In the current case of lost sales the amount of backorders is zero. Figure 3.1then gives a nice interpretation of the inventory position in the lost sales case. The on hand stock corresponding to the states in the Markov chain are given above the states, i.e. it is the (actual) stock level. The states in the Markov representation, i.e. the values in the circles, can be seen as the number of orders due-in. It is the ordered stock that is not received yet. It is easy to verify Equation3.1with BO = 0 from Figure3.1. The on hand stock plus the stock due-in is always equal to the base stock level.

We will now derive expressions for the expected CSD and the expected number of items on hand, due-in and in backorder. These expressions are needed in latter stages of the thesis and will also be a guidance for the reader in the build-up to the combined lost sales and backorder case. We first need to introduce the steady state probabilities, denoted πnfor n = 0, 1, 2, . . .,

that give the probability that the system is in state n at a particular moment in time. They can be derived for the lost sales model using the following equations:

πn−1m = πnnµ, for n = 1, 2, . . . , S. (3.2)

1

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Rewriting Equation3.2gives: πn = λ nπn−1 = λ n λ (n − 1)πn−2 = · · · = λ n n!π0, (3.3)

for n = 1, 2, . . . , S and with λ defined as λ = m_µ. The steady state probabilities πn should

add up to one:

S

X

n=0

πn= 1. (3.4)

It follows from Equations3.3and 3.4that: 1 π0 = S X n=0 λn n! ⇔ π0 = 1 PS n=0 λ n n! . (3.5)

Combining Equations3.3and 3.5gives us the steady state probabilities:

πn= λn n! PS j=0λ j j! , for n = 1, 2, . . . , S. (3.6)

The steady state probability of stockout is πS, which is called the Erlang loss formula in

queuing theory. The lost sales model is also called an Erlang loss system. Notice that multiplying both numerator and denominator by e−λ results in an expression for the steady state probabilities based on terms from a Poisson distribution with rate λ. This observation is convenient for the evaluation of lost sales models.

The expected CSD, E[β], is the sum of all steady state probabilities of the states where the actual stock level is positive. The dependency of E[β] on S is not in the notation of E[β] for notational convenience. The expected CSD can be calculated using Equation3.7:

E[β] = π0+ π1+ · · · + πS−1= S−1

X

n=0

πn= 1 − πS. (3.7)

It is also possible to derive the expected number of items on hand, due-in and in backorder. We write out the expressions to show how these entities can be used for the evaluation of a given base stock policy.

We know that E[BO] = 0 by definition of the lost sales model. Notice that the following equation holds:

nπn= λπn−1. (3.8)

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= S−1 X n=0 (S − n)πn = S S−1 X n=0 πn− S−1 X n=1 nπn = S S−1 X n=0 πn− λ S−2 X n=0 πn. (3.9)

The expected amount of items due-in is:

E[DI] = π1+ 2π2+ · · · + SπS = S X n=1 nπn = λ S−1 X n=0 πn = λ(1 − πS). (3.10)

The same kind of expressions will be derived for a case with one backorder class in Section3.2.

3.2 Backordering

Figure 3.2 shows the actual stock levels and the corresponding Markov states for the back-ordering model. In comparison with the lost sales model the states S + 1, S + 2, . . . are added. The corresponding on hand stock is zero. The inventory position in Equation3.1is still equal to S since the number of backorders for states S + 1, S + 2, . . . is respectively 1, 2, . . .. The number of backorders is not written down in Figure3.2.

The backordering model in Figure3.2is often referred to as M/M/∞/∞ queuing model. The meaning of this notation is explained in Section3.1. The transition probabilities between the states are given above and below the arrows in Figure 3.2. Steady state probabilities are needed to derive expressions for the expected CSD and the expected number of items on hand, due-in and in backorder. The steady state probabilities for the backordering model can be derived using Equation3.2for n ∈ N. This gives again the same πn but now for n ∈ N:

πn= λn n!π0. (3.11) State Stock level 0 S - m µ 1 S − 1 - m 2µ 2 S − 2 -

· · ·

- S − 1 1 - m Sµ S 0 m (S + 1)µ - S + 1 0 -

Spare Parts Inventory Control at Central Warehouses