Inventory models under uncertainty: An adaptive approach
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.
scientific article; zbMATH DE number 3965787
scientific article