Placing a Finite Size Facility with a Center Objective on a Rectangular Plane with Barriers Avijit Sarkar, Rajan Batta, and Rakesh Nagi ∗ Department of Industrial Engineering, 420 Bell Hall University at Buffalo (SUNY), Buffalo, NY 14260, USA August 2005 Abstract This paper addresses the finite size 1-center placement problem on a rectangular plane in the presence of barriers. Barriers are regions in which both facility location and travel through are prohibited. The feasible region for facility placement is subdivided into cells along the lines of Larson and Sadiq (1983). To overcome complications induced by the center (mini- max) objective, we analyze the resultant cells based on the cell corners. We study the problem when the facility orientation is known a priori. We obtain domination results when the facility is fully contained inside 1, 2 and 3-cornered cells. For full containment in a 4-cornered cell, we formulate the problem as a linear program. However, when the facility intersects gridlines, analytical representation of the distance functions becomes challenging. We study the diffi- culties of this case and formulate our problem as a linear or nonlinear program, depending on whether the feasible region is convex or nonconvex. An analysis of the solution complexity is presented along with an illustrative numerical example. Keywords: 1-center placement, finite size facility location, barrier, rectangular plane 1 Introduction Location problems which impose restrictions on locating new facilities and/or travel through are typically referred to as constrained or restricted location problems. Such problems have the fol- lowing two topographical properties. (1) The new facilities cannot be located within certain pre- described restricted areas in the plane. (2) It is not always necessary that any two points in the plane would be “simply communicating,” i.e., the minimum travel distance between any two points in the plane may be made longer by the presence of the restricted regions. * To whom all correspondence should be addressed. E-mail: nagi@buffalo.edu 1