Layout of facilities with some fixed points

This paper deals with the location of facilities or “movable” points on a planar area, on which there already exist fixed points. The minimax criterion for optimality is used and distances among points are assumed to be rectilinear. Two very efficient algorithms for the solution of the problem are presented. One is based on a univariate search, and the other on a steepest descent method. Some computational results are presented. INTRODUCTION We deal in this paper with the location of points on a plane. When we have freedom of choosing the location of all points, then the problem is known as the layout problem, or the quadratic assignment problem [ 51. When all points but one are fixed in place and when the maximum distance to all fixed points is to be minimized, then the problem is known as the one-center problem; when the sum of all distances to the fixed points is to be minimized it is known as the one-median problem [ 131. Here we assume that most points are fixed in place. Problems with this assumption include the multifacility minimax problem, when the greatest weighted distance in the system is to be minimized [6], and the multifacility minisum problem where the weighted sum of distances is minimized [ 131. We investigate a version of the final problem. In this paper, each distance is rectilinear (also known as the !I, distance). This is the distance measure when connections or travel are allowed only along horizontal or vertical directions. PROBLEM DEFlNlTION We must locate II points on a “board.” We assume that r points are fixed, and s = n r points are “free points” in that their sites need to be determined. Also, a nonnegative weight is associated with any pair of points. Typically this weight is zero when there are no connections between the two points. The connection can be weighted by any appropriate positive multiplier for flexibility in the model. Let (xi, yi) be the location of point i for i = 1, , . . ,n. For simplicity of notation we assume that the last r points are fixed (their locations given). Let wi, be the weight between points i andjfori,j= l,... ,n. Note that wi, = 0 when both i and j are fixed. The weighted distance tZvi Drezner is an associate Professor of Business Administration at The University of Michigan-Dearborn. He has a BSc. in Mathematics and a D.Sc. in Computer Science from the Technion, Haifa, Israel. He has recently published over 30 papers in Operations Research, Management Science, IIE Transactions, Naval Research Logistics Quarterly, and other joirnals. %eoree 0. Wesolowskv is a Professor of Management Science and Chairman of the Management Science and Info;mat& Systems Area of McMaster University in Hamilton, Ontario, Canada. He has a B.Sc. from the University of Toronto, an MBA from the University of Western Ontario and a Ph.D. in Business administration from the University of Wisconsin. His research interests include location models, inventory problems, and microcomputer graphics algorithms. 604 ZVI DREZNER and GEORGE 0. WESOLOWSKY between points i and j is Fij (X, Y): Kj (X3 Y) = Wij [ I Xi xj I + I Yi Yj II * The minimax problem is Minimize,,, {F(X, Y)}, where (2) F(X, Y) = maxi,j (F,}. (1) As is known, problem (2) can be decomposed into two one-dimensional problems. Lemma 1: I Xl x2 I + I Yl Y2 I = max{lxl fyi -x2-y21,1x, -Y, --2+~211. Prooj If the signs of x, x2 and y, y, are the same, then I x, x2 I + I y, y, 1 = I x1 x2 + y, y, I and I x1 y, x2 + y, I is smaller because it is equal to 11x1 -x21-b, -Y,II. A similar argument holds for the case when the two signs are opposite to each other, and the lemma is easily verified. The following transformation decomposes problem (2) into two independent onedimensional problems: