MULTIPLE FACILITY LOCATION ANALYSIS PROBLEM WITH WEIGHTED EUCLIDEAN DISTANCE Dleep R. Sule and Anuj A. Davalbhakta Lousana Tech Unversty ABSTRACT Ths paper presents a new graphcal technque for cluster formaton n multple facltes locaton analyss problem wth weghted Eucldean dstance norm. There are two facets of the problem; locaton of facltes and allocaton of customers. The objectve s to mnmze the maxmum weghted dstance traveled wthn clusters. Known parameters are Cartesan coordnates of customer locatons, ther demand weghts, and a number of facltes to be placed. The new procedure ensures the optmum soluton. INTRODUCTION Multple facltes locaton analyss begns wth formaton of clusters of customers so that servces can be rendered effcently and cost effectvely from faclty to all customers n that set of customers. In ths paper, we present new graphcal technque developed to form the clusters of customers. Locaton of each faclty can then be determned usng Vector Method (Davalbhakta and Sule, 2003) developed earler. Multple facltes locaton analyss s a long-standng topc of research and lterature search n ths area results n several methods and heurstcs to solve dfferent types of the problem. Brmberg et al. (2000) have presented extensve emprcal study of mprovements and comparsons of varous old and recent heurstcs and algorthms, whch nclude alternatve locaton-allocaton, projecton, Tabu search, p-medan, genetc search and varous versons of varable neghborhood search. Ths paper underlnes the fact that the problem has been studed for long tme and yet contnues to attract the attenton of researchers because of ts ubqutous nature. The problem consdered n ths paper s to determne the locatons of n facltes and to allocate the customers to them so that demands of all the customers are fulflled through the mnmum possble travel. Same problem has been addressed by Cooper (1961) where he has presented a heurstc for as many as 10 customer locatons. For more customer locatons, however, he has solved the mathematcal equatons to come up wth exact solutons. Sule (2000) n hs book on logstcs of facltes locaton and allocaton has also addressed the same problem and author has specfed analytcal method to solve the problem. However, qute often the problem consdered the travel between the new facltes n addton to that of between customers and facltes. Several examples of ths type are also found durng the bblographcal search (Love, Wesolowsky and Kraemer 1973), (Elznga, Hearn, Randolph 1976), (Charalambous 1981) (Brandy, Rosenthal and Young 1983). Mathematcally, the equatons n ths case also consder the demand weghts of facltes and Eucldean dstance between them n addton to the parameters presented n equaton (1) later n ths paper. Other dfferent aspects of ths problem have also been studed such as mnmum coverng ellpse problem where a mnmum certan dstance has to be mantaned
between two facltes. Douglas and Papayanopoulos (1991) have employed nteractve graphcal method whch produces near-optmal solutons. A decson analyss approach adopted by Current, Ratck, ReVelle (1998) demonstrates analyss of varous decson makng parameters that affect the total number of facltes to be placed n dynamc faclty locaton analyss. We have however, coned our research to the statc faclty locaton and have consdered that the demands of customers n a cluster can be fulflled by the faclty located n respectve cluster. Durng our research, we also compared our method wth the work done by Levn and Ben-Israel (2002), who have surveyed and compared varous technques avalable to determne the facltes locaton and then assgnng the customers to the facltes. Remanng layout of ths paper ncludes the problem statement and mathematcal formulaton followed by the llustratve example along wth summary and concluson. PROBLEM STATEMENT Consder m facltes are to be placed wthn n customers. Locaton of each customer s expressed as ( X a, Ya ),( X b, Yb ) K( X n, Yn ). Customers are assgned weghts based on ther demands. Let W be the demand weght of customer (where = a, b, c n) In case of weghted mnmax faclty locaton analyss problem, total cost s the functon of demand weghts of customers and dstances traveled between customer locatons and the faclty. It s expressed as: 2 2 Mn W [( X X ) + ( Y Y ) (1) Where X Y = X CoOrdnateOfNewFaclty = Y CoOrdnateOfNewFaclty METHOD Ths new graphcal method of cluster formaton begns wth plottng all the customer locatons on graph followed by connectng all the customer locatons on the perphery n such a way that all the customer locatons le ether on the boundary or wthn t. The permeter of thus formed fgure s calculated and s then dvded nto as many fractons as the number of facltes to be placed (m), startng from the customer locaton closest to the orgn (0,0) of the Cartesan coordnate system. Customer locatons at the fractons are then treated as the orgn for the clusters. It s lkely not to have a customer locaton stuated exactly at the poston specfed by the fracton value. In that case, locaton on the boundary closest to the fracton value s selected as the orgn. Once the orgns are determned, weghted Eucldean dstances of rest of the customers from orgns are calculated and then based on mnmum weghted Eucldean dstance crteron, the clusters are formed. The formulaton developed durng ths research work takes nto account the average weghted dstance of every customer locaton enterng the cluster from every other member of the cluster. The customer locatons selected n the cluster are dscarded from further analyss n order to avod any duplcaton.
Steps to be followed are summarzed as follows: 1. Plot all the customer locatons on graph 2. Connect the customers on perphery makng sure that all the locatons are ether on boundary or wthn t. 3. Startng from the pont closest to the orgn of coordnate system (0, 0); dvde the permeter nto m equal fractons. 4. Exstng locatons at these fractonal values (dvson ponts) are termed as orgns. If a dvson pont does not concde exactly wth the locaton of an exstng customer then the locaton of an exstng customer closest to t on the perphery s selected as an orgn. These form the ntal members of the assocated clusters. 5. Check each locaton to determne n whch cluster t should be assgned. Ths s done by checkng ts average weghted Eucldean dstance assocated wth the customers that are presently assgned to each cluster. The dstance s calculated as follows: W j W d j d = W + W j Where, (2) = AlreadyExstngCustomerLocatonsInTheCluster j = EnterngLocaton 6. Jon the customer wth mnmum weghted dstance amongst all the clusters to the assocated clusters. Delete the customer from further evaluatons at ths stage. 7. Repeat steps 5 and 6 tll all the customers are assgned to an approprate cluster. 8. The optmum faclty locaton n each cluster s determned by applyng Vector method [6]. ILLUSTRATIVE EXAMPLE Consder two facltes to be placed to serve 12 customers. Customer locatons and ther demand weghts are gven n Table 1. Table 1: Customer Locatons and ther weghts Customer X Coordnate Y Coordnate Weght A 20 46 3.0 B 15 28 2.0 C 26 35 3.0 D 50 20 2.0 E 45 15 2.0 F 1 6 2.0 G 5 9 4.0 H 12 8 4.5 I 10 2 2.5 J 11 18 5.5 K 6 13 6.0 L 1 2 3.5
All the customer locatons are plotted on graph and then customers on perphery are joned together to form a boundary as shown n Fgure 1. Fgure 1: Customer Locatons Permeter of the boundary s 141.3894 unts. As m = 2, the fracton at whch the orgns should be located s 70.69471 unts. Frst orgn of the cluster s L (1, 2), whle the other orgn s D (50, 20), closest customer to the dvson pont. Weghted dstances are calculated usng formula (2) and are shown n Table 2. Table 2: Cluster Formaton Cycle 1: Cluster 1 Contents: L and cluster 2 contents: D From L (1, 2): Cycle 1 Customers: A B C D E F G H I J K L Weghted Dstance: 77.4 37.6 66.9 66.4 58.4 5.1 15.0 24.7 13.1 40.4 26.7 -- From D (50, 20): Cycle 1 Customers: A B C D E F G H I J K L Weghted Dstance: 47.6 35.9 33.9 -- 7.1 50.9 61.8 55.2 48.7 57.3 66.8 66.4 Customer locaton F s selected as the locaton enterng the cluster orgnated from L. Cycle 2: Cluster 1 Contents: L & F and cluster 2 contents: D Cluster 1 calculatons: Customers: A B C E G H I J K Weghted Dstance: 90.5 41.5 78.2 66.7 16.1 29.8 15.9 48.6 31.0 Cluster 2 calculatons: Customers: A B C D E G H I J K Weghted Dstance: 47.6 35.9 33.9 -- 7.1 61.8 55.2 48.7 57.3 66.8 Customer locaton E s selected as a locaton enterng cluster orgnated from D. Cycle 3: Cluster 1 Contents: L & F and cluster 2 contents: D & E
Contnung the calculatons n the smlar manner the fnal cluster formaton s obtaned as follows: Table 4: Resultng Clusters Cluster 1 Customers X Coordnate Y Coordnate Weght B 15 28 2.0 F 1 6 2 G 5 9 4 H 12 8 4.5 I 10 2 2.5 J 11 18 5.5 K 6 13 6 L 1 2 3.5 Cluster 2 A 20 46 3.0 C 26 35 3.0 D 50 20 2.0 E 45 15 2.0 Cluster formaton s shown n the Fgure 2. Fgure 2: Cluster Formaton Facltes locaton analyss n each cluster s performed usng the method suggested earler whch s based on vector algebra (Davalbhakta and Sule, 2003). Subsequently, faclty locaton for cluster 1 s at (7.68, 10.96) wth the optmum cost of $ 211.52. Whle for cluster 2, the optmum faclty locaton s at (32.81, 31.29) wth the cost of $163.61. CONCLUSION New graphcal method has been devsed to form the clusters of customer locatons n a multple faclty locaton analyss problem wth weghted mnmax Eucldean dstance. When we compared our method wth the work done by Levn and Ben-Israel, t occurred that both the methods present the same results for the problem llustrated n ther paper. However, Levn and Ben-Israel have consdered that all the customers have equal
demands and hence have not consdered the demand weghts n ther heurstc. On the other hand, method proposed n ths paper takes nto account dfferent demand weghts of customers as vtal factor n cluster formaton as not only the dstance between the faclty and customer but the demand of customer also contrbutes to the total cost. REFERENCES 1. Brady, S.D., Rosenthal, R.E., and Young, D. (1983). Interactve Graphcal Mnmax Locaton of Multple Facltes wth General Constrants. AIIE Transactons 15: 242-254. 2. Brmberg, J.; Hansen, P.; Mladenovc, N. and Tallard, E.D. (2000). Improvements and comparson of heurstcs for solvng the uncapactated multsource Weber problem. Operatons research 48: 444-460. 3. Charalambous, C. (1981). An teratve algorthm for the multfaclty mnmax locaton problem wth Euc1dean dstances. Naval Research Logstcs 28: 325-337. 4. Cooper, L. (1963). Locaton-Allocaton Problems. Operatons Research 11: 331-342. 5. Current, J.; Ratck, S. and ReVelle, C. (1998). Dynamc faclty locaton when the total number of facltes s uncertan: A decson analyss approach. European Journal of Operatonal research 110: 597-609. 6. Davalbhakta, A.D. and Sule D. R. (2003). Internatonal Journal of Industral Engneerng. Accepted and In Prnt. 7. Douglas, M. I. and Papayanopoulos, Lee (1991). Mnmax locaton of two facltes wth mnmum separaton. Interactve graphcal solutons. Journal of the Operatonal Research Socety 42: 685-694. 8. Elznga, J.; Hearn, D. and Randolph, W.D. (1976). Mnmax multfaclty locaton wth Eucldean dstance. Transportaton Scence 10: 321-336. 9. Levn, Y. and Ben-Israel, A. (2002). Computers and Operatons Research. Accepted and n Prnt. 10. Love, R.F.; Wesolowsky, G. O. and Kraemer, S.A. (1973). Internatonal Journal of Producton research 11: 37-45 11. Sule, D. R. (2001). Logstcs of Faclty Locaton and Allocaton. Marcel Dekker Inc., NY.