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1 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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Page 1: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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

Page 2: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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

Page 9: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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).

Page 14: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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.

L(Qi,Qij ,λ) = Ri(Qi,Qij )i

n

∑ + λH,

Page 15: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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

Page 16: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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Further decomposition (SPij)

Max Rij = zi pi − si Qij − zi( )[ ] f zi( )dzi + piQij f zi( )dziQij

∫0

Qij∫{ } − cijQij − gi zi −Qij( ) f zi( )dziQi

∫ − λhiQij

Page 17: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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Solution to SPij

The solution to the unconstrained problem is given as follows:

(9)

Page 18: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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 .

Page 19: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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

Qij* = F −1(

pi + gi − cij − λhipi + gi + si

)

dijL ≤Qij

* ≤ dijU

Qij+ = dij

U

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Proposition 2

The solution obtained from Algorithm A is optimal to the subproblem (SPi).

Page 21: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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)’

Page 26: MINLP and Lagrangian heuristic for the newsvendor problem ...liberti/ewminlp/zhang.pdf · 1 MINLP and Lagrangian heuristic for the newsvendor problem with supplier discounts Guoqing

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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   …


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