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MINLP and Lagrangian heuristic for the newsvendor problem with
supplier discounts
Guoqing Zhang
Department of Industrial and Manufacturing Systems Engineering University of Windsor
Windsor, Canada
April 12, 2010
EWMINLP Marseille, France
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1. Introduction
A single period inventory problem: • classical newsboy (or newsvendor) model • a popular strategy for dealing with fixed prices and
uncertain demand • applied to procuring roses by a flower shop for
Valentine Day
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Classical Newsboy model
Parameters: x = Demand (rv): density function f(x), cdf F(x) = Overage cost (loss of excess supply) = Underage cost (loss of profits for under
supply)
The optimal policy:
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Constrained newsboy model with discounts
The problem addressed: Multi-product Budget constraint All-unit quantity discount from suppliers
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2. Modelling: notations
Indices:
i = 1,..., n: index of products, where n is the total number of products
ki = the number of quantity discounts for product i offered by a supplier
j = 1,..., ki: index of quantity segment j for product i offered by a supplier.
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Notations (2)
Parameters: pi = unit sales revenue of product i hi = the budget consumed per unit of product i H = the budget limit of the vendor cij = the unit prices of product i after discount on discount segment j = the lower bound of the quantity of product i on discount segment j = the upper bound of the quantity of product i on discount segment j zi = the random variable of the demand for product i fi(zi) = the probability density function followed by the demand of product i gi = the estimated understocking cost (the loss of goodwill) of one unit of
product i si = the estimated salvaging value of one overstocking unit of product i
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Discount relationships
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Decision variables : the amount of raw material i purchased
from suppliers : the amount of raw material i purchased
on quantity discount segment j : 1 if the retailer buys product i at price
level j; otherwise 0
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Models: Objective
Maximize
€
R = zi pi − si Qi − zi( )[ ] f zi( )dzi + piQi f zi( )dziQi
∞
∫0
Qi∫{ }i=1
n
∑
− cijQijj=1
ki
∑i=1
n
∑ − gi zi −Qi( ) f zi( )dziQi
∞
∫i=1
n
∑
(1)
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Budget Constraint
(2)
(3)
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Quantity discount constraints
(3)
(4)
(5)
(6) (7)
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3. Solution Method: Lagrangian heuristic Relax the budget constraint to construct the
following Lagrangian dual problem:
S.t. constraints (2) to (7)
(8)
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Relaxed problem
With a given value of , the Lagrangian relaxed problem is
s.t. constraints (2) to (7).
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Decomposition The Lagrangian relaxed function can be expressed:
where Ri is defined in the next slide Then, the relaxed problem can be decomposed into the following sub-problems, each of them is correspondent to product i.
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L(Qi,Qij ,λ) = Ri(Qi,Qij )i
n
∑ + λH,
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Subproblem (SPi)
€
Max Ri = zi pi − si Qi − zi( )[ ] f zi( )dzi + piQi f zi( )dziQi
∞
∫0
Qi∫{ } − cijQij
j=1
ki
∑ − gi zi −Qi( ) f zi( )dziQi
∞
∫ − λhiQi
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Further decomposition (SPij)
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Max Rij = zi pi − si Qij − zi( )[ ] f zi( )dzi + piQij f zi( )dziQij
∞
∫0
Qij∫{ } − cijQij − gi zi −Qij( ) f zi( )dziQi
∞
∫ − λhiQij
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Solution to SPij
The solution to the unconstrained problem is given as follows:
(9)
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Proposition 1
Let and be the optimal solutions of problem (SPij) with and without bound constraints, respectively.
Then we have:
(i) If , then .
(ii) If , then .
(iii) If , then .
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Algorithm A: solving sub-problems (SPi) Step 1. Starting from the lowest price, i.e., j = ki
Step 2. Evaluate with
If the solution is realizable, i.e., then , and go to Step 4; else if then else .
Step 3. Let j = j-1. If j = 0, go to the next step; otherwise go back Step 2.
Step 4 Compare the objective values of all solutions we have for SPij.We have
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Qij* = F −1(
pi + gi − cij − λhipi + gi + si
)
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dijL ≤Qij
* ≤ dijU
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Qij+ = dij
U
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Proposition 2
The solution obtained from Algorithm A is optimal to the subproblem (SPi).
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Subgradient algorithm for dual problem
Use subgradient algorithm to solve the Lagrangian dual problem
Use subgradiant approach to find a good range of the Lagrangian multiplier, then employ bisection method to
accelerate search process.
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4. Computational Results
Test Problems:
N: 5 ~ 2000; K: 2~5 Use GAMS to implement on a 1.10Ghz Pentium M
PC
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Solutions
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The solution and running time comparison
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5. Extension: Multi-constraints
Same model except multi-capacity constraints
(2)’
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Extension: Multi-constraints
Similar approach: the two propositions can be extended
Multiple Lagrangian multipliers New algorithms to produce lower bounds Standard subgradient method
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Computational results
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References Abdel-Malek L, Montanari R, Meneghetti D, The capacitated newsboy problem with random yield:
The Gardener Problem. International Journal of Production Economics 2008;115 (1); 113-127 Khouja M, The newsboy problem with multiple discounts offered by suppliers and retailers. Decision
Sciences 1996;27; 589-599 Khouja M, The newsboy problem under progressive multiple discounts. European Journal of
Operational Research 1995;84; 458-466 Lau HS, Lau AHL, The newsstand problem: A capacitated multiple-product single-period inventory
problem. European Journal of Operational Research 1996;94; 29–42 Matsuyama K. (2006) The multi-period newsboy problem, European Journal of Operational Research
170(1), 170-188 Moon I, Silver E, The multi-item newsvendor problem with a budget constraint and fixed ordering
costs. Journal of Operational Research Society 2000;51; 602–608 Pan, K., Lai, K.K., Liang, L. and Leung, S.C.H. (2009) Two-period pricing and ordering policy for
the dominant retailer in a two-echelon supply chain with demand uncertainty, Omega 37 (4), 919-929 Vairaktarakis GL, Robust multi-item Newsboy models with a budget constraint. International Journal
of Production Economics 2000;66; 213–226 Zhang, he multi-product newsboy problem with supplier quantity discounts and a budget constraint,
to appear European Journal of Operational Research, 2010 …