aggregation of fuzzy opinions under group decision-making based on similarity and distance.pdf

8/14/2019 AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING BASED ON SIMILARITY AND DISTANCE.pdf

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Jrl Syst Sci & Complexity (2006) 19: 6371

AGGREGATION OF FUZZY OPINIONS UNDER

GROUP DECISION-MAKING BASED ON

SIMILARITY AND DISTANCEChengguo LU Jibin LAN Zhongxing WANG

Received: 6 December 2004 / Revised: 7 September 2005

Abstract In this article, a new method for aggregating fuzzy individual opinions into a group consensus

opinion is proposed. To obtain the aggregation weights of each individual opinion, a consistency index

of each expert with the other experts is introduced based on similarity and distance. The importance of

each expert is also taken into consideration in the process of aggregation. Finally, a numerical example

is presented to illustrate the efficiency of the procedure.Key words Consistency degree, fuzzy individual opinions, fuzzy numbers, group consensus opinion,

group decision-making.

1 Introduction

Under the circumstance that a fuzzy decision-making problem involves multiple actors,there may arise situations of conflict and agreement among the experts because of differentvalues and information systems. Hence, how to find a reasonable method to aggregate eachindividual opinion into a group consensus opinion is an important issue in a group decision-making environment. Up to now, some aggregation methods have been proposed to solve thegroup decision making problems in an acceptable way[17], but most of them are based on

the fuzzy preference relations[36]

. Hsu and Chen[8]

proposed a similarity aggregation method(SAM) to combine the individual opinions. They introduced a similarity index to measurethe consistency of each expert with the other experts. However, there are several problemsin Hsu and Chens work. First, they assumed that the opinions of all experts represented byfuzzy numbers should have a common intersection at some -level cut, [0, 1]. Otherwise,

it will not work. For example, let us consider three fuzzy numbersA,B, andCranging from2 to 3, 4 to 5, and 6 to 8, respectively. According to Hsu and Chens method, the degree ofsimilarity betweenA andB is zero; the degree of similarity betweenA andC is also equal tozero. However, these two similarities are obviously different. To avoid this disjoint situation,Hsu and Chen suggested that Delphis method should be used to modify each experts opinion.But it will distort the opinions of the experts to some extent. Moreover, if the supports of fuzzy

Chengguo LU

School of Mathematics and Information Science, Guangxi University, Nanning530004, China.

Jibin LANSchool of Economics and Management, South West Jiao-tong University, Chengdu610031, China.

Zhongxing WANG

School of Mathematics and Information Science, Guangxi University, Nanning530004, China.

Email: [email protected].


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64 CHENGGUO LU JIBIN LAN ZHONGXING WANG

numbers do not intersect, we cant conclude that the opinions represented by fuzzy numbersdo not intersect[9]. For example, two experts assign different supports to an alternative undera criterion ranging from 0.78 to 0.80 and 0.81 to 0.83, respectively. The supports are disjoint,but their opinions are very close and the similarity between the opinions should not equal zero.The second problem of Hsu and Chens work is that the degrees of similarity between fuzzyopinions are determined by the proportion of the consistent area to the total area only, and

they did not consider the supports of the consistent area and the total area (leading to a lossof information), and this is discussed in Section 3.

In this article, we propose a consistency aggregation method (CAM) to combine fuzzyindividual opinions. This method is based on similarity and distance because they are indicesequally important to the comparison of fuzzy opinions. By the method, the sum of weightedconsistency among aggregated consensus and the group consensus opinion is obtained. Thenan algorithm is presented to determine the aggregation weights of each individual opinion.

This article is organized as follows. In Section 2, Hsu and Chens similarity aggregationmethod (SAM) is described in brief. In Section 3, we give an improved method to measure thesimilarity between fuzzy numbers, and a modified distance measure is also defined as anotherindex to compare fuzzy numbers. Then we combine the similarity and distance measures toobtain the weights of each opinion. In Section 4, a numerical example is presented to illustrateour consistency aggregation method (CAM). In Section 5, concluding remarks are presented.

2 Preliminaries

A fuzzy number is a fuzzy set in the real line and is completely defined by its membershipfunction ((x) : R [0, 1]). For computational purposes, the fuzzy number is often strict toboth normal and convex.

Normality: sup{(x)} = 1, x R. This requirement means that there is at least onepoint in a real line with the maximum membership value equal to 1.

Convexity: {x1+ (1 )x2} min{(x1), (x2)}, x1, x2 R, [0, 1]. This meansthe points of the real line with the highest membership values are clustered around a giveninterval (or point).

For convenience, the fuzzy number is often represented by a trapezoidal fuzzy number. A

trapezoidal fuzzy number can be denoted by a 4-tuple (a1, a2, a3, a4) where a1 a2 a3 a4.Whena2 = a3the fuzzy number is called a triangular fuzzy number. The condition that a1 = a2and a3 = a4 implies a closed interval. Ifa1 = a2 = a3 = a4, we obtain a crisp real number.

LetRi = (ai, bi, ci, di) be a positive trapezoidal fuzzy number representing ith experts sub-jective estimate to an alternative under a criterion. Now the key issue is to construct an aggre-gation functionFto combineRi (i= 1, 2, , n) into a group opinionR= F( R1,R2, ,Rn).Before the description of our method, Hsus aggregation method (SAM) is briefly described.

In Hsu and Chens article, they calculated the average agreement degree of each expertEi(i= 1, 2, , n) by averaging the degrees of similarity with respect to other experts:

A(Ei) = 1

n 1

nj=1,j=i

S( Ri,Rj), (1)where

S( Ri,Rj) = x(min{ Ri(x), Rj (x)})dxx

(max{ Ri(x), Rj (x)})dx . (2)It is a similarity measure function by Zwick et al.[10], which means the proportion of theconsistent area (

x

(min{ Ri(x), Rj (x)})dx) to the total area (x(max{ Ri(x), Rj (x)})dx).


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AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING 65

Then the aggregation weight of the ith expert is given by

RADi= A(Ei)nj=1

A(Ej). (3)

Without considering the importance of theith expert, the aggregation result is therefore definedas R= F( R1,R2, ,Rn) = n

i=1

RADiRi, (4)

where

is the fuzzy multiplication operator[11].

3 New Aggregation Method Based on Similarity and Distance

Hsu and Chens work is based on the idea that the weight of an experts opinion should belarger if his opinion is closer to the other opinions. But there are some problems as we havepointed out earlier. To solve these problems, Lee[9] proposed an optimal aggregation method(OAM) on the basis of the idea that the weight of an experts opinion should be larger if the

distances between his opinion and other opinions are smaller. In fact, neither Hsu and Chenswork nor Lees work can be efficient enough because distance and similarity are dual importantindices for comparing fuzzy individual opinions. Any one of them can reflect only one aspect ofthe issue while leading to the loss of other information. Therefore, we propose a new consistencyaggregation method (CAM) based on similarity and distance.

3.1 Similarity Between Fuzzy Numbers

Hsu and Chen employed Zwicks similarity measure function to calculate the similaritybetween fuzzy numbers. However, each element in the universe may have a different importance.Let us study an example:

Example 3.1 There are three fuzzy numbersA= (1, 3, 5, 7),B = (2, 3, 5, 6),C= (1, 4, 4, 7)(see Fig. 1).

Figure 1 Three fuzzy numbersA,B, andCUsing Hsu and Chens similarity measure, Eq. (2), the degrees of similarity ofA andB,AandCareS( A,B) = S( A,C) = 34 .However, one can see from Fig. 1 that these two similaritiesshould not be equal because the supports of the intersection ofA andB are larger than thesupports of the intersection ofAandC. To avoid this situation we need to consider the weight


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of each element x R so that we have the following weighted similarity measures. For fuzzynumbersRiandRj , assume the weight of eachx R is w(x), 0 w(x) 1. Then the weightedsimilarity betweenRi andRj is defined as

Sw(

Ri,

Rj) =

x

(wmin(x)min{ Ri(x), Rj (x)})dx

x(wmax(x)max{ Ri(x), Rj (x)})dx, (5)

where wmin(x) and wmax(x) denote the weight functions of the consistent area and the totalarea, respectively. In general, let wmin(x) = min{ Ri(x), Rj (x)}, wmax(x) = max{ Ri(x), Rj (x)} because min{ Ri(x), Rj (x)} and max{ Ri(x), Rj (x)} are membership functions ofthe consistent area and total area, respectively. Thus the improved similarity is given as

Sw( Ri,Rj) =x

(min{ Ri(x), Rj (x)})2dxx

(max{ Ri(x), Rj (x)})2dx . (6)It is easy to prove that Sw( Ri,Rj) satisfies the following properties:(SP1) 0 Sw( Ri,Rj) 1;(SP2) Sw(

Ri,

Rj) = 1 if and only if

Ri =

Rj ;

(SP3) Sw( Ri,Rj) = Sw( Rj ,Ri);(SP4) Sw( Ri,Rj) Sw( Ri,Rk) and Sw( Rj ,Rk) Sw( Ri,Rk) ifRiRj Rk.Let us again consider Example 3.1. Now using our similarity measure Eq. (6) we obtain

Sw( A,B) = 45

Sw( A,C) = 35

.

The result is fair and reasonable.

3.2 Distance Between Fuzzy Numbers

Distance is an important concept in fuzzy set theory and is also a significant index of thecomparison of fuzzy numbers. Many distance measure methods have been proposed up to now,a survey of such distances can be found in papers of Dubois et al. [12], and Heilpern[13]. Wesynthesize the main characteristics of these methods and classify them as two categories as:

The first category takes no consideration of the membership function of fuzzy numbers,

such as the distance used by Tong [14] and Hausedorff metric. The advantages of these methodsare the convenience of operating and ease of understanding. However, it will lead to the loss ofinformation for the negligence of the supports of fuzzy numbers.

The second category is characteristic of taking the membership function into account. Inthese methods, the Hamming metric and the Euclidean metric are most often used. For anytwo fuzzy numbersA andB with membership functions A and B, respectively, we have (see[15]) the Hamming distancedH( A,B)

dH( A,B) = x

| A(x) B(x)|dx (7)and the Euclidean distancedE( A,B)

dE( A,B) = x( A(x) B(x))2dx. (8)These formulas are straightforward generalizations of distances with the membership functions.However, these methods do not always operate. In the case that the intersection between twofuzzy numbers is empty, no conclusion will be drawn. Let us see the following example.


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Example 3.2 There are three fuzzy numbersA = (1, 2.5, 3.5, 4),B = (5, 6, 7, 8),C =(9, 10, 11, 12). Using the Hamming distance Eq. (7), the distance betweenA andB and thedistance betweenA andCare calculated as follows:

dH(

A,

B) = SA+ SB, dH(

A,

C) =SA+ SC,

whereSA, SB, andSCimply the areas ofA,B, andC, respectively. Since SB =SC, it follows:dH( A,B) = dH( A,C).

By analogy, we have dE( A,B) = dE( A,C).But it is obvious that the distance betweenA andB should be smaller than the distance

betweenA andC. To overcome this disadvantage, we introduce a distance between two sets Aand B

dinf(A, B) = inf{d(a, b), a A, b B}, (9)

where d is the usual metric. For trapezoidal fuzzy numbersA = (a1, a2, a3, a4) andB =(b1, b2, b3, b4),

dinf( A,B) = inf{d(a, b), a [a1, a4], b [b1, b4]}. (10)Considering the Hamming distance is a linear distance and is easy to operate, now we define

a new distance measure between fuzzy numbersAandB on the basis of the Hamming distancedH( A,B) and the distance dinf( A,B) as follows:D( A,B) = 1

2

dH( A,B) + dinf( A,B)= 1

2

x

| A(x) B(x)|dx+ dinf( A,B). (11)Note that this distance can be regarded as the Hamming distance when dinf( A,B) = 0, butit does overcome the shortcoming of the Hamming distance. To illustrate the efficiency of thisdistance method, now consider Example 3.2 again. By our distance measure Eq. (11), we have

D( A,B) = 12

(SA+ SB+ 1), D( A,C) =12 (SA+SC+ 5)> D( A,B).This coincides with our intuition.

For the fuzzy numbersRi=(ai, bi, ci, di)(i = 1, 2, , n) of each experts opinion, we cal-culate the distance d( Ri,Rj) between each pair ofRi andRj , then select the largest distanceD( Rp,Rq) = max

i,jD( Ri,Rj). We divide each distance by D( Rp,Rq) so that we obtain the

following normalized distance:

d( Ri,Rj) = D( Ri,Rj)D( Rp,Rq) . (12)

This distance measure fulfills the following properties:(DP1) 0 d( Ri,Rj) 1;(DP2) d( Ri,Rj) = 0 if and only ifRi=Rj ;(DP3) d(

Ri,

Rj) = d(

Rj ,

Ri);

(DP4) d Ri,Rj) d( Ri,Rk) and d( Rj ,Rk) d( Ri,Rk) ifRiRj Rk.3.3 Aggregation Method

LetR1,R2, ,Rnbe nfuzzy numbers representing each experts opinion for an alternativeunder a given criterion. Since we have obtained the similarity Sw( Ri,Rj) and the distance


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d( Ri,Rj) fori, j = 1, 2, , n, now we define a new consistency measure between fuzzy opinionsRi andRj :r( Ri,Rj) = Sw( Ri,Rj) + (1 )(1 d( Ri,Rj)), (13)

where [0, 1] is the weight of Sw(

Ri,

Rj), which reflects the relative importance degree

between the similarity and the distance with respect to the decision maker. 1 is the weight

ofd( Ri,Rj). The basic idea of this aggregation method is that the larger Sw( Ri,Rj) and thesmaller d( Ri,Rj), the larger consistency degree r( Ri,Rj) between fuzzy opinionsRi andRj .Since r( Ri,Rj) is the linear combination ofSw( Ri,Rj) and d( Ri,Rj), it follows

(RP1) 0 r( Ri,Rj) 1;(RP2) r( Ri,Rj) = 1 if and only ifRi=Rj ;(RP3) r( Ri,Rj) = r( Rj ,Ri);(RP4) r Ri,Rj) r( Ri,Rk) and r( Rj ,Rk) r( Ri,Rk) ifRiRj Rk.In practice, group decision-making is highly influenced by the degrees of the importance

of participants. For example, there are some experts such as the managers of a companywith authority and some experts who are more experienced than the others. So an effectiveaggregation method should consider the relative importance weight of each expert. For no lossof generality, let the degree of importance ofith expert be ei (0 ei 1), and

ni=1

ei = 1. (14)

Then the weighted consistency degree of each expert Ei is given as

C(Ei) =nj=1

r( Ri,Rj)ej . (15)Thus the aggregation weight of each expert Ei is calculated by

w(Ei) = C(Ei)n

j=1C(Ej)

. (16)

The aggregation result is therefore defined as

R= F( R1,R2, ,Rn) = ni=1

w(Ei)Ri, (17)

where

is the fuzzy multiplication operator[11].Now we summarize our consistency aggregation method (CAM) and give an algorithm.

3.4 Algorithm CAM

Step 1: For each expert Ei(i= 1, 2, , n), he/she constructs a fuzzy numberRi(i= 1, 2, , n)to represent his/her opinion to the alternative under a given criterion.

Step 2: Calculate the similarity Sw( Ri,Rj) between each pair of experts by Eq. (6).Step 3: Calculate the distance d( Ri,Rj) between each pair of experts by Eq. (12).Step 4: Given [0, 1], calculate the consistency degree r( Ri,Rj) between each pair of experts

by Eq. (13).

Step 5: Select the degree of importance ei of each expert Ei, then calculate the weighted con-sistency degree C(Ei) of each expert Ei by Eq. (15).


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Step 6: Calculate the aggregation weight w(Ei) of expert by Eq. (16).Step 7: Aggregate each fuzzy opinion into a group fuzzy opinion by Eq. (17).

Our consistency aggregation method (CAM) also preserves some important properties asfollows.

Property 3.1 (Agreement preservation[1]) If

Ri =

Rj for all i, j, then

R=

Rj. That is,

if all opinions of experts are identical the aggregation result is the common opinion.

Property 3.2(Order independence[1]

) The result of CAM would not be influenced by thedifferent order with which individual opinions are combined. That is, if{(1), (2), , (n)} is apermutation of{1, 2, , n}, thenR= F( R1,R2, ,Rn) =F( R(1),R(2), ,R(n)).

Property 3.1 and Property 3.2 are consistency equipments.Property 3.3 Let the uncertainty measure H( Ri) of individual opinionRi be defined as

the area under its membership function[1]

H( Ri) = +

Ri(x)dx, (18)then the uncertainty measure defined in Eq. (18) satisfies the following equation:

H( R) = ni=1

w(Ei) H( Ri). (19)This means that the uncertainty of aggregation result is between the uncertainties of all

experts, i.e., mini

H( Ri) H( R) maxi

H( Ri). Referring to Example 4.1, H( R1) = 2, H( R2) =2.25, H( R3) = 2.75, so H( R) = 2.322 is between H( R1) and H( R3).

Property 3.4[8] The common intersection of all experts opinions is concluded in the final

aggregation result. That is, ni=1

RiR.Proof Let cut ofRi beRi = [ai, bi], then n

i=1

Ri = [a, b], wherea = max

iai, b

= mini

bi.

Since R= ni=1

w(Ei)Ri,

it follows R = ni=1

w(Ei)Ri =

ni=1

(w(Ei) ai),

ni=1

(w(Ei) bi)

.

Noticeni=1

(w(Ei) ai) a

= maxi

ai,

ni=1

(w(Ei) bi) b

= mini

bi.

We have proved this property.

Property 3.5 If ni=1

Ri = , the consistent group opinionR also can be derived.When the opinions are disjoint, we can measure the distance d(Ri, Rj) (for all i, j) among

them, then calculate the consistency degree r( Ri,Rj) between each pair of experts by Eq. (13)so that aggregation process can be continued.


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4 Numerical Example

Example 4.1[8] Consider a group decision-making problem with three experts. Theopinions of each expert are given as three positive trapezoidal fuzzy numbers (see Fig. 2):

R1 = (1, 2, 3, 4),

R2 = (1.5, 2.5, 3.5, 5),

R3 = (2, 2.5, 4, 6).

Figure 2 Aggregation result of fuzzy opinionsR1,R2, andR3Now use our consistency aggregation method (CAM) to study this problem.

Step 1: It is given above.Step 2: Calculate the similarity Sw( Ri,Rj) for i, j = 1, 2, 3 as follows:

Sw( R1,R1) = 1, Sw( R1,R2) = 12 , Sw( R1,R3) = 13 ;Sw( R2,R1) = 12 , Sw( R2,R2) = 1, Sw( R2,R3) = 23 ;Sw( R3,R1) = 13 , Sw( R3,R2) = 23 , Sw( R3,R3) = 1.

Step 3: Calculate the distance d(Ri, Rj) for i, j = 1, 2, 3 as follows:

d( R1,R1) = 0, d( R1,R2) = 59 , d( R1,R3) = 1;d( R2,R1) = 59 , d( R2,R2) = 0, d( R2,R3) = 49 ;d(

R3,

R1) = 1, d(

R3,

R2) =

49

, d(

R3,

R3) = 0.

Step 4: Select = 12

, calculate the consistency degreer( Ri,Rj) for i, j = 1, 2, 3 as follows:r( R1,R2) = 1, r( R1,R2) = 0.472, r( R1,R3) = 0.167;r( R2,R1) = 0.472, r( R2,R2) = 1, r( R2,R3) = 0.611;r( R3,R1) = 0.167, r( R3,R2) = 0.611, r( R3,R3) = 1.

Step 5: Let the degrees of importance of three experts be e1 = 0.42, e2 = 0.25, e3 = 0.33,respectively, and we obtain: C(E1) = 0.593, C(E2) = 0.649, C(E3) = 0.553.

Step 6: Calculate the aggregation weight w(Ei) of expert Eias follows: w(E1) = 0.330, w(E2) =0.361, w(E3) = 0.309.

Step 7: Aggregate each fuzzy opinion into a group fuzzy opinion as

R= 3i=1

w(Ei)Ri = (1.489, 2.334, 3.489, 4.979).

Referring to Fig. 2, we can see the second opinionR2 is close to the other opinions, so theaggregation weight ofR2 is the largest, and the consensus opinionR is therefore close toR2. The consistency degree of the opinionR3 is the smallest of three ones so thatR3 is lessimportant, which is fair and reasonable. However, in Hsu and Chens result, the consensusdegree coefficient ofR2 (CDC2 = 0.33) is smaller than CDC1 = 0.34 althoughR2 is closer tothe consensus opinionRthanR1. This does not coincide with peoples intuition. Furthermore,


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the uncertainty of our aggregation result (H( R) = 2.322) is smaller than the uncertainty of Hsuand Chens aggregation result in case 1 (H( R) = 2.341) and case 2 (H( R) = 2.33). The widthof our aggregation result (width( R) = 3.489) is also smaller than the width of Hsu and Chensaggregation result in case 1 (width( R) = 3.519) and case 2 (width( R) = 3.495).5 Conclusions

In this article, we define a similarity measure, and a distance measure is also given to studythe consistency of one expert to the other experts. To deal with the situation when opinions aredisjoint, we introduce a consistency index of each individual opinion on the basis of similarityand distance. The basic idea of our aggregation method is that the aggregation weight of theopinion should be larger if the degrees of similarity between the experts opinion and the otheropinions are larger and the degrees of distance between his opinion and the other opinions aresmaller. The importance of each expert is also considered in the procedure of our method.Finally, a numerical example shows that our method is rather efficient.

References

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