Some issues on properties of the extended IOWA ... - Semantic Scholar
Expert Systems with Applications 38 (2011) 7059–7066
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Some issues on properties of the extended IOWA operators in fuzzy group decision making Jian Wu ⇑, Qing-wei Cao School of Business Administration, Zhejiang Normal University, Jinhua 321004, PR China
a r t i c l e
i n f o
Keywords: Group decision-making Fuzzy preference relation Consistency: IOWA operator
a b s t r a c t Chiclana et al. [Chiclana, F., Herrera-Viedma, E., Herrera, F., Alonso, S. (2007). European Journal of Operational Research 182, 383–399] provided some IOWA operators for aggregating the individual fuzzy preference elations. The aim of this work is further to study their desired properties under group decision making problem with fuzzy preference relations. First, it is proved that the collective preference relations obtained by these cases of IOWA operators verified the reciprocity and the consistency properties. Next, the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP) provide the same priorities of alternatives by utilizing the IOWA operators as aggregation procedure and the row arithmetic mean method (RAMM) as prioritization procedure. Additionally, the Consistency IOWA (CIOWA) operator guarantees that the group consistency degree is no less than the arithmetic mean of all the individual consistency degree. Finally, two illustrative numerical examples are used to verify the developed approaches. 2010 Elsevier Ltd. All rights reserved.
1. Introduction In the group decision making (GDM) problems, there are often two kinds of preference relations: multiplicative preference relations (Barzilai & Golany, 1994; Bryson, 1996; Escobar, Aguarón, & Moreno-Jiméenez, 2004; Satty, 1980; Xu, 2000; Xu & Wei, 2000) and fuzzy preference relations (Chang & Wang, 2009; Chiclana, Herrera, & Herrera-Viedma, 1998, 2001; Wang & Chang, 2007; Wang & Fan, 2007). After constructing preference relations, the decision makers (DMs) then need to aggregate the preference relation into a collective one, which reflects all of the properties contained in all individual preferences. Therefore, it is very important to establish some properties to be verified by such preference relations for GDM problems. One of these properties is so-called consistency property. The consistency measure is a very important problem in decision making using preference relations. The lack of consistency in decision making can lead to inconsistent conclusions. But, there usually arise situations of inconsistency among preference of DMs. Thus, finding a group consistency to present a common opinion of group is an important issue under group decision environment (Bordogna, Fedrizzi, & Pasi, 2002; Bryson, 1996; Choudhury, Shankar, & Tiwari, 2006; Herrera-Viedma,
⇑ Corresponding author. Tel.: +86 579 82298567. E-mail address: [email protected] (J. Wu). 0957-4174/$ - see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.007
Herrera, & Chiclana, 2002; Kacprzyk, Fedrizzi, & Nurmi, 1992; Wang & Lin, 2009). As a result, there is one problem must be solved, that is to say, how to choose an aggregation operator to achieve a higher degree of consistency of the collective preference relations. The Induced Ordered Weighted Averaging (IOWA) was introduced by Yager and Filev (1999), which is suitable to deal with GDM problems with fuzzy preference relations. Recently, Chiclana, Herrera-Viedma, Herrera, and Alonso (2007) presented some particular IOWA operators: the Consistency IOWA (C-IOWA) operator, which applied the ordering of the argument values based upon the consistency of the information sources; and the Importance IOWA (I-IOWA) operator, which applied the ordering of the argument values based upon the importance of the information sources. It is worthy of being mentioned that the C-IOWA operator aggregates individual preferences in such a way that more importance is placed on the most consistent one, which is very reasonable and meaningful. But, the knowledge on the use of these IOWA operators as aggregation method is still quite limited. For example, we still will ask the following questions: (1) If all the individual preferences are acceptable consistency, can the collective fuzzy preference relation by these IOWA operators be considered acceptable consistency? (2) To what degree of consistency can these IOWA operators improve in the collective fuzzy preference relation?
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The main aim of this article is to further study some desired properties of these IOWA operators. We introduced a consistency measure of fuzzy preference relations. Using this consistency measure, we proved that if the individual judgement matrixes have an acceptable consistency then the C-IOWA collective judgement matrix (C-IOWACJM) also is of acceptable consistency. Moreover, our result guarantees that the consistency of C-IOWACJM is smaller than the arithmetic mean of all the individual consistency. The I-IOWA operator also has similar properties. In order to do that, this work is set out as follows. Section 2 briefly reviews some basic concepts such as the IOWA, C-IOWA and I-IOWA operators. Section 3 defines the concept of consistency degree of fuzz preference relations. In Section 4, we study the desired properties of these IOWA operators in fuzzy group decision making. Section 5 provides two illustrative examples. Finally, in Section 6 we draw our conclusions.
2. Preliminaries: IOWA, C-IOWA and I-IOWA operators We start this section by making a generalization of the concepts of IOWA, C-IOWA and I-IOWA operators, which will be used throughout this paper. Yager and Filev (1999) presented an induced OWA (IOWA) operator in which the ordering of the ai(i 2 n) is induced by other variables ui(i 2 n) called the order inducing variables, where ai and ui are the components of the OWA pairs hui, aii(i 2 n). Definition 1. An IOWA operator of dimension n is a mapping, n UGW : Rþ ! Rþ , to which a set of weights or a weighting vector is P associated, W = (w1, w2, . . . , wn)T, wj 2 [0, 1] and nj¼1 wj ¼ 1, and it is defined to aggregate the set of second arguments of list of two tuples {hu1, a1i, . . . , hun, ani}, given on the basis of a positive ratio scale, according to the following expression: G fW ðhu1 ; a1 i; . . . ; hun ; an iÞ ¼
n X
wj bj ;
ð1Þ
der-inducing variable to induce the ordering of the argument values before their aggregation and presented the Consistency IOWA (C-IOWA) operator as follows: Definition 3. If a set of DMs D = {d1, d2, . . . , dm} provides preference about a set of alternatives X = {x1, x2, . . . , xn} by means of fuzzy preference relations {R(1), . . . , R(l), . . . , R(m)}, R(l) 2 R, then a C-IOWA operator is an IOWA operator in which its orderinducing values is the set of consistency index values {CI(R)(1), . . . , CI(R(l)), . . . , CI(R(m))}.
3. The measure of consistency index of fuzzy preference relations In group decision environment, the problem of consistency itself includes two problems (Herrera-Viedma, Herrera, Chiclana, & Luque, 2004) (i) when an DM, consider individually, is said to be consistent and, (ii) when can a whole group of decision makers be considered consistent. This section focuses on the first problem. We firstly introduce the concepts of the additive transitive fuzzy preference relations. Then we present the consistency index of fuzzy preference relations. In the following section, we will focus on the second problem. Definition 4. Let X = {x1, x2, . . . , xn} be a finite set of alternatives. If the DM gives his/her preference relation information on X by means of a preference relation R = (rij)nn, where
rij P 0;
r ij þ r ji ¼ 1;
r ii ¼ 0:5 for all i; j 2 N:
and rij denotes the preference degree or intensity of the alternative xi over xj, then R is called a fuzzy preference relation.
j¼1
where w = (w1, w2, . . . , wn)T is a weighting vector, such that Pn j¼1 wj ¼ 1; wj 2 ½0; 1; bj is the ai value of the IOWA pair having the jth largest ui, and ui in hui, aii is referred to as the order inducing variable and ai as the argument variable. Obviously, one key issue in the IOWA operator theory is to determine the order inducing variable ui, which applies the ordering of the argument values. In some cases, each DM has an importance degree associated with them so that we call this a heterogeneous GDM problem (Dubois & Koning, 1991; Dubois, Fargier, & Prade, 1997; Herrera, Herrera-Viedma, & Verdegay, 1998).The importance degree variable can be used as the order-inducing variable to induce the ordering of the argument values before their aggregation. Chiclana et al. (2007) presented the Importance IOWA (I-IOWA) operator, which applies the ordering of the argument values based upon the importance of the information sources.
Definition 5 (Tanino, 1984). Let R = (rij)nn be a fuzzy preference relation, then R is called an additive transitive fuzzy preference relation, if the following additive transitivity is satisfied:
rij ¼ r ik rjk þ 0:5;
for all i; j; k 2 N:
Definition 6. If we utilize the row arithmetic mean method ðlÞ ðlÞ (RAMM), then can get the priority vector xðlÞ ¼ x1 ; x2 ; . . . ; ðlÞ T (l) xn Þ of the fuzzy preference relation R , where
xðlÞ i ¼
n 1X ðlÞ r ; n j¼1 ij
i ¼ 1; 2; . . . ; n;
l ¼ 1; 2; . . . ; m:
Definition 7. Let A = (aij)nn 2 R and b = (bij)nn 2 R, then we define the distance between A and B as follows: n X n 1X jaij bij j: n i¼1 j¼1
Definition 2. If a set of DMs D = {d1, d2, . . . , dm} provides preference about a set of alternatives X = {x1, x2, . . . , xn} by means of fuzzy preference relations {R(1), . . . , R(l), . . . , R(m)}, and each have an importance degree l(dk) 2 [0, 1], associated to him or her, then an IIOWA operator is an IOWA operator in which its order-inducing values is the set of importance degree.
dðA; BÞ ¼
But in a homogeneous GDM problem (Herrera & Herrera-Viedma, 1997), each DM has equal importance degree. However, the DMs always can have a consistency index (CI) value associated with them. Chiclana et al. (2007) used the consistency as the or-
Theorem 1. Let A = (aij)nn 2 R and B = (bij)nn 2 R, then
ð2Þ
Obviously, the smaller the value of distance degree d(A, B), the closer of the fuzzy preference relations A and B.
(1) d(A, B) P 0; (2) d(A, B) = 0 if and only if A and B are perfectly consistent.
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Proof n X n 1X tjaij bij j P 0: (1) dðA; BÞ ¼ n i¼1 j¼1
ð3Þ
(2) Necessity. If d(A, B) = 0, then aij = bij for all i, j 2 N. Therefore, A and B are perfectly consistent. (3) Sufficiency. If A and B are perfectly consistent, then aij = bij for all i, j 2 N. Thus, we have aij bij = 0 for all i, j 2 N. Therefore, d(A, B) = 0. h In decision making problems based on fuzzy preference relations, the study of consistency is associated the transitivity property. Herrera-Viedma et al. (2004) gave a characterization of the consistency property defined by the additive transitivity property of a fuzzy preference relation Rk ¼ ðrkij Þ:
r kij þ r kjl þ r kil ¼
3 ; 2
8i; j; l 2 f1; . . . ; ng:
Using this characterization method, a procedure to construct a consistent reciprocal fuzzy preference relation R on X = {x1, . . . , xn, n P 2} from n 1 preference values {r12, r23 . . . , rn1n} presents the following steps: _
_
1. R ¼ ð r ij Þ such that:
8 if r > < ij jðiþ1Þ if r ij ¼ ðr iiþ1 þ r iþ1iþ2 . . . rj1j Þ 2 > : _ 1 r ij if
i 6 j 6 i þ 1; i þ 1 < j;
In this section, we implement the C-IOWA operator and the IIOWA operator to aggregate individual fuzzy preference relations in GDM problems, and then study their desired properties. 4.1. The Consistency IOWA (C-IOWA) operator In a homogeneous GDM problem, the DMs have equal importance. However, each DM always can have a Consistency Index (CI) value associated with them, which measures the level of consensus between individual preferences and group preference. Thus, the more consistency is the information provided by the DM, the higher the weighting value should be placed on that information. In this section, we implement the C-IOWA operator to aggregate individual preferences in such a way the greater weight is given the most consistency one. Then we study the reciprocity and consistency properties of the C-IOWACJM, which is obtained by using C-IOWA operator. Definition 9. If R(1), . . . , R(l), . . . , R(m) are the fuzzy preference rela_ _ tions provided by m DMs, then the C-IOWACJM R ¼ ð r ij Þnn is expressed as follows: _ R ¼ C-IOWA hCIðRð1Þ Þ; Rð1Þ i; hCIðRð2Þ Þ; Rð2Þ i; ; hCIðRðmÞ Þ; RðmÞ i ¼ C-IOWA hCIðRðrð1ÞÞ Þ; Rðrð1ÞÞ i; hCIðRðrð2ÞÞ Þ; Rðrð2ÞÞ i; ; hCIðRðrðmÞÞ Þ; RðrðmÞÞ i ¼ ðRðrð1ÞÞ crð1Þ Þ þ ðRðrð2ÞÞ crð2Þ Þ þ . . . þ ðRðrðmÞÞ crðmÞ Þ; ðrð1ÞÞ ðrð2ÞÞ ðrðmÞÞ r ij ¼ r ij crð1Þ þ rij crð2Þ þ þ ðaij crðmÞ Þ
j < i:v
ð5Þ
_
b could have entries not in the interval [0, 1], but in But the matrix R _ _ _ an interval [ a, 1 + a], being a ¼ minf r ij ; r ij 2 Rg . In such a case, Herrera-Viedma et al. (2004) presented a transfor- mation function which preserves reciprocity and additive consistency, that is a function f : ½a; 1 þ a ! ½0; 1, verifying (a) (b) (c) (d)
4. The properties of IOWA operators in fuzzy group decision making
f(a) = 0. f(1 + a) = 1. f(x) + f(1 x) = 1, "x 2 [ a, 1 + a]. f ðxÞ þ f ðyÞ þ f ðzÞ ¼ 32 ; 8x; y;z 2 ½a;1 þ a such that x þ y þ z ¼ 32.
2. The consistent fuzzy preference relation P is obtained as _ P ¼ f ðRÞ.
¼
m Y ðrðlÞÞ ðaij crðlÞ Þ;
ð6Þ
l¼1
where (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n) such that CI(R(r(l1))) P CI(R(r(l))) (r(l))
and
cr(l1) P cr(l) for all l = 2, . . . , m;
(r(l))
), R i is the two tuple with CI(R(r(l))) the lth biggest value hCI(R in the set {CI(R(1)), . . . , CI(R(m))}; c = (cr(1), cr(2), . . . , cr(m))T is a P weighting vector, such that m l¼1 crðlÞ ¼ 1; crðlÞ 2 ½0; 1. Yager (2003) proposed a procedure to determine the weighting vector associated to an IOWA operator. In this case, each comment ðlÞ ðlÞ in the aggregation consists of a triple ðpij ; ul ; v l Þ : pij is the argument value to aggregate, ul is the importance weight value associated to ðlÞ pij , and vl is the order inducing value. Thus, the aggregation is
This consistency index has a definite physical implication and reflects the deviation degree between the fuzzy preference relation R(l) and its corresponding consistent matrix P(l). In this article, we define the concept of consistency index based on the distance between R(l) and its corresponding consistent matrix P(l) as follows.
n X ð1Þ ðnÞ rðlÞ IOWAQ pij ; . . . ; pij ¼ wl pij ;
Definition 8. Suppose R(1), . . . , R(l), . . . , R(m) be the fuzzy preference relations provided by m DMs and P(1), . . . , P(l), . . . , P(m) be their corresponding consistent matrix, then we define a measure of consistency index (CI) of the fuzzy preference relation R(l) as follows:
P where SðlÞ ¼ lk¼1 urðkÞ , and r is the permutation such that ur(l) in rðlÞ ðpij ; urðlÞ ; v rðlÞ Þ is the lth largest value in the set {v1, . . . , vn}. Q is a function: [0, 1] ? [0, 1] such that Q(0) = 0, Q(1) = 1 and if x > y then Q(x) P Q(y) (Yager & Kacprzyk, 1997). In our case, we propose to use the consistency values associated to each one of the DM both as a weight associated to the argument and as the order inducing values ui = vi = CI(R(i)). Thus, the ordering of the preference values is first induced by the ordering of the DMs from most to least consistency one, and the weights of the C-IOWA operator is obtained by applying the above Eq. (10), with reduces to
CIðRðlÞ Þ ¼ 1 dðRðlÞ ; PðlÞ Þ:
ð4Þ
Obviously, the closer CI(R(l)) is to 1 the more consistent the information provided by the DM d(l), and thus more importance should be placed on that information. When using this consistency index, we can obtain some desired properties of C-IOWA operator (see Theorem 6)
l¼1
with
wl ¼ Q
crðlÞ ¼ Q
SðlÞ Sðl 1Þ Q SðnÞ SðnÞ
SðrðlÞÞ Sðrðl 1ÞÞ Q ; SðrðnÞÞ SðrðnÞÞ
ð7Þ
ð8Þ
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J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
Pl
ðrðkÞÞ Þ, and k¼1 CIðR ðrðlÞÞ ðrðlÞÞ (r(l)) ) in ðr ij ; CIðR Þ; CIðRðrðlÞÞ ÞÞ CI(R (r(1)) (r(n))
where SðrðlÞÞ ¼
ðrð1ÞÞ r ij ¼ rij crð1Þ þ rijðrð2ÞÞ crð2Þ þ . . . þ rðijrðmÞÞ crðmÞ P 0 crð1Þ þ 0 crð2Þ þ . . . þ 0 crðmÞ ¼ 0; _ _ ðrð1ÞÞ ðrð1ÞÞ þ r ji crð1Þ þ rijðrð2ÞÞ þ rjiðrð2ÞÞ crð2Þ þ . . . r ij þ r ji ¼ r ij ðrðmÞÞ ðrðmÞÞ þ r ij þ r ji crðmÞ _
r is the permutation such that is the lth largest value in the
set {CI(R ), . . . , CI(R )}. In an aggregation process, we consider that the weighting value of DMs should be implemented in such a way that the effect from those DMs who are less consistency is reduced, and therefore the above is obtained if the linguistic quantifier Q verifiers that the most the consistency of an DM the higher the weighting value of that DM in the aggregation, i.e.:
¼ crð1Þ þ crð2Þ þ . . . þ crðmÞ ¼ 1; ðrð1ÞÞ crð1Þ þ rðiirð2ÞÞ crð2Þ þ . . . þ rðiirðmÞÞ crðmÞ r ii ¼ rii 1 1 1 1 crð1Þ þ crð2Þ þ . . . þ crðmÞ ¼ : ¼ 2 2 2 2
_
CIðRðrð1ÞÞ Þ P CIðRðrð2ÞÞ Þ P . . . P CIðRðrðnÞÞ Þ P 0 ) crð1Þ P crð2Þ . . . P crðnÞ P 0:
_
Thus, R ¼
Theorem 2. Assuming the parameterized family of RIM quantifiers P Q(r) = ra, a P 0, if a 2 [0, 1] and SðrðlÞÞ ¼ lk¼1 CIðRðrðkÞÞ Þ, then cr(l) P cr(l+1), for all l = 1, 2, . . . , m
_ r ij
is also a fuzzy preference relation.
nn
(ii) Since all the R(1), R(2), . . . , R(m) are consistent, i.e., then
1 ðlÞ ðlÞ ðlÞ r ij ¼ r ik þ r kj ; 2
for all l ¼ 1; 2; . . . ; m i; j 2 N:
Thus Proof. If a 2 [0, 1], then the function Q(r) = ra is concave and we can have
_
_
r ik þ r kj ¼
m m X X ðrðlÞÞ ðrðlÞÞ rik crðlÞ þ r kj crðlÞ l¼1
ðlÞÞ Suppose T l ¼ SðSðrrðnÞÞ , and SðrðlÞÞ ¼
crðlÞ
Pl
ðrðkÞÞ Þ, k¼1 CIðR
then
SðrðlÞÞ Sðrðl 1ÞÞ ¼Q Q ¼ Q ðT l Þ Q ðT l1 Þ SðrðnÞÞ SðrðnÞÞ
and
crðlþ1Þ ¼ Q
_
¼ r ij þ _
1 2
and thus, R is also consistent, which completes the proof of Theorem 3. h
Sðrðl þ 1ÞÞ SðrðlÞÞ Q ¼ Q ðT lþ1 Þ Q ðT l Þ SðrðnÞÞ SðrðnÞÞ
Thus, we can obtain
crðlÞ P crðlþ1Þ which completes the Theorem 2.
l¼1
m m X X 1 ðrðlÞÞ ðrðlÞÞ ðrðlÞÞ ¼ r ik þ rkj crðlÞ ¼ rij þ c 2 rðlÞ l¼1 l¼1
Q ðT l Þ QðT l1 Þ P QðT lþ1 Þ Q ðT l Þ:
h
In GDM models with fuzzy preference assessments, it is usually assumed that the fuzzy preference relations to express the judgments are reciprocal. The C-IOWA operator is able to maintain both the reciprocity and the consistency properties in the collective fuzzy preference relation. To study these desired properties, we derive the following theorems.
As regards group decision making, the analytic hierarchy process (AHP) considers two different approaches: the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP). For the RGMM prioritization procedure and the WGMM aggregation procedure, Barzilai & Golany (1994) proved that both aggregation approaches (AIJ and AIP) provide the same priorities of alternatives. We also can draw an analogous conclusion when use the row arithmetic mean method (RAMM) as the prioritization procedure and C-IOWACJM as the aggregation procedure. Definition 11. Denote R(l) 2 R be the fuzzy judgement matrix provided by the lth DM when comparing n alternatives, xðlÞ ¼ ðlÞ
ðlÞ
ðlÞ
ðlÞ
ðx1 ; x2 ; . . . ; xn ÞT as its priority vector, P ðlÞ ¼ ðpij Þnn as the _
_ _
_ T
1
n
corresponding consistent matrix; x ¼ ðx; x; . . . ; x Þ as the priorTheorem 3. Let R(1), R(2), . . . , R(m) be fuzzy preference relations ðlÞ provided by m DMs, where RðlÞ ¼ _ ðr ij Þnn ; ðl ¼ 1; 2; . . . ; m; i; j ¼ _ 1; 2; . . . ; nÞ, then their C-IOWACJM R ¼ ð r ij Þnn is also a fuzzy preference relation, where _
ED E D E ð2Þ ðmÞ ; CIðRð2Þ Þ; r ij ; . . . ; CIðRðmÞ Þ; r ij D ED E D E ðrð1ÞÞ ðrð2ÞÞ ðrðmÞÞ ; CIðRðrð2ÞÞ Þ; r ij ; . . . ; CIðRðrðmÞ Þ; r ij ¼ CI-IOWA CIðRðrð1ÞÞ Þ; r ij ðrð1ÞÞ ¼ r ij crð1Þ þ rðijrð1ÞÞ crð2Þ þ . . . þ rðijrðmÞÞ crðmÞ
r ij ¼ CI-IOWA
D
ð1Þ
CIðRð1Þ Þ; r ij
and
r ij P 0;
_
ity vector of C-IOWACJM R , and P ¼ ðpij Þnn as the corresponding _
consistent matrix of R . Theorem 4. Using the C-IOWACJM as the aggregation procedure, the weighting vector c ¼ ðcrð1Þ ; crð2Þ ; . . . ; crðmÞ ÞT ; crðl1Þ P crðlÞ ; Pm l¼1 crðlÞ ¼ 1, and the RAMM as the prioritization procedure, it holds that the AIJ and the AIP provides the same priorities of alternatives. Proof. Let xðlÞ ¼ ðx1 ; x2 ; . . . ; xn ÞT be the priority of the individ_ ual judgement matrix R(l) and x ¼ ðx1 ; x2 ; . . . ; xn ÞT be the group priorities, then we can obtain ðlÞ
rij þ rji ¼ 1;
1 r ii ¼ ; for all i; j 2 N: 2 _
Furthermore, if all the R(1), R(2), . . . , R(m) are consistent, then R is also consistent. Proof (i) Since R(1), R(2), . . . , R(m) are fuzzy preference relations, we have rij P 0; rij þ r ji ¼ 1 rii ¼ 12 ; for all i; j 2 N, and then
2
_
ðlÞ
ðlÞ
xðAIPÞ ¼ C-IOWA hCIðRð1Þ Þ; xð1Þ i; hCIðRð2Þ Þ; xð2Þ i; ; hCIðRðmÞ Þ; xðmÞ i
¼ C-IOWA hCIðRðrð1ÞÞ Þ; xðrð1ÞÞ i; hCIðRðrð2ÞÞ Þ; xðrð2ÞÞ i; ; hCIðRðrðmÞÞ Þ; xðrðmÞÞ i ¼ xðrð1ÞÞ crð1Þ þ xðrð2ÞÞ crð2Þ þ þ xðrðmÞÞ crðmÞ
xi ðAIPÞ ¼
m X l¼1
xiðrðlÞÞ crðlÞ
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and n n X m m n X X _ 1X 1X 1 ðrðlÞÞ ðrðlÞÞ r xi ðAIJÞ ¼ r ij ¼ r crðlÞ ¼ crðlÞ n j¼1 n j¼1 l¼1 ij n ij j¼1 l¼1
¼
m X
!
crðlÞ xiðrðlÞÞ :
Since
X X m m ðrðlÞÞ ðrðlÞÞ ðrðlÞÞ ðrðlÞÞ r ij pij crðlÞ 6 pij crðlÞ : r ij l¼1 l¼1 Then
l¼1
_
Thus
CIðR Þ P 1
xi ðAIPÞ ¼ xi ðAIJÞ:
m X
¼1
This completes the proof of the Theorem 4. h
n X n X m 1X ðrðlÞÞ ðrðlÞÞ pij crðlÞ r ij n i¼1 j¼1 l¼1
l¼1
As mentioned above, the C-IOWA operator aggregates individual preferences into a collective preference in such a way the greater weight is given the most consistency one. We want to see how the C-IOWA operator improves consistency in the collective fuzzy preference relation. Thus we derive the following theorems. Lemma 1 (Liu & Chen, 2004). For ordered vector x = (x1, x2, . . . , xn), x1 P x2 P, . . . , P xn, and weights w = (w1, w2, . . . , wn), then vector a1, a2, . . ., an, ai P 0(i = 1, 2, . . . , n). If w1 P w2 P P wn, then
a1 w1 x1 þ a2 w2 x2 þ þ an wn xn a1 w1 þ a2 w2 þ þ an wn P ða1 x1 þ a2 x2 þ þ an xn Þ: a1 þ a2 þ þ an
¼
ð9Þ
ð10Þ
with equality if and only if x1 = x2 = = xn Theorem 5. For ordered vector x = (x1, x2, . . . , xn), x1 P x2 P P xn, and weights
wi ¼ 1; then
i¼1
ðx1 þ x2 þ þ xn Þ w1 x1 þ w2 x2 þ þ wn xn P : n
crðlÞ CIðRðlÞ Þ:
Since CI(R(r(l))) P CI(R(r(l+1))) and cr(1) P cr(2) P P cr(m) Then, from Theorem 5, we have m X
crðlÞ CIðRðrðlÞÞ Þ P
m m 1 X 1 X CIðRðrðlÞÞ Þ ¼ CIðRðlÞ Þ: m l¼1 m l¼1
Thus _
n X
m X l¼1
CIðR Þ P
a1 w1 x1 þ a2 w2 x2 þ þ an wn xn a1 w1 þ a2 w2 þ þ an wn ða1 x1 þ a2 x2 þ þ an xn Þ 6 a1 þ a2 þ þ an
crðlÞ
l¼1
l¼1
If w1 6 w2 6 6 wn, then
w ¼ ðw1 ; w2 ; . . . ; wn Þ; if w1 P w2 P . . . P wn and
¼
m X
n X n 1X ðrðlÞÞ ðrðlÞÞ pij r ij n i¼1 j¼1 ! n X n 1X ðrðlÞÞ ðrðlÞÞ 1 pij r ij n i¼1 j¼1
crðlÞ
m 1 X CIðRðlÞ Þ: m l¼1
We complete the proof of the Theorem 6.
h
Corollary 1. If the individual fuzzy judgements _ R(1), R(2), . . . , R(m) are of acceptable consistency, then the C-IOWACJM R is also acceptable consistency, that is to say,
CIðRðlÞ Þ P s;
_
8l ¼ 1; . . . ; m ) CIðRÞ P s;
ð13Þ
where s is the threshold for acceptable consistency.
ð11Þ
_
Corollary 2. The consistency degree of R is greater than the minimum of the consistency degree of R(l), which holds that
Proof. Omitted. h
_
CIðR Þ P Minl¼1;...;m fCIðRðlÞ Þg:
ð14Þ
(l)
Definition 12. Let CI(R ) be a measure of the consistency of matrix _ R(l), and_ CIðR Þ be a measure of the consistency of the collective matrix R .
4.2. The importance IOWA (I-IOWA) operator
Theorem 6. Suppose R(1), R(2), . . . , R(m) be the fuzzy preference relations provided by m DMs when comparing n alternatives with the corresponding weighting vector c = (cr(1), cr(2), . . . , cr(m))T, cr(l1) P cr(l), Pm l¼1 crðlÞ ¼ 1. Using the C-IOWACJM as the aggregation procedure and the RAMM as the prioritization procedure, it holds that:
In a heterogeneous GDM problem (Dubois et al., 1997), each expert has an importance degree associated with them. Chiclana et al. (2007) developed the I-IOWA operator, which used this importance degree variable as the order-inducing variable to induce the ordering of the argument values before their aggregation. In this section, we study the reciprocity and consistency properties of the I-IOWACJM, which is obtained by using I-IOWA operator.
_
CIðR Þ P
m 1 X CIðRðlÞ Þ: m l¼1
ð12Þ
Proof. By Definition 11 and Eq. (4), we have _
_ _
CIðR Þ ¼ 1 dðR; P Þ ¼ 1
n X n _ 1X _ r ij p ij n i¼1 j¼1
n X n X m m X 1X ðrðlÞÞ ðrðlÞÞ ¼1 rij crðlÞ pij crðlÞ n i¼1 j¼1 l¼1 l¼1 n X n X m 1X ðrðlÞÞ ðrðlÞÞ ¼1 r ij pij crðlÞ : n i¼1 j¼1 l¼1
Definition 13. If a set of DMs D = {d1, d2, . . . , dm} provides preference about a set of alternatives X = {x1, x2, . . . , xn} by means of fuzzy preference relations {R(1), R(2), . . . , R(m)}, whose associated imporPm tance degree k ¼ ðk1 ; k2 ; . . . ; km Þ; l¼1 kl ¼ 1 0 6 kl 6 1, then the I-IOWACJM R ¼ ðr ij Þnn is expressed as follows:
R ¼ I IOWA hk1 ; Rð1Þ i; hk2 ; Rð2Þ i; ; hkm ; RðmÞ i ¼ ðRð1Þ k1 Þ þ ðRð2Þ k2 Þ þ . . . þ ðRðmÞ km Þ; m X ðlÞ rij ¼ aij kl ; l¼1
ð15Þ ð16Þ
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J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
In GDM models with fuzzy preference assessments, it usually is assumed that the fuzzy preference relations to express the judgments are reciprocal. The I-ILOWA operator also is able to maintain both the reciprocity and consistency properties in the collective fuzzy preference relation. To study these desired properties, we derive the following theorems.
and n n X m m n X X 1X 1X 1 ðlÞ ðlÞ r ij ¼ r xi ðAIJÞ ¼ r kl ¼ kl n j¼1 n j¼1 l¼1 ij n ij j¼1 l¼1
! ¼
m X
xðlÞ i kl :
l¼1
Thus
xi ðAIPÞ ¼ xi ðAIJÞ: Theorem 7. Let {R(1), R(2), . . . , R(m)} be fuzzy preference relations ðlÞ provided by m DMs, where RðlÞ ¼ ðrij Þnn , (l = 1, 2, . . . , m; i, j = 1, 2, . . . , n), then their I-IOWACJM R ¼ ðrij Þnn is also a fuzzy preference relation, where
r ij ¼
m X
ðlÞ
r ij kl
l¼1
andrij P 0; rij þ r ji ¼ 1; r ii ¼ 12, for all i, j 2 N. Furthermore, if all the R(1), R(2), . . . , R(m) are consistent, then R is also consistent.
This completes the proof of the Theorem 8.
h
Theorem 9. Suppose R(1), R(2), . . . , R(m) be the fuzzy preference relations provided by m DMs when comparing n alternatives with the corresponding weighting vector c = (cr(1), cr(2),. . . ; crðmÞ ÞT ; crðl1Þ P crðlÞ ; Pm l¼1 crðlÞ ¼ 1. Using the I-IOWACJM as the aggregation procedure and the RAMM as the prioritization procedure, it holds that:
CIðRÞ P
m X
kl CIðRðlÞ Þ:
ð17Þ
l¼1
Proof (i) Since R(1), R(2), . . . , R(m) are fuzzy preference relations, we have rij P 0, rij + rji = 1 rii ¼ 12 ; for all i; j 2 N, and then
r ij ¼
m X
ðlÞ
r ij kl P
l¼1
m X
m X
ðlÞ
r ij kl þ
¼
m X
m X
ðlÞ
r ji kl ¼
l¼1
m X ðlÞ ðlÞ ðr ij þ rji Þ kl l¼1
kl ¼ 1;
l¼1 m X
ðlÞ
aii kl ¼
l¼1
Since
m X 1 1 kl ¼ : 2 2 l¼1
X m m X ðlÞ ðlÞ ðlÞ ðlÞ r ij pij kl 6 r ij pij kl : l¼1 l¼1
Thus, R ¼ ðr ij Þnn is also a fuzzy preference relation. (ii) Since all the R(1), . . . , R(l), . . . , R(m) are consistent, i.e., ðlÞ
1 ðlÞ ðlÞ ¼ rik þ r kj ; 2
rik þ r kj ¼
m X
ðlÞ rik
þ
Then
for all l ¼ 1; 2; . . . ; m i; j 2 N:
Then ðlÞ r kj
l¼1
n X n 1X r ij pij n i¼1 j¼1
n X n X m m X 1X ðlÞ ðlÞ rij kl pij kl ¼1 n i¼1 j¼1 l¼1 l¼1 n X n X m 1X ðlÞ ðlÞ ¼1 r ij pij kl : n i¼1 j¼1 l¼1
0 kl ¼ 0;
l¼1
rij
CIðRÞ ¼ 1 dðR; PÞ ¼ 1
l¼1
r ij þ r ji ¼
r ii ¼
Proof. By Definition 14 and Eq. (4), we have
CIðRÞ P 1
m X 1 1 ðlÞ kl ¼ r kj þ r ij þ kl ¼ 2 2 l¼1
and thus, R is also consistent, which completes the proof of Theorem 7. h Definition 14. Denote R(l) 2 R be the fuzzy judgement matrix provided by the l-th DM when comparing n alternatives, xðlÞ ¼ T ðlÞ ðlÞ ðlÞ xðlÞ as its priority vector, P ðlÞ ¼ pij as the cor1 ; x2 ; . . . ; xn nn
¼ ðx 1; x 2; . . . ; x n ÞT as the priority responding consistent matrix; x ij Þnn as the corresponding convector of I-IOWACJM R, and P ¼ ðp sistent matrix of R.
n X n X m 1X ðlÞ ðlÞ r ij pij kl n i¼1 j¼1 l¼1
m X
¼1
l¼1
¼
m X
kl
l¼1
kl ð
n X n 1X ðlÞ ðlÞ r ij pij n i¼1 j¼1
n X n 1X ðlÞ ðlÞ 1 r ij pij n i¼1 j¼1
!
We complete the proof of the Theorem 9.
¼
m X
kl CIðRðlÞ Þ:
l¼1
h
Corollary 3. If the individual fuzzy judegements R(1), R(2), . . . , R(m) are of acceptable consistency, then the I-IOWACJM R is also acceptable consistency, that is to say,
CIðRðlÞ Þ P s;
8l ¼ 1; . . . ; m ) CIðRÞ P s;
ð18Þ
where s is the threshold for acceptable consistency. Theorem 8. Using the I-IOWACJM as the aggregation procedure, the Pm weighting vector k ¼ ðk1 ; k2 ; . . . ; km ÞT ; l¼1 kl ¼ 1, the RAMM as the prioritization procedure, it holds that the AIJ and the AIP provides the same priorities of alternatives.
ðlÞ 1 ;
ðlÞ 2 ;...; (l)
ðlÞ n
CIðRÞ P Minl¼1;...;m fCIðRðlÞ Þg:
T
Proof. Let x ¼ x x x be the priority of the individ ¼ ðx 1; x 2; . . . ; x n ÞT be the group ual judgement matrix R and x priorities, then we can obtain ðlÞ
xðAIPÞ ¼ I-IOWA hk1 ; xð1Þ i; hk2 ; xð2Þ i; ; hkm ; xðmÞ i ¼ xi ðAIPÞ ¼
m X l¼1
m X l¼1
xðlÞ i kl
Corollary 4. The consistency degree of R is greater than the minimum of the consistency degree between R(l), which holds that
xðlÞ kl
ð19Þ
5. Illustrative examples Example 1. In order to see how the C-IOWA operator woks in practice, let us consider the example used in Chiclana et al. (2007). In this example, there are four DMs D = {d1, d2, d3, d4}, and a set of four alternatives X = {x1, x2, x3, x4}. Assume that these DMs provide the following fuzzy preference relations on the set of alternatives:
J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
3
2
ð1Þ
R
0:5 0:3 0:7 0:1 6 0:7 0:5 0:6 0:6 7 7 6 ¼6 7; 4 0:3 0:4 0:5 0:2 5
R
ð2Þ
0:5 0:4 0:6 0:2 6 0:6 0:5 0:7 0:4 7 7 6 ¼6 7; 4 0:4 0:3 0:5 0:1 5
0:9 0:4 0:8 0:5 2
0:5 0:5 0:7 0:0
3
2
Using the DMs’ importance degree k to induce the ordering of these fuzzy preference relations to be aggregated, we can obtain the following collective fuzzy preference relation R ¼ ðr ij Þnn and its corresponding consistent matrix P :
0:8 0:6 0:9 0:5 3
2
6 0:5 0:5 0:8 0:4 7 6 7 Rð3Þ ¼ 6 7; 4 0:3 0:2 0:5 0:2 5
0:5 0:4 0:7 0:8
R ¼ I-IOWAk hk1 ; Rð1Þ i; hk2 ; Rð2Þ i; hk3 ; Rð3Þ i; hk4 ; Rð4Þ i 2 3 0:5000 0:3850 0:6800 0:3500 6 0:6150 0:5000 0:5800 0:4250 7 6 7 ¼6 7; 4 0:3200 0:4200 0:5000 0:1450 5
3
6 0:6 0:5 0:4 0:3 7 6 7 Rð4Þ ¼ 6 7: 4 0:3 0:6 0:5 0:1 5
1:0 0:6 0:8 0:5
0:2 0:7 0:9 0:5
By using the above procedure, we can obtain four consistent matrices as follows:
2
P
ð1Þ
0:5 6 0:7 6 ¼6 4 0:6 0:9
0:3 0:4 0:1
3
0:5 0:6 0:3 7 7 7; 0:4 0:5 0:2 5
2
P
0:7 0:8 0:5 3
2
ð2Þ
0:5 6 0:6 6 ¼6 4 0:4 0:8
0:4 0:6 0:2
0:7 0:9 0:5
0:5 0:5 0:8 0:5 0:5 5=12 6 0:5 0:5 0:8 0:5 7 6 7=12 0:5 7 6 6 ð4Þ Pð3Þ ¼ 6 7; P ¼ 6 4 0:2 0:2 0:5 0:2 5 4 2=3 7=12 0:5 0:5 0:8 0:5
1:0
1=3
0:0
CIðRð2Þ Þ ¼ 0:95;
CIðRð3Þ Þ ¼ 0:65;
CIðRð4Þ Þ ¼ 0:26:
Rðrð2ÞÞ ¼ Rð1Þ ;
Rðrð3ÞÞ ¼ Rð3Þ ;
Rðrð4ÞÞ ¼ Rð4Þ ;
Pðrð1ÞÞ ¼ Pð2Þ ;
Pðrð2ÞÞ ¼ Pð1Þ ;
Pðrð3ÞÞ ¼ Pð3Þ ;
Pðrð4ÞÞ ¼ P ð4Þ ;
Using Eq. (8) with Q(r) = r1/2, we get the following weights
crð1Þ ¼ 0:61; crð2Þ ¼ 0:19; crð3Þ ¼ 0:15; crð4Þ ¼ 0:05; _
_
Then, the C-IOWACJM R1 and its corresponding consistent matrix P 1 are calculated as:
0:3960 0:6390 0:1810
3
7 6 6 0:6040 0:5000 0:6810 0:4330 7 7 R1 ¼ 6 6 0:3610 0:3190 0:5000 0:1340 7; 5 4
_
0:8190 0:5670 0:8660 0:5000 2
0:5000
0:3968 0:5787 0:2160
4 X
CIðRðlÞ Þkl ¼ 0:59;
This confirms the conclusion of Theorem 9.
6. Conclusions
Rðrð1ÞÞ ¼ Rð2Þ ;
0:5000
CIðRÞ ¼ 0:69 >
l¼1
and the judgement matrices R(1), R(2), R(3), R(4) and the corresponding consistent matrices P(1), P(2), P(3), P(4) are reordered as follows, respectively:
2
According to Definition 14 and Eq. (4), we can obtain that
0:5
According to Eq. (4), we can calculate the consistency degree CI(R(l)), l = 1, 2, 3, 4:
CIðRð1Þ Þ ¼ 0:70;
hk1 ; P ð1Þ i; hk2 ; Pð2Þ i; hk3 ; Pð3Þ i; hk4 ; Pð4Þ i 3 0:3908 0:4767 0:1450 0:5000 0:5858 0:2542 7 7 7: 0:4142 0:5000 0:1683 5
0:8550 0:7458 0:8317 0:5000 3
5=12 1=12 7 7 7: 0:5 1=6 5
11=12 5=6
0:6500 0:5750 0:8550 0:5000 P ¼ I IOWAk 2 0:5000 6 0:6092 6 ¼6 4 0:5233
3
0:5 0:7 0:3 7 7 7; 0:3 0:5 0:1 5
2
7065
3
6 0:6032 0:5000 0:6818 0:3192 7 6 7 P1 ¼ 6 7: 4 0:4213 0:3182 0:5000 0:1373 5
_
0:7840 0:6808 0:8627 0:5000
In this article, we have studied the properties of the IOWA operatrs in the aggregation of fuzzy prefernce relations in GDM problems. We have shown that the collective preference relations obtained by these cases of IOWA operators verified the reciprocity and consistency properties. Then, it is proved that the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP) provide the same priorities of alternatives by utilizing the row arithmetic mean method (RAMM) as prioritization procedure and the IOWA operators as aggregation procedure. By using the distance between R(l) and its corresponding consistent matrix P(l), we present the consistency index of fuzzy preference relations. Using this consistency measure, we proved that the C-IOWA and the I-IOWA operator can improve consistency degree in the collective fuzzy preference relation. Finally, a theoretic basis has been developed for the application of these cases of IOWA operators in fuzzy group decision making. However, many decision making progress takes place in which the information is not precisely known. The decision makers cannot estimate their preference with an exact numerical value, but with an interval number or a linguistic viable (Wu, Li, & Li, 2009; Xu, 2004). In our future research, we plan to extend the IOWA operators to deal with these inexact numerical values in GDM problems and study their properties. Acknowledgements
_ 4 1X 0:95 þ 0:70 þ 0:65 þ 0:26 ¼ 0:64; CI R1 ¼ 0:90 > CIðRðlÞ Þ ¼ 4 l¼1 4
This work was supported by Humanities and Social Science Projects of Chinese Ministry of Education under the Grant No. 10YJC630277 and Zhejiang Provincial Natural Science Foundation of China under the Grant No. Y6080215.
Then, we can find that this result is in accordance with Theorem 6.
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Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Some issues on properties of the extended IOWA operators in fuzzy group decision making Jian Wu ⇑, Qing-wei Cao School of Business Administration, Zhejiang Normal University, Jinhua 321004, PR China
a r t i c l e
i n f o
Keywords: Group decision-making Fuzzy preference relation Consistency: IOWA operator
a b s t r a c t Chiclana et al. [Chiclana, F., Herrera-Viedma, E., Herrera, F., Alonso, S. (2007). European Journal of Operational Research 182, 383–399] provided some IOWA operators for aggregating the individual fuzzy preference elations. The aim of this work is further to study their desired properties under group decision making problem with fuzzy preference relations. First, it is proved that the collective preference relations obtained by these cases of IOWA operators verified the reciprocity and the consistency properties. Next, the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP) provide the same priorities of alternatives by utilizing the IOWA operators as aggregation procedure and the row arithmetic mean method (RAMM) as prioritization procedure. Additionally, the Consistency IOWA (CIOWA) operator guarantees that the group consistency degree is no less than the arithmetic mean of all the individual consistency degree. Finally, two illustrative numerical examples are used to verify the developed approaches. 2010 Elsevier Ltd. All rights reserved.
1. Introduction In the group decision making (GDM) problems, there are often two kinds of preference relations: multiplicative preference relations (Barzilai & Golany, 1994; Bryson, 1996; Escobar, Aguarón, & Moreno-Jiméenez, 2004; Satty, 1980; Xu, 2000; Xu & Wei, 2000) and fuzzy preference relations (Chang & Wang, 2009; Chiclana, Herrera, & Herrera-Viedma, 1998, 2001; Wang & Chang, 2007; Wang & Fan, 2007). After constructing preference relations, the decision makers (DMs) then need to aggregate the preference relation into a collective one, which reflects all of the properties contained in all individual preferences. Therefore, it is very important to establish some properties to be verified by such preference relations for GDM problems. One of these properties is so-called consistency property. The consistency measure is a very important problem in decision making using preference relations. The lack of consistency in decision making can lead to inconsistent conclusions. But, there usually arise situations of inconsistency among preference of DMs. Thus, finding a group consistency to present a common opinion of group is an important issue under group decision environment (Bordogna, Fedrizzi, & Pasi, 2002; Bryson, 1996; Choudhury, Shankar, & Tiwari, 2006; Herrera-Viedma,
⇑ Corresponding author. Tel.: +86 579 82298567. E-mail address: [email protected] (J. Wu). 0957-4174/$ - see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.007
Herrera, & Chiclana, 2002; Kacprzyk, Fedrizzi, & Nurmi, 1992; Wang & Lin, 2009). As a result, there is one problem must be solved, that is to say, how to choose an aggregation operator to achieve a higher degree of consistency of the collective preference relations. The Induced Ordered Weighted Averaging (IOWA) was introduced by Yager and Filev (1999), which is suitable to deal with GDM problems with fuzzy preference relations. Recently, Chiclana, Herrera-Viedma, Herrera, and Alonso (2007) presented some particular IOWA operators: the Consistency IOWA (C-IOWA) operator, which applied the ordering of the argument values based upon the consistency of the information sources; and the Importance IOWA (I-IOWA) operator, which applied the ordering of the argument values based upon the importance of the information sources. It is worthy of being mentioned that the C-IOWA operator aggregates individual preferences in such a way that more importance is placed on the most consistent one, which is very reasonable and meaningful. But, the knowledge on the use of these IOWA operators as aggregation method is still quite limited. For example, we still will ask the following questions: (1) If all the individual preferences are acceptable consistency, can the collective fuzzy preference relation by these IOWA operators be considered acceptable consistency? (2) To what degree of consistency can these IOWA operators improve in the collective fuzzy preference relation?
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J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
The main aim of this article is to further study some desired properties of these IOWA operators. We introduced a consistency measure of fuzzy preference relations. Using this consistency measure, we proved that if the individual judgement matrixes have an acceptable consistency then the C-IOWA collective judgement matrix (C-IOWACJM) also is of acceptable consistency. Moreover, our result guarantees that the consistency of C-IOWACJM is smaller than the arithmetic mean of all the individual consistency. The I-IOWA operator also has similar properties. In order to do that, this work is set out as follows. Section 2 briefly reviews some basic concepts such as the IOWA, C-IOWA and I-IOWA operators. Section 3 defines the concept of consistency degree of fuzz preference relations. In Section 4, we study the desired properties of these IOWA operators in fuzzy group decision making. Section 5 provides two illustrative examples. Finally, in Section 6 we draw our conclusions.
2. Preliminaries: IOWA, C-IOWA and I-IOWA operators We start this section by making a generalization of the concepts of IOWA, C-IOWA and I-IOWA operators, which will be used throughout this paper. Yager and Filev (1999) presented an induced OWA (IOWA) operator in which the ordering of the ai(i 2 n) is induced by other variables ui(i 2 n) called the order inducing variables, where ai and ui are the components of the OWA pairs hui, aii(i 2 n). Definition 1. An IOWA operator of dimension n is a mapping, n UGW : Rþ ! Rþ , to which a set of weights or a weighting vector is P associated, W = (w1, w2, . . . , wn)T, wj 2 [0, 1] and nj¼1 wj ¼ 1, and it is defined to aggregate the set of second arguments of list of two tuples {hu1, a1i, . . . , hun, ani}, given on the basis of a positive ratio scale, according to the following expression: G fW ðhu1 ; a1 i; . . . ; hun ; an iÞ ¼
n X
wj bj ;
ð1Þ
der-inducing variable to induce the ordering of the argument values before their aggregation and presented the Consistency IOWA (C-IOWA) operator as follows: Definition 3. If a set of DMs D = {d1, d2, . . . , dm} provides preference about a set of alternatives X = {x1, x2, . . . , xn} by means of fuzzy preference relations {R(1), . . . , R(l), . . . , R(m)}, R(l) 2 R, then a C-IOWA operator is an IOWA operator in which its orderinducing values is the set of consistency index values {CI(R)(1), . . . , CI(R(l)), . . . , CI(R(m))}.
3. The measure of consistency index of fuzzy preference relations In group decision environment, the problem of consistency itself includes two problems (Herrera-Viedma, Herrera, Chiclana, & Luque, 2004) (i) when an DM, consider individually, is said to be consistent and, (ii) when can a whole group of decision makers be considered consistent. This section focuses on the first problem. We firstly introduce the concepts of the additive transitive fuzzy preference relations. Then we present the consistency index of fuzzy preference relations. In the following section, we will focus on the second problem. Definition 4. Let X = {x1, x2, . . . , xn} be a finite set of alternatives. If the DM gives his/her preference relation information on X by means of a preference relation R = (rij)nn, where
rij P 0;
r ij þ r ji ¼ 1;
r ii ¼ 0:5 for all i; j 2 N:
and rij denotes the preference degree or intensity of the alternative xi over xj, then R is called a fuzzy preference relation.
j¼1
where w = (w1, w2, . . . , wn)T is a weighting vector, such that Pn j¼1 wj ¼ 1; wj 2 ½0; 1; bj is the ai value of the IOWA pair having the jth largest ui, and ui in hui, aii is referred to as the order inducing variable and ai as the argument variable. Obviously, one key issue in the IOWA operator theory is to determine the order inducing variable ui, which applies the ordering of the argument values. In some cases, each DM has an importance degree associated with them so that we call this a heterogeneous GDM problem (Dubois & Koning, 1991; Dubois, Fargier, & Prade, 1997; Herrera, Herrera-Viedma, & Verdegay, 1998).The importance degree variable can be used as the order-inducing variable to induce the ordering of the argument values before their aggregation. Chiclana et al. (2007) presented the Importance IOWA (I-IOWA) operator, which applies the ordering of the argument values based upon the importance of the information sources.
Definition 5 (Tanino, 1984). Let R = (rij)nn be a fuzzy preference relation, then R is called an additive transitive fuzzy preference relation, if the following additive transitivity is satisfied:
rij ¼ r ik rjk þ 0:5;
for all i; j; k 2 N:
Definition 6. If we utilize the row arithmetic mean method ðlÞ ðlÞ (RAMM), then can get the priority vector xðlÞ ¼ x1 ; x2 ; . . . ; ðlÞ T (l) xn Þ of the fuzzy preference relation R , where
xðlÞ i ¼
n 1X ðlÞ r ; n j¼1 ij
i ¼ 1; 2; . . . ; n;
l ¼ 1; 2; . . . ; m:
Definition 7. Let A = (aij)nn 2 R and b = (bij)nn 2 R, then we define the distance between A and B as follows: n X n 1X jaij bij j: n i¼1 j¼1
Definition 2. If a set of DMs D = {d1, d2, . . . , dm} provides preference about a set of alternatives X = {x1, x2, . . . , xn} by means of fuzzy preference relations {R(1), . . . , R(l), . . . , R(m)}, and each have an importance degree l(dk) 2 [0, 1], associated to him or her, then an IIOWA operator is an IOWA operator in which its order-inducing values is the set of importance degree.
dðA; BÞ ¼
But in a homogeneous GDM problem (Herrera & Herrera-Viedma, 1997), each DM has equal importance degree. However, the DMs always can have a consistency index (CI) value associated with them. Chiclana et al. (2007) used the consistency as the or-
Theorem 1. Let A = (aij)nn 2 R and B = (bij)nn 2 R, then
ð2Þ
Obviously, the smaller the value of distance degree d(A, B), the closer of the fuzzy preference relations A and B.
(1) d(A, B) P 0; (2) d(A, B) = 0 if and only if A and B are perfectly consistent.
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Proof n X n 1X tjaij bij j P 0: (1) dðA; BÞ ¼ n i¼1 j¼1
ð3Þ
(2) Necessity. If d(A, B) = 0, then aij = bij for all i, j 2 N. Therefore, A and B are perfectly consistent. (3) Sufficiency. If A and B are perfectly consistent, then aij = bij for all i, j 2 N. Thus, we have aij bij = 0 for all i, j 2 N. Therefore, d(A, B) = 0. h In decision making problems based on fuzzy preference relations, the study of consistency is associated the transitivity property. Herrera-Viedma et al. (2004) gave a characterization of the consistency property defined by the additive transitivity property of a fuzzy preference relation Rk ¼ ðrkij Þ:
r kij þ r kjl þ r kil ¼
3 ; 2
8i; j; l 2 f1; . . . ; ng:
Using this characterization method, a procedure to construct a consistent reciprocal fuzzy preference relation R on X = {x1, . . . , xn, n P 2} from n 1 preference values {r12, r23 . . . , rn1n} presents the following steps: _
_
1. R ¼ ð r ij Þ such that:
8 if r > < ij jðiþ1Þ if r ij ¼ ðr iiþ1 þ r iþ1iþ2 . . . rj1j Þ 2 > : _ 1 r ij if
i 6 j 6 i þ 1; i þ 1 < j;
In this section, we implement the C-IOWA operator and the IIOWA operator to aggregate individual fuzzy preference relations in GDM problems, and then study their desired properties. 4.1. The Consistency IOWA (C-IOWA) operator In a homogeneous GDM problem, the DMs have equal importance. However, each DM always can have a Consistency Index (CI) value associated with them, which measures the level of consensus between individual preferences and group preference. Thus, the more consistency is the information provided by the DM, the higher the weighting value should be placed on that information. In this section, we implement the C-IOWA operator to aggregate individual preferences in such a way the greater weight is given the most consistency one. Then we study the reciprocity and consistency properties of the C-IOWACJM, which is obtained by using C-IOWA operator. Definition 9. If R(1), . . . , R(l), . . . , R(m) are the fuzzy preference rela_ _ tions provided by m DMs, then the C-IOWACJM R ¼ ð r ij Þnn is expressed as follows: _ R ¼ C-IOWA hCIðRð1Þ Þ; Rð1Þ i; hCIðRð2Þ Þ; Rð2Þ i; ; hCIðRðmÞ Þ; RðmÞ i ¼ C-IOWA hCIðRðrð1ÞÞ Þ; Rðrð1ÞÞ i; hCIðRðrð2ÞÞ Þ; Rðrð2ÞÞ i; ; hCIðRðrðmÞÞ Þ; RðrðmÞÞ i ¼ ðRðrð1ÞÞ crð1Þ Þ þ ðRðrð2ÞÞ crð2Þ Þ þ . . . þ ðRðrðmÞÞ crðmÞ Þ; ðrð1ÞÞ ðrð2ÞÞ ðrðmÞÞ r ij ¼ r ij crð1Þ þ rij crð2Þ þ þ ðaij crðmÞ Þ
j < i:v
ð5Þ
_
b could have entries not in the interval [0, 1], but in But the matrix R _ _ _ an interval [ a, 1 + a], being a ¼ minf r ij ; r ij 2 Rg . In such a case, Herrera-Viedma et al. (2004) presented a transfor- mation function which preserves reciprocity and additive consistency, that is a function f : ½a; 1 þ a ! ½0; 1, verifying (a) (b) (c) (d)
4. The properties of IOWA operators in fuzzy group decision making
f(a) = 0. f(1 + a) = 1. f(x) + f(1 x) = 1, "x 2 [ a, 1 + a]. f ðxÞ þ f ðyÞ þ f ðzÞ ¼ 32 ; 8x; y;z 2 ½a;1 þ a such that x þ y þ z ¼ 32.
2. The consistent fuzzy preference relation P is obtained as _ P ¼ f ðRÞ.
¼
m Y ðrðlÞÞ ðaij crðlÞ Þ;
ð6Þ
l¼1
where (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n) such that CI(R(r(l1))) P CI(R(r(l))) (r(l))
and
cr(l1) P cr(l) for all l = 2, . . . , m;
(r(l))
), R i is the two tuple with CI(R(r(l))) the lth biggest value hCI(R in the set {CI(R(1)), . . . , CI(R(m))}; c = (cr(1), cr(2), . . . , cr(m))T is a P weighting vector, such that m l¼1 crðlÞ ¼ 1; crðlÞ 2 ½0; 1. Yager (2003) proposed a procedure to determine the weighting vector associated to an IOWA operator. In this case, each comment ðlÞ ðlÞ in the aggregation consists of a triple ðpij ; ul ; v l Þ : pij is the argument value to aggregate, ul is the importance weight value associated to ðlÞ pij , and vl is the order inducing value. Thus, the aggregation is
This consistency index has a definite physical implication and reflects the deviation degree between the fuzzy preference relation R(l) and its corresponding consistent matrix P(l). In this article, we define the concept of consistency index based on the distance between R(l) and its corresponding consistent matrix P(l) as follows.
n X ð1Þ ðnÞ rðlÞ IOWAQ pij ; . . . ; pij ¼ wl pij ;
Definition 8. Suppose R(1), . . . , R(l), . . . , R(m) be the fuzzy preference relations provided by m DMs and P(1), . . . , P(l), . . . , P(m) be their corresponding consistent matrix, then we define a measure of consistency index (CI) of the fuzzy preference relation R(l) as follows:
P where SðlÞ ¼ lk¼1 urðkÞ , and r is the permutation such that ur(l) in rðlÞ ðpij ; urðlÞ ; v rðlÞ Þ is the lth largest value in the set {v1, . . . , vn}. Q is a function: [0, 1] ? [0, 1] such that Q(0) = 0, Q(1) = 1 and if x > y then Q(x) P Q(y) (Yager & Kacprzyk, 1997). In our case, we propose to use the consistency values associated to each one of the DM both as a weight associated to the argument and as the order inducing values ui = vi = CI(R(i)). Thus, the ordering of the preference values is first induced by the ordering of the DMs from most to least consistency one, and the weights of the C-IOWA operator is obtained by applying the above Eq. (10), with reduces to
CIðRðlÞ Þ ¼ 1 dðRðlÞ ; PðlÞ Þ:
ð4Þ
Obviously, the closer CI(R(l)) is to 1 the more consistent the information provided by the DM d(l), and thus more importance should be placed on that information. When using this consistency index, we can obtain some desired properties of C-IOWA operator (see Theorem 6)
l¼1
with
wl ¼ Q
crðlÞ ¼ Q
SðlÞ Sðl 1Þ Q SðnÞ SðnÞ
SðrðlÞÞ Sðrðl 1ÞÞ Q ; SðrðnÞÞ SðrðnÞÞ
ð7Þ
ð8Þ
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J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
Pl
ðrðkÞÞ Þ, and k¼1 CIðR ðrðlÞÞ ðrðlÞÞ (r(l)) ) in ðr ij ; CIðR Þ; CIðRðrðlÞÞ ÞÞ CI(R (r(1)) (r(n))
where SðrðlÞÞ ¼
ðrð1ÞÞ r ij ¼ rij crð1Þ þ rijðrð2ÞÞ crð2Þ þ . . . þ rðijrðmÞÞ crðmÞ P 0 crð1Þ þ 0 crð2Þ þ . . . þ 0 crðmÞ ¼ 0; _ _ ðrð1ÞÞ ðrð1ÞÞ þ r ji crð1Þ þ rijðrð2ÞÞ þ rjiðrð2ÞÞ crð2Þ þ . . . r ij þ r ji ¼ r ij ðrðmÞÞ ðrðmÞÞ þ r ij þ r ji crðmÞ _
r is the permutation such that is the lth largest value in the
set {CI(R ), . . . , CI(R )}. In an aggregation process, we consider that the weighting value of DMs should be implemented in such a way that the effect from those DMs who are less consistency is reduced, and therefore the above is obtained if the linguistic quantifier Q verifiers that the most the consistency of an DM the higher the weighting value of that DM in the aggregation, i.e.:
¼ crð1Þ þ crð2Þ þ . . . þ crðmÞ ¼ 1; ðrð1ÞÞ crð1Þ þ rðiirð2ÞÞ crð2Þ þ . . . þ rðiirðmÞÞ crðmÞ r ii ¼ rii 1 1 1 1 crð1Þ þ crð2Þ þ . . . þ crðmÞ ¼ : ¼ 2 2 2 2
_
CIðRðrð1ÞÞ Þ P CIðRðrð2ÞÞ Þ P . . . P CIðRðrðnÞÞ Þ P 0 ) crð1Þ P crð2Þ . . . P crðnÞ P 0:
_
Thus, R ¼
Theorem 2. Assuming the parameterized family of RIM quantifiers P Q(r) = ra, a P 0, if a 2 [0, 1] and SðrðlÞÞ ¼ lk¼1 CIðRðrðkÞÞ Þ, then cr(l) P cr(l+1), for all l = 1, 2, . . . , m
_ r ij
is also a fuzzy preference relation.
nn
(ii) Since all the R(1), R(2), . . . , R(m) are consistent, i.e., then
1 ðlÞ ðlÞ ðlÞ r ij ¼ r ik þ r kj ; 2
for all l ¼ 1; 2; . . . ; m i; j 2 N:
Thus Proof. If a 2 [0, 1], then the function Q(r) = ra is concave and we can have
_
_
r ik þ r kj ¼
m m X X ðrðlÞÞ ðrðlÞÞ rik crðlÞ þ r kj crðlÞ l¼1
ðlÞÞ Suppose T l ¼ SðSðrrðnÞÞ , and SðrðlÞÞ ¼
crðlÞ
Pl
ðrðkÞÞ Þ, k¼1 CIðR
then
SðrðlÞÞ Sðrðl 1ÞÞ ¼Q Q ¼ Q ðT l Þ Q ðT l1 Þ SðrðnÞÞ SðrðnÞÞ
and
crðlþ1Þ ¼ Q
_
¼ r ij þ _
1 2
and thus, R is also consistent, which completes the proof of Theorem 3. h
Sðrðl þ 1ÞÞ SðrðlÞÞ Q ¼ Q ðT lþ1 Þ Q ðT l Þ SðrðnÞÞ SðrðnÞÞ
Thus, we can obtain
crðlÞ P crðlþ1Þ which completes the Theorem 2.
l¼1
m m X X 1 ðrðlÞÞ ðrðlÞÞ ðrðlÞÞ ¼ r ik þ rkj crðlÞ ¼ rij þ c 2 rðlÞ l¼1 l¼1
Q ðT l Þ QðT l1 Þ P QðT lþ1 Þ Q ðT l Þ:
h
In GDM models with fuzzy preference assessments, it is usually assumed that the fuzzy preference relations to express the judgments are reciprocal. The C-IOWA operator is able to maintain both the reciprocity and the consistency properties in the collective fuzzy preference relation. To study these desired properties, we derive the following theorems.
As regards group decision making, the analytic hierarchy process (AHP) considers two different approaches: the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP). For the RGMM prioritization procedure and the WGMM aggregation procedure, Barzilai & Golany (1994) proved that both aggregation approaches (AIJ and AIP) provide the same priorities of alternatives. We also can draw an analogous conclusion when use the row arithmetic mean method (RAMM) as the prioritization procedure and C-IOWACJM as the aggregation procedure. Definition 11. Denote R(l) 2 R be the fuzzy judgement matrix provided by the lth DM when comparing n alternatives, xðlÞ ¼ ðlÞ
ðlÞ
ðlÞ
ðlÞ
ðx1 ; x2 ; . . . ; xn ÞT as its priority vector, P ðlÞ ¼ ðpij Þnn as the _
_ _
_ T
1
n
corresponding consistent matrix; x ¼ ðx; x; . . . ; x Þ as the priorTheorem 3. Let R(1), R(2), . . . , R(m) be fuzzy preference relations ðlÞ provided by m DMs, where RðlÞ ¼ _ ðr ij Þnn ; ðl ¼ 1; 2; . . . ; m; i; j ¼ _ 1; 2; . . . ; nÞ, then their C-IOWACJM R ¼ ð r ij Þnn is also a fuzzy preference relation, where _
ED E D E ð2Þ ðmÞ ; CIðRð2Þ Þ; r ij ; . . . ; CIðRðmÞ Þ; r ij D ED E D E ðrð1ÞÞ ðrð2ÞÞ ðrðmÞÞ ; CIðRðrð2ÞÞ Þ; r ij ; . . . ; CIðRðrðmÞ Þ; r ij ¼ CI-IOWA CIðRðrð1ÞÞ Þ; r ij ðrð1ÞÞ ¼ r ij crð1Þ þ rðijrð1ÞÞ crð2Þ þ . . . þ rðijrðmÞÞ crðmÞ
r ij ¼ CI-IOWA
D
ð1Þ
CIðRð1Þ Þ; r ij
and
r ij P 0;
_
ity vector of C-IOWACJM R , and P ¼ ðpij Þnn as the corresponding _
consistent matrix of R . Theorem 4. Using the C-IOWACJM as the aggregation procedure, the weighting vector c ¼ ðcrð1Þ ; crð2Þ ; . . . ; crðmÞ ÞT ; crðl1Þ P crðlÞ ; Pm l¼1 crðlÞ ¼ 1, and the RAMM as the prioritization procedure, it holds that the AIJ and the AIP provides the same priorities of alternatives. Proof. Let xðlÞ ¼ ðx1 ; x2 ; . . . ; xn ÞT be the priority of the individ_ ual judgement matrix R(l) and x ¼ ðx1 ; x2 ; . . . ; xn ÞT be the group priorities, then we can obtain ðlÞ
rij þ rji ¼ 1;
1 r ii ¼ ; for all i; j 2 N: 2 _
Furthermore, if all the R(1), R(2), . . . , R(m) are consistent, then R is also consistent. Proof (i) Since R(1), R(2), . . . , R(m) are fuzzy preference relations, we have rij P 0; rij þ r ji ¼ 1 rii ¼ 12 ; for all i; j 2 N, and then
2
_
ðlÞ
ðlÞ
xðAIPÞ ¼ C-IOWA hCIðRð1Þ Þ; xð1Þ i; hCIðRð2Þ Þ; xð2Þ i; ; hCIðRðmÞ Þ; xðmÞ i
¼ C-IOWA hCIðRðrð1ÞÞ Þ; xðrð1ÞÞ i; hCIðRðrð2ÞÞ Þ; xðrð2ÞÞ i; ; hCIðRðrðmÞÞ Þ; xðrðmÞÞ i ¼ xðrð1ÞÞ crð1Þ þ xðrð2ÞÞ crð2Þ þ þ xðrðmÞÞ crðmÞ
xi ðAIPÞ ¼
m X l¼1
xiðrðlÞÞ crðlÞ
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J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
and n n X m m n X X _ 1X 1X 1 ðrðlÞÞ ðrðlÞÞ r xi ðAIJÞ ¼ r ij ¼ r crðlÞ ¼ crðlÞ n j¼1 n j¼1 l¼1 ij n ij j¼1 l¼1
¼
m X
!
crðlÞ xiðrðlÞÞ :
Since
X X m m ðrðlÞÞ ðrðlÞÞ ðrðlÞÞ ðrðlÞÞ r ij pij crðlÞ 6 pij crðlÞ : r ij l¼1 l¼1 Then
l¼1
_
Thus
CIðR Þ P 1
xi ðAIPÞ ¼ xi ðAIJÞ:
m X
¼1
This completes the proof of the Theorem 4. h
n X n X m 1X ðrðlÞÞ ðrðlÞÞ pij crðlÞ r ij n i¼1 j¼1 l¼1
l¼1
As mentioned above, the C-IOWA operator aggregates individual preferences into a collective preference in such a way the greater weight is given the most consistency one. We want to see how the C-IOWA operator improves consistency in the collective fuzzy preference relation. Thus we derive the following theorems. Lemma 1 (Liu & Chen, 2004). For ordered vector x = (x1, x2, . . . , xn), x1 P x2 P, . . . , P xn, and weights w = (w1, w2, . . . , wn), then vector a1, a2, . . ., an, ai P 0(i = 1, 2, . . . , n). If w1 P w2 P P wn, then
a1 w1 x1 þ a2 w2 x2 þ þ an wn xn a1 w1 þ a2 w2 þ þ an wn P ða1 x1 þ a2 x2 þ þ an xn Þ: a1 þ a2 þ þ an
¼
ð9Þ
ð10Þ
with equality if and only if x1 = x2 = = xn Theorem 5. For ordered vector x = (x1, x2, . . . , xn), x1 P x2 P P xn, and weights
wi ¼ 1; then
i¼1
ðx1 þ x2 þ þ xn Þ w1 x1 þ w2 x2 þ þ wn xn P : n
crðlÞ CIðRðlÞ Þ:
Since CI(R(r(l))) P CI(R(r(l+1))) and cr(1) P cr(2) P P cr(m) Then, from Theorem 5, we have m X
crðlÞ CIðRðrðlÞÞ Þ P
m m 1 X 1 X CIðRðrðlÞÞ Þ ¼ CIðRðlÞ Þ: m l¼1 m l¼1
Thus _
n X
m X l¼1
CIðR Þ P
a1 w1 x1 þ a2 w2 x2 þ þ an wn xn a1 w1 þ a2 w2 þ þ an wn ða1 x1 þ a2 x2 þ þ an xn Þ 6 a1 þ a2 þ þ an
crðlÞ
l¼1
l¼1
If w1 6 w2 6 6 wn, then
w ¼ ðw1 ; w2 ; . . . ; wn Þ; if w1 P w2 P . . . P wn and
¼
m X
n X n 1X ðrðlÞÞ ðrðlÞÞ pij r ij n i¼1 j¼1 ! n X n 1X ðrðlÞÞ ðrðlÞÞ 1 pij r ij n i¼1 j¼1
crðlÞ
m 1 X CIðRðlÞ Þ: m l¼1
We complete the proof of the Theorem 6.
h
Corollary 1. If the individual fuzzy judgements _ R(1), R(2), . . . , R(m) are of acceptable consistency, then the C-IOWACJM R is also acceptable consistency, that is to say,
CIðRðlÞ Þ P s;
_
8l ¼ 1; . . . ; m ) CIðRÞ P s;
ð13Þ
where s is the threshold for acceptable consistency.
ð11Þ
_
Corollary 2. The consistency degree of R is greater than the minimum of the consistency degree of R(l), which holds that
Proof. Omitted. h
_
CIðR Þ P Minl¼1;...;m fCIðRðlÞ Þg:
ð14Þ
(l)
Definition 12. Let CI(R ) be a measure of the consistency of matrix _ R(l), and_ CIðR Þ be a measure of the consistency of the collective matrix R .
4.2. The importance IOWA (I-IOWA) operator
Theorem 6. Suppose R(1), R(2), . . . , R(m) be the fuzzy preference relations provided by m DMs when comparing n alternatives with the corresponding weighting vector c = (cr(1), cr(2), . . . , cr(m))T, cr(l1) P cr(l), Pm l¼1 crðlÞ ¼ 1. Using the C-IOWACJM as the aggregation procedure and the RAMM as the prioritization procedure, it holds that:
In a heterogeneous GDM problem (Dubois et al., 1997), each expert has an importance degree associated with them. Chiclana et al. (2007) developed the I-IOWA operator, which used this importance degree variable as the order-inducing variable to induce the ordering of the argument values before their aggregation. In this section, we study the reciprocity and consistency properties of the I-IOWACJM, which is obtained by using I-IOWA operator.
_
CIðR Þ P
m 1 X CIðRðlÞ Þ: m l¼1
ð12Þ
Proof. By Definition 11 and Eq. (4), we have _
_ _
CIðR Þ ¼ 1 dðR; P Þ ¼ 1
n X n _ 1X _ r ij p ij n i¼1 j¼1
n X n X m m X 1X ðrðlÞÞ ðrðlÞÞ ¼1 rij crðlÞ pij crðlÞ n i¼1 j¼1 l¼1 l¼1 n X n X m 1X ðrðlÞÞ ðrðlÞÞ ¼1 r ij pij crðlÞ : n i¼1 j¼1 l¼1
Definition 13. If a set of DMs D = {d1, d2, . . . , dm} provides preference about a set of alternatives X = {x1, x2, . . . , xn} by means of fuzzy preference relations {R(1), R(2), . . . , R(m)}, whose associated imporPm tance degree k ¼ ðk1 ; k2 ; . . . ; km Þ; l¼1 kl ¼ 1 0 6 kl 6 1, then the I-IOWACJM R ¼ ðr ij Þnn is expressed as follows:
R ¼ I IOWA hk1 ; Rð1Þ i; hk2 ; Rð2Þ i; ; hkm ; RðmÞ i ¼ ðRð1Þ k1 Þ þ ðRð2Þ k2 Þ þ . . . þ ðRðmÞ km Þ; m X ðlÞ rij ¼ aij kl ; l¼1
ð15Þ ð16Þ
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In GDM models with fuzzy preference assessments, it usually is assumed that the fuzzy preference relations to express the judgments are reciprocal. The I-ILOWA operator also is able to maintain both the reciprocity and consistency properties in the collective fuzzy preference relation. To study these desired properties, we derive the following theorems.
and n n X m m n X X 1X 1X 1 ðlÞ ðlÞ r ij ¼ r xi ðAIJÞ ¼ r kl ¼ kl n j¼1 n j¼1 l¼1 ij n ij j¼1 l¼1
! ¼
m X
xðlÞ i kl :
l¼1
Thus
xi ðAIPÞ ¼ xi ðAIJÞ: Theorem 7. Let {R(1), R(2), . . . , R(m)} be fuzzy preference relations ðlÞ provided by m DMs, where RðlÞ ¼ ðrij Þnn , (l = 1, 2, . . . , m; i, j = 1, 2, . . . , n), then their I-IOWACJM R ¼ ðrij Þnn is also a fuzzy preference relation, where
r ij ¼
m X
ðlÞ
r ij kl
l¼1
andrij P 0; rij þ r ji ¼ 1; r ii ¼ 12, for all i, j 2 N. Furthermore, if all the R(1), R(2), . . . , R(m) are consistent, then R is also consistent.
This completes the proof of the Theorem 8.
h
Theorem 9. Suppose R(1), R(2), . . . , R(m) be the fuzzy preference relations provided by m DMs when comparing n alternatives with the corresponding weighting vector c = (cr(1), cr(2),. . . ; crðmÞ ÞT ; crðl1Þ P crðlÞ ; Pm l¼1 crðlÞ ¼ 1. Using the I-IOWACJM as the aggregation procedure and the RAMM as the prioritization procedure, it holds that:
CIðRÞ P
m X
kl CIðRðlÞ Þ:
ð17Þ
l¼1
Proof (i) Since R(1), R(2), . . . , R(m) are fuzzy preference relations, we have rij P 0, rij + rji = 1 rii ¼ 12 ; for all i; j 2 N, and then
r ij ¼
m X
ðlÞ
r ij kl P
l¼1
m X
m X
ðlÞ
r ij kl þ
¼
m X
m X
ðlÞ
r ji kl ¼
l¼1
m X ðlÞ ðlÞ ðr ij þ rji Þ kl l¼1
kl ¼ 1;
l¼1 m X
ðlÞ
aii kl ¼
l¼1
Since
m X 1 1 kl ¼ : 2 2 l¼1
X m m X ðlÞ ðlÞ ðlÞ ðlÞ r ij pij kl 6 r ij pij kl : l¼1 l¼1
Thus, R ¼ ðr ij Þnn is also a fuzzy preference relation. (ii) Since all the R(1), . . . , R(l), . . . , R(m) are consistent, i.e., ðlÞ
1 ðlÞ ðlÞ ¼ rik þ r kj ; 2
rik þ r kj ¼
m X
ðlÞ rik
þ
Then
for all l ¼ 1; 2; . . . ; m i; j 2 N:
Then ðlÞ r kj
l¼1
n X n 1X r ij pij n i¼1 j¼1
n X n X m m X 1X ðlÞ ðlÞ rij kl pij kl ¼1 n i¼1 j¼1 l¼1 l¼1 n X n X m 1X ðlÞ ðlÞ ¼1 r ij pij kl : n i¼1 j¼1 l¼1
0 kl ¼ 0;
l¼1
rij
CIðRÞ ¼ 1 dðR; PÞ ¼ 1
l¼1
r ij þ r ji ¼
r ii ¼
Proof. By Definition 14 and Eq. (4), we have
CIðRÞ P 1
m X 1 1 ðlÞ kl ¼ r kj þ r ij þ kl ¼ 2 2 l¼1
and thus, R is also consistent, which completes the proof of Theorem 7. h Definition 14. Denote R(l) 2 R be the fuzzy judgement matrix provided by the l-th DM when comparing n alternatives, xðlÞ ¼ T ðlÞ ðlÞ ðlÞ xðlÞ as its priority vector, P ðlÞ ¼ pij as the cor1 ; x2 ; . . . ; xn nn
¼ ðx 1; x 2; . . . ; x n ÞT as the priority responding consistent matrix; x ij Þnn as the corresponding convector of I-IOWACJM R, and P ¼ ðp sistent matrix of R.
n X n X m 1X ðlÞ ðlÞ r ij pij kl n i¼1 j¼1 l¼1
m X
¼1
l¼1
¼
m X
kl
l¼1
kl ð
n X n 1X ðlÞ ðlÞ r ij pij n i¼1 j¼1
n X n 1X ðlÞ ðlÞ 1 r ij pij n i¼1 j¼1
!
We complete the proof of the Theorem 9.
¼
m X
kl CIðRðlÞ Þ:
l¼1
h
Corollary 3. If the individual fuzzy judegements R(1), R(2), . . . , R(m) are of acceptable consistency, then the I-IOWACJM R is also acceptable consistency, that is to say,
CIðRðlÞ Þ P s;
8l ¼ 1; . . . ; m ) CIðRÞ P s;
ð18Þ
where s is the threshold for acceptable consistency. Theorem 8. Using the I-IOWACJM as the aggregation procedure, the Pm weighting vector k ¼ ðk1 ; k2 ; . . . ; km ÞT ; l¼1 kl ¼ 1, the RAMM as the prioritization procedure, it holds that the AIJ and the AIP provides the same priorities of alternatives.
ðlÞ 1 ;
ðlÞ 2 ;...; (l)
ðlÞ n
CIðRÞ P Minl¼1;...;m fCIðRðlÞ Þg:
T
Proof. Let x ¼ x x x be the priority of the individ ¼ ðx 1; x 2; . . . ; x n ÞT be the group ual judgement matrix R and x priorities, then we can obtain ðlÞ
xðAIPÞ ¼ I-IOWA hk1 ; xð1Þ i; hk2 ; xð2Þ i; ; hkm ; xðmÞ i ¼ xi ðAIPÞ ¼
m X l¼1
m X l¼1
xðlÞ i kl
Corollary 4. The consistency degree of R is greater than the minimum of the consistency degree between R(l), which holds that
xðlÞ kl
ð19Þ
5. Illustrative examples Example 1. In order to see how the C-IOWA operator woks in practice, let us consider the example used in Chiclana et al. (2007). In this example, there are four DMs D = {d1, d2, d3, d4}, and a set of four alternatives X = {x1, x2, x3, x4}. Assume that these DMs provide the following fuzzy preference relations on the set of alternatives:
J. Wu, Q.-w. Cao / Expert Systems with Applications 38 (2011) 7059–7066
3
2
ð1Þ
R
0:5 0:3 0:7 0:1 6 0:7 0:5 0:6 0:6 7 7 6 ¼6 7; 4 0:3 0:4 0:5 0:2 5
R
ð2Þ
0:5 0:4 0:6 0:2 6 0:6 0:5 0:7 0:4 7 7 6 ¼6 7; 4 0:4 0:3 0:5 0:1 5
0:9 0:4 0:8 0:5 2
0:5 0:5 0:7 0:0
3
2
Using the DMs’ importance degree k to induce the ordering of these fuzzy preference relations to be aggregated, we can obtain the following collective fuzzy preference relation R ¼ ðr ij Þnn and its corresponding consistent matrix P :
0:8 0:6 0:9 0:5 3
2
6 0:5 0:5 0:8 0:4 7 6 7 Rð3Þ ¼ 6 7; 4 0:3 0:2 0:5 0:2 5
0:5 0:4 0:7 0:8
R ¼ I-IOWAk hk1 ; Rð1Þ i; hk2 ; Rð2Þ i; hk3 ; Rð3Þ i; hk4 ; Rð4Þ i 2 3 0:5000 0:3850 0:6800 0:3500 6 0:6150 0:5000 0:5800 0:4250 7 6 7 ¼6 7; 4 0:3200 0:4200 0:5000 0:1450 5
3
6 0:6 0:5 0:4 0:3 7 6 7 Rð4Þ ¼ 6 7: 4 0:3 0:6 0:5 0:1 5
1:0 0:6 0:8 0:5
0:2 0:7 0:9 0:5
By using the above procedure, we can obtain four consistent matrices as follows:
2
P
ð1Þ
0:5 6 0:7 6 ¼6 4 0:6 0:9
0:3 0:4 0:1
3
0:5 0:6 0:3 7 7 7; 0:4 0:5 0:2 5
2
P
0:7 0:8 0:5 3
2
ð2Þ
0:5 6 0:6 6 ¼6 4 0:4 0:8
0:4 0:6 0:2
0:7 0:9 0:5
0:5 0:5 0:8 0:5 0:5 5=12 6 0:5 0:5 0:8 0:5 7 6 7=12 0:5 7 6 6 ð4Þ Pð3Þ ¼ 6 7; P ¼ 6 4 0:2 0:2 0:5 0:2 5 4 2=3 7=12 0:5 0:5 0:8 0:5
1:0
1=3
0:0
CIðRð2Þ Þ ¼ 0:95;
CIðRð3Þ Þ ¼ 0:65;
CIðRð4Þ Þ ¼ 0:26:
Rðrð2ÞÞ ¼ Rð1Þ ;
Rðrð3ÞÞ ¼ Rð3Þ ;
Rðrð4ÞÞ ¼ Rð4Þ ;
Pðrð1ÞÞ ¼ Pð2Þ ;
Pðrð2ÞÞ ¼ Pð1Þ ;
Pðrð3ÞÞ ¼ Pð3Þ ;
Pðrð4ÞÞ ¼ P ð4Þ ;
Using Eq. (8) with Q(r) = r1/2, we get the following weights
crð1Þ ¼ 0:61; crð2Þ ¼ 0:19; crð3Þ ¼ 0:15; crð4Þ ¼ 0:05; _
_
Then, the C-IOWACJM R1 and its corresponding consistent matrix P 1 are calculated as:
0:3960 0:6390 0:1810
3
7 6 6 0:6040 0:5000 0:6810 0:4330 7 7 R1 ¼ 6 6 0:3610 0:3190 0:5000 0:1340 7; 5 4
_
0:8190 0:5670 0:8660 0:5000 2
0:5000
0:3968 0:5787 0:2160
4 X
CIðRðlÞ Þkl ¼ 0:59;
This confirms the conclusion of Theorem 9.
6. Conclusions
Rðrð1ÞÞ ¼ Rð2Þ ;
0:5000
CIðRÞ ¼ 0:69 >
l¼1
and the judgement matrices R(1), R(2), R(3), R(4) and the corresponding consistent matrices P(1), P(2), P(3), P(4) are reordered as follows, respectively:
2
According to Definition 14 and Eq. (4), we can obtain that
0:5
According to Eq. (4), we can calculate the consistency degree CI(R(l)), l = 1, 2, 3, 4:
CIðRð1Þ Þ ¼ 0:70;
hk1 ; P ð1Þ i; hk2 ; Pð2Þ i; hk3 ; Pð3Þ i; hk4 ; Pð4Þ i 3 0:3908 0:4767 0:1450 0:5000 0:5858 0:2542 7 7 7: 0:4142 0:5000 0:1683 5
0:8550 0:7458 0:8317 0:5000 3
5=12 1=12 7 7 7: 0:5 1=6 5
11=12 5=6
0:6500 0:5750 0:8550 0:5000 P ¼ I IOWAk 2 0:5000 6 0:6092 6 ¼6 4 0:5233
3
0:5 0:7 0:3 7 7 7; 0:3 0:5 0:1 5
2
7065
3
6 0:6032 0:5000 0:6818 0:3192 7 6 7 P1 ¼ 6 7: 4 0:4213 0:3182 0:5000 0:1373 5
_
0:7840 0:6808 0:8627 0:5000
In this article, we have studied the properties of the IOWA operatrs in the aggregation of fuzzy prefernce relations in GDM problems. We have shown that the collective preference relations obtained by these cases of IOWA operators verified the reciprocity and consistency properties. Then, it is proved that the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP) provide the same priorities of alternatives by utilizing the row arithmetic mean method (RAMM) as prioritization procedure and the IOWA operators as aggregation procedure. By using the distance between R(l) and its corresponding consistent matrix P(l), we present the consistency index of fuzzy preference relations. Using this consistency measure, we proved that the C-IOWA and the I-IOWA operator can improve consistency degree in the collective fuzzy preference relation. Finally, a theoretic basis has been developed for the application of these cases of IOWA operators in fuzzy group decision making. However, many decision making progress takes place in which the information is not precisely known. The decision makers cannot estimate their preference with an exact numerical value, but with an interval number or a linguistic viable (Wu, Li, & Li, 2009; Xu, 2004). In our future research, we plan to extend the IOWA operators to deal with these inexact numerical values in GDM problems and study their properties. Acknowledgements
_ 4 1X 0:95 þ 0:70 þ 0:65 þ 0:26 ¼ 0:64; CI R1 ¼ 0:90 > CIðRðlÞ Þ ¼ 4 l¼1 4
This work was supported by Humanities and Social Science Projects of Chinese Ministry of Education under the Grant No. 10YJC630277 and Zhejiang Provincial Natural Science Foundation of China under the Grant No. Y6080215.
Then, we can find that this result is in accordance with Theorem 6.
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