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. Carlos BispoMulti-echelon Inventory Conference, June 20012 Outline  Problem setting  Control policy class  Previous work  Framework  Capacity management  Main results and limitations  Smooth priorities  Results  Conclusions

3. Carlos BispoMulti-echelon Inventory Conference, June 20013 Problem Setting - I  Multiple Capacitated Machines  Each machine has a finite capacity;  M machines with C m, for m = 1, …, M.  Multiple Products  Each product is characterized by an external stochastic demand;  P products with E[d p ] and cv p, 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.

4. Carlos BispoMulti-echelon Inventory Conference, June 20014 Problem Setting - II  Periodic Review  I n+1 = I n + P n - (P n ) -  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

5. Carlos BispoMulti-echelon Inventory Conference, June 20015 Control Policy Class - I  The system state can be also described by the echelon inventories.  E n = I n + (E n ) -  Defined for each product at each buffer.  Define an Echelon Base Stock for each echelon inventory.  z kmp for all k, m, p  k indexes the visit number, m indexes the machine, p indexes the product.  Produce the difference between the EBS and the actual echelon inventory.  f n,0 = z - E n

6. Carlos BispoMulti-echelon Inventory Conference, June 20016 Control Policy Class - II  Bound by feeding inventory  f n = min{f n,0, (I n ) + }.  Production decisions are functions of f n.  Ideally, P n should equal f n.  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?

7. Carlos BispoMulti-echelon Inventory Conference, June 20017 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

8. Carlos BispoMulti-echelon Inventory Conference, June 20018 Previous work - Capacity management  Static management No Sharing;  Divide each C m into K*P slots, C kmp - No Sharing; Partial Sharing;  Divide each C m into K slots, C km - Partial Sharing; Total Sharing.  No static capacity split - Total Sharing.  Dynamic management  Linear Scaling Rule  Linear Scaling Rule - P n = f n * min{1, C km /  p . f n };  Priority Rule;  Equalize Shortfall Rule;  Other?...

9. Carlos BispoMulti-echelon Inventory Conference, June 20019 Previous work - Main results Partial Sharing  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). Total Sharing  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.

10. Carlos BispoMulti-echelon Inventory Conference, June 200110 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...

11. Carlos BispoMulti-echelon Inventory Conference, June 200111 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  P 1n = . f n * min{1, C km /  p . . f n };  P 2n = (1-  ). f n * min{1, (C km  p P 1n )  p .(1-  ). f n };  P n = P 1n + P 2n  The new set of parameters, , will determine the adequate priority/degree of importance of each product.

12. Carlos BispoMulti-echelon Inventory Conference, June 200112 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.?

13. Carlos BispoMulti-echelon Inventory Conference, June 200113 Results - II Optimal cost as a function of the priority weights The optimal priority weights are  1 = 0.414 and  2 = 0!!!  1 = 0  2 = 0 cost = 348.18  1 = 0  2 = 1 cost = 462.95  1 = 1  2 = 0 cost = 340.62  1 = 1  2 = 1 cost = 348.18  1 =0.4  2 = 0 cost = 330.30

14. Carlos BispoMulti-echelon Inventory Conference, June 200114 Results - III  Single machine, producing three different products  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  h i = 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

15. Carlos BispoMulti-echelon Inventory Conference, June 200115 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.