1. 1 HEURISTICS FOR DYNAMIC SCHEDULING OF MULTI-CLASS BASE-STOCK CONTROLLED SYSTEMS Bora KAT and Zeynep Müge AVŞAR Department of Industrial Engineering Middle East Technical University Ankara TURKEY

2. 2 OUTLINE Two-class base-stock controlled systems Related work in the literature Analysis Model to maximize aggregate fill rate Solution approach Structure of the optimal dynamic (state-dependent) scheduling policy Heuristics to approximate the optimal policy Numerical Results Symmetric case (equal demand rates) Asymmetric case Optimality of the policy to minimize inventory investment subject to aggregate fill rate constraint to maximize aggregate fill rate under budget constraint Conclusion and future work

3. 3 SYSTEM CHARACTERISTICS Single facility to process items Exponential service times Poisson demand arrivals No set-up time Backordering case Preemption allowed Each item has its own queue managed by base-stock policies Performance measure: aggregate fill rate over infinite horizon

4. 4 TWO-CLASS BASE-STOCK CONTROLLED SYSTEM items of type i in process backorders of type i base-stock level for type i demand rate for type i service rate items of type i in stock

5. 5 RELATED WORK IN THE LITERATURE Zheng and Zipkin, 1990. A queuing model to analyze the value of centralized inventory information. Ha, 1997. Optimal dynamic scheduling policy for a make-to-stock production system. van Houtum, Adan and van der Wal, 1997. The symmetric longest queue system. Pena-Perez and Zipkin, 1997. Dynamic scheduling rules for a multi-product make-to-stock queue. Veatch and Wein, 1996. Scheduling a make-to-stock queue: Index policies and hedging points. de Vericourt, Karaesmen and Dallery, 2000. Dynamic scheduling in a make-to-stock system: A partial charact. of optimal policies. Wein, 1992. Dynamic scheduling of a multi-class make-to-stock queue. Zipkin, 1995. Perf. analysis of a multi-item production-inventory system under alternative policies. Bertsimas and Paschalidis, 2001. Probabilistic service level guarantees in make-to-stock manufacturing systems. Glasserman, 1996. Allocating production capacity among multiple products. Veatch and de Vericourt, 2003. Zero Inventory Policy for a Two-Part-Type Make-To-Stock Production System.

6. 6 RELATED WORK IN THE LITERATURE Zheng and Zipkin, 1990. A queuing model to analyze the value of centralized inventory information. symmetric case:identical demand (Poisson) and service (exponential) rates, identical inventory holding and backordering costs base-stock policy employed, preemption allowed main results on the LQ (longest queue) system closed form steady-state distribution of the difference between the two queue lengths closed form formulas for the first two moments of the marginal queue lengths a recursive scheme to calculate joint and marginal distributions of the queue lengths it is analytically shown that LQ is better than FCFS discipline under the long-run average payoff criterion alternative policy: specify (2S-1) as the maximum total inventory to stop producing imposing a maximum of S on each individual inventory  -policy as an extension of LQ for the asymmetric case, extension of the recursive scheme to calculate steady-state probabilities

7. 7 RELATED WORK IN THE LITERATURE Ha, 1997. Optimal dynamic scheduling policy for a make-to-stock production system. two item types allowing preemption, demand (Poisson) and service (exponential) rates, and inventory holding and backordering costs to be different perf. criterion: expected discounted cost over infinite horizon equal service rates: characterizing the optimal policy by two switching curves (base-stock policy, together with a switching curve, for a subset of initial inv. levels) different service rates: optimal to process the item with the larger  index when both types are backordered heuristics: static priority (  ) rule dynamic priority ( modified  and switching) rules

8. 8 RELATED WORK IN THE LITERATURE van Houtum et al., 1997. The symmetric longest queue system. symmetric multi-item case: identical demand (Poisson) and service (exponential) rates, base-stock policy employed, not preemptive performance measure is fill rate (the cost formulation used is the same) investigating (approximating) the performance of the LQ policy with two variants: threshold rejection and threshold addition (to find bounds)

9. 9 MODEL

10. 10 SOLUTION APPROACH Value iteration to solve for long-run avg. payoff No truncation

11. 11 OPTIMAL SCHEDULING POLICY: Symmetric, Finite-Horizon Case

12. 12 OPTIMAL SCHEDULING POLICY: Symmetric, Infinite-Horizon Case

13. 13 OPTIMAL SCHEDULING POLICY: Symmetric Case  S 1 =S 2 =9  S 1 =S 2 =9

14. 14 STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case Cost: c(n 1,n 2 )

15. 15 STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case Region I and Region III and Region IV and Region II and none in stockout LQ to avoid stockout for the item with higher risk both types in stockout SQ to eliminate stockout for more promising type type 1 in stockout B(n 1 ): threshold to be away from region III and to reach region I type 2 in stockout B(n 1 ): threshold to be away from region III and to reach region I

16. 16 HEURISTICS TO APPROXIMATE OPTIMAL POLICY: Symmetric Case (to approximate curve B) Best performance by heuristic 2. Heuristics 1 and 2 perform almost equally well. Heuristic 1  process type 1  process type 2 Heuristic 2  process type 1  process type 2

17. 17 HEURISTICS TO APPROXIMATE OPTIMAL POLICY: Symmetric Case Heuristic 3 Heuristic 4Heuristic 5 performs better than heuristics 3 and 4 for large . steady-state probability distribution by Zheng-Zipkin’s algorithm in LQ region and then proceeding recursively in SQ region.

18. 18 NUMERICAL RESULTS: Symmetric Case Fill Rate (%)

19. 19 NUMERICAL RESULTS: Symmetric Case 

20. 20 NUMERICAL RESULTS: Symmetric Case 

21. 21 THREE TYPES OF ITEMS: Symmetric Case Simulation results for LQ policy and Heuristic 2

22. 22 THE OPTIMAL SCHEDULING POLICY: Asymmetric Case  S 1 =S 2 =8  S 1 =S 2 =8

23. 23 HEURISTIC 1 TO APPROXIMATE OPTIMAL POLICY: Asymmetric Case (to approximate curves A and B) Region I Region II  process type 1  process type 2 Region III  process type 1  process type 2  process type 1  process type 2 Curve A approximated for regions I and III is the diagonal when 1 = 2.

24. 24 HEURISTIC 2 TO APPROXIMATE OPTIMAL POLICY: Asymmetric Case Region I Region II  process type 1  process type 2 Region III  process type 1  process type 2  process type 1  process type 2

25. 25 NUMERICAL RESULTS: Asymmetric Case Fill Rate (%) Instead of LQ policy, delta policy (Zheng-Zipkin): for 1 > 2, process type 2 when n 2 -n 1 > .

26. 26 NUMERICAL RESULTS: Asymmetric Case

27. 27 NUMERICAL RESULTS: Asymmetric Case

28. 28 NUMERICAL RESULTS: Asymmetric Case Weighted Cost Function Fill Rate (%) Indices used for the heuristics are multiplied by the respective w i. Heur-mult: index1 is multiplied also by (1-  ). (adjustment for high  )

29. 29 OPTIMAL POLICY for, Lower expected inventory levels under heuristic policies when  is not too small. Minimum Base-Stock Levels to satisfy Target Fill Rate

30. 30 COMPARISON OF THE POLICIES in terms of fill rate, exp. backorders and inventories

31. 31 COMPARISON OF THE POLICIES in terms of fill rate, exp. backorders and inventories

32. 32 MODEL TO INCORPORATE SET-UP TIME f: minimum cost while processing items of type 1 g : minimum cost while processing items of type 2 sf : minimum cost while setting up the facility for type 1 sg : minimum cost while setting up the facility for type 2  set-up rate

33. 33 CONCLUSION Summary multi-class base-stock controlled systems numerical investigation of the structure of the optimal policy for maximizing the weighted average of fill rates optimal policy for smaller than LQ (symmetric case),  (asymmetric case), FCFS policies give smaller expected inventory when  is not too small accurate heuristics adapted for extensions: asymmetric case, more than two types of items disadvantage: not that easy to implement compared to LQ,  and FCFS policies Future Work optimizing base-stock levels set-up time type-dependent processing time instead of working with aggregate fill rate how to determine the values of ?