...customers from the services provided the facility. Since the weight used in the calculation of the objective function usually represents the number of customers assigned to each demand point, the proportion of coverage is also calculated by weight. However, it is possible to have a second set of weights to be used for the calculation of the coverage proportion.

Serving only a proportion of the customers is a common situation when the facilities are unable to serve all customers and is similar to the concept of capacitated facility location (Love et al., 1988). However, in standard capacitated models every customer is served by a facility because the number of facilities to be located is a decision variable. In our problem we keep the number of facilities to be located fixed and assume that, "unfortunately", some customers will not be served. The objective function of the problem we discuss is minimizing the transportation cost.

A similar model to the one considered in the paper is by Brimberg and ReVelle (1998) where the simple plant location problem (Balinski, 1965; Jucker and Carlson, 1976; Aiken, 1985; Galvao, 1993; ReVelle and Laporte, 1997) is investigated when there is no restriction that all demand must be satisfied. The rationale of the model of Brimberg and ReVelle (1998) is that not all demand is profitable and thus may not need to be served. The problem is formulated as a bi-objective model with two criteria: (i) minimizing total cost; and (ii) minimizing the demand that is not satisfied. The objective function is a convex combination of the two criteria. When varying the weights in the objective function a set of efficient solutions is identified.

The model we consider is an m-median problem (Hakimi, 1964) where the number of facilities is fixed and the focus is on the transportation cost (unlike for the plant location problem where the number of facilities is a decision variable and the focus is on the total cost which includes also the cost of opening new facilities). The rationale behind our model is that with the planned number of new facilities not all the demand can be served. Therefore a service level constraint is added that states that at least a pre-determined proportion [alpha] of customers need to be serviced. As mentioned in Brimberg and ReVelle (1998), with the objective function of minimizing the convex combination of the total cost and total demand unserved, not all efficient points of the bi-objective model are guaranteed to be found. Therefore for a specific proportion [alpha] the optimal solution is not guaranteed to be obtained with the model developed in Brimberg and ReVelle (1998). In this paper we also discuss for the special cas e of one facility the problem on the plane and show that the problem is essentially a Knapsack problem (Nemhauser and Garfinkel, 1972; Papadimitriou and Steiglitz, 1982) on a network.

There are a few other models which allow for serving only part of the demand. Daskin and Owen (1999) consider the p-center and set covering models where only part of the demand is covered. Drezner (1981) suggested an efficient algorithm for the one-cover problem whose objective is to cover as much of the demand as possible within a given radius. The extension of the one-cover to the m-cover problem is discussed by Watson-Gandy (1982). Drezner and Wesolowsky (1994) proposed the opposite objective of finding the minimum possible demand within a circle or a rectangle of a given size. Sherali and Rizzo (1991) discussed uncapacitated p-median problems on a chain graph with a continuum of link demands. Two cases were discussed, one when total supply exceeds total demand and another when total supply is less than total demand.

In the next section we consider the problem of locating one facility on the plane. In Section 3 we deal with the problem of locating a general number of facilities on a network. Computational results for the problems on the plane and on a network are included in Section 4. We conclude the paper in Section 5 where we present conclusions and suggest future research.

2. The planar problem

Let n demand points be located at ([a.sub.1], [b.sub.i]) for i = 1,...,n. Demand point i has two associated weights: [w.sub.i] is the weight for the cost function, and [v.sub.i] is the weight for calculating cover. Of course, one might have [v.sub.i] = [w.sub.i]. Let the location of the new facility be at (x,y). The distance between demand point i and the new facility is denoted by [d.sub.i](x,y). The distance can be measured by any metric (such as Euclidean, rectilinear, general [l.sub.p], or other).

Let [z.sub.i] be a binary variable where

[z.sub.i] = {1 when...

NOTE: All illustrations and photos have been removed from this article.



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