Facility Locations with the Manhattan Metric in the Presence of Barriers to Travel

Abstract

This paper considers the optimal location of p facilities in the plane, under the assumption that all travel occurs according to the Manhattan (or rectilinear or I1) metric in the presence of impenetrable barriers to travel. Facility users are distributed over a finite set of demand points, with the weight of each point proportional to its demand intensity. Each demand point is assigned to the closest facility. The objective is to locate facilities so as to minimize average Manhattan travel distance to a random demand. We show that an optimal set of facility locations can be drawn from a finite set of candidate points, all of which are easy to determine.

Placing a finite size facility with a center objective on a rectangular plane with barriers

“…A grid construction procedure in the presence of barriers that divides the feasible region F into cells has been established in the work of Larson and Sadiq [10]. The same procedure is followed here.…”

“…Larson and Li [9] developed an efficient algorithm for determining the shortest feasible rectangular path between two points in the presence of polygonal barriers. Discretization results for the p-median problem in the presence of arbitrarily shaped barriers under the rectangular distance metric were obtained by Larson and Sadiq [10]. The authors introduced a grid construction procedure that splits the feasible region into cells.…”

“…Batta et al [2] extended the results of [10] to include regions that prohibited facility location but allowed travel through. Aneja and Parlar [1] and Butt and Cavalier [3] developed heuristics for the 1-median problem in the presence of polygonal barriers under the l p distance metric.…”

“…Nandikonda et al [14] address the 1-center problem with arbitrarily shaped barriers under the rectangular distance metric. The authors divide the feasible region into cells as outlined by [10]. To overcome complications due to the center objective, they introduce a new concept to classify cells based on their cell corners.…”

“…In Section 2, we describe and define our problem. In Section 3, we briefly revisit the grid construction procedure of [10] and the concept of ''Equal Travel Time Lines'' (ETTLs), as established in [2]. We then study and establish some new properties of ETTLs in Section 4.…”