Optimal axis orientation for rectilinear minisum and minimax location ZVI DREZNER 1 and GEORGE O. WESOLOWSKY 2 1 Department of Management Science/Information Systems, California State University-Fullerton, Fullerton, CA 92834, USA E-mail: zdrezner@fullerton.edu 2 Faculty of Business, McMaster University, Hamilton, Ontario L8S-4M4, Canada E-mail: wesolows@mcmail.CIS.McMaster.CA Received January 1996 and accepted January 1998 This paper considers the rectilinear minisum and minimax location problems from a dierent point of view in that the orientation of the axes, which de®ne the distances, is now also to be optimized. This corresponds to the situation where the grid of roads or aisles which connects the demand points to the facility, can be designed at the same time as the location of the facility is chosen. Optimization methods are presented for both problems. 1. Introduction The minisum problem (also known as the Weber prob- lem) is one of the most basic problems in location theory. It concerns the problem in which there is a set of demand points on a plane and the site of a facility is to be chosen so that the sum of weighted distances from the facility to the demand points is minimized. This usually corresponds to minimizing the sum of transportation costs, although there have been many other interpretations and applica- tions of the problem [1]. In the minimax problem on the other hand the facility is located so that the smallest of the weighted distances between the facility and any de- mand point is as small as possible. A common interpr- etation is that we are locating an emergency facility, and the maximum weighted distance is the ``worst'' service being provided. There are many dierent ways of de®ning the distances within these two problems. The earliest approach was simply to use the Euclidean or straight line distance. In this case, the distances are independent of the orientation of the axes with respect to which we de®ne the co-ordi- nates of the points. However, straight line travel is seldom possible in practical situations. Many dierent distance measures and metrics are currently being applied to the problem [1,2] and many of these approaches are axis dependent. In this paper we will deal with rectilinear distances, where travel is possible only in directions par- allel to one of the axes; this is often used as an approxi- mation for travel in a rectangular grid of aisles or streets. In this case the distances are certainly axis dependent and so is the optimal location of the facility. When the facility is to be placed when the grid (axes) are already laid out, the rectilinear minisum and minimax problems have well-known solutions. The minisum problem reduces to ®nding two weighted medians using the X and Y co-ordinates separately, and the minimax problem has a variety of possible solution methods, in- cluding linear programming as is discussed by Love, et al. [1]. It has long been recognized [3,4] that the choice of axes is important in location problems; with a consensus viewpoint being that the correct choice of axes should correspond to some existing underlying grid or structure in the location space. Our approach, in contrast, is that we consider a situ- ation in which the planning of the location system is at such a fundamental level that the orientation of the un- derlying grid is part of the optimization process. This would occur when a production facility or an urban system is being designed ``from scratch''. For example, consider a planned station for a robot which moves on a rectilinear grid and needs to service a given set of points. In addition to the best location for the robot the direction of the grid needs to be determined. The objective can be either to minimize the expected time of service, or the maximum time of service that may determine the cycle of the operation. Both minisum and minimax objectives are applicable in this case. Similar problems occur when a set of oil drilling or mining sites need to be serviced. The location of demand points is ®xed, but the rectilinear road system connecting the service station to the sites is to be determined including its axis orientation. There are other situations that lead to this problem. Consider a plant with heavy machines which require service. Moving 0740-817X Ó 1998 ``IIE'' IIE Transactions (1998) 30, 981±986