MATHEMATICAL FOUNDATIONS OF METHODS FOR SOLVING CONTINUOUS PROBLEMS OF OPTIMAL MULTIPLEX PARTITIONING OF SETS

Abstract

Context. The research object is the process of placing service centers (e.g., social protection services, emergency supply storage) and allocating demand for services continuously distributed across a given area. Mathematical models and optimization methods for location-allocation problems are presented, considering the overlap of service zones to address cases when the nearest center cannot provide the required service. The relevance of the study stems from the need to solve problems related to territorial distribution of logistics system facilities, early planning of preventive measures in potential areas of technological disasters, organizing evacuation processes, or providing primary humanitarian assistance to populations in emergencies.
Objective. The rational organization of a network of service centers to ensure the provision of guaranteed service in the shortest possible time by assigning clients to multiple nearest centers and developing the corresponding mathematical and software support.
Method. The concept of a characteristic vector-function of a k-th order partition of a continuous set is introduced. Theoretical justification is provided for using the LP-relaxation procedure to solve the problem, formulated in terms of such characteristic functions. The mathematical framework is developed using elements of functional analysis, duality theory, and nonsmooth optimization.
Results. A mathematical model of optimal territorial zoning with center placement, subject to capacity constraints, is presented and studied as a continuous problem of optimal multiplex partitioning of sets. Unlike existing models, this approach describes distribution processes in logistics systems by minimizing the distance to several nearest centers while considering their capacities. Several propositions and theorems regarding the properties of the functional and the set of admissible solutions are proven. Necessary and sufficient optimality conditions are derived, forming the basis for methods of optimal multiplex partitioning of sets.
Conclusions. Theoretical findings and computational experiment results presented in the study confirm the validity of the developed mathematical framework, which can be readily applied to special cases of the problem. The proven propositions and theorems underpin computational methods for optimal territorial zoning with center placement. These methods are recommended for logistics systems to organize the distribution of material flows while assessing the capacity of centers and the fleet of transportation vehicles involved.