Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

Journal of Intelligent&Fuzzy Systems27(2014)2969–2985

DOI:10.3233/IFS-141256

IOS Press

2969

Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

Huchang Liao a,Zeshui Xu b,?and Meimei Xia c

a Antai College of Economics and Management,Shanghai Jiao Tong University,Shanghai,China

b Business School,Sichuan University,Chengdu,Sichuan,China

c Institute of Information System,Beijing Jiaotong University,Beijing,China

Abstract.Interval-valued intuitionistic fuzzy preference relation(IVIFPR)is an important structure in representing fuzzy informa-tion comprehensively.This paper focuses on the multiplicative consistency of the IVIFPR.Some concepts,such as the approximate multiplicative consistent IVIFPR,the perfect multiplicative consistent IVIFPR and the acceptable multiplicative consistent IVIFPR are de?ned.Then,a desirable property of multiplicative consistent IVIFPR is investigated.Two algorithms are developed to con-struct the approximate/perfect multiplicative consistent IVIFPR.Since inconsistent IVIFPR is common but unreasonable in deriving the priorities of an IVIFPR,an iterative procedure is proposed to improve the consistency of an inconsistent IVIFPR. Furthermore,a convergent approach is developed for group decision making with IVIFPRs.Several numerical examples are given to illustrate the validity and applicability of the algorithms and procedures.

Keywords:Group decision making,interval-valued intuitionistic fuzzy preference relation,multiplicative consistency

1.Introduction

Intuitionistic fuzzy set(IFS)[1]has turned out to be powerful in handling vagueness and uncertainty in decision making.It is characterized by a membership function,a nonmembership function and a hesitancy function.Since IFS was conceived to alleviate some of the drawbacks of FS,many scholars have paid great attention to it[2–12].However,due to the complex-ity and uncertainty of modern society,people may not express the membership degree or the non-membership degree or hesitancy degree in exact number but in value ranges.To model such situation,Atanassov and Gargov [13]generalized IFS into interval-valued intuitionistic fuzzy set(IVIFS),which is more suitable for describ-ing uncertain evaluation information.Many researchers have investigated IVIFS in the setting of decision mak-ing[14–19].Atanassov[14]de?ned some different ?Corresponding author.Zeshui Xu,Business School,Sichuan University,Chengdu610064,China.E-mails:xuzeshui@https://www.360docs.net/doc/f110456803.html,; liaohuchang@https://www.360docs.net/doc/f110456803.html,(Huchang Liao).operators over IVIFSs and investigated their properties. By introducing the interval-valued intuitionistic judg-ment matrix and its score matrix and accuracy matrix, Xu and Chen[15]developed an approach to group decision making(GDM)with interval-valued intuition-istic judgment matrices.As for the incomplete IVIFPRs whose preference information of the decision makers is partly unknown,Xu and Cai[16]proposed two pro-cedures to extend acceptable incomplete IVIFPRs to complete IVIFPRs.Xu and Yager[17]proposed a sim-ilarity measure to depict the degree of agreement among group of experts with IVIFPRs.When dealing with multi-attribute decision making(MADM)problems where individual assessments are provided as IVIFSs, the weight information of attributes may be incomplete. To solve this problem,Wang et al.[18]derived a lin-ear model to determine the weights of attributes.Chen et al.[19]determined the relative importance of criteria with IVIFPR by using an optimization model,and then calculated the aggregated value for each alternative in a MADM problem.

1064-1246/14/$27.50?2014–IOS Press and the authors.All rights reserved

2970H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

In the process of decision making,expert usually expresses his/her preferences by comparing each pair of alternatives and then constructs a preference rela-tion.Preference relation is the most common tool in representing people’s preference information.Up to now,several types of preference relations have been introduced,including multiplicative preference rela-tion[20],fuzzy preference relation[21],linguistic preference relation[22],intuitionistic fuzzy prefer-ence relation(IFPR)[4,7],interval-valued intuitionistic fuzzy preference relation(IVIFPR)[15],intuitionistic multiplicative preference relation[23],hesitant pref-erence relation[24],and so on[25].Since this paper focuses on the IFPR and IVIFPR,below we only roughly review these two kinds of preference relations. Xu[4]de?ned the concepts of IFPR and incomplete IFPR.Xu et al.[5]developed some algorithms to esti-mate the missing elements of an incomplete IFPR.Liao and Xu[6]proposed the IF-PROMETHEE method based on the IFPR.Furthermore,Xu and Liao[7]devel-oped the intuitionistic fuzzy AHP method.A novel de?nition of multiplicative consistency for IFPR was proposed by Liao and Xu[8].The IFPR was also imple-mented into GDM circumstances[9–12].The concept of IVIFPR was introduced by Xu and Chen[15].They also gave an approach to GDM with IVIFPRs.Xu and Cai[16]discussed the incomplete IVIFPRs.The inves-tigation on consistency of a preference relation is an important research topic in decision https://www.360docs.net/doc/f110456803.html,ck-ing of consistency of a preference relation may lead to unreasonable results[8,24].Moreover,in practi-cal application,perfect consistent preference relation is too hard to construct,especially when the number of alternatives is too large.

Some scholars have paid attention to the consistency of a preference relation.The consistency can be per-formed to ensure pairwise comparisons be logical rather than random.Saaty[20]?rstly derived a consistency ratio from the maximum eigenvalue of a multiplicative preference relation,and pointed out that a multiplica-tive preference relation is of acceptable consistency if its consistency ratio is less than0.1.He also presented that it is dif?cult to obtain such a preference relation, especially when it has a high order.Hence,some soft consistent methods were proposed.As to IFPR,Szmidt and Kacprzyk[9]de?ned the intuitionistic fuzzy core and the consensus winner,and aggregated the individual IFPRs into a social fuzzy preference relation by using the fuzzy majority rule equated with a fuzzy linguistic quanti?er.Liao and Xu[8]proposed the multiplicative consistency of IFPR based on the membership degree and nonmembership degree of each intuitionistic fuzzy preference directly.With regard to the IVIFPR,as far as we known,little research has been done on this topic.In order to derive a scienti?c and ef?cient decision result, it is necessary to develop some methods to improve the consistency of an IVIFPR.In this paper,we shall focus on this topic and investigate the consistency of an IVIFPR and give some algorithms to improve the consistency of an IVIFPR.Moreover,an approach for GDM with IVIFPRs will be investigated.

The rest of this paper is set out as follows:Sec. 2reviews some basic concepts and studies the prop-erty of IVIFPR.Sec.3proposes two algorithms to obtain the perfect or approximate multiplicative con-sistent IVIFPR.An iterative algorithm is also given to improve the inconsistent IVIFPR.In Sec.4,a conver-gent approach is developed for GDM with IVIFPRs. The paper ends in Sec.5.

2.Multiplicative consistency of IVIFPR

In the process of decision making,expert is usually needed to provide his/her preferences over alternatives according to pairwise comparisons and then constructs a preference relation.In addition,it is possible that he/she is not sure about the evaluation values.In such cases,it is suitable to express the preferences in interval-valued intuitionistic fuzzy values and then construct an IVIFPR.Xu and Chen[15]gave the de?nition of IVIFPR as follows:

De?nition 1.[15]Let?R=

?r(x i,x j)

n×n be a preference relation on the set X,with ?r(x i,x j)=(?μ(x i,x j),?v(x i,x j),?π(x i,x j))for all i,j=1,2,...,n.?μ(x i,x j)denotes the degree range that expert prefers alternative x i to x j,while?v(x i,x j) means the degree range that expert prefers alternative x j to x i.?π(x i,x j)is called degree range of indeter-minacy or hesitancy.For simplicity,denote?μ(x i,x j),?v(x i,x j),?π(x i,x j),?r(x i,x j)and?R(X)as?μij,?v ij,?πij,?r ij and?R,respectively,i.e.,?R=(?r ij)n×n with ?r ij=(?μij,?v ij,?πij)(i,j=1,2,...,n)and satis?es:?μij=[?μL ij,?μU ij]?[0,1],?v ij=[?v L ij,?v U ij]?[0,1],?πij=[?πL ij,?πU ij]?[0,1],?μji=?v ij,?v ji=?μij,

?πL ij=1?(?μU ij+?v U ij),?πU ij=1?(?μL ij+?v L ij),

?μii=?v ii=[0.5,0.5],μU ij+v U ij≤1,

i,j=1,2,...,n

(1)

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation2971

whereμL ij andμU ij indicate,respectively,the lower and upper bounds of?μij,v L ij and v U ij indicate,respectively, the lower and upper bounds of?v ij.Then?R is called an

interval-valued intuitionistic fuzzy preference relation (IVIFPR).

In particular,if?μL ij=?μU ij=μij,?v L ij=?v U ij=v ij and ?πL ij=?πU ij=πij,for all i,j=1,2,...,n,then the IVIFPR?R reduces to the IFPR R=(r ij)n×n,with r ij= (μij,v ij,πij).Ifμij+v ij=1,for all i,j=1,2,...,n, the IFPR R reduces to a fuzzy preference relation. De?nition2.[16]Let?R=(?r ij)n×n be an IVIFPR, where?r ij=(?μij,?v ij,?πij)(i,j=1,2,...,n),then?R is called an incomplete IVIFPR,if some of its elements are unknown,and the others can be determined by the expert,which satisfy(1).Let? be the set of all known elements in?R.

Considering the relations among all the three compo-nents:?πL ij=1?(?μU ij+?v U ij)and?πU ij=1?(?μL ij+?v L ij), we here and also thereafter denote?r ij by its two former components(?μij,?v ij)for brevity.

As presented in the Introduction,the consistency of an IVIFPR is very important in deriving a reasonable result for a decision making problem.Two questions are raised regarding to the consistency of an IVIFPR, which are:(1)how to judge whether the IVIFPR is con-sistent or not;(2)how to adjust or repair the inconsistent IVIFPR until it is of acceptable consistency.

As to the?rst problem,the concept of consistency has been traditionally de?ned in terms of different transitivity conditions,such as weak transitivity, max-max transitivity,max-min transitivity,restricted max-min transitivity,restricted max-max transitivity, additive transitivity,and multiplicative transitivity[8, 26].Among these transitivity properties,the weak tran-sitivity is the minimum requirement condition to?nd out whether a preference relation is consistent or not. The max-max transitivity is better than the max-min transitivity;however,the max-max transitivity can’t be veri?ed under reciprocity.Neither the restricted max-min transitivity nor the restricted max-max transitivity implies reciprocity.Both the additive transitivity and the multiplicative transitivity imply reciprocity[7,8]. Nonetheless,the additive transitivity is in con?ict with the[0,1]scale used for providing the preference values [23].Therefore,in this paper,we investigate the con-sistency of an IVIFPR from the point of multiplicative transitivity.

To simplify the presentation,let us start our discussion from the study of fuzzy preference relation,a type of easy but widely used preference structure in decision making[21].The fuzzy preference rela-tion can be denoted as C=(c ij)n×n,where c ij∈[0,1], c ij+c ji=1,c ii=0.5,for i,j=1,2,...,n.The mul-

tiplicative transitivity[26]is an important property of C=(c ij)n×n,shown as:

c ij c jk c ki=c ik c kj c ji(2)

where c ij denotes a ratio of preference intensity for the alternative x i to that for x j,in another words,x i is c ij times as good as x j,and c ij∈[0,1],for all i,j=1,2,...,n.

In case where(c ik,c kj)/∈{(0,1),(1,0)},Equation (2)is equivalent to the following[27]:

c ij=

c ik c kj

c ik c kj+(1?c ik)(1?c kj)

(3)

and if(c ik,c kj)∈{(0,1),(1,0)},then we stipulate c ij=0.

Motivated by the above idea,the concept of multi-plicative transitivity for IVIFPR is introduced:

De?nition 3.Let?R=(?r ij)n×n be an IVIFPR, where?r ij=(?μij,?v ij),?μij=[μL ij,μU ij]?[0,1],?v ij= [v L ij,v U ij]?[0,1],and?μii=?v ii=[0.5,0.5],for i,j=1,2,...,n.Then?R is called of multiplicative transitivity,if

μL ij=

?

???

???

0,(μL ik,μL kj)∈{(0,1),(1,0)}

μL ikμL kj

μL ikμL kj+(1?μL ik)(1?μL kj)

,otherwise, for all i

?

???

???

0,(μU ik,μU kj)∈{(0,1),(1,0)}

μU ikμU kj

μU ikμU kj+(1?μU ik)(1?μU kj)

,otherwise

, for all i

?

???

???

0,(v L ik,v L kj)∈{(0,1),(1,0)}

v L ik v L kj

v L ik v L kj+(1?v L ik)(1?v L kj)

,otherwise

, for all i

2972H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

v U ij=?

???

???

0,(v U ik,v U kj)∈{(0,1),(1,0)}

v U ik v U kj

v U ik v U kj U ik)(1?v U kj)

,otherwise

,

for all i

De?nition4.If?R has the property of multiplicative transitivity,we say?R is a perfect multiplicative consis-tent IVIFPR.

Theorem1.For an IVIFPR?R,Equations(4)–(7)are equivalent to the following:

?μL ij=

j?i?1

j?1

k=i+1

?μL ik?μL kj

j?i?1

j?1

k=i+1

?μL ik?μL kj+j?i?1

j?1

k=i+1

(1??μL ik)(1??μL kj)

,i

?μU ij=

j?i?1

j?1

k=i+1

?μU ik?μU kj

j?i?1

j?1

k=i+1

?μU ik?μU kj+j?i?1

j?1

k=i+1

(1??μU ik)(1??μU kj)

,i

?v L ij=

j?i?1

j?1

k=i+1

?v L ik?v L kj

j?i?1

j?1

k=i+1

?v L ik?v L kj+j?i?1

j?1

k=i+1

(1??v L ik)(1??v L kj)

,i

?v U ij=

j?i?1

j?1

k=i+1

?v U ik?v U kj

j?i?1

j?1

k=i+1

?v U ik?v U kj+j?i?1

j?1

k=i+1

(1??v U ik)(1??v U kj)

,i

Proof.For(?μL ik,?μL kj)∈{(0,1),(1,0)},i

evidence that Equation(8)is equivalent to Equation

(4).Thus,we only prove the case that(?μL ik,?μL kj)/∈

{(0,1),(1,0)},i

holds for all i

?μL ij=

?μL ik?μL kj

?μL ik?μL kj+(1??μL ik)(1??μL kj)

?(1??μL ij)?μL ik?μL kj=?μL ij(1??μL ik)(1??μL kj),

?(1??μL ij)j?i?1

j?1

k=i+1

(?μL ik?μL kj)=(?μL ij)j?i?1

j?1

k=i+1

(1??μL ik)(1??μL kj)

?(1??μL ij)j?i?1

j?1

k=i+1

(?μL ik?μL kj)=?μL ij j?i?1

j?1

k=i+1

(1??μL ik)(1??μL kj)

??μL ij=

j?i?1

j?1

k=i+1

?μL ik?μL kj

j?i?1

j?1

k=i+1

?μL ik?μL kj+j?i?1

j?1

k=i+1

(1??μL ik)(1??μL kj)

(12)

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation2973 Conversely,let

?μL ij=

j?i?1

j?1

k=i+1

?μL ik?μL kj

j?i?1

j?1

k=i+1

?μL ik?μL kj+j?i?1

j?1

k=i+1

(1??μL ik)(1??μL kj)

,i

Firstly,we prove the following equation:

?μL ij=

j?i?1

l=0

?μL(i+l)(i+l+1)

j?i?1

l=0

?μL(i+l)(i+l+1)+

j?i?1

l=0

(1??μL(i+l)(i+l+1))

,i

Let p=j?i,then Equation(13)turns to

?μL i(i+p)=

p?1

l=0

?μL(i+l)(i+l+1)

p?1

l=0

?μL(i+l)(i+l+1)+

p?1

l=0

(1??μL(i+l)(i+l+1))

,i

Now we prove Equation(14)by using mathematical induction on n.

1)When p=2,obviously,Equation(14)holds.

2)Suppose that it is true for p=n(n>2),i.e.,

?μL i(i+n)=

n?1

l=0

?μL(i+l)(i+l+1)

n?1

l=0

?μL(i+l)(i+l+1)+

n?1

l=0

(1??μL(i+l)(i+l+1))

(15)

In what follows,we prove it is true for p=n+1, here j=i+n+1.

From Equation(15),and when k=i+1,i+ 2,...,i+n,we can easily get

?μL ik=

k?i?1

l=0

?μL(i+l)(i+l+1)

k?i?1

l=0

?μL(i+l)(i+l+1)+

k?i?1

l=0

(1??μL(i+l)(i+l+1))

,k=i+1,i+2,...,i+n

and

?μL k(i+n+1)=

i+n?k

l=0

?μL(k+l)(i+l+1)

i+n?k

l=0

?μL(k+l)(i+l+1)+

i+n?k

l=0

(1??μL(k+l)(i+l+1))

,k=i+1,i+2,...,i+n

Let

?E ik=k?i?1

l=0

?μL(i+l)(i+l+1)+

k?i?1

l=0

(1??μL(i+l)(i+l+1)),k=i+1,i+2,...,i+n

and

?F k(i+n+1)=

i+n?k

l=0

?μL(k+l)(i+l+1)

+

i+n?k

l=0

(1??μL(k+l)(i+l+1)),

k=i+1,i+2,...,i+n

2974H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation Then we can obtain

?μL i(i+n+1)=

n

i+n

k=i+1

?μL ik?μL k(i+n+1)

n

i+n

k=i+1

?μL ik?μL k(i+n+1)+n

i+n

k=i+1

(1??μL ik)(1??μL k(i+n+1))

=

n

i+n

k=i+1

?

??

?

k?i?1

l=0

?μL

(i+l)(i+l+1)

?E ik

i+n?k

l=0

?μL

(k+l)(i+l+1)

?F k(i+n+1)

?

??

?

n

i+n

k=i+1

?

??

?

k?i?1

l=0

?μL

(i+l)(i+l+1)

?E ik

i+n?k

l=0

?μL

(k+l)(i+l+1)

?F k(i+n+1)

?

??

?+n

i+n

k=i+1

?

??

?1?

k?i?1

l=0

?μL

(i+l)(i+l+1)

?E ik

?

??

?

?

??

?1?

i+n?k

l=0

?μL

(k+l)(i+l+1)

?F k(i+n+1)

?

??

?

Since

???

??1?k?i?1

l=0

?μL(i+l)(i+l+1)

?E ik

?

??

??

?

??

??1?

i+n?k

l=0

?μL(k+l)(i+l+1)

?F k(i+n+1)

?

??

??=

k?i?1

l=0

(1??μL(i+l)(i+l+1))

?E ik

i+n?k

l=0

(1??μL(k+l)(i+l+1))

?F k(i+n+1)

=

n

l=0

(1??μL(i+l)(i+l+1))

?E ik?F k(i+n+1),k=i+1,i+2,...,i+n

and

k?i?1 l=0?μL(i+l)(i+l+1)

?E ik

i+n?k

l=0

?μL(k+l)(i+l+1)

?F k(i+n+1)

=

n

l=0

?μL(i+l)(i+l+1)

?E ik?F k(i+n+1),t=i+1,i+2,...,i+n

we have

?μL i(i+n+1)=

n

i+n

k=i+1

n

l=0

?μL(i+l)(i+l+1)

n

i+n

k=i+1

n

l=0

?μL(i+l)(i+l+1)

+n

i+n

k=i+1

n

l=0

(1??μL(i+l)(i+l+1))

=

n

l=0

?μL(i+l)(i+l+1)

n

l=0

?μL(i+l)(i+l+1)+

n

l=0

(1??μL(i+l)(i+l+1))

Thus,it is true for p=n+1.Let ?μL ik=

k?i?1

l=0

?μL(i+l)(i+l+1)

k?i?1

?μL(i+l)(i+l+1)+

k?i?1

(1??μL(i+l)(i+l+1))

,i

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation2975

?μL kj=

j?k?1

l=0

?μL(k+l)(k+l+1)

j?k?1

l=0

?μL(k+l)(k+l+1)+

j?k?1

l=0

(1??μL(k+l)(k+l+1))

,k

and suppose

?M ik=k?i?1

l=0

?μL(i+l)(i+l+1)+

k?i?1

l=0

(1??μL(i+l)(i+l+1)),i

?N kj=j?k?1

l=0

?μL(k+l)(k+l+1)+

j?k?1

l=0

(1??μL(k+l)(k+l+1)),k

Then we have

?μL ik?μL kj

?μL ik?μL kj+(1??μL ik)(1??μL kj)

=

k?i?1

l=0

?μL

(i+l)(i+l+1)

?M ik

j?k?1

l=0

?μL

(k+l)(k+l+1)

?N kj

k?i?1

l=0

?μL

(i+l)(i+l+1)

?M ik

j?k?1

l=0

?μL

(k+l)(k+l+1)

?N kj

+

?

??

?1?

k?i?1

l=0

?μL

(i+l)(i+l+1)

?M ik

?

??

?

?

??

?1?

j?k?1

l=0

?μL

(k+l)(k+l+1)

?N kj

?

??

?

=

j?i?1

l=0

?μL(i+l)(i+l+1)

j?i?1

l=0

?μL(i+l)(i+l+1)+

j?i?1

l=0

(1??μL(i+l)(i+l+1))

=?μL ij,i

Combining Equation(12)and(16),it follows that Equation(4)and(8)are equivalent.Similarly,we can prove the other part of Theorem1. Theorem1shows an important property of the per-fect multiplicative consistent IVIFPR.We can use this property to check whether an IVIFPR is perfect multiplicative consistent or not.However,the perfect multiplicative consistent IVIFPR is sometimes too ideal for an expert to construct in practice.In the follow-ing,we would introduce the concepts of approximate multiplicative consistent IVIFPR and acceptable mul-tiplicative consistent IVIFPR to model general cases. For the inconsistent IVIFPR,some algorithms should be proposed to adjust or repair it.3.Multiplicative consistency repairing

procedure for inconsistent IVIFPR

3.1.Construct an approximate multiplicative

consistent IVIFPR with n?1known

judgments

When constructing an IVIFPR,expert may be unwill-ing or unable to provide his/her preferences over some of the alternatives due to time pressure,lacking of knowledge,individual emotion or limited expertise related to problem.In such a case,an incomplete IVIFPR is furnished.How to estimate the missing values of an incomplete IVIFPR turns out to be an

2976H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation important issue.For an incomplete IVIFPR,Xu and

Cai[16]proved that in order to estimate all the missing

values,there should exist at least one known element

(except diagonal elements)in each line or each col-

umn of the incomplete IVIFPR?R,i.e.,there exists at

least n?1judgments provided by the expert.In other

words,each one of the alternatives is compared at least

one time.The unknown element?μij in an incomplete

IVIFPR?R can be obtained indirectly according to the

known element?μik and?μkj in? .Inspired by the proof

of Theorem1,an algorithm is proposed to construct the

approximate multiplicative consistent IVIFPR with the

least judgments(i.e.,n?1judgments):

Algorithm1.

Step1.For j>i+1,letˉ?r ij=(ˉ?μij,ˉ?v ij),where

ˉ?μL ij =

j?i?1

k=0

?μL(i+k)(i+k+1)

j?i?1

k=0

?μL(i+k)(i+k+1)+

j?i?1

k=0

(1??μL(i+k)(i+k+1))

,j>i+1(17)

ˉ?μU ij =

j?i?1

k=0

?μU(i+k)(i+k+1)

j?i?1

k=0

?μU(i+k)(i+k+1)+

j?i?1

k=0

(1??μU(i+k)(i+k+1))

,j>i+1(18)

ˉ?v L ij =

j?i?1

k=0

?v L(i+k)(i+k+1)

j?i?1

k=0

?v L(i+k)(i+k+1)+

j?i?1

k=0

(1??v L(i+k)(i+k+1))

,j>i+1(19)

ˉ?v U ij =

j?i?1

k=0

?v U(i+k)(i+k+1)

j?i?1

k=0

?v U(i+k)(i+k+1)+

j?i?1

k=0

(1??v U(i+k)(i+k+1))

,j>i+1(20)

Step2.For j=i+1,letˉ?r ij=?r ij.

Step3.For j

De?nition5.The IVIFPR?R obtained from Algorithm 1is called an approximate multiplicative consistent IVIFPR.

Now,we give a simple numerical example to illus-trate how to build the approximate multiplicative consistent IVIFPR.

Example1.Assume that the expert compares only two pairs of alternatives corresponding to a decision making problem with three alternatives x i(i=1,2,3), and determines the preference information over(x1,x2) and(x2,x3)as?r12=([0.3,0.4],[0.2,0.3])and?r23=

([0.4,0.5],[0.1,0.4]).

Firstly,by Equation(1),we can easily construct an 3×3incomplete IVIFPR with the least judgments as follows:

ˉ?R=?

??

([0.5,0.5],[0.5,0.5])([0.3,0.4],[0.2,0.3])x

([0.2,0.3],[0.3,0.4])([0.5,0.5],[0.5,0.5])([0.4,0.5],[0.1,0.4])

x([0.1,0.4],[0.4,0.5])([0.5,0.5],[0.5,0.5])

?

??

Using Algorithm1,we have

ˉ?μL

13

=?μ

L

12

·?μL23

?μL12·?μL23+(1??μL12)·(1??μL23)

=0.222

Similarly,we can get

ˉ?μU

13

=0.400,ˉ?v L13=0.027,ˉ?v U13=0.222.

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation 2977

Thus,ˉ?

r 13=([0.222,0.400],[0.027,0.222]),and furthermore,ˉ?

r 31=([0.027,0.222],[0.222,0.400]).Therefore,we obtain an approximate multiplicative

consistent IVIFPR:

ˉ?R

=???([0.500,0.500],[0.500,0.500])([0.300,0.400],[0.200,0.300])([0.222,0.400],[0.027,0.222])

([0.200,0.300],[0.300,0.400])([0.500,0.500],[0.500,0.500])([0.400,0.500],[0.100,0.400])([0.027,0.222],[0.222,0.400])

([0.100,0.400],[0.400,0.500])

([0.500,0.500],[0.500,0.500])

???3.2.Construct a perfect multiplicative consistent

IVIFPR with more known judgments Algorithm 1constructs an approximate multiplicat-ive consistent IVIFPR,but it only considers a special case where there are only n ?1known judgments in the off-diagonal of an incomplete IVIFPR.In the following,we further give an algorithm to obtain a per-fect multiplicative consistent IVIFPR with more known judgments.Algorithm 2.

Step 1.For j >i +1,let ˉ?r ij =(ˉ?μij ,ˉ?

v ij ),where ˉ?μL ij =

j ?i ?1

j ?1

k =i +1

?μL ik ?μL kj

j ?i ?1

j ?1 k =i +1

?μL ik ?μL kj +

j ?i ?1

j ?1

k =i +1(1??μL ik )(1??μL kj )

,j >i +1

(21)

ˉ?μU ij =

j ?i ?1

j ?1 k =i +1

?μU ik ?μU kj

j ?i ?1

j ?1 k =i +1

?μU ik ?μU kj +

j ?i ?1

j ?1

k =i +1(1??μU ik )(1??μU kj )

,

j >i +1(22)

ˉ?v L ij =

j ?i ?1

j ?1

k =i +1

?v L ik ?v L kj

j ?i ?1

j ?1 k =i +1

?v L ik ?v L kj +

j ?i ?1

j ?1

k =i +1(1??v L ik )(1??v L kj )

,

j >i +1(23)

ˉ?v U ij =

j ?i ?1

j ?1

k =i +1

?v U ik ?v U kj

j ?i ?1

j ?1 k =i +1

?v U ik ?v U kj +

j ?i ?1

j ?1

k =i +1

(1??v U ik )(1??v U kj )

,

j >i +1(24)

Step 2.For j =i +1,let ˉ?r ij =?r ij .

Step 3.For j

Comparing Algorithm 1with Algorithm 2,we can easily ?nd that the former may get a multiplicative con-sistent IVIFPR more quickly as it only considers part of the preference information,but the latter can get much more objective information.

3.3.Iterative algorithm for repairing inconsistent

IVIFPR The approximate multiplicative consistent IVIFPR can be determined via Algorithm 1and Algorithm 2.However,in general,a multiplicative consistent IVIFPR is too ideal to achieve.As to the inconsistent IVIFPR,we need to ?nd some methods to repair it.

De?nition 6.Let ?R

be an IVIFPR on a ?xed set X ={x 1,x 2,...,x n },then ?R

is an acceptable multiplicative consistent IVIFPR,if

d 1(?R ,ˉ?R

)<τ(25)

2978H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

where d (?R ,ˉ?R

)is the distance measure between the given IVIFPR ?R

and its corresponding perfect or approximate multiplicative consistent IVIFPR ˉ?

R which can be calculated by Algorithm 1or Algorithm 2,and τis the consistency threshold which is determined by the expert.

Very often,the IVIFPR ?R

constructed by expert is of unacceptable multiplicative consistency.Below we pro-pose an iterative algorithm to improve the consistency of an inconsistent IVIFPR.Algorithm 3

Step 1.Suppose p is the number of iterations,N is the maximum number of iteration,and τis the consistency threshold.Let σ=1 N be the iteration step.Let p =1,and construct the perfect (or approximate)multiplica-tive consistent IVIFPR ˉ?

R from ?R (p )by Algorithm 1(or Algorithm 2).

Step 2.Calculate the distance d 1(ˉ?R ,?R (p ))between ˉ?

R and ?R

(p ),where d 1(ˉ?R ,?R (p ))=14(n ?1)(n ?2)

n ?1 i =1n j =i +1

ˉ?μ

L ij ??μL (p )ij + ˉ?μU ij ??μU (p )ij + ˉ?v L ij ??v L (p )ij + ˉ?v U ij ??v U (p )ij + ˉ?πL ij +?ˉ?πL (p )ij ˉ?πU ij ?ˉ?πU (p )ij

(26)If d 1(ˉ?

R ,?R (p ))<τ,then output ?R (p );Otherwise,go to the next step.

Step 3.Construct the fused IVIFPR ??R (p )=(??r (p )ij )n ×n

by using

??μL (p )ij

=(?μL (p )ij

)1?pσ(ˉ?μL ij )

pσ

(?μL (p )ij )1?σ(ˉ?μL ij )pσ+(1??μL (p )ij )1?pσ(1?ˉ?μL ij

)pσ,i,j =1,2,...,n (27)

??μU (p )ij

=

(?μU (p )ij

)1?pσ(ˉ?μU ij )

pσ

(?μU (p )ij )1?pσ(ˉ?μU ij )pσ+(1??μU (p )ij )1?pσ(1?ˉ?μU ij )

pσ,i,j =1,2,...,n (28)

??

νL (p )ij =

(?νL (p )ij

)1?pσ(ˉ?νL ij )

pσ

(?νL (p )ij )1?pσ(ˉ?νL ij )pσ+(1??νL (p )ij )1?pσ(1?ˉ?

νL ij )pσ,i,j =1,2,...,n (29)

??

νU (p )ij =

(?νU (p )ij

)1?pσ(ˉ?νU ij )

pσ

(?νU (p )ij )1?pσ(ˉ?νU ij )pσ+(1??νU (p )ij )1?pσ(1?ˉ?νU ij )

pσ,i,j =1,2,...,n (30)

Let ?R

p +1=??R p and p =p +1,then go to Step 2.Step 4.End.

With Algorithm 3,an improved multiplicative con-sistent IVIFPR ?R

(p )is obtained.Theorem 2.Algorithm 3is convergent.

Proof.We actually repair the inconsistent IVIFPR ?R

(p )through ??R (p ?1)and the deviation between ˉ?

R and ??R (p ?1).As N is the given maximum iteration number,then after p =N iterations of calculation,we can obtain pδ=1

thus ?R (p )=ˉ?

R with the consistency level τ.That is to say,Algorithm 3is convergent. The following example can illustrate Algorithm 2and Algorithm 3.There is one thing on which we have to emphasize:the hesitancy degree is very important in calculating the deviation between two IVIFPRs [28]and we shall not ignore it.

Example 2.Assume that the expert gives an IVIFPR as follows:?R

(1)=?

?

????

([0.5,0.5],[0.5,0.5])

([0.3,0.4],[0.2,0.5])([0.6,0.7],[0.1,0.2])([0.4,0.6],[0.3,0.4])

([0.2,0.5],[0.3,0.4])([0.5,0.5],[0.5,0.5])([0.3,0.6],[0.2,0.4])([0.1,0.3],[0.5,0.7])([0.1,0.2],[0.6,0.7])([0.2,0.4],[0.3,0.6])([0.5,0.5],[0.5,0.5])([0.4,0.4],[0.2,0.3])([0.3,0.4],[0.4,0.6])([0.5,0.7],[0.1,0.3])

([0.2,0.3],[0.4,0.4])

([0.5,0.5],[0.5,0.5])

?

?

?

?

??From Equations (4)–(8),we know that ?R

is not multiplicative consistent.Then Algorithm 3can be

used to improve the consistency of ?R

.Let τ=0.1and σ=0.8.First of all,via Algorithm 2,the perfect mul-tiplicative consistent IVIFPR is constructed.

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation 2979

ˉ?R =????([0.500,0.500],[0.500,0.500])([0.300,0.400],[0.200,0.500])([0.155,0.500],[0.063,0.400])([0.179,0.400],[0.077,0.333])([0.200,0.500],[0.300,0.400])([0.500,0.500],[0.500,0.500])([0.300,0.600],[0.200,0.400])([0.222,0.500],[0.063,0.222])([0.063,0.400],[0.155,0.500])([0.200,0.400],[0.300,0.600])([0.500,0.500],[0.500,0.500])([0.400,0.400],[0.200,0.300])([0.077,0.333],[0.179,0.400])([0.063,0.222],[0.222,0.500])([0.200,0.300],[0.400,0.400]])([0.500,0.500],[0.500,0.500])

??

??.

Let p =1.According to Equation (26),the deviation

is calculated:d 1(ˉ?

R ,?R (1))=0.3675>0.1.According to Algorithm 3,it is needed to construct the fused

IVIFPR ??

R (1).By Equations (44)–(48),we have ??R (1)=????([0.500,0.500],[0.500,0.500])([0.300,0.400],[0.200,0.500])([0.218,0.542],[0.069,0.354])([0.214,0.439],[0.104,0.346])([0.200,0.500],[0.300,0.400])([0.500,0.500],[0.500,0.500])([0.300,0.600],[0.200,0.400])([0.191,0.458],[0.103,0.303])([0.069,0.354],[0.218,0.542])([0.200,0.400],[0.300,0.600])([0.500,0.500],[0.500,0.500])([0.400,0.400],[0.200,0.300])([0.104,0.346],[0.214,0.439])([0.103,0.303],[0.191,0.458])([0.200,0.300],[0.400,0.400])([0.500,0.500],[0.500,0.500])

??

??

Let ?R

(2)=??R (1),then from Equation (26),it fol-lows d 1(ˉ?

R ,?R (2))=0.0585<0.1,which implies ?R 2is of acceptable consistency.

4.An approach for group decision making with IVIFPRs

A GDM problem with IVIFPRs can be

described as follows:Suppose that m experts

e l (l =1,2,...,m )provide their individual IVIFPRs ?R l =(?r ijl )n ×n (l =1,2,...,m )over alternatives

x 1,x 2,...,x n ,where ?r ijl =(?μijl ,?v ijl ),?μijl =

[?μL ijl ,?μ

U ijl ]?[0,1],?v ijl =[?v L ijl ,?v U ijl ]?[0,1],?πijl =[?πL ijl ,?πU ijl ]?[0,1],?μjil =?v ijl ,?v jil =?μijl ,

?πL ijl =1?(?μU ijl +?v U ijl ),?πU ijl =1?(?μL ijl +?v L ijl ),?μiil =?v iil =[0.5,0.5],μU ijl +v U

ijl ≤1,

for all i,j =1,2,...,n ,and ω=(ω1,ω2,...,ωm )T is the weighting vector of the experts e l (l =1,2,...,m )with m l =1ωl =1and 0≤ωl ≤1,l =1,2,...,m .

The aim of GDM is to select the most disable solution

from a set of candidate alternatives.In the process of group decision making,we need to check whether the individual preference relations provided by the experts are consistent or not,and then to repair the inconsistent

ones until they are acceptable.After that,all the individ-ual IVIFPRs can be aggregated into an overall IVIFPR so as to select the most desirable alternative(s).Moti-vated by Xia and Xu [3],we propose a symmetric interval-valued intuitionistic fuzzy weighted averaging (SIVIFW A)operator to fuse the individual IVIFPRs ?R l =(?r ijl )n ×n (l =1,2,...,m )into a collected one.

De?nition 7.For a given collection of IVIFSs ?r =(?r 1,?r 2,...,?r n ),w =(w 1,w 2,...,w n )T being the weighting vector of ?r i (i =1,2,...,n ),where n i =1w i =1and 0≤w i ≤1,if SIVIFWA:I n

→I ,and SIVIFWA w (?r 1,?r 2,...?r n )= n i =1(?μL ?r i )w i n i =1(?μL ?r i )w i + n i =1(1??μL ?r i )w i , n i =1

(?μU ?r i

)w i n i =1(?μU ?r i )w i + n i =1

(1??μU ?r i )w i , n i =1(?v L ?r i )w i

n i =1(?v L ?r i )w i + n i =1(1??v L ?r i )w i , n i =1(?v U ?r

i )w i n i =1(?v U ?r i )w i + n i =1(1??v U ?r i )w i

i =1,2,...,n (31)then we call the function SIVIFW A a sym-metric interval-valued intuitionistic fuzzy weighted

averaging (SIVIFW A)operator.Especially,if w =(1 n,1 n,...,1 n )T ,then the SIVIFWA operator reduces to a symmetric interval-valued intuitionistic fuzzy averaging (SIVIFA)operator.

Theorem 3.Let ?R l =(?r ijl )n ×n (l =1,2,...,m )be m

individual IVIFPRs,then their fusion ?R =(?r ij )n ×n is

also an IVIFPR,where ?r ij =

m

l =1(?μL ijl )

ωl

m l =1(?μL ijl )ωl + m

l =1(1??μL ijl

)ωl , m

l =1(?μU ijl )

ωl

m l =1(?μU ijl )ωl + m l =1(1??μU ijl )

ωl ,

m

l =1(?v L ijl )ωl

m l =1(?v L ijl )ωl + m l =1(1??v L ijl )ωl , m l =1(?v U ijl

)ωl m l =1(?v U ijl )ωl + m l =1(1??v U ijl )

ωl i,j =1,2,...,n

(32)

2980H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation Proof.We can easily obtain that

0≤?v U ij=

m

l=1

(?v U ijl)ωl

m

l=1

(?v U ijl)ωl+

m

l=1

(1??v U ijl)ωl

≤1

1+

m

l=1

(1

?v U ijl?1)ωl

,

≤1

1+

m

l=1

(1

(1??μU ijl)?1)ωl

≤

m

l=1

(1??μU ijl)ωl

m

l=1

(1??μU ijl)ωl

m

l=1

(?μU ijl)ωl

≤1,i,j=1,2,...,n

Then

?μU ij+?v U ji=

m

l=1

(?μU ijl)ωl

m

l=1

(?μU ijl)ωl+

m

l=1

(1??μU ijl)ωl

+

m

l=1

(?v U jil)ωl

m

l=1

(?v U jil)ωl+

m

l=1

(1??v U jil)ωl

≤

m

l=1

(?μU ijl)ωl

m

l=1

(?μU ijl)ωl+

m

l=1

(1??μU ijl)ωl

+

m

l=1

(1??μU jil)ωl

m

l=1

(1??μU jil)ωl

m

l=1

(?μU ijl)ωl

=1,

i,j=1,2,...,n

Hence,Theorem3holds. Based on Theorem3and Equations(4)–(7),the fol-lowing interesting result is obtained:

Theorem4.If all individual IVIFPRs?R l=(?r ijl)n×n (l=1,2,...,m)are multiplicative consistent,then their fused IVIFPR?R=(?r ij)n×n is also multiplicative consistent.

Proof.Let?R l=(?r ijl)n×n(l=1,2,...,m)be multi-plicative consistent,and

?U L

ikjl

=?μL ikl?μL kjl+(1??μL ikl)(1??μL kjl),i

?V L

ik

=

m

l=1

(?μL ikl)ωl+

m

l=1

(1??μL ikl)ωl,i

?W L

kj

=

m

l=1

(?μL kjl)ωl+

m

l=1

(1??μL kjl)ωl,k

?μL ij=

m

l=1

(?μL ijl)ωl

m

l=1

(?μL ijl)ωl+

m

l=1

(1??μL ijl)ωl

=

m

l=1

((?μL ikl?μL kjl)

?U L

ikjl

)ωl

m

l=1

((?μL ikl?μL kjl)

?U L

ikjl

)ωl+

m

l=1

(1?(?μL ikl?μL kjl)

?U L

ikjl

)ωl

=

m

l=1

(?μL ikl?μL kjl)ωl

m

l=1

(?μL ikl?μL kjl)ωl+

m

l=1

((1??μL ikl)(1??μL kjl))ωl

,i

On the other hand,

?μL ij=

?μL ik?μL kj

?μL ik?μL kj+(1??μL ik)(1??μL kj)

=

(

m

l=1

(?μL ikl)ωl

m

l=1

(?μL kjl)ωl)

(?V L ik?W L kj)

(

m

l=1

(?μL ikl)ωl

m

l=1

(?μL kjl)ωl)

(?V L ik?W L kj)+(1?

m

l=1

(?μL ikl)ωl

?V L

ik

)(1?

m

l=1

(?μL kjl)ωl

?W L

kj

)

=

m

l=1

(?μL ikl)ωl

m

l=1

(?μL kjl)ωl

m

l=1

(?μL ikl)ωl

m

l=1

(?μL kjl)ωl+

m

l=1

(1??μL ikl)ωl

m

l=1

(1??μL kjl)ωl

=

m

l=1

(?μL ikl?μL kjl)ωl

m

(?μL?μL)l

m

((1??μL)(1??μL))l

,i

Hence,

?μL ij=

?μL ik?μL kj

?μL ik?μL kj+(1??μL ik)(1??μL kj)

,

i

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation2981 In a similar way,we can get

?μU ij=

?μU ik?μU kj

?μU ik?μU kj+(1??μU ik)(1??μU kj)

,i

?νL ij=

?v L ik?v L kj

?v L ik?v L kj+(1??v L ik)(1??v L kj)

,i

?νU ij=

?v U ik?v U kj

?v U ik?v U kj U ik)(1??v U kj)

,i

which denotes that?R is a multiplicative consistent IVIFPR.Therefore,Theorem4holds.

It should be noted that there are many differ-ent aggregation operators for IVIFSs existing in the literatures.However,as shown by the proof processes of

Theorem4,only the SIVIFW A operator proposed in this paper can guarantee that the fused IVIFPR?R is multiplicative consistent on condition that all individual IVIFPRs?R l(l=1,2,...,m)are multiplicative con-sistent.

Based on the above analysis,we propose an approach for group decision making with IVIFPRs.Algorithm4.

Step1.Let?R(p)l=?R l and p=1.Construct the perfect (or approximate)multiplicative consistent IVIFPRs ˉ?R(p)

l

=(ˉ?r(p)ijl)n×n(l=1,2,...,m)from?R(p)l= (?r(p)ijl)n×n(l=1,2,...,m)by Algorithm1(or Algorithm2).

Step 2.Aggregate all the individual multiplicative consistent IVIFPRˉ?R(p)l=(ˉ?r(p)ijl)n×n into a collective multiplicative consistent IVIFPRˉ?R(p)=(ˉ?r(p)ij)n×n by the SIVIFW A operator,where

ˉ?r(p) ij =

?

?

?

?

m

l=1

(?μL(p)

ijl

)ωl

m

l=1

(?μL(p)

ijl

)

ωl+ m

l=1

(1??μL(p)

ijl

)

ωl,

m

l=1

(?μU(p)

ijl

)

ωl

m

l=1

(?μU(p)

ijl

)

ωl+ m

l=1

(1??μU(p)

ijl

)

ωl

?

?,

m

l=1

(?v L(p)

ijl

)ωl

m

l=1

(?v L(p)

ijl

)ωl+

m

l=1

(1??v L(p)

ijl

)ωl

,

m

l=1

(?v U(p)

ijl

)ωl

m

l=1

(?v U(p)

ijl

)ωl+

m

l=1

(1??v U(p)

ijl

)ωl

i,j=1,2,...,n(33)

Step3.Calculate the deviation between each individual

multiplicative consistent IVIFPRˉ?R(p)l and the collective

multiplicative consistent IVIFPRˉ?R(p),i.e., d2(ˉ?R(p)l,ˉ?R(p))=

1

4(n?1)(n?2)

n?1

i=1

n

j=i+1

ˉ?μL(p)

ijl

?ˉ?μL(p)

ij

+

ˉ?μU(p)

ijl

?ˉ?μU(p)

ij

+

ˉ?v L(p)

ijl

?ˉ?v L(p)

ij

+

ˉ?v U(p)

ijl

?ˉ?v U(p)

ij

+

ˉ?πL(p)

ijl

?ˉ?πL(p)

ij

+

ˉ?πU(p)

ijl

?ˉ?πU(p)

ij

(34)

If d2(ˉ?R(p)l,ˉ?R(p))≤τ?,for all l=1,2,...,m,where

τ?is the consistency threshold,then go to Step5;Oth-

erwise,go to the next step.

Step 4.Letˉ?R(p+1)

l

=(ˉ?r(p+1)

ijl

)n×n,whereˉ?r(p+1)

ijl

=

(ˉ?μ(p+1)

ijl

,ˉ?v(p+1)

ijl

),and

ˉ?μL(p+1) ijl =

(ˉ?μL(p)

ijl

)1?pη(ˉ?μL(p)

ij

)pη

(ˉ?μL(p)

ijl

)

1?pη

(ˉ?μL(p)

ij

)

pη+(1?ˉ?μL(p)

ijl

)

1?pη

(1?ˉ?μL(p)

ij

)

pη

,i,j=1,2,...,n,l=1,2,...,m(35)

ˉ?μL(p+1) ijl =

(ˉ?μL(p)

ijl

)1?pη(ˉ?μL(p)

ij

)pη

(ˉ?μL(p)

ijl

)1?pη(ˉ?μL(p)

ij

)pη+(1?ˉ?μL(p)

ijl

)1?pη(1?ˉ?μL(p)

ij

)pη

,i,j=1,2,...,n,l=1,2,...,m(36)

ˉ?v L(p+1) ijl =

(ˉ?v L(p)

ijl

)1?pη(ˉ?v L(p)

ij

)pη

(ˉ?v L(p)

ijl

)1?pη(ˉ?v L(p)

ij

)pη+(1?ˉ?v L(p)

ijl

)1?pη(1?ˉ?v L(p)

ij

)pη

,i,j=1,2,...,n,l=1,2,...,m(37)

ˉ?v U(p+1) ijl =

(ˉ?v U(p)

ijl

)1?pη(ˉ?v U(p)

ij

)pη

(ˉ?v U(p)

ijl

)1?pη(ˉ?v U(p)

ij

)pη+(1?ˉ?v U(p)

ijl

)1?pη(1?ˉ?v U(p)

ij

)pη

,i,j=1,2,...,n,l=1,2,...,m(38)

2982H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

Let p =p +1,then go to step 2.

Step 5.Let ?R =ˉ?

R (p ),and employ the SIVIFA operator to fuse all the interval-valued intuition-istic fuzzy preference values ?r ij =(?μij ,?v ij )(j =1,2,...,n )corresponding to alternative x i into the overall interval-valued intuitionistic fuzzy preference value ?r i =(?μi ,?v i ),where ?r i = n j =1(?μL ij )1/n n j =1(?μL ij )1/n + n j =1(1??μL ij )1/n , n j =1

(?μU ij )1/n n j =1(?μU ij )1/n + n j =1

(1??μU ij )1/n , n j =1(?v L ij )1/n n j =1(?v L ij )1/n + n j =1(1??v L ij )1/n , n j =1(?v U ij

)1/n n j =1(?v U ij )1/n + n j =1(1??v U ij )1/n i =1,2,...,n (39)Step 6.Rank all the objects corresponding to the meth-ods given by Wang et al.[18].

Step 7.End.

Similar to Algorithm 3,it can be seen that Algo-rithm 4is also convergent.To illustrate Algorithm 4,

we now consider a group decision making problem

that concerns the evaluation and selection of the suit-able suppliers in the supply chain management (adapted

from [19]).

Example 3.As the sound supply chain management can reduce supply chain risk,optimize business process,

maximize revenue,and help an enterprise to maintain

a dominant position especially in the globalized and increasingly competitive markets,how to choose the

best suitable supplier has become a crucial issue.But,

actually,it is very dif?cult to do so due to the complex and con?icting criteria and the limited knowledge of the experts,as well as some emotional factors.Here we use

our procedure presented above to solve this problem.

A high-tech company which manufactures electronic

products intends to choose a supplier of USB con-nectors.There are four alternative suppliers,denoted

as x i ,(i =1,2,3,4),respectively.Four criteria are

considered,that is,?nance (ξ1),performance (ξ2),tech-nique (ξ3),and organizational culture (ξ4).In order

to make the best choice,a committee comprising three experts e l (l =1,2,3)(whose weights are ω=(0.4,0.3,0.3)T )is found.After comparing pairs of enterprises with respect to the four criteria ξj (j =1,2,3,4),the experts e l (l =1,2,3)give their prefer-ences using interval-valued intuitionistic fuzzy values,and then obtain the IVIFPRs as follows:?R

1=????([0.5,0.5],[0.5,0.5])([0.4,0.7],[0.1,0.2])([0.5,0.6],[0.2,0.3])([0.3,0.5],[0.2,0.4])([0.1,0.2],[0.4,0.7])([0.5,0.5],[0.5,0.5])([0.4,0.5],[0.1,0.2])([0.6,0.7],[0.1,0.3])([0.2,0.3],[0.5,0.6])([0.1,0.2],[0.4,0.5])([0.5,0.5],[0.5,0.5])([0.3,0.4],[0.5,0.6])([0.2,0.4],[0.3,0.5])([0.1,0.3],[0.6,0.7])([0.5,0.6],[0.3,0.4])([0.5,0.5],[0.5,0.5])

?

?

?

?

?R

2=????([0.5,0.5],[0.5,0.5])([0.2,0.3],[0.4,0.5])([0.5,0.8],[0.1,0.2])([0.2,0.4],[0.1,0.3])([0.4,0.5],[0.2,0.3])([0.5,0.5],[0.5,0.5])([0.5,0.8],[0.1,0.2])([0.3,0.5],[0.2,0.3])([0.1,0.2],[0.5,0.8])([0.1,0.2],[0.5,0.8])([0.5,0.5],[0.5,0.5])([0.4,0.6],[0.1,0.4])([0.1,0.3],[0.2,0.4])([0.2,0.3],[0.3,0.5])([0.1,0.4],[0.4,0.6])([0.5,0.5],[0.5,0.5])

??

??

?R

3=????([0.5,0.5],[0.5,0.5])([0.4,0.5],[0.2,0.3])([0.6,0.7],[0.1,0.2])([0.5,0.7],[0.2,0.3])([0.2,0.3],[0.4,0.5])([0.5,0.5],[0.5,0.5])([0.4,0.6],[0.2,0.4])([0.7,0.8],[0.1,0.2])([0.1,0.2],[0.6,0.7])([0.2,0.4],[0.4,0.6])([0.5,0.5],[0.5,0.5])([0.6,0.7],[0.1,0.3])([0.2,0.3],[0.5,0.7)

([0.1,0.2],[0.7,0.8])([0.1,0.3],[0.6,0.7])([0.5,0.5],[0.5,0.5])

??

??

Let ?R (0)l =?R l .By Algorithm 2,we construct the perfect multiplicative consistent IVIFPRs ˉ?R (0)l (l =

1,2,3).ˉ?R (0)1=????([0.500,0.500],[0.500,0.500])([0.400,0.700],[0.100,0.200])([0.308,0.700],[0.012,0.059])([0.160,0.609],[0.012,0.086])([0.100,0.200],[0.400,0.700])([0.500,0.500],[0.500,0.500])([0.400,0.500],[0.100,0.200])([0.222,0.400],[0.100,0.273])([0.012,0.059],[0.308,0.700])([0.100,0.200],[0.400,0.500])([0.500,0.500],[0.500,0.500])([0.300,0.400],[0.500,0.600])([0.012,0.086],[0.160,0.609])([0.100,0.273],[0.222,0.400])([0.500,0.600],[0.300,0.400])([0.500,0.500],[0.500,0.500])

????

H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation2983

ˉ?R (0)

2

=

?

??

?

([0.500,0.500],[0.500,0.500])([0.200,0.300],[0.400,0.500])([0.200,0.632],[0.069,0.200]])([0.143,0.720],[0.008,0.143]) ([0.400,0.500],[0.200,0.300])([0.500,0.500],[0.500,0.500])([0.500,0.800],[0.100,0.200])([0.400,0.857],[0.012,0.143]) ([0.069,0.200],[0.200,0.632])([0.100,0.200],[0.500,0.800])([0.500,0.500],[0.500,0.500])([0.400,0.600],[0.100,0.400]) ([0.008,0.143],[0.143,0.720])([0.012,0.143],[0.400,0.857])([0.100,0.400],[0.400,0.600])([0.500,0.500],[0.500,0.500])

?

??

?

ˉ?R (0)

3

=

?

??

?

([0.500,0.500],[0.500,0.500])([0.400,0.500],[0.200,0.300])([0.308,0.600],[0.059,0.222])([0.400,0.778],[0.007,0.109]) ([0.200,0.300],[0.400,0.500])([0.500,0.500],[0.500,0.500])([0.400,0.600],[0.200,0.400])([0.500,0.778],[0.027,0.222]) ([0.059,0.222],[0.308,0.600])([0.200,0.400],[0.400,0.600])([0.500,0.500],[0.500,0.500])([0.600,0.700],[0.100,0.300]) ([0.007,0.109],[0.400,0.778])([0.027,0.222],[0.500,0.778])([0.100,0.300],[0.600,0.700])([0.500,0.500],[0.500,0.500])

?

??

?By Equation(33),we fuse the individual multi-

plicative consistent IVIFPRsˉ?R(0)l(l=1,2,3)into the collective IVIFPRˉ?R(0):

ˉ?R (0)=

?

??

?

([0.500,0.500],[0.500,0.500])([0.332,0.521],[0.195,0.308])([0.272,0.651],[0.033,0.130])([0.210,0.698],[0.009,0.108]) ([0.195,0.308],[0.332,0.521])([0.500,0.500],[0.500,0.500])([0.430,0.631],[0.124,0.251])([0.349,0.679],[0.036,0.213]) ([0.033,0.130],[0.272,0.651])([0.124,0.251],[0.430,0.631])([0.500,0.500],[0.500,0.500])([0.416,0.553],[0.211,0.447]) ([0.009,0.108],[0.210,0.698])([0.036,0.213],[0.349,0.679])([0.211,0.447],[0.416,0.553])([0.500,0.500],[0.500,0.500])

?

??

?Then,via Equation(34),we can obtain the

deviation between each individual multiplicative con-sistent IVIFPRˉ?R(0)l and the collective IVIFPRˉ?R(0), which are:d2(ˉ?R(0)1,ˉ?R(0))=0.193,d2(ˉ?R(0)2,ˉ?R(0))= 0.123,d2(ˉ?R(0)3,ˉ?R(0))=0.156.As all d2(ˉ?R(0)l,ˉ?R(0))> 0.1,we need to repair these individual IVIFPRs.Let

η=0.5,and according to Equations(35)–(38),the new individual IVIFPRsˉ?R(1)l(l=1,2,3)are constructed:

ˉ?R (1)

1

=

?

??

?

([0.500,0.500],[0.500,0.500])([0.365,0.614],[0.141,0.250])([0.290,0.676],[0.020,0.088])([0.184,0.655],[0.010,0.096]) ([0.141,0.250],[0.365,0.614])([0.500,0.500],[0.500,0.500])([0.415,0.567],[0.111,0.225])([0.281,0.543],[0.061,0.242]) ([0.020,0.088],[0.290,0.676])([0.111,0.225],[0.415,0.567])([0.500,0.500],[0.500,0.500])([0.356,0.476],[0.341,0.524]) ([0.010,0.096],[0.184,0.655])([0.061,0.242],[0.281,0.543])([0.341,0.524],[0.356,0.476])([0.500,0.500],[0.500,0.500])

?

??

?

ˉ?R (1)

2

=

?

??

?

([0.500,0.500],[0.500,0.500])([0.261,0.406],[0.287,0.400])([0.234,0.642],[0.048,0.162])([0.174,0.709],[0.009,0.124]) ([0.287,0.400],[0.261,0.406])([0.500,0.500],[0.500,0.500])([0.465,0.723],[0.111,0.225])([0.374,0.776],[0.021,0.175]) ([0.048,0.162],[0.234,0.642])([0.111,0.225],[0.465,0.723])([0.500,0.500],[0.500,0.500])([0.409,0.577],[0.147,0.423]) ([0.009,0.124],[0.174,0.709])([0.021,0.175],[0.374,0.776])([0.147,0.423],[0.409,0.577])([0.500,0.500],[0.500,0.500])

?

??

?

ˉ?R (1)

3

=

?

??

?

([0.500,0.500],[0.500,0.500])([0.365,0.511],[0.198,0.304])([0.290,0.626],[0.044,0.171])([0.296,0.740],[0.008,0.109]) ([0.198,0.304],[0.365,0.511])([0.500,0.500],[0.500,0.500])([0.415,0.616],[0.158,0.321])([0.423,0.731],[0.031,0.218]) ([0.044,0.171],[0.290,0.626])([0.158,0.321],[0.415,0.616])([0.500,0.500],[0.500,0.500])([0.508,0.630],[0.147,0.370]) ([0.008,0.109],[0.296,0.740])([0.031,0.218],[0.423,0.731])([0.147,0.370],[0.508,0.630])([0.500,0.500],[0.500,0.500])

?

??

?Then,the collective IVIFPRˉ?R(1)is

ˉ?R (1)=

?

??

?

([0.500,0.500],[0.500,0.500])([0.332,0.521],[0.195,0.308])([0.272,0.651],[0.033,0.130])([0.210,0.698],[0.009,0.108]) ([0.195,0.308],[0.332,0.521])([0.500,0.500],[0.500,0.500])([0.430,0.631],[0.124,0.252])([0.349,0.677],[0.036,0.213]) ([0.033,0.130],[0.272,0.651])([0.124,0.252],[0.430,0.631])([0.500,0.500],[0.500,0.500])([0.416,0.553],[0.211,0.447]) ([0.009,0.108],[0.210,0.698])([0.036,0.213],[0.349,0.677])([0.211,0.447],[0.416,0.553])([0.500,0.500],[0.500,0.500])

?

??

?

Thus,we can calculate the deviation between each

individual IVIFPRˉ?R(1)l and the collective IVIFPRˉ?R(1):

d2(ˉ?R(1)1,ˉ?R(1))=0.091,d2(ˉ?R(1)2,ˉ?R(1))=0.077,

d2(ˉ?R(1)3,ˉ?R(1))=0.073.As all d2(ˉ?R(1)l,ˉ?R(1))<0.1,let

?R=ˉ?R(1),and then employ the SIVIFA operator to fuse

all the interval-valued intuitionistic fuzzy preference

values?r ij=(?μij,?v ij)corresponding to the criterion

ξj into the overall interval-valued intuitionistic fuzzy

values:

2984H.Liao et al./Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation

?r1=([0.320,0.595],[0.085,0.230]),

?r2=([0.359,0.529],[0.185,0.359]),

?r3=([0.195,0.333],[0.344,0.557]),

?r4=([0.089,0.286],[0.361,0.610]). Finally,we use the comparison laws of interval-valued intuitionistic fuzzy values given by Wang et al.[18]to rank?r i(i=1,2,3,4),and then s(?r1)= 0.6,s(?r2)=0.344,s(?r3)=?0.373,s(?r4)=?0.596. Thus,?r1 ?r2 ?r3 ?r4.Therefore,the ranking of the suppliers is x1 x2 x3 x4.

5.Conclusions

In this paper,the multiplicative consistency of IVIFPR has been investigated.With studying the desir-able property of the multiplicative consistent IVIFPR, we have proposed two algorithms,in which the for-mer procedure is used to construct the approximate multiplicative consistent IVIFPR with only known off-diagonal elements,while the latter is to build the perfect one in general case with much more known evaluation information.Subsequently,we have devel-oped a convergent iterative procedure to improve the multiplicative consistency of an inconsistent IVIFPR.

A convergent iterative algorithm has been proposed for group decision making with IVIFPRs.The numerical examples given in the paper have shown that our algo-rithms and procedures are effective in solving decision making problems with IVIFPR.

Acknowledgments

The authors would like to thank the anonymous ref-erees for their insightful and constructive comments and suggestions.The work was supported in part by the National Natural Science Foundation of China(No. 61273209),the Excellent PhD Thesis Foundation of Shanghai Jiao Tong University(No.20131216),and the Scholarship from China Scholarship Council(No. 201306230047).

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