Abstract
This paper addresses the problem of the optimal location of new facilities within an existing service network to alleviate the burden on current enterprises and meet the increasing demand for services. The study considers two critical aspects: 1) the "capacities" of service centers, which determine the maximum number of services or products each center can offer, and 2) the option for customers to be served by one of the k nearest service centers. Alongside the strategic location of new centers, the model also involves the reallocation of service areas, defining zones of responsibility for all facilities in the network. The primary goal is to minimize the total distance between all customers and the k nearest service centers. Typical "service center–customer" relationships include enterprises and consumers, post offices and clients, or medical testing sites and patients, etc. The mathematical model for territorial segmentation is formulated as an Optimal Multiplex Partitioning of Continuum Sets (OMPCS) problem. This approach enables the creation of overlapping service zones, unlike traditional models that result in first-order partitioning where zones are mutually exclusive and operate as territorial monopolies. We present and implement numerical algorithms for solving OMPCS problems in software. The model examples demonstrate how the strategic location of new facilities and the redistribution of service areas across all network entities can effectively reduce the load on existing centers.