SUMMARY
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 itstransportation 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 theoptimal 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 organizationof collection of a certain resource and its delivery to processing points, as well as further transportation of the product received to places of destination.Conclusions. The results of the computational experiments testify to the correctness of the developed algorithms operation for solving two-stage optimal set partitioning problems with additional couplings. Furthermore, it is confirmed the feasibility of formulating such problems when it is necessary to determine the location of new objects in a given territory, considering the multistageraw material resource distribution process. Further research is subject to the theoretical justification of the convergence of the iterative process realized in the proposed algorithm for solving continuous problems of OPS with additional couplings. In future, the development of software to solve such problems with the involvement of GIS-technologies is planned.