MODELLING AND REALIZING A MULTIDIMENSIONAL RESOURCE ALLOCATION PROBLEM WITH TIME WINDOW CONSTRAINTS

Abstract

The multi-product resource allocation problem is considered, involving the distribution of a finite set of resources, supplied by various vendors, to a finite set of destinations, subject to two-sided time window constraints on resource utilization. The relevance of the considered resource allocation problem is substantiated by an extremely wide range of practical applications, both in its classical formulation and in new modifications, which are constantly necessitated by the dynamic external environment. An analysis of scientific literature has shown that the primary focus in modeling is placed on accounting for resource delivery time, while a significantly smaller number of studies are dedicated to solving multi-product resource allocation problems in the specified formulation. A mathematical model of the problem is constructed as a multi-dimensional optimization problem of geometric programming, specifically, a problem of optimal placement of rectangular objects, which model the destination requests, in a multi-dimensional resource space generated by the total supply of the vendors. The main characteristics of the proposed mathematical model were identified, including substantiating the property of separability of the mathematical formulation, which allowed the original mathematical model to be represented as a finite set of single-product multi-dimensional optimization problems of rectangular placement in the resource space. The goal of this work is to construct a mathematical model and, based on it, conduct a numerical study of the resource allocation problem considering two-sided constraints on resource utilization time. The methodical basis is grounded in the separability property of the objective function of the problem, and consequently, the possibility of representing the resulting single-product problems as optimal rectangular placement tasks. Conditions under which the problem may not have a solution are determined, and a method for finding an approximate solution to the problem is proposed, along with an assessment of the penalty for violating certain time window constraints. The proposed tools for the main research problem were implemented by creating a software simulator in c# using the visual studio visual design environment.