Inventory models under uncertainty: An adaptive approach
type: Item
instance of: scholarly article
title: Inventory models under uncertainty: An adaptive approach
zbMATH Open document ID: 0598.90034
DOI: 10.1016/0378-4754(86)90025-X
published in: Mathematics and Computers in Simulation
publication date: 1986-01-01T00:00:00Z
review text: The authors consider the following nonlinear programming (NLP) problem: \[ (1)\quad \min g_ 0(x)=\min \int \psi_ 0(x,y)f_ y(y,x)dy=\min E[\psi_ 0(x,Y)] \] \[ subject\quad to\quad g_ j(x)=\int \psi_ j(x,y)f_ y(x,y)dy=E[\psi_ j(x,Y)]\leq 0,\quad j=1,...,M, \] where \(x\in X\subset {\mathbb{R}}^ n\), \(y\in D\subset {\mathbb{R}}^ m\), \(\psi_ j(x,Y)\), \(j=0,1,...,M\) are given functions, and \(f_ y(y,x)\) is a probability density function (pdf) depending on a vector of parameters x. The pdf \(f_ y(y,x)\) is assumed unknown but a sample \(Y_ 1,...,Y_ N\) from it is available. To find the approximate solution of this NLP problem (the exact solution is not available since \(f_ y(y,'x)\) is unknown) the sample \(Y_ 1...Y_ N\) is used directly in an adaptive procedure called stochastic approximation in which the optimal solution \(x^*\) of (1) is approximated iteratively, i.e., step by step. Several stochastic optimization models which can be fitted in the framework of the NLP problem (1) are studied and adaptive stochastic approximation procedures approximate the optimal solution \(x^*.\) A number of inventory models are considered: an unconstrained single period one commodity model without and with setup cost (off-line and on- line, with and without memory) and a constrained single and multiple commodity single period model. Numerical techniques and some numerical results are included to illustrate the theory.
Mathematics Subject Classification ID: 90B05
Mathematics Subject Classification ID: 90C90
Mathematics Subject Classification ID: 91A60
Mathematics Subject Classification ID: 93C40
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 3965787
zbMATH Keywords: adaptive control
zbMATH Keywords: approximate solution
zbMATH Keywords: stochastic approximation
author: Reuven Y. Rubinstein
author: Joseph Kreimer
reviewed by: A. S. I. Zinober
MaRDI profile type: Publication
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full work available at URL: 0378-4754(86)90025-x
OpenAlex ID: W2138214895
description: scientific article; zbMATH DE number 3965787