A modelling framework for solving restricted planar location problems using phi-objects

This paper presents a general modeling framework for restricted facility location problems with arbitrarily shaped forbidden regions or barriers, where regions are modeled using phi-objects. Phi-objects are an efficient tool in mathematical modeling of 2D and 3D geometric optimization problems, and are widely used in cutting and packing problems and covering problems. The paper shows that the proposed modeling framework can be applied to both median and center facility location problems, either with barriers or forbidden regions. The resulting models are either mixed-integer linear or nonlinear programming formulations, depending on the shape of the restricted region and the considered distance measure. Using the new framework, all instances from the existing literature for this class of problems are solved to optimality. The paper also introduces and optimally solves a realistic multi-facility problem instance derived from an archipelago vulnerable to earthquakes. This problem instance is significantly more complex than any other instance described in the literature.