Standardization problem: Resource Allocation in a Network

1. Mathematical model of the standardization problem. Among problems of operation research standardization problems take important place, i.e. problems of choosing optimal parameters and the number of items (products) belonging to an uniform collection. The term “uniform collection ” means a set of items applying for the same goal, but having different parameters. For example, for the uniform collection of electrical measuring devices one could choose current power, accuracy, sensitivity as such parameters; for construction materials--mass, width, length, solidity and so on. Let each item of some uniform collection be characterized by k parameters. Let A and B denote the sets of the all parameters vectors ()β β β1 2,,..., k for items supply and items demand, respectively. Let us consider a mathematical model for the standardization problem. Let the sets A and B consist of m and n elements, respectively. A qualification matrix ()Q qij m n = × contains information about the possibilities of replacement of some items type by another. One item of the ith type [ ] {} ()i m m ∈ = 12,,..., can replace qij items of the jth type [] ()j n ∈. The replacement of items of the jth type by items of ith type is impossible, if qij=0. Taking into account the replacement possibilities it is necessary for each type []i m∈ to determine the amount yi of items that should be produced in order to satisfy items demand ()b b bn1 2,,..., and to maximize total profit (or to minimize production costs). Thus the mathematical model of the standardization problem has the following structure. SP: f yi i i