A heuristic method for the multi-item inventory problem with stochastic and substitutable demands
There are n-substitutable products. The demand for the i-th product is a random variable \(x_ i\) with a known density function \(f_ i(x_ i)\). The order quantities \(u_ i\), \(i=1,2,...,n\) denote the decision variables. The unit purchase costs and the sale prices (for each item) are known. If demand for item i is unsatisfied \((x_ i-u_ i>0)\) and a customer observes the excess in item j \((x_ j-u_ j<0)\) then he substitutes item i for item j. The substitution relations are described by the given coefficients \(a_{ji}\). For fixed \((u_ n,...,u_ 1)\) let \(I^ k_ 1=[0,u_ k)\), \(I^ k_ 2=[u_ k,\infty)\) denote the intervals and let \(i_ k\in \{1,2\}\) for \(k=n\), n-1,...,1. Assume that \(P(i_ n,i_{n-1},...,i_ 1)\) expresses the profit corresponding to the decision \((u_ n,...,u_ 1)\) and the realizations of random demands \((x_ n,...,x_ 1)\) with \(x_ k\in I^ k_{i_ k}\), \(k=n,n-1,...,1.\) The expected profit for a single period is of the following form: \[ P(u_ n,...,u_ 1)=\sum_{i_ n,...,i_ 1\in \{1,2\}}\int_{I^ n_{i_ n}}\times...\times \int_{I^ 1_{i_ 1}}p(i_ n,...,i_ 1)f_ n...f_ 1dx_ n...dx_ 1. \] The problem is to find \((u^*_ n,...,u^*_ 1)\) which maximize the expected profit. The author presents a heuristic rule which is based on the exact solution for two product case.
scientific article; zbMATH DE number 4010183