MULTIPLE CRITERIA DECISION MAKING USING VIKOR METHOD. APPLICATION IN IRRIGATION NETWORKS IN THE THESSALONIKI PLAIN

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1 Proceedngs of the 13 th Internatonal Conference on Envronmental Scence and Technology Athens, Greece, 5-7 September 2013 MULTIPLE CRITERIA DECISION MAKING USING VIKOR METHOD. APPLICATION IN IRRIGATION NETWORKS IN THE THESSALONIKI PLAIN C. ΤZIMOPOULOS, D. ZORMPA AND C. EVANGELIDES Laboratory of Hydraulcs and Envronmental Management Arstotle Unversty of Thessalonk, 54124, GREECE E-mals: EXTENDED ABSTRACT A multcrtera model s developed for analyzng the plannng strateges for reducng future socal, economc and envronmental costs. The developed multcrtera decson-makng procedure conssts of generatng alternatves, establshng and rankng crtera, assessng crtera weghts, and applcaton of the compromse rankng method. The paper emphaszes compromse based on cooperatveness as a soluton of a decson problem n water resources plannng. A decson makng process commtted to consensus utlzng compromse soluton as a consensus proposal s presented. The presented methodology s utlzed to obtan the optmal rrgaton network among a fnte number of networks n the area of Thessalonk plan. The VIKOR method s appled here to determne compromse soluton wth noncommensurable and conflctng crtera, ncludng economc, envronmental and socal crtera. It focuses on selectng and rankng from a set of alternatves, and determnes compromse solutons for a problem wth conflctng crtera, whch can help the decson makers to reach a fnal decson. The VIKOR method s based on L 1, (Manhattan dstance) and L, (Chebyshev dstance) metrcs, where 1, 2,..., J s the number of alternatves. The soluton norm Q s ntroduced as a lnear combnaton of the L1, and L, norms and the fnal soluton s mn Q, where mn mn L, L 1, represents the maxmum group utlty (maorty rule) and represents the mnmum ndvdual regret of the opponent. As an applcaton, a fnte number of networks n the area of Thessalonk plan s selected for obtanng the optmal rrgaton network. The networks should be graded for certan evaluaton crtera n order to be ncorporated n the Economc Plan of the GOEB Thessalonk (General Organsaton of Hydraulcs Feld Thessalonk Lagada). The crtera have been selected for each alternatve and as optmal soluton was consdered the network satsfyng mostly the selected crtera. Besdes, the mportance of each crteron s consdered and the method A.H.P. (Analytc Herarchy Process) proposed by Saaty was appled for the assessment of the weghts. Keywords: Mult-crtera analyss, compromse programmng, VIKOR method, Thessalonk plan, rrgaton networks, A.H.P.

2 1. INTRODUCTION Land and waters n the world are under ncreasng pressure from the contnuous growth n demand for many dfferent purposes, and the allocaton of water n the rver basn s a complex management problem, wth condtons that may foster conflcts. Conflct over the management of a shared water resource arses mostly because of dfferng obectves among dfferent nterest groups. The agrcultural sector n Greece s the largest water consumer and essentally t has the lon's share n the management of water. At the begnnng of the second half of the 20th century started all modern land reclamaton proects. The organzed collectve rrgaton networks n Greece were manly bult durng the decade of 1960 (Konstantndes, 1989; Balot, 2009). They are characterzed by age but also by an old technology and the desgn does not take nto account the factor of protecton of both water and the envronment. It s dffcult today to characterze the functonng of an rrgaton network, takng nto account only one crteron. For the management of rrgaton water and repar of rrgaton systems, knowledge of a set of crtera s requred that wll help the sustanable dsposal of water. The desgn of an rrgaton management system s characterzed by: a) a large degree of uncertanty, b) sophstcated subect desgn wth often multdmensonal obectves, c) dffculty n determnng the ndvduals or groups that contrbute to the decson and d) sophstcated structure alternatves whch combne n sequence several elementary actons and plannng horzons (Netto et al., 1996). To solve a problem for MCA, many methods were developed, but n ths artcle the Vkor method s used, as analyzed by Oprcovc and Tzeng (2002; 2004; 2006) and Oprcovc (2009). In the area of water resources management, management problems have been addressed by Ducksten and Oprcovc (1980); Oprcovc (1993); Maraveas (1998); Karasavvds (2003), Baka (2006); Zargham (2006); Schau and Wu (2006 ); Karasavvds (2009) etc. 2. MCDM Multcrtera optmzaton (MCO) s consdered as the process of determnng the best feasble soluton accordng to establshed crtera whch represent dfferent effects. However, these crtera usually conflct wth each other and there may be no soluton satsfyng all crtera smultaneously. Thus, the concept of Pareto optmalty was ntroduced for a vector optmzaton problem (Pareto, 1896; Kuhn and Tucker, 1951; Zadeh, 1963). Pareto optmal solutons have the characterstc that, f one crteron s to be mproved, at least one other crteron has to be made worse. In such cases, a system analyst can ad the decson makng process by makng a comprehensve analyss and by lstng the mportant propertes of the Pareto optmal (nonnferor) solutons. However, n engneerng and management practce there s a need to select a fnal soluton to be mplemented. An approach to determne a fnal soluton as a compromse was ntroduced by Yu (1973), and other dstance-based technques have also been developed (Chen and Hwang, 1992). Many papers have proposed analytcal models as ads n conflct management stuatons. Among the numerous approaches avalable for conflct management, one of the most prevalent s multcrtera decson makng. Multcrtera decson makng (MCDM) may be consdered as a complex and dynamc process ncludng one manageral level and one engneerng level (Ducksten and Oprcovc, 1980). The manageral level defnes the goals, and chooses the fnal optmal alternatve.

3 The multcrtera nature of decsons s emphaszed at ths manageral level, at whch publc offcals called decson makers have the power to accept or reect the soluton proposed by the engneerng level. These decson makers, who provde the preference structure, are off lne from the optmzaton procedure done at the engneerng level. Very often, the preference structure s based on poltcal rather than only techncal crtera. In such cases, a system analyst can ad the decson makng process by makng a comprehensve analyss and by lstng the mportant propertes of nonnferor and/or compromse solutons (Yu, 1973). The engneerng level of the MCDM process defnes alternatves and ponts out the consequences of choosng any one of them from the standpont of varous crtera. Ths level also performs the multcrtera rankng of alternatves. 3. VIKOR method The VIKOR method was developed for multcrtera optmzaton of complex systems. It determnes the compromse rankng-lst, the compromse soluton, and the weght stablty ntervals for preference stablty of the compromse soluton obtaned wth the ntal (gven) weghts. Ths method focuses on rankng and selectng from a set of alternatves n the presence of conflctng crtera. It ntroduces the multcrtera rankng ndex based on the partcular measure of closeness to the deal soluton (Oprcovc, 1998). Assumng that each alternatve s evaluated accordng to each crteron functon, the compromse rankng could be performed by comparng the measure of closeness to the deal alternatve. The multcrtera measure for compromse rankng s developed from the L p-metrc used as an aggregatng functon n a compromse programmng method (Yu, 1973; Zeleny, 1982). The varous J alternatves are denoted as a 1; a 2;... ; a. For alternatve a, the ratng of the aspect s denoted by f,.e., f s the value of crteron functon for the alternatve a ; n s the number of crtera. Development of the VIKOR method started wth the followng form of L p-metrc: 1 n p p p, /, 1 p 1 L w f f f f Wthn the VIKOR method L 1, =S and L, ; =1.2.. = R are used to formulate rankng measure. The soluton obtaned by mn S s wth a maxmum group utlty ( maorty rule), and the soluton obtaned by mn R s wth a mnmum ndvdual regret of the opponent. The compromse soluton F c s a feasble soluton that s the closest to the deal F, and compromse means an agreement establshed by mutual concessons, as s llustrated n c c Fgure 1 by f f f and f f f

4 Fgure 1. Ideal and compromse soluton. The compromse rankng algorthm VIKOR has the followng steps: (1) Determne the best f and the worst f values of all crteron functons, = 1,2,..., n; f f max f, f mn f f the -th functon represents a beneft; max f, f mn f f the -th functon represents a cost. (2) Compute the values S and R, = 1,2,...,J, by the relatons n / S w f f f f 1 R max w f f / f f where w are the weghts of crtera, expressng the DM s preference as the relatve mportance of the crtera. (3) Compute the values Q, = 1,2,...,J, by the relaton where S 1 Q v S S S S v R R R R mns, S max S, R mn R, R max R and v s ntroduced as a weght for the strategy of the maorty of crtera (or the maxmum group utlty ), whereas 1 v s the weght of the ndvdual regret. (4) Rank the alternatves, sortng by the values S, R and Q n decreasng order. The results are three rankng lsts. (5) Propose as a compromse soluton the alternatve [F (1) ] whch s the best ranked by the measure Q (mnmum) f the followng two condtons are satsfed: C 1. Acceptable Advantage : (2) (1) Q( F F ) DQ where: F (2) s the alternatve wth second poston n the rankng lst by Q and

5 DQ 1/ ( m 1). C 2. Acceptable stablty n decson makng : The alternatve F (1) must also be the best ranked by S or/and R. Ths compromse soluton s stable wthn a decson makng process, whch could be the strategy of maxmum group utlty (when v > 0.5 s needed), or by consensus v 0.5, or wth veto (v < 0.5). Here, v s the weght of decson makng strategy of maxmum group utlty. If one of the condtons s not satsfed, then a set of compromse solutons s proposed, whch conssts of: Alternatves F (1) and F (2) f only the condton C 2 s not satsfed. Alternatves F (1), F (2),..., F (M) f the condton C 1 s not satsfed. The F (M) s ( M ) (1) Q F Q F DQ for maxmum M (the postons of determned by the relaton these alternatves are n closeness ). 4. CHOICE OF OPTIMAL IRRIGATION NETWORK 4.1. Descrpton of problem The study area conssts of 5 rrgaton networks of the plan of Thessalonk, specfcally: a) F 1 = the rrgaton network of Agos Athanasos, b) F 2 = the rrgaton network of Brachas, c) F 3 = the rrgaton network of Malgara d) F 4 = the rrgaton network of Chalastras and e) F 5 = the rrgaton network of Chalcedonas. A common feature of fve rrgaton networks (optons / scenaros) s that they are suppled wth water from the rver Axos to serve the water needs of crops. Addtonally, the constructon took place durng the frst phase of land reclamaton proects n the plan of Thessalonk durng the decade of 1960 (Konstantnds, 1989). Therefore, the problem s reduced to the calbraton and classfcaton of these networks n terms of fnancal plannng. To fnd the deal soluton to the problem, t s necessary to establsh crtera that wll cover both the economc-envronmental and productve part of the evaluaton of each scenaro. Thus, we use the followng basc crtera: a) X 1 = productvty per acre based on the crop, b) X 2 = cost of producton, c) X 3 = cost of mantenance, d) X 4 = crop needs n water, e) X 5 = products avalablty. The calbraton crtera was partally made usng the A.H.P method (Analytc Herarchy Process) whch was ntroduced n 1980 by Saaty, for both scenaros and for weghts as shown n Table 1. The worst score s 1 and the maxmum 9. The man reason ths calbraton was used was to acheve as much obectfcaton of compensatory weghts and crtera as possble. Table 1. Par-wse comparson scale for AHP preferences Numercal ratng Verbal udgments of preferences Numercal ratng Verbal udgments of preferences 9 Extremely 5 Strongly 8 Very strongly to extremely 4 to strongly 7 Very strongly 3 6 Strongly to very strongly 2 to moderately 1

6 4.2. Evaluaton of the problem The F (J, I) s the matrx whose elements ndcate the value of dfferent scenaros for dfferent crtera. The ndex J s for dfferent scenaros (alternatves), whle the ndex I for the dfferent crtera. Shows that the F (J, I) the 'calbraton' or evaluaton of each scenaro based on varous crtera. Table 2 shows the numercal calbraton for the study area, whle Table 3 gves the calbraton logc problem. Table 2. Decson matrx X 1 X 2 X 3 X 4 X 5 F F F F F Table 3. Logcal evaluaton of problem X 1 X 2 X 3 X 4 X 5 F 1 F 2 F 3 F 4 F 5 Very strongly to extremely Strongly to very strongly Very strongly to strongly Strongly to moderately to moderately to moderately to moderately Very strongly to extremely to strongly Strongly Extremely For fndng the weghts by Saaty (1980) the normalzaton of problem ntally took place and for ths reason the program Matlab was used, whch resolved the 5x5 matrx (Table 4), and calculated: a) the maxmum prncpal value λ max = and b) the egenvector or anchor vector, whch was consdered the same for each alternatve actvty, where W 1 = 0.919, W 2 = , W 3 = , W 4 = and W 5 = Ths vector s normalzed and weghts were: w,1=0.584, w,2 =0.219, w,3 = 0.107, w,4 =0.049, w,5 =0.041 και 5 W, =1. 1 Table 4. Dmensonless assessment of the problem X 1 X 2 X 3 X 4 X 5 F F F F F For the numercal computaton of dstances, for dfferent weghts and p, drafted the program "Comprom" programmng language Vsual Fortran (Tzmopoulos, 2013). Table 5 shows the dstances of alternatves n relaton to the optmum pont for p = 1 and p = 19.

7 The value p = 19 s qute large value, beyond whch the norm s constant and therefore apples: ( ). L 19 L Table 5. Classfcaton by VIKOR method S sortng S R sortng R Q sortng Q F F F F F Accordng to Table VIKOR rankng n the rankng s as follows: F 1 F4 F2 F3 F5 Under the condton C 1 we have: (2) (1) Q( F ) Q( F ) DQ ( ) > 1/(5-1)=0.25. Accordng to the second condton rankngs of S, R concdes wth the classfcaton of Q and thus fulflled the condton C 2. The rrgaton network of Agos Athanasos outperforms the two norms n relaton to the other four. 5. CONCLUSIONS The VIKOR method ntroduces the rankng ndex based on the partcular measure of closeness to the deal soluton. The VIKOR method used lnear normalzaton, addtonal to the normalzed value n the VIKOR method does not depend on the evaluaton unt of a crteron functon. Summarzng all the above, the method leads the user to fnd the deal soluton and the prortzaton of rrgaton to the needs of the mantenance program, thus enablng the recpent of the decson n ths case n GOEB Thessalonk make the ratonal plannng of mantenance of rrgaton networks, rankng the dfferent networks dependng on ther mportance. REFERENCES 1. Baka M., (2006) Water Resources Management Methods of Multcrtera Analyss (Compromse Programmng) and Applcaton n Watershed Rver Nestos. Master thess, School of Rural and Surveyng Engneerng, AUTH, Balot B., (2009) Management of rrgaton systems by means of lnear programmng. Applcaton to Irrgaton Network Gefyras. Master thess, School of Rural and Surveyng Engneerng, AUTH, Chen J. and Hwang L., (1992) Fuzzy Multple Attrbute Decson Makng: Methods and Applcatons. Sprnger-Verlag, Berln. 4. Ducksten, L. and Oprcovc, S., (1980) Multobectve optmzaton n rver basn development. Water Res. Res., 16(1),

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