Complementary Approaches for a Clustering Problem. Edson L.F. Senne*, Luiz A.N. Lorena**

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1 Copleeary Approaches for a Cluserig Proble Edso L.F. See*, Luiz A.N. Lorea** *FEG/UNESP - Uiversidade Esadual Paulisa Faculdade de Egeharia - Deparaeo de Maeáica Guaraigueá, SP - Brazil e-ail: elfsee@feg.uesp.br **LAC/INPE - Isiuo Nacioal de Pesquisas Espaciais Av. dos Asroauas, Caixa Posal São José dos Capos, SP - Brazil e-ail: lorea@lac.ipe.br Absrac The lagragea/surrogae relaxaio has bee explored as a faser copuaioal aleraive o radiioal lagragea heurisics. This paper discusses wo approaches for usig lagragea/surrogae heurisics o a classical cluserig proble: he p-edia proble, ha is, how o locae p faciliies (edias) o a ework such as he su of all he disaces fro each verex o is eares faciliy is iiized. Key words: Liear Prograig, Neworks, Opiizaio Techiques, Heurisics.

2 . Iroducio The search for p-edia odes o a ework is a classical cluserig proble. The obecive is o locae p faciliies (edias) such as he su of he disaces fro each dead poi o is eares faciliy is iiized. The proble is well kow o be NP-hard (Garey ad Johso, 979) ad several heurisics have bee developed for p-edia probles. The cobied used of lagragea relaxaio ad subgradie opiizaio i a prial-dual viewpoi was foud o be a good soluio approach o he proble (Beasley, 985). By oher had, colu geeraio is a powerful ool for solvig large scale liear prograig probles ad has bee a aural choice i several applicaios, such as he well-kow cuig-sock proble, vehicle rouig ad crew schedulig (Gilore ad Goory, 96, 963; Desrochers ad Souis, 989; Desrochers e al., 992; Vace e al., 994). However, he use of colu geeraio o solve p-edia probles was o sufficiely explored. The iiial aeps appear o be he oes of Garfikel e al. (974) ad Swai (974), ha repor covergece probles, eve for sall isaces, whe he uber of edias is sall copared o he uber of cadidae pois i he ework. This observaio was also cofired laer by Galvão (98). The soluio of large-scale isaces usig a sabilized approach is repored by du Merle e al. (999). The lagragea/surrogae relaxaio has bee explored recely as a faser copuaioal aleraive o radiioal lagragea heurisics. This ew relaxaio has bee used i a uber of applicaios, such as ulidiesioal kapsack probles (Freville e al., 990), se coverig probles (Lorea ad Lopes, 994), geeralized assige probles (Lorea ad Narciso, 996), ad also p-edia probles (See ad Lorea, 2000). The lagragea/surrogae relaxaio ad he radiioal colu geeraio approach ca be cobied o accelerae ad sabilize prial ad dual bouds obaied usig he reduced cos selecio. I is well kow he equivalecies of he Dazig-Wolfe decoposiio (Dazig ad Wolfe, 960), colu geeraio ad lagragea relaxaio opiizaio. Solvig a liear prograig by Dazig-Wolfe decoposiio is he sae as solvig he lagragea by Kelley's cuig plae ehod (Kelley, 960). However, as observed before, i ay cases a sraighforward applicaio of colu geeraio ay resul i slow covergece. The lagragea/surrogae relaxaio ca be used o sabilize ad accelerae he colu geeraio process, providig he selecio of ew producive colus.

3 This paper discusses hese wo approaches of usig lagragea/surrogae relaxaio o solve p-edia probles: (i) cobied wih subgradie opiizaio i a prial-dual viewpoi, ad (ii) cobied wih he colu geeraio process for liear prograig probles. The paper is orgaized as follows. Secio 2 preses he lagragea/surrogae relaxaio cobied wih subgradie opiizaio heurisic. Secio 3 discusses he cobiaio of lagragea/surrogae relaxaio ad colu geeraio process. The copuaioal ess, coduced wih probles fro he lieraure, are preseed i he Secio 4. We coclude ha hese wo approaches of usig lagragea/surrogae relaxaio o solve p- edia probles are copleeary ad ca be pu o work ogeher. 2. The Lagragea/Surrogae Relaxaio ad Subgradie Opiizaio proble: The p-edia proble ca be odeled as he followig biary ieger prograig v(p) = i = d i x i subec o x i = ; N () x ii = p (2) xi x ii ; i, N (3) x i {0,}; i, N (4) where: [di]x is a syeric cos (disace) arix, wih dii = 0, i; [xi]x is he allocaio arix, wih xi = if ode i is allocaed o ode, ad xi = 0, oherwise; xii = if ode i is a edia ad xii = 0, oherwise; p is he uber of faciliies (edias) o be locaed; is he uber of odes i he ework, ad N = {,..., }. Cosrais () ad (3) esure ha each ode is allocaed o oly oe ode i, which us be a edia. Cosrai (2) deeries he exac uber of edias o be locaed (p), ad (4) gives he ieger codiios. defied by: For a give R + ad 0, he lagragea/surrrogae relaxaio of (P) ca be

4 v(lsp ) = i ( d i )x i + subec o (2), (3) ad (4). = = The proble (LSP ) is solved cosiderig iplicily cosrai (2) ad decoposig for idex i, obaiig he followig probles: i = ( d i ) x subec o (3) ad (4). i Each proble is easily solved by leig: βi = { i ( 0, di - ) }, ad choosig I as he idex se of he p salles βi (here cosrai (2) is iplicily cosidered). The, a soluio x i o he proble (LSP ) is: = ad for all i, x ii, = 0, if i I oherwise x i, if i I ad d i < 0 = 0, oherwise The lagragea/surrogae soluio is give by: v(lsp ) = β x i ii + =. The ieresig characerisic of relaxaio (LSP ), is ha for = we have he usual lagragea relaxaio usig he uliplier. For a fixed uliplier, he bes value for ca resul of a lagragea dual: v( D ) = ax 0 v(lsp ). A search procedure for fidig a approxiae value for is show i (See ad Lorea, 2000). The cobied use of lagragea/surrogae ad subgradie opiizaio is give by he followig algorih: Algorih LSSH (lagragea/surrogae subgradie heurisic) Give 0, 0;

5 Se lb = -, ub = + ; Repea Solve relaxaio (LSP ) obaiig x ad v(lsp ); Obai a feasible soluio xf ad updae vf accordigly; Updae lb = ax [lb, v(lsp )]; Updae ub = i [ub, vf]; Se i g = - Σ x, i N; i Updae he sep size θ; Se i = ax { 0, i + θ. g }, i N; Uil (soppig ess). i The iiial used is = i{d }, i N. The sep sizes used are: θ = (ub - lb)/ g 2. i N i The corol of paraeer is he Held ad Karp (97) classical corol. I akes 0 2, begiig wih = 2 ad halvig wheever lb o icreases for 30 successive ieraios. The soppig ess used are: uber of ieraios greaer ha 000, 0.005, ad (ub lb) <. Soluio x is o ecessarily feasible o (P), bu he se I ideify edia odes ha ca be used o produce feasible soluios o (P). Two heurisics are used o ake x prial feasible. The firs calculaes he upper boud a each ieraio of LSSH while is o halved. This heurisic siply akes: v = (i d ). The secod, as suggesed by Beasley (993), f is a ierchage heurisic which is used whe is updaed o /2. i I i 3. The Lagragea/Surrogae Relaxaio ad Colu Geeraio The p-edia proble ca be also odeled as he followig se pariio proble: v(spp) = Mi = c x subec o A x = = (5)

6 = x = p (6) x {0,} where: S = {S,S,...,S } is a se of subses of N; A = [ai]x is a arix wih ai = if 2 i S, ad ai = 0, oherwise; ad c = Mi d i S k S ik. This forulaio is foud i Mioux (987). The sae forulaio ca be obaied fro he proble (P) applyig he Dazig- Wolfe decoposiio cosidered by Garfikel e al. (974) ad Swai (974). If S is he se of all subses of N, he forulaio ca give a opial soluio o he p-edia proble. Bu he uber of subses ca be huge, ad a parial se of colus should be cosidered. Proble (SPP) is also kow as he resriced aser proble i he colu geeraio coex (Barhar e al., 998). (SPP): I his paper we cosider he followig liear prograig se coverig relaxaio of v(scp) = Mi = c x subec o A x = (7) = x = p x [0,]. Observe ha d i 0, i, ad (5) ca be replaced wih (7) i he liear odel. The lagragea/surrogae relaxaio is iegraed o he colu geeraio process rasferrig he ulipliers ( =,...,) of proble (SCP) o he proble Max v(ls P ). 0 The edia wih salles coribuio o v[ Max v(ls P ) ] (ad allocaed o-edias) resuls o be he oe seleced o produce he icoig colu o he sub-proble: v(sub P) = Mi Mi N y {0,} i 0 ( d. ) y i i.

7 The reduced cos (for = ) is rc = v(sub -P) - α ad rc < 0 is he codiio for icoig colus, bu i well kow (Barhar e al., 998) ha, for =,...,, all he correspodig colus y saisfyig: Mi ( d i ) yi yi {0,} pool of colus, acceleraig he colu geeraio process. < α, ca be added o he The cobied use of lagragea/surrogae ad colu geeraio process is give by he followig algorih: Algorih LSCG (lagragea/surrogae colu geeraio heurisic). Se a iiial pool of colus o (SCP); 2. Solve (SCP) obaiig he duals prices, =,...,, ad α; 3. Solve approxiaely a local lagragea/surrogae dual Max v(ls P ), reurig he correspodig colus of (Sub P); 0 4. Apped o (SCP) he colus y saisfyig Mi ( d i ) yi yi {0,} < α ; 5. If o colus are foud i sep 4 he sop; 6. Perfor ess o reove colus ad reur o sep 2. Noe ha, by seig =, he algorih LSCG gives he radiioal colu geeraio process. 4. Copuaioal Tess The approaches discussed above were prograed i C ad ru o a Su Ulra 30 worksaio. Firs, we have cosidered isaces draw fro OR-Library (Beasley, 990) for which opial soluios are kow. The resuls are repored i he able below. Table repors he resuls for LSSH ad LSCG algorihs ad coais: he uber of odes i he ework ad he uber of edias o be locaed; he opial ieger soluio for he isace;

8 p_gap = 00 * (v(scp) opial) / opial, ha is, he perceage deviaio fro opial o he bes prial soluio value v(scp) foud; d_gap = 00 * (opial v(ls P )) / opial, ha is, he perceage deviaio fro opial o he bes relaxaio value v(ls P ) foud; he oal copuaioal ie (i secods). Table : Copuaioal resuls for OR-Library isaces LSSH LSCG p opial soluio p_gap d_gap oal ie p_gap d_gap oal ie Table shows ha he cobied use of lagragea/surrogae ad colu geeraio ca be very ieresig, especially for large-scale probles. The resuls of Table also show ha for a give uber of odes, saller he uber of edias i he isace, harder is he proble o be solved usig he colu geeraio approach. The opposie occurs for lagragea/surrogae approach cobied wih subgradie search ehods, i.e., he isaces for which he uber of edias is abou a hird of he uber he odes see o be easy o LSCG ad hard o LSSH. The copuaioal ess for a large-scale isace draw fro he TSPLIB, copiled by Reiel (998), cofir hese coecures. Table 2 shows he resuls for he Pcb3038 isace

9 (3038 odes). We ca oe ha as he uber of edia icreases, he perforace of LSCG iproves i such a way ha, for p = 500, i is beer ha LSSH. I his able p_gap ad d_gap are calculaed as follows: p_gap = 00 * (v(scp) bes kow soluio) / bes kow soluio d_gap = 00 * (bes kow soluio v(ls P )) / bes kow soluio Table 2: Copuaioal resuls for OR-Library isaces LSSH LSCG p bes kow soluio p_gap d_gap oal ie p_gap d_gap oal ie Coclusio The cobied use of lagragea/surrogae relaxaio wih sugbradie opiizaio (LSSH) ad wih colu geeraio (LSCG) see o be a good approach o solve p-edia cluserig probles. I ers of copuaioal perforace we have oed ha, for a give uber of odes (), he os ie-cosuig probles for LSSH correspod o p = /3. By oher had, for a give uber of odes, he copuaioal perforace of LSCG iproves as he uber of edia icreases. So, hese wo approaches have copleeary copuaioal behavior ad ca be pu o work ogeher. Tha will be ipora o develop decisio suppor syses for large-scale daa obaied fro geographical iforaio syses. Ackowledges: The auhors ackowledge Fudação de Aparo à Pesquisa do Esado de São Paulo - FAPESP (process 99/ ) ad Coselho Nacioal de Desevolvieo Cieífico e Tecológico - CNPq (processes /88-6 ad /89-5) for parial fiacial research suppor. Refereces Barhar, C.; Johso, E.L.; Nehauser, G.L.; Savelsbergh, M.W.P.; Vace, P.H. (998) Brach-ad-Price: Colu Geeraio for Solvig Huge Ieger Progras, Operaios Research, 46 :

10 Beasley, J.E. (985) A oe o solvig large p-edia probles, Europea Joural of Operaioal Research, 2 : Beasley, J.E. (990) OR-Library: Disribuig es probles by elecroic ail. Joural Operaioal Research Sociey, 4 : Beasley, J.E. (993) Lagragea heurisics for locaio probles, Europea Joural of Operaioal Research, 65 : Dazig, G.B.; Wolfe, P. (960) Decoposiio priciple for liear progras. Operaios Research, 8 : 0-. Desrochers, M.; Desrosiers, J.; Soloo, M. (992) A New Opiizaio Algorih for he Vehicle Rouig Proble wih Tie Widows, Operaios Research, 40 : Desrochers, M.; Souis, F. (989) A Colu Geeraio Approach o he Urba Trasi Crew Schedulig Proble, Trasporaio Sciece, 23 : -3. du Merle, O.; Villeeuve, D.; Desrosiers, J.; Hase, P. (999) Sabilized colu geeraio. Discree Maheaics, 94 : Freville, A.; Lorea, L.A.N.; Plaeau, G. (990) New subgradie algorihs for he 0- ulikapsack Lagragea ad surrogae duals, Pre-publicaio # 90. LIPN-Uiversie Paris Nord. Galvão, R.D. (98) A Noe o Garfikel, Neebe ad Rao s LP Decoposiio for he p- Media Proble. Trasporaio Sciece, 5 (3) : Garey, M.R.; Johso, D.S. (979) Copuers ad iracabiliy: a guide o he heory of NPcopleeess, W.H. Freea ad Co., Sa Fracisco. Garfikel, R.S.; Neebe, W.; Rao, M.R. (974) A Algorih for he M-edia Locaio Proble. Trasporaio Sciece, 8 : Gilore, P.C.; Goory, R.E. (96) A liear prograig approach o he cuig sock proble. Operaios Research, 9 : Gilore, P.C.; Goory, R.E. (963) A liear prograig approach o he cuig sock proble - par II. Operaios Research, : Held, M.; Karp, R.M. (97) The ravelig-salesa proble ad iiu spaig rees: par II, Maheaical Prograig, : Kelley, J.E. (960) The Cuig Plae Mehod for Solvig Covex Progras, Joural of he SIAM, 8 : Lorea, L.A.N.; Lopes, F.B. (994) A surrogae heurisic for se coverig probles, Europea Joural of Operaioal Research, 79/ :

11 Lorea, L.A.N.; Narciso, M.G. (996) Relaxaio Heurisics for Geeralized Assige Proble. Europea Joural of Operaioal Research, 9 : Mioux, M. (987) A Class of Cobiaorial Probles wih Polyoially Solvable Large Scale Se Coverig/Se Pariioig Relaxaios. RAIRO, 2 (2) : Reiel, G. (998) See, E.L.F.; Lorea, L.A.N. (2000) Lagragea/Surrogae Heurisics for p-media Probles. I Copuig Tools for Modelig, Opiizaio ad Siulaio: Ierfaces i Copuer Sciece ad Operaios Research, M. Lagua ad J.L. Gozalez-Velarde (eds.) Kluwer Acadeic Publishers, Swai, R.W. (974) A Paraeric Decoposiio Approach for he Soluio of Ucapaciaed Locaio Probles. Maagee Sciece, 2 : Vace, P.H.; Barhar, C.; Johso, E.L.; Nehauser, G.L. (994) Solvig Biary Cuig Sock Probles by Colu Geeraio ad Brach-ad-Boud, Copuaioal Opiizaio ad Applicaios, 3 : -30.