System-focused spare parts management for capital goods by TKI Dinalog

System-focused spare parts management for capital goods

This thesis is part of the Ph.D. thesis series of the Beta Research School for Operations Management and Logistics (onderzoeksschoolâ&#x20AC;&#x201C;beta.nl) in which the following universities cooperate: Eindhoven University of Technology, Maastricht University, University of Twente, VU Amsterdam, Wageningen University and Research, and KU Leuven. A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-4674-9

This research has been funded by The Netherlands Organisation for Scientific Research, ASML, and Dutch Railways.

System-focused spare parts management for capital goods

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op donderdag 24 januari 2019 om 16:00 uur

door

Erwin van Wingerden

geboren te Dordrecht

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: 1e promotor: 2e promotor: leden:

prof.dr. I.E.J. Heynderickx prof.dr.ir. G.J.J.A.N. van Houtum dr. T. Tan prof.dr. F. Langerak prof.dr. A.A. Syntetos (Cardiff University) dr. M.C. van der Heijden (Universiteit Twente) prof.dr. L.M. Maillart (University of Pittsburgh)

Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

Acknowledgments During my Master, I wanted nothing else then to start working at a company and leave university behind me. If someone would have told me that I would spend six more years at a different university after my Master, I would not have believed it. But then I got aware of the PDEng and after having done a PDEng for two years, I apparently was still in for four more years of doing research at the TU/e. These four years have known many ups and downs and more than once did I think that I would never make it this far. Luckily, you all have made it possible that I came this far by supporting me. This thesis would not have been possible without all your support and I want to thank everyone for that. I would like to thank Geert-Jan for having given me the opportunity to do a PhD, but most of all for supporting me along the way by giving valuable comments and feedback. I also want to thank Tarkan for always being there to give me feedback and for all the nice discussions we have had. Without the supervision of the two of you I would not be writing this acknowledgment at this moment. I also would like to thank everyone involved in this NWO-TOP project for the fruitful discussions we had. Bob, Fred, Martijn, Mehmet, and Ruud in particular. I would like to thank Aris, Fred, Lisa and Matthieu for being part of the committee and taking the time to read my thesis and provide me with valuable feedback. I also want to thank Lisa for her involvement in my work during her stay in Eindhoven. During my years in Eindhoven I really enjoyed the board game nights and all the social activities. For that I want to thank everyone joining the activities and with whom I have spend time together, taking our minds off from work for a few minutes. In particular I want to thank Afonso, Gero, and Joost to organize the boardgames (with me). I want to thank Laura for being one of the most frequent hosts of all these board game nights. The board game nights became more popular over the years where at

6 some point I still remember Joost and I discussing whether to send a reminder or not to promote the board game night, because it could just happen that there wouldn’t be sufficient capacity. I hope there will be sufficient capacity for me to join the board game nights to come and perhaps we will invent a board game together some day. I want to thank Dalia for being my buddy during the first years of my PhD from day 1. Perhaps this was due to both having done a PDEng and then started doing a PhD that we had this connection from the start. I have enjoyed our time together and we can say that at least one of our goals was reached (making it to 100). I want to thank Chiel for having shared an office with me, even though it was only for a short time period. Perhaps my level of chess would have been improved if you were my office mate for a longer time period. I want to thank Alireza, with whom I have shared an office for most of my PhD. Over the years we had many fruitful discussion, not only about our work but also on many other topics. I even learned a little bit of Persian over the years which I will never forget. Although in the beginning I thought you might get annoyed by all the conversations I had with other PhD students, but in the end I could not have wished for a better office mate. Next I want to thank Loe, my partner in ’crime’. I always liked figuring out the next OPAC activity together with you, no matter how difficult the hints were. I am sure that José will never forget the ’onbeperkt speculaas’. Hints became much more difficult in the years thereafter. I also want to thank Qianru, but not for insulting me every day at the office. Despite that, I always enjoyed your company and will not forget the (way too large) dinners you made for me. And who knows, perhaps one day you will be able to find a proper badminton match. I would like to thank Jolanda who made me feel at home from the moment I started my PDEng. Furthermore, I would like to thank Claudine, Christel and José for the nice conversations and support they provided me throughout the years with all the appointments and busy schedules. Last but not least I want to thank my family and friends for their support throughout these years but perhaps most of all for their distraction they provided. In particular I want to thank Aafke for making the beautiful cover design and Femke for her unconditional support during all these years. Femke, soon we will start a whole new chapter with the three of us!

Contents 1 Introduction 1.1 Service logistic networks . . . . . . . . . . . . . . . . . . . . . 1.2 The exploitation phase . . . . . . . . . . . . . . . . . . . . . . 1.3 Inventory control of spare parts . . . . . . . . . . . . . . . . . 1.4 Research positioning and objectives . . . . . . . . . . . . . . . 1.4.1 Spare parts management for the initial phase . . . . . 1.4.2 Spare parts management during the stationary phase . 1.5 Contributions and outline of the thesis . . . . . . . . . . . . .

. . . . . . .

2 Spare parts management under double demand uncertainty 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The effect of more variable demand rates . . . . . . . . . . . . . . . 2.3.1 Example distributions for uncertain demand rates . . . . . . 2.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Optimization procedure in a multi-item setting . . . . . . . . 2.4.2 Impact of double demand uncertainty in a multi-item setting 2.4.3 The impact of ignoring double demand uncertainty . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Multi-period spare parts planning under stochastic rameter uncertainty 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model Description . . . . . . . . . . . . . . . . . . . 3.3 Three-stage dynamic program . . . . . . . . . . . . . 3.3.1 Third stage optimization . . . . . . . . . . . 3.3.2 Second stage optimization . . . . . . . . . . . 3.3.3 First stage optimization . . . . . . . . . . . . 3.4 The benefit of reliable demand rate information . . . 7

. . . . . . .

1 2 6 7 9 9 11 14

. . . . . . . . .

17 17 21 24 28 29 29 31 33 36

demand and pa. . . . . . .

. . . . . . .

39 39 42 48 49 52 54 55

CONTENTS

3.5

3.4.1 Potential gains by investing in more information . . . . . . . . 3.4.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 The impact of an emergency warehouse in a two-echelon spare network 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Evaluation procedure . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Local Evaluation Procedure . . . . . . . . . . . . . . . . . 4.3.2 Central Evaluation Procedure . . . . . . . . . . . . . . . . 4.3.3 Overall evaluation procedure . . . . . . . . . . . . . . . . 4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The benefit of an emergency warehouse . . . . . . . . . . . . . . 4.5.1 Optimization of the base stock levels . . . . . . . . . . . . 4.5.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Generalized design approach to obtain a near-optimal cation for a multi-item inventory control problem 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Design aspects of a ABC classification . . . . . . . . . . 5.3.1 Classification criteria . . . . . . . . . . . . . . . . 5.3.2 Number of classes . . . . . . . . . . . . . . . . . 5.3.3 Class sizes . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Target setting . . . . . . . . . . . . . . . . . . . . 5.4 Numerical experiment . . . . . . . . . . . . . . . . . . . 5.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Choosing the best classification criteria . . . . . 5.4.3 The impact of class sizes on performance . . . . 5.4.4 ABC classification target setting . . . . . . . . . 5.5 Conclusions and suggestions for further research . . . .

56 56 58

parts . . . . . . . . . . .

. . . . . . . . . . .

63 63 68 71 73 76 79 81 87 87 88 92

. . . . . . . . . . .

ABC classifi. . . . . . . . . . . . .

. . . . . . . . . . . . .

6 Conclusion 6.1 Research questions revisited . . . . . . . . . . . . . . . . . . . 6.1.1 Spare parts management for the initial phase . . . . . 6.1.2 Spare parts management during the stationary phase . 6.2 Ideas for future research . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . .

95 95 100 101 101 102 103 104 106 106 108 109 113 116

. . . .

119 119 119 122 124

CONTENTS 6.2.1 6.2.2 6.2.3 6.2.4

9 Dealing with double demand uncertainty in complex networks . The transition from initial phase to the exploitation phase . . . Efficient multi-item optimization procedures . . . . . . . . . . . Mimicking system-focused inventory control in spare parts networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 125 125 125

A Single-item and multi-item inventory models

127

B Evaluating an infinite investment

129

Bibliography

133

Summary

143

About the author

147

Introduction Imagine you are waiting on a platform for your next train to go to an important meeting. Unfortunately, you get informed that the train you wanted to take is no longer able to drive safely as a result of a broken part. This does not only cause you to miss this important meeting, but also many other passengers are faced with unpleasant consequences. To keep the delays for their passengers to a minimum, the Dutch Railways (NS) have additional trains which they can use to replace the train that has stopped working properly. However, if too many trains are out of service, the impact of an additional defective train is much bigger as it is likely to disrupt the schedule for the entire day and thus causes a delay for many of their passengers. For many capital goods, such as the trains described above, downtime of the capital good has big consequences and one would prefer to keep the downtime to a minimum. Another example where downtime has large consequences is the case of lithography machines produced by ASML. These machines are installed in semiconductor factories of ASMLâ&#x20AC;&#x2122;s customers. When such a lithography machine is defective, this has a big impact on the production process of the customer. Capital goods such as trains and lithography machines generally have a life cycle consisting of 5 phases; see Figure 1.1. In the first phase, the needs and requirements of the capital goods are determined based on the requirements of the market and available knowledge. In the second phase, the system is designed, after which the capital goods will be produced. After the production phase, the exploitation phase is reached, which is the key phase since the capital goods are then used to deliver the required service or products. Finally, in the last phase the capital goods are no longer being used and get disposed. 1

1. INTRODUCTION

Figure 1.1: Life cycle of a capital good

Two key performance indicators of capital goods are the availability on the one hand and the total costs of ownership (TCO) on the other hand. The availability of capital goods is measured as the time a capital good is running properly divided by the total required time the capital good should be running. The TCO for a capital good consists of all the costs that occur during all 5 phases denoted in Figure 1.1. In the case of e.g. trains, these costs consist of the purchasing costs (which include the costs for the design and production), the disposal costs, as well as the maintenance and service logistics costs during the exploitation phase. Capital goods generally are used for a long time period, and although they require a significant investment, the purchasing cost of capital goods is only a fraction of the TCO. An equal or even larger part of the TCO consists of the maintenance and service logistic costs, which are the costs involved to keep the capital goods operating. An important aspect of these costs are the costs involved for keeping spare parts, which are used to replace broken parts and thereby can significantly reduce the downtime. In addition, downtime costs, which are the costs involved when a machine is not available, are often very high (see e.g. Ă&#x2013;ner et al., 2007; van Dongen, 2011). In the example of the trains, the NS may have to compensate their passengers for their delay, pay overtime for employees that cannot make their original schedule or even has to arrange alternative modes of transport as a result of a defective train. A company would generally prefer to have both lower TCO as well as less downtime. However, in general there is a tradeoff between the TCO and the availability or downtime as depicted in Figure 1.2. In order to reach the desired availability, given the design of a capital good, it is key to have an appropriate service network to provide the maintenance necessary.

1.1

Service logistic networks

To perform the required maintenance for capital goods, a company has to rely on many different resources, such as spare parts, tools, and service engineers. A service network generally consist of one or more locations where capital goods can be maintained, where spare parts are kept on stock and one or more repair shops which

1.1. SERVICE LOGISTIC NETWORKS

Figure 1.2: Downtime versus TCO

are responsible of repairing the broken parts. These locations and facilities ensure that the required maintenance resources are at the right place at the right to enable keeping downtime to a minimum. The costs involved with the organization of these service networks can be huge, especially due to the large value of spare parts that are kept on stock. One example is the service network of ASML. For already one of the machine types alone, ASML owns spare parts used worth over fifty millions of euro (Van Aspert, 2014). Another example is that of commercial airlines, for which it has been estimated that the total value of all spare parts on stock are worth over $40 billion (Harrington, 2007), which shows the importance of the decision on how much spare parts to stock as this can be a large fraction of the TCO. Typically, one can distinguish two types of service networks in practice; Original Equipment Manufacturer (OEM) networks and user networks. The NS and military organizations are examples of user networks, who are responsible for the maintenance and service logistics of their own systems. ASMLâ&#x20AC;&#x2122;s network is an example of an OEM network where ASML is responsible for the maintenance and service logistics of their systems sold. Figures 1.3, and 1.4 show an archetypical example of a user network and OEM network respectively, where the installed base denotes the total number of systems or capital goods that needs to be maintained.

1. INTRODUCTION

Figure 1.3: Representation of a typical user network (cf. Basten and van Houtum, 2014)

Both types of networks generally have stocking locations at a central level, as well as at a local level nearby the customers or where capital goods are being used. For user networks local warehouses are generally located within one region or country, whereas for OEM networks these warehouses can be spread all over the world. Where user networks generally rely on their internal repair shops to repair most of their broken parts, it is common for an OEM network to outsource the repairs, although nowadays it is increasingly more common for user networks to also outsource the repair of the broken parts. In the example of NS a train which is no longer working has consequences for many passengers, and only if too many trains are out of service, the impact will be really big. Thus from a maintenance point of view, one would like to keep the number of times that too many trains are out of service to a minimum. When only a few trains have to wait for a spare part, this will not immediately have big consequences on the overall performance. The NS would thus manage the maintenance and service logistics in such a way, that the risk of having too many defective trains is kept to a minimum. For OEM networks it is becoming increasingly more common to offer

1.1. SERVICE LOGISTIC NETWORKS

Figure 1.4: Representation of a typical OEM network (cf. Basten and van Houtum, 2014)

1. INTRODUCTION

Figure 1.5: Life cycle of the installed base during the exploitation phase

full service contracts together with the capital goods (see Cohen et al., 2006; Oliva and Kallenberg, 2003). Examples of OEMs that commonly close service contracts are e.g. ASML, IBM, OcĂŠ and Vanderlande. These service restrictions ensure that the OEMs provide the resources necessary to meet the desired service level and are generally strict. Whenever the OEMs cannot meet these service levels, they incur penalty costs.

1.2

The exploitation phase

The service networks described focus on the maintenance and service logistics during the Exploitation phase (see also Figure 1.1). During this phase the capital goods are actually being used and parts can break down, which need to be repaired and replaced. Within the exploitation phase, one can distinguish three different phases; the initial phase, the stationary phase, and the end-of-life phase. Depending on the phase, the installed base may be changing over time, as can be seen in Figure 1.5 During the initial phase, the installed base is growing and there is a very high level of uncertainty about the failure behavior as many parts have not been used before. Demand estimates are generated via judgemental forecasting, which is generally not very reliable. As a result one not only has to deal with the uncertainty of the failure

1.3. INVENTORY CONTROL OF SPARE PARTS

behavior of the parts, also the rate at which parts fail are uncertain. In other words, companies are dealing with double demand uncertainty. Only having a stochastic demand distribution may thus not be significant to deal with all the uncertainty. This double demand uncertainty makes it even more difficult and important to make good decisions on the amount of spare parts that should be kept on stock as both the risks of stocking too much as well as obsolescence are larger in this case. When the installed base is no longer growing, and is more or less stable, one reaches the stationary phase. During this phase the company generally can make more reliable forecasts of the demand rates for all parts, and has a fully functioning service network. However, the service networks are becoming increasingly more complex by having multiple locations from where spare parts can be supplied from as well as the use of lateral and emergency shipments, which denote shipments from other locations than the local warehouse dedicated for the maintenance of that capital good (see also Figure 1.4). Optimizing maintenance decisions for these maintenance networks becomes increasingly more complex. The last phase within the exploitation phase occurs when a system is being replaced by a newer system, or when a system is no longer required. The installed base starts to decrease and the company scales back the maintenance and service logistics costs in such a way that they can still obtain the desired availability while not making unnecessary costs.

1.3

Inventory control of spare parts

During the life cycle of a system, regardless of the type of service network or phase a system is in, having enough spare parts on stock is key to enable a swift repair of the capital goods. Whenever a capital good is not working, and there is no spare part available, this causes unavailability. It is then not possible to get the capital good up and running, regardless of the tools and engineers that are also required. However, keeping spare parts on stock results in costs, i.e. opportunity and warehousing costs (often called holding costs), and costs due to obsolescence of parts. Spare parts become obsolete when the size of the installed base phases out or when a better version of the component is being developed and parts are not used anymore. Thus if one decides to keep too many parts on stock, this results in very high holding costs. One would thus prefer to have the least amount of holding and obsolescence costs and to have the system availability as high as possible (as indicated by the arrows in Figure 1.2).

1. INTRODUCTION

In order to achieve a high availability against minimum holding costs, one can make use of so-called â&#x20AC;&#x153;system-focused spare parts managementâ&#x20AC;?, where stock levels of the spare parts are chosen such that the inventory holding costs are kept as low as possible while still ensuring the desired system availability. This is also referred to as multi-item inventory models or a so-called system approach (Sherbrooke, 1968). The main idea of the system approach is that one is only interested in the overall availability of the capital goods and not in the availability of the individual parts. When a capital good is unavailable due to the unavailability of an cheap or expensive part, the impact is the same. In other words, it is thus beneficial to stock cheap parts at large quantities, especially those parts that have a higher demand rate, and stock less of the most expensive parts that are demanded less frequently. By doing so, one can significantly reduce the total holding costs, whereas the system availability remains the same or even increases (Van Houtum and Kranenburg, 2015). By further improving on system-focused spare parts management, costs can be further decreased and better decisions can be made.

For system-focused spare parts models it is common to manage the inventory using a base stock policy, also known as (S â&#x2C6;&#x2019; 1, S) policy (see e.g. Alfredsson and Verrijdt (1999); Ozkan et al. (2015); Drent and Arts (2018)). In this case broken parts comes in from the customer and a new part is immediately send to the customer. The broken part is then directly send into repair or a new part is ordered at the supplier when the part can not be repaired. An alternative could be to batch the repairs for economies of scale, which could potentially reduce the costs. However, waiting for these batches introduces additional risks with respect to the downtime of the capital goods, for which the costs are likely to exceed the transport costs. Moreover, as there are commonly many different parts being sent to the supplier periodically (daily, weekly, monthly), consolidation of shipments still occurs. Therefore, the costs of ordering are often not taken into consideration when considering service networks as the availability is more important.

For system-focused spare parts management, the models that are applicable depend strongly on the phase the capital good is in and there is no single model that can be applied for all phases. In Section 1.4 we focus on the initial phase and the stationary phase. For each of these phases we provide a main research question that we want to answer throughout this thesis. In Section 1.5 we summarize our contributions and give an outline of this thesis.

1.4. RESEARCH POSITIONING AND OBJECTIVES

1.4

Research positioning and objectives

In this section we further elaborate on the initial phase as well as the stationary phase, position our work relative to the existing literature, and present our research questions and objectives.

1.4.1

Spare parts management for the initial phase

When considering the initial phase of a capital good, besides the recent contribution of Martinetti et al. (2017), hardly any literature is available. However, during this initial phase, when the capital goods are being introduced to the field, companies already have to make decisions on the base stock levels (Hu et al., 2018). As this decision also influences all future decisions on the base stock levels, it is important for a company to choose the base stock levels in the best possible way. Most components are known to have random failures, and are often assumed to follow a Poisson demand process. Unfortunately, when a new capital good is introduced, there are many new parts that have not been used before. The rate at which these parts fail is estimated based on expert opinions as there is no historical demand information. Based on these expert opinions, the company has to decide upon the base stock levels. As the demand rate for new parts are often highly uncertain, the actual demand rate can be different from the expected demand rate. This makes it more difficult to decide upon the desired base stock level. When a company decides to ignore this uncertainty of the demand rate, it is likely that the desired service level the company wants to obtain is not met. This results in more downtime and thus higher costs. On the other hand, if the company decides to stock much more than necessary, the company incurs unnecessary amounts of holding costs and is likely to be left with large amounts of excess stock. An example of a company that faces these issues is ASML. As there are many developments, it is logical that new lithography machines contain many new parts. For parts that were already used in previous lithography machines, ASML is able to make reasonable estimates. However, there is a large amount of uncertainty regarding the demand rates of new parts. This uncertainty can be reduced but this brings costs as this requires man-hours from the engineering department to get better information on the failure behaviour of these parts. Therefore, we would like to give an answer to the following main research question:

1. INTRODUCTION

Main Research Question 1 How to deal with double demand uncertainty within spare parts inventory control in the initial phase and how much to invest in more reliable demand rate information? As spare parts fail randomly, it is common to model this uncertainty using a distribution. However, it is then often assumed that the average demand is known and correct. When this is not the case, as in many cases where there is little information available, one might use a distribution assuming the parameter(s) are known. When one uses a distribution with known parameter(s), the decision maker ignores the uncertainty in the demand rate itself. It is likely better to include the double demand uncertainty to allow a better decision making. We develop a method that allows the decision maker to take this double demand uncertainty into consideration and thereby allow for better decision making and to limit the impact of assuming perfect information on the average demand. Research Objective 1 Develop a model to incorporate the double demand uncertainty into the spare parts management. Once one knows how to take the double demand uncertainty into consideration, we are interested in the potential benefits of taking this demand rate uncertainty into consideration. Therefore, our second research objective is to investigate the impact of double demand uncertainty on the availability by comparing to a decision model that ignores the double demand uncertainty. Research Objective 2 Investigate the availability when applying a decision model that ignores the double demand uncertainty. In Chapter 2 we provide a model in which the decision on how much stock to keep is made taking into account the double demand uncertainty. As we want to give an answer to the question on whether and how much to invest in more reliable demand rate information, we need a model that can take information updates into consideration. When a company spends man-hours in getting more reliable demand rate information, this information becomes available at a later point in time, during which it is still possible to adjust the base stock levels. Moreover, as parts that are no longer kept on stock could be sold back, it is also likely that at this stage, the parts are still more valuable and thus this may potentially reduce the risk and cost of obsolescence. This brings us to our next research objective:

1.4. RESEARCH POSITIONING AND OBJECTIVES

Research Objective 3 Develop a model that supports the decision maker for investments in more reliable information on the demand rate and show for which parts the company can benefit the most from such an investment. In Chapter 3 we provide a model that extends the model of Chapter 2 by including the possibility to invest in more reliable information and adjust the base stock level at a later time period when there is a more reliable forecast.

1.4.2

Spare parts management during the stationary phase

Once capital goods have been in use for some time, and the installed base is more or less stable, a company is faced with different challenges compared to the challenges at the start of the exploitation phase. At this point in time, the company generally has reliable predictions of the demand rate, and thus it is safer to assume that the demand rate is known and given for the decision making. A significant amount of work has already been done regarding multi-item inventory models and the system approach for this phase (see also Basten and van Houtum, 2014). In this thesis we focus on applying system-focused inventory control for twoechelon service networks, such as the networks of ASML and the NS, and on easier methods to enable more companies to apply system-focused inventory control. Our main research question for the stationary phase is as follows: Main Research Question 2 i How to incorporate general structures for emergency and lateral shipments within spare parts inventory control models for two-echelon networks in the stationary phase; ii How to mimic system-focused inventory control by methods that are easier to adopt? When capital goods are spread out over different regions, especially when costs of downtime are very high, one commonly has multiple warehouses where the spare parts are stored in close proximity of the customers, also referred to as local warehouses. When a part breaks down, a spare part can be obtained from a nearby local warehouse. These warehouses are all replenished from a single location in the network, also called the central warehouse. When a part has broken down, and there is a stockout at the nearest local warehouse, it is of key importance to get a spare part to the customer as soon as possible. In order

1. INTRODUCTION

to ensure this, it is common to obtain spare parts from other locations, also known as lateral transshipments (when the spare part is obtained from another local warehouse) or emergency shipments (when the spare part is obtained from the central warehouse). There may however be multiple reasons why an emergency shipment from the central warehouse is preferred over a lateral transshipment. The distance to the central warehouse may be shorter compared to the distance of local warehouses, or one is able to obtain the parts faster from the central warehouse, which could be due to the fact that local warehouses may not be equipped for a fast handling of these lateral or emergency shipments. Next to this, when a shipment involves a shipment between two different countries, there may also be issues with government regulations. As a result, it may be the case that not all local warehouses allow for lateral transshipments to all other local warehouses and that the central warehouse is preferred over lateral transshipments. In the case of NS the local warehouses are located at sites where the maintenance is being done. Whenever one of these local warehouses does not have the required component it is preferably shipped from the central warehouse instead of other local warehouses as it takes less time to obtain a part from the central warehouse. Therefore, in order to deal with these issues, it is possible to incorporate the use of a so-called emergency warehouse. This warehouse is able to efficiently and quickly ship parts to all other warehouses when needed and does not have a dedicated customer. As the benefit of this emergency warehouse may depend on the structure of the network, we want to find out for which network structures this emergency warehouse has the most benefit. One of the first papers about lateral transshipments is the work of Allen (1958). Since then, much research has been done with respect to the use of lateral and emergency shipments (see Paterson et al. (2011) for an overview). Later work with respect to twoechelon networks followed with the work of Muckstadt and Thomas (1980), who only consider emergency shipments from the central warehouse. Alfredsson and Verrijdt (1999) are one of the first that consider a two-echelon network with both emergency and lateral transshipments, although this work requires a long computation time. As the order at which emergency shipments are handled may differ per company, the model should be flexible in terms of the sequence at which other locations are considered in the case of a stockout at the location where the demand has arrived. Therefore, we develop a new general model which allows us to evaluate the perfor-

1.4. RESEARCH POSITIONING AND OBJECTIVES

mance of many different two-echelon networks. Making use of this model, we investigate the potential benefit of an emergency warehouse, or in other words the benefit of keeping a separate stock dedicated to supply the customers in case of an emergency. Research Objective 4 Develop a procedure to evaluate the performance, in terms of expected waiting time for a part, for a problem setting with multiple local warehouses and an emergency warehouse in a two-echelon network with lateral and emergency shipments that can be done in any order. Research Objective 5 Investigate the benefit of an emergency warehouse. Unfortunately, not all companies have the required knowledge, or can afford the costs involved with applying a system approach such as IT and control costs, to apply the developed methods on spare parts management that are available from the literature. Especially many small to medium sized companies may not be able to manage such a large amount of SKUs using multi-item inventory models. Therefore, we provide a means to mimic system-focused inventory control using methods that are easier to apply. One of these possibilities is to make use of a so-called ABC classification to manage the spare parts inventories. When applying an ABC classification, companies categorize their inventory into multiple classes and manage each SKU within a class in the same way. For different classes, one may have different target service levels or follow different types of inventory policies (e.g., more advanced policies for expensive SKUs than for other SKUs). In this way companies are basically able to differentiate between different SKUs while still having a simple inventory control policy. These ABC classifications are easier to implement when dealing with many SKUs (Buxey, 2006). Companies would thus be able to come closer to a system approach using an ABC classification and significantly reduce the holding costs. When we look in the existing literature, ABC classifications generally do not consider the resulting overall service level, and thus a system approach. Only the work of Teunter et al. (2010) considers the overall service level as a result of using an ABC classification. However, they assume a strictly positive base stock level for each SKU. When applying a system approach, it is common to have SKUs for which the company does not keep any spare parts on stock. If one would thus be able to design an ABC classification such that is able to come close to the results of a system approach, this would allow an easier implementation of the main idea of the system approach. As the service level per class of parts is only one aspect, there are also other important aspects that eventually results in an ABC classification. We consider four different aspects that together design the ABC classification; service level per class, number of

1. INTRODUCTION

classes, number of parts in each class, and classification criteria. Research Objective 6 Develop a method to design an ABC classification that considers all important aspects to incorporate the idea of a system approach. As an ABC classification is determined not only by the ranking of SKUs, which determines the importance and hence the desired service level for each SKU within a class, it is key to consider all aspects that together design the ABC classification and know the impact of these different aspects. Research Objective 7 Investigate the importance of the different aspects of an ABC classification when trying to obtain results similar to a system approach. Among others we show that the design of the class sizes when trying to apply a system approach with an ABC classification is of great importance.

1.5

Contributions and outline of the thesis

In this thesis, we discuss on how one should apply system-focused spare parts management for different problem settings. In Chapter 2 we consider the setting as described in Section 1.4.1. We model demand to follow a Poisson distribution with an unknown demand rate and present a way to model this uncertainty. We show for a multiitem setting under a service level requirement, that costs always increase when the uncertainty of the demand rate parameter increases. Next, we show in a numerical experiment that in order to ensure the desired availability, up to four times the initial investment in spare parts can be required due to the uncertainty of the demand rates. This value will be even larger when there is more double demand uncertainty. Further, if one would ignore the uncertainty of the demand rates, the availability would be much lower than desired and one can thus expect a poor performance involving possible high downtime costs. In Chapter 3 we extend the work presented in Chapter 2 by considering the option to invest in information, which potentially allows for better decision making. We first derive analytical results proving that a higher double demand uncertainty also leads to higher expected costs. We perform numerical experiments with which we show that especially for expensive parts with more double demand uncertainty there is a potential benefit of investing in more reliable demand rate information with expected

1.5. CONTRIBUTIONS AND OUTLINE OF THE THESIS

cost differences of over 15 percent. The benefits are most likely even larger when there is more uncertainty regarding the demand rate. In Chapter 4 we consider a service network as described in Section 1.4.2, inspired by ASML and the NS. This network consists of multiple capital goods spread over different regions, with a local warehouse dedicated to one or more groups of capital goods. There is an option to get the part from a different location in the case there is a stockout at the nearest local warehouse. One of these options is the use of an emergency warehouse, which is an warehouse solely used to supply parts to any region whenever a stockout has occurred. We present an accurate approximate procedure to evaluate such a complex network for a single spare part. Using simulation we show that this approximate evaluation procedure is accurate, especially for higher availability levels, which are the availability levels a company is interested in. We optimize the base stock levels via a smart enumeration procedure, to show the benefit of the emergency warehouse. We show that savings up to 30 percent can be obtained via an emergency warehouse. Note that although the models in Chapter 3 and 4 are for a single part with holding and penalty costs, by applying Lagrangian relaxation we can end up with a solution for the multi-item problem (see Appendix A). In this case, the models in Chapters 3 and 4 are thus representing the so-called decentralized problems. Finally, in Chapter 5 we propose a design that incorporates four important aspects to obtain an ABC classification that captures the idea of a system approach. We show that using our generalized design approach, it is possible to come very close to the results of a system approach. This makes it possible for small to medium sized companies that do not have sufficient knowledge to implement a system approach, to obtain close to optimal results by using an ABC classification. Moreover, we show the importance of having the right class sizes when designing an ABC classification to capture the idea of a system approach, as choosing wrong class sizes results in solutions that are far from optimal.

Spare parts management under double demand uncertainty 2.1

Introduction

Whenever an OEM introduces new capital goods there are many struggles to overcome when starting to use these capital goods. Especially regarding the maintenance of capital goods there is a lot of uncertainty. Due to the high costs of capital goods in general it is common to have very high service level targets in place, or strict agreements with the customers regarding the fraction of time the capital goods should be running. When capital goods are being used for multiple years, spare parts fail randomly, but the rate at which the spare parts fail are better predictable. This allows for good decision making on the amount of spare parts to stock, despite the uncertainty that exists in the demand process itself. For a new product introduction this is much more difficult since, in addition, the demand rates are highly uncertain. During the introduction, estimates of the demand rates are commonly based upon expert knowledge and limited qualitative research. As a result, it may not be wise to only use the expectation of the demand rate to make spare parts inventory decisions. It is better to take the uncertainty in the demand rate estimations into account when deciding upon the inventory levels. If one would ignore this uncertainty, one may get much more downtime due to a lack of spare parts. Especially during the introduction of capital goods, it is very unsatisfactory for customers as well as the OEM when downtime is too high. 17

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

In this chapter we model the uncertainty about the actual demand rate in addition to having the common uncertainty of a demand process. We consider a situation where parts of multiple stock keeping units (SKUs) for multiple identical machines are provided from a single centrally located stock point. As this is the only location from where these parts can be supplied, customers have to wait for a spare part in the case of a stockout. As the availability of the system is of key importance, there is a constraint on the expected number of backorders for all SKUs together, which is directly related to having a system availability constraint (Van Houtum and Kranenburg, 2015). Demand follows a Poisson process but the exact value of the demand rate is not known to the decision maker. We assume that the demand rates are drawn from known and given distribution. However, not for every component the amount of uncertainty is similar, and thus this distribution may be part-dependent. One of the questions we would like to answer is whether a better predictability of the failure rates, i.e., more reliable predictions of the demand rates, always leads to a better performance in terms of costs for parts that have the same estimated mean. There are many papers that study the impact of stochastic demand in the case of inventory holding and backordering costs where the variability of the demand distribution is different. Gerchak and Mossman (1991) analyze the effect of a more variable demand distribution, using the definition as introduced by Ross (1983), for a single period, single item newsvendor problem. They show that only after a certain threshold value the optimal order quantity increases but it may also decrease in some cases. Ridder et al. (1998) also consider a single item, single period newsvendor problem with continuous demand. They show that simply having a larger demand variance under the same expectation may even lead to lower costs. These papers consider a single period and are only looking at the distribution of the demand during that period. We however consider a multi-item setting where we have an infinite horizon with continuous review. Demand for all parts follows a Poisson process but the failure rate corresponding to this Poisson process is uncertain and follows from known and given distributions. Other papers that study the variability of the demand distribution itself and the impact on inventory are Jemai and Karaesmen (2005); Xu et al. (2010). A stream of literature that is closely related to demand rate uncertainty is the stream of demand forecasting and in particular the stream of forecasting intermittent demand (see e.g. Croston, 1972; Teunter and Duncan, 2009; Syntetos et al., 2015; Babai et al., 2018). In intermittent demand forecasting, the demand rate is predicted using only a

2.1. INTRODUCTION

limited amount of data. These methods estimate the frequency of demand occurrence as well as the size of the demand during an occurrence and as such can be considered a different form of double demand uncertainty. When applying these methods as an input for inventory control, these methods are shown to result in service levels that are often below the target service level. Prak et al. (2017) showed safety stocks may end up 30 percent too low and provides a corrected procedure to adjust for this. However, as all these methods require at least some amount of historical demand information they cannot be applied for the situation we are considering, where there is no historical demand information. As we consider an infinite horizon, the leadtime demand distribution is important to measure the steady-state performance of the system. The uncertainty of the demand rate appears to be similar to having uncertainty on the lead time. Hence, our problem is also related to the problem where the impact of uncertainty in the lead time is analyzed in infinite-horizon inventory models. Bagchi et al. (1986) investigate the impact of lead time variability on stockouts and stockout risk. They numerically show, based on a case study, that it is important to take the lead time variability into consideration. Feeney and Sherbrooke (1966) consider a Poisson demand process but assume i.i.d lead times (they assume an exogenous parallel supply process). This means that orders that are placed later may arrive earlier at the stockpoint. They show that the system performance of any base stock policy then only depends on the mean value of the lead time. There are also many papers that consider inventory models with uncertain lead times under the assumption of sequential deliveries (see e.g Song et al., 2010; Zipkin, 1986; Kamath and Pakkala, 1999a; Nahmias, 1979; Ehrhardt, 1984; Chen and Yu, 2005). See Svoronos and Zipkin (1991), and Zipkin (1991) for an extensive discussion about the difference between sequential and parallel supply processes. Song (1994) investigates the effect of lead time uncertainty when having sequential deliveries. She assumes that demand follows a Poisson process and analyzes the impact of a more variable lead time by comparing two problem instances with identical demand distributions but different lead time distributions. She considers the case where each problem instance is identical in expectation of the lead time but differs in the variability of the lead time, using the definition of increasing convex ordering (see e.g. Ross, 1983; Whitt, 1985). She shows that when the expectation is the same, a more variable lead time leads to a more variable lead time demand, and thus to higher costs for a given base stock level as well as for the optimal base stock level. In this chapter we show for our model that a more variable demand rate parameter has a similar effect as a more variable lead time in the model of Song (1994). Therefore, we obtain similar results as Song (1994).

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

Another stream of literature is the stream of research concerning the use of statistical methods to predict the demand. A commonly used statistical method is the use of a Bayesian approach to predict the failure rate (see e.g. Kamath and Pakkala, 1999b; Hayes, 1969). Unfortunately these methods commonly require sufficient amounts of data. Another option in the case of limited data is presented by Akcay et al. (2011). Even though less data is necessary, all these methods require at least some historical data. Unfortunately, such data is not available for the problem that we study because the capital good has not been used in the field before, there usually is no statistical evidence about the demand rate and thus these methods would not be applicable to our problem. Ge et al. (2018) apply a similar approach as we do, only in a different context. They consider the situation where a multi-component system with a linear structure is designed. For each component, a choice between multiple designs is made, where the failure rate of each design is uncertain. For each design, a distribution is given for the failure rate. They are able to make better design decisions by incorporating the impact of different designs on the failure rate. Given this setting, they propose an approximate evaluation procedure which ensures that the down time of the customer does not exceed the predetermined level. In this Chapter we give an answer to the Research Objectives 1 and 2, as introduced in Chapter 1: Research Objective 1 Develop a model to incorporate the double demand uncertainty into the spare parts management. Research Objective 2 Investigate the availability when applying a decision model that ignores the double demand uncertainty. Our main contributions in this chapter are as follows. First, we show for a multi-item problem with Poisson demand processes, when the expected demand rates are the same, more variability of the demand rate parameters, in terms of increasing convex ordering, always leads to higher costs when minimizing inventory holding costs subject to an expected mean number of backorders constraint. Although one may have expected this result, there are examples where more uncertainty may actually lead to lower costs Ridder et al. (1998). Secondly, we show the equivalence between lead time uncertainty in the model of Song (1994), and uncertainty of the demand rate parameters in our model. Both uncertainties are equal in terms of the impact

2.2. MODEL DESCRIPTION

on the lead time demand distribution. Next, we show that a greedy heuristic results in solutions on the efficient frontier for the problem we are considering. Making use of this, we show numerically that if the demand rate follows a Gamma distribution, where for each item the variance of this Gamma distribution is such that each item has the same squared coefficient of variation, the relative holding costs increase almost linearly in the squared coefficient of variation. Moreover, when demand rates are larger, the impact of the additional demand uncertainty is much bigger, and if the lead time is longer, the relative impact of the additional uncertainty is even larger. The latter also hold when modeling the uncertainty using a Uniform distribution. Finally, we find that if a company would decide to ignore the additional uncertainty in the demand rate, and simply take the expectation, this leads to poor solutions. In Section 2.2 we give the description of our model and the inventory problem that we want to solve. Then, in Section 2.3 we show the equivalence between demand rate uncertainty and lead time uncertainty, and show that higher variability always results in higher costs. In Section 2.4 we present our numerical results, which we use for our second research objective. We show how costs increase in terms of the additional uncertainty for different scenarios, and show the impact of ignoring the demand rate uncertainty on the solution obtained. Finally, we give our conclusions in Section 2.5.

2.2

Model description

In this section, we introduce three models we use throughout the remainder of this chapter, the basic model, the uncertain demand rate model, and the uncertain lead time model. For the basic model we consider a single warehouse where several repairable stock keeping units (SKUs) are kept on stock. The set of SKUs is denoted by I, and the number of SKUs is denoted by |I|. For convenience, the SKUâ&#x20AC;&#x2122;s are numbered i = 1, 2, . . . |I|. For each SKU i â&#x2C6;&#x2C6; I, demand occurs according to a Poisson process with rate mi . The deterministic repair lead time, the fixed time it takes for a broken part send to the supplier to be repaired, for SKU i, denoted by ti (> 0), is known and given. Demand is fulfilled immediately if possible and otherwise backordered and fulfilled as soon as possible. The inventory is managed by a continuous review, base stock policy with base stock level Si . The inventory holding costs are denoted by hi (> 0). As the spare parts are repaired when a broken part is sent to the maintenance shop, the total amount of parts does not decrease and we thus have a one time acquisition of the SKUs. We are interested in the long term average costs.

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

Let us denote the lead time demand by Xi , which is Poisson distributed with mean mi ti . Notice that Xi is integer valued. The distribution of the stock on hand, OHi , and the number of backordered demands, BOi , at an arbitrary point in time is then given by: (P ∞ if x = 0; y=Si P {Xi = y} P {OHi = x} = P {Xi = Si − x} if x ∈ N, x ≤ Si . (P Si if x = 0; y=0 P {Xi = y} P {BOi = x} = P {Xi = x + Si } if x ∈ N. From these we can obtain the expected number of backorder EBOi (Si ), for all Si ∈ N0 : ∞ X EBOi (Si ) = (x − Si )P {Xi =x} x=Si +1

(2.1)

= mi ti − Si +

Si X

(Si − x)P {Xi =x} .

x=0

Then the average inventory holding costs for SKU i under basestock level Si are Ci (Si ) = E[hi (Si − Xi )+ ] = hi

Si X

(Si − x)P {Xi = x}.

(2.2)

x=0

By taking the sum of all individual costs, one can calculate the total costs, denoted by C(S), where S = (S1 , . . . , S|I| ) denotes a vector consisting of all base stock levels. The total costs are calculated as follows: X C(S) = Ci (Si ). (2.3) i∈I

Note that for expensive repairable parts, there may still be a lot of value for the broken parts, and thus one also pays holding costs for the total number parts in the pipeline. Although we do not consider this, it is easy to adjust the cost function, and the results shown throughout this chapter still hold in this case. Because one is interested in the availability of a capital good, one common service measure is the aggregate mean number of backorders (Kranenburg and Van Houtum, 2009). The mean number of backorders is a common service measure used when dealing with inventory control of critical spare parts. Other service metrics that could be interesting besides the mean number of backorders would be the mean waiting time, which is closely related to the mean number of backorders, and the fill rate. The latter is mainly interesting when it is particularly important that a part is on stock

2.2. MODEL DESCRIPTION

when demand occurs. See Section 2.7 of Van Houtum and Kranenburg (2015) for a more detailed explanation of different service metrics. The aggregate mean number of backorders in steady state is calculated as follows: EBO(S) =

EBOi (Si )

(2.4)

i∈I

Hence in mathematical terms our optimization problem is as follows: (P)

min subject to

C(S) EBO(S) ≤ EBOobj Si ∈ N0 , ∀ i ∈ I

where EBOobj is our maximum level for the mean number of backorders. Our second model, the uncertain demand rate model is similar to the basic model except the demand rates are not given. We still have the same expectation of the demand rate, mi , but its exact value follows from a distribution. In other words, in our second model the demand rate mi is multiplied by yi , which is a realization of the random variable Yi ≥ 0 with probability density function fyi . In the case of the uncertain demand rate model, the lead time demand distribution for SKU i, Xiud , follows a Poisson distribution with unknown mean yi mi ti and is calculated as follows: P Xiud = x =

∞

Z 0

(umi ti )x −umi ti e fyi (u)du, x ∈ N0 . x!

(2.5)

Let EBOiud (Si ), and Ciud (Si ) respectively denote the expected number of backorders for the uncertain demand rate model and costs. Formulas for these functions are the same as equations (2.1) and (2.2) except that we replace Xi by Xiud . Our third model, the uncertain lead time model is similar to the basic model except that we have uncertainty in the lead time. We assume sequential delivery of the SKUs, thus orders may not cross (see Zipkin, 1991, for more information about sequential and parallel supply systems). The lead time ti is multiplied by zi which is a realization of the random variable Zi ≥ 0 with probability density function fzi . In the case of the uncertain lead time model, the lead time demand distribution for SKU i, Xiul , follows a Poisson distribution with unknown mean mi zi ti and is calculated as follows: P Xiul = x =

Z 0

∞

(mi vti )x −mi vti e fzi (v)dv, x ∈ N0 . x!

(2.6)

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

Let EBOiul (Si ), Ciul (Si ) respectively denote the expected number of backorders and the costs for the uncertain lead time model which are calculated as in equations (2.1) and (2.2) except that Xi is now replaced by Xiul .

2.3

The effect of more variable demand rates

In this section we prove that both the costs as well as the expected number of backorders are increasing in the variability of the demand rate. A first key observation we d need to make is that if fyi =fzi , i ∈ I, then the lead time demand Xiud in the uncertain demand rate model is the same as the lead time demand Xiul in the uncertain lead time model.

Proposition 2.1. Suppose fyi = fzi , then Xiud = Xiul for all i ∈ I Proof. Let x ∈ N0 : ∞

(umi ti )x −umi ti e fyi (u)du x! 0 Z ∞ (mi uti )x −mi uti = e fzi (u)du x! 0 = P Xiul =x

P Xiud =x =

To study the effect of the demand rate variability, we use the definition of variability that has been used for comparison of the variability of two random variables in other papers (Whitt, 1985; Song, 1994). Let u(w) be a real function defined on an ordered set U of the real line. Let H(u) be the number of sign changes of u(w). Graphically, this is the number of times u(w) crosses the w-axis when w ranges over the entire set U . More rigorously we have H(u) = sup H[u(w1 ), u(w2 ), . . . , u(wk )], where the supremum is extended over all sets {w1 , w2 , . . . , wk } with wi ∈ U for all i, w1 < w2 < · · · < wk , k is arbitrary but finite, and H(x1 , x2 , . . . , xk ) is the number of sign changes of the sequence x1 , . . . , xk , zero terms being discarded.

Definition 2.3.1. Consider two random variables V1 and V2 with the same mean E[V1 ] = E[V2 ], having probability density functions v1 and v2 . Suppose V1 and V2

2.3. THE EFFECT OF MORE VARIABLE DEMAND RATES

are either both continuous or both discrete. We say V1 is more variable than V2 , denoted V1 ≥var V2 , if H(v1 − v2 ) = 2 with sign sequence +, −, +. That is, v1 crosses v2 exactly twice, first from above and then from below. Examples of distributions under which Definition 2.3.1 holds are: 1. v1 is Gamma (Weibull) distributed with shape parameter α, v2 is Gamma (Weibull) distributed with shape parameter β, where β < α, and the scale parameters are chosen such that the distributions are equal in expectation. 2. v1 is Uniform (a1 , b1 ), v2 is Uniform (a2 , b2 ), where a1 < a2 , b1 > b2 , and a1 + b1 = a2 + b2 See Song (1994) for more examples for which Definition 2.3.1 holds. Let us now introduce the definition of increasing-convex ordering.

Definition 2.3.2. For two random variables V1 and V2 , V1 ≥ic V2 if and only if E[f (V1 )] ≥ E[f (V2 )] for all nondecreasing convex functions f.

(2.7)

When we have two nonnegative variables with identical mean, Ross (1983) shows that Equation (2.7) is equivalent to a stronger condition: Lemma 2.3.1. (Ross, 1983). If V1 and V2 are nonnegative random variables such that E[V1 ] = E[V2 ], then V1 ≥ic V2 if and only if E[f (V1 )] ≥ E[f (V2 )] for all convex functions f.

(2.8)

Let us now introduce two different systems j = 1, 2, for the uncertain demand model. Each system is equal in number and characteristics of the SKUs except from the uncertainty of the demand rates. Let Yi,j denote the random variable for SKU i for ud system j. Let Xi,j denote the lead time demand distribution of the uncertain demand model for SKU i under system j. The corresponding expected holding costs for each SKU, the total expected holding costs, and expected number of backorders ud are denoted by Ci,j (Si ), Cjud (S), and EBOjud (S) respectively.

Proposition 2.2. Let i ∈ I. Suppose we have Yi,1 ≥var Yi,2 and E[Yi,1 ] = E[Yi,2 ] ud ud ud ud then Xi,1 ≥ic Xi,2 and E[Xi,1 ] = E[Xi,2 ].

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

Proof. Let us consider two different instances of the uncertain lead time model, d d Zi,1 = Yi,1 and Zi,2 = Yi,2 for all i ∈ I. For the uncertain lead time model we know that, by making use of Proposition 4.10 of Song (1994), Zi,1 ≥var Zi,2 implies ul ul ul ul Xi,1 ≥ic Xi,2 and E[Xi,1 ] = E[Xi,2 ] ∀ i ∈ I.

Making use of Proposition 2.1 we are then able to show that d

ud ul ul ud Xi,1 = Xi,1 ≥ic Xi,2 = Xi,2 ∀i∈I ud ul ul ud E[Xi,1 ] = E[Xi,1 ] = E[Xi,2 ] = E[Xi,2 ]∀i∈I

Using this result we prove that both the costs as well as the expected number of backorders are increasing when the demand rate is more variable. Let us now introduce the following definition: Definition 2.3.3. Let f (x) be a function on Z, and x0 ∈ Z. (i) f (x) is decreasing for x ≥ x0 if ∆f (x) = f (x + 1) − f (x) ≤ 0,

x ≥ x0 ;

(ii) f (x) is increasing for x ≥ x0 if ∆f (x) = f (x + 1) − f (x) ≥ 0,

x ≥ x0 ;

(iii) f (x) is convex for x ≥ x0 if ∆2 f (x) = ∆f (x + 1) − ∆f (x) ≥ 0,

x ≥ x0 ;

Using Definition 2.3.3, we show that the costs are convex in Si Proposition 2.3. For any Si ∈ N0 , i ∈ I, Ciud (Si,j ), j ∈ {1, 2}, is convex in Si . Proof. First let us take the first order differences of Ciud (Si ), Si ∈ N0 : ∆Ciud (Si ) = Ciud (Si + 1) − Ciud (Si ) SX Si i +1 X = hi (Si + 1 − x)P {Xiud =x} − hi (Si − x)P {Xiud =x} x=0

= hi = hi

Si X x=0 Si X x=0

≥ 0.

(Si + 1 − x)P {Xiud =x} − hi

x=0 Si X x=0

P {Xiud =x}

2.3. THE EFFECT OF MORE VARIABLE DEMAND RATES

This shows that the costs are increasing on its whole domain. Further, ∆2 Ciud (Si ) = ∆Ciud (Si + 1) − ∆Ciud (Si ) SX Si i +1 X = hi P {Xiud =x} − hi P {Xiud =x} x=0

x=0

hi P {Xiud =x}

= ≥ 0.

In a similar fashion one can also show that the costs functions Ciud and Ciul are convex. ud ud ud ud Proposition 2.4. Let Xi,1 ≥ic Xi,2 and E[Xi,1 ] = E[Xi,2 ] for all i ∈ I, then ud ud (Si ) for all i ∈ I Ci,1 (Si ) ≥ Ci,2

EBO1ud (S) ≥ EBO2ud (S) Proof. Observe that for each i ∈ I R(Si , u) = hi [Si − u]+ is convex in u as shown in Proposition 2.3. Then, by Lemma 2.3.1, for each fixed S, ud ud ud ud Ci,1 (Si ) = E[R(Si , Xi,1 )] ≥ E[R(Si , Xi,2 )] = Ci,2 (Si ), i ∈ I

This result implies ( EBO1ud (S)

−

EBO2ud (S)

mi ti − Si +

SX i −1 x=0

i∈I

( −

mi ti − Si +

SX i −1

(Si − x)P

ud Xi,2 =x

)

x=0

i∈I

) ud (Si − x)P Xi,1 =x

(S −1 i X

i∈I

x=0

ud (Si − x)P {Xi,1 =x} −

SX i −1

) ud (Si − x)P {Xi,2 =x}

x=0

ud ud X Ci,1 (Si ) − Ci,2 (Si ) i∈I

≥0

Let S∗1 and S∗2 be the optimal base stock levels that satisfy the EBO constraint for the uncertain demand rate model for systems 1 and 2 respectively. From Proposition 2.4 it is clear that any feasible solution S∗1 is also feasible for the less variable system since

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

EBO1ud (S∗1 ) ≥ EBO2ud (S∗1 ). Using Proposition 2.4 this implies C1ud (S∗1 ) ≥ C2ud (S∗2 ). To summarize, we can combine Propositions 2.2, and 2.4 to show that: ud ud ud ud ] for all i ∈ I, then ] = E[Yi,2 and E[Yi,1 ≥var Yi,2 Theorem 2.5. Let Yi,1 ud ud ud ud 1. Xi,1 ≥ic Xi,2 and E[Xi,1 ] = E[Xi,2 ] for all i ∈ I |I|

2. C1ud (S) ≥ C2ud (S) for all S ∈ N0

|I|

3. EBO1ud (S) ≥ EBO2ud (S) for all S ∈ N0 4. C1ud (S∗1 ) ≥ C2ud (S∗2 )

2.3.1

Example distributions for uncertain demand rates

In this section we consider two distributions to model the uncertainty of the demand rate parameters and get expressions of the lead time demand distribution. We first look at Uniformly distributed demand rates and then at Gamma distributed demand rates. Although we only consider these two distributions, any distribution that satisfies Definition 2.3.1 can be used.

Uniformly distributed demand rates Suppose Yi,j is uniformly distributed on the interval [1 − αi,j , 1 + αi,j ) with 0 < αi,j ≤ 1, where αi,j represent the amount of uncertainty for system j. Then one can show that (after some algebra) ud P {Xi,j = x}

1 [Ex+1,ti ((1 + αi,j )mi ) − Ex+1,ti ((1 − αi,j )mi )] , 2αi,j mi ti

where Ek,λ stands for the distribution function of an Erlang distribution with k ∈ N phases and scale parameter λ > 0, i.e., Ek,λ (u) = 1 −

k−1 X j=0

(λu)j −λu e , u ≥ 0. j!

The corresponding probability density function is given by ek,λ (u) =

(λk uk−1 ) −λu e , u ≥ 0. (k − 1)!

Note that it holds that if αi,1 ≥ αi,2 , then Yi,1 ≥var Yi,2 .

2.4. NUMERICAL EXPERIMENTS

Gamma distributed demand rates Suppose mi,j Yi,j ∼ Gamma(ai,j , bi,j ), with ai,j being the shape parameter and bi,j the scale parameter. These parameters result in corresponding µi,j , and σi,j which denote the mean and variance respectively. When demand follows a Poisson process, for which the rate of this Poisson process follows a Gamma distribution, after some algebra it is possible to show that the demand during this period follows a Negative Binomial distribution with shape parameters ri,j , and pi,j , where ri,j = a, for all i ∈ I, j ∈ {1, 2} pi,j =

1 , for all i ∈ I, j ∈ {1, 2}. 1+b

The probability density function of the Negative Binomial distribution is as follows: ud P {Xi,j = x} =

Γ(ri,j + x) r p i,j (1 − pi,j )x I0,1,... (x) Γ(x + 1)Γ(ri,j ) i,j

2 2 Note that it holds that if σi,1 ≥ σi,2 , then Yi,1 ≥var Yi,2 . Moreover, note that for any t > 0, tYi,j ∼ Gamma(a, tb).

2.4

Numerical experiments

In this section we present our numerical results to show the impact of the uncertainty of the demand rate. We first introduce our multi-item problem that we want to solve. We show that a greedy heuristic leads to solutions on the efficient frontier for our problem setting. Secondly, we assume for our numerical experiments that the demand rates are either Gamma or uniformly distributed and consider different scenarios to get an indication of the impact of the additional uncertainty. Based on these scenarios we show the impact of ignoring the uncertainty in the demand rates as well.

2.4.1

Optimization procedure in a multi-item setting

We consider a multi-item problem where there is uncertainty about the failure rate of the parts, which is equal to the uncertain demand rate model described in Section 2.2. The problem (P’) we are interested in is as follows: (P’)

min subject to

PSi C ud (S) = x=0 C ud (Si ) PSi i ud EBO (S) = x=0 EBOiud (Si ) ≤ EBOobj Si ∈ N0 , ∀ i ∈ I

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

In order to solve this problem, we make use of a greedy heuristic as described by Van Houtum and Kranenburg (2015). In order to make sure we obtain efficient solutions, we need to ensure that the costs are increasing and convex, and the expected number of backorders are decreasing and convex. Even though this algorithm generates efficient solutions, this algorithm works best for a sufficiently large number of SKUs. When the number of SKUs is suffiently large, the solutions generated by the greedy algorithm are known to be close to optimality. In Section 2.2 we showed that the costs are increasing and convex on the whole domain. Therefore, we only need to show that the expected number of backorders are decreasing and convex. Proposition 2.6. For any Si ∈ N0 , i ∈ I, EBOiud (Si ) is decreasing and convex for Si ∈ N0 . Proof. First let us take the first order differences of EBOiud (Si ), Si ∈ N0 : ∆EBOiud (Si ) = EBOiud (Si + 1) − EBOiud (Si ) ∞ X =− P {Xiud =x} x=Si +1

= −(1 −

Si X

P {Xiud =x})

x=0

≤0 This shows that the expected number of backorders are decreasing on its whole domain. Further, ∆2 EBOiud (Si ) = ∆EBOiud (Si + 1) − ∆EBOiud (Si ) = P {Xiud =Si + 1} ≥0 which shows that EBOiud (Si ) is convex on its whole domain. Hence, the problem P 0 is separable and the functions EBOiud (Si ) and Ciud (Si ) are convex on their whole domain. As a result we can generate a set of efficient solutions using a greedy algorithm. We start with a solution by setting Si = 0 for each SKU i ∈ I, which is an efficient solution as it has the lowest possible costs C ud (S) = 0. The decrease in EBOiud (Si ), which is equal to −∆EBOi (Si ), divided by the increase in costs, dePSi noted by ∆Ciud (Si )(= hi x=0 P {Xiud = x}), is denoted by Γi . The SKU which has

2.4. NUMERICAL EXPERIMENTS

the highest value of Γi is increased by one, which results in a new efficient solution. This process is repeated until the objective has been met. Algorithm 1 gives an overview of the greedy algorithm applied to obtain efficient solutions. Algorithm 1 Greedy algorithm Step 1. Set Si := 0, for all i ∈ I, and S = (0, 0, . . . , 0); C ud (S) := 0 EBOiud (Si ) := mi ti . PSi PSi P {Xiud = x}) : P {Xiud = x})/(hi x=0 Step 2. Γi := (1 − x=0 k := arg max{Γi ; i ∈ I}; S := S + ek ;

Step 3. Compute C ud (S) and EBOud (S); If EBOud (S) ≤ EBOobj , then stop. Else, goto Step 2.

2.4.2

Impact of double demand uncertainty in a multi-item setting

To show how the uncertainty of the demand rate impacts the relative growth of the (near) optimal expected holding costs, we create a test bed with 32 different instances, each instance having 5 scenarios to vary the uncertainty of the demand rate parameter. Let j ∈ {0, 1, 2, 3, 4} denote the scenarios. For this numerical experiment we assume all SKUs i ∈ I have the same lead time, t. For each instance, we draw 250 parts with a random expected demand rate per month, mi , and holding costs per month, hi . These values do not differ between the different scenarios within each instance. For each SKU i ∈ I the double demand uncertainty is modeled either using the Gamma distribution or Uniform distribution. For the instances with Gamma distributed demand rates, 2 the variance of the demand rate, denoted by σi,j , is set to obtain a desired squared coefficient of variation of the demand rate, denoted by CVj2 , which is done as follows: 2 σi,j = CVj2 m2i

(2.9)

2 Thus SKUs that have a higher value of mi also have higher values of σi,j . For the instances with uniformly distributed demand rates, the demand rate for SKU i follows

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

a Uniform distribution U ((1 − αj )mi , (1 + αj )mi ), where αj determines the level of double demand uncertainty. We consider two distributions to draw the expected demand rates, two distributions to draw the holding costs of the SKUs, two different values of the lead time t, and two different values for the objective EBOobj . We do this for both Gamma distributed demand rates as well as uniformly distributed demand rates, which brings us to the 32 different instances. We have 5 scenarios of each instance, by varying the uncertainty of the demand rate parameter, leading to a total of 160 experiments. The base stock levels are set using the greedy algorithm. We repeat each experiment ten times and calculate the average costs over these ten repetitions. Let Cj∗∗ denote the average expected costs for system j. We are interested in the relative increase in holding costs when comparing a scenario with double demand uncertainty to the scenario without double demand uncertainty. Let ∆Cj represent the relative near optimal holding costs increase which is calculated as follows

∆Cj = (Cj∗∗ − C0∗∗ )/C0∗∗ , j ∈ {1, 2, 3, 4}. The results of the experiments can be found in Tables 2.1 and 2.2 for Gamma and uniformly distributed demand rates respectively. Based on these results we make a few interesting observations. First of all, when we use Equation (2.9) to set the variance of the demand rates for each SKU, the average expected holding costs of the solutions increase linearly with respect to the value of CVj2 under Gamma distributed demand rates. The same also holds for instances with longer lead times, although the impact of the uncertainty is much bigger in this case, with differences over 1800%. Under uniformly distributed demand rates, costs also increase when increasing the uncertainty, although not as much as for the Gamma distribution and costs also do not increase linearly. Another interesting observation is that the difference in the cost structure of the SKUs does not significantly influence the relative cost increase for different values of demand rate uncertainty. However, when the difference between prices of parts are larger, one is able to compensate a small part of the uncertainty by applying a system approach. Finally, for the scenarios where demand rates are larger on average, costs also increase significantly in order to compensate for the additional uncertainty.

2.4. NUMERICAL EXPERIMENTS

2.4.3

The impact of ignoring double demand uncertainty

In this Section we investigate the impact of ignoring the uncertainty in the demand rate for the first 16 instances as presented in Table 2.1. We first derive the base stock levels obtained for scenario 0, under which there is no demand rate uncertainty, S∗∗ 0 . We then calculate EBOj (S∗∗ ) for the scenarios under which we do have demand rate 0 uncertainty. The results of the actual expected number of backorders when ignoring the demand rate uncertainty, EBOj (S∗∗ 0 ), are presented in Table 2.3. Based on the results in Table 2.3 we see that ignoring the uncertainty leads to significantly bigger values of the expected number of backorders. If we would have to increase the expected holding costs by 23.5% (see Table 2.1, instance 9, scenario 1) we get an EBO value that is 390% larger than the objective. As the EBO values obtained while ignoring the additional uncertainty are much larger than the objective, one can expect solutions that will result in long down times of the machines, for which costs are likely to exceed the additional costs of the base stock levels by far.

hi U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19)

mi U(0, 1) U(0, 1) U(0, 10) U(0, 10) U(0, 1) U(0, 1) U(0, 10) U(0, 10) U(0, 1) U(0, 1) U(0, 10) U(0, 10) U(0, 1) U(0, 1) U(0, 10) U(0, 10)

Instance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 1 1 1 3 3 3 3 1 1 1 1 3 3 3 3

t 1 1 1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

EBOobj CV12 = 0.25 +20.5% +20.2% +116.2% +116.0% +46.9% +46.7% +258.4% +258.0% +23.5% +23.2% +122.0% +121.6% +50.3% +50.3% +266.8% +266.1%

â&#x2C6;&#x2020;Cj CV22 = 0.5 +40.7% +40.4% +224.7% +224.1% +92.5% +92.1% +484.9% +483.9% +46.0% +45.6% +236.3% +235.5% +99.6% +99.3% +503.2% +501.8% CV32 = 1 +80.4% +79.7% +435.1% +434.0% +182.4% +181.6% +919.3% +917.3% +90.2% +89.5% +459.0% +457.4% +196.3% +195.5% +958.7% +955.8%

CV42 = 2 +158.6% +157.4% +848.8% +846.5% +360.3% +358.7% +1769.0% +1764.8% +176.7% +175.5% +897.6% +894.3% +387.6% +386.0% +1851.9% +1846.1%

Table 2.1: Relative holding costs for different levels of uncertainty under Gamma distributed demand rates

U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19) U(5, 15) U(1, 19)

U(0, 1) U(0, 1) U(0, 10) U(0, 10) U(0, 1) U(0, 1) U(0, 10) U(0, 10) U(0, 1) U(0, 1) U(0, 10) U(0, 10) U(0, 1) U(0, 1) U(0, 10) U(0, 10)

Instance

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1 1 1 1 3 3 3 3 1 1 1 1 3 3 3 3

1 1 1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

EBOobj α1 = 0.433 CV12 = 0.0625 +4.7% +4.7% +22.6% +22.7% +10.4% +10.4% +44.2% +44.2% +5.4% +5.2% +21.2% +21.2% +10.3% +10.3% +39.6% +39.8%

∆Cj α2 = 0.6125 α3 = .866 2 CV2 = 0.125 CV32 = 0.25 +9.0% +16.7% +9.0% +16.4% +37.7% +60.1% +37.8% +60.5% +18.5% +31.4% +18.5% +31.4% +69.8% +107.0% +70.2% +107.8% +9.9% +16.9% +9.6% +16.8% +34.3% +53.6% +34.4% +53.8% +17.8% +29.1% +17.8% +29.2% +61.6% +93.2% +61.9% +93.6% CV42

α4 = 1 = 0.3333 +21.1% +20.8% +72.3% +72.8% +38.5% +38.6% +126.9% +127.9% +20.8% +20.7% +63.9% +64.2% +35.3% +35.3% +109.9% +110.5%

Table 2.2: Relative holding costs for different levels of uncertainty under uniformly distributed demand rates

2.5

2. SPARE PARTS MANAGEMENT UNDER UNCERTAINTY

Conclusions

Modeling the uncertain demand rate using a distribution instead of a point estimate allows companies to take the additional uncertainty they are facing when introducing capital goods into consideration. For a multi-item stocking problem with backordering and a constraint on the expected number of backorders we show that an increase of the uncertainty of the demand rate always leads to higher costs. Moreover, we show using numerical experiments that when demand rates follow from a Gamma distribution the relative costs of the near-optimal solution multi-item problem increase almost linearly in the squared coefficient of variation. However, the slope of these relative costs increases depend mainly on the expected demand rates, as well as the lead time relative to these demand rates. In the case of uniformly distributed demand rates the increase in costs is less significant and does not show the same linear relationship between the coefficient of variation and costs. Finally, we show for Gamma distributed demand rates what the impact would be if a company would ignore the additional uncertainty, and show this leads to poor solutions that are far from desirable. Although we allow for many possible distributions for the double demand uncertainty, we only considered Gamma and uniformly distributed demand rates. Even though one is able to model the double demand uncertainty explicitly using our model, one still needs to decide upon the distribution to use to model this uncertainty, as this may also impact the solution. Moreover, it would be interesting to find out how much a company can benefit if one has the possibility to reduce the double demand uncertainty. We go into further detail about this in Chapter 3

2.5. CONCLUSIONS

Table 2.3: EBO values when ignoring the demand rate uncertainty Instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

EBOobj 1 1 1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

CV12 = 0.25 2.16 2.12 30.63 29.70 5.69 5.51 174.96 170.88 0.39 0.38 14.21 13.77 1.57 1.53 114.07 111.64

EBOj (S∗∗ 0 ) CV22 = 0.5 CV32 = 1 3.64 7.06 3.54 6.83 76.00 164.98 74.10 161.93 12.91 29.73 12.49 28.88 382.02 720.17 375.59 711.85 0.92 2.56 0.88 2.45 45.75 118.99 44.64 116.94 5.12 15.92 4.96 15.52 290.00 603.76 285.67 597.7

CV42 = 2 14.32 13.88 306.41 302.49 62.06 60.69 1189.20 1180.10 7.10 6.85 248.95 246.04 41.37 40.59 1063.00 1056.00

Multi-period spare parts planning under stochastic demand and parameter uncertainty 3.1

Introduction

During the initial phase of a capital good, the demand rates of the components of this capital good are commonly based upon expert knowledge and/or limited qualitative research as there have not been any observed failures. However, when the demand rates are highly uncertain, simply using the expected demand rates may result in high unavailability which may thereby result in unnecessary high costs, as shown in Chapter 2. Therefore, it is best not to ignore the demand rate uncertainty the OEM is facing. In Chapter 2 we address this uncertainty by modeling demand as a Poisson process, where the demand rate is drawn from a distribution instead of assuming this to be a given constant value. In Chapter 2 the decision on the base stock level used is only taken once at the start of this initial phase. By incorporating the double demand uncertainty, on average better decisions are made. In this chapter we continue on the problem discussed in Chapter 2. This chapter is related to the same literature and can be positioned accordingly. In addition to Chapter 2 we also consider the case where one can invest in more reliable demand rate information. This information is generally obtained by having an engineer spending time and effort into obtaining a better estimation. This possibility to get more reliable estimates or demand rate information 39

3. MULTI-PERIOD SPARE PARTS PLANNING

is related to other fields within the literature, in particular the literature on Advance Demand Information (ADI). This ADI consists of information of (part of) the future demand, which can be used to make better inventory decisions (see e.g. Ă&#x2013;zer (2011)). Tan et al. (2007) consider the case where there is imperfect advance demand information and reveal conditions under which having imperfect ADI leads to the largest cost reduction. Other papers that look into the impact of imperfect ADI are e.g. Benjaafar et al. (2011) and Topan et al. (2018). In this chapter we consider a model where as opposed to having imperfect demand information concerning the future, we have a more reliable estimate of the demand rate. In both cases, one is interested in how much this value of information is, even though this information may still be imperfect. It is common that before the capital goods are actually being used, a decision has to be made already on how many spare parts need to be kept on hand. These decisions are commonly based on the information that is available during this design phase, which is generally very limited and based on expert knowledge. If the OEM waits for a long enough time period to forecast the demand rates, this potentially would allow the OEM to make better stocking decisions by taking the average demand over the last years. However, this generally would take too much time, especially since demand rates in general are rather low. After this long time period, it may be difficult to produce additional parts, as the production of the capital goods themselves may already have been ceased. Naturally this problem could be addressed by overstocking at the start of the initial phase, but this will not only incur very high inventory holding costs as these are typically expensive parts, but it also brings in a high risk of obsolescence as well, since these components are typically very specific. Accordingly, having high base stock levels at first and reducing them at a later point in time is also a costly option, as it may be difficult to sell these spare parts. As a result, the amount of money the OEM would get back can be much less than the cost of manufacturing. Whenever the OEM would thus decide to stock too much, this would not only involve high holding costs, but also the costs involved when the OEM needs to sell unnecessary parts. If a company would have a more accurate estimation of the demand rate, the company may be able to adjust the base stock level based on the more reliable estimate. At this earlier point in time, parts may still have more value when selling back excess stock, which would be the case if the demand rate turns out to be lower than expected initially. This would potentially reduce the costs of obsolescence and thus reduce the expected maintenance costs. One option to allow an earlier final decision making is to spend time and effort into getting more information about the actual demand rates. In many cases this generally would require engineers to spend time and effort in in-

3.1. INTRODUCTION

vestigating the demand rate behavior. Sources of these demand rate information can come from different methods, such as failure mode effect analysis (FMEA), Availability Assessment (AVAS), and Design Verification. After the engineers are done with the investigation, the company has a better and more reliable predictability which they can use to update the base stock levels at an earlier point in time. Although costs are involved in this process, depending on how difficult it is to obtain information about the demand rate of the component, as well as how much information the OEM wants to obtain about this component, a decrease in the double demand uncertainty would result in better decision making. In this chapter we give an answer to Research Objective 3: Research Objective 3 Develop a model that supports the decision maker for investments in more reliable information on the demand rate and show for which parts the company can benefit the most from such an investment. In order to give an answer to this question, and thereby answer research objective 3, we model the problem such that the OEM is able to make better decisions at the start of the introduction phase by incorporating possible future demand rate information changes as a result of the investment in information. We present a two period inventory model in which there is the opportunity to invest in different amounts of information at the start of the first period to allow for a better decision making in the second period. Although it could be possible to consider multiple periods and incorporate the information obtained from the observed failures, this information is rather limited compared to the information obtained by the investigation of the engineers and not taken into consideration. We consider a single repairable component, which is part of a new capital good that is launched. We assume demand for this component follows according to a Poisson process. There is a limited amount of information available about the demand rate for this component, and thus we have a distribution of the demand rate based on expert knowledge. Whenever the component fails, the capital good is no longer working, which results in downtime costs for the customer. The component is stored in a single location, and whenever there is a stockout, the customer has to wait until the part becomes available. To model the double demand uncertainty we use the same logic as we applied in Chapter 2. At the start of the introduction, the OEM has the option to invest in information,

3. MULTI-PERIOD SPARE PARTS PLANNING

where the amount of information retrieved is mainly depending on the time spend on getting more information. A higher investment results in less uncertainty concerning the demand rate at a later point in time. This reduced amount of uncertainty allows the OEM to adjust the stock level of the component accordingly. However, whenever the base stock level is reduced, this means the company has overstocked in the first period, and thus there are costs involved as the OEM will only get back a fraction of the original price. The main contribution of this chapter is that we provide a means to investigate whether it can be interesting for a company to invest in more information. We model the possibility to invest in having less demand rate uncertainty as a two period inventory model, where in the first period one has to decide upon the investment as well as the initial base stock level. At the start of the second period, one can adjust the base stock level based on more reliable demand rate information as a result of the investment. In order to solve this problem, we model the problem as a three-stage stochastic dynamic program, where we first decide upon the investment, and then on the base stock levels at the start of the first and second period respectively. We derive analytical expressions for the optimal base stock levels for both periods and derive a lower bound on the total expected costs. Using enumeration, we are able to find the optimal investment. In Section 3.2 we present our model for the corresponding two-period problem. We solve this model by formulating the model as a three-stage dynamic program in Section 3.3. In Section 3.3.1 we derive an expression for the optimal second period base stock level. In Section 3.3.2 we derive an expression of the first period optimal base stock level. Then in Section, 3.3.3 we derive the optimal investment. We present our numerical results where we show the potential of more information for different instances in Section 3.4. Finally, we give our conclusions in Section 3.5.

3.2

Model Description

We consider the setting where one group of identical machines is supported by a single stockpoint. We consider a single critical repairable component, also denoted as stock keeping unit (SKU). Moreover, we distinguish two periods with length T1 and T2 for the first and second period respectively. In case of a failure of a component, the corresponding machine stops working and a spare component is supplied from the stockpoint when available. The failed part is returned to the stockpoint and imme-

3.2. MODEL DESCRIPTION

Figure 3.1: Timeline

diately sent into repair. The repair takes a deterministic time L. In other words, the stock is controlled by a continuous-review basestock policy with base stock level S1 and S2 for respectively the first and second period. Demand is backordered in case of a stockout. The demand of the SKU follows a Poisson process with demand rate λtrue . As a result, the amount of stock in the pipeline is Poisson distributed with rate Lλtrue . We assume that L is much smaller than T1 and T2 , i.e. L << Ti , i = 1, 2. The exact value of λtrue is not known to the decision maker at the start of the first period. We assume it is only known that its value is anywhere in between a1 and b1 with equal probability. In other words, the demand rate is drawn from U(a1 , b1 ). At the start of the first period it is possible to invest man-hours of the engineering department to retrieve more information regarding the demand rate. For simplification, we assume fixed levels of investment, denoted by investment level I ∈ N0 , resulting in different uncertainty levels in the second period. For each investment level I there are costs involved denoted by C(I). For a higher investment level, the costs increase but the company also obtains more information about the demand rate at the start of the second period. Figure 3.1 represents a timeline consisting of the decisions of both periods that are made at the different points in time for an OEM. At the start of each period one has to decide on how much stock to keep on hand, with base stock levels S1 and S2 for the first and second period respectively. At the start of the second period, the OEM gets updated bounds for the demand rate parameter, denoted by a2 and b2 . These bounds depend on the investment level chosen at the start of the first period. We assume that depending on the investment level there is a fixed number of possible intervals of equal length that the OEM can

3. MULTI-PERIOD SPARE PARTS PLANNING

(a) Interval construction with I = 0

(b) Interval construction with I = 1

Figure 3.2: Interval construction for different investment levels

obtain in the second period. Every investment level increases the number of possible intervals by exactly one, starting at a single interval if the OEM decides not to invest (thus having the same bounds as at the start of the first period). The set of possible second period intervals is denoted by N (I) = {1, 2, . . . , I + 1}. For convenience we number the intervals from left to right, where the first interval starts at a1 , and the last interval ends at b1 . Let a2 (n, I) and b2 (n, I) denote the start and end respectively of the n-th interval under an investment level I At the start of the second period, the OEM obtains the values corresponding to the interval that contains the value of λtrue , thus a2 (n, I) ≤ λtrue ≤ b2 (n, I), n ∈ N (I), I ∈ N0 . Figure 3.2a shows the relationship between the investment level I and the constructed intervals. For example, if we have an investment level of I = 2, and an initial interval (a1 , b1 ) = (1, 4), there are 3 possible intervals in the second period, (1, 2), (2, 3), (3, 4). For any given interval, each value of λtrue within this interval is even likely. Thus,

3.2. MODEL DESCRIPTION

in other words, λtrue is considered to be drawn from a new uniform distribution U(a2 (n, I), b2 (n, I)). Based on the way that we construct the intervals, each second period interval, n ∈ N (I), I ∈ N0 , then has the following size: b2 (n, I) − a2 (n, I) =

b1 − a1 . I +1

For each investment level I, we calculate a2 (n, I) and b2 (n, I) as follows: a2 (n, I) = a1 + (n − 1) b2 (n, I) = a1 + n

b1 − a1 , I +1

b1 − a 1 , I +1

I ∈ N0 ; n ∈ N (I). I ∈ N0 ; n ∈ N (I).

Let ψn denote the probability that λtrue is within interval n. As λtrue has an equal chance to obtain any value between b1 and a1 , and we know the size of each interval n for a given investment level I, ψn (I) is calculated as follows: ψn (I) =

b2 (n, I) − a2 (n, I) 1 = b1 − a1 I +1

Note that the OEM can thus obtain every possible interval with equal probability. The size of the second period interval is decreasing in I and as I goes to infinity, b2 (n, I) − a2 (n, I) goes to 0, which means that in the case of infinite investment levels, there would be no more uncertainty about the demand rate, thus one would exactly know the value of λtrue at the start of the second period. Although this means having perfect demand rate information, one still has the uncertainty stemming from the demand distribution itself. There are different costs associated with keeping spare parts on stock. Firstly, the company has to obtain the spare parts at the beginning of the period if it wants to increase the base stock level. The purchasing costs of a part are denoted by c, thus the total costs to obtain the necessary spare parts, including the possible increase at the start of the second period, are cS1 + c(S2 − S1 )+ . At the end of the first period, the company can decide to decrease the base stock level, and thus sell one or more components. At the end of the second period, the spare parts are no longer used, and thus the OEM sells all components. Generally, the company gets back less than what it initially paid for the purchase of the spare parts. Let α1 c, and α2 c (0 ≤ α2 ≤ α1 ≤ 1) denote the amount of money that the OEM gets back for selling the stock at the end of period 1 and period 2, respectively.

3. MULTI-PERIOD SPARE PARTS PLANNING

Thus we get α1 c(S1 − S2 )+ back at the end of the first period, and α2 cS2 at the end of the second period. Thus for every part sold at the end of the first and second period, we have a difference between the purchase and sale of c(1 − α1 ) and c(1 − α2 ). For ease of notation, we introduce a new parameter denoting this difference in the price for obtaining the parts and selling the parts, denoted by w1 and w2 , which are as follows: w1 = c(1 − α1 ), w2 = c(1 − α2 ). The total costs as a result of this difference are then w1 (S1 − S2 )+ and w2 S2 at the end of the first and second period respectively. Besides these costs, the OEM also incurs holding costs (the costs for physically storing the parts and opportunity costs) per time unit per part that is somewhere within the pipeline or on hand at the stockpoint, denoted by h. The total holding costs for the first and second period are then denoted by hT1 (S1 + EBO1 (S1 )) and hT2 (S2 + EBO2 (S2 )) respectively, where EBO1 (S1 ) and EBO2 (S2 ) denote the expected number of backorders in the steady state for the first and second period respectively. We assume that during the entire first and second period, the system is in the steady state. Note that as we consider a finite time period, we actually underestimate the performance as we start with a positive stock level. After a time interval equal to the lead time L we can however use the steady-state calculation as we look at the demand during the lead time. As only during a time interval with length L at the start of the period the calculations are not accurate, and given that the length of the period is sufficiently large compared to the lead time L, the impact of this assumption is expected to be small. For each moment of time, for each demand that is backordered, the OEM incurs a penalty cost p1 or p2 for the first and second period respectively. Depending on the agreements made with the customer, these penalty costs may differ, although in general these penalty costs are very large as downtime of the capital good has a big impact for the customers. Beside these costs, there are also costs involved for the repair of the broken parts. Every broken part is sent to the repair shop to be repaired, which involves certain repair costs. However, as we cannot influence the rate at which parts fail, and thus the

3.2. MODEL DESCRIPTION

rate at which parts will be send to the repair shop, we ignore these costs in our model. Let X1 , and X2 denote the amount of stock in the pipeline at an arbitrary point in time for period 1 and 2 respectively when one does not exactly know the value of λtrue . X1 then has the following distribution (see also Chapter 2): Z

P {X1 = x} = a1

(uL)x e−uL du, (b1 − a1 )x!

x ∈ N0 .

X2 has a similar distribution but with parameters a2 (n, I), and b2 (n, I) instead of a1 and b1 . For a given investment level and interval we calculate X2 as follows: Z b2 (n,I) (uL)x P {X2 (n, I) = x} = e−uL du, x ∈ N0 . (3.1) (b (n, I) − a (n, I))x! 2 2 a2 (n,I) Let EBO1 (S1 ), and EBO2 (S2 , n, I) represent the expected number of backorders at an arbitrary point of time during the first and second period. Given the base stock levels, the interval n, and investment level I, these are calculated as follows with S1 , S2 , I ∈ N0 , n ∈ N (I) (see also Chapter 2): ∞ X

EBO1 (S1 ) =

(x − S1 )P {X1 = x}

x=S1 +1

∞ X

(x − S1 )P {X1 = x} +

S1 X

(S1 − x)P {X1 = x}

x=0

= E[X1 ] − S1 +

S1 X

(S1 − x)P {X1 = x}

x=0

EBO2 (S2 , n, I) =

S1 X a1 + b1 (S1 − x)P {X1 = x}, L − S1 + 2 x=0

∞ X

(x − S2 )P {X2 (n, I) = x}

x=S2 +1

∞ X

(x − S2 )P {X2 (n, I) = x} +

x=0

= E[X2 ] − S2 +

S2 X

(S2 − x)P {X2 (n, I) = x}

x=0 S2 X

(S2 − x)P {X2 = x}

x=0

S2 X a2 (n, I) + b2 (n, I) L − S2 + (S2 − x)P {X2 (n, I) = x}, 2 x=0

where E[Xi ], i = 1, 2, is the average demand during the lead time.

3. MULTI-PERIOD SPARE PARTS PLANNING

At the start of the first period the decision maker has to decide on the investment level I, as well as the base stock level S1 . At the start of the second period, the base stock level, S2 , is decided upon, making use of the new information. Any excess stock is disposed of involving unit costs w1 . At the end of the second period, all excess stock is disposed off at unit costs w2 . Depending on the base stock level chosen for the first period, the company faces the following expected holding costs and backordering costs, denoted by c1 (S1 ): c1 (S1 ) = T1 hS1 + T1 (p1 + h)EBO1 (S1 ). In the second period, not only the costs for holding and backordering of the spare parts are incurred, but also the costs as a result of selling excess stock at either the beginning or at the end of the second period. We ignore the fact that the value of money may decrease over time, although it is possible to incorporate these costs as well. The total costs for the second period, for a given interval, investment level and base stock levels, denoted by c2 (S1 , S2 , n, I), are calculated as follows: c2 (S1 , S2 , n, I) = T2 hS2 + T2 (p2 + h)EBO2 (S2 , n, I) + w1 (S1 − S2 )+ + w2 S2 (3.2) The objective then is to minimize the total expected costs.

3.3

Three-stage dynamic program

In order to solve the model described in Section 3.2 we have three different decisions that need to be taken. At the start of the first period, one has to decide upon the investment level I as well the initial base stock level S1 . Then after T1 time periods, one has to determine the base stock level S2 , but this time with possible more information as a result of the investment. For the analysis we assume that one first decides upon the investment level I before one decides upon the base stock level S1 . We then formulate the problem as a three-stage dynamic program. At the first stage we decide upon the investment level I ∈ N0 . Based on the investment level we decide upon the first period base stock level S1 in the second stage. In the third stage we decide upon the second period base stock level, S2 , given I, S1 and the interval n ∈ N (I) that contains λtrue . At the start of the third stage the values of S1 , I and n are known. The direct costs for the third stage consist of the holding and backordering costs in the second period, and the costs of selling excess stock at either the start or end of the second period and

3.3. THREE-STAGE DYNAMIC PROGRAM

depend solely on the value of S2 . The minimum expected costs for this third stage, denoted as V3 (S1 , n, I), are as follows:

V3 (S1 , n, I) = min c2 (S1 , S2 , n, I) S2 ∈N0

(3.3)

For the second stage, one has already decided upon the investment level I, and the decision variable is the base stock level for the first period S1 . The direct costs for this stage consist of the holding and backordering costs over the first period, denoted by c1 (S1 ). Besides the direct costs, one also has the expected total costs for the third stage, which depend on the value of S1 as well. The minimum expected costs for the second stage, denoted by V2 (I) are then as follows: h i X V2 (I) = min c1 (S1 ) + ψn (I)V3 (S1 , n, I) S1 ∈N0

(3.4)

n∈N (I)

For the first stage, we start with an empty state, and need to decide upon the investment level that minimizes the total expected costs over both periods. The direct costs consist of the costs for investing in more information, denoted by C(I). Moreover, one wants to minimize the expected total costs for the second stage given this investment level. The optimality equation for our problem is then as follows: h i V1 (∅) = min C(I) + V2 (I) I∈N0

(3.5)

In the following sections we derive properties for the second and third stage optimization. For the first stage optimization, we use enumeration to find the optimal investment level.

3.3.1

Third stage optimization

In order to solve the third stage problem, as presented by Equation (3.3), one needs to find the optimal base stock level S2∗ that minimizes c2 (S1 , S2 , n, I), where S1 , n, I are known and given. In order to find the optimal second period base stock level, we first show that c2 (S1 , S2 , n, I) is convex in S2 (cf. Definition 2.3.3). Proposition 3.1. Suppose S1 , n, and I are given. Then c2 (S1 , S2 , n, I) is convex in S2 ∈ N0 .

3. MULTI-PERIOD SPARE PARTS PLANNING

Proof. The first order difference of c2 (S1 , S2 , n, I) for S2 is given by: â&#x2C6;&#x2020;c2 (S1 , S2 , n, I) = c2 (S1 , S2 + 1, n, I) â&#x2C6;&#x2019; c2 (S1 , S2 , n, I) = w1 (S1 â&#x2C6;&#x2019; S2 â&#x2C6;&#x2019; 1)+ + T2 h(S2 + 1) + T2 (p2 + h) â&#x2C6;&#x2019;1 +

SX 2 +1

! (S2 + 1 â&#x2C6;&#x2019; x)P {X2 (n, I)=x}

x=0

+ w2 (S2 + 1) â&#x2C6;&#x2019; w1 (S1 â&#x2C6;&#x2019; S2 )+ â&#x2C6;&#x2019; T2 hS2 â&#x2C6;&#x2019; T2 (p2 + h)

S2 X

! (S2 â&#x2C6;&#x2019; x)P {X2 (n, I)=x}

â&#x2C6;&#x2019; w2 S2

x=0

= w1 (S1 â&#x2C6;&#x2019; S2 â&#x2C6;&#x2019; 1)+ â&#x2C6;&#x2019; w1 (S1 â&#x2C6;&#x2019; S2 )+ + T2 h + T2 (p2 + h)

S2 X

P {X2 (n, I)=x} â&#x2C6;&#x2019; T2 (p2 + h) + w2

x=0

It is easy to see that the first order difference is increasing as a function of S2 . This shows that c2 (S1 , S2 , n, I) is convex on its whole domain. Using Proposition 3.1, for a given investment level and interval one can derive an expression for the optimal second period base stock level by finding the lowest base stock level, S2 , for which â&#x2C6;&#x2020;c2 (S1 , S2 , n, I) â&#x2030;Ľ 0: T2 p2 + w1 1{x<S1 } â&#x2C6;&#x2019; w2 S2â&#x2C6;&#x2014; (S1 , n, I) = min x|F2 (x) â&#x2030;Ľ T2 (p2 + h)

(3.6)

where F2 (x) represents the cumulative distribution function of X2 (n, I), which for a given investment level and interval can be calculated using Equation 3.1. From Equation (3.6) it becomes clear that there is a relationship between the first period base stock level S1 and S2â&#x2C6;&#x2014; (S1 , n, I) due to the term â&#x20AC;&#x153;w1 1{x<S1 } â&#x20AC;?. When S1 is large, there may be an incentive to keep more stock in the second period, as the stock has already been bought. This can be seen from the threshold value, which shifts to the right as S1 increases. One can show that given the interval and investment level, there is a lower bound as well as an upper bound on the optimal second period base stock level, denoted by S2LB (n, I), and S2U B (n, I) respectively. These bounds are calculated as follows: T2 p2 â&#x2C6;&#x2019; w2 S2LB (n, I) = min x|F2 (x) â&#x2030;Ľ , I â&#x2C6;&#x2C6; N0 , n â&#x2C6;&#x2C6; N (I) T2 (p2 + h) T2 p2 + w1 â&#x2C6;&#x2019; w2 S2U B (n, I) = min x|F2 (x) â&#x2030;Ľ , I â&#x2C6;&#x2C6; N0 , n â&#x2C6;&#x2C6; N (I) T2 (p2 + h)

3.3. THREE-STAGE DYNAMIC PROGRAM

Figure 3.3: Example relation between S1 and S2∗ (S1 , n, I) Whenever S1 is smaller than S2LB (n, I), and we would increase S1 by one, it has no effect on the decision to keep S2LB (n, I) parts in the second period. One still decides to keep a basestock level of S2LB (n, I) during the second period. For values of S1 ≥ S2U B (n, I), an increase of S1 does also not influence the second period decision. In this case it is never good to have more stock than S2U B (n, I). Thus regardless of the increase of S1 , the second period optimal base stock level remains S2U B (n, I). When S2LB (n, I) ≤ S1 < S2U B (n, I), and the base stock level S1 is increased by one, there is also a change in the second period base stock level S2 . In particular, in this case the second period base stock level is always equal to the first period base stock level. By choosing to keep the same base stock level until the end of the second period, the OEM incurs a cost of w2 instead of w1 . Moreover, the OEM also has more holding costs as a result of the increase of the second period base stock level. However, the OEM incurs less backordering costs as there are more spare parts. The decrease in backordering costs is larger than the increase in the other costs. To conclude, we thus have the following relationship between S1 and S2∗ (S1 , n, I): Proposition 3.2. Suppose S1 , n, and I are given. Then the optimal sescond period

3. MULTI-PERIOD SPARE PARTS PLANNING

base stock level S2∗ (S1 , n, I) is as follows:  LB LB   S2 (n, I), 0 ≤ S1 < S2 (n, I) S2∗ (S1 , n, I) = S1 , S2LB (n, I) ≤ S1 < S2U B (n, I)   UB S2 (n, I), S2U B (n, I) ≤ S1

(3.7)

Figure 3.3 graphically shows this relation. We make use of this relationship between the first and second period base stock level to get an expression for the optimal first period base stock level under a given investment level.

3.3.2

Second stage optimization

In this section we show that the cost function we want to minimize is convex in S1 . We first show that the direct costs for the first period, denoted by c1 (S1 ), are convex in S1 .

Proposition 3.3. c1 (S1 ) is convex in S1 ∈ N0 . Proof. After some rewriting we obtain the following first order difference ∆c1 (S1 ) = c1 (S1 + 1) − c1 (S1 ) = T1 h(S1 + 1) + T1 (p1 + h)EBO(S1 + 1) − T1 hS1 − T1 (p1 + h)EBO(S1 ) SX 1 +1 a1 + b1 L − S1 − 1 + (S1 + 1 − x)P {X1 = x} = T1 h + T1 (p1 + h) 2 x=0 ! S1 X a1 + b1 (S1 − x)P {X1 = x} − T1 (p1 + h) L − S1 + 2 x=0

= T1 (p1 + h)

S1 X

P {X1 = x} − T1 p1

x=0

As ∆c1 (S1 ) is increasing in S1 , c1 (S1 ) is convex in S1 . Next, we show that for any n ∈ N (I), V3 (S1 , n, I) is convex in S1 ∈ N0 . Proposition 3.4. V3 (S1 , n, I) is convex in S1 ∈ N0 , for each given I ∈ N0 and n ∈ N (I). Proof. Note that by making use of Equation (3.7) we can show the following:

3.3. THREE-STAGE DYNAMIC PROGRAM

1. 0 ≤ S1 < S2LB (n, I): We know in this case that S2∗ (S1 + 1, n, I) = S2∗ (S1 , n, I) = S2LB (n, I). By rewriting, we then obtain that ∆V3 (S1 , n, I) = V3 (S1 + 1, n, I) − V3 (S1 , n, I) = c2 (S1 , S2∗ (S1 + 1, n, I), n, I) − c2 (S1 , S2∗ (S1 , n, I), n, I) = c2 (S1 , S2LB (n, I), n, I) − c2 (S1 , S2LB , n, I) =0 2. S2U B (n, I) ≤ S1 : We know in this case that S2∗ (S1 + 1, n, I) = S2∗ (S1 , n, I) = S2U B (n, I). We then obtain the following first order difference: ∆V3 (S1 , n, I) = V3 (S1 + 1, n, I) − V3 (S1 , n, I) = c2 (S1 + 1, S2∗ (S1 + 1, n, I), n, I) − c2 (S1 , S2∗ (S1 , n, I), n, I) = c2 (S1 + 1, S2U B (n, I), n, I) − c2 (S1 , S2U B (n, I), n, I) = w1 3. S2LB (n, I) ≤ S1 < S2U B (n, I): In this case we know that S2∗ (S1 , n, I) = S1 and S2∗ (S1 + 1, n, I) = S1 + 1. Using this result and by rewriting we obtain ∆V3 (S1 , n, I) = V3 (S1 + 1, n, I) − V3 (S1 , n, I) = c2 (S1 + 1, S2∗ (S1 + 1, n, I), n, I) − c2 (S1 , S2∗ (S1 , n, I), n, I) = c2 (S1 + 1, S1 + 1, n, I) − c2 (S1 , S1 , n, I) = T2 (p2 + h) −1 +

SX 1 +1

! (S1 + 1 − x)P {X2 (n, I)=x}

x=0

+ w2 (S1 + 1) + T2 h(S1 + 1) − T2 hS1 − w2 S1 ! S1 X − T2 (p2 + h) (S1 − x)P {X2 (n, I)=x} x=0

= T2 (p2 + h)

S1 X

P {X2 (n, I) = x} − T2 p2 + w2

x=0

From these equations it becomes clear that for each domain, the first order differences are increasing in S1 . In order to show that the first order differences are increasing on the whole domain, it suffices to show the following: ∆V3 (S2U B (n, I) − 1, n, I) ≤ w1 ,

3. MULTI-PERIOD SPARE PARTS PLANNING 0 ≤ ∆V3 (S2LB (n, I), n, I).

For the first condition, consider the case where one would decide not to change the second period base stock level in the case S1 := S1 + 1. In this case, one would always incur additional cost of w1 . Thus it is clear that for any S1 ∈ N0 , ∆V3 (S1 , n, I) ≤ w1 . Next, to show that ∆V3 (S1LB (n, I), n, I) ≥ 0, we thus need to show that S2LB (n,I)

T2 (p2 + h)

P {X2 (n, I) = x} − T2 p2 + w2 ≥ 0

x=0

We know that S2∗ (S1 , n, I) = S2LB (n, I), and from Equation 3.6, we derive that in order for S2∗ (S1 , n, I) to be optimal, the following must hold: F2 (S2LB (n, I)) ≥

T2 p2 − w2 . T2 (p2 + h)

Hence, 0 ≤ ∆V3 (S2LB (n, I), n, I). It thus becomes clear that ∆V3 (S1 , n, I), is increasing on its whole domain, and is thus convex in S1 . Let V̂2 (I, S1 ) denote the total expected costs for the second stage under a given value of I and S1 , which is calculated as follows: X V̂2 (I, S1 ) = c1 (S1 ) + ψn (I)V3 (S1 , n, I) n∈N (I)

Proposition 3.5. V̂2 (I, S1 ) is convex in S1 ∈ N0 under a given I ∈ N0 . Proof. Note that since ψn (I) ≥ 0, and from Propositions (3.3), and (3.4) we know that c1 (S1 ) and V3 (S1 , n, I) are convex in S1 , the function for total expected costs is a linear combination of convex functions, hence this function is convex.

3.3.3

First stage optimization

In this section we determine the optimal investment level I ∗ that minimizes the total expected first stage costs V1 (I), hence provides a solution for our overall problem. Unfortunately, this cost function is not convex in I, and thus we cannot guarantee that when we find a local minimum, this is also a global minimum. Therefore, we enumerate over the values of I until we can guarantee that the costs will never be smaller than the best solution found so far. We start with an investment level of I = 0

3.4. THE BENEFIT OF RELIABLE DEMAND RATE INFORMATION

and increase the investment level until we can ensure that we cannot find a better solution. Let V̂1 (I) denote the minimum expected costs when deciding upon interval I in the first stage, and let Ṽ1 (I) denote the smallest value of V̂1 (I) found during the enumeration. These are calculated as follows: V̂1 (I) = C(I) + V2 (I) Ṽ1 (I) = min V̂1 (x) x=0,...,I

We can ensure that it not possible to further improve on the best solution when the following holds: C(I) ≥ Ṽ1 (I) Making use of this stopping criteria when enumerating over I, as well as the convexity properties derived in Sections 3.3.1 and 3.3.2 to determine the optimal base stock levels, the optimal investment level I ∗ can be determined. Algorithm 2 describes how one finds the optimal investment level, denoted by I ∗ , based on the following enumeration procedure:

Algorithm 2 Enumeration procedure to find optimal investment level Step 1. Let I := 0 Step 2. Calculate V̂1 (I), Ṽ1 (I). Step 3. If C(I) ≥ Ṽ1 (I) : I ∗ = min{x|V̂1 (x) = Ṽ1 (I)}, stop. Else, I := I + 1, go to Step 2.

3.4

The benefit of reliable demand rate information

In this section we show the benefit of investing in more reliable demand rate information. We first derive the expected costs when we would have perfect demand rate information in the second period, which is similar to having an infinite investment level without any costs. We compare these costs with the costs if we would decide not to invest in information at all. This difference can be seen as the potential gains by investing in more reliable information.

3. MULTI-PERIOD SPARE PARTS PLANNING

We then show, using numerical experiments, in what situations the potential gains are the largest, and thus it would be more worthwhile to invest in more reliable information.

3.4.1

Potential gains by investing in more information

In order to find the potential gains of information, we need to derive the lowest possible value for V2 (I) and compare this value with V2 (0). We also consider the case where there is perfect demand rate information in the second period, thus V2 (∞). Using Equation 3.4, we can show that V2 (∞) is calculated as follows:

Z h V2 (∞) = min c1 (S1 ) + S1 ∈N0

V30 (S1 , x) i dx b1 − a1

(3.8)

where V30 (S1 , x) denotes the optimal second period costs when the demand rate is known to be exactly x and is calculated similar to V3 (S1 , n, I). Given the value of V2 (∞), we then obtain the following upper bound on the potential gains: J U B = V2 (0) − V2 (∞)

(3.9)

As one also has costs involved in investing in more information, one would never be able to reduce the total expected costs by this amount, and therefore this value can be seen as an upper bound on the potential gains of more reliable demand rate information. Unfortunately, calculating V2 (∞) requires evaluating an infinite number of possible third stage scenarios. In Appendix B we give an approach on how one can numerically calculate V2 (∞). As an alternative one might also take a sufficiently large I such that the second period intervals are small enough in order to get a reasonable estimate of V2 (∞).

3.4.2

Numerical experiments

In order to find out what parameters have the biggest impact on whether it is interesting to invest in information, measured by the potential gains J U B , we use a factorial design and apply our model on 48 instances.

3.4. THE BENEFIT OF RELIABLE DEMAND RATE INFORMATION

First, we consider two different scenarios for the length of the first and second period. In the first case, T1 = 26 weeks and T2 = 260 weeks. For the other case, the length of the first period is twice as large, although the total length over both periods stays the same. So (T1 , T2 ) ∈ {(26, 260), (52, 234)}. Secondly, we have the costs of selling excess stock at the end of the first and second period. We assume the costs of a spare part are either 5000 or 50000 euro, thus c ∈ {5, 000, 50, 000}. Thirdly, we vary the amount of money the OEM gets back at the end of the second period. We consider α1 = 0.8, α2 = 0.2 and α1 = 0.9, α2 = 0.1. Moreover, we take C(I) = 1000 ∗ I. We assume that every investment level corresponds to a little more than a working day of an engineer, which costs around 750 euro per day. Next, we are interested in the holding costs and penalty costs. We start with holding costs of h = 0.004 ∗ c per part per week, corresponding to roughly 20 percent holding costs of the unit price per year per part. For the numerical experiments, we have identical penalty costs for both periods (p1 = p2 ). Depending on the capital good, costs of having backordered parts can differ. Therefore we consider a case where having backordered parts is relatively less expensive, p1 = p2 = 25, 000 per week, and a case where having backordered parts is relatively expensive, p1 = p2 = 50, 000 per week. Finally, we consider different values for the initial uncertainty of the demand rate. We keep the expectation the same to make the results better comparable to each other, but the variance is different for the different instances. The expectation of the demand rate is equal to 0.25 per week, and a1 ∈ {0, 0.1, 0.2}. This results in b1 ∈ {0.5, 0.4, 0.3} respectively. We keep the lead time fixed, which takes around one month (L = 4 weeks), throughout all the different scenarios. The maximum potential benefit of more reliable demand rate information is represented by B max and calculated as follows: B max = (V2 (0) − V2 (∞))/V2 (0).

3. MULTI-PERIOD SPARE PARTS PLANNING

The expected average benefit, denoted by B avg is then calculated as follows: B avg = (V2 (0) − V1 (∅))/V2 (0). Table 3.1 shows the expected costs when one decides not to invest, V2 (0), when one invests the optimal investment level I ∗ , V1 (∅), as well as the expected costs when one would have no double demand uncertainty in the second period, V2 (∞). Let S2min and S2max represent the smallest second period base stock level and the largest second period base stock level respectively over all possible intervals given the investment level I ∗ and first period base stock level S1∗ . From the results it is clear that if there is more uncertainty regarding the demand rate parameter, it becomes more interesting to invest in information. In the instances where the uncertainty is very limited, it is hardly ever beneficial to invest in more information. When we consider the instances with more uncertainty, it is clear that in particular for the more expensive parts one decides to invest more upon the double demand uncertainty. This is mainly due to the fact that the costs for investing in information are relatively low. However, when we consider the potential benefit as a percentage, there are no large differences between the different instances. An interesting observations is however that S1∗ in many cases is close or equal to S2max , especially for scenarios with the largest double demand uncertainty. It is also clear that the length of the periods influence the benefit, having a shorter second period makes the relative benefit smaller. This is mainly due to the fact that the time one can benefit from more accurate demand rate information is shorter. In the instances where T1 = 52, the first period optimal base stock levels are sometimes larger than when T1 = 26. When the first period is relatively small, one allows for a higher risk of having a stockout compared to when the first period is larger. Also the costs w1 and w2 have a similar relationship, as a larger w1 results in smaller values of S1∗ since it involves higher obsolescence costs at the end of the first period. When a company would thus have to decide upon parts for which it is most interesting to invest in information, one should aim for the most expensive parts that have a large amount of uncertainty. For these parts, the benefit of having more reliable demand rate information is likely to be the largest.

3.5

Conclusions

In this chapter we consider a problem companies may face at the start of an introduction of the capital good. The company has unreliable estimates of the demand rate,

3.5. CONCLUSIONS

Table 3.1: Numerical experiment to investigate benefit of more reliable demand rate information Inst 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

T1 26

T2 260

c 5000

h 20

w1 1000

w2 4000

p1 = p2 25000

50000

500

4500

25000

50000

200

10000

40000

25000

50000

5000

45000

25000

50000

234

5000

1000

4000

25000

50000 500

4500 25000

50000

200

10000

40000

25000

50000

5000

45000

25000

50000

a1 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2

b1 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3 0.5 0.4 0.3

V2 (0) 64591 60571 54268 70630 62820 59932 67591 63188 56768 73856 65820 62432 497503 447137 423094 541110 505010 457116 517503 467137 443094 566110 525010 477116 64591 60571 54268 70630 62820 59932 67591 63188 56768 73856 65820 62432 497503 447137 423094 541110 505010 457116 517503 467137 443094 566110 523691 477116

V1 (∅) 58792 56770 54268 62161 60744 59067 60691 59062 56768 64277 63244 61707 426768 426530 423094 464978 463104 457116 440657 443530 440546 478978 481437 477116 59364 57313 54268 63112 60902 59233 61198 59313 56768 64862 63402 61733 433302 428591 423000 470191 467828 457116 446404 445591 439667 484191 483175 477116

I∗ 3 2 0 3 1 1 2 2 0 3 1 1 8 4 0 4 2 0 8 4 2 4 2 0 2 2 0 3 1 1 2 2 0 3 1 1 3 4 2 4 2 0 7 4 2 4 2 0

S1∗ 6 5 5 6 6 5 6 6 5 7 6 6 4 4 4 5 4 4 4 4 4 5 4 4 6 6 5 7 6 6 6 6 5 7 6 6 4 4 4 5 4 4 5 4 4 5 5 4

S2min 3 4 5 3 5 5 3 4 5 3 5 5 1 3 4 2 3 4 1 3 3 2 3 4 3 4 5 3 5 5 3 4 5 3 5 5 2 3 3 2 3 4 1 3 3 2 3 4

S2max 7 6 5 7 6 6 7 6 5 7 6 6 5 5 4 6 5 4 5 5 4 6 5 4 7 6 5 7 6 6 7 6 5 7 6 6 5 5 4 6 5 4 5 5 4 6 5 4

V2 (∞) 54086 54675 54253 57673 58167 57996 55834 56922 56757 59677 60485 60636 417070 421394 420714 456302 459775 456757 430878 437364 438306 470198 478769 477003 54812 55205 54263 58734 58465 58210 56542 57163 56759 60393 60765 60630 424114 423214 420307 461914 464894 456999 436920 438943 437600 475610 480741 477069

B max 16.26% 9.73% 0.03% 18.34% 7.41% 3.23% 17.39% 9.92% 0.02% 19.20% 8.11% 2.88% 16.17% 5.76% 0.56% 15.67% 8.96% 0.08% 16.74% 6.37% 1.08% 16.94% 8.81% 0.02% 15.14% 8.86% 0.01% 16.84% 6.93% 2.87% 16.35% 9.54% 0.02% 18.23% 7.68% 2.89% 14.75% 5.35% 0.66% 14.64% 7.94% 0.03% 15.57% 6.04% 1.24% 15.99% 8.43% 0.01%

B avg 8.98% 6.28% 0.00% 11.99% 3.30% 1.44% 10.21% 6.53% 0.00% 12.97% 3.91% 1.16% 14.22% 4.61% 0.00% 14.07% 8.30% 0.00% 14.85% 5.05% 0.58% 15.39% 8.30% 0.00% 8.09% 5.38% 0.00% 10.64% 3.05% 1.17% 9.46% 6.13% 0.00% 12.18% 3.67% 1.12% 12.90% 4.15% 0.02% 13.11% 7.36% 0.00% 13.74% 4.61% 0.77% 14.47% 7.97% 0.00%

3. MULTI-PERIOD SPARE PARTS PLANNING

and has to make spare parts stocking decisions based on this limited and unreliable forecast. We consider the case where the company is able to invest in information by having engineers perform additional analysis. When this information becomes available, which is considered the start of the second period, the company is able to adjust the base stock level which has been set at the start of the first period based on the new information. At the end of the second period, when the parts are no longer required, or at the start of the second period when the base stock level is reduced, parts are being sold back to the supplier. These costs are generally higher at the end of the second period compared to the first period as the parts also have become less useful for the supplier at this time. For the analysis, we model the problem as a three-stage dynamic program. In the first stage one decides upon how much to invest in more reliable information. The more time a company invest in information, the more reliable the demand rate information at the start of the second period. Once the company has decided upon the amount of time that should be invested on more reliable demand rate information, one has to decide upon the initial base stock level at the second stage. Finally in the third stage, based on the new demand rate information and the current base stock level, the company decides on the base stock level for the second period. We first derive analytical results for the third stage, where we show that the costs function is convex in the second period base stock level, and hence we can find an expression for the optimal second period base stock level. Also for the second stage, we are able to derive analytical results where we show that the costs are convex in the first period base stock level. As for the investment level we cannot show convexity, we use enumeration to find the optimal investment level. Based on these results, we perform numerical experiments to show the impact of the different parameters on the potential benefits of being able to invest in information. We find that investing in information is most beneficial when there is a sufficient amount of uncertainty, and the parts are relatively expensive. We modeled the uncertainty of the distribution using a Uniform distribution. A downfall of the Uniform distribution is that for our numerical experiments there is a limit on the maximum coefficient of variation one can model. When we would have been able to start with higher values of the coefficient of variation, larger differences are to be expected in terms of the benefit of being able to invest in more reliable information. Finally, we want to make a remark that for the numerical experiments it is possible

3.5. CONCLUSIONS

to also consider different distributions and apply a similar approach as we have done throughout this chapter. In this case, the intervals can be constructed in a similar way, except that the probability to obtain a certain interval should be corresponding to probability distribution of the original double demand uncertainty. One can then take the probability distribution within this interval, and normalize this such that the probabilities add up to one.

The impact of an emergency warehouse in a two-echelon spare parts network 4.1

Introduction

In this Chapter, motivated by the examples of ASML and NS mentioned in Chapter 1, we consider a two-echelon network. This network consists of local warehouses where demand arrives, a central warehouse and an emergency warehouse, which allows for the possibility to stock spare parts which are only sent to the customers in the case of an emergency request. The emergency warehouse does not necessarily have to be the fastest option, as this may depend on the structure of the network. Therefore, when a demand arrives at a local warehouse that does not have stock on hand, a preference list is followed, which states the order at which warehouses are requested to deliver the spare part, including both local warehouses, as well as the emergency warehouse. This preference list is given and known, as in practice companies determine this preference list based on the speed at which these locations can deliver the parts. When none of these warehouses can deliver the part, the part is obtained from the supplier directly. Our goal is to give an answer to the following research objectives: Research Objective 4 Develop a procedure to evaluate the performance, in terms of expected waiting time for a part, for a problem setting with multiple local warehouses 63

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

and an emergency warehouse in a two-echelon network with lateral and emergency shipments that can be done in any order. Research Objective 5 Investigate the benefit of an emergency warehouse. By giving an answer to these research objectives, this chapter contributes to the literature on spare parts inventory control in multi-echelon networks. This literature has a long and rich history. For a two-echelon network managed by a base stock policy without lateral and emergency shipments, Sherbrooke (1968) introduces the METRIC (Multi-Echelon Technique for Repairable Item Control) approach to estimate the expected number of backorders at the local warehouses. Sherbrooke (1968) approximates the replenishment lead times for the local warehouses by independent and deterministic lead times. This method has been developed further by Graves (1985), where a negative binomial distribution is fitted to the first two moments of the pipeline stock. This approximation was shown to give more accurate results for the expected number of backorders than the results of the METRIC approach. Gravesâ&#x20AC;&#x2122; method has been generalized to multi-indenture systems later on by Rustenburg et al. (2004). Muckstadt and Thomas (1980) provide an approximate evaluation procedure and focus on the optimization of the base stock levels by comparing centralized and decentralized decision making. Besides the evaluation, Sherbrooke (1968) develops a heuristic optimization method to set the base stock levels to minimize the total holding costs while keeping the total expected number of backorders under the target for the whole system. For a similar problem, but without a constraint per local warehouse, Wong et al. (2007) develop multiple heuristics. Andersson and Melchiors (2001) consider a two-echelon system where there are no backorders at the local warehouses, but demand is lost instead. They develop an accurate heuristic to determine a cost-effective base stock policy, based on the METRIC approximation. Muckstadt and Thomas (1980) extend the work of Sherbrooke (1968) by considering emergency shipments. They introduce a heuristic optimization of the base stock levels in order to compare the benefit of centralized decision making over decentralized decision making. Ozkan et al. (2015) develop a fast approximate evaluation procedure that was shown to outperform the approximation of Muckstadt and Thomas (1980). They use the observation that emergency requests are treated differently by the central warehouse than replenishment requests and improve the accuracy of their approximation by taking this into account explicitly. In this chapter we make use of this idea presented at Ozkan et al. (2015) for our approximation. Although all above methods consider two-echelon networks and thus take into account the impact of the stock of the central warehouse on the

4.1. INTRODUCTION

performance of the local warehouses, these methods do not take the use of lateral and/or emergency shipments into consideration. For two-echelon networks where emergency shipments are sent only from the supplier to the local warehouses, and the central warehouse is faced by both replenishment requests as well as direct customer demand, AxsĂ¤ter et al. (2004) develops a heuristic method using a critical level at the warehouse to differentiate the demand streams based on their priorities. However, the use of lateral transshipments is not taken into consideration, thus this model is only suitable when local warehouses are located far apart from each other. For a multi-echelon network with lateral transshipments, AxsĂ¤ter (1990) develops an approximate evaluation based on the METRIC approach to estimate the replenishment lead time from the central warehouse to the local warehouses. The local warehouses are divided in groups, between which the use of lateral transshipments is possible. However, the use of emergency shipments is not possible. Grahovac and Chakravarty (2001); Boucherie et al. (2017) consider a two-echelon network, where there is a possibility of lateral transshipments from other local warehouses as well as an emergency shipment from the central warehouse, similar to the network we consider in this chapter. However, unlike many other papers with lateral and/or emergency shipments, the central warehouse is checked first whenever there is no stock at the local warehouse. Only if this is not possible, they look for a lateral transshipments at other local warehouses. This may be the case when distances between local warehouses are large or similar with respect to the distance to the central warehouse. Moreover, an emergency shipment from the supplier is not considered. Alfredsson and Verrijdt (1999) consider a two-echelon network with both lateral and emergency shipments. They assume the use of full pooling between local warehouses, thus representing the case that a lateral transshipment can be provided from each local warehouse to any other local warehouse, and if this is not possible they make use of an emergency shipment from the central warehouse or supplier. Making use of the full pooling assumption, they aggregate the total demand of the local warehouses in order to calculate the fraction of demand satisfied by emergency shipments by modeling the problem as a two-dimensional Markov process. They numerically compute the limiting distribution of the Markov process, which is shown to give accurate results, and even exact in the case that all local warehouses are identical. Unfortunately, this method is very time-consuming, even for small problem instances. Moreover, local warehouses in practice are generally not identical. In this chapter, we not only tackle the problem of complexity but also allow for a more general net-

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

work structure, including the use of an emergency warehouse and flexibility with respect to the preferred sequence at which demand is handled in case of a stockout at a local warehouse. The above mentioned papers are thus special cases of our problem. Another stream of literature is the stream about single-echelon, multi-location networks with lateral and emergency shipments. Kutanoglu (2008) propose an iterative scheme for the evaluation in the case there is full pooling. As key performance indicator for the company, they use time-based service constraints. Kutanoglu and Mahajan (2009) look at a similar network, where they allow for prioritizing the warehouses for the lateral transshipments, and propose an implicit enumeration-based method to find the minimum-cost stock levels. Wong et al. (2005) develop a heuristic optimization method which is built on exact evaluations via Markov processes for a similar network. As the central warehouse is assumed to have ample stock, the emergency shipment can always be provided by the central warehouse. A similar system is studied by Kranenburg and Van Houtum (2009), where it is also possible to have a form of partial pooling. They introduce main warehouses, which are local warehouses which can provide other local warehouse by means of a lateral transshipment, and regular warehouses which do not provide lateral transshipments to other warehouses. They introduce an approximate evaluation procedure based on modeling lateral transshipments as Poisson overflow processes, similar to AxsĂ¤ter (1990). In order to minimize the holding costs subject to a waiting time constraint, they develop an efficient greedy heuristic. This work has been implemented at ASML and has shown that just a few main warehouses can be sufficient to get most of the gains of lateral transshipments. For a more general system with lateral and emergency shipments with customers requesting spare parts, van Wijk et al. (2012) develop an approximate heuristic where it is possible to define a preference list at which local warehouses are consulted for each customer. The method of Reijnen et al. (2009) is a special case of the evaluation of van Wijk et al. (2012). The use of this preference list is similar to our case. For a complete overview of studies related to lateral transshipments, see Paterson et al. (2011). Although all of the above literature consider a certain form of lateral transshipments, they do not all allow for the same amount of flexibility, or take the central warehouse into consideration. Finally, we look at the stream of literature that looks at the use of dedicated stock for emergency shipments, which has similarities to the use of an emergency warehouse. AxsĂ¤ter et al. (2013) consider a single-echelon, multi-location network where they introduce an emergency warehouse. This support warehouse is used to provide the part whenever a local warehouse runs out of stock, but does not directly face demand.

4.1. INTRODUCTION

However, the central warehouse is not taken into consideration. Howard et al. (2015) look at a similar network, including a central warehouse with ample capacity. The customers can get the part from the support warehouse or the central warehouse in the case the local warehouse is out of stock when a request occurs. Making use of the pipeline information they achieve cost-efficient policies for requesting emergency shipments. van Wijk et al. (2013) study a multi-location inventory problem with a so-called quick response warehouse and derive the optimal policy for when to make use of the quick response stock. Although these papers consider the use of an emergency warehouse, the use of their emergency warehouse is different to this chapter. These emergency warehouses are located in the field and may not be able to serve all warehouses whereas our emergency warehouse is located nearby the central warehouse at a more central location and is always able to deliver any other warehouse. Moreover, in these papers the finite stock at the central warehouse and/or the use of lateral transshipment between local warehouses are not considered. We contribute to the literature by introducing the use of an emergency warehouse, which is a warehouse dedicated for fast emergency shipments without having its own external demand, for a two-echelon network with lateral and emergency shipments. We consider different networks for which certain sequences of lateral transshipments and/or an emergency shipment from the emergency warehouse can be applied. We show that cost savings of over 30 percent may occur compared to the case where we do not have an emergency warehouse (see Section 4.5). Next to this, we provide an accurate and efficient evaluation procedure for two-echelon networks that can make use of both lateral and emergency shipments as well as take this emergency warehouse into consideration. Due to the complexity of the network and the evaluation of the performance measures, we resort to an approximate evaluation procedure instead of exact analysis. We test our approximate evaluation procedure using simulation, and show that our approximation is accurate for the instances that we are mainly interested in, namely instances with relatively high system availability levels. Finally we allow a very general structure for the use of emergency and lateral transshipments, as described above. The variety of structures as assumed in Grahovac and Chakravarty (2001), Muckstadt and Thomas (1980) and Alfredsson and Verrijdt (1999) are all special cases of our general structure. As our approximate evaluation procedure is flexible, fast, and efficient, this work can also be easily applied in practice. The remainder of this chapter is organized as follows. In Section 4.2, we give a for-

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

mal description of the model. In Section 4.3 we present our approximate evaluation procedure. This procedure consists of two parts, a local evaluation procedure and a central evaluation procedure. We perform a numerical study on a variety of different scenarios in Section 4.4 where we compare our evaluation procedure with simulation in order to measure its accuracy for different settings. Then in Section 4.5 we apply a smart enumeration procedure to obtain insights on the usefulness of the emergency in warehouse. Finally, we conclude our chapter in Section 4.6.

4.2

Model Description

We consider a single-item, two-echelon inventory model with a central warehouse (CW), an emergency warehouse, and one or more local warehouses (LW). Let J = {1, 2, . . . , |J|} be the set of local warehouses with |J|≥ 1. The index of the emergency warehouse is denoted by |J|+1. The emergency warehouse, a warehouse without its own external demand dedicated to ship spare parts as fast as possible when requested, is located nearby the central warehouse at a strategic location such that spare parts can be put on transport by several different transport modes. It is also possible that the emergency warehouse is a separate warehouse within the central warehouse that can handle such an emergency request much faster compared to the central warehouse. Demand at each local warehouse is assumed to arrive according to a Poisson process with a constant rate µj , j ∈ J. The emergency warehouse (EW) does not face direct customer demand. The inventory in the network is controlled by a base stock policy. Hence, if a warehouse (a local warehouse, the emergency warehouse or the central warehouse) fulfills a demand, the inventory levels drop by one unit, and immediately a replenishment order is placed. In spare parts inventory systems, availability of (critical) spare parts is of crucial importance, as lack of spare parts might result in system down time. The cost associated with these down times (contractually or otherwise) typically overweigh the economies of scale cost benefit of batching in shipment or ordering (for the same SKU), as the risk of downtime increases while waiting to batch with lower stock. Consequently, in spare part inventory systems it is very common to use a base stock policy at all warehouses, both in theory and in practice (see e.g. Muckstadt, 2005; Sherbrooke, 2004; Van Houtum and Kranenburg, 2015). Also at ASML, such a base stock policy is applied. Let K = J ∪ {|J|+1}. We denote the base stock level as Sk , k ∈ K ∪ {0}, where S = (S0 , S1 , S2 , . . . , S|J|+1 ). Whenever there is a request at a local warehouse and there is no stock on hand at that local warehouse, there is a fixed sequence at which other local warehouses and

4.2. MODEL DESCRIPTION

the emergency warehouse are checked to see if they can deliver the spare part. This can be any sequence, although it is commonly based upon the time it takes to get the part from that warehouse. Let vj , j ∈ J, be the array consisting of the sequence at which warehouses are checked for an emergency shipment (for convenience of notation both lateral transshipments and emergency shipments are called emergency shipments from here on), with vj (i) the ith warehouse in the array (i = 1, 2, . . . , pj ), with pj ≥ 1 representing the maximum number of warehouses that is checked in case of a stockout. The warehouses in this sequence can consist of both local warehouses as well as the emergency warehouse. If the warehouse where the demand arrives does not have stock, it will check warehouses vj (1) up and to vj (pj ). When none of the local warehouses or the emergency warehouse have stock, the central warehouse, denoted by index 0, is checked for an emergency shipment. If the central warehouse has stock on hand it will supply the spare part and immediately request a replenishment at the supplier. When there is also no stock at the central warehouse, an emergency shipment will be requested from the supplier, denoted by index −1. A graphical representation of the order at which the possible shipment options are used when demand arrives at local warehouse 1, given the example sequence v1 = (1, |J|+1, |J|) is presented in Figure 4.1.

Figure 4.1: Example order of demand fullfillment when demand arrives at warehouse 1, with v1 = (1, |J|+1, |J|)

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

We assume that the the supplier has ample stock, and the supply lead time from the supplier to the central warehouse is denoted by t0 > 0. The deterministic transportation time for a replenishment from the central warehouse to local warehouse j is denoted by tj . Notice that because the emergency warehouse is in close proximity of the central warehouse, the replenishment lead time from the central warehouse to the emergency warehouse is assumed to be zero (t|J|+1 = 0). Let us introduce the set Q = K ∪ {−1, 0}. As we are mainly interested in the expected time each customer has to wait for a spare part, and the costs involved to provide a part, which depends on where the spare part is delivered from, we need to calculate θq,j , the fraction of demand of local warehouse j, j ∈ J, that is served from location q, q ∈ Q. Note that by definition for each local warehouse j ∈ J it holds P that q∈Q θq,j = 1. Whenever warehouse q ∈ Q is used to fulfill the demand which arrives at local warehouse j ∈ J, there are also costs involved for the emergency shipment and waiting time from location q to j, denoted by cem q,j . These costs consist of two different cost factors. First of all, this includes the costs for fast transportation and handling, denoted by cship q,j . Secondly, and most important, this includes the costs for the additional standstill as a result of not having the part available at local warehouse j. For every hour the machine is not operating there are costs involved denoted by cdown . j em The time it takes in hours to obtain a part from warehouse q to j is denoted by tq,j . As a result, the total costs involved when a part is delivered from warehouse q to warehouse j is calculated as follows: ship down em cem tq,j q,j = cq,j + cj

Whenever the demand can be fulfilled by the local warehouse where the demand has em arrived, we assume cem j,j = 0. As such, cq,j denotes the additional emergency costs compared to fulfilling the demand from the local warehouse directly. Next to the emergency costs, for each demand that is sent to the customer from warehouse q there are costs to replenish warehouse q denoted by crep q . In the case of the supplier, these costs are zero. In the case of the central warehouse, it involves costs for replenishing the central warehouse. In the case of a local or emergency warehouse, it involves both the costs of a replenishment to the central warehouse as well as the local or emergency warehouse. The total costs are then denoted as follows:   X X X rep  Ĉ(S) = hSk + µj  θq,j (cem q,j + cq ) , k∈K∪{0}

j∈J

q∈Q

where h represents the holding costs rate for each spare part kept on stock. Note that

4.3. EVALUATION PROCEDURE

for the holding costs we assume that the costs of the pipeline inventory are included, which are the costs over the time between placing the order at the supplier and receiving the actual part at the central warehouse and the time it takes to replenish a local warehouse from the central warehouse. We apply the same holding costs rate for every warehouse as value added from transport is minor. When holding costs rate differ between warehouses, for instance due to value added, it is possible to make the holding costs rate location dependent by changing h to hk in the formula above. If there would be no lateral or emergency shipments, and thus all demand is satisfied by the local warehouses where demand arrived directly, one would observe the following replenishment costs: C rep =

µj crep j

j∈J

These costs C rep form a constant factor, i.e. they are independent of the basestock policy. Hence, we can substract them from the total costs function Ĉ(S) and then obtain the following new costs function C(S): C(S) = Ĉ(S) − C rep  =

X k∈K∪{0}

hSk +

X j∈J

µj 

 X

rep  θq,j (cem − q,j + cq )

q∈Q

µj crep j

X k∈K∪{0}

hSk +

X j∈J

µj 

θq,j (4.1)

q∈Q

j∈J

 =

 X

cq,j θq,j  ,

q∈Q

− crep where cq,j = crep + cem q q,j . The overall goal is to minimize the total costs C(S). j Using the emergency warehouse may also bring additional costs in practice, such as costs for the additional storage or employee costs to handle the separate storage area, which we do not take into consideration. As a company has to manage multiple parts, the costs corresponding to operating such an emergency warehouse are difficult to translate to costs for a single part as these costs are generally shared over all parts. As a result of this assumption, the costs of a solution when being able to use an emergency warehouse can never exceed the costs of the solutions where one is not able to use an emergency warehouse.

4.3

Evaluation procedure

In this section we explain our approximate evaluation procedure to obtain the systems performance for given base stock levels. We avoid numerical evaluations of Markov

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

processes as in the procedure of Alfredsson and Verrijdt (1999), because that leads to too long computation times to be applicable in practice. Instead, we decouple all locations and incorporate their dependencies in an appropriate way. A fast performance evaluation is then obtained by iteratively analyzing individual locations and updating of the dependencies.

As the central warehouse has a finite stock, whenever the central warehouse runs out of stock and a replenishment request comes in, the central warehouse has to wait for a replenishment from the supplier before it can send the part to the local warehouse or emergency warehouse. By adding the average waiting time for a part at the central warehouse in the steady state, W0 , to the fixed transport time, we estimate the average replenishment lead time from the central warehouse as follows: treg k = t k + W0 ,

k â&#x2C6;&#x2C6; K.

(4.2)

To apply our approximate evaluation algorithm, we decouple the network into two parts as shown in Figure 4.2 making use of Equation (4.2). By decoupling the problem into two separate parts, which do still depend on each other, we obtain two smaller evaluation procedures which can be evaluated efficiently.

The Local Evaluation Procedure, which is further explained in Section 4.3.1, consists of the evaluation of the local warehouses and the emergency warehouse. For the evaluation we consider the emergency warehouse as another warehouse without its own external demand process, which can always be checked for an emergency shipment by all other local warehouses. Given the average replenishment lead time from the central warehouse, treg k , we evaluate the performance of the local warehouses and the emergency warehouse. The output of this evaluation procedure is used as an input for the Central Evaluation Procedure.

The Central Evaluation Procedure, which is further explained in Section 4.3.2, consists of the evaluation of the central warehouse. For this procedure we are interested in the average waiting time, which is used as an input for the Local Evaluation Procedure and depends on the ratio between replenishment and emergency requests. As these two evaluation procedures depend on the output of the other evaluation procedure we propose an iterative procedure in Section 4.3.3, where we combine the two evaluation procedures into a single evaluation procedure.

4.3. EVALUATION PROCEDURE

Figure 4.2: Network decoupling and interaction

4.3.1

Local Evaluation Procedure

For the Local Evaluation Procedure we adapt the evaluation procedure of Reijnen et al. (2009) to include the use of the emergency warehouse. The approach of Reijnen et al. (2009) is shown to be efficient and usable for large problem instances (see also van Wijk et al., 2012). Demand arrives at the local warehouses only with rate Mj,j = Âľj , j â&#x2C6;&#x2C6; J. The local warehouses try to deliver the part from stock and if not possible, they try to obtain the part from another warehouse. Firstly, warehouse j is checked. If warehouse j does not have stock this results in so-called overflow demand from warehouse j to warehouse vj (1). Let us now introduce the following notation which we use in our Local Evaluation Procedure:

Mk,j : Mk : βk :

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

The demand rate for warehouse k, k ∈ K, originating from local warehouse j, j ∈ J Total demand rate for warehouse k, k ∈ K, including P overflow demand (= j∈J Mk,j ) The fraction of total demand that warehouse k ∈ K is able to directly deliver from stock

For the evaluation, we assume that the warehouses are independent and we know βj , then an approximation of the average demand rate for warehouse vj (1), originating from warehouse j, is: Mvj (1),j = (1 − βj )Mj,j . (4.3) Using the fraction of demand that is satisfied by warehouse vj (1), βvj (1) , the demand warehouse vj (2) faces, originating from warehouse j, is on average Mvj (2),j = (1 − βvj (1) )Mvj (1),j . For 2 ≤ i ≤ pj these demand rates can be expressed as follows: Mvj (i),j = (1 − βvj (i−1) )Mvj (i−1),j

(4.4)

As the emergency warehouse can be checked by any local warehouse, the emergency warehouse is included in every sequence vj . See Figure 4.3 for a graphical representation of the overflow of demand where the emergency warehouse is at the end of the sequence. We assume that the overflow demand stream follows a Poisson process, then the total demand at warehouse k also follows a Poisson process with rate Mk . Under this approximation we can evaluate each location independently of each other. In practice, there is however a correlation between the overflow demand streams. As we have emergency shipments, the number of parts in the pipeline is at most Sk for local warehouse k. As a result the behavior of the number of parts on order for location k is as the number of jobs in an M |G|c|c queue with c = Sk parallel servers. This queue is also called an Erlang loss system. The fill rate can thus be calculated using known results from the Erlang loss system. The Erlang loss probability is given by: ρc /c! , L(c, ρ) = Pc x x=0 ρ /x! where ρ represents the offered load. The fill rate can thus be calculated as 1 minus the fraction of time that all servers are occupied, with Sk representing the number of servers, and treg k Mk , the demand during the leadtime arriving at warehouse k, representing the offered load: βk = 1 − L(Sk , treg k Mk )

k ∈ K,

(4.5)

4.3. EVALUATION PROCEDURE

Figure 4.3: Illustration of overflow demand

Using the fill rate and the demand rate, we calculate the fraction of demand for warehouse j that is served by warehouse k: Î¸k,j =

Î˛k Mk,j Âľj

k â&#x2C6;&#x2C6; K, j â&#x2C6;&#x2C6; J.

To determine the values of Mk , Mk,j and Î˛k we use an iterative procedure. For the first iteration we assume there is no overflow demand at all, thus Mj,j = Âľj and Mvj (i),j = 0 for 1 â&#x2030;¤ i â&#x2030;¤ pj , j â&#x2C6;&#x2C6; J. For each warehouse k â&#x2C6;&#x2C6; K we determine Mk , and calculate Î˛k using Equation (4.5), where treg is given. After this step we make an k improved estimation of the demand using Equation (4.4). These steps are repeated until Mk does not change more than , with small. Although we cannot proof these values converge, they did converge in all experiments conducted. The Local Evaluation Procedure is described in Algorithm 3.

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

Algorithm 3 Local Evaluation Procedure Step 1. Initialization Mj,j := Âľj Mk,j := 0 P Mk := jâ&#x2C6;&#x2C6;J Mk,j Î˛k := 1 â&#x2C6;&#x2019; L(Sk , treg k Mk )

â&#x2C6;&#x20AC;j â&#x2C6;&#x2C6; J â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K, j â&#x2C6;&#x2C6; J, k 6= j â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K

Step 2. Compute Î˛k , and Mk until each Mk does not change more than Mvj (i),j = (1 â&#x2C6;&#x2019; Î˛vj (iâ&#x2C6;&#x2019;1) )Mvj (iâ&#x2C6;&#x2019;1),j â&#x2C6;&#x20AC;j â&#x2C6;&#x2C6; J and 1 â&#x2030;¤ i â&#x2030;¤ pj P Mk = jâ&#x2C6;&#x2C6;J Mk,j â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K reg Î˛k = 1 â&#x2C6;&#x2019; L(Sk , tk Mk ) â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K Step 3. Calculate the performance Î˛ M k â&#x2C6;&#x2C6; K, j â&#x2C6;&#x2C6; J. Î¸k,j = k Âľjk,j

4.3.2

Central Evaluation Procedure

In this section we introduce the Central Evaluation Procedure which evaluates the performance of the central warehouse. We estimate the average waiting time at the central warehouse, W0 , which we need in order to estimate the average replenishment lead times to the local warehouses and the emergency warehouse. We split this evaluation procedure into two cases; the case where we make use of the emergency warehouse (S|J|+1 > 0), and the case where we do not make use of the emergency warehouse (S|J|+1 = 0). If we do not make use of the emergency warehouse, i.e. if we have a base stock level of zero at the emergency warehouse, then two different demand streams arrive at the central warehouse; replenishment requests and emergency orders of demand that could not be satisfied by local warehouses. If we make use of the emergency warehouse, then the central warehouse only sees replenishment requests. This is due to the fact that the transportation time to the emergency warehouse is zero, hence; if the emergency warehouse does not have stock, neither does the central warehouse. In both cases we assume we know the fill rates of each local warehouse and the emergency warehouse as well as the demand rates for each warehouse. Central Evaluation when the emergency warehouse is not used In the case we do not have an emergency warehouse, or do not decide to make use of it, we make use of the idea of Ozkan et al. (2015). However, as we have lateral

4.3. EVALUATION PROCEDURE

transshipments between the local warehouses in our model, we have to adapt the overall method of Ozkan et al. (2015). As long as the central warehouse has stock on hand, all demand that arrives at the local warehouses is satisfied by the central warehouse, either by a replenishment or by providing an emergency shipment. Thus the demand rate of the central warehouse as long as it has stock on hand is M0 =

µj .

(4.6)

j∈J

When the central warehouse does not have stock on hand the replenishment requests still arrive at the central warehouse, as these requests are backordered. As the emergency requests will not be backordered but supplied by the supplier directly instead, the central warehouse does not observe these requests. As a result, the total demand rate for the central warehouse when it does not have stock on hand is equal to the demand rate for replenishment requests, which is the rate at which demand is satisfied by the local warehouses: X M00 = M k βk . (4.7) k∈K

We model the inventory level at the central warehouse as a birth-death process, i.e. a continuous-time Markov process with states x ≤ S0 . We pretend for the analysis purposes that the leadtime t0 of the supplier is exponential with the same mean, i.e. by exponential times with rate µ0 = t10 . Moreover, the possible states for the central warehouse are truncated by setting the maximum number of backorders to P S = k∈K Sk as there can never be more backorders at the central warehouse. The steady state probabilities then satisfy the following equations: ( M0 0 −x)µ0 πx+1 , −S ≤ x < 0 πx = (S0M 0 0 ≤ x < S0 (S0 −x)µ0 πx+1 , By expressing everything as a function of πS0 , πS0 follows from the normalization. The mean number of backorders and the mean waiting time (using Little’s law) are then: −1 X B0 = (−x)πx x=−S

W0 =

B0 M00

Given this mean waiting time, we update the replenishment lead time for each warehouse k ∈ K by Equation (4.2). The fraction of demand that is fulfilled by an

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

emergency shipment from the central warehouse is approximated as follows: θ0,j =

β0 (1 − βvj (pj ) )Mvj (pj ),j , µj

where we look at the fraction of demand that is not delivered by the last warehouse in the sequence of warehouse j, and where β0 represents the fill rate at the central warehouse, which is calculated as follows: β0 =

S0 X

πx .

x=1

As demand is satisfied by either one of the local warehouses, the emergency warehouse, the central warehouse or the supplier through the use of an emergency shipment, we Ppj know that i=1 θvj (i),j + θ0,j + θ−1,j = 1, for all j ∈ J. We then get the following expression for the fraction of demand that is satisfied by an emergency shipment from the supplier: pj X θ−1,j = 1 − θ0,j − θvj (i),j (4.8) i=1

Central Evaluation when using the emergency warehouse When we make use of the emergency warehouse, there is a small change due to the relation between the central warehouse and the emergency warehouse. Because the transport time from the central warehouse to the emergency warehouse is zero, we know that as long as the central warehouse has stock, so does the emergency warehouse. As a result we know that as long as the central warehouse has stock on hand, all demand arriving at the local warehouses is provided by either the local warehouses and/or the emergency warehouse. Therefore we know the demand rate for the central warehouse as long as it has stock on hand follows Equation (4.6). Whenever the central warehouse does not have stock on hand, only the replenishment requests arrive at the central warehouse as the emergency requests are directly delivered by the supplier. In this case the demand rate is equal to Equation (4.7). We thus make use of the same equations as in Section 4.3.2 to calculate the expected waiting time, W0 . Because we know that whenever the emergency warehouse does not have stock on hand, neither does the central warehouse, we know that the central warehouse never delivers the part by an emergency request to the local warehouses. As a result the fraction of demand satisfied by the central warehouse by an emergency shipment is

4.3. EVALUATION PROCEDURE

as follows: θ0,j = 0. After this step we use Equation (4.8) to estimate the fraction of total demand delivered by the supplier. Central Evaluation algorithm A complete overview of the Central Evaluation Procedure is described in Algorithm 4. In this procedure, the calculation of θ0,j , and θ−1,j is omitted, because that calculation is only needed at the end of the overall evaluation procedure. Algorithm 4 Central Evaluation Procedure Step 1. Initialization W0 := 0 P M0 := j∈J µj P S := k∈K Sk Step 2. Compute W0 P M00 := k∈K Mk βk  M00  , −S̄ ≤ x < 0  − x)µ0 πx := (S0 M 0   , 0 ≤ x < S0 (S0 − x)µ0 P−1 B0 := x=−S̄ (−x)πx B0 W0 := M 0 0

Note that in the special case that no stock is kept at the central warehouse, thus S0 = 0, the central warehouse places an order at the supplier to replenish the warehouse as soon as a replenishment request arrives. As the supplier lead time, t0 , is fixed, the replenishment lead time is then as follows: treg k = tk + t0 , which is then no longer an approximation but exact. Therefore we use this expression in the case we keep no stock at the central warehouse.

4.3.3

Overall evaluation procedure

In this section we describe the overall evaluation procedure where we combine Algorithm 3 and 4 into a single algorithm. As each of the two evaluation procedures relies

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

on the output of the other procedure, we use an iterative procedure. We start by selecting a starting value of zero for the waiting time at the central warehouse and initializing the necessary variables. Then we apply the Local Evaluation procedure, followed by applying the central warehouse procedure. Then if the difference between the waiting time of central warehouse compared to the previous run, or the initialized value, is smaller than with small, the iterative procedure is finished and we only need to determine the fraction of demand that is satisfied by each location. Also here we are not able to proof convergence, although each experiment conducted by us the values converged.

The overall evaluation procedure is described in Algorithm 5.

Algorithm 5 Overall Evaluation Procedure Step 1. Initialization W0 := 0 P M0 := jâ&#x2C6;&#x2C6;J Âľj treg = t k + W0 k Mj,j := Âľj Mk,j := 0 P Mk := jâ&#x2C6;&#x2C6;J Mk,j Î˛k := 1 â&#x2C6;&#x2019; L(Sk , treg k Mk )

â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K â&#x2C6;&#x20AC;j â&#x2C6;&#x2C6; J â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K, j â&#x2C6;&#x2C6; J, k 6= j â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K

Step 2. Apply Step 2 of the Local Evaluation Procedure described in Algorithm 3

Step 3. Apply Step 2 of the Central Evaluation Procedure described in Algorithm 4

Step 4. Repeat Step 2 and Step 3 until W0 does not change more than Step 5. Finalization Î˛k Mk,j Âľj Î˛0 (1â&#x2C6;&#x2019;Î˛vj (pj ) )Mvj (pj ),j Âľj

Î¸k,j

If S|J|+1 = 0 :

Î¸0,j

If S|J|+1 > 0 :

Î¸0,j Î¸â&#x2C6;&#x2019;1,j

:= 0 Ppj := 1 â&#x2C6;&#x2019; Î¸0,j â&#x2C6;&#x2019; i=1 Î¸vj (i),j

â&#x2C6;&#x20AC;k â&#x2C6;&#x2C6; K, j â&#x2C6;&#x2C6; J â&#x2C6;&#x20AC;j â&#x2C6;&#x2C6; J â&#x2C6;&#x20AC;j â&#x2C6;&#x2C6; J â&#x2C6;&#x20AC;j â&#x2C6;&#x2C6; J

4.4. NUMERICAL RESULTS

Table 4.1: Detailed order for emergency shipments for |J|= 8 LW 1 2 3 4 5 6 7 8

4.4

No-Part {1,9,2,3,4} {2,9,3,4,1} {3,9,4,1,2} {4,9,1,2,3} {5,9,6,7,8} {6,9,7,8,5} {7,9,8,5,6} {8,9,5,6,7}

No-Full {1,9,2,3,4,5,6,7,8} {2,9,3,4,1,6,7,8,5} {3,9,4,1,2,7,8,5,6} {4,9,1,2,3,8,5,6,7} {5,9,6,7,8,1,2,3,4} {6,9,7,8,5,2,3,4,1} {7,9,8,5,6,3,4,1,2} {8,9,5,6,7,4,1,2,3}

Part-No {1,2,3,4,9} {2,3,4,1,9} {3,4,1,2,9} {4,1,2,3,9} {5,6,7,8,9} {6,7,8,5,9} {7,8,5,6,9} {8,5,6,7,9}

Part-Full {1,2,3,4,9,5,6,7,8} {2,3,4,1,9,6,7,8,5} {3,4,1,2,9,7,8,5,6} {4,1,2,3,9,8,5,6,7} {5,6,7,8,9,1,2,3,4} {6,7,8,5,9,2,3,4,1} {7,8,5,6,9,3,4,1,2} {8,5,6,7,9,4,1,2,3}

Full-No {1,2,3,4,5,6,7,8,9} {2,3,4,1,6,7,8,5,9} {3,4,1,2,7,8,5,6,9} {4,1,2,3,8,5,6,7,9} {5,6,7,8,1,2,3,4,9} {6,7,8,5,2,3,4,1,9} {7,8,5,6,3,4,1,2,9} {8,5,6,7,4,1,2,3,9}

Numerical results

In this section we test our approximate evaluation procedure on its accuracy by comparing the results with results obtained by simulation, which we consider to be the exact results. We consider 72 different instances where we look at networks with either 8 or 16 local warehouses. We consider two different demand rates per warehouse, a demand rate of 0.02 per week representing slow moving parts, and a demand rate of 0.1 per week representing faster moving parts. Moreover, we consider five different sequences at which the demand is fulfilled in case the local warehouse where demand arrives does not have stock on hand. First of all, we consider the sequence “No-No", where we do not have any lateral transshipments and the emergency warehouse is requested as a first resort when the local warehouse does not have any stock on hand. The second sequence we consider is the “No-Partial" sequence. In this case, first the emergency warehouse is checked and if this warehouse is not able to deliver, all local warehouses in the same region are checked for a lateral transshipment. For the sequence “No-Full" we also first look at the emergency warehouse and then for a lateral transshipment. In this case all other local warehouses can be checked for a lateral transshipment. Figure 4.4a represents the network structure with locations for which these three sequences makes sense. Next, we have the network structure as presented in 4.4b. One possible sequence for such a structure is the “Partial-No" structure, where we first check for lateral transshipment at the local warehouses in the same region and then consider the emergency warehouse. For the sequence “Partial-Full" we have a similar order, except that we also look at the other remaining local warehouses after checking the emergency warehouse as this might still be faster than getting a part from the supplier. Our last sequence “Full-No" represents the case where we first look at all local warehouses for a lateral transshipment and at the emergency warehouse only in case none of these

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

(a) No-No, No-Part, No-Full network structure

(b) Part-No, Part-Full network structure

Figure 4.4: Network structures which require a different sequence. Local warehouses are denoted by triangles and the central and emergency warehouse are denoted together by the star symbol

local warehouses are able to deliver the part. Figure 4.4c gives a possible network structure for which this sequence is appropriate. Table 4.1 gives the detailed order for these sequences for each warehouse in the case of 8 local warehouse, consisting of two regions. Note that for 16 warehouses the order is similar, as we then have two regions of 8 local warehouses instead of 4. Moreover, we consider different combinations of base stock levels, which altogether gives us a total of 72 instances. The simulation software used to obtain the exact results is Omnet++. For each instance we use a warm-up period of 50,000 years, followed by a period of 2,500,000 years. For each instance, we did 25 replications. The results for the instances with 8 local warehouses are presented in Table 4.2, and the results for the instances with 16 local warehouses are presented in Table 4.3, together with a 99 percent confidence interval. Because the emergency warehouse can be in between lateral transshipments, we split up the total fraction of demand satisfied by lateral transshipments into the fraction of lateral transshipments before consulting the emergency warehouse, αj1 , and the fraction of lateral transshipments after consulting the emergency warehouse, αj2 . Let e be the index denoting the emergency warehouse in the sequence vj , we then calculate αj1 and αj2 as follows:

4.4. NUMERICAL RESULTS

αj1

e−1 X

θvj (x),j

x=2

αj2

pj X

θvj (x),j

x=e+1

Note that in some sequences we do not have lateral transshipments or only before or after consulting the emergency warehouse. As αj2 or αj1 then do not exist, this is denoted by “N/A". From the tables we can observe that in general our evaluation procedure is accurate as the differences with the simulation results are very limited. Especially if we look at the higher fill-rates and/or higher demand rates, the evaluation procedure turns out to be more accurate. This is as expected, because our assumption that overflow demand also follows a Poisson distribution is more reasonable for these instances. However, it should be noted that especially for the case that the fill rate of the local warehouses are lower, in the range of 0 to 60 percent say, differences tend to become larger although this should not be a big issue as generally the service provided to the customers is on the higher range with expected fill rates over 90 percent in general due to the high downtime costs. Moreover, the computation time per instances is in the range of 1 to 70 milliseconds. For instances with more local warehouses, as well as for instances with higher demand rates, the run time increases. The number of iterations needed between the central and local evaluation procedure has a big influence on the run time. We found that the number of iterations needed is between 1 and 16. We can conclude that our evaluation procedure is accurate and fast enough if we would apply it on real life symmetric instances. Next to the symmetric instances, we also consider a smaller set of asymmetric instances, where the demand rate and base stock levels differ per local warehouse to analyse whether our approximation is still accurate. Table 4.4 gives an overview of the demand rates and base stock levels, considering a scenario with 8 warehouses. We use the same sequences as provided in Table 4.1 and the same lead times as we did for the symmetric instances. The results for the asymmetric instances are shown in Table 4.5. The results show the average absolute error, measured by the difference between the outcome of the approximation and simulation. Based on these results, we can conclude that the evaluation method remains accurate.

Inst.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

0.1

0.02

µj

1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1

S|J|+1 No-No No-Partial No-Full Partial-No Partial-Full Full-No No-No No-Partial No-Full Partial-No Partial-Full Full-No No-No No-Partial No-Full Partial-No Partial-Full Full-No No-No No-Partial No-Full Partial-No Partial-Full Full-No No-No No-Partial No-Full Partial-No Partial-Full Full-No No-No No-Partial No-Full Partial-No Partial-Full Full-No

Sequence

βj Appr. 0.7511 0.7198 0.7190 0.6648 0.6646 0.6604 0.8118 0.8006 0.8005 0.7644 0.7644 0.7637 0.9587 0.9584 0.9584 0.9554 0.9554 0.9554 0.8295 0.7573 0.7554 0.7206 0.7203 0.7167 0.8977 0.8916 0.8916 0.8650 0.8650 0.8649 0.7659 0.6361 0.6222 0.5781 0.5718 0.5523 Sim. 0.7571 0.7187 0.7145 0.6738 0.6709 0.6633 0.8223 0.8011 0.7993 0.7689 0.7677 0.7641 0.9568 0.9557 0.9557 0.9516 0.9516 0.9516 0.8290 0.7452 0.7326 0.7125 0.7022 0.6932 0.8964 0.8782 0.8769 0.8461 0.8453 0.8421 0.7685 0.6422 0.6110 0.6012 0.5749 0.5603 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.0009 0.0008 0.0007 0.0007 0.0007 0.0008 0.0007 0.0007 0.0007 0.0006 0.0007 0.0007 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0003 0.0004 0.0005 0.0005 0.0005 0.0005 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004 0.0003 0.0004 0.0004 0.0003 0.0004 0.0004

αj1 Appr. N/A N/A N/A 0.3226 0.3228 0.3394 N/A N/A N/A 0.2325 0.2325 0.2363 N/A N/A N/A 0.0446 0.0446 0.0446 N/A N/A N/A 0.2733 0.2736 0.2833 N/A N/A N/A 0.1347 0.1347 0.1351 N/A N/A N/A 0.3902 0.3946 0.4461 Sim. N/A N/A N/A 0.2921 0.2923 0.3281 N/A 0N/A N/A 0.2150 0.2152 0.2328 N/A N/A N/A 0.0483 0.0483 0.0484 N/A N/A N/A 0.2611 0.2638 0.2980 N/A N/A N/A 0.1477 0.1481 0.1568 N/A N/A N/A 0.3259 0.3242 0.3986 ± 0.0004 ± 0.0003 ± 0.0003

± 0.0003 ± 0.0003 ± 0.0004

± 0.0004 ± 0.0004 ± 0.0005

± 0.0005 ± 0.0005 ± 0.0005

± 0.0005 ± 0.0006 ± 0.0007

± 0.0006 ± 0.0006 ± 0.0008

αj2 Appr. N/A 0.1055 0.1085 N/A 0.0003 N/A N/A 0.0434 0.0438 N/A 0.0000 N/A N/A 0.0036 0.0036 N/A 0.0000 N/A N/A 0.1780 0.1824 N/A 0.0004 N/A N/A 0.0182 0.0182 N/A 0.0000 N/A N/A 0.2560 0.2824 N/A 0.0068 N/A Sim. N/A 0.1195 0.1330 N/A 0.0092 N/A N/A 0.0679 0.0737 N/A 0.0037 N/A N/A 0.0097 0.0097 N/A 0.0000 N/A N/A 0.1825 0.2063 N/A 0.0162 N/A N/A 0.0445 0.0470 N/A 0.0012 N/A N/A 0.2258 0.2778 N/A 0.0438 N/A ± 0.0002

± 0.0004 ± 0.0004

± 0.0000

± 0.0002 ± 0.0002

± 0.0001

± 0.0003 ± 0.0004

± 0.0000

± 0.0002 ± 0.0002

± 0.0001

± 0.0004 ± 0.0005

± 0.0002

± 0.0006 ± 0.0006

θ|J|+1,j Appr. 0.1616 0.1723 0.1725 0.0123 0.0123 0.0002 0.1495 0.1557 0.1557 0.0031 0.0031 0.0000 0.0378 0.0380 0.0380 0.0000 0.0000 0.0000 0.0576 0.0621 0.0622 0.0057 0.0057 0.0000 0.0867 0.0902 0.0902 0.0003 0.0003 0.0000 0.0906 0.0949 0.0951 0.0254 0.0266 0.0016 Sim. 0.1394 0.1489 0.1498 0.0241 0.0246 0.0055 0.1189 0.1257 0.1261 0.0121 0.0123 0.0020 0.0341 0.0346 0.0346 0.0001 0.0001 0.0000 0.0502 0.0556 0.0560 0.0117 0.0124 0.0033 0.0707 0.0756 0.0757 0.0050 0.0051 0.0008 0.0767 0.0838 0.0845 0.0267 0.0294 0.0127

Table 4.2: Results for |J|= 8, t0 = 20, tj = 3 for all j ∈ J, t|J|+1 = 0

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.0005 0.0005 0.0005 0.0003 0.0003 0.0001 0.0005 0.0004 0.0005 0.0002 0.0002 0.0001 0.0003 0.0003 0.0003 0.0000 0.0000 0.0000 0.0001 0.0002 0.0002 0.0001 0.0001 0.0000 0.0002 0.0002 0.0002 0.0001 0.0001 0.0000 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001

θ−1,j Appr. 0.0873 0.0024 0.0000 0.0003 0.0000 0.0000 0.0387 0.0003 0.0000 0.0000 0.0000 0.0000 0.0035 0.0000 0.0000 0.0000 0.0000 0.0000 0.1129 0.0026 0.0000 0.0004 0.0000 0.0000 0.0156 0.0000 0.0000 0.0000 0.0000 0.0000 0.1435 0.0130 0.0003 0.0063 0.0002 0.0000 Sim 0.1035 0.0129 0.0027 0.0100 0.0030 0.0031 0.0588 0.0053 0.0009 0.0040 0.0011 0.0011 0.0091 0.0000 0.0000 0.0000 0.0000 0.0000 0.1208 0.0167 0.0051 0.0147 0.0054 0.0055 0.0328 0.0017 0.0030 0.0012 0.0003 0.0003 0.1548 0.0482 0.0267 0.0462 0.0277 0.0284 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.0007 0.0002 0.0001 0.0002 0.0001 0.0001 0.0004 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002

Inst.

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

0.1

0.02

µj

1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Sequence

βj Appr. 0.7216 0.6506 0.6505 0.6027 0.6027 0.6024 0.7573 0.7071 0.7070 0.6642 0.6642 0.6641 0.9466 0.9453 0.9453 0.9409 0.9409 0.9409 0.7901 0.6452 0.6450 0.6172 0.6172 0.6168 0.8315 0.7387 0.7387 0.7159 0.7159 0.7159 0.9273 0.9135 0.9135 0.9062 0.9062 0.9062 Sim. 0.7224 0.6487 0.6472 0.6053 0.6044 0.6022 0.7597 0.7035 0.7027 0.6651 0.6647 0.6635 0.9458 0.9437 0.9437 0.9388 0.9388 0.9388 0.7898 0.6243 0.6150 0.5953 0.5881 0.5825 0.8309 0.7178 0.7150 0.6917 0.6898 0.6875 0.9265 0.9063 0.9064 0.8971 0.8970 0.8970 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.0008 0.0007 0.0007 0.0006 0.0006 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0008 0.0003 0.0003 0.0003 0.0002 0.0002 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0004 0.0004 0.0004 0.0003 0.0003 0.0003 0.0002 0.0003 0.0002 0.0003 0.0003 0.0003

αj1 Appr. N/A N/A N/A 0.3967 0.3967 0.3976 N/A N/A N/A 0.3356 0.3356 0.3359 N/A N/A N/A 0.0591 0.0591 0.0591 N/A N/A N/A 0.3823 0.3823 0.3832 N/A N/A N/A 0.2841 0.2841 0.2841 N/A N/A N/A 0.0938 0.0938 0.0938 Sim. N/A N/A N/A 0.3866 ± 0.0006 0.3868 ± 0.0006 0.3968 ± 0.0007 N/A N/A N/A 0.3306 ± 0.0008 0.3308 ± 0.0008 0.3362 ± 0.0008 N/A N/A N/A 0.0612 ± 0.0002 0.0612 ± 0.0002 0.0612 ± 0.0002 N/A N/A N/A 0.3914 ± 0.0003 0.3941 ± 0.0004 0.4140 ± 0.0003 N/A N/A N/A 0.3039 ± 0.0003 0.305 ± 0.0003 0.3117 ± 0.0003 N/A N/A N/A 0.1029 ± 0.0003 0.1029 ± 0.0003 0.1030 ± 0.0003

αj2 Appr. N/A 0.2283 0.2285 N/A 0.0000 N/A N/A 0.1651 0.1652 N/A 0.0000 N/A N/A 0.0125 0.0125 N/A 0.0000 N/A N/A 0.3221 0.3225 N/A 0.0000 N/A N/A 0.2254 0.2254 N/A 0.0000 N/A N/A 0.0584 0.0584 N/A 0.0000 N/A Sim. N/A 0.2323 0.2367 N/A 0.0025 N/A N/A 0.1770 0.1793 N/A 0.0011 N/A N/A 0.0183 0.0183 N/A 0.0000 N/A N/A 0.3340 0.3502 N/A 0.0099 N/A N/A 0.2442 0.2494 N/A 0.0026 N/A N/A 0.0687 0.0687 N/A 0.0000 N/A ± 0.0000

± 0.0002 ± 0.0002

± 0.0001

± 0.0003 ± 0.0004

± 0.0001

± 0.0004 ± 0.0003

± 0.0000

± 0.0002 ± 0.0002

± 0.0000

± 0.0008 ± 0.0008

± 0.0001

± 0.0008 ± 0.0008

θ|J|+1,j Appr. 0.1136 0.1210 0.1210 0.0006 0.0006 0.0000 0.1207 0.1278 0.1278 0.0002 0.0002 0.0000 0.0414 0.0422 0.0422 0.0000 0.0000 0.0000 0.0314 0.0325 0.0325 0.0005 0.0005 0.0000 0.0348 0.0359 0.0359 0.0000 0.0000 0.0000 0.0268 0.0281 0.0281 0.0000 0.0000 0.0000 Sim. 0.1073 0.1156 0.1158 0.0058 0.0059 0.0006 0.1093 0.1178 0.1178 0.0032 0.0033 0.0002 0.0373 0.0380 0.0380 0.0000 0.0000 0.0000 0.0303 0.0324 0.0325 0.0052 0.0055 0.0011 0.0328 0.0351 0.0352 0.0022 0.0022 0.0003 0.0234 0.0249 0.0249 0.0000 0.0000 0.0000

Table 4.3: Results for |J|= 16, t0 = 20, tj = 3 for all j ∈ J, t|J|+1 = 0

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.0004 0.0006 0.0005 0.0001 0.0001 0.0000 0.0005 0.0005 0.0005 0.0001 0.0001 0.0000 0.0002 0.0002 0.0002 0.0000 0.0000 0.0000 0.0002 0.0001 0.0001 0.0000 0.0001 0.0000 0.0001 0.0002 0.0001 0.0001 0.0000 0.0000 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000

θ−1,j Appr. 0.1648 0.0001 0.0000 0.0000 0.0000 0.0000 0.1220 0.0000 0.0000 0.0000 0.0000 0.0000 0.0119 0.0000 0.0000 0.0000 0.0000 0.0000 0.1785 0.0002 0.0000 0.0000 0.0000 0.0000 0.1336 0.0000 0.0000 0.0000 0.0000 0.0000 0.0459 0.0000 0.0000 0.0000 0.0000 0.0000 Sim 0.1703 0.0034 0.0003 0.0023 0.0004 0.0000 0.1310 0.0017 0.0001 0.0011 0.0002 0.0001 0.0169 0.0000 0.0000 0.0000 0.0000 0.0000 0.1799 0.0093 0.0023 0.0080 0.0024 0.0024 0.1363 0.0028 0.0004 0.0022 0.0004 0.0005 0.0501 0.0000 0.0000 0.0000 0.0000 0.0000 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.0006 0.0001 0.0000 0.0001 0.0000 0.0000 0.0006 0.0001 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0001 0.0000 0.0000 0.0001 0.0001 0.0003 0.0001 0.0000 0.0001 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000

Inst.

73 74 75 76 77 78

µj 0.02 0.02 0.1 0.04 0.08 0.02 0.02 0.04

Sj 1 2 4 2 3 4 1 3

No-No No-Part No-Full Part-No Part-Full Full-No

Sequence

βj Average Abs. Error 0.42 % 0.62 % 0.63 % 0.76 % 0.76 % 0.78 %

αj1 Average Abs. Error N/A N/A N/A 0.71 % 0.71 % 0.78 %

αj2 Average Abs. Error N/A 1.50 % 1.53 % N/A 0.02% N/A

θ|J|+1,j Average Abs. Error 1.01 % 0.96 % 0.96 % 0.05 % 0.05 % 0.00 % %

θ−1,j Average Abs. Error 1.16 % 0.02 % 0.00 % 0.01 % 0.00 % 0.00 %

Table 4.5: Results for asymmetric instances, |J|= 8, t0 = 20, tj = 3 for all j ∈ J, t|J|+1 = 0

Warehouse j 1 2 3 4 5 6 7 8

Table 4.4: Asymmetric instance description

86 4. THE IMPACT OF AN EMERGENCY WAREHOUSE

4.5. THE BENEFIT OF AN EMERGENCY WAREHOUSE

4.5

The benefit of an emergency warehouse

In this section we show the benefit of having an emergency warehouse. We first calculate the optimal base stock levels, including the use of an emergency warehouse. Then we calculate the optimal base stock level for the same instance, but where we do not have an emergency warehouse. We then compare the costs of these two solutions to get insights about the benefit of having an emergency warehouse. We describe our optimization procedure to determine the base stock levels in Section 4.5.1. Because an incremental heuristic like the greedy heuristic does not necessarily lead to the optimal solution, we resort to a smart enumeration procedure. In Section 4.5.2 we get insights about the benefit of having an emergency warehouse by comparing the results for different instances.

4.5.1

Optimization of the base stock levels

In this section we describe the smart enumeration procedure that we use to set the optimal base stock levels for both cases where we do and do not have an emergency warehouse, assuming the evaluation procedure is correct. The problem we want to optimize is as follows:  min C(S) =

X k∈K∪{0}

hSk +

X j∈J

µj 

 X

cq,j θq,j 

q∈Q

Sk ∈ N0 for all k ∈ K ∪ 0, where cj,j = 0. Let us define C ∗ (l) as the lowest costs over all feasible solutions P|J|+1 with a total stock of exactly l, thus k=0 Sk = l. We then start with l = 0, and increase l to find the lowest overall costs. To determine at which value of l we can be certain that an increase would never lead to a better solution anymore we first define a lower bound on the total costs. Let us introduce β̂j (Sj ) = 1−L(Sj , tj µj ) as the upper bound on the fill-rate as we ignore waiting time from the central warehouse and do not include overflow demand from other local warehouses due to lateral transshipments. Let us first introduce the set Z(l) that consists of all possible solutions that satisfy P P k∈K∪0 Sk = l. Moreover, note that q∈Q\{j} θq,j = (1 − βj (S)) by definition. We

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

define CLB (l), a lower bound on the total costs as:   X µj (1 − β̂j (Sj )) min {cq,j } CLB (l) = hl + min  S∈Z(l)

q∈Q\{j}

j∈J  X CLB (l) ≤ hl + min  µj (1 − βj (Sj )) min {cq,j } S∈Z(l)

q∈Q\{j}

j∈J

   X X µj  cq,j θq,j  = C ∗ (l) ≤ hl + min  S∈Z(l)

j∈J

q∈Q

Using known results that L(c, ρ) is decreasing as a function of c, see Karush (1957), we can show that CLB (l) is convex and we are able to find the optimal values for Sk given l. Moreover, let u denote the value that minimizes CLB (u). Having this lower bound on the total costs, we know that we can stop the enumeration as soon as C ∗ ≤ CLB (l + 1), and l ≥ u, where C ∗ is the best solution found so far during the enumeration procedure, and which is set at a very high value initially. We then obtain the following smart enumeration procedure to obtain the optimal base stock levels, S∗ :

Algorithm 6 Smart enumeration procedure Step 1. Let l = 0, C ∗ = ∞

Step 2. For each S with S∗ = S

P|J|+1 j=0

Sk = l, compute C(S). If C(S) ≤ C ∗ , C ∗ = C(S), and

Step 3. If C ∗ ≤ CLB (l), and l > u, stop. Else, l = l + 1 and go to Step 2

In this procedure, solutions are evaluated by the approximate evaluation procedure. Hence we cannot guarantee that we do find optimal solutions, although it is likely the result are close-to-optimal as our approximate evaluation procedure is shown to be accurate.

4.5.2

Comparison

We apply the enumeration procedure on a number of different network structures and demand and cost settings. For all cases we assume the central warehouse is located

4.5. THE BENEFIT OF AN EMERGENCY WAREHOUSE

in The Netherlands. First we look at a network structure that is suitable for the “No-No”, “No-Part” and “No-Full” sequence; see Figure 4.4a. Table 4.6 describes the cost structure for this network. The costs are motivated by the time it takes for a warehouse from that location to the other location and multiplied by a fictive cost per hour for having downtime of the system. Note that the costs can be scaled to any number and still obtain the same solution as long as the cost ratio stays the same. For this network local warehouses are located relatively close to the central and thus emergency warehouse. We assume two warehouses to be located in England and two warehouses to be located in Germany. For such a structure it is interesting to first go for an emergency shipment instead of a lateral transshipment. For the lateral transshipments we can distinguish two cases, the case where we only retrieve a part from the warehouse of the same country, and the case where we also allow lateral transshipments from Germany to England and the other way around. Note that the difference in costs between the emergency warehouse and central warehouse is mainly due to the fact that parts are received faster when send from the emergency warehouse compared to the central warehouse. This can have multiple reasons, for example, parts can be picked faster, there may be more opportunities to send the part, or agreements have been made with customs for parts in this warehouse, as well as the strategic location of the emergency warehouse (i.e. nearby an airport). The cost structure in the second network (see Figure 4.4b), representing the “Part-No” and “Part-Full”sequence is described by Table 4.7. For this network local warehouses are further away from the central warehouse, which makes lateral transshipments more interesting than an emergency shipment. However, the local warehouses are divided over two regions for which the distance is further than the distance to the central and emergency warehouse, thus it is not interesting to first go for a lateral transshipment at all the local warehouses. For this case we assume there are two local warehouses located in the East coast of the USA and two local warehouses in Taiwan. Our final network structure represents the “Full-No” sequence and is not further presented in this chapter because there were no instances for which the emergency warehouse gave any benefit. When we have a group of relatively closely related warehouses and the central warehouse is further away it makes sense to first go for a lateral transshipment at any of the other warehouses before going for an emergency shipment. An example could be having all local warehouses in a different country than the central warehouse. For each network structure the sequence is based on these costs. However, in the case of partial lateral transshipments only one of the other local warehouses with the lowest positive costs is considered.

4. THE IMPACT OF AN EMERGENCY WAREHOUSE Table 4.6: Network structure for No-Part and No-Full sequence

cq,j Local Warehouse 1 Local Warehouse 2 Local Warehouse 3 Local Warehouse 4 Emergency Warehouse Central Warehouse Supplier

Local Warehouse 1 0 3000 4500 3750 2250 5250 27000

Local Warehouse 2 3000 0 3750 4500 2250 5250 27000

Local Warehouse 3 4500 3750 0 3000 2250 5250 27000

Local Warehouse 4 3750 4500 3000 0 2250 5250 27000

Table 4.7: Network structure for No-Part and No-Full sequence cq,j Local Warehouse 1 Local Warehouse 2 Local Warehouse 3 Local Warehouse 4 Emergency Warehouse Central Warehouse Supplier

Local Warehouse 1 0 4500 18000 18000 7500 10500 27000

Local Warehouse 2 4500 0 18000 18000 7500 10500 27000

Local Warehouse 3 18000 18000 0 3000 10500 13500 27000

Local Warehouse 4 18000 18000 3000 0 10500 13500 27000

We considered a large variety of different parameter settings to apply the enumeration procedure. For each of the scenarios we take the replenishment lead time from the supplier to the central warehouse t0 equal to 12 weeks. The transport time from the central warehouse to the local warehouses (except the emergency warehouse for which this is 0) tj is equal to 1 week. We vary the demand per week µj and holding costs rate per week h as well. In Table 4.8 we present the different scenarios, where S∗ , and C(S∗ ) represent the optimal base stock levels and corresponding costs when we have an emergency warehouse, respectively. S∗∗ , and C(S∗∗ ) represent the optimal base stock levels and corresponding costs when we do not have an emergency warehouse. Based on these results we see that the possibility of using the emergency warehouse, and thus keep stock separate for this, does not always lead to a large cost reduction. Especially when a lateral transshipment is more interesting cost wise, and the spare parts are not very expensive this rarely will lead to using the emergency warehouse as it is preferable to have more stock at the local warehouses. However, if a shipment from the emergency warehouse is less expensive than lateral transshipments, or when lateral transshipments are not possible, there is a large benefit. We even see scenarios for which cost savings can be over 30 percent.

4.5. THE BENEFIT OF AN EMERGENCY WAREHOUSE

Table 4.8: Differences in costs between having an emergency warehouse or not Scenario 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

µj 0,02 0,02 0,02 0,1 0,1 0,1 0,2 0,2 0,2 0,02 0,02 0,02 0,1 0,1 0,1 0,2 0,2 0,2 0,02 0,02 0,02 0,1 0,1 0,1 0,2 0,2 0,2 0,02 0,02 0,02 0,1 0,1 0,1 0,2 0,2 0,2 0,02 0,02 0,02 0,1 0,1 0,1 0,2 0,2 0,2

h 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200 150 500 1200

Sequence No-No No-No No-No No-No No-No No-No No-No No-No No-No No-Part No-Part No-Part No-Part No-Part No-Part No-Part No-Part No-Part No-Full No-Full No-Full No-Full No-Full No-Full No-Full No-Full No-Full Part-No Part-No Part-No Part-No Part-No Part-No Part-No Part-No Part-No Part-Full Part-Full Part-Full Part-Full Part-Full Part-Full Part-Full Part-Full Part-Full

With emergency warehouse S∗ C(S∗ ) (1,0,0,0,0,2) 693,7 (1,0,0,0,0,1) 1505,2 (0,0,0,0,0,0) 2160 (5,1,1,1,1,2) 1874,7 (3,0,0,0,0,3) 4780 (1,0,0,0,0,2) 7910,4 (8,2,2,2,2,2) 3013,7 (7,0,0,0,0,4) 8359,8 (3,0,0,0,0,3) 14544 (1,0,0,0,0,2) 693,7 (1,0,0,0,0,1) 1505,2 (0,0,0,0,0,0) 2160 (5,1,1,1,1,1) 1771,2 (3,0,0,0,0,3) 4780 (1,0,0,0,0,2) 7910,4 (8,2,2,2,2,1) 2885,7 (7,1,1,1,1,2) 8101,4 (3,0,0,0,0,3) 14544 (1,0,1,1,0,0) 645,3 (1,0,0,1,0,0) 1483,2 (0,0,0,0,0,0) 2160 (4,1,1,1,1,1) 1671,8 (2,1,1,1,1,1) 4405,5 (1,0,0,0,0,2) 7910,4 (7,2,2,2,2,1) 2795,1 (6,1,1,1,1,2) 7574,8 (3,0,0,0,0,3) 14544 (0,1,1,1,1,0) 784,5 (0,0,1,0,1,0) 1838 (0,0,0,0,0,0) 2160 (5,1,2,1,2,0) 1916,3 (4,1,1,1,1,0) 5159,6 (1,0,1,0,1,0) 9505,3 (10,2,2,2,2,0) 2982,2 (7,2,2,2,2,0) 8741,9 (4,1,1,1,1,0) 17283 (0,1,1,1,1,0) 757,3 (0,0,1,0,1,0) 1829,7 (0,0,0,0,0,0) 2160 (3,2,2,2,2,0) 1909,5 (4,1,1,1,1,0) 5248 (1,0,0,0,1,1) 9579,1 (10,2,2,2,2,0) 2983,2 (7,2,2,2,2,0) 8752,1 (5,1,1,1,1,1) 17925

Without emergency warehouse S∗∗ C(S∗∗ ) (1,1,1,1,1) 886,9 (2,0,0,0,0) 1751,2 (0,0,0,0,0) 2160 (7,1,1,1,1) 2113,9 (5,1,1,1,1) 5625,6 (3,0,0,0,0) 10185 (12,2,2,2,2) 3278,6 (11,1,1,1,1) 9535,8 (6,1,1,1,1) 18442 (0,1,1,1,1) 773,41 (0,0,0,0,0) 2160 (0,0,0,0,0) 2160 (6,1,1,1,1) 1845,2 (4,1,1,1,1) 5006,6 (1,0,1,0,1) 9340,4 (10,2,2,2,2) 2927,4 (7,2,2,2,2) 8608 (5,1,1,1,1) 16919 (1,0,1,1,0) 645,3 (1,0,0,1,0) 1483,2 (0,0,0,0,0) 2160 (5,1,1,1,1) 1707,4 (3,1,1,1,1) 4446 (1,0,1,1,1) 8321,9 (8,2,2,2,2) 2807,3 (7,1,2,1,2) 7862,8 (4,1,1,1,1) 15452 (0,1,1,1,1) 784,5 (0,0,1,0,1) 1838 (0,0,0,0,0) 2160 (5,1,2,1,2) 1916,3 (4,1,1,1,1) 5159,6 (1,0,1,0,1) 9505,3 (10,2,2,2,2) 2982,2 (7,2,2,2,2) 8741,9 (4,1,1,1,1) 17283 (0,1,1,1,1) 757,3 (0,0,1,0,1) 1829,7 (0,0,0,0,0) 2160 (3,2,2,2,2) 1909,5 (4,1,1,1,1) 5248 (1,0,1,0,1) 9794,1 (10,2,2,2,2) 2983,2 (7,2,2,2,2) 8752,1 (5,1,2,1,2) 18302

∆C 21,78% 14,05% 0% 11,32% 15,03% 22,33% 8,08% 12,33% 21,14% 10,31% 30,31% 0% 4,01% 4,53% 15,31% 1,42% 5,89% 14,04% 0% 0% 0% 2,09% 0,91% 4,94% 0,43% 3,66% 5,88% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 2,2% 0% 0% 2,06%

4. THE IMPACT OF AN EMERGENCY WAREHOUSE

Because the base stock levels have to be integer, it can occur that a small change in the (cost) parameters may lead to a very different outcome. It might thus be that a small change makes the emergency warehouse less beneficial than before or the other way around. These differences are even bigger when there are less options in terms of lateral transshipments. Moreover, when lateral transshipments are cheaper the emergency warehouse becomes less beneficial. The emergency warehouse is especially interesting because of its fast shipments, and thus lower costs for the emergency shipment compared to a shipment from the central warehouse, as well as the ability to deliver the part to every other local warehouse, thus having a pooling effect. If it is then possible to obtain a part by a lateral transshipment from any other local warehouse, there is no longer a benefit in terms of pooling for the emergency warehouse. However, if parts are very expensive and it is thus too expensive to stock parts locally at multiple locations, the emergency warehouse might still pay off due its fast shipments. Overall it can be concluded that having an emergency warehouse can definitely pay off, although one should consider the structure of the network before deciding on whether it will pay off or not to have this emergency warehouse. If it faster to obtain parts from other local warehouses compared to the emergency warehouse, it becomes less interesting to have an emergency warehouse, although the emergency warehouse is often organized as such to allow for fast emergency shipments, whereas for local warehouses this is not generally the case. Whereas having local warehouses spread over the continent or world and a centrally located emergency warehouse most likely will be beneficial, unless parts are cheap enough to have abundant amounts of stock at each local warehouse to compensate the downtime costs. As a company generally has to deal with a large variety of different parts with different prices and failure rates, the emergency warehouse could then be used for those parts for which using an emergency warehouse gives the most benefit.

4.6

Conclusions

In this chapter we consider a two-echelon spare parts network with lateral and emergency shipments, and introduce the use of an emergency warehouse, a warehouse that is able to ship parts fast to any local warehouses in the network in the case of a stockout. We allow for a very general structure at which the lateral and emergency shipments are handled, which enables us to model network structures considered in the literature before as well. We first derive an approximate evaluation procedure

4.6. CONCLUSIONS

that allows us to evaluate the overall performance. By means of simulation we show that the evaluation procedure is accurate. Based on our accurate evaluation procedure we derive a lower bound on the optimal costs, as well as a smart enumeration procedure to find a close-to-optimal solution. We then compare the case where we have the emergency warehouse available to us to the case where we do not have this option available. By comparing the costs of the optimal solutions, we find that costs difference of over 30% are possible. Even though our evaluation procedure is fast, one might want to even further improve the speed of the evaluation. The main reason of the calculation time is due to the iterations between the central warehouse and the local warehouses. If one would be able to adjust this procedure such that this might either converge faster or may no longer need any iterations, this would possibly further reduce the calculation time although likely at the cost of a slightly less accurate result.

Generalized design approach to obtain a near-optimal ABC classification for a multi-item inventory control problem 5.1

Introduction

In Chapter 1 we mentioned that not all companies are able to apply a system approach due to a lack of knowledge or cannot afford the investment needed. As a practical solution, companies may resort to the use of ABC classifications (see e.g. Ramanathan, 2006; Hatefi and Torabi, 2015; Flores and Whybark, 1987; Chu et al., 2008; Eilon and Mallya, 1985; Liu et al., 2015; Cavalieri et al., 2008). However, the classification of SKUs is typically made using rules of thumb: e.g. 20% of SKUs that have the highest annual dollar volume are grouped under Class A. Such oversimplification is prone to significant suboptimization. In this chapter we emphasize that this can be improved by a -still simple- "generalized" ABC classification approach, in which the classification is made more carefully. In particular, we attempt to provide to give an answer to our main research objectives by making use of three data sets, stemming from three different companies: Research Objective 6 Develop a method to design an ABC classification that considers all important aspects to incorporate the idea of a system approach.

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

Research Objective 7 Investigate the importance of the different aspects of an ABC classification when trying to obtain results similar to a system approach. When considering to use an ABC classification there are four aspects that together determine the performance. First, as different SKUs have different characteristics and can be of different importance to the company, SKUs can be ranked based on one or more criteria. There have been many authors that propose different criteria to rank the SKUs. The most common criterion is the annual dollar volume (see e.g. Silver et al., 1998; Stevenson, 2007; Nahmias, 1997). However, this criterion does not set the relation with the resulting inventory costs. As inventory costs can be a large fraction of the total costs, there is a shift in the focus of ABC classifications where inventory costs are taken into consideration when deciding how to rank the SKUs (Ramanathan (2006), Bacchetti and Saccani (2012), Bacchetti et al. (2013), Zhou and Fan (2007), Teunter et al. (2010), Ng (2005), Babai et al. (2015), Hollier and Vrat (1978), Zhang et al. (2001) and Ernst and Cohen (1990)). Teunter et al. (2017) even use a ranking to directly assign a service level to the different SKUs, which can be considered as an alternative to ABC classifications. Multi-criteria classifications, where SKUs are classified into three classes based on multiple criteria, have also been proposed (see e.g. Flores et al., 1992). Moreover, in practice we commonly observe two-dimensional classifications where each SKU is classified twice based on each criterion and the final classification is obtained by taking the combination of the previous two classifications. In this way, often nine classes are being used (see e.g. Duchessi et al., 1988; Flores and Whybark, 1987). See Figure 5.1 for an example of a two-dimensional classification. Although these classifications are applied in practice, they have not been compared with respect to their resulting inventory costs before. The second aspect we consider is the number of classes used to classify the SKUs. Although most of the previously mentioned literature considers three classes only, it is possible to extend this to a larger number of classes. A larger number of classes would lead to a reduction of inventory investment costs and increase in performance as three classes may not capture the differences between SKUs sufficiently well (Lee, 2002). However, an increase of the number of classes also makes it more difficult to manage the SKUs as it becomes more complicated to determine the optimal targets per class as well as the proper class sizes. Based on our data, we show that increasing the number of classes results in lower inventory costs, and that six classes are sufficient to obtain near-optimal results. Thirdly, one needs to decide on the class sizes and thus how many SKUs should be put in each class. For ABC classifications with three classes, a common rule of thumb

5.1. INTRODUCTION

Figure 5.1: Example of Price and Demand classification

is to have about 20% of the SKUs in class A, 30% in class B and 50% in class C, considering three classes (see e.g. Jacobs et al., 2009; Slack et al., 2007; Stevenson, 2007; Nahmias, 1997). However, when classes are managed by setting different service levels per class, there are no clear guidelines on how to manage each class of SKUs. Van Kampen et al. (2012) state that it is likely to obtain a better performance with more classes, which does bring a higher complexity, but that is still unclear how to manage the class sizes in combination with the number of classes. There are authors that argue that class A SKUs should have the highest service target to avoid frequent backlogs (see e.g. Armstrong, 1985; Stock and Lambert, 2001), whereas other authors claim that class C SKUs should have the highest service as dealing with stockouts is not worth the effort (see e.g. Knod and Schonberger, 2001). Moreover, these suggestions are not considering the relative size of each class together with the desired service or classification criteria used. Millstein et al. (2014) propose a method to determine class sizes, which maximizes profit under a budget constraint. Also, Tsai and Yeh (2008) propose a grouping method that considers ordering and holding costs under deterministic demand. However, neither of these papers consider the class sizes in the case of stochastic demand and under an aggregate service level constraint. To the best of our knowledge, besides this rule of thumb - which is also used by Teunter et al. (2010) - there is no literature on this topic in relation with inventory costs under an aggregate service constraint. In this chapter we show that this rule of thumb can give far from optimal results and that the class size should depend on the aggregate fill rate target.

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

Finally, one needs to decide on how to manage each class of parts. In this chapter this is done by setting a target fill rate for each class. The fill rate is a commonly used service metric, in particular in cases where it is important the SKUs are on hand when the demand arrives. Although most literature only considers the classification criteria used to classify the SKUs and does not relate to the resulting inventory costs, there are a few authors that do consider the resulting inventory costs and also use the fill rate as service measure (see e.g. Teunter et al., 2010; Babai et al., 2015). The first authors to combine target setting with a classification are Teunter et al. (2010). However, they determine class sizes using a rule of thumb and assume that each SKU is put on stock as they only consider strictly positive target fill rates, which is common in some fields, but not common for spare parts. As our focus is solely on the overall performance from an inventory cost perspective, and because we make use of spare parts data, stocking all SKUs may lead to additional costs, and thus suboptimal results. Especially when there are large cost differences between the SKUs it is not desirable to stock all SKUs (Sherbrooke, 1968). As these four aspects cannot be considered independently of each other (e.g. the target service level of a class depends on how many classes there are), we examine each combination of all four aspects together, which to the best of our knowledge has not been done before. We consider three possible classification criteria, including two-dimensional classifications which have not been compared before, vary the number of classes, look at the impact of different class sizes, and set the optimal target fill rate per class. This approach is applied on three different data sets consisting of real life spare parts data of three different companies, with different characteristics. We compare the outcome of the classification with the optimal solution, which stands for having a separate class for each SKU (see e.g Sherbrooke, 1968; Van Houtum and Kranenburg, 2015). This work falls into the category â&#x20AC;&#x153;Inventory controlâ&#x20AC;? within the framework of Driessen et al. (2010). We do not consider the other processes within this framework. The main contribution of this chapter is the proposed generalized approach to obtain a simple ABC classification that takes all four aspects jointly into account. Even though Teunter et al. (2010) present an approach to combine three of those aspects, they do not consider the optimization of the different class sizes. However, not taking this into consideration most likely leads to using wrong class sizes, which can have a big impact on the performance and result in solutions that are far from optimal. Cost differences up to 290% are found when we compare the class sizes we find with those determined by using the common rule of thumb in which 20% of the SKUs

5.1. INTRODUCTION

are in class A, 30% in class B and 50% in class C (cf. Teunter et al., 2010). This shows the importance of taking the class sizes into consideration when designing a ABC classification. Using our structured scheme, where we take all four aspect into consideration, we are able to design a classification that is able to meet the aggregate fill rate target while getting near optimal results. We use enumeration to find the optimal class sizes, as well as for the target setting. As we have an aggregate service constraint and need to decide on the target fill-rate per class, both decisions are depending on each other which makes it difficult to get a simple expression or rule of thumb. We are able to show that the decision depends on the target aggregate fill-rate. Secondly, we present an algorithm to determine the target fill-rate per class more efficiently. Our algorithm significantly decreases the calculation time for our classification scheme, whereas the cost increase on average is less than 2%. Especially for sensitivity analysis or when the number of SKUs in the system is large, this algorithm can be very useful. Our third contribution is that we use real life spare parts data and show how one can apply our approach on different classification criteria found in the literature. Using this data we show that the resulting inventory investment costs can be within one percent of the optimal solution without bringing the expense of a more complicated inventory control. Finally, even though we apply our approach to a selection of criteria and number of classes based on recent literature,our approach is not restricted to these criteria and number of classes, but can be widely applied to other criteria, and number of classes. It is even possible to apply our approach to different demand distributions and inventory policies. The remainder of this chapter is organized as follows: In Section 5.2 we introduce the model description. In Section 5.3 we give our generalized approach on how to design a ABC classification scheme that considers all of the four aspects jointly. In Section 5.4 we first explain our real life data set of spare parts, which we use for our numerical experiment. We then present the optimality gap of the alternatives that we consider and we conduct some analysis to obtain more insights regarding how to set the right class sizes and propose an algorithm to set the target fill rates per class in a fast way. We conclude this chapter and give some recommendations for future research in Section 5.5.

100

5.2

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

Model

We consider a single warehouse in which several SKUs are kept on stock. Whenever a SKU is requested, it is immediately delivered from stock on hand, and backordered and delivered as soon as possible when there is no stock on hand. The set of SKUs is denoted by I; the number of SKUs is |I|∈ N := {1, 2, . . .}. For notational convenience, the SKUs are numbered as 1, . . . , |I|. For each SKU the stock is controlled by a continuous-review basestock policy, with basestock level Si for SKU i. For each SKU i ∈ I, we assume that demand follows an independent Poisson process with a constant rate λi (≥ 0), which is particularly plausible in the maintenance and spare parts environment, but also in some retail environments. The total demand rate for P all SKUs is denoted by Λ = i∈I λi . The replenishment lead time Li (> 0) is assumed to be constant. Each SKU has a price of ci , and the total inventory investment costs are given by X C(S) = ci Si , (5.1) i∈I

where S = (S1 , . . . , S|I| ) denote the vector consisting of all basestock levels. The fill rate for item i, in steady state, is denoted by βi (Si ). The (aggregate) fill rate in steady state is as follows: X λi β(S) = βi (Si ). (5.2) Λ i∈I

Note that the calculation for the fill rate is as follows (see also Van Houtum and Kranenburg, 2015) SX i −1 βi (Si ) = P {Xi = x} , (5.3) x=0

P {Xi = x} =

(λi Li )x −λi Li e , x ∈ N0 . x!

(5.4)

The target aggregate fill rate for β(S) is given by β o . The objective is to minimize the inventory investment costs subject to an aggregate fill rate constraint. In mathematical terms the optimization problem is as follows: (A)

min subject to

C(S) β (S) ≥ β o Si ∈ N0 , ∀ i ∈ I

As managing each SKU individually can be very complex, especially for a large number of SKUs, companies often resort to the use of ABC classifications. In the next section, we show how we design a ABC classification to obtain minimum inventory

5.3. DESIGN ASPECTS OF A ABC CLASSIFICATION

101

investment costs against the aggregate fill rate target. Although we made some assumptions on the demand process and the inventory control policy our approach can be easily applied with other demand processes and/or inventory control policies as this merely changes the evaluation of the fill rate and costs. The same steps of our approach can still be applied to design the ABC classification.

5.3

Design aspects of a ABC classification

In this section we propose a generalized design of a ABC classification scheme. As the four different aspects cannot be considered separately, we consider each combination and their resulting inventory investment costs. We describe the four aspects in the same order as we apply our scheme. We first describe the different classification criteria we consider in this research. Then we explain the number of classes we consider for each of the four different classification criteria. Thirdly, we explain how we set the class sizes. Finally, we describe how we set the target service level for each class.

5.3.1

Classification criteria

When considering a classification for SKUs, we first need to define the classification criteria that enable us to rank the SKUs. Once we have ranked the SKUs we can take the other three aspects into consideration. There has been extensive studies on how to classify SKUs, and recently there were also authors that relate the classification with the inventory control (see e.g. Babai et al., 2015; Teunter et al., 2010; Ng, 2005). The annual dollar volume - i.e. annual demand times price - (ADV) criterion, is the traditional criterion for ABC classifications and found in many textbooks (see e.g. Silver et al., 1998; Nahmias, 1997). However, if we follow the idea of a system approach (Sherbrooke, 1986), parts with a high price end up with a lower service level, whereas parts with a high demand end up with a higher service level. Therefore, ADV is a poor construction in terms of inventory investment costs as demand and price work in opposite directions as under the system approach. This was also observed by Teunter et al. (2010) and they proposed a classification based on Demand/Price (D/P), which captures the idea of a higher service for cheap and fast moving SKUs. They found that this classification criterion was the best performing criterion among the criteria they considered.

102

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

In practice, two-dimensional classifications based on price and demand are often seen to manage the inventory (see e.g Duchessi et al., 1988; Flores and Whybark, 1987), but they have not been compared before. This two-dimensional classification classifies SKUs based on both price and demand. We compare this two-dimensional classification to the previous mentioned methods. Note that we also compared some other classification criteria; leadtime demand, price, and leadtime and price. We omit them in our exposition as they did not give better results or further insights. Therefore, we present the results and findings of the following three criteria: 1. Annual Dollar volume (ADV) (one-dimensional) This single criterion classification is the most common criterion. The demand rate of each SKU is multiplied with the price of each SKU and then ordered based on this annual dollar volume. The SKUs are ranked in descending order, where SKUs with the largest ADV are considered class A SKUs. 2. Demand/Price (D/P) (one-dimensional) This classification is proposed by Teunter et al. (2010), and shown to give the best performance related to inventory costs. The SKUs are classified based on their demand rate divided by their price. They are then ranked in descending order, where the top group of SKUs, consisting of fast moving and cheap SKUs, are considered class A SKUs which should get the highest service level. 3. Price and Demand (P&D) (two-dimensional) This classification is often found in practice: Each SKU is classified based on both price and demand. See Figure 5.1 for an example of this classification. Note that for the D/P classification criteria, demand and price work in the opposite direction to determine the class to which each SKU belongs. This is not the case for the ADV and the P&D classification where both demand and price work in the same direction. It would thus be possible to have the fast moving and cheap SKUs in class C instead of A when there are three classes as well. We still use the same order as Teunter et al. (2010), although it should be noted that it does not make a difference on our results as we do not rely on fixed class sizes.

5.3.2

Number of classes

When considering ABC classifications, often three classes per dimension are considered: Class A, class B, and class C. However, it is also possible to consider different numbers of classes when classifying SKUs. As less classes will most likely end up with higher investment costs, it makes the problem less complex. Increasing the number will most likely give better results because there are more options available, but this

5.3. DESIGN ASPECTS OF A ABC CLASSIFICATION

103

will increase the complexity of the problem. Therefore, there is a trade-off between performance and manageability. Graham (1987) and Silver et al. (1998) mention that up to six classes can be used as more classes increases the complexity. Using six classes, Teunter et al. (2010) showed that good results can already be obtained. For the two-dimensional classification, a matrix approach is common with 9 classes Duchessi et al. (1988). Having a 4 by 4 matrix would lead to 16 classes which becomes too difficult and costly to manage. In our numerical experiment we vary the number of classes between 2 and 6 for the one-dimensional classifications. For the two-dimensional classification we consider a 2 by 2 matrix and a 3 by 3 matrix. Moreover, we add the results of using only a single class to manage all SKUs, thus not applying any differentiation between the SKUs at all. By considering these different class sizes, we obtain in total 13 combinations of ABC classification criteria and number of classes. Given the number of classes, we introduce a set of classes K where classes are numbered (1, . . . , |K|).

5.3.3

Class sizes

For each of the above mentioned combinations, we need to decide on the size of each class, which in turn determines the class each SKU will be placed in. There has not been any research about the best class sizes for ABC classifications, when considering inventory costs. Typically a rule of thumb for the ADV classification is used (see e.g. Jacobs et al., 2009), in which 20% of the SKUs are class A, 30% class B, and 50% class C. Teunter et al. (2010) also use this rule of thumb for their D/P classification. One of our goals in this chapter is to find out whether different class sizes have an impact on the inventory investment costs. Therefore, we set threshold values to determine in which class each SKU is placed. For example, in the case we have three classes and SKUs are ranked based on their price and the price of a SKU is larger than all threshold values, it will be placed in class C (unlike the logic of the ADV classification). If the price is smaller than the smallest threshold value it will be placed in class A. When there are more than three classes, or multiple dimensions, there are also more threshold values but the logic remains the same. Based on our findings, which we present later, the best class sizes are not the same for each data set in our numerical experiment, indicating that it is not straightforward to find the optimal class sizes or a good rule of thumb. Therefore, we enumerate over

104

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

all possible combinations of class sizes. We first rank all the SKUs according to the classification criteria. Then we decide on the number of SKUs to have in each class, where class A contains the higher ranked SKUs. By considering all possible combinations of class sizes, we find the combination of class sizes that eventually gives the best solution. For example, if there are two classes and five parts, there are six different combinations as one can have either zero, one, two, three, four, or five SKUs in class A and the number of SKUs in the class B is five minus the number of SKUs in the first class. The number of possible combinations of class sizes does however scale exponentially in the number of classes when applying this method. Once we know the number of SKUs in each class, the class to which SKU i simply follows by the ranking of the SKUs.

5.3.4

Target setting

Given the set of classes K and the class to which each SKU i ∈ I belongs, noted by ki , we need to decide on how to manage each class of SKUs. It is common to set different service targets for each class, θk , to manage each class (Teunter et al., 2010). Although there are also other ways to manage the inventory of each SKU in a class, for instance by choosing Si such that the distance to the service level target of that SKU’s class, θki , is minimized, we follow the more common approach of Teunter et al. (2010), where the fill rate has to be greater than or equal to the target for each SKU in that class. We thus need to decide on θk , k ∈ K, the target item fill rate per class. We set the value of Si , i ∈ I, equal to the lowest integer that satisfies this fill rate target, θki , for SKU i. The corresponding problem (C) is then as follows: (C)

min subject to

C(S) β (S) ≥ β o Si = min{x|βi (x) ≥ θki , x ∈ N0 }, θk ∈ [0, 1) ,

∀i∈I ∀k∈K

Different than what Teunter et al. (2010) does, we allow to have zero stock of SKUs. Sherbrooke (1968) already showed that when we are mainly concerned with the overall performance it is sometimes better to take more risk for the very expensive SKUs because their inventory investment costs are very high, and reduce the risk on many other cheap SKUs. To obtain the solution that minimizes the inventory investment costs of problem (C), we first look at each class separately, as the classes are independent due to the fact

5.3. DESIGN ASPECTS OF A ABC CLASSIFICATION

105

that the class to which each SKU belongs is known at this stage. Let Sk = {Si }iâ&#x2C6;&#x2C6;I, ki =k , be the vector of basestock levels for each SKU i in class k, and X Î&#x203A;k = Îťi , iâ&#x2C6;&#x2C6;I, ki =k

be the total demand rate for class k. The aggregate fill rate for class k, Î˛ k (S), and the inventory investment costs for class k, C k (Sk ), are calculated as follows: X Îťi Î˛i (Si ) Î˛ k (Sk ) = Î&#x203A;k iâ&#x2C6;&#x2C6;I, ki =k

C k (Sk ) =

ci Si .

iâ&#x2C6;&#x2C6;I, ki =k

The following is applied for each class k â&#x2C6;&#x2C6; K, with being a small positive value close to zero: 1. Set Î¸k := 0 Si = 0 â&#x2C6;&#x20AC; ki = k, i â&#x2C6;&#x2C6; I Sk = {Si }iâ&#x2C6;&#x2C6;I, ki =k k k k k Compute n C (S ) and Î˛ (S ) o k k k k k Ek := S , C (S ), Î˛ (S ), Î¸k 2. Î¸k :=min(Î˛i (Si ) : i â&#x2C6;&#x2C6; I, ki = k) For all i â&#x2C6;&#x2C6; I, ki = k : If Î˛i (Si ) = Î¸k , then Si := Si + 1 Sk = {Si }iâ&#x2C6;&#x2C6;I, ki =k k 3. Compute Cn (Sk ) and Î˛ k (Sk ); o Ek = Ek â&#x2C6;Ş Sk , C k (Sk ), Î˛ k (Sk ), Î¸k

4. If Î˛ k (Sk ) â&#x2030;Ľ 1 â&#x2C6;&#x2019; , then stop, else go to 2 Note that in Step 2, we look at the minimum fill rate over all SKUs in that class. Using this approach, we do not need to set pre-defined target item fill rates for each class or concern about the stepsize we have to take, as we only need to know which SKUs have the lowest fill rate, and increase the basestock level of all SKUs for which this is the case. The resulting solution, is thus the solution for which the target item fill rate is equal to the minimum fill rate. After this procedure, we have a solution vector per class, Ek , consisting of the basestock levels, costs, fill rate, and target fill rate. We then enumerate over all different combinations of solutions per class, Î¸k , and look for the combination that gives the lowest inventory investment costs while satisfying the aggregate fill rate target.

106

5.4

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

Numerical experiment

In this section we apply our approach using empirical data, which enables us to find which overall classification gives the best result for our current problem (A). In Section 5.4.1 we give an overview of the data we use for our numerical experiment. We show that the D/P classifications gives near-optimal results with only a few classes in Section 5.4.2. Then in Section 5.4.3 we show the impact of the class sizes on the performance. Finally we introduce a greedy ABC algorithm that reduces the calculation time significantly in Section 5.4.4.

5.4.1

Data

To apply our generalized approach to design a ABC classification we use three spare parts data sets from three different companies. Data set 1 consists of spare parts usage for the maintenance of Ground Support Equipment (GSE) at an airport. The GSE consists of a wide range of vehicles consisting of passenger cars up to specialised equipment such as de-icing vehicles used to remove ice off the wings of airplanes. Although there are many expensive parts that are not frequently used, the data also contains fast moving spare parts that are not expensive. The largest majority of parts can be delivered within 3 weeks. Data set 2 consists of spare parts usage for a company involved in producing and maintaining lithography machines. Compared to the other two companies, the demand for the spare parts are much lower. Moreover, many parts are expensive, with costs up to over a million euro. Besides these high costs and low demand rates, the time it takes to obtain the part from the supplier are relatively high compared to the other two companies, with an average leadtime of 110 days. Data set 3 consists of spare parts used for the maintenance of professional printing machines. These parts have the highest usage of the three different companies on average, there are also many parts that are both expensive, as well as fast moving. See Figure 5.2, of the price and demand relationship for data sets 2 and 3. From this figure, we clearly see that company 3 has parts that are both expensive, as well as fast moving (the parts within the red circle), whereas this is not the case for company 2, which has much more slower moving parts, which can be very expensive. An overview of the characteristics of the data of all three companies is presented in Table 5.1.

5.4. NUMERICAL EXPERIMENT

107

(a)

(b)

Figure 5.2: Relation between price and demand for data sets 2 (a) and 3 (b)

108

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS Table 5.1: Data characteristics

Data set 1

Data set 2

Data set 3

5.4.2

Demand per year Price ( e) Leadtime (days) Demand per year Price ( e) Leadtime (days) Demand per year Price ( e) Leadtime (days)

Min 0.10 0.01 1.00 0.0003 0.01 5.18 1.00 0.01 14.02

Average 8.53 78.61 5.97 0.4329 4,819.89 106.84 26.82 142.51 28.18

Max 610.50 10,950 111.00 110.29 1,068,444 320.29 297 4,381.59 42.01

Choosing the best classification criteria

In this section we present our findings regarding the inventory investment costs for the different classifications. For our numerical study, we choose to look at aggregate fill rate targets of 80%, 90%, and 97%, representing a low, medium and high service. We have also looked at different targets in between, but the results were similar and therefore we only present the results for these three different aggregate fill rate targets. Note that our methodology is not restricted to a certain service level or measure. In Table 5.2 we present our findings regarding the inventory investment costs for the different classifications of all three data sets. We represent the costs as a percentage of the system approach, which is considered the optimal solution. To apply the system approach, we use the algorithm as described in Section 2.4.2 of Van Houtum and Kranenburg (2015). The results regarding the ADV are in line with the results of Teunter et al. (2010): we clearly see that the ADV classification gives very poor results. We did not show the findings for the ADV classification for 5 and 6 classes as they only give slightly better results, but still far from optimal. An explanation for this is that both fast moving SKUs as well as expensive SKUs can be put in the same class of SKUs according to this criterion. However, from the inventory investment cost perspective, we prefer to stock less expensive SKUs over expensive SKUs, because expensive SKUs bring higher inventory investment costs. Moreover, it is more interesting to stock fast moving SKUs over slow moving SKUs because the contribution to the the aggregate fill rate of fast moving SKUs is larger. The classification that leads to the lowest inventory investment costs is the D/P classification. The D/P classification captures the trade-off between the contribution to

5.4. NUMERICAL EXPERIMENT

109

the aggregate fill rate and costs very well, therefore the SKUs in the same class will end up with similar fill rate targets as if we apply a system approach. Figure 5.3 shows the inventory levels for each SKU for data set 2 and 3 for the optimal solution and confirms this thought. We find that by applying the D/P classification with six classes, and setting the best class sizes and target fill rates, the resulting inventory investment costs are within a few percent of the optimal solution for all three data sets. The P&D classification is also able to capture this trade-off, although it does not directly capture this trade-off for each SKU. Although we see that the costs for the Price and Demand classification are higher compared to the D/P classification, the resulting costs are still very reasonable, especially compared to using no classification or the ADV classification. For data set 3 it even gets within 6 percent of the optimal solution. This is mainly due to the fact that this data set has parts that are both fast moving as well as expensive. Using the Price and Demand classification, these parts can be handled separately, thereby providing more flexibility in this region. When we look at the number of classes used for the D/P classifications, we clearly see that when only three classes are considered, already a significant gain with respect to two classes can be obtained. Whenever the number of classes is increased further, the performance increases but more limited. Therefore, we expect that having four classes will be sufficient to get near optimal results.

5.4.3

The impact of class sizes on performance

When considering the size of each class, we would like to know what the best division of class sizes is. As the D/P criterion gives the best overall results, from now on we consider the impact of the class sizes only for this criterion. Besides the approach to have fixed class sizes, i.e. 20% of the SKUs in class A, 30% in class B and 50% in class C (Teunter et al., 2010), with class A consisting of the cheaper SKUs and class C of the expensive SKUs, there has not been any literature about the proper class sizes in relation with the inventory costs. For ease of exposition, we consider the case that there are three classes from now on, although the results also hold for larger or fewer number of classes. Based on our numerical experiments, a key observation is that there are large inventory investment cost differences by only changing the class sizes. In the best solutions found for our data sets, we observe that there is always one class for which SKUs are not stocked. This is always class C, for which the value of D/P is the smallest, thus consisting of the expensive and/or slow moving SKUs. Because we consider spare

110

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

(a)

(b)

Figure 5.3: Optimal base stock levels for data sets 2 (a) and 3 (b)

One class

Price and Demand

Demand/Price

Annual Dollar Volume

Classification

80% 3,098 2,753 2,749 19 12 9 9 7 40 23 3,300

2 3 4 2 3 4 5 6 4 (2x2) 9 (3x3)

1,278 1,217 1,216 38 4 2 1 1 26 12 1,294

90% 330 329 327 47 6 3 1 1 26 22 342

97%

Data set 1

Number of classes

Inventory investment costs (% over optimal) 1,595 1,595 1,595 44 9 4 4 2 74 23 1,628

80% 670 665 665 43 15 10 7 7 50 16 711

90% 229 224 223 53 6 2 1 0 25 14 328

97%

Data set 2

Table 5.2: Inventory investment costs compared to optimal solution

87 79 77 7 2 1 1 1 5 2 132

80%

79 70 69 42 6 4 3 2 25 6 120

90%

38 35 34 17 12 10 5 2 21 6 61

97%

Data set 3

112

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

parts data, these SKUs have very high costs, and thus a very large impact on the total inventory costs as soon as the company decides to stock these SKUs. Therefore, in order to achieve a high availability on a system level, the low availability for these SKUs are compensated with higher availability of less expensive parts. However, if the number of SKUs in class C is taken to be too large, it would no longer possible to compensate the lower availability, even with ample stock of all other SKUs in all other classes. As a result all SKUs in class C have to be stocked as it becomes necessary to increase the target for this class from zero to a positive value to reach a feasible solution. We call this a “forced presence” of stock whenever this occurs. Using our data sets we find that whenever a forced presence of stock occurs, we obtain poor solutions with very high holding costs. The impact of the forced presence does however depend on the differences in costs of the SKUs, as the bigger the cost difference, the bigger the impact. Making use of this insight of forced presence of stock, it is possible to prevent having poor class sizes. Whenever class C becomes too large, we need to make this class smaller to prevent having a forced presence of stock. Whenever the following equation does not hold, one needs to make class C smaller: X i∈I,ki 6=C

λi ≥ βo. ∆

(5.5)

From this equation we see that the forced presence of stock depends not only on the class sizes, but also on the overall availability target, β o . If we have a very high availability target, it is more likely to obtain a forced presence and we thus need to make class C smaller compared to the case where we have a low availability target. By taking the forced presence into consideration, we improve on current methods such as the commonly used rule of thumbs that do not consider the forced presence of stock (see e.g. Teunter et al., 2010). Depending on the data set and the overall availability target, using the commonly used rule of thumb may easily lead to a forced presence of stock, and thus a poor solution. Figure 5.4 shows the fraction of total demand in this class using the best target setting for different aggregate fill rate targets when there are three classes. We find that the class size of the class consisting of the most expensive SKUs, is always close to the maximum allowed size that still satisfies Equation 5.5. We then clearly observe the increase in inventory costs due to the forced presence of stock. When it is not possible to consider all combinations of class sizes, this equation can be used to check beforehand whether it will lead to a forced presence of stock. In Table 5.3 we present our findings when we would not use our approach to consider the different class sizes but use the commonly used rule of thumb instead, as suggested by Teunter et al. (2010), to divide the SKUs over the classes. We compare these

5.4. NUMERICAL EXPERIMENT

113

50000 45000

Inventory investment costs

40000 35000

80 percent aggregate fill rate

30000 25000

90 percent aggregate fill rate

20000

97 percent aggregate fill rate

15000 10000

5000 0 0,00%

5,00%

10,00% 15,00% Percentage of demand in class C

20,00%

25,00%

Figure 5.4: Impact of different class sizes on investment costs for data set 1

findings with our approach, where we also consider the class sizes. We present the results of 3 and 4 classes, although we also considered 5 and 6 classes but these showed similar results to having 4 classes. An important finding is that if the target fill rates are set optimally, but class sizes are not carefully considered in the design of the classification, huge cost differences are possible, with cost differences over 300% higher than our solution. This difference is mainly explained by the forced presence of class C SKUs. Note that if it is required to have strictly positive basestock levels (so that all SKUs are kept on stock), we do always have forced presence of stock and thus setting the best class size would have less impact, although these solutions would always be more costly as we have to stock these expensive parts.

5.4.4

ABC classification target setting

In order to apply our approach, we make use of enumeration to determine the optimal target setting for each class, as described in Section 5.3.4. Unfortunately, when the number of classes is increasing, the run time is increasing exponentially; for the D/P classification with five classes and an aggregate fill rate target of 90% the run times are around 45 minutes. Although these calculations are not done on a regular

114

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

Table 5.3: Inventory investment costs over optimal for data set 1 based on D/P classification Number of classes 3 3 4 4

best rule of thumb best rule of thumb

Target aggregate fill rate 80% 90% 97% 12 4 6 180 22 302 9 2 3 14 60 301

basis, and thus may not have a big impact, companies may be interested in sensitivity analysis to see how the total inventory investment costs change with respect to the aggregate fill rate in order to set their goals. Therefore, we propose the greedy ABC algorithm to set the target fill rates for each class based on getting the “biggest bang for the buck”. We increase θk with a small step size, θstep , for the class where we get the biggest increase in aggregate fill rate per euro invested.

This ratio, Γk , is defined as the difference in aggregate fill rate after increasing θk , divided by the difference in inventory investment costs. Because we do not have to calculate all solutions for each class and use enumeration, this results in significantly reduced run times. Algorithm 7 gives an overview of the greedy algorithm for the ABC classification.

The run-times using this algorithm are around 0.15 seconds when there are two classes and up to around 980 seconds for six classes on a Pentium Intel(R) Core(TM) i52520M CPU with 4GB memory.

We find that savings in calculation time of around factor 20 are obtained for five classes, and increasing for larger numbers of classes. The increase in costs compared to the optimal target setting, over all different targets per class with more than three classes, is less than two percent on average. Thus for a larger number of classes or a larger number of SKUs, the greedy ABC algorithm can significantly speed up the calculation time of our approach whereas the cost increase is limited.

5.4. NUMERICAL EXPERIMENT

Algorithm 7 Greedy ABC algorithm 1. Set θk := 0 for all k ∈ K Si = 0 for all i ∈ I S = (S1 , S2 , . . . , S|I| ) Compute C(S) and β(S) 2. For all k ∈ K: (a) For all i ∈ I: If ki = k: vi = min {x|βi (x) ≥ θki + θstep , x ∈ N0 } − Si else, vi = 0 (b) vk = (v1 , v2 , . . . , v|I| ) Γk := (β(S + vk )) − β(Sk ))/(C(S + vk ) − C(S)) 3. m :=arg max{Γk : k ∈ K} S := S + vk θk :=min{βi (S) : i ∈ I, ki = k} for all k ∈ K 4. Compute C(S) and β( S); If β k (S) ≥ β o , then stop, else go to 2

115

116

5.5

5. GENERALIZED DESIGN APPROACH FOR ABC CLASSIFICATIONS

Conclusions and suggestions for further research

In this chapter we propose a generalized approach on how to design a ABC classification to minimize the inventory investment costs by combining four different key aspects: Classification criteria, number of classes, class sizes, target fill rates per class. Using our structured approach on empirical data we show that it is possible to obtain a solution within a few percent of the optimal solution by using D/P as classification criterion. We show that this classification criteria captures the idea of the system approach very well, and that even though the Demand and Price classification is commonly used in practice and gives decent results, especially for one of the data sets, it is still outperformed by the D/P classification. Another key insight is that it is of great importance to have the proper class sizes, as using a rule of thumb as proposed by Teunter et al. (2010) leads to costs that are far from optimal, with costs differences of over 280% compared to the optimal costs. This large difference is mainly explained by the forced presence of stock of expensive and slow moving SKUs, although there are also significant gains when there is no forced presence of stock. When using a simple rule of thumb, it is important that the target aggregate fill rate is taken into consideration when determining the class sizes. We also propose a greedy ABC algorithm which significantly reduces run times, whereas the cost increase of using this algorithm is only limited to a few percent on average. Although the design of a classification is not something done often in practice, it does allow for fast sensitivity analysis to support decision making using our approach. For further research it may be interesting to find simple rules to determine proper class sizes instead of enumeration, which may further reduce the run time of our approach. Moreover, we used a basestock policy for each class to manage the inventory, whereas it might be interesting to apply different policies for different classes. Especially if transport costs are involved for each order. By including these costs and batching for one or more of the classes, it would be interesting to see whether the D/P classification would still perform best. Note that although we did not investigate this problem, our methodology can still be applied as the difference lies only in the evaluation of the performance and costs. Another interesting point for further research is the target setting. We assumed that each SKU at least has to meet the target for its class. However, it might be interesting

5.5. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

117

to see how the performance of the different classifications would change if we would set the basestock levels such that the difference with the target is minimized instead. This may also reduce the impact of forced presence of stock as not all parts have to be stocked at once in this case.

Conclusion In this thesis we focused on applying system-focused spare parts management for capital goods. By applying system-focused spare parts management one is able to keep downtime to a minimum while also being able to reduce inventory holding costs. In this thesis we developed different system-focused inventory models, applicable for either the initial phase or the stationary phase of the life cycle of capital goods. In Chapters 2 and 3 we developed models to manage the spare parts inventory of capital goods during the initial phase of the life cycle. For these models we studied how one can incorporate the double demand uncertainty one has during this initial phase. In Chapter 4 we developed an approximate evaluation procedure to evaluate the performance of two-echelon networks with lateral and emergency shipments. In Chapter 5 we developed a methodology to mimic the use of a system-approach. In the remainder of this chapter we revisit our research question and objectives and point out interesting points for future research.

6.1

Research questions revisited

In this thesis we focused on two main research questions, one research question for the initial phase and a second research question for the stationary phase. These research questions led us to more specific research objectives. In the following sections we briefly revisit the main research questions and our research objectives.

6.1.1

Spare parts management for the initial phase

At the start of a life cycle, there is a considerable amount of uncertainty on the demand rate of spare parts, which makes system-focused spare parts inventory man119

120

6. CONCLUSION

agement more complicated. The more uncertainty on the demand rate, the more difficult managing the spare parts inventory will be. There is a higher risk of stocking either too much or too little. Reducing the uncertainty on the demand rate can therefore potentially reduce these risks and thereby the costs. This led us to our first main research question: Main Research Question 1 How to deal with double demand uncertainty within spare parts inventory control in the initial phase and how much to invest in more reliable demand rate information? We divided the main research question into smaller research objectives. Our first focus is on how one should decide upon the base stock levels for the spare parts as the double demand uncertainty will impact the decisions that need to be taken. This led us to our first research objective: Research Objective 1 Develop a model to incorporate the double demand uncertainty into the spare parts management. In Chapter 2 we developed a multi-item inventory model to incorporate double demand uncertainty by assuming the demand rate follows from a given and known distribution. We were able to show analytically that a more variable demand rate distribution always leads to a lower system availability under given base stock levels and hence higher costs when a given target availability has to be met. We use our model to numerically show the impact of the double demand uncertainty on the total expected costs. Costs are shown to increase significantly when there is more double demand uncertainty. Double demand uncertainty in combination with having longer lead times and larger expected demand rates may result in cost increases of over 18 times the costs without double demand uncertainty. Thus when base stocks are not increased to hedge against this uncertainty, this is likely to impact the availability. This led to our second research objective: Research Objective 2 Investigate the availability when applying a decision model that ignores the double demand uncertainty. By applying our model we investigated the availability if one decides to make decisions without taking the double demand uncertainty into consideration. We de-

6.1. RESEARCH QUESTIONS REVISITED

121

termined the base stock levels, ignoring the double demand uncertainty, and then evaluated the expected performance using our model in Chapter 2. Based on the results we showed that even under small variation levels, very large differences between the target availability and the expected availability were obtained. Therefore, one may expect significant downtime costs when ignoring the uncertainty.

As the solutions of the model developed in Chapter 2 always lead to higher base stock levels when there is more uncertainty, a company would prefer to keep this increase of the base stock level as a result of the double demand uncertainty to a minimum. One option to do this is to get more information on the demand rate to reduce the double demand uncertainty. This led us to our third research objective:

In Chapter 3 we extended the model of Chapter 2 by modeling the possibility to invest in more reliable information. At the same time, we switched to a single-item setting that facilitates easier analysis. Although there is a possibility to invest in information, this information becomes available at a later point in time. We incorporated that the decision maker has to make a decision on the base stock level before this information is available. Later on, the decision maker is able to change the base stock level. If the base stock level is decreased, the excess spare parts stock is returned to the supplier for an amount less than the price for new items.

We modeled the problem as a three-stage stochastic problem and derived analytical expressions for the optimal base stock levels. We use these expressions to find the potential savings of investing in more reliable information on the demand rate. We apply the model developed in Chapter 3 on different instances where we vary the price of the part, downtime costs, demand rate uncertainty, the length of the periods, and the cost of returning parts for a lower price. For these instances we found savings up to 16 % compared to the expected costs without investing in more reliable information. Investing in more reliable information on the demand rate is found to be most attractive for expensive parts and when there is more uncertainty on the demand rate.

122

6.1.2

6. CONCLUSION

Spare parts management during the stationary phase

For the stationary phase, we defined the following main research objective: Main Research Question 2 i How to incorporate general structures for emergency and lateral shipments within spare parts inventory control models for two-echelon networks in the stationary phase; ii How to mimic system-focused inventory control by methods that are easier to adopt? In order to given an answer to (i), we focused on developing a fast and efficient algorithm for a two-echelon network with emergency and lateral transshipments where it is possible to distinguish regular replenishment orders from emergency orders at the central warehouse. This led us to the following research objective: Research Objective 4 Develop a procedure to evaluate the performance, in terms of expected waiting time for a part, for a problem setting with multiple local warehouses and an emergency warehouse in a two-echelon network with lateral and emergency shipments that can be done in any order. In Chapter 4 we developed a general model which allows the decision maker to evaluate the performance of a two-echelon network with lateral and emergency shipments, including the possibility to keep stock at a centrally located emergency warehouse used only to provide emergency shipments. We also developed an approximate evaluation procedure with which the performance of a given solution can be evaluated in around 1 to 70 milliseconds, which is sufficiently fast for real-life instances. We showed that our evaluation procedure is accurate, especially for instances with high target service levels. Research Objective 5 Investigate the benefit of an emergency warehouse. Using the model we developed in Chapter 4 we are able to show the impact of the emergency warehouse, and show the potential benefit. We show numerically that the use of an emergency warehouse can reduce up to 30 percent of the total expected costs. The emergency warehouse has the largest benefit when local warehouses are relatively far apart from each other compared to the distance to the emergency warehouse. When local warehouses are close to each other, lateral transshipments are

6.1. RESEARCH QUESTIONS REVISITED

123

relatively more interesting, resulting in hardly any benefit of the emergency warehouse. When we look back at our main research question 2 (i), we developed a model that focused on a efficient evaluation of two-echelon networks by incorporating the emergency warehouse. It may have been possible to develop a more accurate evaluation procedure, for instance by modeling the problem as a Markov process. This would however lead to larger computational complexity and may not be applicable for use in practice. Similarly, it could have been an alternative to use a critical level policy at the central warehouse instead of the emergency warehouse, but that would have complicated the analysis. To address main research question 2 (ii), we made use of an ABC classification scheme. An ABC classification enables dividing the parts into different categories and then managing the parts based on targets set per class. Once the classes have been determined, the responsibility can be divided and targets can be set for each part. When inputs change, one can still determine the base stock levels based on the classes that were made and the targets set per class. Thus in order to determine how to decide upon a good ABC classification, the following research objective needs to be reached: Research Objective 6 Develop a method to design an ABC classification that considers all important aspects to incorporate the idea of a system approach. In Chapter 5, for a single-location setting, we developed a generalized approach to use an ABC classification to manage the spare parts inventory. We incorporated the idea of system-focused inventory control by combining four different important aspects of an ABC classification. The four different aspects of an ABC classification considered are: the criteria used to rank parts, the number of classes, the class sizes, and the service level target per class. By combining these four different aspects, the idea of system-focused inventory control can be mimicked and near optimal results can be obtained, while the computational and managerial burden of implementing a system approach is avoided. As we consider different aspects in the ABC classification, it is important to show how each of these aspects impacts the results and give an answer to our last research objective: Research Objective 7 Investigate the importance of the different aspects of an ABC classification when trying to obtain results similar to a system approach.

124

6. CONCLUSION

By applying our methodology numerically on real-life spare parts data, we showed that class sizes can have a big impact on the performance. For the best solutions it is common to have no stock for one of the classes. In our case this class is the class with the most expensive parts or parts that have the lowest demand rates. If one would not determine the appropriate class sizes one must always stock all parts in the class with most expensive parts to obtain a feasible solution, which we call a forced presence of stock. This forced presence of stock results in a large increase in total costs and therefore should be avoided. Also other aspects such as the ranking of parts, the number of classes, the target setting per class, and especially the right combination of these different aspects allow obtaining results that are close to optimal.

6.2 6.2.1

Ideas for future research Dealing with double demand uncertainty in complex networks

In Chapters 2 and 3 we considered taking the double demand uncertainty into consideration to improve spare parts decision making when spare parts are kept at a single stock point. These models are one of the first models that deal with system-focused spare parts management during the initial phase of the life cycle of capital goods. In practice one may have to deal with more complex network structures. For further research, one might consider a model that allows incorporating the double demand uncertainty for these more complex network structures. The double demand uncertainty then has an influence on all locations. When the demand rate is significantly higher or lower than expected, this often holds for all locations. There is thus a strong correlation of the demand rates between the different locations. An example of a direction one could consider is to adjust the calculation in Chapter 4 such that instead of having a point estimate of the demand rate as an input, one could include the double demand uncertainty at each local warehouse. Using the correlation of the demand rate, it might be possible to simplify calculations.

6.2. IDEAS FOR FUTURE RESEARCH

6.2.2

125

The transition from initial phase to the exploitation phase

In Chapters 2 and 3 we considered spare parts inventory control at the initial phase in the exploitation phase. We modeled the uncertainty using a distribution and assumed this distribution will not change at all (Chapter 2) or changes only at one particular point in time based on expert information (Chapter 3). However, in practice there will be actual demand of the parts which makes it possible to also include historical demand information into the distribution for the demand rates. As such, the information available gradually changes from high levels of uncertainty towards having almost no uncertainty in the demand rates as commonly assumed in the stationary phase. It would be interesting to combine our model with a model that is able to capture the demand rate information based on historical demands observed from the field, such as the use of Bayesian updating procedures. By combining these models, one may be able to model the transition from the initial phase to the stationary phase.

6.2.3

Efficient multi-item optimization procedures

In Chapter 4 we developed a fast evaluation procedure which can be used to determine a networks performance based on the given base stock levels. However, for the optimization of the base stock levels we used a smart enumeration procedure to determine the optimal base stock levels for a single part. For a single part this procedure is sufficiently fast, however if we would want to solve a multi-item problem, one needs to resort to faster heuristics which still lead to close-to-optimal solutions. We expect that due to the interaction between the central warehouse and the local warehouses, a greedy heuristic may not always lead to close-to-optimal solutions. As a result, other fast and efficient methods need to be developed.

6.2.4

Mimicking system-focused inventory control in spare parts networks

In this thesis we considered applying an ABC classification to mimic system-focused inventory control for a single location. It might also be possible to apply an ABC classification in a two-echelon network by either decoupling the central and local warehouses or by decoupling the parts themselves.

126

6. CONCLUSION

For the first approach, decoupling the central and local warehouses, one can use an ABC classification to manage the spare parts at the central warehouse. Once these spare partsâ&#x20AC;&#x2122; inventory levels have been set, it is possible to apply existing systemfocused models for single echelon models with lateral and emergency shipments by approximating the replenishment lead time. Another approach that could be interesting is to use an ABC classification to set a service level target per part. Based on this service level for each part, one can then determine the optimal base stock levels over the spare parts network using the model we developed in Chapter 4. However, this may only work well when service levels and parts usage for the different local warehouses are relatively similar. How to develop such an ABC classification in a spare parts network to mimic systemfocused inventory control is not straightforward and therefore may be an interesting topic for future research.

Appendix A

Single-item and multi-item inventory models In this section, we show the relationship between single-item and multi-item inventory models. As an example, let J denote the set of warehouses where spare parts are kept on stock, and let I denote the set of stock keeping units (SKUs). The inventory at each location is managed with a base stock level Si,j , i ∈ I, j ∈ J. Let S denote the vector of base stock levels for all SKUs i ∈ I at all warehouses j ∈ J and let Si denote a vector of base stock levels for SKU i at all locations j ∈ J. A common service measure used is the overall expected number of backorders at warehouse j, EBOj (S). The overall expected number of backorders at warehouse j is the sum over the expected number of backorders of each individual part at that warehouse, denoted by EBOi,j (Si ): EBOj (S) =

EBOi,j (Si ), j ∈ J

i∈I

Let Ci,j (Si ) denote the total holding costs for SKU i at warehouse j. Moreover, let P P C(S) = j∈J i∈I Ci,j (Si,j ) denote the total costs for the entire network. Let J loc represent the set of warehouse where demand arrives from the customers or systems. The objective would then be to minimize the total costs while satisfying the constraint on the overall expected number of backorders for all locations j ∈ J loc , denoted by EBOjobj j ∈ J loc . (A)

min subject to

C(S) EBOj (S) ≤ EBOjobj j ∈ J loc Si,j ∈ N0 , ∀ i ∈ I, j ∈ J 127

128 APPENDIX A. SINGLE-ITEM AND MULTI-ITEM INVENTORY MODELS

One can make use of the Lagrange relaxation technique to solve this problem (see e.g. Porteus, 2002; Van Houtum and Kranenburg, 2015). Let λ = {λ1 , λ2 , λ...,|J|loc }. The Lagrange function is then defined as follows: ! X X obj λj L(S, λ) = C(S) + EBOi,j (Si ) − EBOj . j∈J loc

i∈I

This Lagrange function can be rewritten as follows: X X λj EBOjobj , L(S, λ) = Li (Si , λ) − i∈I

j∈J loc

where Li (Si , λ) =

(Ci,j (Si ) + λj EBOi,j (Si ))

j∈J

is the decentralized Lagrangian function for SKU i. By introducing penalty costs for having backorders, the overall problem can be divided into smaller decentralized problems which can be solved more easily. One then obtains different solutions for problem (A) by varying the values of λ. By applying the dual formulation of the problem, one is able to set λ in such a way that a (close-to-)optimal solution is obtained for Problem (A).

Appendix B

Evaluating an infinite investment When I goes to infinity the demand rate Îťtrue is known exactly at the start of the third stage. Given the value of Îťtrue , the demand during the lead time is Poisson distributed with rate Îťtrue L, with L being the deterministic lead time with corresponding cumulative distribution function F (x, Îťtrue ), x â&#x2C6;&#x2C6; N0 . Note that under perfect information of the demand rate, the optimal second period S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , Îťtrue ) depends on both the value of Îťtrue as well as the first period base stock level S1 and is calculated as follows: T2 p2 + w1 1{x<S1 } â&#x2C6;&#x2019; w2 S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , Îťtrue ) = min x|F (x, Îťtrue ) â&#x2030;Ľ , T2 (p2 + h)

(B.1)

Fortunately, we can evaluate the total expected costs in a different way due to the integrality of the base stock levels. Because of the integrality of the base stock levels there may be multiple values of Îťtrue and S1 that lead to the same value of S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , Îťtrue ). In other words, consider second period intervals a02 (k) and b02 (k), for any k â&#x2C6;&#x2C6; {1, 2, . . . , K} that satisfy the following condition: S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , a02 (k)) = S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , b02 (k)),

k â&#x2C6;&#x2C6; {1, 2, . . . , K}.

(B.2)

Algorithm 8 explains how one can construct these intervals such that all intervals together cover the interval (a1 , b1 ) and have the property as described in Equation (B.2). Let represents a very small value. After applying Algorithm 8, one has to take into consideration that the sizes of the intervals may differ. The probability to 129

130

APPENDIX B. EVALUATING AN INFINITE INVESTMENT

Figure B.1: Possible outcome when constructing the intervals to obtain a lower bound on total expected costs

obtain interval k, denoted by Ď&#x2C6;k0 (S1 ) is proportional to the size of the initial interval, and calculated as follows:

Ď&#x2C6;k0 (S1 ) :=

b02 (k) â&#x2C6;&#x2019; a02 (k) , b1 â&#x2C6;&#x2019; a1

k â&#x2C6;&#x2C6; {1, 2, . . . , K}, S1 â&#x2C6;&#x2C6; N0

(B.3)

Figure B.1 graphically represents a possible outcome. In this example there are three intervals and the optimal second period base stock level is either 1,2 or 3 for any value within interval 1,2, and 3 respectively.

Algorithm 8 Interval construction Step 1. k := 1 a02 (k) := a1 Step 2. Using Equation (B.1): b02 (k) := min {b1 , max {x|S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , a02 (k)) = S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , x)}} Step 3. If b02 (k) < b1 : k := k + 1, a02 (k) := b02 (k â&#x2C6;&#x2019; 1) + , Go to Step 3. Else: K := k, stop.

As the optimal second period base stock level within each interval is always the same, and as every value of Îťtrue within this interval is equally likely to occur in the third

131 stage, it can be shown that Z

V30 (S1 , x) dx â&#x2030;&#x2C6; b1 â&#x2C6;&#x2019; a1

Ď&#x2C6;k0 (S1 ) T2 hS2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , a02 (k))

kâ&#x2C6;&#x2C6;{1,2,...,K}

(B.4)

+ w1 (S1 â&#x2C6;&#x2019; S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , a02 (k)))+ + w2 S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , a02 (k)) + T2 (p2 + h)EBO2 (S2â&#x2C6;&#x2014;â&#x2C6;&#x2014; (S1 , a02 (k)), a02 (k), b02 (k))

where, EBO2 (S2 , a02 (k), b02 (k)) represents the expected number of backorders in the second period when the demand rate is uniformly distributed between a02 (k) and b02 (k), under a base stock level S2 . This is calculated as follows:

EBO2 (S2 , a02 (k), b02 (k))

! S2 X a02 (k) + b02 (k) 0 0 (S2 â&#x2C6;&#x2019; x)P {X(a2 (k), b2 (k)) = x} , L â&#x2C6;&#x2019; S2 + 2 x=0

where X(a02 (k), b02 (k)) represents the probability density function of the demand during the lead time when the demand rate is uniformly distributed between a02 (k) and b02 (k). In the latter case, we do not have to calculate an infinite number of possible scenarios, however, the number and lengths of the intervals may depend on S1 . Therefore, we enumerate over S1 until we can ensure that these intervals do not change anymore and costs can no longer decrease. From Equation (B.1) it becomes clear that the largest value of the second period base 0 stock level, given Îťtrue , denoted by S2U B (Îťtrue ) is as follows: T2 p2 + w1 â&#x2C6;&#x2019; w2 U B0 S2 (Îťtrue ) = min x|F (x, Îťtrue ) â&#x2030;Ľ . T2 (p2 + h) 0

As b1 is the largest possible value of Îťtrue , we can show that when S1 > S2U B (b1 ), the third stage intervals are independent of the base stock level decided upon in the second stage. Next to this, one needs to ensure that the second stage direct costs do not decrease when increasing S1 , as this might also lead to a lower total expected costs for the second stage. Using Proposition 3.3, we know that the direct costs are convex in S1 . Thus if we take the smallest value of S1 such that c1 (S1 + 1) > c1 (S1 ), we can ensure that the first period costs will no longer decrease.

132

APPENDIX B. EVALUATING AN INFINITE INVESTMENT

When both conditions are met, we know that the total expected costs for the second stage cannot decrease when further increasing S1 . Let S1U B denote the maximum value of these two conditions, which is calculated as follows: n o 0 S1U B = max S2U B (b1 ), min {x|c1 (x + 1) > c1 (x)} . For each value of 0 ≤ S1 ≤ S1U B we then construct the intervals for the second periods using Algorithm 8 and calculate the corresponding probabilities for each interval using Equation B.3. Given these, we can calculate V3 (S1 , n, ∞) making use of the relation described in Equation B.4. By then taking the smallest value of the total expected costs over all values of 0 ≤ S1 ≤ S1U B , we approximately obtain the value of V2 (∞). Algorithm 9 describes the steps taken to calculate V2 (∞). Algorithm 9 Calculation of V2 (∞) Step 1. Set V2 (∞) := ∞, S1 = 0 Step 2. Apply Algorithm 8 to obtain a02 (k), b02 (k), k ∈ {1, 2, . . . , K} Step 3. Calculate ψk0 (S1 ) using Equation (B.3) Step 4. Calculate total expected costs: n Rb V2 (∞) := min V2 (∞), c1 (S1 ) + a11 Step 6. If S1 > S1U B : Stop. Else: S1 := S1 + 1, go to Step 2.

V30 (S1 ,x) b1 −a1 dx

Bibliography Akcay, A., Biller, B., Tayur, S., 2011. Improved inventory targets in the presence of limited historical demand data. Manufacturing & Service Operations Management 13 (3), 297–309. Alfredsson, P., Verrijdt, J., 1999. Modeling emergency supply flexibility in a twoechelon inventory system. Management Science 45 (10), 1416–1431. Allen, S., 1958. Redistribution of total stock over several user locations. Naval Research Logistics Quarterly, 51–59. Andersson, J., Melchiors, P., 2001. A two-echelon inventory model with lost sales. International Journal of Production Economics 69 (3), 307–315. Armstrong, D., 1985. Sharpening inventory management. Harvard Business Review (December), 42–43,46–48,50–51,54,58. Axsäter, S., 1990. Modelling emergency lateral transshipments in inventory systems. Management Science 36, 1329–1338. Axsäter, S., Howard, C., Marklund, J., 2013. A distribution inventory model with transshipments from a support warehouse. IIE Transactions 45:3, 309–322. Axsäter, S., Kleijn, M., De Kok, T., 2004. Stock rationing in a continuous review two-echelon inventory model. Annals of Operations Research 126, 177–194. Babai, M., Dallery, Y., Boubaker, S., Kalai, R., 2018. A new method to forecast intermittent demand in the presence of inventory obsolescence. International Journal of Production Economics. Babai, M., Ladhari, T., Lajili, I., 2015. On the inventory performance of multi-criteria 133

134

BIBLIOGRAPHY

classification methods: empirical investigation. International Journal of Production Research 53:1, 279–290. Bacchetti, A., Plebani, F., Saccani, N., Syntetos, A., 2013. Empirically-driven hierarchical classification of stock keeping units. International Journal of Production Economics 143, 263–274. Bacchetti, A., Saccani, N., 2012. Spare parts classification and demand forecasting for stock control: Investigating the gap between research and practice. Omega 40, 722–737. Bagchi, U., Hayya, J., Chu, C., 1986. The effect of leadtime variability: the case of independent demand. Journal of operations management 6, 159–177. Basten, R., van Houtum, G., 2014. System-oriented inventory models for spare parts. Surveys in Operations Research and Management Science 19, 34–55. Benjaafar, S., Cooper, W., Mardan, S., 2011. Production-inventory systems with imperfect advance demand information and updating. Naval Research Logistics 58 (2), 88–106. Boucherie, R. J., van Houtum, G.-J., Timmer, J., van Ommeren, J.-K., 2017. A twoechelon spare parts network with lateral and emergency shipments: A product-form approximation. Probability in the Engineering and Informational Sciences, 1–20. Buxey, G., 2006. Reconstructing inventory management theory. International Journal of Operations and Production Management 26 (9), 996–1012. Cavalieri, S., Garetti, M., Macchi, M., Pinto, R., 2008. A decision-making framework for managing maintenance spare parts. Production Planning & Control 19 (4), 379–396. Chen, F., Yu, B., 2005. Quantifying the value of leadtime information in a singlelocation inventory system. Manufacturing & service operations management 7 (2), 144–151. Chu, C., Liang, G., Liao, C., 2008. Controlling inventory by combining abc analysis and fuzzy classification. Computers and Industrial Engineering 55, 841–851. Cohen, M., Agrawal, N., V., A., 2006. “winning in the aftermarket”. Harvard Business

BIBLIOGRAPHY

135

Review May issue, 129–138. Croston, J., 1972. Forecasting and stock control for intermittent demands. Operational Research Quarterly 23 (3), 289–303. Drent, M., Arts, J., 2018. Expediting in two-echelon spare parts inventory systems. Working paper. Driessen, M., Arts, J., van Houtum, G., Rustenburg, W., Huisman, B., 2010. Maintenance spare parts planning and control: A framework for control and agenda for future research. Beta working paper 325. Duchessi, P., Tayi, G., Levy, J., 1988. A conceptual approach for managing of spare parts. International Journal of Physical Distribution and Materials Management 18 (5), 8–15. Ehrhardt, R., 1984. (s,s) policies for a dynamic inventory model with stochastic leadtimes. Operations Research 32, 121–132. Eilon, S., Mallya, R. V., 1985. An extension of the classical ABC inventory control system. Omega 13 (5). Ernst, R., Cohen, M., 1990. Operations related groups (orgs): A clustering procedure for production/inventory systems. Journal of Operations Management 9 (4), 574– 598. Feeney, G., Sherbrooke, C., 1966. The (s-1,s) inventory policy under compound poisson demand. Management Science 12 (5), 391–411. Flores, B., Olson, D., Dorai, V., 1992. Management of multicriteria inventory classification. Mathematical and Computer Modelling 16, 71–82. Flores, B., Whybark, D., 1987. Implementing multiple criteria ABC analysis. Journal of Operations Management 7 (1-2), 79–85. Ge, Q., Peng, H., Van Houtum, G., Adan, I., 2018. Reliability optimization for series systems under uncertain component failure rates in the design phase. International Journal of Production Economics 196 (2), 163–175. Gerchak, Y., Mossman, D., 1991. On the effect of demand randomness on inventories and costs. Operations Research 40 (4), 804–807.

136

BIBLIOGRAPHY

Graham, G., 1987. Distribution Inventory Management for the 1990s. Inventory Management Press, Richardson, Texas. Grahovac, J., Chakravarty, A., 2001. Sharing and lateral transshipments of inventory in a supply chain with expensive low-demand items. Management Science 47, 579– 594. Graves, S., 1985. A multi-echelon inventory model for a repairables item with onefor-one replenishment. Management Science 31 (10), 1247–1256. Harrington, L., 2007. From just in case to just in time. Air Transport World, 77–80. Hatefi, S., Torabi, S., 2015. A common weight linear optimization approach for multicriteria abc inventory classification. Advances in Decision Sciences 2015. Hayes, R., 1969. Estimation problems in inventory control. Management Science 15 (11), 686–701. Hollier, R., Vrat, P., 1978. A proposal for classification of inventory systems. Omega 6, 277–279. Howard, C., Marklund, J., Tan, T., Reijnen, I., 2015. Inventory control in a spare parts distribution system with emergency stocks and pipeline information. Manufacturing and Service Operations Management 17 (2), 142–156. Hu, Q., Boylan, J., Chen, H., Labib, A., 2018. Or in spare parts management: A review. European Journal of Operational Research 266, 395–414. Jacobs, F., Chase, R., Aquilano, N., 2009. Operations and Supply Management. McGraw-Hill, New York. Jemai, Z., Karaesmen, F., 2005. The influence of demand variability on the performance of a make-to-stock queue. European Journal of Operational Research 164 (1), 195–205. Kamath, K. R., Pakkala, T., 1999a. A bayesian approach to a dynamic inventory model under an unknown demand distribution. Computers & Operations Research 29, 403–422. Kamath, K. R., Pakkala, T., 1999b. A bayesian approach to a dynamic inventory model under an unknown demand distribution. Computers & Operations Research

BIBLIOGRAPHY

137

29, 403–422. Karush, W., 1957. A queueing model for an inventory problem. Operations Research 5, 693–703. Knod, E., Schonberger, R., 2001. Operations Management: Meeting Customer Demands. 7th ed. McGraw-Hill, New York. Kranenburg, A., Van Houtum, G., 2009. A new partial pooling structure for spare parts networks. European Journal of Operational Research 199 (3), 908–921. Kutanoglu, E., 2008. Insights into inventory sharing in service parts logistics systems with time-based service levels. Computers and Industrial Engineering 54, 341–358. Kutanoglu, E., Mahajan, M., 2009. An inventory sharing and allocation method for a multi-location service parts logistics network with time-based service levels. European Journal of Operational Research 194, 728–742. Lee, C., 2002. Demand chain optimization: Pitfalls and key principles. NONSTOP’s “supply chain management seminar”, White Paper Series http://www.idii.com/wp/nonstop_dco.pdf. Liu, J., Liao, X., Zhao, W., Yang, N., 2015. A classification approach based on the outranking model for multiple criteria abc analysis. Omega. Martinetti, A., Braaksma, A., Ziggers, J., van Dongen, L., 2017. On the initial spare parts assortment for capital assets: A structured approach aiding initial spare parts assortment decision-making (saisad). In: Advances in Through-life Engineering Services. Springer, pp. 429–442. Millstein, M., Yang, L., H., L., 2014. Optimizing abc inventory grouping decisions. International Journal of Production Economics 148, 71–80. Muckstadt, J., 2005. Analysis and algorithms for service part supply chains. Springer: Berlin. Muckstadt, J., Thomas, L., 1980. Are multi-echelon inventory methods worth implementing in systems with low-demand-rate items? Management Science 26, 483–494. Nahmias, S., 1979. Simple approximations for a variety of dynamic leadtime lost-sales inventory models. Operations Research 27 (5), 904–924.

138

BIBLIOGRAPHY

Nahmias, S., 1997. Production and operations analysis. Irwin, Boston. Ng, W., 2005. A simple classifier for multiple criteria abc analysis. European Journal of Operational Research 177, 344–353. Oliva, R., Kallenberg, R., 2003. Managing the transition from products to services. International Journal of Service Industry Management 14 (2), 160–172. Öner, K., Kiesmüller, G., Van Houtum, G., 2007. Life cycle costs measurement of complex systems manufactured by an engineer-to-order company. In: Qui, R.G., Russell, D.W., Sullivan, W.G., Ahmad, M. (eds.) The 17th International Conference on Flexible Automation and Intelligent Manufacturing, FAIM, Philadelphia, 589– 596. Özer, Ö., 2011. Inventory management: information, coordination, and rationality. In: Advances in Through-life Engineering Services. Springer, Boston, pp. 321–365. Ozkan, E., Van Houtum, G., Serin, Y., 2015. A new approximate evaluation method for two-echelon inventory systems with emergency shipments. Annals of Operations Research 224, 147–169. Paterson, C., Kiesmüller, G., Teunter, R., Glazebrook, K., 2011. Inventory models with lateral transshipments: A review. European Journal of Operational Research 210, 125–136. Porteus, E., 2002. Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford. Prak, D., Teunter, R., Syntetos, A., 2017. On the calculation of safety stocks when demand is forecasted. European Journal of Operational Research 256 (2), 454–461. Ramanathan, R., 2006. ABC inventory classification with multiple-criteria using weighted linear optimization. Computers and Operations Research 33 (3), 695–700. Reijnen, I., Tan, T., Van Houtum, G., 2009. Inventory planning for spare parts networks with delivery time requirements. Working paper. Ridder, A., Van der Laan, E., Salomon, M., 1998. How larger demand variability may lead to lower costs in the newsvendor problem. Operations Research 46 (6), 934–936.

BIBLIOGRAPHY

139

Ross, S., 1983. Stochastic Processes. John Wiley, New York. Rustenburg, W., van Houtum, G. J., Zijm, W., 2004. Exact and approximate analysis of multi-echelon, multi-indenture spare parts systems with commonality. In: Shantikhumar, J., Yao, D., Zijm, W. (Eds.), Stochastic modeling and optimization of manufacturing systems and supply chains. Boston: Kluwer Academic, Ch. 7, pp. 143–176. Sherbrooke, C., 1968. Metric: A multi-echelon technique for recoverable item control. Operations Research 16 (1), 122–141. Sherbrooke, C., 1986. Vari-metric: Improved approximations for multi-indenture, multi-echelon availability models. Operations Research 34 (2), 311–319. Sherbrooke, C., 2004. Optimal inventory modeling of systems: Multi-echelon techniques. Wiley. Silver, E., Pyke, D., Peterson, R., 1998. Inventory management and production planning and scheduling. 3rd ed. John Wiley & Sons. Slack, N., Chambers, S., Johnston, R., 2007. Operations Management. Prentice Hall, Harlow. Song, J., 1994. The effect of leadtime uncertainty in a simple stochastic inventory model. Management Science 40 (5), 603–613. Song, J., Zhang, H., Hou, Y., Wang, M., 2010. The effect of lead time and demand uncertainties in (r,q) inventory systems. Operations Research 58 (1), 68–80. Stevenson, W., 2007. Operations Management. McGraw-Hill, New York. Stock, J., Lambert, D. M., 2001. Strategic Logistics Management. 4th ed. IrwinMcGraw Hill, New York. Svoronos, A., Zipkin, P., 1991. Evaluation of one-for-one replenishment policies for multiechelon inventory systems. Management Science 37, 68–83. Syntetos, A. A., Babai, M. Z., Gardner Jr, E. S., 2015. Forecasting intermittent inventory demands: simple parametric methods vs. bootstrapping. Journal of Business Research 68 (8), 1746–1752.

140

BIBLIOGRAPHY

Tan, T., Güllü, R., Erkip, N., 2007. Modelling imperfect advance demand information and analysis of optimal inventory policies. European Journal of Operational Research 177 (2), 897–923. Teunter, R., Babai, M., Syntetos, A., 2010. ABC Classification: Service Levels and Inventory Costs. Production and Operations Management 19, 343–352. Teunter, R., Duncan, L., 2009. Forecasting intermittent demand: a comparative study. Journal of the Operational Research Society 60, 321–329. Teunter, R., Syntetos, A., Babai, M., 2017. Stock keeping unit fill rate specification. European Journal of Operational Research 259, 917–925. Topan, E., Tan, T., Van Houtum, G.-J., Dekker, R., 2018. Using imperfect advance demand information in lost-sales inventory systems with the option of returning inventory. IISE Transactions 50 (3), 246–264. Tsai, C., Yeh, S., 2008. A multiple objective particle swarm optimization approach for inventory classification. International Journal of Production Economics 114, 656–666. Van Aspert, M., 2014. Design of an integrated global warehouse and field stock planning concept for spare parts. van Dongen, L., 2011. Maintenance engineering: Instandhouding van verbindingen, inaugural lecture, university of twente. Van Houtum, G. J., Kranenburg, A. A., 2015. Spare Parts Inventory Control under System Availability Constraints. Springer, New York. Van Kampen, T., Akkerman, R., Van Donk, D., 2012. Sku classification: a literature review and conceptual framework. International Journal of Operations and Production Management 32, 850–876. van Wijk, A., Adan, I., Van Houtum, G. J., 2012. Approximate evaluation of multilocation inventory models with lateral transshipments and hold back levels. European Journal of Operational Research 218 (3), 624–635. van Wijk, A., Adan, I., Van Houtum, G. J., 2013. Optimal allocation policy for a multi-location inventory system with a quick response warehouse. Operations

BIBLIOGRAPHY

141

Research Letters 41 (3), 305–310. Whitt, W., 1985. Uniform conditional variability ordering of probability distributions. Journal of Applied Probability 22, 619–633. Wong, H., Kranenburg, B., Van Houtum, G. J., Cattrysse, D., 2007. Efficient heuristics for two-echelon spare parts inventory systems with an aggregate mean waiting time constraint per local warehouse. OR Spektrum 29, 699–722. Wong, H., van Houtum, G. J., Cattrysse, D., van Oudheusden, D., 2005. Simple, efficient heuristics for multi-item, multi-location spare parts systems with lateral transshipments and waiting time constraints. Journal of the Operational Research Society 56, 1419–1430. Xu, M., Chen, Y., Xu, X., 2010. The effect of demand uncertainty in a price-setting newsvendor model. European Journal of Operational Research 207 (2), 946–957. Zhang, R., Hopp, W., Supatgiat, C., 2001. Spreadsheet implementable inventory control for a distribution centre. Journal of Heuristics 7 (2), 185–203. Zhou, P., Fan, L., 2007. A note on multi-criteria ABC inventory classification using weighted linear optimization. European Journal of Operational Research 182 (3), 1488 – 1491. Zipkin, P., 1986. Stochastic leadtimes in continuous time inventory models. Naval Research Logistics Quarterly 33, 763–774. Zipkin, P., 1991. Evaluation of base-stock policies in multiechelon inventory systems with compound-poisson demands. Naval Research Logistics 38, 763–774.

Summary System-focused spare parts management for capital goods When a capital good, such as the lithography machine of ASML, used in a 24 hour production process is out of order for a period of time, this has a big impact on the entire production process and will cost the production company a large amount of money. Also for many other capital goods, the costs of not being able to operate are often considerable. A proper management of the maintenance of capital goods is important to make sure downtime of these capital goods is kept to a minimum. Capital goods generally consist of many different critical parts which cause the machine to stop working when a part has broken down. In case this happens, it is generally not easy nor the fastest option to repair the broken part on site, and therefore the part is preferably replaced by a spare part to bring the capital good back up and running as fast as possible. However, simply stocking an ample amount of stock is also not desirable as this results in high holding costs as parts can be very expensive. Therefore, one generally prefers to have the highest availability of capital goods while at the same time minimizing the inventory holding costs of spare parts. In this thesis, we focus on developing models with this aim in mind. We consider models that are applicable for the initial phase of a capital good as well as models for the stationary phase of a capital good. To make stocking decisions, a company generally has estimates on the expected lifetimes of critical components, and hence on the demand rates, but these demand rates may be highly uncertain. This holds in particular during the initial phase of a newly designed system. This uncertainty in the demand rates leads to a double demand uncertainty: the common uncertainty that one also has under known demand rates and the additional uncertainty because of the uncertainty in the demand rates. When the rate is higher than expected, this involves high downtime costs whereas if the rate is lower than expected this involves unnecessary holding costs. In Chapter 2 we provide a model that is able to take the double demand uncertainty into consideration 143

144

BIBLIOGRAPHY

for the decision making. We show that, depending on the setting and distribution used to model the uncertainty, costs can increase to over 18 times the costs of the solution under known demand rates. More importantly, when one decides not to take the double demand uncertainty into consideration when deciding upon the base stock levels it is likely that this will result in a lot of downtime. Having shown the benefit of more reliable demand rate information, we investigated how the decisions should change if there is a possibility to invest in more reliable demand rate information in Chapter 3. We consider a finite horizon that models the first years of the exploitation phase of a capital good, where decisions are taken at the start of the horizon on the initial base stock level and the investment in more reliable demand information and on the adjustment of the base stock level at a later time point. The problem is modeled as a three-stage dynamic program and we derive analytical results to obtain the optimal base stock levels. We then numerically determine the optimal investment in more reliable demand rate information. Based on these results, we show that the price of a part and the level of uncertainty have the largest influence on how much can potentially be saved by investing in more reliable demand rate information. Another interesting observation we made is that the optimal base stock level at the start of the time horizon, when there is still a large amount of double demand uncertainty, is often rather high in order to avoid large downtime costs. In Chapter 4 we introduced a model for two-echelon spare parts networks where we included the use of an emergency warehouse. We included the use of an emergency warehouse to distinguish between a regular replenishment order and emergency request at the central warehouse. The emergency warehouse is used to keep a number of spare parts separate which are only used in case of an emergency request. We developed an approximate evaluation procedure to evaluate the performance of this network. This approximate evaluation procedure has not only shown to be accurate but also applicable to many different network structures. Based on the approximate evaluation procedure we investigated the benefit of an emergency warehouse. We show that in particular when local warehouses are far apart from each other, and thus lateral transshipments are not very attractive, there is more benefit in using an emergency warehouse. When local warehouses are located closely to each other and there is plenty of opportunity for lateral transshipments, there is hardly any benefit for the use of such an emergency warehouse. Finally, in Chapter 5 we developed a generalized approach to determine a classification scheme that enables companies to manage spare parts inventory in a simple way while still getting close-to-optimal. We show that getting close to system-focused

BIBLIOGRAPHY

145

inventory control is possible by taking the following four aspects into consideration: the class sizes, number of classes, ranking criteria, and the service level targets per class. We also show that if one of these aspects is not chosen properly, this may lead to poor results.

About the author Erwin van Wingerden was born in Dordrecht on November 13, 1987. He received his BSc and MSc in Econometrics and Management Science from the Erasmus University in Rotterdam in 2011 and 2012, respectively. His MSc thesis titled â&#x20AC;&#x153;Getting a better grip on forecasting intermittent partsâ&#x20AC;? was done under the supervision of Rommert Dekker and Rob Basten during his internship at Gordian Logistics Experts. After finishing his MSc, he started the PDEng traineeship at the Eindhoven University of Technology, which he finished in 2014. His final project at KLM Equipment Services was done under the supervision of Geert-Jan van Houtum. From February 2014 to March 2018 he was a PhD candidate in Eindhoven working on an NWO-TOP research project in collaboration with ASML and NS Services and Maintenance under the supervision of Tarkan Tan and Geert Jan van Houtum. During his last year as a PhD candidate he also collaborated with Lisa Maillart from the University of Pittsburgh on Chapter 3.

147