A Method for Group Decision-making with Uncertain Preference Ordinals Based on Probability Matrix

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1 65 A pubcao of CHEMICA ENGINEERINGTRANSACTIONS VO. 51, 016 Gue Edo:Tcu Wag, Hogyag Zag, e Ta Copyg 016, AIDIC Sevz S..., ISBN ; ISSN Te Iaa Aocao of Ceca Egeeg Oe a DOI: /CET A Meod fo Goup Deco-ag w cea Pefeece Oda Baed o Pobaby Max Mg Jag a, Xaowe a, Ce You a,hapg Re* b a Scoo of Bue,Ma Noa vey.zagzou,363000,p.r.ca b Scoo of Sofwae, Jagx vey of Scece ad Tecoogy, Nacag, , P.R. Ca cap@163.co I pape, a ew eod o ove e goup deco-ag pobe popoed, wc e pefeece foao o aeave povded by deco ae e fo of ucea pefeece oda. T pape fy gve wo ew defo o e pobaby a e aeave aed eac poo. Te, ode o poce ucea pefeece oda, wo ew defo ae ued epecvey o couc a deco ax e fo of pobabe. O ba, a weg pobaby ax ad a coecve pobaby ax o aeave w egad o a poo ae couced. Fay, a opzao ode bu baed o e coecve pobaby ax, ad e ag of aeave ca be obaed by ovg e ode. 1. Ioduco Mupe cea deco ag (MCDM) a dcpe aed a uppog deco ae wo faced w ueou ad cofcg aeave o ae a opa deco (Pedycz, 013; Mada e a., 015). We goup deco-ag (GDM) deco-ag goup cog of upe ebe. Mupe cea goup deco-ag (MCGDM) pobe vove a e of feabe aeave a ae evauaed o e ba of upe, cofcg ad o-coeuae cea by a goup of dvdua. Meawe, a pobe w exeve eoeca ad pacca bacgoud dua egeeg (Soa e a., 015; Weayae, e a. 016; Ravda, 016).I e ea wod, due o e copexy ad uceay of decoag pobe, e ao of cogo, e eao accuace ad ac of deco ae owedge, e pefeece ag o ag oda of aeave povded by deco ae aybe e fo of ucea pefeece oda. Howeve, few appoace o ove MCGDM pobe w pefeece ag o ucea pefeece oda ca be foud e exg eaue. Te exg appoace ave ade o ove e GDM pobe w ucea pefeece oda o aeave. T pape vegae e MCDM pobe, wee e pefeece foao o aeave povded by deco ae e fo of ucea pefeece oda. I ode o ove ee pobe Fa e a. (010) gave evea defo o ucea pefeece oda, couced a deco ax e fo of pobabe, ad bu a opzao ode baed o e coecve pobaby ax. Fa appoac ca ove e GDM pobe effecvey. Howeve, egaded a eac ag poo of a ucea pefeece oda o aeave a e ae pobaby Fa appoac bu o e w aw of ua cogo. Fo exape, uppog a ucea pefeece oda of a aeave o eva [3,7], e deco ae ofe e pobaby of e aeave aed 5 poo ge a 3 o 7 poo (Xu, 005). I fac, e coe a pefeece oda o e owe boud ad uppe boud of a ucea pefeece oda, e ae e pobaby a e aeave aed e coepodg poo. A e ae e, e coe a pefeece oda e cee of a ucea pefeece oda, e age pobaby a e aeave aed e coepodg poo. Focug o e above obec, pape w fy pove Fa ode by gvg wo ew defo o e pobaby a e aeave aed eac poo, wc aed by Wag ad Xu (008). Te, o poce ucea pefeece oda, a ax e fo of pobabe couced. Baed o e pobaby ax, a weg pobaby ax ad a coecve pobaby ax o aeave w egad o Peae ce ace a: Jag M.., X.W., You C., Re H.P., 016, A eod fo goup deco-ag w ucea pefeece oda baed o pobaby ax, Ceca Egeeg Taaco, 51, DOI: /CET

2 66 a poo ae couced. Fueoe, a opzao ode bu baed o e coecve pobaby ax, ad e ag of aeave ca be obaed by ovg e ode.. Peae I eco, we w oduce oe bac cocep eaed o upe cea goup deco ag pobe, wc e pefeece foao o aeave povded by deco ae e fo of ucea pefeece oda. Defo 1. (Fa e a, 010) e Z be e e of pove ege. A ucea pefeece oda expeed, 1,,, wee, 1,, Z,, ad ae e owe boud ad uppe boud of.fo pcy, we expe a,. Rea 1. e deoe e ube of a aeave a GDM aay a we a e oa ube of ag poo. e M 1,,, be e e of a e ag poo, wee 1,,, deoe a e ag poo e 1, d,,, epecvey. If, M epee a e ag poo of a aeave e,e e ae, e bee e coepodg aeave w be. Tu, fo ucea pefeece oda,,, M. Rea. Code a ucea pefeece oda,.e u 1, e u deoe e ube of pobe ag poo ad ao vewed a e uceay degee of. Tu, e geae u, e geae e uceay degee of w be. I upe cea goup deco-ag (MCGDM) poce, e S be a fe e of aeave C c c c,,, 1 e of cea,,,, 1 be a 1,,, e weg veco of cea c ( 1,, ),wee 0, 1, e E e e e be e e of deco ae,,, 1 ae, w 0, 1,,, 1 1 be e wegg veco of deco 1.Suppoe e deco ae e 1,, povde e pefeece fo aeave e fo of ucea pefeece oda, e, e foowg defo ae obaed. Defo. (You e a., 013) e Z be e e of pove ege. A ucea pefeece oda, wee, 1,, Z, expeed, 1,,, dcae e pefeece foao a e aeave afe e cea c gve by e deco ae e, ad ae e owe boud ad uppe boud of, 1,,,, 1,,,, 1,,,. Epecay f oda. Fo pcy, we expe a,., e educe o a ag Rea 3. e deoe e ube of a aeave a MCGDM aay a we a e oa ube of M 1,,, be e e of a e ag poo, wee 1,,, deoe a e ag poo. e ag poo e 1, d,,, epecvey, f, M epee a e ag poo of a aeave e,e e ae, e bee e coepodg aeave w be. Tu, fo ucea pefeece oda,,, M, e ae aeave w be. If,,, M o, e bee e ag poo of e a ucea pefeece oda o a aeave povded by e deco ae. Fa e a (010) egaded a e aeave coud be aed poo, 1,, w e ae poby, ad gave e defo of e pobaby veco. Defo 3. (Fa e a, 010) e, (, M ) be a ucea pefeece oda o a aeave povded by e deco ae. Te, e pobaby veco o 1 by p p, p,, p ad e eee of p ae gve by epeeed

3 67 0, 1,,, 1; 1, =, 1,, p ; u 0, = 1,,,. (1) Wee p ( 1,,, ) deoe e pobaby a e aeave aed e poo, afe 1 p 1 ad 0 p 1, 1,,,. Defo 4. e, (, M ) be a ucea pefeece oda o a aeave povded by e deco ae, e, e pobaby a e aeave aed eac poo, by: v C 1 1,, 1,, epeeed, 1 () wee v 0, v 1,ad 0 0 C1 C0 1.Ta, e aeave coud be aed poo, 1,, w 1 poby 1 C. Defo 5. e, (, M ) be a ucea pefeece oda o a aeave povded by e deco ae, e, e pobaby a e aeave aed eac poo, by: u e v,, 1,, u e wee u, u, 1. epeeed Rea 4. I eay o pove a v Defo 4 a e foowg we-ow popee:1) v yeca,.e., v v 1 ( 1,,, ). ) e pobaby veco v ( v1, v,, v ) T,we 1,wc afy (3) v v v v v v v (4) we,wc afy v v v v = v v v (5) We v Defo 5 a a ae popey a v. Code a ucea pefeece oda,,, M ad ovaed by Defo 4 ad Defo 5, we ave e foowg defo. Defo 6. e,,, M.Te, e pobaby veco o 1 by,,, p p p p ad e eee of p ae gve by ~ ~ p epeeed 0, 1,,, 1; c,, 1,,,, 1,, 1; 0, 1,,,. (6)

4 68 Wee deoe p deoe e pobaby a e aeave aed e poo, uc a p 1ad 0 p 1, 1,,,. 1 Defo 7. e,,, M,, 1,, 1. 1 epeeed by p p, p,, p ad e eee of p ae gve by Te, e pobaby veco o 0, 1,, 1; u e p,, 1,,, 1; u e 0, 1,,,. (7) wee p deoe e pobaby a e aeave aed e poo, uc a 0 p 1, 1,,,. 1 p 1 ad Rea 5. If fo ucea pefeece oda,,, M,.e., educed o a ag 1 oda, e e eee of pobaby veco p p, p,, p ~ ~ p o ae gve by 0, 1,,, 1; 1, ; 0, 1,,,. (8) Defo 8. e 1,,, 1,,,,, p 1 be ucea pefeece oda ad p p p p p, p,, p be e coepodg pobaby veco. e w w1, w,, w be a weg veco, wee w deoe e weg of uc a w 1ad 0 w 1, 1,,,. Te, e ovea pobaby 1 veco o p, p,, p epeeed by p 1 p, p,, p ad e eee of p ae gve by p w p, 1,,, 0 1 (9) 3. Te popoed appoac I eco, pape w pee a adg eod fo MCGDM pobe w ucea pefeece oda. Fy, gve a bef decpo of e MCGDM pobe w ucea pefeece oda. Te, a pobaby ax, e vog foao ax, e coecve vog foao ax ad a opzao ode ae couced. Fay, a ago fo deeg e ag poo of eac aeave gve. S,,, C c, c,, c ( ) be a e of cea, woe e be a fe e of aeave, 1 weg veco w w w w 1,,,, wee w 0( 1,, ), w 1 ad E e1, e,, e ( ) be e e of deco ae woe weg veco Suppoe e deco ae e 1,, 1 1 1,,,,wee 0, 1,,,ad 1. povde e pefeece fo aeave e fo of ucea pefeece oda,.e.,,,, M, wee deoe e pefeece foao a e aeave afe e cea c gve by e deco ae e, 1,,,, 1,,,, 1,,,.Te pobe coceed pape o a aeave o o eec e o deabe aeave() aog a 1

5 fe e S 1 e p p, p,, p baed o ucea pefeece oda be pobaby veco o ucea pefeece oda.te eod decbed a foow:.i ca be deeed accodg o Defo.6 o Defo.7, wee p deoe e pobaby a cea c aed e poo, uc a p 1 ad 0 p 1, 1,,, ; 1,,,. Fo e coveece of aay, e 1 deco ax e fo of pobabe baed o p couced a foow: p c c c p1 p p p p p p p 1 p p (10) g Eq.(10), e eee of e ow of e pobaby ax P ad e weg veco w ae aggegaed o fo e weg pobaby veco o aeave,wc ae e weg of e cea c 1,,, o codeao, ad gve by 1 q w p,, 1,,, ; 1,,, Toug Eq.(11), ca be eay ee a q 1.Baed o e obaed veco q 1,,,,e weg pobaby ax Q ( ) Q q 1 q 1 ca be couced,.e., 1 1 q1 q1 q 1 1 q q q 1 q q q (1) Te eee of e weg pobaby ax Q ad veco ae aggegaed o fo e coecve pobaby veco o aeave 1,.e.,,,, (11), wee deoe e coecve eu by a e deco ae a aeave aed e poo ad gve by 1 Baed o veco, pape couc e foowg coecve pobaby ax,.e., = q,, 1,,,. 1 (13) Baed o coecve pobaby ax, pape aep o deveop a eod o deee e ag poo of eac aeave a oe aeave oy aed oe ag poo. Te decpo of eod gve beow. e b, 1,,, be 0-1vaabe, wee b 1 epee a aeave aed e poo ad b 0, oewe. Te oa pobaby a aeave ae aed poo ca be expeed a 1 1 b, wee 1 b 1( 1,,, ) ad 1 b 1( 1,,, ).To a aeave o eec e be aeave(), we ca couc e foowg opzao ode ax z= 1 1 b Subec o b 1, 1,,,, b 1, 1,,,, b 0 o 1,, 1,,, (14)

6 630 Exg aeaca opzao ofwae ca be ued o ove ode (14). I uay, we gve a ago o deee e ag poo of aeave ad ep ae peeed a foow: ep1: Cacuae pobaby veco p by Eq.(7) o Eq.(8) baed o, 1,,, ; 1,,,. ep: Couc pobaby ax P p baed o p, 1,,, ; 1,,,. ep3: Couc e weg pobaby axq q by Eq.(11). ep4: Couc coecve pobaby ax baed oq q. ep5: Bud e opzao ode (14) baed o ax ad ove by Hugaa eod. ep6: Deee e ag poo of eac aeave baed o e obaed opa ouo() of ode (14) ad ecod e pobaby of ag poo of aeave baed o ax. 4. Cocuo I upe cea goup deco-ag uao a e deco ae ca o gve e exac vaue, e deco ae ay be uaby expeed w pefeece oda. Focug o pobe, pape popoe a ew eod o ove e MCDM pobe a e pefeece foao e fo of ucea pefeece oda. I pove e eod popoed by Fa, ad oe e w e aw of geea ua cogo. F, deveop wo oa dbuo-baed eod o deee e pobaby a e aeave aed eac poo. Te, ode o poce ucea pefeece oda, a ax e fo of pobabe couced. Fueoe, a weg pobaby ax ad a coecve pobaby ax o aeave w egad o ag poo ae couced. Fay, a opzao ode bu baed o e coecve pobaby ax, ad e ag of aeave ca be obaed by ovg e ode. Te eod popoed pape ay ao be ued MCDM pobe w oe pefeece foao foa, e.g., eva ad e-vaue, ucea guc vaabe, ec. Acowedge T pape wa uppoed by Sof Scece Poec of Fua Naua Scece Foudao (o. 016R0074). Refeece Fa Z. P., u Y., 010, A appoac o ove goup deco-ag pobe w oda eva ube, IEEE Taaco o Sye, Ma ad Cybeec, Pa B: Cybeec, 40(5), DOI: /TSMCB Fa Z. P., Yue Q., Feg B., u Y., 010, A appoac o goup deco-ag w ucea pefeece oda, Copue ad Idua Egeeg, 58(1), 51-57, DOI: / Mada, A., Juo, A., Zavada, E. K. 015, Fuzzy upe cea deco-ag ecque ad appcao-two decade evew fo 1994 o 014, Expe Sye w Appcao, 4(8), , DOI: /.ewa Pedycz W. 013, Aay ad Deg of Iege Sye, Gaua Copug, CRC Pe, Boca Rao, Fac. Ravda, A. R. 016, Mupe Cea Deco Mag Suppy Ca Maagee, CRC Pe, Boca Rao, Fac. Soa, A., Hewage, K., Reza, B., Sadq, R. 015, Mupe aeode u-cea deco-ag e coex of Mucpa Sod Wae Maagee: A evew. Wae Maagee, 35, , DOI: /.waa Wag Y., Xu Z. S., 008, A ew eod of gvg OWA weg, Maeac Pacce ad Teoy, 38(3), Weayae, W. K., Aaage,. S., De Sva, S. S. 016, Appcao of a upe-cea deco ag appoac fo eecg o-peea eevo fo cuue-baed fey deveope: Cae udy fo S aa, Aquacuue, 459, 6-35, DOI: /.aquacuue Xu Z. S. 005, A Ovevew of eod fo deeg OWA weg, Ieaoa Joua of Iege Sye, 0: , DOI: /.0097