Lower and upper bounds for the continuous single facility location problem in the presence of a forbidden region and travel barrier

Abstract

In this paper, we investigate FRB, which is the single facility Euclidean location problem in the presence of a (non-)convex polygonal forbidden region where travel and location are not permitted. We search for a new facility’s location that minimizes the weighted Euclidean distances to existing ones. To overcome the non-convexity and non-differentiability of the problem’s objective function, we propose an equivalent reformulation (RFRB) whose objective is linear. Using RFRB, we limit the search space to regions of a set of non-overlapping candidate domains that may contain the optimum; thus we reduce RFRB to a finite series of tight mixed integer convex programming sub-problems. Each sub-problem has a linear objective function and both linear and quadratic constraints that are defined on a candidate domain. Based on these sub-problems, we propose an efficient bounding-based algorithm (BA) that converges to a (near-)optimum. Within BA, we use two lower and four upper bounds for the solution value of FRB. The two lower and two upper bounds are solution values of relaxations of the restricted problem. The third upper bound is the near-optimum of a nested partitioning heuristic. The fourth upper bound is the outcome of a divide and conquer technique that solves a smooth sub-problem for each sub-region. We reveal via our computational investigation that BA matches an existing upper bound and improves two more.