The two-stage distributing of material flows in the transport-logistics system of fuel and energy complex is considered. The structural elements of such system are mines (centers of the first stage), extracting coal from various mineral deposits in a certain area, and enterprises that consume or process coal (centers of the second stage). A presented method for solving this problem is based on elements of the theory of continuous linear problems of optimal set partitioning, duality theory, and methods for solving linear programming problems of transport type. The optimal solution of the two-stage location-allocation problem is obtained in an analytical form, which contains parameters that are the optimal solution of the auxiliary finite-dimensional optimization problem with a non-differentiable objective function. Therefore, the part of numerical algorithm is non-differentiable optimization method – modification of Shor’s r-algorithm. The results of computational experiments solving model problems confirm the correctness of the presented method and algorithm. It is demonstrated the synergistic effect obtained from formulation of continuous problems of optimal partitioning sets with additional constraints. It is showed, how important to take into account the multi-stage distributing of raw materials when it is necessary to locate new transport-logistics system objects in a given territory.

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...

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Purpose is to elaborate innovative and computationally efficient algorithm to solve a problem of two-stage allocation of the resource occupying continuously the specified area as well as to demonstrate the behaviour of the corresponding software developed with the application of advanced geoinformation resources. Methods. The paper involves mathematical models of continuous problems of optimal set partitioning with additional connections to describe two-stage problems of the material resource location-allocation. Methodological approach to the solution of such problems is based on the idea of their reducing to the problem of infinite-dimensional mathematical programming for which it is possible to obtain optimal solution in the analytical form with the help of the duality theory apparatus. Findings. Mathematical and algorithmic apparatus to solve continuous problems applied for the fuel and energy complex enterprises has been developed making it possible to obtain partitioning of the deposit area into the zones, which are allocated to the first-stage enterprises exclusively. The algorithm operation is demonstrated in terms of the model problem solution. It has been defined that the benefit of such an approach is in the reducing of the infinite-dimensional programming problem to the problem of finite-dimensional nonsmoth optimization since the obtained computational formulas contain the parameters which determination requires solving the auxiliary problem of the nondifferentiable function optimization. Originality. Contrary to the previously developed one, the proposed algorithm does not stipulate solution of the linear programming problem of transport type at each step of the iteration process. Such a problem is solved only once to find the volumes of product transportation between the first-stage and second-stage enterprises after defining all the optimal solution components. Practical implications. Software implementation of the algorithm on the basis of the advanced geoinformation technologies and resources, in terms of the solution of raw material flow allocation, makes it possible to reduce total costs for the management of material flows and their accompanying service flows throughout the whole logistic chain beginning from the flow origin up to its arrival to the end user.

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