Optimal Inventory-Backorder Tradeoff in an Assemble-to-Order System with Random Leadtimes

1. Optimal Inventory-Backorder Tradeoff in an Assemble-to-Order System with Random Leadtimes Yingdong Lu – IBM T.J. Watson Research Center Jing-Sheng Song – University of California, Irvine David Yao – Columbia University

2. Outline • The Assemble-to-Order System • Model formulation • Properties of optimal solution • Solution techniques • Numerical results • Conclusion

3. Problem Background • Assemble-to-order • Mass customization: Dell, Compaq, Ford • Only keep component inventory • Final product is assembled after an order is realized • Optimal tradeoff between inventory and service • Service measure • Average number of product backorders • E[B] = Average # of customer orders waiting • Proportion to average customer waiting time

4. The Assemble-to-Order System SuppliersComponents Products Backorders (Items) (Customer demands) L1 1 Q121 Q122 L2 l12 2 QK2 QK1 lK QKm Lm m

5. The Demand Model(multivariate compound Poisson process) • m different components • Overall demand: Poisson process with rate l • Type-K demand: requires only the components in K K = any subset of {1,…, m} QKi = required number of units of component i in K qK = probability a demand is of type-K SqK = 1 • Aggregate demand of component i: Compound Poisson process with rate li = lS{i: in K} qK

6. Other Modeling Assumptions • The leadtimes for each component are i.i.d. random variables • Li has distribution Gi • Base-stock policies (order-up-to policies) • si = base-stock level for item i • FCFS • An order is backlogged if it is not yet completely filled. • Committed inventory • If we have some items in stock but not others that are requested by an order, we put aside those available items as committed inventory for that order.

7. The Optimization Problem minimize E[B(s1, …, sm)] subject to c1s1+…+cmsm< C where B = total number of customer backorders For any demand type K, let BK = type-K backorders = number of type-K orders not yet completely satisfied Then, B = SK BK

8. Solution Properties Let si* be the optimal base-stock level for component i. If ci> cj, li E[Li] <lj E[Lj], and li<lj , then si* < sj*. Example: If i and j have the same cost, and are always requested together, then the one with longest leadtime has higher optimal base-stock level.

9. Solution Techniques • Surrogate the objective function by simple lower and upper bounds • Both the upper- and lower-bound problems share similar structures, which can be solved by • an exact network flow algorithm (but the number of arcs grows exponentially in the number of components) • faster greedy heuristic algorithms • Numerical results show that the heuristic algorithm is effective.

10. The Supply Subsystem Suppliers X1 Xm X2 QK2 Q121 Q122 QKm QK1 l12 lK Arrivals (Replenishment Orders)

11. The Lower Bound Xi = outstanding orders of component i in steady state, has a Poisson distribution with mean liE[Li] Bi = number of component i backorders = [Xi - si ]+ BK,i = number of type-K backorders that have component i backlogged E[BK,i] lK E[Bi]/li BK = number of type-K backorders = maxi e K {BK,i} E[BK] > maxi e K {E[BK,i]} maxi e K {lK E[Bi]/li} E[B] = average total number of backorders = S KE[BK] >S Kmaxi e K {lK E[Bi]/li}  

12. Surrogate Problem I: Lower Bound • Using the lower bound to approximate the objective function, we obtain the following surrogate problem • After a change of variables, the problem becomes

13. Solving Surrogate Problem I • Hierarchy structure: • There exists a complete ordering of all the order types (the subsets). For any K and (i,j), such that i, j belong to K, but not any set lower than K, we have zi=zj • The hierarchy structure enables us to devise • an exact shortest-path algorithm (but still slow for large problems) • a much faster greedy-type heuristic K1 K2 K3

14. Surrogate Problem II: Upper Bound • Applying Lai-Robins inequality, we have the following surrogate problem:

15. 6 differentiating items 1. built-in zip drive 2. standard hard drive 3. high-profile hard drive 4. DVD-Rom drive 5. standard processor 6. high-profile processor 6 major demand types {2,5} {3,5} {1,2,5} {1,3,6} {1,3,4,5} {1,3,4,6} A Personal Computer Example