Minimax Location Problems With Nonlinear Costs*

Minimax location problems have rece ived considerable attention in the literature as models for locating fac ilities tha t are to provide emerge ncy or convenient service to a set of ex isting facilities. In most of these problems the re is given a set of ex isting facilities whose locations are represented as points in some space, and new fac ility locations are also to be spec ified as points in that space . A distance func tion is chosen to represe nt the travel di stance between the new and ex isting fac ility locations . The minimax objective is to locate the new fac ilities so that the maximum di stance, or a fun ction of di stance, between the new and existing fac ility locations is minimized. Different problems may be spec ified by the choice of the space and of the dista nce function used . Most of the problems in the literature may be placed into one of two classes: those using a norm-derived di stance in the space R" for some n , and those on a network using network distances. Also, diffe rent problems may be spec ified by the choice of the cost-representing fun ctions of travel distance. These func tions a re refe rred to genericall y as "cost" funct ions, but they may measure cost, time or some othe r fo rm of inconvenience. This paper considers one-fac ility minimax location problems that permit quite ge neral cost functions, namely any continuous (strictly) increasing fun c tion of the travel di stance. In the prev ious cons iderations of minimax location problems, cost fun ctions were assumed to be linea r, or in many cases the identity function. The initial formul ation and analysis a re given for a problem in R" using norm-derived distances ; however, the results obtained also hold for problems on a ne twork and are disc ussed subsequentl y. In addition, for minimax location problems using rectilinear distance, this paper extends prev ious solution procedures to these more general cost fun ctions in the space R2. For the problem on a tree network , previous solution procedures are also extended to the general cost functions. Most of the literature on minimax location problems has appeared in the las t ten years, although one ve rsion of the problem was first formul ated in 1857 as a " minimum covering sphere problem." A brief historical review is given by Elzinga and Hearn [6]1 who deal with the case involving the Eucl idean distance in R" and the identity cost function. Elzinga and Hearn [5] and Nair and Chandrasekaran [23] develop additional solution procedures for the same problem in R2. A multi-facility version of the Euclidean distance problem in R2 is considered as a convex programming problem by Love, Wesolowsky and Kraemer [20). Geometrical properties motiva te solutions by Elzinga and Hearn [5] and by Francis [8] to a one-fac ility rectilinear distance problem inR2 using the identity cost func tion. Wesolowsky [24] , and Dea ring and Francis [2] solve a multi-fac ility version of the rectilinear distance problem in R2 . One-fac ility minimax location problems on a network with linear cost fun ctions were first cons idered by Hakimi [12], [13] and subsequently by Goldman [11] , Dearing and Francis [3] and Handle r [16]. For tree ne tworks the problem has been solved by Goldman [11], Handler [15], [16] , Halfin [14], and Dearing and Franc is [3). A generalization for locating several new fac ilities, called the "p-center" problem, has been considered by Christofid es and Viola [1] , Minieka [22], Garfinkel et a l. [10] , and H andler [16] . Multi-fac ility minimax pro blems on tree networks are studied by Dearing, Francis , and Lowe [4].