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Smooth Priorities for Make-to-Stock Inventory Control

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. Outline. Problem setting Control policy class Previous work Framework Capacity management

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Smooth Priorities for Make-to-Stock Inventory Control

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  1. 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

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

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

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

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

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

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

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

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

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

  11. Smooth priorities • Key motivation • IPA is valid to LSR • What changes to introduce in the LSR, keeping it smooth, that will incorporate the concept of priority and will improve its performance? • One answer • Two phase LSR • P1n=  . fn * min{1, Ckm/Sp . . fn}; • P2n= (1-). fn * min{1, (Ckm-SpP1n)/Sp .(1-). fn}; • Pn = P1n + P2n • The new set of parameters, , will determine the adequate priority/degree of importance of each product. Multi-echelon Inventory Conference, June 2001

  12. Results - I • Some preliminary tests • One single machine producing two products for which we know what is the best priority order. • Priority to product 1. • Load is 80%. • If the best priority order is the best way of controlling such a system then we would expect 1 = 1 and 2 = 0. • 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. • Is it convex, smooth, etc.? Multi-echelon Inventory Conference, June 2001

  13. a1= 0 a2= 0 cost = 348.18 a1= 0 a2= 1 cost = 462.95 a1= 1 a2= 0 cost = 340.62 a1= 1 a2= 1 cost = 348.18 a1=0.4 a2= 0 cost = 330.30 Results - II Optimal cost as a function of the priority weights The optimal priority weights are 1 = 0.414 and 2 = 0!!! Multi-echelon Inventory Conference, June 2001

  14. Results - III • Single machine, producing three different products • E[d1] = 8, cv1 = ¼, b1 = 100 • E[d2] = 12, cv2 = ½, b2 = 40 • E[d3] = 20, cv3 = 1, b3 = 20 • hi = 10, for i = 1, 2, 3 • 1 = 2 = 3 = 1 • From earlier studies we know that product 1 should have higher priority, then product 2, and then 3. • Running the optimization we got • 1 = 0.523, 2 = 0.363, 3 = 0.006 Multi-echelon Inventory Conference, June 2001

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

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