Analysis of Supply Contracts with Total Minimum Commitment

Yehuda Bassok and Ravi Anupindi

presented by

Zeynep YILDIZ

Outline

Introduction Model and Analysis Computational Study Conclusion

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

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)

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

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

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

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

Model and Analysis – Notation

Model and Analysis – Notation

Model and Analysis

Ct(It,Kt,Qt) = cQt + L(It + Qt) + EDt{C*

t-1(It-1,Kt-1)}

Purchasingcost

Holding & Shortage cost

Total expected costFrom period t

Through 1

Optimal cost from period t-1 through 1

Ct*(It,Kt) = minQt

Ct(It, Kt, Qt)

C0(I0,0) 0

where It-1 = It + Qt - t

Kt-1= (Kt - Qt)+

Model and Analysis

The optimization problem

Model and Analysis

The Single Period Problem Assume K10

Min C1 (I1, K1, Q1) = cQ1 + L(I1+Q1)

s.t. Q1≥ K1

Cost function is convex in Q1

S1 is obtained from standard newsboy problem

Q1 =S1-I1 if S1-I1≥K1

K1 o.w

Model and Analysis

The Two-Period Problem Assume K2>0

Model and Analysis

Proposition 1

1. The function C1*(I1,K1) is convex with

respect to I1,K1.

2. The function C2(I2,K2,Q2) is convex in I2,K2,Q2.

Model and Analysis

Q2 < K2 (K1>0) Constrained problem in last period

Q2 ≥ K2 (K1=0) Standard newsboy problem in last period

Model and Analysis

(P1)

MaxQ2 {cQ2+L(I2+Q2)+E {C1

*(I1,K1)}}

s.t. Q2<K2

Q2≥0. (P2)

MaxQ2 {cQ2+L(I2+Q2)+E {C1

*(I1,0)}}

s.t. Q2≥K2

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 (SM) 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 St-1,…,S1; in period t

Model and Analysis

Solution structure for unconstrained version of (P1) and (P2)

Proposition 21. The unconstrained solution of (P1) is order-up-to

SM, that is, Q2*=(SM-I2)+, where SM=F-1(/+h).

2. The unconstrained solution of (P2) is order-up-to S2 where S2 is the optimal order-up-to level of the two period standard newsboy problem.

3. S2≤ SM

Model and Analysis

Proposition 31. Assuming that I2≤SM, 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 Q2

*=K2.

2. If I2≥SM then Q2*=0.

Model and Analysis

Model and Analysis

The N-Period Problem Proposition 41. The function Ct

*(It,Kt) is convex with respect to It, Kt for t=1,…,N-1.

2. The cost function Ct(It,Kt,Qt) is convex in It, Kt, Qt for t=1,…,N.

Proposition 5At every period t, t=2,…,N, if Kt>0 then there are two critical levels SM and St such that

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

Computational Study

Effect of Coefficient of Variation of Demand

Computational Study

Effect of Price Discount

Computational Study

Effect of Shortage Cost

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