Optimal Control of One-Warehouse Multi-Retailer Systems with Discrete Demand M.K. Doğru A.G. de Kok G.J. van Houtum Department of Technology Management, Technische Universiteit Eindhoven Eindhoven, Netherlands

System Under Study One warehouse serving N retailers, external supplier with ample stock, single item Retailers face stochastic, stationary demand of the customers Backlogging, No lateral transshipments Centralized control  single decision maker, periodic review Operational level decisions: when & how much to order 2 S 0 warehouse C C C.... N 2 1 retailers

Literature Clark and Scarf [1960] –Allocation problem –Decomposition is not possible, balance of retailer inventories –Optimal inventory control requires solving a multi-dimensional Markov decision process: Curse of dimensionality –Solution is state dependent Eppen and Schrage [1981] –W/h cannot hold stock (cross-docking point) –Base stock policy, optimization within the class –Balance assumption (allocation assumption)

Literature Federgruen and Zipkin [1984a,b] –Balance assumption –Optimality results for finite horizon problem, w/h is a cross-docking point –Optimality results for infinite horizon problem with identical retailers and stock keeping w/h Diks and De Kok [1998] –Extension of optimality results to N-echelon distribution systems Literature on distribution systems is vast –Van Houtum, Inderfurth, and Zijm [1996] –Axsäter [2003] 4

Literature Studies that use balance assumption: Eppen and Schrage [1981], Federgruen and Zipkin [1984a,b,c], Jönsson and Silver [1987], Jackson [1988], Schwarz [1989], Erkip, Hausman and Nahmias [1990], Chen and Zheng [1994], Kumar, Schwarz and Ward [1995], Bollapragada, Akella and Srinivasan [1998], Diks and De Kok [1998], Kumar and Jacobson [1998], Cachon and Fisher [2000], Özer [2003] 5

Motivation Optimality results up to now are for continuous demand distributions This study aims to extend the results to discrete demand distributions Why discrete demand? –It is possible to handle positive probability mass at any point in the demand distribution, particularly at zero. –Intermittent (lumpy) demand 6

System Under Study W/h orders from an external supplier; retailers are replenished by shipments Fixed leadtimes Added value concept Backordering, penalty cost Objective: Minimize expected average holding and penalty costs in the long-run 7 0 N

Analysis: Preliminaries Echelon stock concept Echelon inventory position = Echelon stock + pipeline stock 8 0 N Echelon inventory position of w/h Echelon stock of w/h Echelon stock of 2 Echelon inventory position of

Analysis: Dynamics of the System

Analysis: Echelon Costs 10

Analysis: Costs attached to a period

Analysis: Optimization Problem 12

Analysis: Allocation Decision Suppose at the time of allocation ( t+l 0 ), the sum of the expected holding and penalty costs of the retailers in the periods the allocated quantities reach their destinations ( t+l 0 +l i ) is minimized. 13 Myopic allocation Balance Assumption: Allowing negative allocations

Analysis: Allocation Decision Example 1: N=3, identical retailers 14 Balanced Allocation is feasible

Analysis: Allocation Decision Example 2: N=3, identical retailers 15 Balanced Allocation is infeasible

Analysis: Balance Assumption Interpretations –Allowing negative allocations –Permitting instant return to the warehouse without any cost –Lateral transshipments with no cost and certain leadtime 16

Under the balance assumption, only depends on the ordering and allocation decisions that start with an order of the w/h in period t. 17 Analysis: Allocation Decision

Analysis: Single Cycle Analysis 18 Retailers: N=2

Analysis: Single Cycle Analysis 19 Allocation Problem –Necessary and sufficient optimality condition –Incremental (Marginal) allocation algorithm – is convex

Analysis: Single Cycle Analysis 20 Warehouse Optimal policy is echelon base stock policy

Infinite Horizon Problem 21

Newsboy Inequalities Existence of non-decreasing optimal allocation functions. Bounding Newsboy Inequalities –Optimal warehouse base stock level –Newsboy inequalities are easy to explain to managers and non- mathematical oriented students –Contribute to the understanding of optimal control 22

Conclusions Under the balance assumption, we extend the decomposition result and the optimality of base stock policies to two-echelon distribution systems facing discrete demands. –Retailers follow base stock policy –Shipments according to optimal allocation functions –Given the optimal allocation functions, w/h places orders following a base stock policy Optimal base stock levels satisfy newsboy inequalities –Distribution systems with cont. demand: Diks and De Kok [1998] We develop an efficient algorithm for the computations of an optimal policy 23

Further Research N-stage Serial System with Fixed Batches –Chen [2000]: optimality of (R,nQ) policies –Based on results from Chen [1994] and Chen [1998] we show that optimal reorder levels follow from newsboy inequalities (equalities) when the underlying customer demand distribution is discrete (continuous)

Further Research Eppen and Schrage [1981], Federgruen and Zipkin [1984a,b,c], Jönsson and Silver [1987], Jackson [1988], Schwarz [1989], Erkip, Hausman and Nahmias [1990], Chen and Zheng [1994], Kumar, Schwarz and Ward [1995], Bollapragada, Akella and Srinivasan [1998], Diks and De Kok [1998], Kumar and Jacobson [1998], Cachon and Fisher [2000], Özer [2003] Doğru, De Kok, and Van Houtum [2004] –Numerical results show that the balance assumption (that leads to the decomposition; as a result, analytical expressions) can be a serious limitation. No study in the literature that shows the precise effect of the balance assumption on expected long-run costs 25

Further Research Optimal solution by stochastic dynamic programming –true optimality gap, precise effect of the balance assumption –how good is the modified base stock policy Model assumptions –discrete demand distributed over a limited number of points –finite support Developed a stochastic dynamic program Partial characterization of the optimal policy both under the discounted and average cost criteria in the infinite horizon –provides insight to the behavior of the optimal policy –finite and compact state and action spaces –value iteration algorithm 26

Preliminary Results: Identical Retailers Test Bed: 72 instances –N=2 –w.l.o.g. –Parameter setting –demand ~ [0,1,2,3,4,5] –LB-UB gap > 2.5% 27

Preliminary Results: Identical Retailers 28 : optimality gap : (UB-LB)/LB*100

Analysis: Single Cycle Analysis 29 Single-echelon: discrete Two-echelon: continuous Two-echelon: discrete