Validating Multiattribute Decision Making Methods for Supporting Group Decisions

Transcription

1 Valdatg Multattrbute Decso Makg Methods for Supportg Group Decsos Chug-Hsg Yeh, Seor Member, IEEE Clayto School of Iformato Techology Faculty of Iformato Techology, Moash Uversty Clayto, Vctora, 3800, Australa Yu-Her Chag Departmet of Trasportato ad Commucatos Maagemet, Natoal Cheg Kug Uversty Taa, 700, Tawa Abstract Dfferet multattrbute decso makg (MADM) methods ofte produce cosstet rakg outcomes for the same problem. I group decso settgs, dvdual rakg outcomes made by dvdual decso makers are ofte cosstet wth the group rakg outcome. To address the cosstecy problem of rakg outcomes, ths paper develops a ew valdato approach for selectg the most vald rakg outcome amog all feasble outcomes. Based o four ormalzato procedures ad three aggregato procedures, e MADM methods are developed to solve the geeral group MADM problem that requres cardal rakg of the decso alteratves. The valdato approach selects the group rakg outcome of a MADM method whch has the hghest cosstecy degree wth ts correspodg dvdual rakg outcomes. A scholarshp studet selecto problem s used to llustrate how the approach works. The approach s applcable to large-scale multattrbute group decso problems where cosstet rakg outcomes ofte est betwee dfferet MADM methods ad betwee dfferet decso makers. Keywords MADM, group decso makg, valdato I. INTRODUCTION Multattrbute decso makg (MADM) has bee wdely used supportg decsos rakg or selectg oe or more alteratves from a fte umber of alteratves wth respect to multple crtera or attrbutes. Qute a few MADM methods have bee developed for a wde varety of decso problems [][2]. These decso problems ca be solved by varous MADM methods. However, there s o best method for the geeral MADM problem, due to the multplcty ad complety of multattrbute decsos. I decso stuatos where cardal rakg of all or a subset of the alteratves s requred, dfferet methods ofte produce cosstet rakg outcomes for the same problem [3][4]. As such there s a eed to apply all the methods avalable to solve a MADM problem ad select the most vald method that best reflects the values of the decso makers [5][6]. Valdato MADM research have bee coducted to address three ma ssues cludg the problem formulato relato to the defto ad scorg of attrbutes, the quatfcato of weghts, ad the selecto of the MADM methods [7]. The results of MADM research suggest that the valdato of MADM methods remas a maor challegg ssue [8]. I ths paper, we focus o the valdato of decso outcomes produced by varous MADM methods for a gve group MADM problem. I ths group MADM problem, a collectve or group decso s to be made for evaluato or selecto of alteratves based o the opos of all dvdual decso makers volved wth respect to a set of coflctg crtera or attrbutes. The ssue of the MADM method selecto has bee addressed from varous decso cotets alog two les of developmet: (a) epermetal comparsos of MADM methods for eamg ther approprateess of use ad/or theoretcal valdty, ad (b) method selecto procedures for specfc characterstcs of the decso problem ad dstct features of avalable methods the form of decso support systems or as geeral selecto prcples [3]. These studes caot result a set of gudeles or practcal decso support systems that eable decso makers to select a proper MADM method for a specfc problem practce. Due to ther mplct ad eplct assumptos, these studes do ot ormally eame the valdty of the decso outcome. To address ths valdato ssue MADM research for supportg group multattrbute decsos, we preset a ew valdato approach. The approach ca select the most vald rakg outcome from a umber of rakg outcomes produced by avalable MADM methods the cotet of group decso makg. The most wdely used theory solvg MADM problems probably s the multattrbute utlty theory or multattrbute value theory (MAVT) [9]. Wth smplcty both cocept ad computato, MAVT-based MADM methods are tutvely appealg to the decso makers practcal applcatos. These methods are partcularly suted to decso problems where a cardal preferece or rakg of the decso alteratves s requred. I addto, these methods are the most approprate quattatve tools for group decso support systems [0][]. As such, ths paper cosders three ma MAVT-based MADM methods whch are applcable to largescale decso problems where the rakg outcomes produced by dfferet methods are most lkely to be sgfcatly dfferet. Despte ther dversty, MAVT-based MADM problems ofte share the followg commo characterstcs: (a) a fte umber of comparable alteratves, (b) multple attrbutes (evaluato crtera) for evaluatg the alteratves, (c) ocommesurable uts for measurg the performace ratg of /08 /$ IEEE CIS 2008

2 the alteratves o each attrbute, ad (d) attrbute weghts for represetg the relatve mportace of each attrbute. The performace ratgs of the alteratves o all attrbutes are to be aggregated wth the attrbute weghts usg a MADM method order to obta a overall preferece value for each alteratve. The resultat overall preferece values provde a cardal rakg of the alteratves. I subsequet sectos, we frst dscuss the geeral MAVTbased group MADM problem together wth avalable methods. We the preset the ew approach for valdatg the rakg outcomes produced by avalable methods. Fally we coduct a emprcal study of a scholarshp studet selecto problem to demostrate the effectveess of the approach. II. THE GROUP MADM PROBLEM The group MADM problem volves a fte set of m decso alteratves A (, 2,..., m), whch are to be evaluated by a group of p decso makers DM k (k, 2,, p) wth respect to a set of attrbutes or crtera C (, 2,...,). These evaluato crtera are measurable quattatvely or assessable qualtatvely, ad are depedet of each other. Assessmets are to be made by each decso maker DM k (k, 2,, p) to determe (a) the weght vector W k (w k, w k 2,, w k ) ad (d) the decso matr X k { k,, 2,, m;, 2,, }. The weght vector W k represets the weghts (relatve mportace) of the attrbutes C (, 2,...,) gve by the decso maker DM k usg a cardal scale. Cardal weghts are usually ormalzed to sum to, order to allow the weght value to be terpreted as the percetage of the total mportace weght. The decso matr X k represets the performace ratgs ( ) of alteratve A (, 2,..., m) wth respect to attrbutes C (, 2,..., ), whch are ether obectvely measured (for quattatve attrbutes) or subectvely assessed by the decso maker DM k (for qualtatve attrbutes) usg cardal values. The cardal values gve the weght vector W k ad the decso matr X k represet the absolute prefereces of the decso maker DM k. These dvdual weght vectors ad the decso matrces are averaged to represet the group weght vector W ad group decso matr X. As such, the group weght vector s gve by where w w W (w, w 2,, w ) () p k k p ;, 2,,. The group decso matr X s gve by p where k 2 X... k m m 2 m p ;, 2,, m;, 2,,. (2) The W () ad X (2) dcate that the opos of all p decso makers o attrbute weghts ad alteratves performace ratgs are weghted equally. Gve the group weght vector W ad the group decso matr X, the obectve of the problem s to rak all the alteratves by gvg each of them a overall preferece value wth respect to all attrbutes. III. MADM METHODS The geeral group MADM problem preseted above ca be solved by MAVT-based methods, such as (a) the smple addtve weghtg (SAW) method, ad (b) the techque for order preferece by smlarty to deal soluto (TOPSIS) ad (c) the weghted product (WP) method. The ma dffereces betwee these methods le (a) the ormalzato procedure for comparg all performace ratgs measured usg ocommesurable uts o a commo scale, ad (b) the aggregato procedure for combg the ormalzed decso matr ad weght vector for obtag a overall preferece value for each alteratve [3]. Due to these structural dffereces, the rakg outcome produced by these methods may ot always be cosstet for a gve decso matr ad weght vector. I fact, the emprcal study preseted ths paper shows that the dvdual rakgs are so dfferet from the group rakgs that the relatve effectveess of the MADM methods used eeds to be eamed to help make ratoal group decsos. A MADM method essece volves two key procedures: ormalzato ad aggregato. I geeral applcatos, quattatve performace ratgs of the alteratves are ofte assessed by dfferet measuremet uts. MADM methods thus use a ormalzato procedure order to make the comparso across performace ratgs uder dfferet uts a decso matr compatble. For eample, SAW uses the ormalzato procedure of lear scale trasformato (ma), ad TOPSIS uses the vector ormalzato procedure. However, there are other ormalzato procedures that ca be used wth MADM methods, depedet of the aggregato procedure used. The followg lsts four avalable ormalzato procedures for MADM methods [2]. A. Normalzato Procedures ) Vector Normalzato (N) Ths procedure dvdes the performace ratgs of each attrbute the decso matr by ts orm. The ormalzed performace ratgs (r ) of the decso matr are calculated as r (3) m 2 The vector ormalzato procedure mples that all attrbutes have the same ut legth of vector. The ma advatage of ths procedure s that every attrbute s measured dmesoless uts, thus makg t easer for ter-attrbute comparsos. The ma dsadvatage s that t does ot lead to a measuremet scale of equal legth because the mmum ad mamum values of the scales are ot equal to each attrbute.

3 Due to a o-lear scale trasformato, a straghtforward comparso s hard to make [2]. 2) Lear Scale Trasformato, Ma-M Method (N2) Ths procedure uses the followg formulas to ormalze the decso matr ( ) for beeft (the larger, the greater the preferece) attrbutes ad cost attrbutes (the smaller, the greater the preferece) respectvely. m, f s a beeft attrbute ma m r (4) ma,f s a cost attrbute ma m where ma ad m are the mamum ad mmum values of the th attrbute respectvely. The advatage of ths ormalzato procedure s that the scale of measuremet rages precsely from 0 to. The lowest ormalzed performace ratg of a attrbute s 0, whle the hghest ormalzed performace ratg s. A possble drawback of ths procedure s that the scale trasformato does ot lead to a proportoal chage performace ratgs [2]. 3) Lear Scale Trasformato - Ma Method (N3) Ths procedure dvdes the performace ratgs of each attrbute by ts mamum value. The ormalzed value of for beeft ad cost attrbutes s gve respectvely as, f s a beeft attrbute ma r (5), f s a cost attrbute ma where ma s the mamum value of the th attrbute. The value of the ormalzed r rages from 0 to, ad the attrbute s more favorable as r approaches. The sgfcace of the scale trasformato s that all performace ratgs are trasformed a lear (proportoal) way, so that the relatve order of magtude of the performace ratgs remas equal [2]. 4) Lear Scale Trasformato - Sum Method (N4) Ths procedure dvdes the performace ratgs of each attrbute C (, 2,, ) by the sum of performace ratgs for that attrbute, as follows: r where s performace ratg for each alteratve A (, 2,..., m) wth respect to attrbute C (, 2,, ). Wth the ormalzed decso matr, a aggregato procedure ca be appled to obta a overall preferece value for each alteratve, o whch the alteratve rakg ca be based. The followg lsts three aggregato procedures avalable MAVT-based MADM methods [3][6]. (6) B. Aggregrato Procedures ) The Smple Addtve Weghtg (SAW) Method The SAW method, also kow as the weghted sum method, s probably the best kow ad most wdely used MADM method [2]. The basc logc of the SAW method s to obta a weghted sum of the performace ratgs of each alteratve over all attrbutes. Wth a ormalzed decso matr (r ) ad a weght vector (w ), the overall preferece value of each alteratve (V ) s obtaed by V w r ;, 2,, m. (7) The greater the value (V ), the more preferred the alteratve (A ). Research results have show that the lear form of tradeoffs betwee attrbutes used by the SAW method produces etremely close appromatos to complcated olear forms, whle matag far easer to use ad uderstad [2]. 2) The Techque for Order Preferece by Smlarty to Ideal Soluto (TOPSIS) The TOPSIS method s based o the cocept that the most preferred alteratve should ot oly have the shortest dstace from the postve deal soluto, but also have the logest dstace from the egatve deal soluto [2]. Wth a ormalzed decso matr (r ) ad a weght vector (w ), the postve deal soluto A + ad the egatve deal soluto A - ca be determed based o the weghted ormalzed performace ratgs (y ) by y w r ;, 2,, m;, 2,,. (8) ( 2 2 A y, y,..., y ); A ( y, y,..., y ) (9) y y + ma y, f m y, f s a beeft attrbute s a cost attrbute m y, f s a beeft attrbute ma y, f s a cost attrbute The dstace (D + ) betwee alteratves A ad the postve deal soluto, ad the dstace (D - ) betwee alteratves A ad the egatve deal soluto ca be calculated respectvely by D + ( + 2 y y) ; D ( y y ), 2,, m. (0) The overall preferece value of each alteratve (V ) s gve by D V ;, 2,, m. () + D + D The greater the value (V ), the more preferred the alteratve (A ). 2 ;

4 The advatages of usg TOPSIS have bee hghlghted by (a) ts tutvely appealg logc, (b) ts smplcty ad comprehesblty, (c) ts computatoal effcecy, (d) ts ablty to measure the relatve performace of the alteratves wth respect to dvdual or all attrbutes a smple mathematcal form, ad (e) ts applcablty solvg varous practcal MAVT-based MADM problems [3]. Despte ts merts comparso wth other MADM methods, the TOPSIS method does ot cosder the relatve mportace (weght) of the dstaces from the postve ad the egatve deal solutos [4]. Ths ssue has bee addressed by modfyg TOPSIS to corporate attrbute weghts the dstace measuremet [3][5]. I the modfed TOPSIS procedure, (8) ad (0) the aggregato procedure are replaced wth (2) ad (3) respectvely, gve as follows: D + y r ;, 2,, m;, 2,,. (2) w ( y y) ; D w ( y y ) ;, 2,, m. (3) I ths study we use the modfed TOPSIS procedure to weght the dstace betwee the alteratve ad the postve (or egatve) deal soluto. Ths modfed TOPSIS procedure reflects the terrelatoshp betwee the dstace ad the correspodg attrbute weght as mpled by the cocept of the degree of optmalty used [3]. 3) The Weghted Product (WP) Method The WP method uses multplcato for coectg attrbute ratgs, each of whch s rased to the power of the correspodg attrbute weght. Ths multplcato process has the same effect as the ormalzato process for hadlg dfferet measuremet uts, as such t requres o ormalzato procedure preseted the prevous secto. The logc of WP s to pealze alteratves wth poor attrbute values more heavly. Wth a decso matr ( ) ad a weght vector (w ), the overall preferece score of each alteratve (S ) s gve by S w ;, 2,, m. (4) where w. w s a postve power for beeft attrbutes ad a egatve power for cost attrbutes. I ths study, for easy comparso wth other methods, the relatve overall preferece value of each alteratve (V ) s gve by where w V ( * ) w ;, 2,, m. (5) * ad 0 V. The greater the value (V ), ma the more preferred the alteratve (A ). C. MADM Methods Combg the four ormalzato procedures (N, N2, N3, ad N4) wth the frst two aggregato procedures (SAW ad modfed TOPSIS) wll result 8 dfferet methods, amed as SAW-N, SAW-N2, SAW-N3, SAW-N4, TOPSIS-N, TOPSIS-N2, TOPSIS-N3, ad TOPSIS-N4 respectvely. These 8 methods ad the WP method ca be used to solve the same geeral group MADM problem that requres cardal rakg of all the alteratves. IV. VALIDATION OF GROUP DECISION OUTCOMES Ofte there are qute a few methods avalable for solvg a specfc MADM problem. Due to ther structural dfferece, these methods may produce cosstet rakg outcomes for a gve weght vector ad decso matr [6]. Despte sgfcat developmets MADM method selecto research, the valdato of decso outcomes remas a ope ssue. Ths s maly due to the fact that the true cardal rakg of alteratves s ot kow [3]. To address ths ssue for the 9 avalable MADM methods preseted, we develop a ew valdato approach for selectg the most vald rakg outcome amog all feasble outcomes the cotet of group decso makg. I group decso makg, the stakeholders or decso makes ofte have dfferet vews of the attrbute weghts ad the alteratves performace ratgs. To reach a compromsed soluto, the values gve by dvdual decso makers for the weght vector ad decso matr are ofte averaged as the group values. As such, the dvdual rakg outcomes based o the values gve by dvdual decso makers may ot be cosstet wth the fal group rakg outcome derved from the averaged group values. As suggested by estg studes [3][7] ad evdeced the emprcal study of ths paper, dfferet MADM methods ofte produce dfferet fal group rakg outcomes for the same group weght vector W ad group decso matr X. Amog all feasble group rakg outcomes, dvdual decso makers would prefer the outcome whch s most cosstet wth ther ow rakg outcome. Ths s the oto o whch the ew valdato approach s based. Wth m decso alteratves A (, 2,..., m) ad p decso makers DM k (k, 2,, p) usg a MADM method, oe group rakg outcome V (, 2,, m) ad p dvdual rakg outcomes V k (k, 2,, p) wll be produced. The cosstecy degree (or the correlato) betwee V ad each V k ca be measured by Pearso s correlato coeffcets or Spearma s rak correlato coeffcets, resultg p correlato coeffcets. The cosstecy degree of the method betwee the group ad dvdual rakg outcomes for the gve problem ca be obtaed as the average of the p correlato coeffcets. The approach wll select the group rakg outcome of a MADM method whch has the hghest cosstecy degree, as compared to that of other methods. That s, the rakg outcome selected wll have the greatest cosstecy wth all dvdual rakg outcomes produced by the same method. Ths mples that the method used s the most vald oe, as the rakg outcome t produces s most acceptable by the decso makers as a whole.

5 V. EMPIRICAL STUDY To llustrate the cosstet rakg outcomes by dfferet MADM methods ad dfferet decso makers, we preset a scholarshp studet selecto problem. To show how the ew valdato approach ca be used to select the most vald rakg outcome for a gve problem, we apply the 9 MADM methods preseted the prevous secto to solve the scholarshp studet selecto problem. The obectve of the scholarshp studet selecto problem s to select applcats for dustry sposored scholarshps based o ther performace o o-academc, qualtatve selecto crtera va a tervew process. The reaso for ecludg the academc crtera s that all the applcats have overcome a cosderable academc hurdle to become elgble for study. Therefore they are epected to be capable of meetg all the specfed academc requremets of the scholarshp. Durg the scholarshp durato of three years, scholarshp studets are requred to work wth dustry sposors for a total perod of oe year uder a dustry-based learg program. Based o comprehesve dscussos wth dustry sposors, a set of eght attrbutes (selecto crtera) relevat to the dustrybased learg program s determed, cludg commuty servces, sports ad hobbes, work eperece, eergy, commucato sklls, busess attrbute, maturty, ad leadershp. I ths scholarshp studet selecto problem, the eght attrbutes are weghted equally, because the three tervewers (decso makers) or the dustry sposors (stakeholders) caot determe other acceptable weghts a far, covcg maer. Ths s le wth the prcple of suffcet reaso [8], whch suggests the use of equal weghts f the decso makers have o reaso to prefer oe attrbute to aother. I addto, o sgle attrbute weghtg method ca guaratee a more accurate result, ad the same decso makers may elct dfferet weghts usg dfferet methods [9][20]. Ths suggests that there s o easy way for determg attrbute weghts ad there are o crtera for determg what the true weght s. I a recet scholarshp ru, 6 applcats atteded the tervew. Ther performace o the eght attrbutes was assessed by three tervewers (decso makers) o a 6-pot Lkert-type scale, ragg from 5 (etremely hgh) to 0 (etremely low). The result of ths tervew process costtuted three dvdual decso matrces (X, X 2 ad X 3 ) ad oe group decso matr X wth m 6 ad 8. The group decso matr X was geerated from X, X 2 ad X 3 usg (2). The group weght vector W used was (0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25) whch satsfes w. By applyg the e MADM methods to the group decso matr X ad the group weght vector W, e group rakg outcomes are obtaed. These e group rakg outcomes are ot cosstet. As a smple llustrato, Table I shows the group rakgs (out of 6) of the frst 0 applcats (A, A 2,, A 0 ) (based o the SAW-N method) usg three of the e methods. For easy comparso, applcats A (, 2,, 6) are deoted order of ther overall preferece value V (, 2,, 6) by the SAW-N method. If there were oly 0 applcats to be selected, A 0 would ot be selected usg TOPSIS-N or WP. For most decso stuatos where the umber of applcats to be selected vares, there wll be some applcats beg cluded usg some methods ad beg ecluded wth other methods. TABLE I. GROUP RANKING COMPARISON OF FIRST 0 APPLICANTS BETWEEN THREE REPRESENTATIVE METHODS SAW-N TOPSIS-N WP A V Rakg V Rakg V Rakg A A A A A A A A A A By applyg each of e MADM methods to the three dvdual decso matrces (X, X 2 ad X 3 ) ad the weght vector W separately, three dvdual rakg outcomes are obtaed. These dvdual rakg outcomes by dvdual decso makers usg each method are ot cosstet wth the group rakg outcome usg the same method. As a smple llustrato, Table II shows the group rakg (out of 6) of the frst 0 applcats (A, A 2,, A 0 ) ad three dvdual rakgs made by the three decso makers (DM, DM 2, ad DM 3 ) usg the SAW-N method. Each of the three decso makers wll select a dfferet set of top 0 applcats ad deed a dfferet set of scholarshp studets whe the umber of applcats to be selected s more tha oe. TABLE II. RANKING COMPARISON OF FIRST 0 APPLICANTS BETWEEN THREE DECISION MAKERS USING THE SAW-N METHOD Group DM DM 2 DM 3 A V Rakg V Rakg V Rakg V Rakg A A A A A A A A A A

6 To select amog the cosstet group rakg outcomes produced by e methods, we apply the valdato approach to each of e MADM methods dvdually. The cosstecy degrees of SAW-N, SAW-N2, SAW-N3, SAW-N4, TOPSIS- N, TOPSIS-N2, TOPSIS-N3, TOPSIS-N4, ad WP usg Pearso s correlato coeffcets (Spearma s rak correlato coeffcets) are 0.8 (0.66), 0.77 (0.59), 0.79 (0.6), 0.80 (0.62), 0.67 (0.5), 0.65 (0.52), 0.59 (0.45), 0.52 (0.46) ad 0.73 (0.4) respectvely. Ths result suggests that the group rakg outcome produced by the SAW-N method should be used, as t s most cosstet wth the vews of dvdual decso makers, thus most acceptable by them. It s oteworthy that the selecto of the group rakg outcome produced by the SAW-N method s ustfable oly for the problem data set used the emprcal study. Dfferet problem data sets may result a dfferet method beg selected. Ths suggests that o sgle best method ca be assumed for the geeral group cardal rakg problem. I solvg a gve group decso problem wth may methods avalable, the valdato approach developed ths paper ca be appled to all avalable methods for detfyg the most vald rakg outcome from the perspectve of all decso makers as a whole. VI. CONCLUSION There are ormally a umber of methods avalable for solvg group MADM problems, defed by a gve weght vector ad decso matr. Icosstet rakg outcomes are ofte produced by dfferet decso makers ad MADM methods. Despte the mportace of valdatg decso outcomes produced by dfferet MADM methods, very few studes have bee coducted to help the decso makers deal wth the cosstecy problem of rakg outcomes ad make vald decsos. I ths paper, we have developed a ew emprcal valdato approach for selectg the most vald group rakg outcome for a gve problem data set. The most vald group rakg outcome s the oe that s most cosstet wth dvdual rakgs made by dvdual decso makers usg a partcular MADM method. We have preseted a scholarshp studet selecto problem to llustrate how the approach ca be used to help select the most vald group rakg outcome for a gve data set. Wth ts smplcty both cocept ad computato, the approach ca be appled geeral group decso problems solvable by compesatory MADM methods. It s partcularly suted to large-scale MADM problems where the decso outcomes produced by dfferet methods dffer sgfcatly. ACKNOWLEDGMENT Ths research was supported by the Natoal Scece Coucl of Tawa (Grat NSC H MY2) ad the Natoal Cheg Kug Uversty (Grat OUA 96-2T-6-9). REFERENCES [] J.S. Dyer, P.C. Fshbur, R.E. Steuer, J. Walleus, ad S. Zots, Multple crtera decso makg, multattrbute utlty theory: The et te years, Maagemet Scece, vol. 38:5, pp , 992. [2] C.L. Hwag ad K.S. Yoo, Multple Attrbute Decso Makg: Methods ad Applcatos. Berl: Sprger-Verlag, 98. [3] C.-H. Yeh, The selecto of multattrbute decso makg methods for scholarshp studet selecto, Iteratoal Joural of Selecto ad Assessmet, vol. :4, pp , [4] S.H. Zaaks, A. Solomo, N. Wshart, ad S. Dublsh, Mult-attrbute decso makg: A smulato comparso of select methods, Europea Joural of Operatoal Research, vol. 07, pp , 998. [5] B.F. Hobbs, V. Chakog, W. Hamadeh, ad E.Z. Stakhv, Does choce of multcrtera method matter? A epermet water resources plag, Water Resources Research, vol. 28, pp , 992. [6] C.-H. Yeh, A problem-based selecto of multattrbute decsomakg methods, Iteratoal Trasactos Operatoal Research, vol. 9, pp. 69-8, [7] E. Berroder ad V. St, A method usg weght restrctos data evelopmet aalyss for rakg ad valdty ssues decso makg, Computers & Operatos Research, vol. 34, pp , [8] T.J. Stewart, Future treds MCDM, Multcrtera Aalyss, J. Clmaco, Ed. Berl: Sprger, 997, pp [9] R. Keeey ad H. Raffa, Decsos wth Multple Obectves, Prefereces ad Value Tradeoffs. New York: Cambrdge Uversty Press, 993. [0] U. Bose, A.M. Davey, ad D.L. Olso, Mult-attrbute utlty methods group decso makg: past applcatos ad potetal for cluso GDSS, Omega, vol. 25:6, pp , 997. [] N.F. Matsatss ad A.P. Samaras, MCDA ad preferece dsaggregato group decso support systems, Europea Joural of Operatoal Research, vol. 30:2, pp , 200. [2] S. Chakraborty ad C.-H. Yeh, Cosstecy comparso of ormalzato procedures multattrbute decso makg, WSEAS Trasactos o Systems ad Cotrol, vol. 2:2, pp , [3] C.-H. Yeh ad Y.-H. Chag, Modelg subectve evaluato for fuzzy group multcrtera decso makg, Europea Joural of Operatoal Research, press. [4] S. Oprcovc ad G.-H. Tzeg, Compromse soluto by MCDM methods: A comparatve aalyss of VIKOR ad TOPSIS, Europea Joural of Operatoal Research, vol. 56:2, pp , [5] H. Deg, C.-H. Yeh ad R.J. Wlls, Iter-compay comparso usg modfed TOPSIS wth obectve weghts, Computers & Operatos Research, vol. 27:0, pp , [6] Y.-H. Chag ad C.-H. Yeh, Evaluatg arle compettveess usg multattrbute decso makg, Omega, vol. 29:5, pp , 200. [7] B.F. Hobbs ad P. Meer, P., Eergy Decsos ad the Evromet: A Gude to the Use of Multcrtera Methods. Bosto: Kluwer Academc Publshers, [8] M.K. Starr ad L.H. Greewood, Normatve geerato of alteratves wth multple crtera evaluato, Multple Crtera Decso Makg, M.K. Starr ad M. Zeley, Eds. New York: North Hollad, 977, pp [9] M. Weber ad K. Borcherdg, Behavoral flueces o weght udgmets multattrbute decso makg, Europea Joural of Operatoal Research, vol. 67, pp. -2, 993. [20] C.-H. Yeh, R.J. Wlls, H. Deg ad H. Pa, Task oreted weghtg multcrtera aalyss, Europea Joural of Operatoal Research, vol. 9:, pp , 999.