Multiple sourcing in single- and multi-echelon inventory systems

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1 Multiple sourcing in single- and multi-echelon inventory systems I n a u g u r a l d i s s e r t a t i o n zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften der Universität Mannheim vorgelegt von Steffen Klosterhalfen aus Mannheim

2 Dekan: Prof. Dr. Hans H. Bauer Erstberichterstatter: Prof. Dr. Stefan Minner Zweitberichterstatter: Prof. Dr. Moritz Fleischmann Tag der mündlichen Prüfung: 31. Mai 2010

3 To my parents and Agathe

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5 Acknowledgements Even though it has been already four and a half years since I have started working on the research that finally resulted in this thesis, it does not feel that long to me. It is commonly known that time flies when you are having stress...or was it fun? For me it has been definitely a mixture of both, with the latter one prevailing. Although very demanding and straining at times, I have really enjoyed my time at the (Department of Logistics at the) University of Mannheim. The reason for this having been such an extraordinary and positive experience in my life is most importantly due to the people I have met, learned from, and worked together with during this period. Without these outstanding individuals I would have never been able to complete this work. First and foremost, I would like to thank my supervisor, Prof. Dr. Stefan Minner. With his great enthusiasm for doing research and his creativity he was the one who aroused my own interest in the research world during my diploma thesis and encouraged me to embark on this exciting journey. I have benefited tremendously from his valuable comments, his innovative approaches to research problems, and his persistent pursuit of achieving results of highest quality. I am deeply thankful to Stefan for teaching me how to conduct research by setting such a great example himself: always viewing problems from different angles, critically analyzing results, and thus deriving sound conclusions. Furthermore, I am most grateful to Prof. Dr. Moritz Fleischmann for reviewing my thesis. Beyond that I have sincerely appreciated the warm welcome at his chair and his readiness to answer questions and discuss problems of various nature at any time. During my research stays at Eindhoven University of Technology and Boston University I have had the opportunity to get to know many interesting people and various V

6 Acknowledgements VI distinguished researchers. Two of them I would like to emphasize here in particular: Prof. Dr. Gudrun Kiesmüller and Professor Sean Willems. It has been a fruitful and exceptional experience for me to collaborate with them on joint research projects. I owe my deepest gratitude to both of them for all the advice they have provided and the progress I have made thanks to them. The close bond I have formed with several (former and present) colleagues at the University of Mannheim significantly enhanced my learning, on and off the academic track. I can deem myself extremely lucky for having been part of such a great team. My special thanks go to Dr. Jan Arnold, Daniel Dittmar, Gabriele Eberhard, Dr. David Francas, Dr. Jörn Grahl, Stefan Hahler, Dr. Mirko Kremer, Dr. Lena Kunze, Ruth Pfitzmann, Volker Ruff, Dr. Holger Stephan, Dr. Sandra Transchel, Dr. Joachim Vaterrodt, Yao Yang, and Caroline Zuber. Moreover, my colleagues and friends from the Dean s Office of the Business School and the Graduate School of Economic & Social Sciences have greatly contributed to an amazing work and social environment. Most importantly, I would like to express my dearest gratitude to my family. During the entire research process, especially my parents and my sister have provided me with unconditional support and encouragement. Knowing that I could always rely on them gave me the strength I needed to achieve this goal. Thanks a million to all of you. It is my habit to save the best for last...and I won t break it here. In this case, the best is you, Agathe. I am eternally grateful for you having been there for me particularly during these stressful last couple of months, when I was finishing my thesis. You have been immensely understanding about me having so little time for you...and you always are about all my other minor and major quirks. You are the best that could have ever possibly happened to me. Mannheim, July 2010

7 Contents Acknowledgements List of Figures List of Tables List of Abbreviations List of Symbols V XII XV XVI XVII 1 Introduction Motivation Research questions Structure and overview Fundamentals and literature review Fundamentals Demand Continuous distributions Discrete distributions Discretized (continuous) distribution Inventory control Inventory level classifications Performance measures for inventory control Single-echelon order-up-to level model with single sourcing Supply network modeling Stockpoint, process, and stage VII

8 Contents VIII Network structures Literature review Single-echelon inventory models with multiple sourcing options Derivation of the optimal policy Parameter determination for a given (non-optimal) policy Multi-echelon inventory models with single sourcing Stochastic-service approach Guaranteed-service approach Comparison and combination of the stochastic- and guaranteed-service approach Multi-echelon inventory models with multiple sourcing options 46 3 Single-echelon inventory model with dual sourcing Introduction Assumptions and notations Inventory control policies Definitions Optimal policy General Markov Decision Process formulation Application to the dual-sourcing inventory problem Non-optimal policies Single-index policy Constant-order policy Dual-index policy Order-splitting policy Summary and implications Comparison of the constant-order and dual-index policy Introduction Theoretical considerations Numerical study Numerical design Computational aspects Results

9 Contents IX Summary and implications Multi-echelon inventory model with dual sourcing Introduction Multi-echelon inventory optimization approaches Common assumptions Stochastic-service approach Optimization model Optimization procedure Guaranteed-service approach Standard optimization model and procedure Extended optimization model and procedure Summary and implications Comparison of the stochastic- and guaranteed-service approach Introduction Theoretical considerations Benefits of the approaches Allocation benefit of the stochastic-service approach Decoupling benefit of the guaranteed-service approach Numerical study Serial systems Divergent systems Convergent systems Summary and implications Combination of the stochastic- and guaranteed-service approach Introduction Serial systems Interface modeling Combination of the approaches Numerical study Divergent systems Convergent systems Summary and implications Guaranteed-service approach with dual sourcing Introduction

10 Contents X Serial and convergent systems Assumptions, notations, and changes to the standard guaranteed-service approach Single-echelon model Multi-echelon model Numerical example Other network structures Extensions Multiple sourcing Optimization of sourcing fractions Other inventory control policies Summary and implications Conclusions and outlook Conclusions Outlook Appendices 233 A Additional figures and tables 234 A.1 Comparison of the constant-order and dual-index policy for µ = A.2 Combination of the stochastic- and guaranteed-service approach B Proofs 242 B.1 Lemma B.2 Lemma B.3 Lemma B.4 Lemma B.5 Lemma B.6 Lemma B.7 Derivation of optimal δ s B.8 Lemma B.9 Lemma B.10 Lemma B.11 Property

11 Contents XI B.12 Property B.13 Lemma B.14 Lemma Bibliography 260

12 List of Figures 1.1 Sequential single-echelon vs. multi-echelon approach Stage structure Serial system Divergent system Convergent system Tree system General (acyclic) system Relationship between single- and dual-sourcing policy costs for L f = Serial system illustration GS approach Expedited items analysis Internal service level as a function of the operating flexibility cost Serial system Optimality share and relative benefit with respect to the processing time Serial system Optimality share and relative benefit with respect to the final-stage service level Serial system Optimality share and relative benefit with respect to the coefficient of variation of period demand Divergent system illustration Divergent system Optimality share and relative benefit with respect to the processing time Divergent system Optimality share and relative benefit with respect to the final-stage service level Divergent system Optimality share and relative benefit with respect to the coefficient of variation of period demand XII

13 List of Figures XIII 4.11 Interface modeling between an SS and a GS subnetwork System illustration with a hybrid-service (HS) stage Optimality share and additional average HS benefit with respect to the internal service-level ranges Optimality share and additional average HS benefit with respect to (a) and (b) Optimality share and additional average HS benefit with respect to (a) and (b) Divergent HS systems Convergent HS systems Dual-sourcing supply network represented by processes and stockpoints Supply network Stage assignment and current workaround Dual-sourcing network vs. assembly network Supply network explaining notations Case 1 Timeline for net stock calculation Case 2 Timeline for net stock calculation Supply network with dual sourcing Example Supply network with dual sourcing Example Sample network with dual sourcing and divergent substructure Sample network with single sourcing and general structure Supply network with multiple suppliers Timeline for the net stock calculation for a stockpoint with three suppliers Dual-sourcing subnetworks c addf -range for a dual-sourcing cost advantage (T f = 5, T s = 12 T f, ν = 45%, η = 250, k = 2.33, CV = 1) Simple supply network with dual sourcing A.1 Optimality share and additional average HS benefit with respect to the internal service-level ranges A.2 Optimality share and additional average HS benefit with respect to final-stage service level A.3 Optimality share and additional average HS benefit with respect to processing-time pattern

14 List of Figures XIV A.4 Optimality share and additional average HS benefit with respect to holding-cost pattern A.5 Optimality share and additional average HS benefit with respect to coefficient of variation of demand A.6 Optimality share and additional average HS benefit with respect to the internal service-level ranges A.7 Optimality share and additional average HS benefit with respect to final-stage service level A.8 Optimality share and additional average HS benefit with respect to processing-time pattern A.9 Optimality share and additional average HS benefit with respect to holding-cost pattern A.10 Optimality share and additional average HS benefit with respect to coefficient of variation of demand B.1 Distribution function F µy,σ Y (solid) and F µz,σ Z (dashed)

15 List of Tables 3.1 Relationship between DIP model and lost-sales model COP-DIP comparison Parameter values Single- vs. dual-sourcing cost for µ = Single- vs. dual-sourcing cost for L = 5, h = 0.5 and 1.0, µ = Single- vs. dual-sourcing cost for L = 10, h = 0.5 and 1.0, µ = CV -effect on TRC COP TRC DIP TRC DIP for (L f = 1, L s = 11), b b+h = 0.95, µ = Quantity details for (L f = 1, L s b = 11), = 0.95, gamma, µ = b+h 4.1 Expediting analysis for the extended GS approach Parameter values Serial system Parameter values for simulation runs Divergent system Parameter values for simulation runs Holding-cost and processing-time patterns HS result overview GS approach with dual sourcing Parameter values c addf -range for a dual-sourcing cost advantage (T f = 5, T s = 12 T f, ν = 45%, η = 250, k = 2.33) Dual-sourcing cost advantage (T f = 5, T s = 12 T f, c adds = 100, ν = 45%, η = 250, k = 2.33) A.1 Single- vs. dual-sourcing cost for µ = A.2 Single- vs. dual-sourcing cost for L = 5, h = 0.5 and 1.0, µ = A.3 Single- vs. dual-sourcing cost for L = 10, h = 0.5 and 1.0, µ = XV

16 List of Abbreviations ANOVA BS COP DIP DP e.g. gam geom GS HS i.e. i.i.d. LS MDP nbin norm n/s P ois P r SIP SS TC TRC vs. w.l.o.g. Analysis of variance Best single-sourcing strategy Constant-order policy Dual-index policy Dynamic program for example gamma geometric guaranteed-service hybrid-service that is identically and independently distributed Lost sales Markov Decision Process negative binomial normal not significant Poisson Probability Single-index policy stochastic-service Total cost Total relevant cost versus without loss of generality XVI

17 List of Symbols Single-echelon model with dual sourcing Demand d t D t D(L) D FD t f(x) F(x) f L F L φ(x) Φ(x) E[D] = µ VAR[D] = σ 2 σ CV = σ µ Γ(κ, θ) demand realization in period t demand random variable of period t demand random variable over L periods maximum feasible demand realization filled demand in period t single-period demand probability density/mass function single-period demand cumulative distribution function L-period demand probability density function L-period demand cumulative distribution function standard normal probability density function standard normal cumulative distribution function single-period demand expectation single-period demand variance single-period demand standard deviation coefficient of variation complete gamma function with shape parameter κ and scale parameter θ Timespans L L f L s L = L s L f (replenishment) lead time lead time when sourcing from the fast supplier lead time when sourcing from the slow supplier lead-time difference XVII

18 List of Symbols XVIII Cost, service, and performance measures c f c s c = c f c s h h e b α target, β target, γ target procurement cost per unit when sourcing from the fast supplier procurement cost per unit when sourcing from the slow supplier procurement cost difference or expediting premium per unit (local) holding cost per unit and period echelon holding cost per unit and period backorder cost per unit and period theoretical target service levels, which are exogenously given α, β, γ achieved service levels Inventory control and replenishment variables B B f B s = B s B f B LS δ f δ s NS t OH t BO t PI t IP t IP f t IPt s local order-up-to level local order-up-to level for replenishments with the fast supplier local order-up-to level for replenishments with the slow supplier difference in the local order-up-to levels order-up-to level in the single-sourcing lost-sales model fraction of demand sourced from the fast supplier fraction of demand sourced from the slow supplier net stock at the end of period t on-hand stock at the end of period t backorders at the end of period t pipeline inventory (i.e. in-transit to the stage) at the beginning of a period before any orders arrive inventory position at the beginning of period t before receipt and placement of any orders fast inventory position at the beginning of period t slow inventory position at the beginning of period t

19 List of Symbols XIX O t Q Q f t Q s t CAP f CAP s SST k(sl) overshoot at the beginning of period t after ordering from the fast supplier constant order quantity (under the COP) fast order quantity of period t slow order quantity of period t capacity limit on the fast order capacity limit on the slow order safety stock level safety factor for a given service level SL Markov Chain p ij P o = [o 1, o 2,...] Z t SSP transition probability from state i to j transition probability matrix steady-state overshoot distribution state vector of period t set of all possible states (state space) Markov Decision Process a action or decision DSP(i) set of all possible actions/decisions in state i (decision space) (a) a policy r(i, a) reward/cost of being in state i and choosing decision a AC((a) ) average cost or gain of a given policy (a) p ij (a) transition probability from state i to j when decision a is chosen π i Y t DC(a) HB(x) steady-state probability of being in state i state vector of period t direct cost of decision a expected holding and backorder cost function

20 List of Symbols XX Multi-echelon model with dual sourcing The notation already introduced for the single-echelon system also applies for the multi-echelon system with an additional stage/stockpoint index. Network i, j a k, i,j J A (i, j) LC(i) N i DS(i) FS DS j (i) N DS(j) i P type P MS K network that runs from stage i to j k i, j -matrix that shows which stages are part of the GS subnetwork/hs stage i, j set of all feasible GS subnetworks/hs stages arc set of the network representation arc from i to j level code of stockpoint i subset of stockpoints that are connected to i on the subgraph with stockpoints that have a higher level as i dual-sourcing subnetwork i, i.e. subset of stockpoints in the subgraph from one dual-sourced stockpoint (e.g. j) to the next i subset of stockpoints that are not part of any dualsourcing subnetwork, including the final stockpoint subset of stockpoints in DS(i), which are predecessors of stockpoint i s supplier j (including j) subset of all stockpoints in the DS(j) upstream of i number of stockpoints with type {SI, DS}, i.e. single or dual sourcing, respectively number of multiple-sourced stockpoints with K suppliers Demand ρ ij UD f UD s demand correlation between the single-period demands of products at stages/stockpoints i and j single-period demand random variable of the fast supplier single-period demand random variable of the slow supplier

21 List of Symbols XXI [UD f ] L [UD s ] L p f L-period demand random variable of the fast supplier L-period demand random variable of the slow supplier probability that an order is placed with the fast supplier Timespans L j,i T i /T j,i T f i T s i ST i ST f i STi s τ i τ i,j M i (replenishment) lead time between stockpoints j and i processing time of stage i/between stockpoints j and i processing time when stockpoint i sources from the fast supplier processing time when stockpoint i sources from the slow supplier (outgoing) service time of stage/stockpoint i service time of the fast supplier of stockpoint i service time of the slow supplier of stockpoint i coverage or net replenishment time of stage/stockpoint i vector of net replenishment times in the network from stage i to j maximum replenishment lead time of stage/stockpoint i Cost, service, and performance measures c OF i c add j,i c cum i ν η α SS i OS type pure operating flexibility cost per unit at stage i cost added when an item moves from stockpoint j to i cumulative cost of an item at stockpoint i holding-cost/interest rate for the underlying base period scalar that converts the COGS in the same time unit as the pipeline and safety stock cost α-service level at stage i, which is used for sizing S i in the SS approach optimality share of the respective approach (type {GS, SS}) within the two pure approaches, GS and SS

22 List of Symbols XXII OS type all RB type optimality share of the respective approach (type {GS, SS, HS}) within all three approaches relative benefit of the respective approach (type {GS, SS, HS}) Inventory control and replenishment variables B i,j vector of local order-up-to levels in the subnetwork from stage i to j S i F [S 1,...S i ] Q (j,i),t δ j,i echelon order-up-to level at stage i echelon distribution function at i (replenishment) quantity that i sources from j in period t sourcing fraction, i.e. fraction of the total order quantity that i sources from supplier j Optimization model x type i,j indicator variable that is 1, if the GS subnetwork (type = GS) or HS stage (type = HS) from i to j is chosen or 0 otherwise c type i,j cost of the on-hand stock in the GS subnetwork (type = GS) or HS stage (type = HS) from i to j Dynamic program z k Z k u k u k = (u 1 k, u2 k, u3 k ) U k (z k ) C SI k one-dimensional state variable of stage k state space of stage k one-dimensional decision variable of stage k three-dimensional decision variable of stage k decision space of stage k within a state z k minimum safety stock cost for the subnetwork with stockpoint set N k in case all items are delivered by a different supplier

23 List of Symbols XXIII C DS k C SIopt k C DSopt k minimum safety stock cost for the subnetwork with stockpoint set N k in case an item is delivered by two suppliers minimum of the sum of the safety stock cost for the subnetwork with stockpoint set N DS(i) k and the total cost of the preceding dual-sourcing subnetworks minimum total cost up to stockpoint k in case k is a dual-sourced stockpoint

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25 1 Introduction 1.1 Motivation According to a study of the Aberdeen Group (see Viswanathan (2007)), inventory management was ranked on top of the list of investments in application-oriented software for companies in Within inventory management, multi-tier or multiechelon inventory optimization was the top priority. This software application area grew by 32% while the overall supply chain management space grew about 7% (see Trebilcock (2009b)). Although in 2008 the market for supply chain management software applications and services saw only a slight increase of 4% over 2007 according to AMR Research (see Trebilcock (2009a)), companies are still putting much emphasis on improving their inventory management activities. 91% of over 170 companies that were surveyed by the Aberdeen Group in 2009 indicated that they have made, or have been asked to provide, recommendations in the past six months to management on how to improve their inventory management processes (see Viswanathan (2009)). An effective inventory management is particularly important in times of economic downturn, like the current global recession. In order to contain cost and free working capital, inventories need to be reduced. On the other hand, there is the risk of losing business in case of insufficient inventories. For the solution of this cost-service trade-off, management more and more often employs advanced software tools to support their decisions (see Ellis et al. (2009)). Traditional inventory management planning processes and software applications have only been capable of managing inventory at the individual site level. Even though a company might plan its inventory levels at several locations of its supply network centrally, the actual inventory optimization is done one location (or echelon) at a time (see Figure 1.1(a)). Such a sequential single-echelon approach completely 1

26 1.1 Motivation 2 neglects interdependencies between the sites, however. Thus, too much inventory might be held or inventory could end up in the wrong place, because aspects such as the following are not taken into account: Is it more costly to hold inventory at an upstream or downstream location? How does the order decision of a downstream location affect the demand process that the upstream location sees? Which level of service should the individual upstream locations provide to their internal customers such that the external customer demand can be satisfied according to the service target there? Lead time External Supplier Optimization site 2 Lead time External Supplier Simultaneous optimization site 1 and 2 Stockkeeping site 2 Stockkeeping site 2 Lead time Optimization site 1 Lead time Stockkeeping site 1 Stockkeeping site 1 Demand Demand External Customer External Customer (a) Sequential single-echelon approach (b) Multi-echelon approach Figure 1.1: Sequential single-echelon vs. multi-echelon approach In contrast, the inventory optimization software tools that have been developed over the past two decades take a holistic approach to inventory management. Such multiechelon approaches consider all locations in the supply network simultaneously, from the external supplier to the end-customer, with the objective of minimizing total inventory cost in the entire system subject to the service, which is to be guaranteed towards the external customer (see Figure 1.1(b)). Thus, the shortcomings of the sequential single-echelon approach are counteracted. It is reported that it is not unusual for a global supply chain to see inventory levels reduced by as much as 15-25% (Ellis et al. (2009)). There are two major drivers for these advances. First, information and computer technology has gone through great improvements mak-

27 1.1 Motivation 3 ing information across the entire supply network available for such a multi-echelon application and enabling the solution of very large and complex problems. Second, the advancement in multi-echelon inventory research in recent years has produced models that can capture and handle a broad variety of real-world problems and problem sizes (see, e.g., Willems (2008)). None of the software vendors such as LogicTools (which is now a part of IBM), Optiant R, SmartOps R, or ToolsGroup explain in detail the algorithms that are implemented in their tools. However, a closer look at the affiliated scientists and their scientific contributions in this area clearly suggests that these models and algorithms represent extensions of the two pioneering contributions to multi-echelon inventory research by Simpson (1958) and Clark and Scarf (1960). Based on these two seminal papers on multi-echelon inventory models in basestock/order-up-to level environments without lot-sizing, two competing research strands have developed over the years. Although they solve the same inventory optimization problem in their core, they make a different assumption with regard to the role of safety stock. The resulting consequences for the material flow in the system coined the terms full-delay and no-delay approaches (van Houtum et al. (1996)), or stochastic- and guaranteed-service approaches (Graves and Willems (2003)). In the stochastic-service (SS) approach, safety stock is assumed to be the only buffer against demand variability. The guaranteed-service (GS) approach, on the other hand, assumes that safety stock is sized to cover demand variability up to a certain level only, i.e. the normal or maximum reasonable demand. All variability exceeding this level is dealt with by other extraordinary countermeasures. Both assumptions are quite strong. In reality, the truth probably lies somewhere in between. A single supply network might consist of stages with low flexibility as well as stages with high flexibility. For the former ones, the SS approach would be the appropriate one to use, whereas the latter ones are appropriate candidates for the application of the GS approach. In both the academic literature and the currently available software applications, the two approaches are mainly treated and implemented as mutually exclusive frameworks, however. Since both approaches yield different results due to the differing underlying assumptions, a practitioner (as a buyer of these products) faces the dilemma of choosing the appropriate approach and thus software tool for the inventory optimization of its supply networks. This

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