Optimizing facility location through the integration of desired and (Semi/Full) obnoxious facilities utilizing rectilinear distance metrics

Facility location decisions in practice rarely involve purely desirable or purely undesirable interactions. Instead, planners must balance accessibility to beneficial services and demand points with adequate separation from hazardous or nuisance-generating facilities. This conflict is particularly critical in dense urban and industrial areas, where improper siting can increase travel costs, environmental exposure, and regulatory or community opposition. To address this challenge, this study proposes an integrated continuous location framework that simultaneously models attraction to desirable facilities and repulsion from obnoxious and semi-obnoxious ones under rectilinear (Manhattan) distance, consistent with grid-based urban movement patterns. Two mixed-integer programming models are developed: a desired–obnoxious model that minimizes the weighted total distance to desirable facilities while maximizing the minimum weighted separation from obnoxious facilities, and a desired–semi-obnoxious model that captures mixed externalities with varying interaction intensities. The models are linearized, and an exact Benders decomposition approach is implemented to enhance computational efficiency and scalability. Numerical experiments demonstrate that explicitly incorporating semi-obnoxious and obnoxious interactions can significantly alter optimal locations compared with traditional attraction-only models. The approach is demonstrated through four applications, including real case studies for locating a medical center in eastern Tehran and evaluating a petrochemical facility site in Khorramabad, providing decision-makers with more defensible and practice-aligned siting recommendations. In the final part of the study, the proposed framework is further extended by introducing an environmentally oriented bi-objective formulation that simultaneously optimizes the location decision and an explicit environmental performance measure.