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HEURISTICS FOR DYNAMIC SCHEDULING OF MULTI -CLASS BASE - STOCK CONTROLLED SYSTEM S

HEURISTICS FOR DYNAMIC SCHEDULING OF MULTI -CLASS BASE - STOCK CONTROLLED SYSTEM S B ora KAT and Zeynep M üge AVŞAR Department of Industrial Engineering Middle East Technical University Ankara TURKEY. OUTLINE. Two- c lass base - s tock controlled systems

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HEURISTICS FOR DYNAMIC SCHEDULING OF MULTI -CLASS BASE - STOCK CONTROLLED SYSTEM S

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  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. OUTLINE • Two-class base-stock controlled systems • Related work in the literature • Analysis • Model to maximize aggregate fill rate • Solution approach • Structure of the optimaldynamic (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 futurework

  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 managedby base-stock policies • Performance measure: aggregatefill rate over infinite horizon

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

  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 andvan 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. 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. 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. 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. MODEL

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

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

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

  13. OPTIMAL SCHEDULING POLICY: SymmetricCase r=0.4, S1=S2=9 r=0.9, S1=S2=9

  14. STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case Cost:c(n1,n2)

  15. STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case Region I and Region II and Region III and Region IV and none in stockout LQ to avoid stockout for the item with higher risk type 1 in stockout B(n1): threshold to be away from region III and to reach region I both types in stockout SQ to eliminate stockout for more promising type type 2 in stockout B(n1): threshold to be away from region III and to reach region I

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

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

  18. NUMERICAL RESULTS: SymmetricCase Fill Rate (%)

  19. NUMERICAL RESULTS: SymmetricCaser = 0.90

  20. NUMERICAL RESULTS: Symmetric Case r = 0.90

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

  22. THE OPTIMAL SCHEDULING POLICY: Asymmetric Case r=0.4, S1=S2=8 r=0.9, S1=S2=8

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

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

  25. NUMERICAL RESULTS: Asymmetric Case Fill Rate (%) Instead of LQ policy, delta policy (Zheng-Zipkin): for 1>2, process type 2 when n2-n1>.

  26. NUMERICAL RESULTS: Asymmetric Case

  27. NUMERICAL RESULTS: Asymmetric Case

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

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

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

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

  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 g : set-up rate

  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 ?

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