Prioritized Multi-Criteria Decision Making Based on the Idea of PROMETHEE

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1 Available olie at Procedia Computer Sciece 17 (2013 ) Iformatio Techology ad Quatitative Maagemet (ITQM2013) Prioritized Multi-Criteria Decisio Makig Based o the Idea of PROMETHEE Xiaoha Yu a *, Zeshui Xu b, Yig Ma c a College of Commuicatios Egieerig, PLA Uiversity of Sciece ad Techology, Najig, , Chia b College of Scieces, PLA Uiversity of Sciece ad Techology, Najig, , Chia c PLA Uiversity of Sciece ad Techology, Najig, , Chia Abstract There may be prioritizatios amog criteria i some practical multi-criteria decisio makig (MCDM) problems, which are called prioritized MCDM oes. The ivestigatio of such a kid of problems beefits the developmet of the MCDM. However, the existig methods of prioritized MCDM caot cover all situatios, so we develop a ew method based o the idea of PROMETHEE i this paper so as to overcome the drawbacks of the existig methods. After determiig the prefereces amog alteratives by compare them i pairs, we costruct a ituitioistic preferece relatio. A liear optimizatio model is the established to derive the rakig order of the alteratives. At legth, a simple example is take to illustrate the feasibility ad practicability of our ew method The Authors. Published by Elsevier B.V. Selectio ad peer-review uder resposibility of the orgaizers of the 2013 Iteratioal Coferece o Iformatio Techology ad Quatitative Maagemet Keywords: multi-criteria decisio makig, prioritizatio, PROMETHEE, ituitioistic preferece relatio; 1. Itroductio Sice it was itroduced i mid-1960 s, multi-criteria decisio makig (MCDM) has bee a hot topic i decisio makig ad systems egieerig, ad bee prove as a useful tool due to its broad applicatios i a umber of practical problems [1,2,3]. I most of existig literatures, the criteria i MCDM problems are commoly regarded as idepedet ad the relatioships amog criteria are seldom discussed. I fact, due to the complexity of MCDM problems i our daily lives, there may be various coectios amog criteria. A possible kid of relatioships amog criteria ca be prioritizatios, as stated by Yager [4,5], ad a typical example ca be the relatioship betwee the criteria of safety ad cost i the case of selectig a bicycle for child. We usually do ot allow a loss i safety to be compesated by a beefit i cost, i.e., tradeoffs betwee safety ad cost are uacceptable i this case. Simply speakig, there is prioritizatio betwee the criteria of safety ad cost, ad safety has a higher priority tha cost. Such a kid of MCDM problems with prioritizatios amog criteria are called prioritized MCDM oes. * Correspodig author. Tel.: address: yua2006@126.com The Authors. Published by Elsevier B.V. Selectio ad peer-review uder resposibility of the orgaizers of the 2013 Iteratioal Coferece o Iformatio Techology ad Quatitative Maagemet doi: /j.procs

2 450 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) Up to ow, some pioeers have already paid their attetios to this topic [4-9], icludig relevat applicatios i Iformatio Retrieval [10] ad preferece votig [11]. Curret researches maily focus o how to aggregate evaluatig iformatio with respect to criteria with prioritizatios amog them, i.e., how to costruct prioritized aggregatio operators. However, if the prioritized aggregatio operators are applied ito the prioritized MCDM problems, there may be some drawbacks. A typical kid of drawbacks ca be the case that the safeties of two bicycles are idetical ad caot reach the requiremet of the cosumer, the the overall evaluatios of these two bicycles (the results of the prioritized aggregatio) are the same ad we caot give out a effective advice for the cosumer which bicycle shall be chose, because accordig to the idea of the prioritized aggregatio operators we will ot take ito accout the cotributios of criteria with lower priority if criteria with higher priority caot reach the requiremets of the decisio maker. Ituitively, i such a case, it is more likely that the bicycle with lower cost will be chose. This is just a special case as so to clarify the imperfectess of the existig prioritized aggregatio operators. I most practical prioritized MCDM problems, it is usually hard to discover such a kid of imperfectess because of larger umbers of criteria ad complex prioritizatios amog them. Therefore, it is ecessary for us to develop some prioritized MCDM methods so as to make up the imperfectess of the prioritized aggregatio operators based method. For this purpose, we propose a ew prioritized MCDM method i this paper i virtue of the idea of PROMETHEE that rak alteratives cocerig the comparisos of the alteratives i pairs. Based o the ew method, a simple example of prioritized MCDM problem illustrates that the drawback of the prioritized aggregatio operators based method ca be well overcome. We believe that the idea of the proposed prioritized MCDM method is ew ad feasible, ad it is well worth developig. 2. Prioritized Multi-Criteria Decisio Makig Problems The classic multi-criteria decisio makig (MCDM) prescribes ways of evaluatig, rakig ad selectig the most favorable alterative from a set of feasible oes which are characterized by multiple, usually coflictig, criteria [1,2]. The fudametal compoets of a MCDM problem are a set of criteria, C { c1, c2,..., c }, of iterest to the decisio maker ad a set of possible alteratives, X { x1, x2,..., x m }, so as to evaluate each alterative ad select the best oe(s). I their pioeerig work o the MCDM, Bellma ad Zadeh [12] suggested that each criterio ca be represeted as a fuzzy subset over the alteratives. I particular, if c j ( j 1,2,..., ) is a criterio, the we ca represet it as a fuzzy subset c j over X such that cj( x i) is the degree to which this criterio is satisfied by the alterative x i, i.e., cj( x i) is the satisfactio degree of x i over c j. Here, we shall assume cj( x i) [0,1] ( i 1,2,..., m; j 1,2,..., ). However, there usually are some kids of iterdepedeces amog criteria i actual MCDM problems. Prioritizatios are just oe kid of them. If there exists the prioritizatios betwee a pair of criteria, for example ck c l ( kl, {1,2,.., }) ( meas prior to i this paper), we take it for grated that the loss of c k caot be compesated by the beefit of c l. If the prioritizatios rather tha other iterdepedeces exist i a MCDM problem, we call it a prioritized MCDM problem, which is formularized as below. Defiitio 1 [4]. I a MCDM problem, if the set of criteria, C { c1, c2,..., c }, ca be partitioed ito q distict prioritized hierarchies, { H1, H2,..., H q }, such that H k H l if k l, where Hk { ck1, ck2,..., ck k } C, C q q k 1H k (i.e., k 1 k), ad Hk H l for kl, {1,2,..., } ( deotes the ull set), the the problem is called a prioritized multi-criteria decisio makig problem. As stated by Yager [4,5], the prioritizatios ca be classified ito two cases: 1) strictly ordered prioritizatios, if the prioritizatios amog the criteria are a strict liear orderig, i.e., each prioritized hierarchy

3 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) has oly oe criterio, ad the umber of prioritized hierarchies is (without loss of geerality, we always have c1 c2... c ); otherwise, 2) the priority orderig is called weakly ordered prioritizatio. By meas of a ordered weighted averagig (OWA) operator, Ya et al. [6] have revealed that the weakly ordered prioritizatios ca be trasformed to the strictly ordered oes through regardig each prioritized hierarchy as a pseudo criterio, thus we will oly discuss the problems of prioritized MCDM with strict prioritized criteria i this paper. Aother importat cocept is the decisio maker s expectatios/requiremets i a prioritized MCDM problem [5,6]. For each criterio c j ( j 1,2,..., ), the decisio maker usually specifies a expectatio j. If the satisfactio degree of a alterative x over c j achieves the correspodig expectatio, i.e., cj( x ) j, the cj ( x ) ca satisfies the decisio maker; otherwise, cj ( x ) caot satisfies the decisio maker, the the decisio maker will ot take ito accout the cotributes of criteria with lower priority tha c j ay loger. A fuctio E :[0,1] [0,1] is itroduced by Yager [5] to formularize this case, where E (0) 0, E (1) 1, ad if y z ad yz, [0,1], the Ey ( ) Ez ( ). I this paper, Ec ( j ( x )), called the expectatio level of cj ( x ), idicates the degree that the satisfactio degree of a alterative x over c j achieves the decisio maker s expectatio. Geerally speakig, because the decisio maker s expectatios are various for differet criteria, we shall take differet fuctios to measure the expectatio levels with respect to respective criteria. For example, E j correspods to c j, ad the the expectatio level of cj ( x ) ca be Ej( cj( x )). If it is ot ambiguous, we simplify Ej( cj( x )) as Ej ( x ) i followig sectios. Up to ow, most of literatures cotribute themselves to how to aggregate satisfactio degrees so as to evaluate a alterative sythetically cocerig multiple criteria with prioritizatios amog them. Such a kid of problems ca be called prioritized multi-criteria aggregatio (PMCA) oes. A commo idea to hadle such problems is to devise prioritized aggregatio operators [4,5]. We assume that a alterative x shall be evaluated by cosiderig a set of criteria, C { c1, c2,..., c }, where c 1 c 2... c, ad each expectatio level Ej ( x ) is obtaied by correspodig fuctio E j :[0,1] [0,1] ( j 1,2,..., ), the the overall satisfactio degree of x ca be calculated by a prioritized aggregatio operator Cx ( ) PA c( x), c( x),..., c( x) j 1 w 1 2 w c ( x) j j (1) where w ( w1, w2,..., w ) T is the weightig vector of criteria, wj E ( ) k 0 k x, ad E0( x ) 1 by covetio. However, this idea is usuitable to be applied ito prioritized MCDM problems. Let us see the followig example. Example 1. The decisio maker wats to select the better alterative from x 1 ad x 2 cocerig three prioritized criteria c1 c2 c 3 i a prioritized MCDM problem. Satisfactio degrees of x 1 ad x 2 with respect to c 1, c 2 ad c 3 are (0.5, 1, 1) ad (0.5, 0, 0) respectively. Accordig to the decisio maker s expectatios, E 1(0.5) 0, the we have E 1 ( x 1 ) E 1 ( x 2 ) 0. I such a case, the weights are the same for x 1 ad x 2, ad ( 1, 2, 3) T T w w w w (1,0,0). By utilizig (1), we derive their overall satisfactio degrees Cx ( 1) Cx ( 2) 0.5, so we caot judge which alterative is better. But ituitively, x 1 is better tha x 2 because c1( x1) c1( x 2), c2( x1) c2( x 2) ad c3( x1) c3( x 2). Therefore, we shall develop a more proper prioritized MCDM method. j 1

4 452 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) PROMETHEE-Based Prioritized Multi-Criteria Decisio Makig I this sectio, we will develop a ew prioritized multi-criteria decisio makig (MCDM) method based o the idea of PROMETHEE. The PROMETHEE (Preferece Rakig Orgaizatio Method for Erichmet Evaluatio) uses the outrakig methodology to rak alteratives [13]. The PROMETHEE is implemeted i four steps: 1) defie preferece fuctio; the preferece fuctio shows the preferece of the decisio maker for a alterative x k with respect to aother alterative x l regardig a criterio; 2) calculate preferece idex; the preferece idex is used to quatitatively compare alteratives i pairs takig all criteria ito accout comprehesively; 3) costruct valued outrakig graph; outgoig ad icomig flows are determied by meas of relevat preferece idices i this step; ad 4) rak alteratives accordig to the valued outrakig graph. The cetral idea of the PROMETHEE is to compare alteratives i pairs regardig criteria firstly oe by oe the comprehesively. I Example 1, the reaso that both x 1 ad x 2 have idetical overall satisfactio degrees is that their satisfactio degrees over c 1 caot achieve the decisio maker s expectatio, i.e., E1( x1) E1( x 2) 0, thus the cotributios of c 2 ad c 3 are igored. If itroducig prefereces by comparig alteratives pairwise just like what the PROMETHEE does, we may cosider effectively the cotributios of criteria with lower priority o matter how the satisfactio degrees with respect to those criteria with higher priority caot achieve the decisio maker s expectatios Method ad Process We assume that the decisio maker wats to select the best oe(s) from a set of alteratives, X { x1, x2,..., x m }, by cosiderig a set of criteria, C { c1, c2,..., c }, i a prioritized MCDM problem, where c1 c2... c. All satisfactio degrees, ci( x j) ( i 1,2,..., m; j 1,2,..., ), is give, ad we have already derived all expectatio levels E ( x ) i accordace with some give iformatio. i j Step 1. Defie the preferece fuctio. The preferece fuctio Pc( x), c( x) j k j l 0, if c ( x ) c ( x ) j k j l c ( x ) c ( x ), if c ( x ) c ( x ) j k j l j k j l 2 P :[0,1] [0,1] ca be defied as (2) where :[0,1] [0,1] is a mootoic icreasig fuctio with (0) 0, j {1,2,..., } ad k l {1,2,..., m }. I (2), Pc ( j( xk), cj( x l)) gives a measure of the preferece of x k over x l regardig ( x) x. I such a case, (2) ca be rewritte as c j. I this paper, Pc( x), c( x) j k j l 0, if c ( x ) c ( x ) j k j l c ( x ) c ( x ), if c ( x ) c ( x ) j k j l j k j l (3) Step 2. Calculate the preferece idex. Firstly, we itroduce a revisio fuctio to take ito accout the ifluece of expectatio levels. The revisio fuctio : [ 1, 1] [0, 1] [0, 1] ca be defied as ( yz, ) y z y z, y 0 (1 y) z, y 0 (4) Because the fuctio E is mootoic icreasig, we have Ej( xk) Ej( x l) if cj( xk) cj( x l), ad Ej( xk) Ej( x l) if cj( xk) cj( x l). Let Ejkl Ej( xk) Ej( x l) ad cjkl cj( xk) cj( x l), the by (4) we have

5 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) E, P c ( x ), c ( x ) jkl j k j l E c E c, if c 0 jkl jkl jkl jkl jkl 0, if c 0 jkl (5) From (5), if Ej( xk) Ej( x l) meas x k is more closer to the decisio maker s expectatio tha x l, the [ Ej kl, P( cj( xk), cj( xl))] P( cj( xk), cj( xl)) c j kl. Especially, if E jkl 1, [ Ejkl, P( cj( xk), cj( x l))] 1 is the maximum. However, if Ej( xk) Ej( x l), the cj( xk) cj( xl) [ Ej kl, P( cj( xk), cj( x l))] 0. Furthermore, the iflueces of criteria with higher priority must be take ito accout. We defie jkl as the iflueces of c s ( s 1,2,..., j 1) to c j, where jkl 1, j 1 j 1 s 1 mi 1 E,1, j 2,3,..., skl (6) I this case, we the calculate the preferece idex of x k over x l by 1 ( x, x ) E, P c ( x ), c ( x ), k l {1,2,..., m} k l j kl j kl j k j l j 1 (7) ( xk, x l) [0,1] gives a measure of the preferece of x k over x l cocerig all criteria, ad the closer to 1, the greater the preferece. Step 3. Costruct ituitioistic preferece relatio (IPR). I [14], the otio of the IPR was itroduced: Defiitio 2 [14]. A ituitioistic preferece relatio (IPR) B o X is represeted by a matrix B ( bij ) m m X X with b ij ( ij, ij ), where b ij is a ituitioistic fuzzy value, composed by the certaity degree ij to which x i is preferred to x j ad the certaity degree ij to which x i is o-preferred to x j, ad 1 ij ij is iterpreted as the hesitatio degree to which x i is preferred to x j. Furthermore, ij ad ij satisfy 0 ij ij 1, ji ij, ji ij ad ii ii 0.5 ( i, j 1,2,..., m ). I the last step, we obtai two preferece idices betwee alteratives x k ad x l, ( xk, x l) ad ( xl, x k). The former is a measure that x k is preferred to x l, ad the latter is a measure that x l is preferred to x k. I other words, ( xl, x k) ca be regarded as a measure that x k is o-preferred to x l. Thus if a IPR ca be costructed based o ( xk, x l) ad ( xl, x k) ( k l {1,2,..., m }), the rakig order of alteratives i X ca be easily derived accordig to some existig method based o the IPR. If it is uambiguous, we deote ( x, x ) as kl ( k l {1,2,..., m }). k l Theorem 1. 0 kl lk 1 for all k l {1,2,..., m }. Proof. We assume that cj( xk) cj( x l), the Ej( xk) Ej( x l) (because E j is a mootoic icreasig fuctio), ad accordig to (3) Pc ( j( xk), cj( xl)) cj( xk) cj( xl) c j kl ad Pc ( j( xl), cj( x k)) 0. By (5), we further get Ejkl, P cj( xk), cj( xl) Ejkl cjkl Ejkl c jkl ad Ejlk, P cj( xl), cj( x k) 0. Because 0 cjkl, E jkl 1, we have 0 [ Ejkl, P( cj( xk), cj( x l))] 1. Accordig to (6), it is obvious 0 jkl, jlk 1, thus 0 E, P c ( x ), c ( x ) E, P c ( x ), c ( x ) 1 j kl j kl j k j l jlk jlk j l j k

6 454 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) E, P c ( x ), c ( x ) E, P c ( x ), c ( x ) j 1 j kl j kl j k j l j lk j lk j l j k j kl Ej kl, P cj( xk), cj( xl) j 1 j 1 j lk Ej lk, P cj( xl), cj( xk) kl lk Therefore, the theorem always holds. Sequetially, we costruct a matrix, B ( b kl ) m m, o the basis of the preferece idices kl ( k l {1,2,..., m }), such that ay elemet i B is a ituitioistic fuzzy value b kl ( kl, kl ) ( kl, lk ) if k l, ad otherwise b (0.5, 0.5) for k 1,2,..., m. Obviously B is a IPR sice all elemets i B satisfy Defiitio 2. kk Step 4. Rak alteratives based o the costructed IPR. I [15], Xu itroduced a method to reveal the differetial priority of multiple objects by meas of their IPR, which will be used i this step to get the rakig vector of alteratives. A rakig vector ca be defied as v ( v1, v2,..., v ) T m, where v k reflects the rakig m degree of the alterative x k, ad v k 0 ( k 1,2,..., m), v k 1. If the IPR B ( b kl ) m m is cosistet, the kl 0.5( vk v l 1) 1 kl, i.e., 0.5( vk v l 1) [ kl,1 kl ] for all k 1,2,..., m 1 ad l k 1,..., m. However, B ( b ) is usually icosistet. I this case, Xu [15] itroduced two kids of deviatio variables kl m m d kl ad d kl, k 1,2,..., m 1 ad l k 1,..., m, so as to relax the above iequatio as kl dkl 0.5( vk v l 1) 1 kl d kl, for all k 1,2,..., m 1 ad l k 1,..., m, where d kl ad d kl are both oegative real umbers. Because kl kl ad kl lk, we have kl dkl 0.5( vk vl 1) 1 lk d kl. Moreover, Xu [15] established a liear optimizatio model: m 1 (M-1) mi D ( d d ) k 1 l k 1 st.. 0.5( v v 1) d 0.5( v v 1) d 1 v 0, i 1,2,..., m i m vi i 1 1 m dkl, dkl 0 k 1,2,..., m 1; l k 1,..., m kl kl k l kl kl k l kl lk Solvig this model, we ca get the optimal rakig vector v ( v1, v2,..., v ) T m, which ca be used to rakig the alteratives. The larger v k, the more precedig i order the alterative x k Illustrative Example I this subsectio, we will take a simple example to illustrate the feasibility of the prioritized MCDM method i the precedig. Example 2. Suppose there are six criteria, C { c1, c2,..., c 6}, ad five alteratives, X { x1, x2,..., x 5}, i a prioritized MCDM problem, where c 1 c 2... c 6. All satisfactio degrees cj( x i) ( i 1,2,...,5; j 1,2,...,6) are give i the followig table: k 1

7 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) Table 1. Satisfactio degrees c j(x i) c 1 c 2 c 3 c 4 c 5 c 6 x x x x x I additio i accordace with the decisio maker s expectatios, we have already obtaied all expectatio levels E ( x ) of correspodig satisfactio degrees as below. j Table 2. Expectatio levels i E j(x i) c 1 c 2 c 3 c 4 c 5 c 6 x x x x x We first cosider the prefereces of x 1 over x 2. By (3), Pc ( j( x1), cj( x 2)) ( j 1,2,...,6) ca be calculated as 0.2, 0, 0, 0.1, 0.3 ad 0 respectively. Ad the by (5), [ Ej 12, P( cj( x1 ), cj( x 2 ))] ( j 1,2,...,6) equal to 1, 0, 0, 0.28, 1 ad 0 respectively. We the have j 12 1, 1, 1, 0.375, ad for j 1,2,...,6 respectively by (6). I this case, the preferece idex of x 1 over x 2 ca be calculated by (7) as Similar to the above process, we ca calculate the preferece idices of x k over x l for all k l {1,2,...,5} : , , , , 21 0, , , 25 0, , , , , 41 0, 42 0, 43 0, 0, 0.128, 0.188, 0.086, Sequetially, a IPR B ( b kl ) 5 5 ca be costructed, where b kl ( kl, lk ) for k l {1,2,...,5} ad b (0.5, 0.5) for k {1,2,...,5}, i.e., kk (0.5, 0.5) (0.247, 0) (0.08, 0.121) (0.605, 0) (0.139, 0.128) (0, 0.247) (0.5, 0.5) (0.024, 0.259) (0.380, 0) (0, 0.188) B ( b kl ) 5 5 (0.121, 0.08) (0.259, 0.024) (0.5, 0.5) (0.568, 0) (0.196, 0.086) (0, 0.605) (0, 0.380) (0, 0.568) (0.5, 0.5) (0, 0.443) (0.128, 0.139) (0.188, 0) (0.086, 0.196) (0.443, 0) (0.5, 0.5) the by solvig the model (M-1) i Matlab, we get the optimal rakig vector v ( v1, v2,..., v ) T m (0.359, 0.132, 0.315, 0.002, 0.192) T. Therefore, the rakig order of alteratives is x 1 x 3 x 5 x 2 x 4, ad the best alterative is x 1.

8 456 Xiaoha Yu et al. / Procedia Computer Sciece 17 ( 2013 ) Besides, similar to Example 1, if we utilize the prioritized aggregatio operator based method to compare x 2 ad x 4 i Example 2, they will be idifferet, which is ot accordat with our ituitive data aalysis of their satisfactio degrees i Table 1 ( x 2 is obviously better tha x 4 ). Our method well overcomes the drawback, so it is more proper to hadle the prioritized MCDM problems tha the prioritized aggregatio operator based method. 4. Cocludig Remarks I this paper, we have developed a ew prioritized multi-criteria decisio makig (MCDM) method based o the idea of PROMETHEE aimig at overcome the drawbacks of existig methods. After comparig alteratives i pairs, we iovatively costructed a ituitioistic preferece relatio, which is a sigificat combiatio of decisio makig techology ad fuzzy theory. By solvig a liear optimizatio model, established o the basis of the ituitioistic preferece relatio, we fially raked the alteratives. However, the proposed method is somewhat complex eve though it ca well deal with the prioritized MCDM problems. Therefore, it is ecessary to improve it i our future work. Ackowledgmets The work was supported i part by the Natioal Natural Sciece Foudatio of Chia (Nos ad ). Refereces [1] Koele P. Multiple attribute decisio makig: a itroductio. Thousad Oaks: Sage Publicatios; [2] Yager RR, Rybalov A. Full reiforcemet operators i aggregatio techiques. IEEE T Syst Ma Cy B 1998; 28: [3] Peg Y, Kou G, Wag G, Shi Y. FAMCDM: a fusio approach of mcdm methods to rak multiclass classificatio algorithms. Omega- It J Maage S 2011; 39: [4] Yager RR. Prioritized aggregatio operators. It J Approx reaso 2008; 48: [5] Yager RR. Modelig prioritized multicriteria decisio makig. IEEE T Syst Ma Cy B 2004; 34: [6] Ya HB, Huyh VN, Nakamori Y, Murai T. O prioritized weighted aggregatio i multi-criteria decisio makig. Expert Syst Appl 2011; 38: [7] Yu XH, Xu ZS. Prioritized ituitioistic fuzzy aggregatio operators. Iform Fusio 2013; 14: [8] Yager RR, Walker CL, Walker EA, A prioritized measure for multi-criteria aggregatio ad its shapley idex, Fuzzy Iformatio Processig Society (NAFIPS), 2011 Aual Meetig of the North America, p [9] Che SM, Wag CH. A geeralized model for prioritized multicriteria decisio makig systems. Expert Syst Appl 2009; 36: [10] da Costa Pereira C, Dragoi M, Pasi G. Multidimesioal relevace: Prioritized aggregatio i a persoalized Iformatio Retrieval settig. Iform Process Maag 2011; 48: [11] Ami GR, Sadeghi H. Applicatio of prioritized aggregatio operators i preferece votig. It J Itell Syst 2010; 25: [12] Bellma RE, Zadeh LA. Decisio-makig i a fuzzy eviromet. Maage Sci 1970; 17: [13] Bras JP, Vicke PH, Mareschal B. How to select ad how to rak projects: the PROMETHEE method. Eur J Oper Res 1986; 24: [14] Xu ZS. Ituitioistic preferece relatios ad their applicatio i group decisio makig. Iform Scieces 2007; 177: [15] Xu ZS. A method for estimatig criteria weights from ituitioistic preferece relatios. Fuzzy Iform Eg 2009; 1: