Download Analysis of Supply Contracts with Total Minimum Commitment

Analysis of Supply Contracts with
Total Minimum Commitment
Yehuda Bassok and Ravi Anupindi
presented by
Zeynep YILDIZ
Outline
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Introduction
Model and Analysis
Computational Study
Conclusion
Introduction
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Traditional Review Models
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In practice…
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No restrictions on purchase quantities
Most flexible contracts
Restrictions on buyer by commitments
In general,
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Common in electronics industry
Family of products
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expressed in terms of a minimum amount of money to
purchase products of the family
Introduction
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In this paper,
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Single-product periodic review
The buyer side
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Total minimum quantity to be purchased over
the planning horizon
Flexibility to place any order in any period
The supplier side
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Price discounts applied to all units purchased
(price discount scheme for commitments)
Introduction – Basic Differences
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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
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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
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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
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Actions taken by the buyer
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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
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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)}
Total expected cost Purchasing
Holding &
From period t
cost
Shortage cost
Through 1
Optimal cost
from period t-1 through 1
Ct*(It,Kt) = minQtCt(It, Kt, Qt)
C0(I0,0)  0
where
It-1 = It + Qt - t
Kt-1= (Kt - Qt)+
Model and Analysis
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The optimization problem
Model and Analysis
The Single Period Problem
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Assume K10
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
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1.
2.
Proposition 1
The function C1*(I1,K1) is convex with
respect to I1,K1.
The function C2(I2,K2,Q2) is convex in
I2,K2,Q2.
Model and Analysis
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Q2 < K2 (K1>0)
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Constrained problem in last period
Q2 ≥ K2 (K1=0)
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Standard newsboy problem in last period
Model and Analysis
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(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
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Until the cumulative purchases exceed the total
commitment quantity (say this happened in
period t), follow a base stock policy with orderup-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
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1.
2.
3.
Solution structure for unconstrained version of
(P1) and (P2)
Proposition 2
The unconstrained solution of (P1) is order-up-to
SM, that is, Q2*=(SM-I2)+, where SM=F-1(/+h).
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.
S2≤ SM
Model and Analysis
Proposition 3
Assuming that I2≤SM, one and only one of the
following conditions holds:
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1.
1.
2.
3.
2.
Problem (P1) has and unconstrained optimal solution that
is feasible. In this case, this solution is also the optimal
solution of Problem (P).
Problem (P2) has an unconstrained optimal solution that
is feasible. In this case, this solution is also the optimal
solution of Problem (P)
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.
If I2≥SM then Q2*=0.
Model and Analysis
Model and Analysis
The N-Period Problem
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Proposition 4
1.
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.
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Proposition 5
At every period t, t=2,…,N, if Kt>0 then there are
two critical levels SM and St such that
Computational Study
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Discounted contracted price against market price
with no restrictions
Percentage savings in total costs
Parameters
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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
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Effect of Coefficient of Variation of Demand
Computational Study
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Effect of Price Discount
Computational Study
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Effect of Shortage Cost
Conclusion
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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