Aggregating fuzzy opinions in the group decision-making environment

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AGGREGATING FUZZY

OPINIONS IN THE GROUP

DECISION-MAKING

ENVIRONMENT

SHYI-MING CHEN

Published online: 29 Oct 2010.

To cite this article: SHYI-MING CHEN (1998) AGGREGATING FUZZY

OPINIONS IN THE GROUP DECISION-MAKING ENVIRONMENT, Cybernetics

and Systems: An International Journal, 29:4, 363-376, DOI:

10.1080/019697298125641

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GROUP DECISION-MAKING ENVIRONMENT

SHYI-MING CHEN

Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

This pape r pre se nts a ne w m ethod for de aling with fuzzy opinion aggre ga-tion in group de cision-m aking proble ms. The propose d me thod has the

s .

following advantage s: 1 The e xpe rts’ estimate s do not ne ce ssarily have a s x s .

com mon inte rse ction at a -leve l cuts, whe re a g 0, 1 . 2 It can pe rform s .

fuzzy opinion aggre gation in a more efficient manne r. 3 It doe s not ne ed to use the De lphi m ethod to adjust trape zoidal fuzzy numbe rs give n by e xpe rts.

s

Some re searche rs B ardossy e t al., 1993; Chen & Lin, 1995; Chen et al., 1989; Hsu & Chen, 1996; Ishikawa et al., 1993; Kacprzyk & Fe drizzi, 1988; Kacprzyk e t al., 1992; Le e, 1996; Nurmi, 1981; Spillman et al.,

.

1980; Tanino, 1984, 1990; Xu & Z hai, 1992; Che n, 1997 have focuse d on the fuzzy opinion aggregation problem in the multicrite ria group

s .

de cision-making MCD M e nvironme nt based on fuzzy set the ory sZ ade h, 1965 to combine the individual opinions of experts, whe re e ach. e xpe rt usually has its own opinion or e stimate d rating unde r e ach crite rion for e ach alte rnative. Thus, to find a group conse nsus function

This work was supporte d in part by the National Scie nce Council, Republic of China, unde r grant NSC 86-2623-E-009-018.

The author would like to thank Profe ssor H. M. Hsu, De partm ent of Industrial Engine ering, National Chiao Tung Unive rsity, for he r e ncourage m ent in this work.

Addre ss corre sponde nce to Profe ssor Shyi-Ming Che n, De partme nt of Compute r and Inform ation Scie nce , National Chiao Tung Unive rsity H sinchu, Taiwan, Republic of China. E-mail: sym che n@ cis.nctu.e du.tw

Cybernetics and Systems: An International Journal, 29:363] 376, 1998

CopyrightQ1998 Taylor & Francis

0196-97 22 ¤98 $12.00 + .00 363

for aggre gating these e stimate s ratings to a common opinion is an important issue in handling multicrite ria group decision-making prob-lems. Be cause in multicriteria de cision making with group de cision-mak-ing problem s, the e stimate s of e xperts of a crite rion for an alte rnative may involve subjective ness, im pre cision, and vague ness, fuzzy set the ory can provide us with a useful way to deal with the fuzzine ss of human judgm ents.

s .

Kacprzyk e t al. 1992 showe d how fuzzy logic with linguistic quanti-s .

fie rs can be use d in group decision making. Tanino 1984 discussed fuzzy prefe rence orde rings in group decision making. Bardossy e t al. s1993 re prese nte d expert opinions or imprecise e stimate s of a physical. variable by using fuzzy numbe rs and de velope d five technique s for combining the se fuzzy numbe rs into a single fuzzy number e stim ate. The guide lines for the choice of com bination te chnique are also

pro-s .

vide d in B ardossy et al. Ishikawa e t al. 1993 propose d the max-min Delphi me thod and fuzzy De lphi me thod via fuzzy inte gration. Xu and

s .

Z hai 1992 pre sented e xte nsions of the analytic hierarchy process in a fuzzy e nvironme nt, where e ach e xpe rt repre se nts it’s subje ctive judg-me nt by an inte rval value rating of each criterion for e ach alte rnative.

s .

Le e 1996 pre sented a method for group de cision making using fuzzy se t theory for e valuating the rate of aggre gative risk in software deve

l-s .

opme nt. Nurmi 1981 prese nte d some approache s to collective de cision s .

making with fuzzy pre fere nce re lations. Hsu and Che n 1996 pre sented s .

a similarity aggregation m ethod SAM for aggregating individual fuzzy opinions into a group fuzzy conse nsus opinion, whe re the e stimate s of e xpe rts are re prese nte d by positive trape zoidal fuzzy numbe rs.

Howe ve r, the re are some drawbacks of the me thod prese nte d in s .

Hsu and Che n 1996 , shown as follows:

1. It re quires that the expe rts’ estimates have a com mon interse ction s x

at some a -le vel cut, whe re a g 0, 1 . If the initial e stimates of the

ith expe rt and the jth e xpert have no interse ction, then it must use

s .

the De lphi method Satty, 1980 or get m ore information to adjust the trapezoidal fuzzy numbe r given by e ach e xpert to obtain a com mon interse ction at the a -le vel cut. Howe ve r, applying the De lphi m ethod to adjust the trape zoidal fuzzy numbe rs given by the expe rts will take a large amount of time to pe rform the operations. 2. It re quires a large amount of tim e to calculate the de gre e of agreem ent betwe en experts’ e stimate s be cause it use s a com plicated

similarity me asure function S to calculate the de gre e of agre em ent

Ä

of the subjective e stimate Ri of expe rt Ei and subje ctive estim ate

Ä

Rj of e xpe rt Ej, s . s .

H

u

t

m in

w

fRÄi u ,fRÄj u

5

/

d u

Ä

Ä

S R

t

i,Rj

/

s s . s .

H

u

t

max

w

fRÄi u ,fRÄj u

5

/

d u

Ä

Ä

whe re Ri and Rj are positive trapezoidal fuzzy num bers and the

Ä

Ä

membership functions of the trape zoidal fuzzy numbe rs Ri and Rj

are fRÄi and fRÄj, re spective ly.

Thus, it is necessary to de velop a ne w me thod for dealing with the fuzzy opinion aggre gation proble m in a more flexible and m ore efficient manner.

In this pape r, we prese nt a ne w m ethod for dealing with the fuzzy opinion aggregation proble m. The propose d me thod can ove rcome the

s .

drawbacks of the one pre se nte d in Hsu and Che n 1996 due to the fact that

1. The e xpe rts’ e stim ates do not necessarily have a common interse c-s x

tion at the a le vel, whe re a g 0, 1 . Thus, it is more flexible than the one pre sented in Hsu and Che n.

2. It doe s not ne e d to use the De lphi me thod to adjust trape zoidal fuzzy numbe rs given by e xpe rts.

3. It can calculate the de gre e of similarity be twe e n the subje ctive estimate s of e xpe rts in a more e fficie nt manne r. Thus, it can pe rform fuzzy opinion aggre gation in a more efficient manne r.

BASIC CONCEPTS OF FUZZY SET THEORY

The theory of fuzzy sets was propose d by Z ade h in 1965. Le t U be the

Ä

v 4

unive rse of discourse, Us u1,u2, . . . ,un . A fuzzy se t A of U is a set v s s .. s s .. s s ..4

of ordere d pairs u1,fAÄ u1 , u2,fAÄ u2 , . . . , un,fAÄ un , where fAÄ

w x is the me mbe rship function of the fuzzy se t A, fAÄ: Uª 0, 1 , and

Ä

s .

fAÄ ui indicates the degree of me mbership of ui in the fuzzy se t A. A

Ä

fuzzy se t A of the universe of discourse U is called a norm al fuzzy set if s .

’ uig U, fAÄ ui s 1. If for all u1,u2 in U,

s s . . s s . s . . s .

fAÄ l u1q 1 y l u2 0 Min fAÄ u1 ,fAÄ u2 1

Ä

then the fuzzy set A is calle d a conve x fuzzy se t. A fuzzy numbe r is a fuzzy subset in the unive rse of discourse U that is both normal and

Ä

conve x. For e xample , Figure 1 shows a fuzzy numbe r A of the unive rse of discourse U. A standardize d fuzzy numbe r is a fuzzy numbe r define d

w x in the universe of discourse U, whe re Us 0, 1 .

Ä

A trape zoidal fuzzy numbe r M of the unive rse of discourse U can

s .

be characte rized by a quadruple a,b,c,d shown in Figure 2.

In the following, we brie fly revie w the de fuzzification technique of

s .

trape zoidal fuzzy numbe rs from Chen 1994, 1996 and Kauffman and

Ä

s .

Gupta 1988 . Conside r the trape zoidal fuzzy numbe r M shown in Figure 3, whe re e is the defuzzification value of the trapezoidal fuzzy

Ä

numbe r M. From Figure 3, we can see that

1_{s . s . s . s .} 1_{s . s . s . s .} by a 1 q ey b 1 s dy c 1 q cy e 1 2 2 1_{s . s . s .} 1_{s . s .} « 2 by a 1 q ey b s 2 dy c q cy e 1 1 s . s . s . s . « ey b y cy e s 2 dy c y 2 by a 1_{s .} 1_{s .} « 2es 2 dy c y 2 by a q bq c F ig u r e 1 . A fuzzy num be r.

F igu r e 2 .A trape zoidal fuzzy numbe r. dy cy bq aq 2bq 2c « 2es 2 aq bq cq d s 2 aq bq cq d s . « es 2 4

SIMILARITY MEASURES

s .

Z wich e t al. 1987 have made a comparative analysis of 19 sim ilarity s .

me asure s among fuzzy se ts. In Che n and Lin 1995 we m ade a compari-son of sim ilarity of fuzzy se ts. In the following, we introduce a me thod for measuring the de gre e of similarity betwe en trapezoidal fuzzy num

-F igu r e 3 . De fuzzification of a trape zoidal fuzzy numbe r.

Ã

Ã

s .

be rs Che n & Lin, 1995 . Le t A and B be two standardize d trape zoidal fuzzy numbe rs,

Ã

s .

As a1,b1,c1,d1

Ã

s .

Bs a2,b2,c2,d2

whe re 0 ( a1( b1( c1 ( d1( 1 and 0 ( a2( b2( c2 ( d2 ( 1. Then the de gre e of sim ilarity be twe e n the standardize d trapezoidal fuzzy

Ã

Ã

numbe rs A and B can be me asure d by the similarity function S, <a1y a2<q <b1y b2<q <c1y c2<q <d1y d2<

Ã

Ã

s . s . S A,B s 1 y 3 4

à Ã

Ã

Ã

s . w x s .

whe re S A, B g 0, 1 . The larger the value of S A,B , the greate r the

Ã

similarity betwe en the standardized trapezoidal fuzzy numbe rs A and

Ã

B.

Ã

Ã

Le t A and B be two standardized trapezoidal fuzzy numbers,

Ã

s .

As a1,b1,c1,d1

Ã

s .

Bs a2,b2,c2,d2

whe re 0 ( a1( b1( c1 ( d1( 1 and 0 ( a2( b2( c2 ( d2 ( 1. Then

Ã

Ã

Ã

Ã

s . s .

it is obvious that S A,B s S B,A .

w x w x

Le t xand y be two real value s, where xg 0, 1 and yg 0, 1 . It is obvious that the real value s xand ycan be repre se nte d by standardize d

s .

trape zoidal fuzzy numbe rs x and y, re spectively, where xs x,x,x,x

s . s .

and ys y,y,y,y . B y applying formula 3 , the degre e of sim ilarity be twe e n the re al value s xand y can be evaluate d as follows:

<xy y<q <xy y<q <xy y<q <xy y< s . S x,y s 1 y 4 < < s . s 1 y xy y 4

It is obvious that this re sult coincide s with the one shown in Che n et al. s1989 ..

Ä

Le t A be a positive trapezoidal fuzzy numbe r in the unive rse of discourse U, where

w x

Us 0 ,m

Ä

s .

As a,b,c,d

and 0 ( a ( b ( c ( d ( m. The n, the positive trapezoidal fuzzy num

-Ä

be r A can be translate d into the standardized trapezoidal fuzzy numbe r

Ã

A shown as follows: a b c d

Ã

As

t

, , ,

/

m m m m

whe re 0 ( a

r

m ( b

r

m ( c

r

m ( d

r

m ( 1 and the mem bership

func-Ã

tion curve of the standardize d trapezoidal fuzzy numbe r A is as shown

Ã

in Figure 4. In this case, the standardize d trapezoidal fuzzy number A is w x

de fined in the unive rse of discourse U, whe re Us 0, 1 .

A NEW METHOD FOR HANDLING FUZZY OPINION

AGGREGATION PROBLEMS

In the following, we prese nt a ne w m ethod for handling fuzzy opinion aggregation proble ms. The algorithm e ssentially is a modification of the

s .

one pre sented in Hsu and Che n 1996 . Le t U be the unive rse of

w x s .

discourse , Us 0,m . Assume that e ach e xpert Ei is 1, 2, . . . ,n

Ä

s .

constructs a positive trape zoidal fuzzy num ber Ris ai,bi,ci,di to re prese nt the subjective e stimate of the rating to a given crite rion and

F ig u r e 4 . A standardiz ed trape zoidal fuzzy numbe r A.

alternative , where 0 ( ai ( bi ( ci ( di ( m. Furtherm ore , assume that

s .

the degre e of importance of export Ei is 1, 2, . . . ,n is wi, whe re

w x n

wig 0, 1 and p is 1wis 1. The algorithm is prese nte d as follows:

Ä

s .

Step 1: Translate e ach trape zoidal fuzzy numbe r Ris ai,bi,ci,di

Ã

given by e xpe rt Ei into standardize d trapezoidal fuzzy number Ri s is 1, 2, . . . ,n., whe re ai bi ci di

Ã

Ris

t

, , ,

/

m m m m s U U U U. s ai ,bi ,ci ,di and 0 ( aUi ( bUi ( cUi ( dUi ( 1. s .

Step 2: Based on formula 3 , calculate the de gre e of agree ment

Ã

Ã

s .

S Ri,Rj of the opinions betwe en each pair of e xpe rts Ei and Ej,

Ã

Ã

s . w x

whe re S Ri,Rj g 0, 1 , 1( i ( n, 1 ( j ( n, and i/ j. s .

Step 3: Calculate the average de gre e of agre eme nt A Ei of e xpert Ei s is 1, 2, . . . ,n., whe re n 1

Ã

Ã

s . s . A Ei s

_p

S R

t

i,Rj

/

6 ny 1 _{js 1} j/ i s .

Step 4: Calculate the re lative degre e of agree me nt RA Ei of e xpe rt Ei s is 1, 2, . . . ,n., whe re s . A Ei s . s . RA Ei s n _{s .} 7 p is 1 A Ei

Step 5: Assum e that the we ight of the de gre es of importance of the e xperts and the we ight of the relative de gree of agre em ent of the

w x w x

e xperts are y1and y2, re spective ly, where y1g 0, 1 and y2g 0, 1 . s .

Calculate the conse nsus de gre e coe fficie nt C Ei of expert Ei s is 1, 2, . . . ,n., whe re

y1 y2

s . s . s .

C Ei s ) wiq ) RA Ei . 8

y1q y2 y1q y2

Ä

Step 6: The aggregation result of the fuzzy opinions is R, where

Ä

s . s . s . s .

Rs C E1 mR1

Å

C E2 mR2

Å

? ? ?

Å

C En mRn 9 ope rators m and

Å

are the fuzzy multiplication ope rator and the fuzzy addition ope rator, re spe ctively.

In the following, we use an e xample to illustrate the fuzzy opinion aggregation proce ss.

Example: Assume that expe rts E1,E2, and E3construct positive trape

-Ä

Ä

Ä

zoidal fuzzy numbers R1, R2, and R3 to re prese nt the subje ctive e stimate of the rating to a given crite rion and alte rnative, re spectively, whe re

Ä

s . R1s 1 , 2 , 3 , 4

Ä

s . R2s 4 , 5 , 6 , 7

Ä

s . R₃s 7 , 8 , 9 , 10

Ä

Ä

Ä

Assume that the trape zoidal fuzzy num bers R1, R2, and R3are define d

w x s .

on the universe of discourse U, whe re Us 0, 10 i.e., m s 10 , and assume that the we ights of the expe rts E1, E2, and E3 are 0.4, 0.4, and

s .

0.2, respe ctively i.e., w1s 0.4, w2s 0.4, and w3s 0.2 . Furthe rmore , assume that the we ight of the degree s of importance of the e xpe rts and the we ight of the relative de gree s of agre eme nt of the expe rts are 0.9

s .

and 0.6, re spectively i.e ., y1s 0.9 and y2s 0.6 . Based on the pro-posed algorithm, we can ge t the following re sult:

wSte p 1 Be causex ms 10, we can translate the trapezoidal fuzzy

Ä

Ä

Ä

numbe rs R1, R2, and R3 into the standardize d trape zoidal fuzzy

Ã

Ã

Ã

numbe rs R1, R2, and R3, re spe ctively, whe re

1 2 3 4

Ã

R1s

t

, , ,

/

10 10 10 10 s . s 0 .1 , 0 .2 , 0 .3 , 0 .4

4 5 6 7

Ã

R2s

t

, , ,

/

10 10 10 10 s . s 0 .4 , 0 .5 , 0 .6 , 0 .7 7 8 9 10

Ã

R3s

t

, , ,

/

10 10 10 10 s . s 0 .7 , 0 .8 , 0 .9 , 1 .0

wSte p 2 B ase d on formula 3 , we can ge t the following re sults:x s . <0 .1y 0 .4 q 0 .2 y 0 .5 q 0 .3 y 0 .6 q 0 .4 y 0 .7< < < < < < <

Ã

Ã

S R

t

1,R2

/

s 1 y 4 s 0 .7

Ã

Ã

S R

t

2,R1

/

s 0 .7 <0 .1y 0 .7 q 0 .2 y 0 .8 q 0 .3 y 0 .9 q 0 .4 y 1 .0< < < < < < <

Ã

Ã

S R

_t

₁,R₃

_/

s 1 y 4 s 0 .4

Ã

Ã

S R

t

3,R1

/

s 0 .4 <0 .4y 0 .7 q 0 .5 y 0 .8 q 0 .6 y 0 .9 q 0 .7 y 1 .0< < < < < < <

Ã

Ã

S R

_t

2,R3

_/

s 1 y 4 s 0 .7

Ã

Ã

S R

_t

₃,R₂

_/

s 0 .7

wSte p 3 Based on formula 6 , the degre e s of agree ment of expe rtsx s . E1, s . s .

E2, and E3 can be evaluate d and are e qual to A E1 , A E2 , and

s .

A E3 , respe ctively, whe re 0 .7q 0 .4 s . A E1 s s 0 .35 2 0 .7q 0 .7 s . A E2 s s 0 .7 2 0 .4q 0 .7 s . A E3 s s 0 .35 2

wSte p 4 Based on formula 7 , the relative de gree s of agre e ment ofx s . s . e xperts E1, E2, and E3 can be e valuated and are equal to RA E1 ,

s . s .

RA E2 , and RA E3 , re spe ctively, whe re 0 .35 s . RA E1 s s 0 .25 0 .35q 0 .7 q 0 .35 0 .7 s . RA E2 s s 0 .5 0 .35q 0 .7 q 0 .35 0 .35 s . RA E3 s s 0 .25 0 .35q 0 .7 q 0 .35

wSte p 5 B ecause the we ights of the expertsx E1, E2, and E3 are 0.4, 0.4,

s .

and 0.2, respectively i.e ., w1s 0.4, w2s 0.4, w3s 0.2 , and be -cause the we ight of the de gree s of importance of the expe rts and the we ight of the re lative degree s of agree me nt of the e xpe rts are

s .

0.9 and 0.6, re spectively i.e ., y₁s 0.9 and y₂s 0.6 , then base d on s .

formula 8 we can ge t the following re sults:

0 .9 0 .6 s . C E1 s ) 0 .4q )0 .25 0 .9q 0 .6 0 .9q 0 .6 s 0 .34 0 .9 0 .6 s . C E2 s ) 0 .4q )0 .5 0 .9q 0 .6 0 .9q 0 .6 s 0 .44

0 .9 0 .6 s .

C E3 s ) 0 .2q )0 .25

0 .9q 0 .6 0 .9q 0 .6 s 0 .22

wSte p 6 Based on formula 9 , we can see that the aggregation result ofx s .

Ä

the fuzzy opinions is the trape zoidal fuzzy numbe r R, where

Ä

s .

Ä

s .

Ä

s .

Ä

Rs C E1 mR1

Å

C E2 mR2

Å

C E3 mR3 s . s . s 0 .34 m 1 , 2 , 3 , 4

Å

0 .44m 4 , 5 , 6 , 7 s .

Å

0 .22m 7 , 8 , 9 , 10 s . s . s 0 .34, 0 .68 , 1 .02, 1 .36

Å

1 .76 , 2 .2 , 2 .64 , 3 .08 s .

Å

1 .54 , 1 .76 , 1 .98 , 2 .2 s . s 3 .64, 4 .64 , 5 .64, 6 .64

The mem bership function curve s of the trapezoidal fuzzy numbe rs

Ä

Ä

Ä

Ä

R1,R2,R3 and the aggre gation re sult R are shown in Figure 5.

CONCLUSIONS

s . In this pape r, we have e xtended the work of Hsu and Chen 1996 to propose a new me thod for de aling with fuzzy opinion aggregation with group decisionmaking proble ms. From the illustrative e xample pre

-F igu r e 5 .Me m be rship functions of R , R , R , and R.1 2 3

se nte d pre viously, we can se e that the propose d me thod can ove rcome s .

the drawbacks of the one pre sented in Hsu and Che n 1996 because

1. The e xpe rts’ e stim ates do not necessarily have a common interse c-s x

tion at a -leve l cuts, whe re a g 0, 1 . Thus, it is more fle xible than the one pre sented in Hsu and Che n.

2. It can calculate the de gre e of similarity be twe e n the subje ctive estimate s of e xpe rts in a more e fficie nt manne r. Thus, it can pe rform fuzzy opinion aggre gation in a more efficient manne r. 3. It doe s not ne e d to use the De lphi me thod to adjust trape zoidal

fuzzy numbe rs given by e xpe rts.

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