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30 9 Vol.30,No.92010 9 SystemsEngineering—Theory&PracticeSept.,2010 :1000-6788(2010)09-1658-07 :C934 :A , Æ( , 410083) , , !#$ .! % &’ (!# $ ,%)(*&+ ’ (, ’)-* .* ’ ! ,+,! /0,-12.3/045,61278,9:783 3; =,45?* ’@6! /A ’B&’.37CD ! EBF GHF. 8 ; ;!#$; Grey-stochasticmulti-criteriadecision-makingapproachbasedonprospecttheoryWANGJian-qiang,ZHOULing(SchoolofBusiness,CentralSouthUniversity,Changsha410083,China)AbstractAsforthegrey-stochasticmulti-criteriadecision-makingproblemsinwhichprobabilitiesandthecriteriavalueofalternativesarebothintervalgreynumbers,adecision-makingapproachbasedonprospecttheoryisproposed.Inthismethod,astheintervalgreynumbercomparisonisdefinedfirstly,aprospectvaluefunctionofintervalgreynumberscanbedefinedandtheprospectvalueofeachalternativearecalculatedbasedonallotheralternativesasthereferencepoint.Afterthataprogrammingmodelwhichsatisfiesthealgorithmofmaximizingdeviationcanbeenacted.Afterthat,thecriteriaweightsareattainedandtheorderofalternativescanbelistedbycomparingtheprospectvalueofeachalternative.Finally,thefeasibilityandvalidityofthisapproachareillustratedbyanexamplecomparedwithgreyrandommulti-criteriadecision-makingapproachbasedonexpectedvalueillustrated.Keywordsmulti-criteriadecision-making;grey-stochastic;prospecttheory;intervalgreynumberproba-bility19: , , , . , , , , , . , ! . . ,#! ! . [1–5]! $ # , $$%% # [6] & ,& # ’ & . [7] [8]$%!# # # & & , & ( #! $ & .;’=:2009-06-08?@A:Æ)( * +,(70771115,70631004);Æ)’ -!* +,(70921001)BCÆD:.!(1963–),,/##$$,(%,0%,)&1*2’)*++,&32)4,--;-((1986–),),*’ ($,)%-+,)&1*2’)*++*+32--. 9 .!,-:+.,52’.6-,7//01*011659! # $-. 8 23. , , 1 /423 950:64 5.7 $1 ;2 ,-. 6 /6$ 002 %, 8!8 93 4:78.! , 950:64 5.7 # -,5; 167 & , , 100! =3 & 8. 8, 1 & .9?9= 1;9 .Tversky Kahneman 1979:27 ,27 & “?@@# 9,@@?# 9” A38B,C27 !&:# A38B& D; A [9−10].E !=34 5;,1B =34 6 1 7. , E 8 9 , , [11−12]. 8, !# $C # , ? ,27 :D F,@& : .2EFGHIJKLM2.1NOPQ=3AG#H; 3A /$ IJ [13]. , B #:B K E C$ , FG“⊗”HD;.I Ja? 3Ja J # ,F ⊗∈[a,a]. L # = @ ,K D J 3J9 # , LMA # @F & @.BR1[14]E⊗∈[a,a],0≤aa, C # , 1) # ⊗ nNF ⊗n=[a,a]n=[an,an];2) # ⊗ nN O:n−1F ⊗1n=[a,¯a]1n=⎧⎨⎩[a1n,¯a−1n],n G ,[a1n,¯a1n]:[−¯a1n,−a1n],n H ;3)∀p,q∈R,pq=0,⊗pq=(⊗p)q.! # G?, P [13] $19M , # $19H& 8.BR2E⊗1∈[a,a],aa,⊗2∈[b,b],bb # ,I⊗1 ⊗2@0.5≤P(⊗1≥⊗2)≤1, J⊗1≥⊗2, F⊗1≥P⊗2;Q ,⊗1⊗2.A ,P(⊗1≥⊗2)=min{la+lb,max(a−b,0)}la+lb(1) ⊗1≥⊗2 $19[2].la=a−a,lb=b−b.1 ,J(1) 0≤P(⊗1≥⊗2)≤1,K P(⊗1≥⊗1)=0.5,P(⊗1≥⊗2)+P(⊗2≥⊗1)=1.BR3R SL$1C= $ ,CK6 $ - # ⊗@J7 , 6! # 1BI 3,K # , J&A SL NM # # SL( J SL)[4], ζ(⊗)D;,AC$ # $TDND16;.D1STUVEWUXYZF[ζ(⊗)GEWHIζ(⊗)⊗1⊗2...⊗i...⊗np(⊗)p1(⊗)p2(⊗)...pi(⊗)...pn(⊗) D1 ,ζ(⊗) # # SL;⊗i SL$1C Ui $;⊗i∈[xi,xi];pi(⊗) SLCUi $. # ;pi(⊗)∈[pi,pi];n ## SL$1C 6 $ .A , ?9 # BCL ? SL, Opi(⊗)@0≤pi≤pi D9[7].2.2\]^_ $ “ $A ” “ ”P- [9],OV=ni=1π(pi)v(xi)(2)1660EFVW2’)OQ 30 A ,π(p) ,G # XJA ,$5 6 ! @JA R; A [15]:π(p)=pγ(pγ+(1−p)γ)1γ(3)v(x) $A , 1 =AH@ $.Tversky KahnemanY7 $A @J FA :v(x)=xα,x≥0−θ(−x)β,x0(4)A Z α β$%D;P6 /Q $FA [I 9,α,β1D;RA \S.θ HD;/Q P6 ]^ JK.θ1D;/QK_.BR4E # ⊗1∈[a,a],aa,⊗2∈[b,b],bb,I # ⊗2 Z?, # ⊗1 4 $A :v(⊗1)=(d(⊗1,⊗2))α,−θ(d(⊗1,⊗2))β,⊗1≥⊗2⊗1⊗2(5)A ,d(⊗1,⊗2)=a−b 2+(a−b)2 S # TN[3].3‘aJKLMLGbcdMefgNOhEB , E A={a1,···,ai,···,am}, E C={C1,···,Cj,···,Cn},K6 9U‘T, LLF W={w1,···,wj,···wn}T,C@PMD9:nj=1wj=1,wj≥0,j=1,2,···,n. ? ,G@D9H[12]. ,E 67s:$1 4 5.,5.E Θ={θ1,···,θk,···,θs},Uk:5.θk50 # # pk(⊗),pk(⊗)∈[pk,pk]. ai Cj $ xij,xij BCL ? # # SL. ai Cj θk5. $ # xijk(⊗),xijk(⊗)∈[xijk,xijk],A 1≤i≤m,1≤j≤n,1≤k≤s.a $; VQD=(xijk)m×n×s. R EA SW : .3N TN :Ui1MH%L . Ob - Lc LX! YZ VP,!xijk(⊗)& MH%L N [16]:!46 $rkij=xijkxmaxjk,xijkxmaxjk (6)A xmaxjk=max1≤i≤m{xijk};! $rkij=xminjkxijk,xminjkxijk (7)A xminjk=min1≤i≤m{xijk}.a $;=s 4 5. d %& VQRk(⊗)=(rkij)m×n,k=1,2,···,s.Ui2 VQ. ,Z? C9 . 1 & .K!AZ?H[L“P6”:“/Q”. Z?3, U] D2= YZ BT YZ . ,G \ Z?,NCe Q :V , [11]7Z? 3 $S% , AR W Z?,]6 6 $ $,;= $VQ. 9 .!,-:+.,52’.6-,7//01*011661O^_J(2) 84,$;6 ,6 S $ :Vij(⊗)=klvrkijl πijl(pk(⊗)),l=1,2,···,m,Kl=i(8)A , 84 _J(5),$; $A :vrkijl =⎧⎨⎩ d rkij,rkljα,rkij≥rklj−θ d rkij,rkljβ,rkijrklj(9)O^ 167P6 /Q. A :πijl(pk(⊗))=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩pγk(⊗)(pγk(⊗)+(1−pk(⊗))γ)1γ,rkij≥rkljpδk(⊗) pδk(⊗)+(1−