Satisfying partial demand in facilities location ODED BERMAN 1 , ZVI DREZNER 2 and GEORGE O. WESOLOWSKY 3 1 Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, M5S 3E6, Canada E-mail: Berman@Rotman.Utoronto.Ca 2 College of Business and Economics, California State University-Fullerton, Fullerton, CA 92834, USA E-mail: zdnezner@fullerton.edu 3 Faculty of Business, McMaster University, Hamilton, Ontario L8S 4M4, Canada E-mail: wesolows@mcmail.cis.mcmaster.ca Received October 2000 and accepted September 2001 In this paper we consider the location of new facilities which serve only a certain proportion of the demand. The total weighted distancesoftheserveddemandisminimized.Weconsidertheproblemintheplaneforthelocationofonefacilityandonanetwork for the location of m-facilities. Some computational experience with these models are reported. 1. Introduction Consider the problem of locating a facility which serves only a portion of the customers. The facility needs to serveatleastaproportion a ofthecustomers,foragiven a.Equivalently,theproblemcanbestatedasexcludinga proportion of 1  a of customers from the services pro- vided by the facility. Since the weight used in the calcu- lation of the objective function usually represents the numberofcustomersassignedtoeachdemandpoint,the proportion of coverage is also calculated by weight. However,itispossibletohaveasecondsetofweightsto be used for the calculation of the coverage proportion. Servingonlyaproportionofthecustomersisacommon situation when the facilities are unable to serve all cus- tomersandissimilartotheconceptofcapacitatedfacility location(Love et al.,1988).However,instandardcapac- itatedmodelseverycustomerisservedbyafacilitybecause thenumberoffacilitiestobelocatedisadecisionvariable. In our problem we keep the number of facilities to be located fixed and assume that, ‘‘unfortunately’’, some customerswillnotbeserved.Theobjectivefunctionofthe problemwediscussisminimizingthetransportationcost. Asimilarmodeltotheoneconsideredinthepaperisby Brimberg and ReVelle (1998) where the simple plant lo- cationproblem(Balinski,1965;JuckerandCarlson,1976; Aiken,1985;Galvao,1993;ReVelleandLaporte,1997)is investigated when there is no restriction that all demand mustbesatisfied.TherationaleofthemodelofBrimberg andReVelle(1998)isthatnotalldemandisprofitableand thus may not need to be served. The problem is formu- lated as a bi-objective model with two criteria: (i) mini- mizing total cost; and (ii) minimizing the demand that is not satisfied. The objective function is a convex combi- nationofthetwocriteria.Whenvaryingtheweightsinthe objectivefunctionasetofefficientsolutionsisidentified. The model we consider is an m-median problem (Hakimi,1964)wherethenumberoffacilitiesisfixedand the focus is on the transportation cost (unlike for the plantlocationproblemwherethenumberoffacilitiesisa decisionvariableandthefocusisonthetotalcostwhich includes also the cost of opening new facilities). The ra- tionalebehindourmodelisthatwiththeplannednumber 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 a of customers need to be serviced. As mentioned in Brimberg and Re- Velle (1998), with the objective function of minimizing the convex combination of the total cost and total de- mandunserved,notallefficientpointsofthebi-objective model are guaranteed to be found. Therefore for a spe- cific proportion a the optimal solution is not guaranteed tobeobtainedwiththemodeldevelopedinBrimbergand ReVelle (1998). In this paper we also discuss for the special case of one facility the problem on the plane and showthattheproblemisessentiallyaKnapsackproblem (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) con- siderthe p-centerandsetcoveringmodelswhereonlypart of the demand is covered. Drezner (1981) suggested an efficient algorithm for the one-cover problem whose ob- jective 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 0740-817X Ó 2002 ‘‘IIE’’ IIE Transactions (2002) 34, 971–978