on reciprocity indexes in the aggregation of fuzzy preference relations using the owa operator

Fuzzy Sets and Systems 159 (2008) 185–192www.elsevier.com/locate/fss

On reciprocity indexes in the aggregation of fuzzy preferencerelations using the OWA operator

Yucheng Donga,b,∗, Hongyi Lib, Yinfeng Xua,c

aDepartment of Management Science, Management School, Xi’an Jiaotong University, Xi’an 710049, PR ChinabFaculty of Business Administration, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

cState Key Lab for Manufacturing Systems Engineering, Xi’an 710049, PR China

Received 27 November 2006; received in revised form 19 June 2007; accepted 19 June 2007Available online 28 June 2007

Abstract

Chiclana, Herrera and Herrera-Viedma studied conditions under which the reciprocity property is maintained in the aggregation ofreciprocal fuzzy preference relations using the OWA operator guided by a relative linguistic quantifier. In this note, we focus on thereciprocity in the aggregation of fuzzy preference relations (that is, the additive reciprocity is not assumed) using the OWA operator.By defining a reciprocity index for measuring the non-reciprocity degree of fuzzy preference relations, we show that a sufficient(but not necessary) condition under which a “collective” fuzzy preference relation, obtained by aggregating a set of “individual”fuzzy preference relations using an OWA operator guided by a linguistic quantifier with parameter a and b, has a reciprocity indexno greater than the largest of the corresponding “individual” reciprocity indexes. Our result is helpful to complete Chiclana et al.’sdecision model [F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decisionmaking based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33–48].© 2007 Elsevier B.V. All rights reserved.

Keywords: Fuzzy preference relations; OWA operator; Linguistic quantifier; Reciprocity

1. Introduction

Fuzzy preference relations are widely used in decision making models [5,13,15,16]. Chiclana et al. [1–3] presenteda notable fuzzy multipurpose decision making model (FMDMM), integrating different preference representations:preference orderings, utility functions, reciprocal fuzzy preference relations and Saaty’s multiplicative preferencerelations [14]. In FMDMM, the decision makers’ preferences about the alternatives are represented by means of thereciprocal fuzzy preference relations. In [4], Chiclana et al. discussed conditions under which the reciprocity property ismaintained in their decision model. Herrera et al. [6,7] presented a corresponding multiplicative multipurpose decisionmaking model, and studied the relationship between the FMDMM and the multiplicative multipurpose decision makingmodel.

In this note, we will define an index for measuring the non-reciprocity degree of fuzzy preference relations. The mainaim of this note is to study conditions under which a “collective” fuzzy preference relation, obtained by aggregating a

∗ Corresponding author. Department of Management Science, Management School, Xi’an Jiaotong University, Xi’an 710049, PR China. Tel.:+86 29 82673492.

E-mail address: [email protected] (Y. Dong).

0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2007.06.010

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186 Y. Dong et al. / Fuzzy Sets and Systems 159 (2008) 185–192

set of “individual” fuzzy preference relations using Yager’s OWA operator [17] guided by a relative linguistic quantifier[9,10,18,19], has a reciprocity index no greater than the largest of the corresponding “individual” reciprocity indexes.This study is helpful to complete Chiclana et al.’s decision model, based on the following two reasons:

(i) Taking the multiplicative preference relation as an example, Koczkodaj and Orlowski [11,12] have questioned thereciprocity assumption of individual preference relations (p. 80 in [12]): “all previous papers, reciprocal aji =1/aij response are assumed such that for comparison of alternatives only n(n − 1)/2 judgements are needed. Thereciprocity condition has never been questioned since it is convenient for computing inverses. However, there isa problem when superfluous assessments are allowed. In practical applications, even comparing the same objectto itself may not always yield 1 (e.g., blind testing of DNA samples, blind tasting of wines, etc.).” Based on thisargument, Koczkodaj and Orlowski present the concept of generalized multiplicative pairwise comparison matrices,in which the reciprocity property is not assumed, and discuss a prioritization method for this kind of preferencerelations. In general, the research progresses in multiplicative preference relations can benefit the research in fuzzypreference relations. Basically, the experts’ preferences about the alternatives are represented by means of thereciprocal fuzzy preference relations in FMDMM. However, inspired by the generalized multiplicative preferencerelations, we can also directly use fuzzy preference relations to express individual preferences in FMDMM. In thisway, it becomes necessary to study the aggregation of fuzzy preference relations using the OWA operator.

(ii) Chiclana et al. [4] have shown that the reciprocity property is not generally preserved in the aggregation of fuzzypreference relations using the OWA operator [17]. That is to say, the “collective” preference relations, obtainedby aggregating a set of “individual” reciprocal fuzzy preference relations using an OWA operator guided by alinguistic quantifier, are in general fuzzy preference relations.

This note is organized as follows. In Section 2, we formally define the reciprocity index for fuzzy preference relationsand introduce the decision making problem. In Section 3, we prove a property on reciprocity indexes in FMDMM. InSection 4, an illustrative example is provided. Concluding remarks and future researches are included in Section 5.

2. Presentation of the problem

There are a set of alternatives X = {x1, . . . , xn}, and a set of fuzzy preference relations {P (1), . . . , P (m)}, whereP (k) = (p

(k)ij )n×n, and p

(k)ij represents the preference degree or intensity of alternative xi over alternative xj for expert

ek . In this note, we use the values of |p(k)ij + p

(k)ji − 1| for measuring the non-reciprocity degree between the pairwise

comparison terms p(k)ij and p

(k)ji . To facilitate the description, we define the following reciprocity indexes.

Definition 1. Let P = (pij )n×n be a fuzzy preference relation. We denote RIij (P ) = |pij +pji −1| as the reciprocityindex for measuring the non-reciprocity degree between the pairwise comparison terms pij and pji . Denote RI(P ) =maxi,j |pij + pji − 1| as the reciprocity index of P .

These reciprocity indexes are simple and intuitive. The higher the values of RIij (P ) and RI(P ), the more the measurednon-reciprocity degree. When RIij (P ) = 0, the pairwise comparison terms, pij and pji , are completely reciprocal.When RI(P ) = 0, P is a reciprocal fuzzy preference relation. Naturally, the set of reciprocal fuzzy preference relationsis a subset of the set of fuzzy preference relations.

In [17], Yager defined the OWA operator as follows:

Definition 2. An OWA operator of dimension m is a mapping � : Rm → R with an associated weight vectorw = (w1, w2, . . . , wm)T such that

m∑k=1

wk = 1

and

�(a1, a2, . . . , am) =m∑

k=1

wkbk,

where bk is the kth largest of {a1, a2, . . . , am}.

Y. Dong et al. / Fuzzy Sets and Systems 159 (2008) 185–192 187

In [18], Yager suggested an interesting way to compute the weights (i.e., wk, k = 1, 2, . . . , m) of the OWA operatorusing linguistic quantifiers, which, in the case of a non-decreasing proportional quantifier Q, is given by this expression:

wk = Q(k/m) − Q((k − 1)/m) ∀k

being the membership function of a non-decreasing proportional quantifier Q, as follows:

Q(x) =

⎧⎪⎨⎪⎩

0, 0�x < a,x − a

b − a, a�x�b,

1, b < x�1,

with a, b ∈ [0, 1]. When it is used a fuzzy linguistic quantifier Q to compute the weights of the OWA operator �, it issymbolized by �Q.

Using an OWA operator �Q, we derive a collective preference relation, P c = (pcij )n×n indicating the global

preference between every pair of alternatives according to the majority of experts opinions, which is represented by Q.In this case,

pcij = �Q(p

(1)ij , . . . , p

(k)ij , . . . , p

(m)ij ) =

m∑k=1

wkq(k)ij ,

where q(k)ij is the kth largest value in the set {p(1)

ij , . . . , p(m)ij }.

If all of P (1), . . . , P (m) are reciprocal fuzzy preference relations, Chiclana et al. [4] give conditions under which P c

is a reciprocal fuzzy preference relation (see Theorem 1).

Theorem 1 (Chiclana et al. [4]). A “collective” fuzzy preference relation P c, obtained by aggregating reciprocal fuzzypreference relations P (1), . . . , P (m) using the OWA operator �Q, is reciprocal iff a + b = 1.

Chiclana et al. also show that a + b = 1 is the necessary and sufficient condition to ensure that P c is reciprocal.Thus, the reciprocity property of P c is not generally preserved in the aggregation using the OWA operator guided bya relative linguistic quantifier. That is to say, P c is a fuzzy preference relation.

In the following section, we will show that the reciprocity index of P c can be maintained under the conditiona + b = 1, i.e.,

a + b = 1 �⇒ RI (P c)� mmax

kRI (P (k)).

It is obvious that the above result is a generalization of Theorem 1. The main aim of this note is to prove the result.Additionally, we will show that a + b = 1 is not a necessary condition to ensure that RI (P c)� maxm

k RI (P (k)).

3. On reciprocity indexes in FMDMM

In this section, we will introduce a property on reciprocity indexes in the aggregation of fuzzy preference relationsusing the OWA operator guided by a relative linguistic quantifier. Before presenting this property, we introduce thefollowing lemmas.

Lemma 1. For any real numbers x1, x2, y1, y2, max{|x1 + y1 − 1|, |x2 + y2 − 1|}� max{|x1 + y2 − 1|, |x2 + y1 − 1|}if x1 �x2 and y1 �y2.

Proof. We distinguish four cases, according to the signs of x1 + y2 − 1 and x2 + y1 − 1.Case A: x1+y2−1�0 and x2+y1−1�0. In this case, because x1 �x2 and y1 �y2, we have x1+y1−1�x1+y2−1�0

and x1 +y1 −1�x2 +y1 −1�0. This implies that max{|x1 +y1 −1|, |x2 +y2 −1|}� max{|x1 +y2 −1|, |x2 +y1 −1|}.Case B: x1 + y2 − 1�0 and x2 + y1 − 1 < 0. In this case, we easily have x1 + y1 − 1�x1 + y2 − 1�0 (i.e.,

|x1 + y1 − 1|� |x1 + y2 − 1|) and x2 + y2 − 1�x2 + y1 − 1 < 0 (i.e., |x2 + y2 − 1|� |x2 + y1 − 1|) when y1 �y2.This also implies that max{|x1 + y1 − 1|, |x2 + y2 − 1|}� max{|x1 + y2 − 1|, |x2 + y1 − 1|}.


Case C: x1 + y2 − 1 < 0 and x2 + y1 − 1�0. In this case, it can be obtained that x2 + y2 − 1�x1 + y2 − 1 < 0(i.e., |x2 + y2 − 1|� |x1 + y2 − 1|) and x1 + y1 − 1�x2 + y1 − 1�0 (i.e., |x1 + y1 − 1|� |x2 + y1 − 1|) when x1 �x2.Then, we have max{|x1 + y1 − 1|, |x2 + y2 − 1|}� max{|x1 + y2 − 1|, |x2 + y1 − 1|}.

Case D:x1+y2−1 < 0 andx2+y1−1 < 0. In this case, becausex1 �x2 andy1 �y2, we havex2+y2−1�x1+y2−1 <

0 and x2+y2−1�x2+y1−1 < 0. This implies that max{|x1+y1−1|, |x2+y2−1|}� max{|x1+y2−1|, |x2+y1−1|}.Summarizing, we have completed this proof. �

Let M = {P (k) = (p(k)ij )n×n|k = 1, 2, . . . , m} be a set of fuzzy preference relations. Let N = {Q(k) =

(q(k)ij )n×n|k = 1, 2, . . . , m} be another set of fuzzy preference relations, where q

(k)ij (i�j) is the kth largest value

in the set {p(1)ij , p

(2)ij , . . . , p

(m)ij } and q

(k)ij (i > j) is the kth smallest value in the set {p(1)

ij , p(2)ij , . . . , p

(m)ij }. Here, we

introduce an algorithm to transform M into N .

ALGORITHMInput: M = {P (k) = (p

(k)ij )n×n|k = 1, 2, . . . , m};

Output: N = {Q(k) = (q(k)ij )n×n|k = 1, 2, . . . , m};

Step 1: Let � = 1 and s = 0. Let F (s) = {F (k),(s) = (f(k),(s)ij )n×n|k = 1, 2, . . . , m}, where f

(k),(s)ij = p

(k)ij ;

Step 2: Let � = 1;Step 3: Let

f(�),(s+1)

ij ={

max{f (�),(s)

ij , f(�+1),(s)

ij } for i�j,

min{f (�),(s)

ij , f(�+1),(s)

ij } for i > j,

f(�+1),(s+1)

ij ={

min{f (�),(s)

ij , f(�+1),(s)

ij } for i�j,

max{f (�),(s)

ij , f(�+1),(s)

ij } for i > j,

and

f(k),(s+1)ij = f

(k),(s)ij for k �= �, � + 1.

Update F (s+1) = {F (k),(s+1) = (f(k),(s+1)ij )n×n|k = 1, 2, . . . , m};

Step 4: Let s = s + 1;Step 5: If ��m − �, then � = � + 1 and go to step 3; otherwise go to next step;Step 6: If � < m, then � = � + 1 and go to step 2; otherwise go to next step;Step 7: Let N = F (s).

Note. This algorithm is based on the bubble sort algorithm. Using the algorithm, we can generate the sequence{F (s)|s = 0, 2, . . . , m(m − 1)/2}, where F (s) = {F (k),(s) = (f

(k),(s)ij )n×n|k = 1, 2, . . . , m}. The main purpose of

introducing this algorithm is to prove Lemma 2, by showing maxmk=1 {RI (F (k),(s))}� maxm

k=1 {RI (F (k),(s+1))}.

Lemma 2.

mmaxk=1

{RI (P (k))}� mmaxk=1

{RI (Q(k))}.

Proof. We have M = F (0) and N = F (m(m−1)/2) from the above algorithm. This means that Lemma 2 can beobtained by showing maxm

k=1 {RI (F (k),(s))}� maxmk=1 {RI (F (k),(s+1))} for s = 0, 1, . . . , m(m − 1)/2 − 1. Let

E(s) = (F (s) ∪ F (s+1)) − (F (s) ∩ F (s+1)). According to the above algorithm, We have E(s) = � or E(s) ={F (�),(s), F (�+1),(s), F (�),(s+1),

F (�+1),(s+1)}, where 1��m − 1. Thus, we consider two cases:Case A: E(s) = �. In this case F (s) = F (s+1). Thus, maxm

k=1 {RI (F (k),(s))} = maxmk=1 {RI (F (k),(s+1))}.


Case B: E(s) = {F (�),(s), F (�+1),(s), F (�),(s+1), F (�+1),(s+1)}, where 1��m. According to the above algorithm,we have

f(�),(s+1)ij =

{max{f (�),(s)

ij , f(�+1),(s)ij } for i�j,

min{f (�),(s)ij , f

(�+1),(s)ij } for i > j

and

f(�+1),(s+1)ij =

{min{f (�),(s)

ij , f(�+1),(s)ij } for i�j,

max{f (�),(s)ij , f

(�+1),(s)ij } for i > j.

We continue to consider two subcases:Subcase B1: (f

(�),(s)ij − f

(�+1),(s)ij ) × (f

(�),(s)j i − f

(�+1),(s)j i )�0 for i�j . Without loss of generality, we assume that

f(�),(s)ij �f

(�+1),(s)ij and f

(�),(s)j i �f

(�+1),(s)j i for i�j . Then we have that

f(�),(s+1)ij = f

(�),(s)ij ,

f(�),(s+1)j i = f

(�+1),(s)j i ,

f(�+1),(s+1)ij = f

(�+1),(s)ij

and

f(�+1),(s+1)j i = f

(�),(s)j i for i < j.

Thus

max{RIij (F(�),(s)), RIij (F

(�+1),(s))} = max{|f (�),(s)ij + f

(�),(s)j i − 1|, |f (�+1),(s)

ij + f(�+1),(s)j i − 1|}

and

max{RIij (F(�),(s+1)), RIij (F

(�+1),(s+1))} = max{|f (�),(s)ij + f

(�+1),(s)j i − 1|, |f (�+1),(s)

ij + f(�),(s)j i − 1|}.

By Lemma 1, it can be shown that

max{|f (�),(s)ij + f

(�+1),(s)j i − 1|, |f (�+1),(s)

ij + f(�),(s)j i − 1|} � max{|f (�),(s)

ij + f(�),(s)j i

−1|, |f (�+1),(s)ij + f

(�+1),(s)j i − 1|}.

Consequently,


(�+1),(s+1))}� max{RIij (F(�),(s)), RIij (F

(�+1),(s))}.

Subcase B2: (f(�),(s)ij − f

(�+1),(s)ij ) × (f

(�),(s)j i − f

(�+1),(s)j i ) < 0 for i�j . Without loss of generality, we assume that

f(�),(s)ij > f

(�+1),(s)ij and f

(�),(s)j i < f

(�+1),(s)j i for i < j . Then we have that

f(�),(s+1)ij = f

(�),(s)ij ,

f(�),(s+1)j i = f

(�),(s)j i ,

f(�+1),(s+1)ij = f

(�+1),(s)ij


and

f(�+1),(s+1)j i = f

(�+1),(s)j i for i < j.

Consequently,


(�+1),(s+1))} = max{RIij (F(�),(s)), RIij (F

(�+1),(s))}.

Summarizing subcases B1 and B2, we have that


(�+1),(s+1))}� max{RIij (F(�),(s)), RIij (F

(�+1),(s))} for i�j.

Consequently,

max{RI (F (�),(s+1)), RI (F (�+1),(s+1))}� max{RI (F (�),(s)), RI (F (�+1),(s))}.

Since max{RI (F (k),(s))| k = 1, 2, . . . , �−1, �+1, . . . , m} = max{RI (F (k),(s+1))| k = 1, 2, . . . , �−1, �+1, . . . , m}in case B, we have

max{RI (F (k),(s))| k = 1, 2, . . . , m}� max{RI (F (k),(s+1))| k = 1, 2, . . . , m}.

Summarizing cases A and B, we have completed this proof. �

Lemma 3 (Chiclana et al. [4]). wk = wm+1−k if a + b = 1.

Based on Lemmas 1, 2 and 3, we present a property on reciprocity indexes in FMDMM (see Theorem 2).

Theorem 2. RI (P c)� maxmk=1 {RI (P (k))} if a + b = 1.

Proof. From the definition of the OWA operator and Definition 1, we have

RIij (Pc) =

∣∣∣∣∣m∑

k=1

wkq(k)ij +

m∑k=1

wm+1−kq(k)j i − 1

∣∣∣∣∣ .If a + b = 1, by Lemma 3, we have that wk = wm+1−k . Consequently,

RIij (Pc) =

∣∣∣∣∣m∑

k=1

wkq(k)ij +

m∑k=1

wkq(k)ji − 1

∣∣∣∣∣�

m∑k=1

(wk|q(k)ij + q

(k)ji − 1|)

� mmaxk=1

|q(k)ij + q

(k)ji − 1|.

This implies that

RI (P c) = maxi,j

Rij (Pc)� m

maxk=1

(maxi,j

|q(k)ij + q

(k)ji − 1|

)= m

maxk=1

{RI (Q(k))}.

From Lemma 2, we have that RI (P (c))� maxmk=1 {RI (P (k))} if a + b = 1. This completes the above proof. �


Table 1The results of the illustrative example

P1 P2 P3 P4 P c P c′

RI (•) 0.1788 0.1880 0.1974 0.1827 0.1305 0.1764

4. Illustrative example

In order to show how these theoretical results work in practice, let us consider the following example. In the example,there are four fuzzy preference relations P1, P2, P3 and P4. They are listed as follows:

P (1) =

⎛⎜⎜⎝

0.4807 0.4734 0.5660 0.38470.6832 0.4916 0.9033 0.06500.2553 0.0203 0.5713 0.13600.7573 0.8792 0.7552 0.4248

⎞⎟⎟⎠ , P (2) =

⎛⎜⎜⎝

0.5716 0.1613 0.4497 0.76660.8832 0.4712 0.3965 0.50930.6431 0.4807 0.5934 0.27690.0454 0.6248 0.5788 0.4338

⎞⎟⎟⎠ ,

P (3) =

⎛⎜⎜⎝

0.4334 0.8460 0.8173 0.65980.1724 0.5327 0.4377 0.96360.2346 0.5918 0.4883 0.33150.5376 0.0201 0.5752 0.5023

⎞⎟⎟⎠ , P (4) =

⎛⎜⎜⎝

0.4086 0.1658 0.6434 0.74390.8282 0.5569 0.9646 0.61680.3223 0.0135 0.5689 0.82830.2987 0.5184 0.3184 0.4453

⎞⎟⎟⎠ .

Without loss of generality, suppose that a = 0.2 and b = 0.8, and the collective preference relation of P1, P2, P3and P4 is P c. We also consider the case of a = 0.3 and b = 0.8. Then the collective preference relation of P1, P2, P3and P4 is P c′

.

P c =

⎛⎜⎜⎝

0.4626 0.3503 0.6095 0.68080.7177 0.5125 0.6722 0.55490.3138 0.2592 0.5652 0.33380.4153 0.5513 0.5703 0.4435

⎞⎟⎟⎠ , P c′ =

⎛⎜⎜⎝

0.4499 0.2884 0.5853 0.66590.6901 0.5060 0.6198 0.50780.2800 0.2038 0.5618 0.28460.3689 0.5111 0.5510 0.4375

⎞⎟⎟⎠ .

Table 1 shows the results of the illustrative example. We find that RI (P c)� max{RI (P (k))|k = 1, 2, . . . , m} (seeTable 1), which is in accordance with Theorem 2. At the same time, we also find that RI (P c′

)� max{RI (P (k))|k =1, 2, . . . , m}, which shows that a + b = 1 is not a necessary condition to ensure that RI (P c)� max{RI (P (k))|k =1, 2, . . . , m}.

5. Conclusions and future research

In this paper, we first introduce a reciprocity index of fuzzy preference relations, and then discuss a property onreciprocity indexes in the aggregation of fuzzy preference relations using the OWA operator. Let P (k) = (p

(k)ij )n×n, k =

1, 2, . . . , m be m fuzzy preference relations. Using the OWA operator guided by a linguistic quantifier Q with theparameters a and b, we derive the collective preference relations P c from P (k) = (p

(k)ij )n×n, k = 1, 2, . . . , m. We have

shown that the reciprocity index of P c is no greater than the largest reciprocity indexes among {P (k) = (p(k)ij )n×n|k =

1, 2, . . . , m} under the condition that a + b = 1. This result is important to further complete the Chiclana et al.’sdecision model.

Inspired by using the three-way transitivity [8] to characterize the consistency of fuzzy preference relations, we cancharacterize reciprocity by using the two-way transitivity, in the sense that if an alternative xi is preferred to alternativexj (i.e., pij �0.5), then alternative xj should be inferior to xi (i.e., pji �0.5). Obviously, the two-way transitivity is aweaker condition than the additive reciprocal property. In our future research, we will study conditions under which thetwo-way transitivity (i.e., (pij − 0.5)(pji − 0.5)�0, ∀i, j ) is maintained when aggregating fuzzy preference relationsusing an OWA operator guided by a relative linguistic quantifier.


Acknowledgements

We are very grateful to the editor and the anonymous referees for their valuable comments and suggestions. Moreover,Yucheng Dong and Yinfeng Xu would like to acknowledge the financial support of grants (no. 70121001, 70471035and 70525004) from NSF of China. And, Hongyi Li would like to acknowledge the financial support of a grant (no.CUHK4443/04H) from the Research Grants Council of the Hong Kong Special Administrative Region, China.

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