An Optimal Two-Stage Allocation of Material Flows in a Transport-Logistic System With Continuously Distributed Resource

Abstract

Context. The object of the research is a two-stage process of material flows allocation in the transport-logistic system, the structural elements of which are enterprises that collect a resource, is been distributed in a certain territory (centers of the first stage), and the enterprises that consume or process this resource. A mathematical model of such process is a two-stage problem of the optimal partitioning of a continual set with the locating of subset centers under additional constraints presented in the paper. Objective. The goal of the work is to ensure the reduction of transport costs in the organization of multi-stage production, the raw material resource of which is distributed in some territory, through the development of appropriate mathematical apparatus and software. The urgency of the work is explained by one of the most pronounced tendencies in extracting and processing branches of industry and agriculture, namely, the creation of territorially-distributed multilevel companies that include dozens of large enterprises and carry out a full cycle of production from raw material harvesting with its integrated use and the product manufacturing to its transportation to end consumers. Method. Mathematical apparatus for two-stage problems of optimal partitioning of sets with additional couplings was developed using the basic concepts of the theory of continuous linear problems of optimal set partitioning, duality theory, and methods for solving linear programming problems of transport type. The research shows that the formulation of a multi-stage transport-logistic problem in a continuous variant (in the form of an infinite-dimensional optimization problem) is expedient when the number of resource suppliers is limited but very large. The application of the developed mathematical apparatus makes it possible to find the optimal solution of the two-stage allocation-distribution problem in an analytic form (the analytic expression includes parameters that are the optimal solution of the auxiliary finite-dimensional optimization problem with a nondifferentiable objective function). The proposed iterative algorithm for solving the formulated problem bases on modification of Shor's r-algorithm and the method of potentials for solving the transport problem. Results. Developed mathematical models, methods and algorithms for solving continuous multi-stage problems for locating enterprises with a continuously distributed resource can be used to solve a wide class of continuous linear location-allocation problems. The presented methods, algorithms and software allow solving several practical problems connected, for example, with the strategic planning in the production, social and economic fields. The theoretical results obtained are been brought to the level of specific recommendations that can be used by state-owned and private enterprises in solving logistics tasks related to the organization of collection of a certain resource and its delivery to processing points, as well as further transportation of the produ...

Solution of the problem to optimize two-stage allocation of the material flows

“…Note that function G1 (τi I , ψ), being in its essence the refined dual functional of problem (1)-( 5) does not depend on variable v (contrary to the dual problem shown in paper [6]). That has become possible owing to the fact that in terms of any admissible fixed first-stage centres τi I , …, τ I N (including optimal ones), optimality criterion of pair   ** ( ),…”

“…The paper is focused on the numerical algorithm of solving two-stage allocation of the resource occupying continuously the specified area based on the mathematical apparatus described above. Contrary to the algorithm developed to solve similar problems in paper [6], that very algorithm does not involve the solution of problem of linear programming of transportation type at each stage of the iteration process. Such a problem is solved only once to find the volumes of product transportation between the first-stage and secondstage enterprises after the defining of the rest of the optimal solution components.…”

Про Один Підхід До Кластеризації Посівних Площ

“…Докладніше підхід, алгоритми вирішення та приклад розв'язку таких задач наведені в [3]. Як можна бачити, такий підхід має аналітичний розв'язок, а існуючі методи та алгоритми вирішення модельних задач дають досить непогані результати розв'язку у порівнянні з дискретними чи напівдискретними методами.…”

An Algorithm for Solving a Two-Stage Continuous-Discrete Location Problem for Medical Logistics Optimization

“…In situations requiring the location of many centers and considering population demand in facility location, continuum approaches to location problems become pertinent. Study [18] focuses on multistage location tasks, evaluating various facility groups and potential locations. This paper underscores the significance of considering continuum scenarios in such tasks and presents a model for two-stage location.…”

“…An algorithm utilizing a genetic approach with priority encoding is developed to address the issue ( 12) - (18). The algorithm's general description is as follows.…”