1. Analysis of Supply Contracts with Total Minimum Commitment Yehuda Bassok and Ravi Anupindi presented by Zeynep YILDIZ

2. Outline Introduction Model and Analysis Computational Study Conclusion

3. Introduction Traditional Review Models  No restrictions on purchase quantities  Most flexible contracts In practice…  Restrictions on buyer by commitments In general,  Common in electronics industry  Family of products expressed in terms of a minimum amount of money to purchase products of the family

4. Introduction In this paper,  Single-product periodic review  The buyer side Total minimum quantity to be purchased over the planning horizon Flexibility to place any order in any period  The supplier side Price discounts applied to all units purchased (price discount scheme for commitments)

5. Introduction – Basic Differences Stochastic environment, whereas most of the quantity discount and total quantity commitment literature assumes deterministic environment Buyer makes a commitment a priori to purchase a minimum quantity

6. Introduction – Basic Contributions Identifies the notion of a minimum commitment over the horizon in a stochastic environment Identifies the structure of the optimal purchasing policy given the commitment and shows its simple and easy to calculate structure The simplicity of the optimal policy structure enables us to evaluate and compare different contracts and to choose the best one

7. Model and Analysis – Assumptions Distribution of demands is known i.i.d.r.v Deliveries are instantaneous Unsatisfied demand is backlogged Setup costs are negligible Purchasing, holding, and shortage costs incurred by the buyer are proportional to quantities and stationary over time Salvage value is 0 High penalty for not keeping the commitment Periods numbered backward

8. Model and Analysis Actions taken by the buyer  At the beginning of each period the inventory and the remaining commitment quantity are observed  Orders are placed  Demand is satisfied as much as possible  Excess inventory is backlogged to the next period Optimal policy that minimize costs for the buyer characterized in terms of  The order-up-to levels of the finite horizon version of the standard newsboy problem with the discounted purchase cost  The order-up-to level of a single-period standard newsboy problem with zero purchase cost

9. Model and Analysis – Notation

11. Model and Analysis C t (I t,K t,Q t ) = cQ t + L(I t + Q t ) + E D t {C * t-1 (I t-1,K t-1 )} Purchasing cost Holding & Shortage cost Total expected cost From period t Through 1 Optimal cost from period t-1 through 1 C t * (I t,K t ) = min Q t C t (I t, K t, Q t ) C 0 (I 0,0)  0 whereI t-1 = I t + Q t -  t K t-1 = (K t - Q t ) +

12. Model and Analysis The optimization problem

13. Model and Analysis The Single Period Problem Assume K 1  0 MinC 1 ( I 1, K 1, Q 1 ) = cQ 1 + L(I 1 +Q 1 ) s.t.Q 1 ≥ K 1 Cost function is convex in Q 1 S 1 is obtained from standard newsboy problem Q 1 = S 1 -I 1 if S 1 -I 1 ≥K 1 K 1 o.w

14. Model and Analysis The Two-Period Problem Assume K 2 >0

15. Model and Analysis Proposition 1 1. The function C 1 * (I 1,K 1 ) is convex with respect to I 1,K 1. 2. The function C 2 (I 2,K 2,Q 2 ) is convex in I 2,K 2,Q 2.

16. Model and Analysis Q 2 0)  Constrained problem in last period Q 2 ≥ K 2 (K 1 =0)  Standard newsboy problem in last period

17. Model and Analysis (P1) Max Q 2 {cQ 2 +L(I 2 +Q 2 )+E {C 1 * (I 1,K 1 )}} s.t.Q 2 <K 2 Q 2 ≥0. (P2) Max Q 2 {cQ 2 +L(I 2 +Q 2 )+E {C 1 * (I 1,0)}} s.t.Q 2 ≥K 2

18. Model and Analysis The structure of the optimal policy  Until the cumulative purchases exceed the total commitment quantity (say this happened in period t), follow a base stock policy with order- up-to level (S M ) until period t+1  After the commitment has been met, follow a base stock policy for the rest of horizon as in a standard newsboy problem with order-up-to levels of S t-1,…,S 1 ; in period t

19. Model and Analysis Solution structure for unconstrained version of (P1) and (P2) Proposition 2 1. The unconstrained solution of (P1) is order-up-to S M, that is, Q 2 * =(S M -I 2 ) +, where S M =F -1 (  /  +h). 2. The unconstrained solution of (P2) is order-up-to S 2 where S 2 is the optimal order-up-to level of the two period standard newsboy problem. 3. S 2 ≤ S M

20. Model and Analysis Proposition 3 1. Assuming that I 2 ≤S M, one and only one of the following conditions holds: 1. Problem (P1) has and unconstrained optimal solution that is feasible. In this case, this solution is also the optimal solution of Problem (P). 2. Problem (P2) has an unconstrained optimal solution that is feasible. In this case, this solution is also the optimal solution of Problem (P) 3. Neither problem (P1) nor (P2) has an optimal unconstrained solution that is feasible. In this case the optimal solution of Problem (P) is Q 2 * =K 2. 2. If I 2 ≥S M then Q 2 * =0.

21. Model and Analysis

22. The N-Period Problem Proposition 4 1. The function C t * (I t,K t ) is convex with respect to I t, K t for t=1,…,N-1. 2. The cost function C t (I t,K t,Q t ) is convex in I t, K t, Q t for t=1,…,N. Proposition 5 At every period t, t=2,…,N, if K t >0 then there are two critical levels S M and S t such that

23. Computational Study Discounted contracted price against market price with no restrictions Percentage savings in total costs Parameters Demand is normal and mean = 100 Coefficient of variation of demand = 0.125, 0.25, 0.5, 1.0 Percentage discount = 5%, 10%, 15% Market price per unit = $1.00 Penalty cost per unit = $25, $40, $50 Holding cost per unit = 25% of purchasing price Number of periods = 10 C programming language

24. Computational Study Effect of Coefficient of Variation of Demand

25. Computational Study Effect of Price Discount

26. Computational Study Effect of Shortage Cost

27. Conclusion Simple and easy to compute Compare different contracts and determine whether the contract is profitable and identify the best one Show effect of commitments, coefficient of variation of demand, percentage discount, and penalty costs on savings