Revista de Sistemas de Iformação da FSM. 8 (20) pp. 3-35 http://www.fsma.edu.br/si/sistemas.html Two Examples of pplicatio of TOPSIS to Decisio Makig Problems Reato. Krohlig, & Talles T.M. de Souza bstract I this work, we apply the Techique for Order Preferece by Similarity to Ideal Solutio (TOPSIS) to two case studies. The first example is cocered with the evaluatio of the best respose alteratives i case of accidet with oil spill i the sea. The secod example cosiders the retal evaluatio of residetial properties. Simulatios results are promisig ad show the feasibility of the techique. Keywords Decisio makig, multi-criteria aalysis, TOPSIS. M I. INTRODUCTION ulticriteria decisio makig problems are usually characterized by a fiite umber of alteratives ad by several criteria (attributes) that are usually coflictig ad a vector of weights that idicates the importace of each criterio. Lots of effort was applied to the developmet of several methodologies to help solve the multicriteria decisio makig problem, with several importat advaces beig made alog the way. techique for decisio makig that is widely used is kow as TOPSIS (Techique for Order Preferece by similarity to Ideal Solutio), which is a used to evaluate de performace of alteratives through their similarity to the ideal solutio [7]. ccordig to this techique, the best alterative is the oe closest to the positive ideal solutio ad most distat from the egative ideal solutio [2, 6]. The ideal positive solutio is the oe that maximizes beefit criteria ad miimizes cost criteria, while the ideal egative solutio is the oe that maximizes cost criteria ad miimizes beefit criteria. That meas that the positive ideal solutio is made of the highest possible values at all beefit criteria, while the egative ideal solutio is made up of the highest possible values i the cost criteria. For the last years, TOPSIS has bee applied to decisio makig problems i Brazil. methodology to evaluate the effect of urbaizatio o water beds beig aggregated through TOPSIS ad ELECTRE TRI was proposed i []. I [5] TOPSIS was used to evaluate the saitatio services i the Velhas River basi i MG. I [0] a method of prioritizig surveillace actios i the presece of pesticides i superficial waters was applied i some regioal basis. I this case, TOPSIS was used to select the hierarchy of basis with higher cotamiatio potetial. I [] TOPSIS was used i decisio support i the choice of trasport route for grai productio. I [4, 5] a methodology to guide strategic decisios o the choice of highest priority projects i Brazil s trasport ifrastructure usig TOPSIS was proposed. I [8, 9] fuzzy TOPSIS was applied to the problem of oil spill i the souther shore of ES ad i [6] it was used for the selectio of hedge fud maagers. I this article our goal is to apply the TOPSIS techique i two case studies: i) to evaluate the best alteratives for cotetio i case of oil spills i Espírito Sato s souther shores. ii) for the selectio of best alteratives i retal evaluatio of residetial properties. The rest of this article is orgaized as follows: i sectio 2 we briefly describe TOPSIS. Results for both studies are preseted i sectio 3, while coclusios ad future directios fialize the article i sectio 4. II. MULTICRITERI DECISION MKING The decisio matrix is made of alterative ad criteria is described by: C... C x K x =... M O M m xm x L m () R.. Krohlig is a professor at the Productio Egieerig Departmet ad djuct Professor at the Graduate Program i Computer Sciece (PPGI) at the Federal Uiversity of Espírito Sato - UFES, v. Ferado Ferrari, s/. Prédio CT VII CEP 29060-270, Vitória, ES (e-mail: krohlig.reato@gmail.com). T.T.M. de Souza is a udergraduate studet at the Computer Egieerig course at the Computer Sciece Departmet of UFES, v. Ferado Ferrari, s/. Prédio CT VII CEP 29060-270, Vitória, ES (e-mail: talles@gmail.com). 3
where, 2,..., m are feasible alteratives,, C, C2,..., C are criteria ad x ij idicates the performace of alterative accordig to criterio C j. The weight vector ( ) W = w, w,..., w is composed by the idividual weights for every criterio C j is held to w j =. Data from matrix i= come from differet sources ad therefore they must be ormalized so that they ca be trasformed ito a o dimesioal matrix ad the compariso amog criteria is possible. I this work, the matrix is ormalized for each criterio C j accordig to the followig formula: xij pij =, com i =,..., m, j =,..., m xij i= This way, the ormalized decisio matrix represets the relative performace of the alteratives ad ca be described by = ( pij ) mx, com i =,..., m, e j =,...,. Now, we describe the multicriteria decisio makig techique TOPSIS.. TOPSIS (Techique for Order Preferece by Similarity to Ideal Solutio) Geerally, the evaluatio criteria ca be classified ito two groups: beefit ad cost. Beefit meas that a higher value is best, while for cost, the opposite is held true. The algorithm to calculate the best alterative accordig to TOPSIS [6] is described by the followig steps: Step : Calculate the positive ideal solutios + (beefits) ad he egative ideal solutios (costs) accordig to the followig formula: + + + + ( p, p2,..., p m ) ( p, p2,..., p m ) = (2) = (3) where + p j = ( max i pij, j J ;mi i pij, j J2 ) (4) ( mi, ; max, 2 ) = (5) p j i pij j J i pij j J where J ad J 2 represet the criterio beefit ad cost, respectively. Step 2: Calculate the Euclidea distaces betwee + (beefits) ad betwee i ad followig formula: 2 d + = w j ( p + j pij ) com i =,..., m. j= i () i ad (cost) usig the (6) 2 d = w j ( p j pij ) com i =,..., m. j= Step 3: Calculate the relative closeess ξ i for each alterative i i relatio to the positive ideal solutio + usig: ξ i di = + di + di I the ext sectio we apply this techique to two case studies. III. EXPERIMENTL RESULTS. Case Study : Oil spill i the souther shore of Espírito Sato The first case study cosiders a problem ivolvig a possible accidet causig a oil spill i the sea at the souther shore of the state of Espírito Sato [3]. The decisio matrix adopted from [8, 9] show i Table is composed by 0 alteratives ad 2 criteria, C beig a cost criterio ad C 2 a beefit criterio, accordig to the termiology used i TOPSIS. We ormalized the decisio matrix as described i the previous sectio. The weight vector is W = ( w, w2 ) = (0.507,0.4893) [8]. The result of the applicatio of TOPSIS is show i Table 2. s we ca see, alterative 8 represeted the best strategy accordig to TOPSIS. B. Case Study 2: Retal evaluatio of residetial properties The secod case study cosiders a problem ivolvig the evaluatio of retal prices i the city of Volta Redoda, RJ. The iformatio o the decisio matrix preseted here are available i [4, 3]. The decisio matrix i Table 3 is made of 5 alteratives ad 8 criteria. The ormalized decisio is show i Table 4. The weight vector is listed i Table 5. The results of the applicatio of TOPSIS are show i Table 6. s we ca see, the alterative 5 is the best oe accordig to TOPSIS. recet study o real state evaluatio usig a hybrid approach combiig TODIM ad eural etworks was proposed i [2]. Similarly, TOPSIS could also be used, for it is also based i distace metrics, i.e., the best alterative is the oe closer to the positive ideal solutio ad farther away from the egative ideal solutio. IV CONCLUSIONS I this paper we applied the decisio support techique kow as TOPSIS (Techique for Order Preferece by Similarity to Ideal Solutio) to two case studies: i) to evaluate the best alteratives for cotetio i case of oil spills i Espírito Sato s souther shores. ii) for the selectio of best alteratives i retal evaluatio of residetial properties. I both case studies ivestigated, we could verify the (7) (8) 32
feasibility of TOPSIS, which is very effective ad very simple i terms of computatioal implemetatio. Oe of its mai advatages is the fact that it does ot require parameter tuig. If the decisio matrices with ucertaities are described by fuzzy umbers, the fuzzy TOPSIS [9] might be used to solve those problems. CKNOWLEDGMENTS Talles T.M. de Souza would like to thak the support of UFES for its scietific stiped. The authors would like to thak the Editor i chief ad the reviewers for the suggestios that helped improve the paper. REFERÊNCIS [] Castro, L.M.. de. Proposição de metodologia para avaliação dos efeitos da urbaização os corpos de água. Tese de Doutorado em Saeameto, Meio mbiete e Recursos Hídricos. Uiversidade Federal de Mias Gerais, Mias Gerais, 2007. [2] Ertuğrul, I., & Karakaşoğlu, N. Performace evaluatio of Turkish cemet firms with fuzzy aalytic hierarchy process ad TOPSIS methods. Expert Systems with pplicatios, 36(), pp. 702-75,2009. [3] Ferreira, J.P. álise de estratégias de resposta a derramameto de óleo pesado o litoral do Espírito Sato utilizado modelagem computacioal, Dissertação de Mestrado, Programa de Pós-Graduação em Egeharia mbietal, Uiversidade Federal do Espírito Sato, Espírito Sato, 2006. [4] Gomes, L.F..M. & Ragel, L..D. applicatio of the TODIM method to the multicriteria retal evaluatio of residetial properties, Europea Joural of Operatioal Research,93, 204-2, 2009. [5] Heller, P.G.B. valiação dos serviços de saeameto de quatro muicípios da bacia hidrográfica do Rio das Velhas/MG. Uma abordagem da dimesão tecológica. Dissertação de Mestrado em Saeameto, Meio mbiete e Recursos Hídricos da Escola de Egeharia. Uiversidade Federal de Mias Gerais. Mias Gerais, 2007. [6] Huag, J. Combiig etropy weight ad TOPSIS method for iformatio system selectio. I Proceedigs of the IEEE Coferece o Cyberetics ad Itelliget Systems, CIS 2008, pp. 28-284, 2008. [7] Hwag, C. L. & Yoo, K. P. Multiple attributes decisio makig methods ad applicatios. Berli: Spriger-Verlag, 98. [8] Krohlig, R.. & Campaharo, V.C. Fuzzy TOPSIS para tomada de decisão multicritério: Uma aplicação para o caso de acidetes com derramameto de óleo o mar, ais do XLI SBPO - Simpósio Brasileiro em Pesquisa Operacioal, SOBRPO, Porto Seguro, B, de -4 de setembro, 2009. [9] Krohlig, R.. & Campaharo, V.C. Fuzzy TOPSIS for group decisio makig: case study for accidets with oil spill i the sea. Expert Systems with pplicatios, 38(4), pp. 490-497, 20. [0] Meezes, C.T. Métodos para priorização de ações de vigilâcia da preseça de agrotóxicos em águas superficiais: Um estudo em Mias Gerais.Dissertação de Mestrado em Saeameto, Meio mbiete e Recursos Hídricos da Escola de Egeharia. Uiversidade Federal de Mias Gerais. Mias Gerais, 2007. [] Mirada, L.M. de. Cotribuição a um modelo de aálise multicritério para apoio à decisão da escolha do corredor de trasporte para escoameto da produção de graéis agrícolas de Mato Grosso. Tese de Doutorado em Egeharia de Trasportes. Uiversidade Federal do Rio de Jaeiro, Rio de Jaeiro, 2008. [2] Moreira, D.S., Silva, R.S. & Ferades,.M.R. Egeharia de avaliações de imóveis apoiada em técicas de aálise multicritério e redes eurais artificiais, Revista de Sistemas de Iformação da FSM,.6, pp. 49-58, 200. [3] Ragel, L..D & Gomes, L.F..M. valiação multicritério de imóveis resideciais: Uso combiado dos métodos UT-CR e TODIM. Ivestigació Operativa, v. XVII, p. 3-27, 2009. [4] Silva, R.B da. álise estruturada das dimesões do desevolvimeto sustetável e a iterdepedêcia com a fução trasporte: Proposta de metodologia para orietar decisões estratégicas a implatação de projetos prioritários para a ifraestrutura de trasporte o Brasil. Tese de Doutorado em Egeharia Oceâica. Uiversidade Federal do Rio de Jaeiro COPPE/UFRJ, Rio de Jaeiro, 2008. [5] Silva, R.B. da & Cavalcati Netto, M.. Uma estrutura de apoio à decisão para orietar a escolha de projetos prioritários para a ifraestrutura de trasporte o Brasil, ais do XLII SBPO - Simpósio Brasileiro em Pesquisa Operacioal, SOBRPO, Beto Goçalves, RS, de 30 de gosto 03 de Setembro, 200. [6] Valladares, C.F.G. O processo para seleção de gestores de fudos de ivestimeto utilizado o método multicritério TOPSIS. Dissertação de Mestrado Profissioalizate em dmiistração. Faculdades Ibmec RJ, 20. Table. Decisio Matrix lteratives C 3 (x0 ) C 3 2 (x0 ) 8.627 5.223 2 9.838 4.023 3 0.374 3.495 4 8.200 5.659 5 5.854 7.989 6 8.08 5.790 7 6.845 7.083 8 5.738 8.238 9 5.858 8.89 0 6.269 7.808 Table 2. lterative ξ i Orderig Classificatio 0.3695 8 2 0.3 9 3 0.0000 0 4 0.465 7 5 0.9568 3 6 0.4859 6 7 0.7584 5 8.0000 9 0.986 2 0 0.8989 4 33
lteratives Table 3. Decisio Matrix [4] Criteria C C2 C3 C4 C5 C6 C7 C8 3 290 3 3 6 4 0 2 4 80 2 2 4 2 0 3 3 347 2 2 5 0 4 3 24 2 3 2 5 4 0 5 5 360 3 4 4 9 6 2 89 2 3 5 0 7 85 4 0 8 5 80 2 3 6 0 9 2 2 2 3 0 6 0 0 0 2 20 3 5 0 4 280 2 2 7 4 2 90 5 2 0 3 2 60 3 3 2 6 4 3 320 3 3 2 8 2 5 4 80 4 6 lteratives Table 4. Normalized decisio matrix [4] Criteria C C2 C3 C4 C5 C6 C7 C8 0.068 0.03 0. 0.075 0.045 0.069 0.74 0 2 0.09 0.064 0.067 0.05 0.045 0.046 0.087 0 3 0.068 0.23 0.033 0.05 0.09 0.057 0.043 0 4 0.068 0.044 0.067 0.075 0.09 0.057 0.74 0 5 0.4 0.27 0. 0. 0.82 0.03 0.043 0.43 6 0.045 0.03 0.067 0.075 0.045 0.057 0.043 0 7 0.023 0.03 0.033 0.025 0.045 0.046 0 0.43 8 0.4 0.028 0.067 0.075 0.045 0.069 0 0.43 9 0.045 0.043 0.067 0.075 0 0.069 0 0 0 0.045 0.042 0.033 0.075 0.045 0.057 0.043 0 0.09 0.099 0.067 0.05 0.09 0.08 0.74 0.43 2 0.023 0.032 0.33 0.025 0.045 0.057 0.087 0 3 0.045 0.057 0. 0.075 0.09 0.069 0.043 0.43 4 0.068 0.3 0. 0.075 0.09 0.092 0.087 0.43 5 0.09 0.064 0.033 0. 0.045 0.069 0.043 0.43 34
Table 5. Weight for each criterio [4] Criteria Weights Normalized Weights C 5 0.25 C2 3 0.5 C3 2 0.0 C4 4 0.20 C5 0.05 C6 2 0.0 C7 0.05 C8 2 0.0 Table 6. lterative Orderig ξ i Classificatio 0.544 7 2 0.3432 0 3 0.385 9 4 0.4547 8 5 6 0.364 2 7 0.2609 8 0.5627 5 9 0.0896 4 0 0.200 3 0.8209 2 2 0 5 3 0.5565 6 4 0.839 3 5 0.6582 4 35