Modeling Gradual and Joint Coverage in Location Problems

Abstract

Location problems are a core area of research within OR/MS and decision sciences, with diverse applications in logistics, facility location, healthcare, defense, energy and transportation. One important feature of location problems is the coverage of demand or service areas by facilities, which can have significant economic, social, and environmental implications. Conventional models often assume binary or deterministic coverage, where a facility either fully covers a demand point or does not cover it at all. Although this simplification is useful for theoretical derivations, back-of-the-envelope calculations, and performance comparison, it overlooks the nuances and complexities of real-world scenarios. In this study, we provide an overview of the modeling challenges in location problems that incorporate gradual and joint coverage, where multiple facilities provide partial and cooperative coverage to demand points. Based on previous studies in this domain, we present mathematical formulations, and discuss techniques for linearization and approximation. As an illustrative example, we discuss a capacitated minimal covering location problem (MCLP) adapted from [21], which aims to determine the location and size of undesirable facilities in a given region. We start by introducing the nonlinear formulation that minimizes the sum of demand covered by those undesirable facilities. Subsequently, we introduce three integer linear programming formulations given in [21], two of which involve linear approximations based on a separable programming approach and a tangent line approximation method, while the third involves an exact reformulation of the problem. We also discuss the impact of linearization approximation errors on solution quality and time. © 2026 by World Scientific Publishing Co.