The multi capacitated clustering problem

Transcription

1 The multi capacitated clusterig problem Bruo de Aayde Prata 1 Federal Uiversity of Ceará, Brazil Abstract Clusterig problems are combiatorial optimizatio problems wi several idustrial applicatios. The variat i which e idividuals have distict types ad e clusters have multiple capacities for each type was ot reported i e available literature. This paper aims at presetig e Multi Capacitated Clusterig Problem (MCCP), a ew variat of e capacitated clusterig problem wi e above metioed characteristics. A 0-1 programmig model is preseted for e proposed variat. Prelimiary experimets lead to e developmet of heuristics for e resolutio of e MCCP i admissible computatioal times. Keywords: Combiatorial Optimizatio; Locatio Theory; Iteger Programmig. 1. Itroductio The clusterig problem is a difficult ad widely studied combiatorial optimizatio problem, wi several real world applicatios. It cosists of defiig a set of clusters wi a maximum homogeeity of elemets i a same group, ad, cocomitatly, a maximum heterogeeity betwee clusters. A Capacitated Clusterig Problem (CCP) appears whe each cluster has a bouded capacity for receive idividuals. Cocerig to capacitated clusterig problems, several approaches have bee reported i e literature. Models ad algorims for e CCP are preseted by Mulvey & Beck (1984), Fraça et. al. (1999), Ibrahim & Christofides (1999), Ahmadi & Osma (2005), Negreiros & Palhao (2006), Chaves & Lorea (2010), Scheurer & Wedolsky (2006), Deg & Bard (2011) ad Yag et al. (2011). I e reviewed literature was ot foud a capacitated clusterig problem i which e clusters had multiple capacities. The Multi Capacitated Clusterig Problem (MCCP) cosists i a capacitated clusterig problem i which each cluster has a give capacity for each type of idividual. If all e idividuals are of e same type, e problem is reduced to e classical capacitated clusterig problem. The existece of multiple types of idividuals, as well as multiple capacities i e facilities, icreases e complexity of is ew variat of e clusterig problem. 1 Assistat Professor. Departmet of Idustrial Egieerig, Federal Uiversity of Ceará. Cotact: baprata@ufc.br. 1

2 A example of a practical applicatio for e MCCP is e clusterig of studets i schools. Each studet lives i a certai address ad studies i a give grade level, ad e schools have capacities for each type of grade level. The problem cosists i allocatig all e studets ito e multi capacitated schools, aimig e miimizatio of e distace betwee e studets ad e schools. Oer practical applicatio for e MCCP is e clusterig of products i facilities, like, for example, a refiery. The products of several types eed to be processed i e plats, sice a give plat has a specific capacity for each type of product. I Figure 1, we have a visualizatio of a istace for e MCCP. This istace cosiders eight idividuals, beig five idividuals of type 1 (i blue) ad ree idividuals of type 2 (i red), ad also two clusters. The first cluster has a capacity of 3 idividuals of type 1 ad 1 idividual of type 2, ad e secod cluster has a capacity of 2 idividuals of type 1 ad 2 idividuals of type 2. A feasible solutio for e istace is preseted i Figure 1. Figure 1. MCCP solutio for a istace wi 8 idividuals ad 2 clusters. This paper aims at presetig e Multi Capacitated Clusterig Problem. A mixed iteger programmig formulatio is proposed for e above metioed problem. 2. Problem statemet I our approach, e idividuals are cosidered wi multiple characteristics ad e clusters have capacities for each type of idividual. I what follows, some otatio is itroduced for e MCCP proposed model. Let D be a matrix represetig e set of distace betwee idividuals ad cadidates to receive a cluster. 2

3 Sets M idividuals N cadidates for cluster K types of cliets P maximum umber of clusters Parameters d i distace betwee e i cliet ad e cluster. c k capacity of e cluster for e k type of cliet. q ik amout of e i cliet for e k type of cliet. I is problem we wat to allocate a idividual for e earest cluster, satisfyig e costrait of e capacity for each cluster, for each type of cliet. The variables of e model are as follows. Biary decisio variables x i 1, 0, if ei oerwise i M, N cliet is allocated to e cluster y 1, 0, N if e oerwise potetialcluster is selected Obective: miimize e total dissimilarity betwee each idividual i e multi capacitated clusters. The MCCP ca ow be formulated as follows. [MCCP] miimize m z = i1 1 d i x i (1) 3

4 subect to: 1 x 1 i = 1,, m. i (2) m qikxi i1 c k = 1,, k= 1,, p. (3) x y 0 i = 1,, m, = 1,, i 1 y i p x i {0,1} i= 1,, m = 1,, y {0,1} = 1,,. (4) (5) (6) (7) The obective fuctio (1) miimizes e distace betwee e idividuals ad e multi capacitated clusters. Costraits (2) impose at a idividual is allocated to oe cluster. Costraits (3) impose e amout of idividuals, for each type, allocated for a give cluster. Costrait (4) prohibits at a idividual is allocated i a cluster which is ot selected. Costrait (5) bouds e maximum umber of clusters. Fially costraits (6) ad (7) defie e scope of e model variables. 3. Coclusios This paper itroduces a ew variat of e capacitated clusterig problems amed multi capacitated clusterig problem. To e best kowledge of e auor, ere is i e literature o oer formulatio for e MCCP. Prelimiary experimets, which are ot reported here, lead to e developmet of heuristics for e resolutio of e MCCP i admissible computatioal times. Refereces Ahmadi, S., Osma, I.H. Greedy radom adaptive memory programmig search for e capacitated clusterig problems. Europea Joural of Operatios Research 2005; 162: Chaves, A.A., Lorea, L.A.N. Clusterig search algorim for e capacitated cetered clusterig problems. Computers & Operatios Research 2010; 37: Deg, Y., Bard, J.F. A reactive GRASP wi pa relikig for capacitated clusterig. Joural of Heuristics 2011; 17: Fraça, P.M., Sosa, N.M., Pureza, V. A adaptive tabu search algorim for e capacitated clusterig. Iteratioal Trasactio i Operatioal Research 1999; 6:

5 Ibrahim, I.H., Christofides, N. Capacitated clusterig problems by hydrid simulated aealig ad tabu search. Iteratioal Trasactio i Operatioal Research 1999; 3: Mulvey J.M., Beck M.P. Solvig capacitated clusterig problems. Europea Joural of Operatios Research 1984; 18: Negreiros, M, Palhao, A. The capacitated cetred clusterig problem. Computers & Operatios Research 2006; 33: Scheuerer, S., Wedolsky, R. A scatter search heuristic for e capacitated clusterig problem. Europea Joural of Operatios Research 2006; 169: Yag, Z., Che, H., Chua, F. A Lagragia relaxatio approach for a large scale ew variat of capacitated clusterig problem. Computers & Idustrial Egieerig 2011; 61: