Iteratoal Joural of Computatoal Itellgece Systems, Vol. 7, o. 6 (December 04, 096-08 Some geometrc aggregato operators based o log-ormally dstrbuted radom varables -Fa Wag School of Scece, Hua Uversty of Techology, Zhuzhou 4007, Cha Ja-Qag Wag School of Busess, Cetral South Uversty, Chagsha 4008, Cha Sheg-Yue Deg School of Scece, Hua Uversty of Techology, Zhuzhou 4007, Cha Receved 9 July 0 Accepted 5 Jauary 04 Abstract The eghted geometrc averagg (WGA operator ad the ordered eghted geometrc (OWG operator are to of most basc operators for aggregatg formato. But these to operators ca oly be used stuatos here the gve argumets are exact umercal values. I ths paper, e frst propose some e geometrc aggregato operators, such as the log-ormal dstrbuto eghted geometrc (LDWG operator, log-ormal dstrbuto ordered eghted geometrc (LDOWG operator ad log-ormal dstrbuto hybrd geometrc (LDHG operator, hch exted the WGA operator ad the OWG operator to accommodate the stochastc ucerta evromet hch the gve argumets are log-ormally dstrbuted radom varables, ad establsh varous propertes of these operators. The, e apply the LDWG operator ad the LDHG operator to develop a approach for solvg mult-crtera group decso mag (MCGDM problems, hch the crtero values tae the form of log-ormally dstrbuted radom varables ad the crtero eght formato s o completely. Fally, a example s gve to llustrate the feasblty ad effectveess of the developed method. Keyords: Mult-crtera decso mag; log-ormal dstrbuto; formato fuso; LDWG operator; LDOWG operator; LDHG operator. Itroducto Iformato aggregato operators play a mportat role mult-crtera decso mag (MCDM. As e o, the eghted geometrc averagg (WGA operator ad the ordered eghted geometrc (OWG operator are to of most commo operators for aggregatg argumets. The WGA operator frst eghts all the gve argumets ad the aggregates all these eghted argumets to a collectve oe. The OWG operator frst reorders all the gve argumets descedg order ad the eghts these ordered argumets, ad fally aggregates all these ordered eghted argumets to a collectve oe. But the WGA operator ad the OWG operator ca oly be used stuatos here the gve argumets are exact umercal values. Correspodg author. Tel.: +86-7-8880594. E-mal: qag@csu.edu.c (Ja-Qag Wag. Co-publshed by Atlats Press ad Taylor & Fracs 096
.F. Wag et al I the last decades, the WGA operator ad the OWG operator have bee exteded to accommodate the fuzzy ad ucerta stuatos. u et al. -5 exteded these to operators to accommodate the stuatos here the put argumets are fuzzy umbers, for example, u proposed the fuzzy ordered eghted geometrc (FOWG operator, u ad Da 4 proposed the ucerta ordered eghted geometrc (UOWG operator, Wag ad Yag 5 proposed the trapezodal fuzzy ordered eghted geometrc (TFOWG operator; u et al. 6- exteded these to operators to accommodate the stuatos here the put argumets are tutostc fuzzy umbers or exteded tutostc fuzzy umbers, for stace, u ad Yager 6 proposed the tutostc fuzzy eghted geometrc (IFWG operator, tutostc fuzzy ordered eghted geometrc (IFOWG operator ad tutostc fuzzy hybrd geometrc (IFHG operator, We 7 proposed the dyamc tutostc fuzzy eghted geometrc (DIFWG operator ad ucerta dyamc tutostc fuzzy eghted geometrc (UDIFWG operator, Ta 8 proposed the geeralzed tutostc fuzzy ordered geometrc averagg (GIFOGA operator, u 9 ad u ad Che 0 developed the terval-valued tutostc fuzzy eghted geometrc (IIFWG operator, tervalvalued tutostc fuzzy ordered eghted geometrc (IIFOWG operator ad terval-valued tutostc fuzzy hybrd geometrc (IIFHG operator, Wag proposed the fuzzy umber tutostc fuzzy eghted geometrc (FIFWG operator, fuzzy umber tutostc fuzzy ordered eghted geometrc (FIFOWG operator ad fuzzy umber tutostc fuzzy hybrd geometrc (FIFHG operator; u et al. -5 exteded these to operators to accommodate the stuatos here the put argumets are lgustc varables or ucerta lgustc varables, for example, u proposed the lgustc eghted geometrc averagg (LWGA operator, lgustc ordered eghted geometrc averagg (LOWGA operator ad lgustc hybrd geometrc averagg (LHGA operator, We proposed the exteded -tuple eghted geometrc (ET-WG operator ad exteded -tuple ordered eghted geometrc (ET-OWG operator, u 4 proposed the ucerta lgustc eghted geometrc mea (ULWGM operator ad ucerta lgustc ordered eghted geometrc (ULOWG operator, We 5 proposed the ucerta lgustc hybrd geometrc mea (ULHGM operator, hch s based o the ULWGM operator ad the ULOWG operator. I addto, they establshed varous propertes of these operators ad appled them to solve MCDM problems. I some MCDM stuatos, the gve argumets tae the form of radom varables uder stochastc ucerta evromet 6-. Hoever, at preset, there s fe research related to the aggregato operators for aggregatg argumets hch are the form of radom varables, except that Wag ad Yag exteded the eghted arthmetc averagg (WAA operator 4, 5 to accommodate the stuatos here the put argumets are ormally dstrbuted radom varables, ad preseted the ormal dstrbuto umber eghted arthmetc averagg (DWAA operator ad the dyamc ormal dstrbuto umber eghted arthmetc averagg (DDWAA operator. I addto, stochastc MCDM problems, ormal dstrbuto th bell-shaped curve s usually used to descrbe the radom varato that occurs the crtero value, ad the crtero value s commoly characterzed by to parameters: the expected value ad the stadard devato 6, 6, 7. Hoever, may measuremets of crtero values sho a more or less seed dstrbuto. Partcularly, seed dstrbutos are commo he expected values are lo, varaces large, ad values caot be egatve. Such seed dstrbutos ofte approxmately ft the log-ormal dstrbuto 8, 9. Moreover, a varable mght be dstrbuted as log-ormally f t ca be thought of as the multplcatve product of a large umber of depedet radom varables each of hch s postve. The logormal dstrbuto s a cotuous probablty dstrbuto of a radom varable hose logarthm s ormally dstrbuted 8, 0, ad t ca model may staces, such as the loss of vestmet rs, the chage prce dstrbuto of a stoc, ad the falure rates product tests ad so o 8, -. Therefore, real-lfe, there are may stochastc MCDM problems hch the crtero values tae the form of log-ormally dstrbuted radom varables. At preset, the stochastc MCDM problems, hch the crtero values tae the form of ormally dstrbuted radom varables, have attracted lots of attetos from researchers 6-. But regardg to the stochastc MCDM problems, hch the crtero values tae the form of log-ormally dstrbuted radom varables, there s stll fe related research. It s orthy of potg out that, may stochastc MCDM problems th log-ormally Co-publshed by Atlats Press ad Taylor & Fracs 097
Some geometrc aggregato operators dstrbuted radom varables, e eed to aggregate the gve log-ormally dstrbuted radom varables to a sgle oe. I such case, formato fuso techques are ecessary. I ths paper, based o the WGA operator ad the OWG operator, e shall develop some e geometrc aggregato operators for aggregatg argumets hch tae the form of log-ormally dstrbuted radom varables, ad apply them to solve mult-crtera group decso mag (MCGDM problems hch the crtero values are the form of log-ormally dstrbuted radom varables ad the crtero eght formato s o completely. I order to do that, ths paper s orgazed as follos. I Secto, e troduce some basc cocepts of logormal dstrbuto. I Secto, e exted the WGA operator ad the OWG operator to accommodate the stuatos here the put argumets tae the form of log-ormally dstrbuted radom varables, propose some e geometrc aggregato operators, such as the log-ormal dstrbuto eghted geometrc (LDWG operator, log-ormal dstrbuto ordered eghted geometrc (LDOWG operator ad log-ormal dstrbuto hybrd geometrc (LDHG operator, ad establsh varous propertes of these operators. I Secto 4, e apply the LDWG operator ad the LDHG operator to develop a approach for solvg the MCGDM problems, hch the crtero values tae the form of log-ormally dstrbuted radom varables ad the crtero eght formato s o completely. I Secto 5, e provde a llustratve example to demostrate the feasblty ad effectveess of the developed method. Fally, e coclude the paper Secto 6.. Prelmares The ormal dstrbuto s a cotuous probablty dstrbuto defed by the follog probablty desty fucto, o as the Gaussa fucto or formally the bell curve 4 : ( x f( x e, x here s the expectato, 0 s the stadard devato, ad s the varace. Geerally, e use ~ (, as a mathematcal expresso meag that s dstrbuted ormally th the expectato ad the varace. The log-ormal dstrbuto s a probablty dstrbuto of a radom varable hose logarthm s ormally dstrbuted 8, that s, f l Y ~ (,, the Y has a log-ormal dstrbuto. If Y s dstrbuted log-ormally th parameters ad, the e rte Y ~ log- (,. For coveece, e call log- (, a log-ormal dstrbuto, ad let be a set of all log-ormal dstrbutos. Defto 8 Let log- (, ad log- (, be to log-ormal dstrbutos, the ( ( log- (, ; a log- a (, a, a 0. It s easy to prove that all operatoal results are stll log-ormal dstrbutos, ad accordg to these to operatoal las, e have the follog. ; ( ( ; ( a a a, a 0 ; ( ( ( a a a a (4 Furthermore, f, a, a 0. log- (, s a log-ormal dstrbuto, the ts expected value ( log ad stadard devato ( log ca be calculated by the follog formulas 8 : log ( e ( ( e e log ( Therefore, by usg the relato betee expectato ad varace statstcs, the follog, e propose a method for the comparso betee to log-ormal dstrbutos, hch s based o the expected value ( log ad the stadard devato ( log. Co-publshed by Atlats Press ad Taylor & Fracs 098
.F. Wag et al Defto Let log- (, ad log- (, be to log-ormal dstrbutos, the ( If log ( log (, the s smaller tha, deoted by ; ( If log ( log (, the If log ( log (, the s equal to, deoted by ; If log ( log (, the s bgger tha, deoted by ; If log ( log (, the s smaller tha, deoted by.. Log-ormal dstrbuto geometrc aggregato operators.. The LDWG ad LDOWG operators I hat follos, based o Defto, e propose some e geometrc aggregato operator for aggregatg log-ormal dstrbuto formato, such as the LDWG operator ad the LDOWG operator. Defto Let log- (, (,,, be a collecto of log-ormal dstrbutos, ad let LDWG:, f LDWG (,,, ( the LDWG s called the log-ormal dstrbuto eghted geometrc operator (LDWG of dmeso, here (,,, T s the eght vector of (,,,, th 0 ad. Especally, f (,,, T, the the LDWG operator s reduced to the log-ormal dstrbuto geometrc averagg (LDGA operator of dmeso, hch s defed as: LDGA (,,, Theorem Let ( (4 log- (, (,,, be a collecto of log-ormal dstrbutos ad (,,, T be the eght vector of (,,,, th 0 ad, the ther aggregated result by usg the LDWG operator s also a log-ormal dstrbuto, ad LDWG (,,, log-, (5 Proof: Obvously, the aggregated result by usg the LDWG operator s also a log-ormal dstrbuto. I the follog, e prove Eq. (5 by usg mathematcal ducto o. ( For : sce log- (, log- (, The LDWG (, log-, log-,. ( If Eq. (5 holds for, that s LDWG (,,, log-,, by Defto, e have LDWG (,,,, log-, log-, The, he log-, Co-publshed by Atlats Press ad Taylor & Fracs 099
Some geometrc aggregato operators log-, log-, log-,...e. Eq. (5 holds for Thus, based o ( ad (, Eq. (5 holds for all, hch completes the proof of Theorem. Example. Let log- (.06, 0.0, log- (., 0.8, log- (.9, 0. ad 4 log- (.9, 0. be four logormal dstrbutos, ad (0.,0., 0.,0.4 T be the eght vector of (,,,4, the LDWG (,,, 4 log- (0..06 0.. 0..9 0.4.9, 0. 0.0 0. 0.8 0. 0. 0.4 0. log- (.986, 0.08. Theorem (Propertes of LDWG Let (,,, be a log- (, collecto of log-ormal dstrbutos, ad (,,, T be the eght vector of (,,,, th [0,] the e have the follog. ( (Idempotecy: If all equal,.e. for all, the ad, (,,, are (,,, LDWG (6 ( (Boudary : LDWG (,,, (7 here m{ }, max{ }. ( (Mootocty: Let (,,, be a for collecto of log-ormal dstrbutos. If all, the (,,, LDWG LDWG (,,, (8 Defto 4. Let log- (, (,,, be a collecto of log-ormal dstrbutos. A log-ormal dstrbuto ordered eghted geometrc (LDOWG operator of dmeso s a mappg LDOWG:, that has a assocated eght vector (,,, T such that 0 ad LDOWG. Furthermore, (,,, (9 ( ( ( here ( (, (,, ( s a permutato of (,,, such that ( ( for all. Especally, f (,,, T, the the LDOWG operator s reduced to the LDGA operator. Smlar to Theorem, e have the follog. Theorem. Let log- (, (,,, be a collecto of log-ormal dstrbutos, the ther aggregated result usg the LDOWG operator s also a log-ormal dstrbuto, ad (,,, log-, LDOWG ( ( (0 here (,,, T s the eght vector related to the LDOWG operator, th 0 ad OWA eghts 5., hch ca be determed smlar to the Co-publshed by Atlats Press ad Taylor & Fracs 00
.F. Wag et al Example. Let log- (.06, 0.0, log- (., 0.8, log- (.9, 0. ad 4 log- (.9, 0. be four logormal dstrbutos. The by Eq. (, e ca calculate the expected values of Sce Thus (,,,4 : log ( 8.0045, log ( 9.646, log ( 4.049, log ( 4 7.04. ( ( ( ( log log log 4 log ( (., 0.8, ( (.06, 0.0, ( 4 (.9, 0., (4 (.9, 0.. Suppose that (0.55,0.45,0.45 0.55 T (derved by the ormal dstrbuto based method [5] s the eght vector of the LDOWG operator, the by Eq. (0, e get LDOWG (,,, 4 log- (0.55.0.45.06 0.45.9 0.55.9, 0.55 0.8 0.45 0.0 0.45 0. 0.55 0. log- (.946, 0.055. The LDOWG operator has the follog propertes smlar to those of the LDWG operator. Theorem 4 (Propertes of LDOWG Let log- (, (,,, be a collecto of log-ormal dstrbutos, ad (,,, T be the eght vector related to the LDOWG operator, th [0,] ad, the e have the follog. ( (Idempotecy: If all equal,.e. for all, the (,,, are (,,, LDOWG ( ( (Boudary : LDOWG (,,, ( here m{ }, max{ } ( (Mootocty: Let. (,,, be a for collecto of log-ormal dstrbutos. If all, the (,,, LDOWG LDOWG (,,, ( (,,, be a (4 (Commutatvty: Let collecto of log-ormal dstrbutos, the (,,, (,,, (,,, LDOWG LDOWG (4 here s ay permutato of (,,,. From Eq. (4, t ca be o that the LDOWG operator has commutatvty property that e desre. It s orth otg that the LDWG operator does ot have ths property. Besdes the above propertes, the LDOWG operator has the follog desrable results. Theorem 5 Let log- (, (,,, be a collecto of log-ormal dstrbutos, ad (,,, T be the eght vector related to the LDOWG operator, th [0,] ad, the ( If (, 0,, 0 T, the (,,, max{ } LDOWG. ( If (0,0,, T, the (,,, m{ } LDOWG. ( If, r 0, ad r, the LDOWG (,,, ( here ( r s the th largest of r (,,,. Co-publshed by Atlats Press ad Taylor & Fracs 0
Some geometrc aggregato operators.. The LDHG operator From Defto ad Defto 4, t ca be o that the LDWG operator eghts oly the log-ormal dstrbutos, hereas the LDOWG operator eghts oly the ordered postos of the log-ormal dstrbutos stead of eghtg the argumets themselves. To overcome ths lmtato, hat follos, e propose a LDHG operator, hch eghts both the gve log-ormal dstrbutos ad ther ordered postos. Defto 5. Let log- (, (,,, be a collecto of log-ormal dstrbutos. A log-ormal dstrbuto hybrd geometrc (LDHG operator of dmeso s a mappg LDHG:, hch has a assocated vector (,,, T th 0 ad, such that LDHG, (,,, ( ( ( s the th (5 here ( largest of eghted log-ormal dstrbutos (,,,, (,,, T s the eght vector of (,,,, th 0 s the balacg coeffcet. Theorem 6. Let ad, ad log- (, (,,, be a collecto of log-ormal dstrbutos ad log- (, (,,,, the (,,, LDHG, log-, ( ( ( ( ( (6 ad the aggregated result derved by usg the LDHG operator s also a log-ormal dstrbuto. Example. Let log- (.6, 0.7, log- (., 0.8, log- (.9, 0. ad 4 log- (.0, 0. be four logormal dstrbutos, ad let (0.,0., 0.,0. T be the eght vector of (,,,4. The by Defto, e ca get the eghted log-ormal dstrbutos: log- (.78, 0.6, log- (.65, 0.6, log- (.68, 0.44, 4 log- (.64, 0.088. By Eq. (, e ca calculate the expected values of (,,,4 : Sce Thus log ( 5.687, log ( 4.57, log (.9904, log ( 4 5.090. ( ( ( ( log log log log 4, ( (.65, 0.6, ( (.68, 0.44, ( (.78, 0.6 (4 4 (.64, 0.088. Suppose that (0.55,0.45,0.45, 0.55 T (determed by the ormal dstrbuto based method 5 s the eght vector of the LDHG operator, the by Eq. (6, t follos that LDHG (,,,, 4 log- (0.55.65 0.45.68 0.45.78 0.55.64, 0.55 0.6 0.45 0.44 0.45 0.6 0.55 0.088 log- (.656, 0.077. Theorem 7. The LDWG operator s a specal case of the LDHG operator. Proof. Let (,,, T, the Co-publshed by Atlats Press ad Taylor & Fracs 0
.F. Wag et al LDHG, (,,, ( ( ( LDWG (,,, hch completes the proof of Theorem 7. Theorem 8. The LDOWG operator s a specal case of the LDHG operator. Proof. Let (,,, T, the,,,,, the LDHG, (,,, ( ( ( ( ( ( LDOWG (,,, hch completes the proof of Theorem 8. Obvously, from Theorem 7 ad Theorem 8, e ca see that the LDHG operator eghts both the gve log-ormal dstrbutos ad the ordered postos of these argumets, reflects the mportace degrees of both the gve log-ormal dstrbutos ad ther ordered postos, ad geeralzes both the LDWG operator ad the LDOWG operator at the same tme. 4. A applcato of the LDWG ad LDHG operators to MCGDM I ths secto, e apply the LDWG ad LDHG operators to MCGDM based o log-ormally dstrbuted radom varables. Let {,,, m } be a dscrete set of feasble alteratves, C { C, C,, C } be the (,,, T be the C (,,, th 0 fte set of crtera, ad eght vector of ad. Let decso maers, ad eght vector of d (,,, t D { d, d,, d t } be the set of e ( e, e,, e T t be the th e 0 ad t e. Assume that the decso maers d (,,, t represet, respectvely, the characterstcs of the alteratves (,,, m th respect to the crtera C (,,, by logormally dstrbuted radom varables log - (,( (,,, m ; ( ( (,,, ;,,, t, ad costruct decso ( ( R ( (,,, t. matrces m Based o the above decso formato, the follog, e propose a practcal procedure to select the most desrable alteratve(s. Step ormalze the decso matrces ( ( R ( m (,,, t. Let C C c b be the set of all beeft crtera ad be the set of all cost crtera, the e ca use the follog formulas to ( ( trasform the decso matrces R ( m to the correspodg ormalzed decso matrces R ( (log - (,( ( ( ( ( m m (,,, t : ( ( max max, m m ( ( ( (, C C ( (, C ( max max{ } b C (7 c C (8 C (9 ote that stadard devato s relatve to expectato, so the Eq. (9 s sutable for all C C. Step Utlze the LDWG operator LDWG,,, ( ( ( ( (0 to aggregate the crtero values of the th colum of ( ( the ormalzed decso matrces R ( m ad derve the dvdual overall values log- (,( of the alteratves ( ( ( Co-publshed by Atlats Press ad Taylor & Fracs 0
Some geometrc aggregato operators (,,, m gve by the decso maers d (,,, t, here ( ( ( (, ( Step Utlze the LDHG operator ( ( ( t LDHG ev,,,, t v v v ( ( ( t ( to derve the collectve overall values log - (, of the alteratves (,,, m, here t ( v (, v v v t t v ( v (,,, T s the eghtg vector of the LDHG operator, th v 0 ( ( ( ( ad t v, log - (,( s the th largest of the eghted log-ormal dstrbutos ( te ( te ( t te (, (,(,,( t 4 5 Table. Decso matrx ( R (ut: te thousads RMB C C C log - (99,9. log - (9,0. log - (68,8.9 log - (48,0.9 log - (46,9.6 4 5 Table. Decso matrx log - (79,7.9 log - (69,8.5 log - (7,6.8 log - (07,7.5 log - (,6.9 ( R (ut: te thousads RMB log - (9,6.6 log - (6,6. log - (0,6.8 log - (66,7. log - (59,7.5 C C C log - (8,9.6 log - (85,0. log- (59, 9. log - (46,0.9 log - (455,9.7 4 5 Table. Decso matrx log - (5,7.6 log - (69,9. log - (5,8.6 log - (09,9. log - (9,8.9 ( R (ut: te thousads RMB log - (4,5.5 log - (8,6. log - (5,6.5 log - (69,7.5 log - (57,8.6 C C C log - (79,9. log - (9,9.8 log - (5,0.6 log - (6,0.6 log - (469,.7 log - (55,7.9 log - (69,9. log - (57,8.6 log - (6,9. log - (09,8.8 log - (,5.7 log - (6,6. log - (,6.7 log - (68,7. log - (58,7.6 Co-publshed by Atlats Press ad Taylor & Fracs 04
.F. Wag et al ( (, (,, ( t s a permutato of (,,, t, ad t s the balacg coeffcet. Step 4 Utlze Eq. ( ad Eq. ( to calculate the expected values ( log ad the stadard devatos ( log of the collectve overall values (,,, m. Step 5 Use Defto to ra all the alteratves (,,, m, ad the select the best oe accordg to the values ( log ad ( log (,,, m. 5. Illustratve example Let us suppose that there s a vestmet compay, hch ats to vest a total amout of moey the best opto (adapted from Ref. 7. There s a pael th fve possble compaes hch to vest the moey: ( s a arms compay; ( s a computer compay; ( s a food compay; (4 4 s a TV compay. The s a auto compay; ad (5 5 crtera to be cosdered the selecto process are the follog: ( C : cost; ( C : et preset value; ad ( C : loss, hose eghtg vector s (0.5,0.7,0.8 T. Three decso maers d (,, (hose eghtg vector s 4 5 Table 4. ormalzed decso matrx R ( C C C log - (0.8797, 0.096 log - (0.89, 0.05 log - (0.958, 0.090 log - (0.80, 0.0 log - (0.758, 0.005 4 5 log - (0.8558, 0.04 log - (0.85, 0.06 log - (0.874, 0.009 log - (0.947, 0.00 log - (0.960, 0.0 Table 5. ormalzed decso matrx R ( log - (0.95, 0.09 log - (0.9559, 0.06 log - (.0000, 0.040 log - (0.78, 0.046 log - (0.876, 0.0444 C C C log - (0.9, 0.005 log - (0.97, 0.07 log - (0.9777, 0.098 log - (0.758, 0.0 log - (0.774, 0.007 log - (.000, 0.06 4 5 log - (0.7699, 0.0 log - (0.85, 0.085 log - (0.776, 0.064 log - (0.9479, 0.079 log - (0.9785, 0.07 Table 6. ormalzed decso matrx R ( log - (0.970, 0.05 log - (0.940, 0.06 log - (0.960, 0.085 log - (0.769, 0.0444 log - (0.880, 0.0509 C C C log - (0.96, 0.096 log - (0.8977, 0.009 log - (0.9696, 0.06 log - (0.7484, 0.049 log - (0.78, 0.04 log - (0.85, 0.08 log - (0.788, 0.064 log - (.0000, 0.085 log - (0.9479, 0.070 log - (0.994, 0.07 log - (0.9559, 0.06 log - (0.9774, 0.096 log - (0.778, 0.040 log - (0.88, 0.0450 Co-publshed by Atlats Press ad Taylor & Fracs 05
Some geometrc aggregato operators e (0.0,0.40,0.0 T evaluate the performace of these compaes (,,, 5 th respect to the crtera C (,, by log-ormally dstrbuted radom varables ( ( (, ad costruct the log - (,( R ( (,, decso matrces ( ( 5 as lsted Table ~ Table. To get the best compay, the follog steps are volved. Step Utlze Eq. (7 ~ Eq. (9 to ormalze the decso matrces R to the ( 5 R ( ( ( ( correspodg decso matrces ( 5 (,, as lsted Table 4 ~ Table 6. ote that the crtero C s beeft crtero ad the crtera C ad C are cost crtera. Step Utlze Eq. (0 to aggregate the crtero values of the th colum of the ormalzed decso ( ( matrces ad derve the dvdual R ( 5 ( overall values of the alteratves (,,, 5 gve by the decso maers d (,, : ( log - (0.8864, 0.057, ( log - (0.8855, 0.059, ( log - (0.97, 0.05, ( 4 log - (0.8547, 0.068, ( 5 log - (0.8495, 0.06, ( log - (0.8790, 0.044, ( log - (0.888, 0.065, ( log - (0.8990, 0.06, ( 4 log - (0.84, 0.08, ( 5 log - (0.869, 0.089, ( log - (0.894, 0.047, ( log - (0.887, 0.06, ( log - (0.954, 0.068, ( 4 log - (0.960, 0.077, ( 5 log - (0.840, 0.08. Step Utlze Eq. ( to derve the collectve overall values of the alteratves (,,, 5, here the eghtg vector of LDHG operator s (0.49,0.54,0.49 T hch s determed by the ormal dstrbuto based method 5 :, log - (0.865, 0.0087 log - (0.860, 0.0096 log - (0.8896, 0.009 4 log - (0.8577, 0.004 5 log - (0.89, 0.00,,,. Step 4 Utlze Eq. ( to calculate the expected values ( log of the collectve overall values (,,, 5 : Thus (.69 log log, (.704 log, (.580 log 4. (.44 (.97 log 5 log ( log ( log ( ( (. log 4 log 5 Step 5 Use Defto to ra all the alteratves (,,, 5 : 4 5. Therefore, the best vestmet eterprse s. 6. Coclusos I ths paper, e have exteded the WGA operator ad the OWG operator to accommodate the stochastc ucerta stuatos here the gve argumets tae the form of log-ormally dstrbuted radom varables, proposed some e geometrc aggregato operators, such as the LDWG operator, the LDOWG operator ad the LDHG operator, ad establshed varous desrable propertes of these operators. Weghts represet dfferet aspects both the LDWG operator ad the LDOWG operator. The LDWG operator eghts oly the log-ormal dstrbutos, the LDOWG operator eghts oly the ordered postos of the log-ormal dstrbutos stead of eghtg the log-ormal dstrbutos themselves, ad both these to operators cosder oly oe of them. The LDHG operator eghts both the gve log-ormal dstrbutos Co-publshed by Atlats Press ad Taylor & Fracs 06
.F. Wag et al ad ther ordered postos, ad geeralzes both the LDWG operator ad the LDOWG operator at the same tme. I addto, the LDOWG operator ad the LDHG operator ca releve the fluece of ufar data o the fal results by assgg lo eghts to those uduly hgh or uduly lo oes. Furthermore, e have gve a applcato of the LDWG operator ad the LDHG operator to MCGDM based o logormally dstrbuted radom varables. Ths paper erches ad develops aggregato operator theory. I the future, e shall cotue org the exteso ad applcato of the developed operators to other domas. Acoledgemets The authors are very grateful to the edtor ad the aoymous referees for ther costructve commets ad suggestos that have led to a mproved verso of ths paper. Ths or as supported by the atoal atural Scece Foudato of Cha (o. 778, 706 ad 674075, ad the Humates ad Socal Scece Foudato of the Mstry of Educato of Cha (o. YJA604 ad YJC6000. Refereces. J. Aczél ad T.L. Saaty, Procedures for sytheszg rato udgmets, Joural of Mathematcal Psychology 7 ( (98 9-0.. Z.S. u ad Q.L. Da, The ordered eghted geometrc averagg operators, Iteratoal Joural of Itellget Systems 7 (7 (00 709-76.. Z.S. u, A fuzzy ordered eghted geometrc operator ad ts applcato fuzzy AHP, Systems Egeerg ad Electrocs 4 (7 (00 -. 4. Y.J. u ad Q.L. Da, Ucerta ordered eghted geometrc averagg operator ad ts applcato to decso mag, Systems Egeerg ad Electrocs 7 (6 (005 08-040. 5..F. Wag ad.j. Yag, Trapezodal fuzzy ordered eghted geometrc operator ad ts applcato to decso mag, Mathematcs Practce ad Theory 4 (0 (0 78-8. 6. Z.S. u ad R.R. Yager, Some geometrc aggregato operators based o tutostc fuzzy sets, Iteratoal Joural of Geeral Systems 5 (4 (006 47-4. 7. G.W. We, Some geometrc aggregato fuctos ad ther applcato to dyamc multple attrbute decso mag the tutostc fuzzy settg, Iteratoal Joural of Ucertaty, Fuzzess ad Koledge-Based Systems 7 ( (009 79-96. 8. C. Ta, Geeralzed tutostc fuzzy geometrc aggregato operator ad ts applcato to mult-crtera group decso mag, Soft Computg 5 (5 (0 867-876. 9. Z.S. u, Methods for aggregatg terval-valued tutostc fuzzy formato ad applcato to decso mag, Cotrol ad Decso ( (007 5-9. 0. Z.S. u ad J. Che, O geometrc aggregato over terval-valued tutostc fuzzy formato, : Proceedgs of Fourth Iteratoal Coferece o Fuzzy Systems ad Koledge Dscovery (FSKD 007, Haou, Cha, 007, pp. 466-47...F. Wag, Fuzzy umber tutostc fuzzy geometrc aggregato operators ad ther applcato to decso mag, Cotrol ad Decso (6 (008 607-6.. Z.S. u, A method based o lgustc aggregato operators for group decso mag th lgustc preferece relatos, Iformato Sceces 66 (-4 (004 9-0.. G.W. We, A method for multple attrbute group decso mag based o the ET-WG ad ET-OWG operators th -tuple lgustc formato, Expert Systems th Applcatos 7 ( (00 7895-7900. 4. Z.S. u, A approach based o the ucerta LOWG ad duced ucerta LOWG operators to group decso mag th ucerta multplcatve lgustc preferece relatos, Decso Support Systems 4 ( (006 488-499. 5. G.W. We, Ucerta lgustc hybrd geometrc mea operator ad ts Applcato to group decso mag uder ucerta lgustc evromet, Iteratoal Joural of Ucertaty, Fuzzess ad Koledge-Based Systems 7 ( (009 5-67. 6. R. Lahdelma, S. Maoe ad P. Salme, Multvarate Gaussa crtera SMAA, Europea Joural of Operatoal Research 70 ( (006 957-970. 7. S.B. Yao ad C.Y. Yue, Approach to stochastc multattrbute decso problems usg rough sets theory, Joural of Systems Egeerg ad Electrocs 7 ( (006 0-08. 8. G.T. Jag, Z.P. Fa ad Y. Lu, Method for multple attrbute decso mag th ormal radom varables, Cotrol ad Decso 4 (8 (009 87-9. 9. Y. Lu, Z.P. Fa ad Y. Zhag, A method for ormal stochastc multple attrbute decso mag cosderg teractos amog attrbutes, Operatos Research ad Magemet Scece 0 (5 (0 0-6. 0. G.T. Jag ad Z.P. Fa, A method for MADM th ormal radom varables, Systems Egeerg-Theory & Practce (7 (0 57-54.. J.Q. Wag ad J. Re, Stochastc mult-crtera decsomag method based o WC-OWA operator, Cotrol ad Decso ( (007 49-4.. J. Re ad Y. Gao, Stochastc mult-crtero decsomag method based o terval operato, Systems Egeerg ad Electrocs ( (00 08-...F. Wag ad.j. Yag, Dyamc stochastc multple attrbute decso mag method th complete certa Co-publshed by Atlats Press ad Taylor & Fracs 07
Some geometrc aggregato operators formato, Systems Egeerg: Theory & Practce 0 ( (00-8. 4. J.C. Harsay, Cardal elfare, dvdualstc ethcs, ad terpersoal comparsos of utlty, Joural of Poltcal Ecoomy 6 (4 (955 09-. 5. A. Se, Collectve Choce ad Socal Welfare, Holde- Day, Sa Fracsco, CA, 970. 6. M. oa, Asprato level approach stochastc MCDM problems, Europea Joural of Operatoal Research 77 ( (007 66-640. 7. K. Zaras, Rough approxmato of a preferece relato by a mult-attrbute domace for determstc, stochastc ad fuzzy decso problems, Europea Joural of Operatoal Research 59 ( (004 96-06. 8. J. Atchso ad J.A.C. Bro, The Logormal Dstrbuto, Cambrdge Uversty Press, Cambrdge, 957. 9. E.L. Cro ad K. Shmzu, Logormal Dstrbutos: Theory ad Applcato, Deer, e Yor, 988. 0. E. Lmpert, W. Stahel ad M. Abbt, Log-ormal dstrbutos across the sceces: eys ad clues, BoScece 5 (5 (00 4-5.. I. Atoou, V.V. Ivaov, V.V. Ivaov, P.V. Zrelov, O the log-ormal dstrbuto of stoc maret data, Physca A: Statstcal Mechacs ad ts Applcatos ( 4 (004 67-68.. R.C. Merto ad P.A. Samuelso, Fallacy of the logormal approxmato to optmal portfolo decsomag over may perods, Joural of Facal Ecoomcs ( (974 67-94.. L.W. Che, Proect Ivestmet Rs Aalyss: Theores ad Methods, Cha Mache Press, Beg, 004. 4. G. McPherso, Statstcs scetfc vestgato: ts bass, applcato ad terpretato, Sprger-Verlag, e Yor, 990. 5. Z.S. u, A overve of methods for determg OWA eghts, Iteratoal Joural of Itellget Systems 0 (8 (005 84-865. Co-publshed by Atlats Press ad Taylor & Fracs 08