Two-Stage Transportation Problem with Unknown Consumer Demands

Abstract

The work investigates a mathematical model of a two-stage transportation problem for finding the most economical plan for the transportation of homogeneous products from suppliers to consumers, where the demands of consumers are unknown, taking into account constraints on their lower and upper bounds. It is an extension of the classic two-stage transportation problem, where products are transported from suppliers to consumers only through intermediate points. Intermediary firms and various storage facilities (warehouses) can be such intermediate points.The relationship of the developed mathematical model with the two-stage continuous-discrete problem of optimal partitioning-distribution, which is characterized by the presence of two stages, is investigated. The problem consists in determining the areas of collection of the continuously distributed resource (raw material) by enterprises of the first stage and the volumes of transportation of the processed product from the enterprises of the first stage to consumers (points of the second stage), in order to minimize the total costs of transportation of the resource from suppliers to consumers through processing points (collection points, storage points).The material of the article is presented in two sections. Section 1 describes the mathematical model of the two-stage transportation problem with unknown consumer demands and provides the necessary and sufficient conditions for the compatibility of the system of linear constraints. It is shown that its special case coincides with the classic two-stage transportation problem.Section 2 provides a description of the model problem of optimal partitioning-distribution for the continuous area Ω and the discrete analog of the model problem. The results of computational experiments for a rectangular area Ω = {x = (x(1), x(2)) : 0 ≤ x(1) ≤ 1, 0 ≤ x(2) ≤ 1} with discretizations by grids 31 × 31 and 500 × 500 are presented. Optimal plans for transportation of processed product from points of the first stage to points of the second stage for both grids were found. The average time spent by the Gurobi solver to solve problems for the second grid, where the number of variables equals 250018 and the number of constraints equals 250009, is a few seconds on modern PCs.