Continuous Maximum Coverage Location Problem with Arbitrary Shape of Service Areas and Regional Demand

This paper addresses the maximum coverage location problem in a generalized setting, where both facilities (service areas) and regional demand are modeled as continuous entities. Unlike traditional formulations, our approach allows for arbitrary shapes for both service areas and demand regions, with additional constraints on facility placement. The key novelty of this work is its ability to handle complex, irregularly shaped service areas, including approximating them as unions of centrally symmetric shapes. This enables the use of an analytical approach based on spatial symmetry, which allows for efficient estimation of the covered area. The problem is formulated as a nonlinear optimization task. We analyze the properties of the objective function and leverage the Shapely library in Python 3.13.3 for efficient geometric computations. To improve computational efficiency, we develop an extended elastic model that significantly reduces processing time. This model generalizes the well-known quasi-physical, quasi-human algorithm for circle packing, extending its applicability to more complex spatial configurations. The effectiveness of the proposed approach is validated through test cases in which service areas take the form of circles, ellipses, and irregular polygons. Our method provides a robust and adaptable solution for various settings of practically interesting continuous maximum coverage location problems involving irregular regional demand and service areas.