Smooth Priorities for MaketoStock Inventory Control Carlos F

Smooth Priorities for Make-to-Stock Inventory Control Carlos F. G. Bispo Instituto de Sistemas e Robótica – Instituto Superior Técnico Technical Univ. of Lisbon - Portugal Carlos Bispo Multi-echelon Inventory Conference, June 2001

Outline Ø Problem setting Ø Control policy class Ø Previous work Ø Framework Ø Capacity management Ø Main results and limitations Ø Smooth priorities Ø Results Ø Conclusions Carlos Bispo Multi-echelon Inventory Conference, June 2001 2

Problem Setting - I Ø Multiple Capacitated Machines Ø Each machine has a finite capacity; Ø M machines with Cm, for m = 1, …, M. Ø Multiple Products Ø Each product is characterized by an external stochastic demand; Ø P products with E[dp] and cvp, for p = 1, …, P. Ø Jumbled and re-entrant flow Ø Each product may have different paths through the system; Ø There can be more than a visit to each machine. Carlos Bispo Multi-echelon Inventory Conference, June 2001 3

Problem Setting - II Ø Periodic Review Ø In+1 = In + Pn - (Pn)- Ø Performance Measures Ø Operational Cost based t Holding cost rates for inventory along the line and end product when positive t Backlog cost rates for end product inventory when negative Ø Service Level based t Type-1 Service: percent of demand served directly from the shelf Ø Decisions & Problem t What are the production amounts at any instant for all products? t Minimize the operational costs and/or satisfy service level constraints Carlos Bispo Multi-echelon Inventory Conference, June 2001 4

Control Policy Class - I Ø The system state can be also described by the echelon inventories. Ø En = In + (En)Ø Defined for each product at each buffer. Ø Define an Echelon Base Stock for each echelon inventory. Ø zkmp for all k, m, p Ø k indexes the visit number, m indexes the machine, p indexes the product. Ø Produce the difference between the EBS and the actual echelon inventory. Ø fn, 0 = z - En Carlos Bispo Multi-echelon Inventory Conference, June 2001 5

Control Policy Class - II Ø Bound by feeding inventory Ø fn = min{fn, 0 , (In)+}. Ø Production decisions are functions of fn. Ø Ideally, Pn should equal fn. Ø However, there are capacity bounds. Ø How are we to determine the production decisions when several products compete for a bounded resource? Ø E. g. , how is capacity shared/allocated? Carlos Bispo Multi-echelon Inventory Conference, June 2001 6

Previous work - Framework Ø Single product flow line Ø Glasserman & Tayur (1994, 1995) t Infinitesimal Perturbation Analysis (IPA) to compute optimal echelon base stock levels t Necessary stability condition shown to be sufficient Ø Multiple product re-entrant flow line Ø Bispo & Tayur (2001) t Need to address how capacity is shared both from a static and dynamic point of view t IPA to compute the optimal echelon base stock levels t Necessary stability condition show to be sufficient, even in the presence of random yield and jumbled flows. t Some technical problems with IPA Carlos Bispo Multi-echelon Inventory Conference, June 2001 7

Previous work - Capacity management Ø Static management Ø Divide each Cm into K*P slots, Ckmp - No Sharing; Ø Divide each Cm into K slots, Ckm - Partial Sharing; Ø No static capacity split - Total Sharing. Ø Dynamic management Ø Linear Scaling Rule - Pn = fn * min{1, Ckm/Sp . fn}; Ø Priority Rule; Ø Equalize Shortfall Rule; Ø Other? . . . Carlos Bispo Multi-echelon Inventory Conference, June 2001 8

Previous work - Main results Ø LSR and ESR are close in performance for Partial Sharing, Sharing and beat PR for a wide variety of parameters. Ø However, there are cases where PR beats both (related to average demand, variance coefficient, and backlog costs). Ø LSR degrades its performance for Total Sharing Ø Other than that ESR is usually the best, unless. . . PR. Ø Some dominance results to determine what is the adequate priority list. Ø Lowest average demand, lowest variance coefficient, highest backlog cost should have higher priority Ø The best costs are always achieved under the Total Sharing. Carlos Bispo Multi-echelon Inventory Conference, June 2001 9

Previous work – main limitations Ø When the weights converting units of products into units of capacity, , are not uniform and the system is re-entrant Ø PR does not generate smooth decisions for Total Sharing. t IPA not applicable!!! Ø ESR does not generate smooth decisions for Total Sharing. t IPA not applicable!!! Ø LSR generates smooth decisions but its performance is not the best. Ø How to determine the adequate priority list in the absence of clear cut dominance criteria? Ø Still a combinatorial problem. . . Carlos Bispo Multi-echelon Inventory Conference, June 2001 10

Smooth priorities Ø Key motivation Ø IPA is valid to LSR Ø What changes to introduce in the LSR, keeping it smooth, that will incorporate the concept of priority and will improve its performance? Ø One answer Ø Ø Ø Two phase LSR P 1 n= . fn * min{1, Ckm/Sp . . fn}; P 2 n= (1 - ). fn * min{1, (Ckm-Sp. P 1 n)/Sp . (1 - ). fn}; Pn = P 1 n + P 2 n The new set of parameters, , will determine the adequate priority/degree of importance of each product. Carlos Bispo Multi-echelon Inventory Conference, June 2001 11

Results - I Ø Some preliminary tests Ø One single machine producing two products for which we know what is the best priority order. t Priority to product 1. t Load is 80%. Ø If the best priority order is the best way of controlling such a system then we would expect 1 = 1 and 2 = 0. Ø Also, with such a small scale problem we can have a glance at how does the cost evolve as a function of the priority weights. t Is it convex, smooth, etc. ? Carlos Bispo Multi-echelon Inventory Conference, June 2001 12

Results - II Optimal cost as a function of the priority weights 1= 0 2= 0 cost = 348. 18 1= 0 2= 1 cost = 462. 95 a 1= 1 a 2= 0 cost = 340. 62 1= 1 2= 1 cost = 348. 18 a 1=0. 4 a 2= 0 cost = 330. 30 The optimal priority weights are 1 = 0. 414 and 2 = 0!!! Carlos Bispo Multi-echelon Inventory Conference, June 2001 13

Results - III Ø Single machine, producing three different products Ø Ø Ø E[d 1] = 8, cv 1 = ¼, b 1 = 100 E[d 2] = 12, cv 2 = ½, b 2 = 40 E[d 3] = 20, cv 3 = 1, b 3 = 20 hi = 10, for i = 1, 2, 3 1 = 2 = 3 = 1 Ø From earlier studies we know that product 1 should have higher priority, then product 2, and then 3. Ø Running the optimization we got Ø 1 = 0. 523, 2 = 0. 363, 3 = 0. 006 Carlos Bispo Multi-echelon Inventory Conference, June 2001 14

Conclusions Ø With the two phase LSR we get a way of estimating the relative importance of each product in a continuous space. t Each [0, 1]. Ø No longer a combinatorial problem. Ø Given that each phase is still an LSR, IPA is valid. Ø The mixed problem has been converted into a non linear program where all variables are real: echelon base stock and priority weights. Ø If all are equal to 1 or to 0, then we get the original LSR. Carlos Bispo Multi-echelon Inventory Conference, June 2001 15