Inverse single facility location problem in the plane with variable coordinates
Nazanin Tour-Savadkoohi, Jafar Fathali
TL;DR
This work addresses the inverse continuous single facility location problem with variable coordinates in the plane, where the goal is to adjust existing client coordinates at minimum cost to make a prescribed point optimal. It introduces two algorithms, ISFLP1 and ISFLP2, that solve a semi-infinite inverse formulation by solving finite subproblems and updating coordinates, with a specialization to the inverse minisum problem where, for squared Euclidean distance, the subproblems become linear programs and the forward minimizer is the center of gravity. The paper provides optimality conditions and convergence insights, and supports the approach with computational experiments across multiple norms and problem instances, showing comparable objective values to existing methods while highlighting trade-offs in speed. The framework extends to other inverse location settings and offers a flexible alternative to modifying weights, enabling coordinate-based adjustments in a range of planning applications.
Abstract
In traditional facility location problems, a set of points is provided, and the objective is to determine the best location for a new facility based on criteria such as minimizing cost, time, and distances between clients and facilities. Conversely, inverse single facility location problems focus on adjusting the problem's parameters at minimal cost to make a specific point optimal. In this paper, we present an algorithm for the general case of the inverse single facility location problem with variable coordinates in a two-dimensional space. We outline the optimality conditions of this algorithm. Additionally, we examine the specific case namely the inverse minisum single facility location problem and test the algorithm on various instances. The results demonstrate the algorithm's effectiveness in these scenarios.