Optimal Control of One-Warehouse Multi-Retailer Systems with Discrete Demand

1. Optimal Control of One-Warehouse Multi-Retailer Systems with Discrete Demand M.K. Doğru • A.G. de Kok • G.J. van Houtum m.k.dogru@tm.tue.nl • a.g.d.kok@tm.tue.nl • g.j.v.houtum@tm.tue.nl Department of Technology Management, Technische Universiteit Eindhoven Eindhoven, Netherlands

2. retailers 1 C warehouse S 0 2 C .... .... N C 2 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

3. ...... 3 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)

4. 4 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]

5. 5 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]

6. 6 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

7. 7 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 1 0 2 ...... N

8. ..... ..... 8 Analysis: Preliminaries 1 • Echelon stock concept • Echelon inventory position = Echelon stock + pipeline stock Echelon inventory position of 2 Echelon stock of w/h 0 2 Echelon stock of 2 Echelon inventory position of w/h ..... N

9. ..... ..... 2 9 Analysis: Dynamics of the System

10. 10 Analysis: Echelon Costs

11. ... 2 11 Analysis: Costs attached to a period

12. 12 Analysis: Optimization Problem

13. 13 Analysis: Allocation Decision • Suppose at the time of allocation ( t+l0 ), the sum of the expected holding and penalty costs of the retailers in the periods the allocated quantities reach their destinations ( t+l0 +li ) is minimized. Myopic allocation Balance Assumption: Allowing negative allocations

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

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

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

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

18. 18 Analysis: Single Cycle Analysis Retailers: N=2

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

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

21. 21 Infinite Horizon Problem

22. 22 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

23. 23 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

24. ...... 24 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).

25. 25 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

26. 26 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

27. 27 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%

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

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