A Multilevel Algorithm for the Network Design Problem

Transcription

1 A Multlevel Algorthm for the Network Desgn Problem Hdeson A. Slva 1,2, Alceu S. Brtto Jr 2, Luz E. S. Olvera 3, Alessandro L. Koerch 2,3 1 Companha Paranaense de Energa(Copel), 2 Pontfíca Unversdade Católca do Paraná, 3 Unversdade Federal do Paraná hdeson@copel.com, alceu@ppga.pucpr.br, lesolvera@nf.ufpr.br, alekoe@ppga.pucpr.br Abstract- Ths paper presents a level-based algorthm to deal wth the network desgn problem. The proposed algorthm performs the desgn of the dfferent network levels smultaneously, n the sense that the nformaton from an ascendng level can be used to optmze the soluton obtaned for a prevous level. The expermental results have shown a sgnfcant cost reducton of 13.16% when the proposed algorthm s compared to a conventonal approach n whch the network levels are processed n separate. In addton, the tme consumed for an engneer to desgn the network used n the experment was reduced. I. INTRODUCTION The Network Desgn Problem (NDP) conssts of desgnng a low cost network by scalng and defnng the devces and the lnks among them n order to attend a known demand whle mnmze the cost. Many proposals to solve such a challengng problem are avalable n the lterature, and NPD s NP-Completeness[1]. The motvaton s related to the potental applcatons n dfferent areas of the human actvty, such as the plannng of telecommuncaton, electrc power, water and gas networks. Most of the avalable methods model the NDP as a Stenertree Problem (STP) and use evolutonary algorthms to search for possble solutons. Huy and Ngha [2] propose an approach based on parallel genetc algorthm that uses a ftness based on a Dstance Network Heurstc (DNH). They acheved promsng results by comparng the solutons obtaned through tests carred out on the OR-Lbrary [3] wth related works based on dfferent meta-heurstcs: a) Esbensen s genetc algorthm wth graph reducton n [4] ; b) greedy randomzed adaptve search procedure (GRASP) n [5] ; c) a parallel approach of the GRASP method (PGRASP) n[6] ; and d) the Tabu Search (TS) n [7]. Dng and Ish n [8] present a dynamc verson of the Stener-tree. Such an approach uses an Onlne Genetc Algorthm (OLGA) and the PRIM algorthm for ftness evaluaton. Promsng results have been acheved when compared wth other ftness functons, such as: the DNH, the Shortest Path Heurstc (SPH) and the Average Dstance Heurstc (ADH). Zhong and Huang [9] propose a new dscrete Partcle Swarm Optmzaton (PSO) algorthm to deal wth Stenertree Problem. They have also compared the obtaned results wth a dfferent approach based on Genetc Algorthm (GA). The expermental results usng the OR-Lbrary have shown that the proposed algorthm s better than both: the GA based verson and the orgnal PSO. Zhan, Zhang and Chung[10] proposes another modfcaton of the PSO algorthm, that speeds up the processng by consderng only the promsng solutons (partcles) for ftness evaluaton. Besdes GA and PSO, other searchng technques have been used to solve the STP such as the Tabu Search[11] [12]. Despte the aforementoned contrbutons to solve the NDP problem, the network desgn s stll an open problem. To explan the reason, let us to focus on the nfrastructure of telecommuncaton networks. Fgure 1 shows a dagram that represents a network wth three levels. In the frst level (last mle) are the clents (demands), whch must be attended by faclty nodes represented by crcles. In the second level, the red squares are faclty nodes, whle the crossed square represents a faclty node of the thrd level of the network. The desgn of the network nfrastructure should address all levels smultaneously. However, the current methods usually consder the desgn of each level ndvdually, such as the solutons proposed for traffc networks [13] [14], urban ralway networks [15], water networks [16], power dstrbuton [17] [18], and telecommuncaton networks [19]. In a complete soluton, the desgn of the last-mle (frst level) where the devces (facltes) necessary to attend a set of clents (demands) s defned should takes nto account the second level where the defnton of the mddle devces necessary to support the access devces of the last-mle s done. Moreover, the desgn of the second level should takes nto account the thrd level where the defnton of the man devces necessary to support everythng s carred out. Fg.1: Dagram of a three-level network structure.

2 The goal of ths paper s to propose a new algorthm to deal wth the NDP that consders smultaneously the optmzaton of multple levels of the network nfrastructure. The proposed algorthm s able to desgn a low cost network provdng the number and the geographc locaton of devces and the lnk among them for each level of the network. The expermental results show that the proposed algorthm provdes a cost reducton n the network project. The paper s organzed as follows: Secton II ntroduces the concepts related to the desgn of telecommuncaton network. Secton III presents the proposed algorthm for network desgn, whle Secton IV shows the experments used to evaluate the proposed method. Conclusons and future works are presented n the last secton. II. THE NETWORK DESIGN PROBLEM The network desgn problem conssts of defnng a for each level of the network: a) the number of devces that wll be nstalled; b) the geographc locaton of each devce; and c) the correct path between devces or between clents and devces whle mnmzng the nfrastructure cost. The number of levels n the nfrastructure may vary accordng to the applcaton. For nstance, n the case of telecommuncaton networks, the number of levels s usually three. The problem can be modeled through a graph representaton. In a smple way, at each level of the nfrastructure we have nodes representng facltes and demands. Ascendng nodes represent facltes from ascendng levels of the network structure. Equaton (1) represents the objectve functon that mnmzes the cost of the entre network mplementaton. It sums up the path length that s necessary to connect demand nodes wth facltes nodes. When there are ascendng nodes, the cost of the correspondng connecton s also takng nto account. Equatons (2) and (3) mpose some constrants to the problem, that only facltes and ascendng nodes that are actvated (status: on) must be consdered n equaton (1). The objectve functon (1) represents the cost of all actvated nodes plus the cost of the path for lnkng nodes. Varables: L: number of levels M l : set of demand nodes of the l th level; N l : set of faclty nodes of the l th level; A l : set of ascendng nodes of the l th level; x j : cost of the th demand to the j th faclty node; y j : cost of the th faclty node to the j th ascendng node; n : bnary status of the th faclty node: (1-on, 0-off); a : bnary status of the th ascendng node: (1-on, 0-off); c l : cost of the th actvated faclty node of the l th level. Objectve Functon: MIN x n + y a + c n (1) M j N j N j A j j N Constrants: n {0,1} N (2) a {0,1} A (3) The elements of the array c are the monetary costs of the correspondng devces. In ths paper, we consder a proportonal cost of the devces of the dfferent levels. The cost ncreases from level one to level three. Durng the experments, we have consdered the cost of devces of the levels one, two and three as 100, 1000 and , respectvely. III. THE MULTILEVEL ALGORITHM The proposed Multlevel Algorthm for Network Desgn (MAND) performs the network desgn ndependently of the number of levels. 1 MAND ( M ) 2 M: set of all nodes of problem; 3 L: number of levels; 4 l: current level; 5 A[l]: set of ascendng nodes per level; 6 N[l]: set of faclty nodes per level; 7 Begn 8 cost_level[l]={max,, Max}; 9 got_mproved[l]={true,, True}; 10 l=1; 11 whle (l < L) 12 whle (Any_Level_s_Improved(Got_Improved)) 13 whle (got_mproved[l] and got_mproved[l+1]) 14 cost = Level_Processng(l); 15 f (cost < cost_level[l]) 16 then cost_level[l]=cost; 17 got_mproved[l]=true; 18 else got_mproved[l]=false; 19 endf 20 cost = Level_Processng(l+1); 21 f (cost < cost_level[l+1]) 22 then cost_level[l+1]=cost; 23 got_mproved[l+1]=true; 24 else got_mproved[l+1]=false; 25 endf 26 endwhle 27 whle (got_mproved[l+2] and got_mproved[l+1]) 28 cost = Level_Processng(l+2); 29 f (cost < cost_level[l+2]) 30 then cost_level[l+2]=cost; 31 got_mproved[l+2]=true; 32 else got_mproved[l+2]=false; 33 endf 34 cost = Level_Processng(l+1); 35 f (cost < cost_level[l+1]) 36 then cost_level[l+1]=cost; 37 got_mproved[l+1]=true; 38 else got_mproved[l+1]=false; 39 endf 40 endwhle 41 endwhle 42 f (got_mproved[l] and l > 1) 43 then l=l-1; 44 else l=l+1; 45 endf 46 endwhle 47 return cost_level[l]; 48 end. Algorthm 1: Multlevel Algorthm for Network Desgn.

3 The L levels of a network are smultaneously processed. The nput data s the set of demands M (clents), whch at ths startng pont contans only the demands for the frst level. Ths data s used for processng the level l n order to fnd an ntal set of facltes to attend the clents (last-mle). Intally, the only nformaton used to compute the cost s the value assocated wth the set of faclty nodes assocated wth the level l. In the next teraton, after the level l+1 has beng processed, t s possble to re-evaluate the facltes defned for the level l by consderng the cost of the ascendng nodes from the level l+1. By consderng the nformaton from ascendng levels, t allows us to search for better soluton consderng the nteracton between adjacent levels. Thus, when the current level s l, t s possble to use nformaton from level l+1, f the current level s l+1, t s possble to use nformaton from level l+2, and so on. Ths allows us to optmze the entre network nfrastructure smultaneously. The procedure Level_Processng(l) s executed for each level. It s responsble for searchng the number and geographc locaton of devces, defne path between devces, and between devces and demands. Insde the Level_Processng dfferent evolutonary algorthms may be used, such as: GA[20] and PSO[21]. For the experments undertaken to evaluate the proposed method, we have used a GA. To fnd the best path between nodes t s necessary a graph search algorthm. The Djkstra algorthm [23] s employed n ths paper. At the end of the proposed algorthm, there s a condtonal control to check f there are more levels for desgnng. If there are more levels, then the loops are repeated. Fgure 2 llustrates the trells of the proposed algorthm. It represents the nteracton between levels. For nstance, the soluton 3 n the frst level (l) s an optmzed verson of the soluton 2, but consderng addtonal nformaton from the soluton B of the second level (l+1). In a smlar way, the soluton D n the second level s an optmzaton of the soluton C, but consderng addtonal nformaton from the soluton I of the thrd level (l+2). The evolutonary method used to desgn the levels was a standard GA based n bt representaton, whch s mplemented n Kanpur Genetc Algorthms Laboratory (KanGAL) [22], wth the followng parameters: Chromosome length: number of geographc coordnates Populaton sze: 60 Generaton: 10,000 Probablty of crossover: 0.95 Probablty of mutaton: Sharng: false Selecton: tournament selecton. The chromosome length s defned based on the number of geographc coordnates whch represent the locatons where the facltes may be nstalled. The GA parameters were obtaned wth from emprcal experments. In addton, to fnd the best path between nodes the Djkstra algorthm [23] was employed. Fg.3: Case of Study A geographc vew of the clents. IV. Fg.2: Trells of the MAND. EXPERIMENTAL RESULTS The MAND was mplemented n C programmng language, and the reported results were obtaned on a PC-Pentum Dual- Core E5300, 2.6 GHz wth 2 Gbytes of memory. Fgure 3 shows the nput data used n our experments: 105 clents (green ponts) and ther geographc postons. Based on that, we have executed two sets of experments. In the frst set of experment (E1), the desgn of the network nfrastructure was carred out by consderng the proposed algorthm, and the three network levels were desgned smultaneously. In the second set of experments (E2), we have performed the desgn of each level n separate the output of the last-mle (frst level) was used to desgn the

4 second level, and so on. For both set of experments, the fnal results represent the average of fve repettons snce the tests nvolve a search strategy for optmzaton of each level of the network. the most expensve levels. We can observe that 19 facltes were defned n frst level to attend the clents, 3 facltes n second level to attend level one and 1 faclty n thrd level to attend the second level. As one can also see n Table I, the proposed algorthm shows a sgnfcant ncreasng n the tme consumng (52%) when compared wth the strategy where each level s processed n separated. The reason s that n the proposed algorthm some levels are computed more than once, snce nformaton from ascendng levels are used n the prevous levels. However, the tme consumed by the proposed algorthm s stll lower than the tme requred by an engneer to desgn the same network aded by a CAD software and usng only maps (about 5 days),.e, wthout the use of an specfc software to ad n the network desgn. The fnal soluton, provded by the MAND, s shown n Fgure 4. As one can see, the nodes (facltes) of level one (purple crcles), attend the clents (green crcles). The nodes of level one are attended by nodes of the second level (red squares); and fnally, the nodes of level two are attended by nodes of the thrd level (orange squares). Fg.4: Example of a soluton found by the proposed MAND TABLE I EXPERIMENTAL RESULTS E1 (MAND Algorthm) (average of 5 runs) E2 (levels n separate) (average of 5 runs) Optmzed Level Parameters $ cost # demands # facltes $cost # demands # facltes 3 8 $ cost # demands 3 8 # facltes 1 1 Total Cost Tme consummed (n mnutes) The average results of the set of experments E1 and E2 are shown n Table I. As one can see, the proposed algorthm provdes a sgnfcant total cost reducton of 13.16%, even showng a hgher cost for the frst level. It s related to the strategy used by the algorthm that computes agan a prevous level usng nformaton from an ascendng level. In fact, t s possble to observe that n some cases t s better to have a hgher cost n the frst levels of the network to save money n V. CONCLUSION AND FUTURE WORKS In ths paper, we have presented a multlevel algorthm to deal wth the network desgn problem. The algorthm performs the desgn of the dfferent levels a network smultaneously, n the sense that the nformaton of ascendng levels s used to process agan a prevous level. The expermental results have shown a sgnfcant cost reducton when the proposed algorthm was compared wth a conventonal approach where the network levels are processed n separate. Based on that, we may conclude that the MAND can be appled n the plannng of real-world nstances of network nfrastructure, adng engneers to save tme and money. Future work wll be done wth the objectves (lke cost and flexblty) of usng mult-objectve algorthms n the MAND. ACKNOWLEDGMENT The authors wsh to thank COPEL (Companha Paranaense de Energa) and CNPq (Conselho Naconal de Desenvolvmemnto Centífco e Tecnológco). REFERENCES [1] D. S.Johnson, J. K. Lenstra, A. H. G. Rnnoov Kan. The Complexty of the Network Desgn Problem. Networks, vol. 8, n. 4. John Wley & Son, New Jersey, 1978, pp [2] N. V. Huy, N. D. Ngha. Solvng Graphcal Stener Tree Problem Usng Parallel Genetc Algorthm. IEEE Internatonal Conference on Research, Innovaton and Vson for the Future, pp [3] J. E. Beasley. OR-LIBRARY: dstrbutng test problems, URL: accessed n January/2010. [4] H. Esbensen. Computng Near-Optmal Solutons to the Stener Problem n Graph Usng a Genetc Algorthm, Networks, vol.26, pp , [5] L. S. Martns, P. Pardalos, M. G. Resende, C. C. Rbero. Greedy Randomzed Adaptve Search Procedures for the Stener Problem n

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