Some properties of the induced continuous ... - Semantic Scholar
Computers & Industrial Engineering 59 (2010) 100–106
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Some properties of the induced continuous ordered weighted geometric operators in group decision making q Jian Wu *, Qing-wei Cao, Jun-lin Zhang School of Business Administration, Zhejiang Normal University, Jinhua 321004, PR China
a r t i c l e
i n f o
Article history: Received 29 August 2009 Received in revised form 10 January 2010 Accepted 18 March 2010 Available online 21 March 2010 Keywords: Group decision making Interval multiplicative preference relation Compatibility ICOWG operator
a b s t r a c t In Computers and Industrial Engineering 56 (2009) 1545–1552, an induced continuous ordered weighted geometric (ICOWG) operator is presented to deal with group decision making (GDM) problems with interval multiplicative preference relations. But, we still do not know whether the ICOWG operator can improve the consensus among a group of decision makers. The aim of this paper is to study some desired properties of the ICOWG operator in GDM problems. Firstly, the concept of Compatibility Degree and Compatibility Index (CI) is defined. We then present the Compatibility Index induced COWG (CI-ICOWG) operator to aggregate interval multiplicative preference relations, which induces the order of argument values based on the Compatibility Index of decision makers (DMs). The main novelty of the CI-ICOWG operator is that it aggregates individual preference relation in such a way that more importance is placed on the most compatibility one. Thus, the CI-ICOWG operator can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. Additionally, if the leading decision maker’s interval multiplicative preference relation P and each of interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then P and the collective judgement matrix (CJM) of Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility. Finally, an illustrative numerical example is used to verify the developed approaches. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In the process of group decision-making, decision maker (DM) usually needs to compare a finite set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g with respect to a single criterion, and constructs preference relations. In general, the preference relations take the form of multiplicative preference relation or fuzzy preference relation, whose elements estimate the dominance of one alternative over another and take the form of exact numerical values. However, many decision making processes take place in an environment, where the information is not precisely known. Thus, the DMs may have vague knowledge about the preference degrees of one alternative over another, and cannot estimate their preference with an exact numerical value, but with an interval number (Wu, Li, & Li, 2009). After constructing preference relations, the DMs then need to aggregate the preference relations into a collective one and rank the given alternatives. However, there usually arise situations of conflict among preference relations of DMs. Therefore, the consensus reaching process is necessary to obtain a final solution (Choudhury, Shankar, & Tiwari, 2006; Herrera-Viedma, Herreq
This manuscript was processed by Area Editor Imed Kacem. * Corresponding author. Tel.: +86 579 82298567. E-mail address: [email protected] (J. Wu).
0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.03.005
ra, & Chiclana, 2002a, Herrera-Viedma, Chiclana, & Herrera, 2002b, 2005, 2007). So, one key issue in the theory of group judgments aggregation is how to choose the aggregation operator, which will affect the group consensus (Dong, Xu, & Li, 2008a, 2008b; Wang & Fan, 2007; Xu, 2000, 2005, 2009). Yager (1988) provided a family of averaging operator called the ordered weighted averaging (OWA) operators, which allows the implementation the concept of fuzzy majority and has been extended extensively in the resolution process of different problems (Chiclana, Herrera, & Herrera-Viedma, 2003; Dubois, Fortemps, & Pirlot, 2001; Herrera, Herrera-Viedma, & Verdegay, 1997; Torra, 1997; Wang & Parkan, 2007; Wang, Luo, & Liu, 2007; Wu, Liang, & Huang, 2007). Recently, Wu et al. (2009) presented the induced continuous ordered weighted geometric (ICOWG) operator to aggregate interval multiplicative preference relations in group decision making (GDM) problems. In particular, we presented the reliability induced COWG (R-ICOWG) operator and the relative consensus degree induced COWG (RCD-ICOWG) operator. The main novelty of the R-ICOWG and RCD-ICOWG operators is that they permit the aggregation of DMs’ preference in such a way that more importance is given to the most consensus one. However, the knowledge on the use of the ICOWG operator as aggregation method is still quite limited. For example, few studies have been focused on the question that whether the ICOWG operator can improve the group Compatibility Degree.
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
This work focuses on resolving the question that how to measure and improve the consensus among a group of decision makers. To do so, we define the concept of Compatibility Degree for two interval multiplicative linguistic preference relations, and give a Compatibility Index. The Compatibility Index (CI) value measures the level of consensus between individual decision maker and the leading decision maker (Xu, 2004). In order to exert the effect of the leading decision maker, it is often required that the group opinion should be close to the one of the leading decision maker. Thus, the more compatibility in the information provided by the DM, the higher the weighting value should be placed on that information. With this Compatibility Index, we present a CI induced COWG (CI-ICOWG) operator to aggregate the interval multiplicative preference relations. The main novelty of this operator is that it aggregates individual preference relations in such a way that more importance is placed on the most compatibility one. Thus, the collective judgement matrix (CJM) by utilizing of the CI-ICOWG operator can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees (see the results of the illustrative example). Additionally, the leading decision maker’s interval multiplicative preference relation P and the CJM of Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, if P and Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility (see Corollary 1). It is noteworthy that the CI-ICOWG operator can be used for aggregating the interval multiplicative preference relations, thus these results are of great importance both theoretically and practically. In order to do that, this work is set out as follows. Section 2 gives a review of the IOWG, COWG and ICOWG operator. In Section 3, we define the concept of Compatibility Degree and Compatibility Index for two interval multiplicative preference relations. Then, the CI-ICOWG operator is presented and its desired properties are studied. Section 4 provides an illustrative example. Finally, in Section 5 we draw our conclusions.
Definition 3 (Wu et al., 2009). Suppose ½ai ; bi ði ¼ 1; 2; . . . ; nÞ is a set of interval numbers. An induced continuous ordered weighted geometric (ICOWG) operator is a mapping: Xþ ! Rþ , which has associated with it an exponential weighting vector w ¼ ðw1 ; Pn w2 ; . . . ; wn ÞT , such that wi 2 ½0; 1 and i¼1 wi ¼ 1, and is defined to aggregate the set of second arguments of list of n two tuples ðhu1 ; ½a1 ; b1 i; . . . ; hun ; ½an ; bn iÞ according to the following expression:
Uðhu1 ; ½a1 ; b1 i; . . . ; hun ; ½an ; bn iÞ ¼ IOWGðhu1 ; g Q ð½a1 ; b1 Þi; . . . ; hun ; g Q ð½an ; bn ÞiÞ ¼ IOWGðhurð1Þ ; g Q ð½arð1Þ ; brð1Þ Þi; . . . ; hurðnÞ ; g Q ð½arðnÞ ; brðnÞ ÞiÞ ¼
n Y
ðg Q ð½arðiÞ ; brðiÞ ÞÞwi
with r : f1; . . . ; ng ! f1; . . . ; ng being a permutation such that urðiÞ P urðiþ1Þ ; 8 i ¼ 1; . . . ; n 1; hurðiÞ ; g Q ½arðiÞ ; brðiÞ i is the two tuples with urðiÞ the ith highest value in the set fu1 ; . . . ; un g, and g Q ½arðiÞ ; brðiÞ is calculated by COWG operator (2). Definition 4 (Yager and Xu, 2006). If a DM provides his/her preference information on x according to the interval multiplicative prefe ¼ ða ~ij Þnn where erence relation A
~ij ¼ ½aij ; aþij ; a
~ji ¼ ½aji ; aþji ; a
aji aþij ¼ aþji aij ¼ 1;
P aji > 0;
aþii ¼ aii ¼ 1;
for all i; j ¼ 1; 2; . . . ; n:
~ij Þ ¼ g Q ð½aij ; aþij Þ ¼ aþij g Q ða
We start this section by making a generalization of the concepts of IOWG, COWG and ICOWG operators.
~ij Þ ¼ g Q ða
UGW ðhu1 ; a1 i; . . . ; hun ; an iÞ ¼
n a
ðbj Þwi
where bj is the jth largest element of the collection of the aggregated objects a1 ; a2 ; . . . ; an . Definition 2 (Yager and Xu, 2006). A continuous ordered weighted geometric averaging (COWG) operator is a mapping g : Xþ ! Rþ which has associated with it a basic unit-interval monotonic (BUM) function: Q : ½0; 1 ! ½0; 1 and is monotonic with the properties: (1) Q ð0Þ ¼ 0; (2) Q ð1Þ ¼ 1; and (3) Q ðxÞ P Q ðyÞ if x P y, such that
g Q ð½a; bÞ ¼ b: þ
0
b
1 ~ji Þ g Q ða
aij
!R 1 dQ ðyÞy dy 0
dy
;
aþij
for all i 6 j;
ð5Þ
where Q is a basic unit-interval monotonic (BUM) function. e¼ Definition 5. If the interval multiplicative preference relation A ~ij Þnn is consistent, where ða
aij ¼ aik aþkj ;
aþij ¼ aþik akj ;
for all i 6 k 6 j
¼ 1; 2; . . . ; n:
ð6Þ
3. The CI-ICOWG and its properties 3.1. The definition of Compatibility Degree and Compatibility Index Let Gn be a set of all n n interval multiplicative preference relations. ~ Þ e ¼ ða e ¼ ðb ~ij Þnn 2 Gn and B Definition 6. Let A ij nn 2 Gn , then the e e Compatibility Degree of A and B is defined as follows:
e BÞ e ¼ Cð A; ;
ð4Þ
ð1Þ
i¼1
R a 1 dQdyðyÞy dy
aþji
e ¼ ðg ða ~ij ÞÞnn the expected multiplicative preferThen we call g Q ð AÞ Q e where g ða ~ij Þ is the expected vaence relation corresponding to A, Q ~ij of the alternative xi over xj , obtained lue of preference degree a by COWG operator (2):
2. Preliminaries: IOWG, COWG and ICOWG operators
Definition 1 (Yager and Filev, 1999). An IOWG operator of dimenn sion n is a mapping, UGW : Rþ ! Rþ , to which a set of weights or a weighting vector is associated, W ¼ ðw1 ; w2 ; . . . ; wn ÞT ; wj 2 ½0; 1 Pn and j¼1 wj ¼ 1, and it is defined to aggregate the set of second arguments of list of two tuples fhu1 ; a1 i; . . . ; hun ; an ig, given on the basis of a positive ratio scale, according to the following expression:
ð3Þ
i¼1
ð2Þ
where X is the set of closed intervals, in which the lower limits of all closed intervals are positive, Rþ is the set of positive real numbers, and [a, b] is an closed interval in Xþ .
n X n 1X þ aij bji þ aþij bji n2 : 2 i¼1 j¼1
ð7Þ
e BÞ, e the greater the CompatObviously, the smaller the value of Cð A; e ibility Degree of the interval multiplicative preference relations A e and B.
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
e ¼ ðaij Þ e Definition 7. Let A nn 2 Gn and B ¼ ðbij Þnn 2 Gn , then
e BÞ e ¼ CIð A;
1 e BÞ e Cð A;
aij ¼ dij ð8Þ
Theorem 1. Let A ¼ ðaij Þnn 2 Gn and B ¼ ðbij Þnn 2 Gn , then e BÞ e P 0; (1) Cð A; e e and B e ¼ 0 if and only if A e are perfectly compatible. (2) Cð A; BÞ Proof (1) Since
" # n X n n X n n X X 1X þ þ þ þ þ a b þ aij bji ¼ aij bji þ aji bij þ aii bii 2 i¼1 j¼1 ij ji i>j j¼1 i¼1 1 nðn 1Þ n2 P 2þn ¼ 2 2 2 and
" # n X n n X n n X 1X 1 X þ þ þ þ a b ¼ a b þ aji bij þ aii bii 2 i¼1 j¼1 ij ji 2 i>j j¼1 ij ji i¼1 1 nðn 1Þ n2 P 2þn ¼ : 2 2 2 Then, we have n X n X þ aij bji þ aþij bji n2 P 0: i¼1
and thus, we can obtain
e DÞ e AÞ e ¼ 0: e ¼ Cð A; Cð A;
e and B. e is called the Compatibility Index of A e e The greater the value of CIð A; BÞ, the greater the Compatibility e and B. e Degree of the interval multiplicative preference relations A
e BÞ e ¼1 Cð A; 2
for all i; j 2 N
j¼1
e BÞ e ¼ 0, then a bþ ¼ a bþ and aþ b ¼ aþ b (2) Necessity. If Cð A; ij ji ji ij ij ji ji ij for all i; j 2 N. Thus, we can obtain aij ¼ bij for all i; j 2 N. e and B e are perfectly compatible. Therefore, A e and B e are perfectly compatible, then aij ¼ bij (3) Sufficiency. If A þ þ for all i; j 2 N. Thus, we have a ij bji ¼ 1 and aij bji ¼ 1 for all e e i; j 2 N. Therefore, Cð A; BÞ ¼ 0. h
3.2. The CI-ICOWG operator to aggregate interval multiplicative preference relations In a homogeneous GDM problem, the DMs have equal importance. However, in this situation, each expert always can have a Compatibility Index (CI) value associated with them, which measures the level of consensus between individual preference relations and the leading decision maker’s preference relation. Thus, the more compatibility is the information provided by the expert, the higher the weighting value should be placed on that information. In this section, we present the CI-ICOWG operator to aggregate interval multiplicative preferences. Then, we analyze the reciprocity, consistent and compatibility properties of the CJM, which is obtained by using of the CI-ICOWG operator. Definition 8. If a set of DMs D ¼ fd1 ; d2 ; . . . ; dm g provides preference about a set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g by means of interval multiplicative preference relations fRð1Þ ; Rð2Þ ; . . . ; RðmÞ g; RðlÞ 2 R, then a CI-IOWG operator is an IOWG operator in which its order-inducing values is the set of Compatibility Index values fCIð1Þ ; . . . ; CIðlÞ ; . . . ; CIðmÞ g. Definition 9. If Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are the interval multiplicative preference relations provided by m DMs, then the CJM _ _ R ¼ ð r ij Þnn is expressed as follows: _
R ¼ CI-IOWGðhCIð1Þ ; Rð1Þ i; hCIð2Þ ; Rð2Þ i; . . . ; hCIðmÞ ; RðmÞ iÞ ¼ CI-IOWGðhCIðrð1ÞÞ ; Rðrð1ÞÞ i; hCIðrð2ÞÞ ; Rðrð2ÞÞ i; . . . ; hCIðrðmÞÞ ; RðrðmÞÞ iÞ ¼ ðRðrð1ÞÞ Þcrð1Þ ðRðrð2ÞÞ Þcrð2Þ . . . ðRðrðmÞÞ ÞcrðmÞ ð9Þ
_ r ij
¼
ðrð1ÞÞ ðr ij Þcrð1Þ
ðrð2ÞÞ ðr ij Þcrð2Þ
ðrðmÞÞ crðmÞ ðr ij Þ
m Y ðrðlÞÞ ¼ ðr ij ÞcrðlÞ ;
ð10Þ
l¼1
e B; e D e 2 Gn , then we have Theorem 2. Let A; e AÞ e ¼ 0, (1) Reflexivity: Cð A; e e e e AÞ, (2) Symmetry: Cð A; BÞ ¼ Cð B; e BÞ e DÞ e ¼ 0 and Cð B; e DÞ e ¼ 0, then Cð A; e ¼ 0. (3) Transitivity: If Cð A;
Proof n X n 1X ða aþ þ aþij aji Þ n2 ¼ 0 2 i¼1 j¼1 ij ji
(1)
e AÞ e ¼ Cð A;
(2)
n X n X þ e BÞ e e ¼ 1 e AÞ Cð A; ða b þ aþij bji Þ ¼ Cð B; 2n2 i¼1 j¼1 ij ji
(3) Since
e BÞ e ¼ 0 and Cð B; e DÞ e ¼ 0; Cð A; from Theorem 1, we have
aij ¼ bij then
and bij ¼ dij
for all i; j 2 N
where ðrð1Þ; rð2Þ; . . . ; rðnÞÞ is a permutation of (1, 2, . . . , n) such that CIðrðl1ÞÞ P CIðrðlÞÞ and crðl1Þ P crðlÞ for all l ¼ 2; . . . ; m; hCIðrðlÞÞ ; RðrðlÞÞ i is the two tuple with CIðrðlÞÞ the lth biggest value in the set fCIð1Þ ; . . . ; CIðmÞ g; c ¼ ðcrð1Þ ; crð2Þ ; . . . ; crðmÞ ÞT is a weighting vector, P such that m l¼1 crðlÞ ¼ 1; crðlÞ 2 ½0; 1. Theorem 3. Let Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be interval multiplicative preferðlÞ ence _relations provided by m DMs, where RðlÞ ¼ ðrij Þnn , then their _ CJM R ¼ ð r ij Þnn is also an interval multiplicative preference_relation. Furthermore, if all Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are consistent, then R is also consistent. Proof (i) Since Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are interval multiplicative preference relations, we have
~rijrðlÞ ¼ ½r rij ðlÞ ; r rij ðlÞþ ; rðlÞþ
¼ r ij
rðlÞ
r ji
¼ 1; . . . ; m
~rjirðlÞ ¼ ½r rji ðlÞ ; ¼ 1;
rðlÞþ
r ij
rðlÞþ
rji
;
rðlÞ
P 0;
P r ij
rðlÞ
r ij
rðlÞþ
rji
for all i; j 2 N; l
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
Pl
Thus m m Y Y _ _ ðrðlÞÞþ crðlÞ ðrðlÞÞ crðlÞ r þij ¼ ðr ij Þ P ðr ij Þ ¼ r ij P 0; l¼1 _ r ij
l¼1 m Y
_þ
r ji ¼
ðrðlÞÞ
r ji
ðrðlÞÞþ
r ji
ðr ij
ðrðlÞÞþ crðlÞ Þ
¼ 1;
ðrðlÞÞ crðlÞ Þ
¼ 1:
l¼1 _þ r ij
m Y
_
r ji ¼
ðr ij
l¼1 _
_
Hence, R ¼ ð r ij Þnn is also an interval multiplicative preference relation. (ii) Since all Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are consistent, we can obtain rðlÞ rðlÞ rðlÞþ rðlÞþ rðlÞþ rðlÞ ¼ r ik r kj ; rij ¼ rik r kj for all l ¼ 1; 2; . . . ; rij m i 6 k 6 j ¼ 1; 2; . . . ; n. Then, we have _
r ij ¼ ¼
m m crðlÞ Y crðlÞ _ Y _ ðrðlÞÞ ðrðlÞÞþ r ik r kj ¼ r ik r þkj ; l¼1
and _þ
r ij ¼
m m crðlÞ Y crðlÞ Y ðrðlÞÞþ ðrðlÞÞþ ðrðlÞÞ r ij ¼ r ik r kj l¼1
¼
ðrðlÞÞþ
r ik
crðlÞ
l¼1
m Y
ðrðlÞÞ
r kj
crðlÞ
_
SðrðlÞÞ Sðrðl 1ÞÞ Q SðrðnÞÞ SðrðnÞÞ
ðrðlÞÞ
in
Pl
ðrðkÞÞ , and k¼1 CI ðrðlÞÞ ðrðlÞÞ ðrij ; CI ; CIðrðlÞÞ Þ is ðrðnÞÞ
where SðrðlÞÞ ¼
ð12Þ
r is the permutation such that the lth largest value in the set
CIðrð1ÞÞ P CIðrð2ÞÞ P P CIðrðnÞÞ P 0 ) crð1Þ P crð2Þ P crðnÞ P 0
l¼1
m Y
g. fCIðrð1ÞÞ ; . . . ; CI 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 compatibility is reduced, and therefore the above is obtained if the linguistic quantifier Q verifiers that the most the compatibility of an expert the higher the weighting value of that expert in the aggregation, i.e.:
l¼1
l¼1
crðlÞ ¼ Q
CI
m m crðlÞ Y crðlÞ Y ðrðlÞÞ ðrðlÞÞ ðrðlÞÞþ r ij ¼ r ik r kj l¼1
where SðlÞ ¼ k¼1 urðkÞ , and r is the permutation such that urðlÞ in rðlÞ ðpij ; urðlÞ ; v rðlÞ Þ is the lth largest value in the set fv 1 ; . . . ; v n g. Q is a function: ½0; 1 ! ½0; 1 such that Q ð0Þ ¼ 0; Q ð1Þ ¼ 1 and if x > y then Q ðxÞ P Q ðyÞ (Yager & Kacprzyk, 1997). In our case, we propose to use the Compatibility Index values associated with each one of the DMs, both as a weight associated to the argument and as the order inducing values ui ¼ v i ¼ CIðiÞ . Thus, the ordering of the preference values is first induced by the ordering of the DMs from most to least compatibility one, and the weights of the CI-IOWG operator is obtained by applying Eq. (11), which reduces to
_
¼ r þik r kj :
l¼1
_
Hence, R is also consistent, which completes the proof of the theorem. h Definition 10. Let Rð1Þ ; . . . ; RðlÞ ; . . . ; RðmÞ be the interval multiplica-
Theorem 4. Assuming the parameterized family of regular increasing monotone (RIM) quantifiers Q ðrÞ ¼ r a ; a P 0. If a 2 ½0; 1 and P SðrðlÞÞ ¼ lk¼1 CIðrðkÞÞ , then crðlÞ P crðlþ1Þ , for all l ¼ 1; 2; . . . ; m.
tive preference relations provided by m DMs, and g Q ðR Þ;
Proof. If a 2 ½0; 1, then the function Q ðrÞ ¼ ra is concave and we can have
g Q ðRð2Þ Þ; . . . ; g Q ðRðmÞ Þ be the corresponding expected multiplicative
QðT l Þ Q ðT l1 Þ P Q ðT lþ1 Þ Q ðT l Þ
ð1Þ
ðlÞ
preference relations, where g Q ðRðlÞ Þ ¼ g Q ðrij Þnn ; l ¼ 1; 2; . . . ; m, then their collective expected multiplicative preference relation _
_
g Q ðR Þ ¼ g Q ð r ij Þnn is expressed as follows: _
g Q ðR Þ ¼ CI-ICOWGðhCIð1Þ ; g Q ðRð1Þ Þi; . . . ; hCIðmÞ ; g Q ðRðmÞ ÞiÞ
ðlÞÞ Suppose T l ¼ SðSðrrðnÞÞ and SðrðlÞÞ ¼
Pl
k¼1 CI
ðrðkÞÞ
, then
SðrðlÞÞ Sðrðl 1ÞÞ crðlÞ ¼ Q Q ¼ Q ðT l Þ Q ðT l1 Þ SðrðnÞÞ SðrðnÞÞ and
¼ CI-IOWGðhCIðrð1ÞÞ ; g Q ðRðrð1ÞÞ Þi; . . . ; hCIðrðmÞÞ ; g Q ðRðrðmÞÞ ÞiÞ
crðlþ1Þ ¼ Q
¼ ðg Q ðRðrð1ÞÞ ÞÞcrð1Þ ðg Q ðRðrð2ÞÞ ÞÞcrð2Þ . . . ðg Q ðRðrðmÞÞ ÞÞcrðmÞ
Sðrðl þ 1ÞÞ SðrðlÞÞ Q ¼ QðT lþ1 Þ QðT l Þ SðrðnÞÞ SðrðnÞÞ
Thus, we can obtain _
g Q ð r ij Þ ¼
m Y
crðlÞ P crðlþ1Þ
rðlÞ
ðg Q ðr ij ÞÞcrðlÞ :
l¼1
which completes the proof of the theorem. h
Before implementing this operator to aggregate individual preference relations, we need to calculate its associated weights. Yager (2003) proposed a procedure to determine the weighting vector associated with an IOWA operator. In this case, each comment in ðlÞ ðlÞ the aggregation consists of a triple ðpij ; ul ; v l Þ, where pij is the argument value to aggregate, ul is the importance weight value associðlÞ ated to pij , and v l is the order inducing value. Thus, the aggregation is ð1Þ
ðnÞ
IOWAQ ðpij ; . . . ; pij Þ ¼
n X
rðlÞ
wl pij ;
l¼1
with
SðlÞ Sðl 1Þ wl ¼ Q Q SðnÞ SðnÞ
ð11Þ
Lemma 1. Xu, 2000Let xi > 0; bi > 0; i 2 N and n Y i¼1
b
xi i 6
n X
Pn
i¼1 bi
¼ 0, then
bi xi
ð13Þ
i¼1
Lemma 2. Liu and Chen, 2004 For ordered vector x ¼ ðx1 ; x2 ; . . . ; xn Þ; x1 P x2 P P xn , and weights w ¼ ðw1 ; w2 ; . . . ; wn Þ, then vector a1 ; a2 ; . . . an ; ai P 0 ði ¼ 1; 2; . . . ; nÞ.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 If w1 6 w2 6 6 wn , then
ð14Þ
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
a1 w1 x1 þ a2 w2 x2 þ þ an wn xn a1 w1 þ a2 w2 þ þ an wn 6 ða1 x1 þ a2 x2 þ þ an xn Þ a1 þ a2 þ þ an
ð15Þ
with equality if and only if x1 ¼ x2 ¼ ¼ xn
_
CðRðlÞ ; PÞ 6 s; 8 l ¼ 1; . . . ; m ) CðR ; PÞ 6 s;
Definition 11. Suppose Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be the interval multiplicative preference relations provided by m DMs when comparing n alternatives, and P be the interval multiplicative preference relation given by a leading decision _maker. Let CðRðlÞ ; PÞ be Compatibil_ ity Degree of RðlÞ and P, and CðR ; PÞ be Compatibility Degree of R and P. From Theorem 1, it can be obtained that RðlÞ is perfectly compatible with P if CðRðlÞ ; PÞ ¼ 0. The smaller the value of CðRðlÞ ; PÞ, the more Compatibility Degree and consensus degree of RðlÞ and P. Theorem 5. Suppose Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be the interval multiplicative preference relations provided by m DMs when comparing n alternatives. Using the CJM as the aggregation procedure, it holds that: _
CðR ; PÞ 6
ð16Þ
n X n Y m m Y 1 X rðlÞ ðrij ÞcrðlÞ ðpji ÞcrðlÞ 2 n i¼1 j¼1 l¼1 l¼1
1 2n2
_
Corollary 2. The Compatibility Degree between R and P is no more greater than the maximum of the Compatibility Degree between RðlÞ and P, which holds that _
CðR ; PÞ 6 Maxl¼1;...;m fCðRðlÞ ; PÞg:
n X
n X
m Y
i¼1
j¼1
l¼1
ðrðlÞÞ þ crðlÞ pji Þ
ðr ij
6
l¼1
m X
relation of the leading decision maker, then we let Cðg Q ðRðlÞ Þ; g Q ðPÞÞ
rðlÞ þ crðlÞ
ðrij
pji Þ
þ
m Y
be Compatibility Degree of g Q ðRðlÞ Þ and g Q ðPÞ, and Cðg Q ðR ; PÞÞbe _
Compatibility Degree of g Q ðR Þ and g Q ðPÞ.
_
Cðg Q ðR ; PÞÞ 6
!
1 C g Q ðRðlÞ Þ; g Q ðPÞ m
pji Þ
Proof. By Eq. (7), we have _
Cðg Q ðR Þ; g Q ðPÞÞ ¼
crðlÞ ðrðijrðlÞÞ pþji Þ
l¼1
ðrðlÞÞþ crðlÞ pji Þ
ðr ij
l¼1
6
m X
Then n X n m m X X 1 X CðR ; PÞ 6 2 c ðrðrðlÞÞ pþji Þ þ crðlÞ ðrijðrðlÞÞþ pji Þ 2n i¼1 j¼1 l¼1 rðlÞ ij l¼1 ! m n X n X 1 X ðrðlÞÞ ðrðlÞÞþ þ crðlÞ ðr pji þ r ij pji Þ ¼ 2n2 i¼1 j¼1 ij l¼1 _
m X
¼
n X n Y m m Y 1 X rðlÞ ðg Q ðr ij ÞÞcrðlÞ ðg Q ðpji ÞÞcrðlÞ 2 n i¼1 j¼1 l¼1 l¼1
¼
n X n Y m crðlÞ 1 X rðlÞ g ðr Þg Q ðpji Þ n2 i¼1 j¼1 l¼1 Q ij
!
From Lemma 1, it can be obtained that
l¼1
l¼1
l¼1
rðlþ1Þ
Since CðR ; PÞ 6 CðR ; PÞ and crðlÞ P crðlþ1Þ Then, from Lemma 2, we have
crðlÞ CðRðrðlÞÞ ; PÞ 6
l¼1
n X n Y m 1 X rðlÞ crðlÞ g ðr Þ g Q ðpji Þ n2 i¼1 j¼1 l¼1 Q ij
m m crðlÞ X Y rðlÞ g Q ðr ij Þg Q ðpji Þ 6 crðlÞ g Q ðrrij ðlÞ Þg Q ðpji Þ
crðlÞ CðRrðlÞ ; PÞ
rðlÞ
m X
n X n _ 1 X g ð r ij Þ g Q ðpji Þ 2 n i¼1 j¼1 Q
¼
crðlÞ ðrðijrðlÞÞþ pji Þ
l¼1
¼
ð19Þ
rðlÞþ crðlÞ
ðr ij
l¼1
and m Y
ð18Þ
Definition 12. Suppose g Q ðRð1Þ Þ; g Q ðRð2Þ Þ; . . . ; g Q ðRðmÞ Þ be the corresponding expected multiplicative preference relations of m DMs, and g Q ðPÞ be the corresponding expected multiplicative preference
From Lemma 1, it can be obtained that m Y
s is the threshold for acceptable compatibility.
Theorem 6. Suppose Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be the interval multiplicative preference relations provided by m DMs when comparing n alternatives. Using the CI-ICOWG operator as the aggregation procedure, it holds that:
n X n n X n Y m _ 1 X 1 X rðlÞ CðR ; PÞ ¼ 2 r ij pji ¼ 2 ðr ÞcrðlÞ pji n i¼1 j¼1 n i¼1 j¼1 l¼1 ij _
¼
where
ð17Þ
_
m 1 X CðRðlÞ ; PÞ m l¼1
Proof. By Eq. (7), we have
¼
Corollary 1. If the individual interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the CJM _ R is also acceptable compatibility, that is to say,
m X l¼1
crðlÞ
m 1 X CðRðrðlÞÞ ; PÞ m l¼1
Then
!
m m 1 X 1 X CðRðrðlÞÞ ; PÞ ¼ CðRðlÞ ; PÞ ¼ m l¼1 m l¼1
_ n X n X m 1 X Cðg Q R ; g Q ðPÞÞ 6 2 c g rrðlÞ g Q pji n i¼1 j¼1 l¼1 rðlÞ Q ij ¼ crðlÞ
m X l¼1
n X n 1 X rðlÞ g Q ðpji Þ g r Q ij n2 i¼1 j¼1
!
Thus m 1 X CðR ; PÞ 6 CðRðlÞ ; PÞ m l¼1 _
which completes the proof of the theorem. h
¼ crðlÞ Cðg Q ðRrðlÞ Þ; g Q ðPÞÞ Since Cðg Q ðRrðlÞ Þ; g Q ðPÞÞ 6 Cðg Q ðRrðlþ1Þ Þ; g Q ðPÞÞ and crðlÞ P crðlþ1Þ Then, from Lemma 2, we have
105
J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106 m X
crðlÞ Cðg Q ðRrðlÞ Þ; g Q ðPÞÞ 6
l¼1
m X
crðlÞ
l¼1
m 1 X C g Q ðRrðlÞ Þ; g Q ðPÞ m l¼1
!
m 1 X C g Q ðRrðlÞ Þ; g Q ðPÞ ¼ m l¼1
¼
1 m
m X
CðRð1Þ ; PÞ ¼ 1:150;
CðRð2Þ ; PÞ ¼ 1:425;
According to Eq. (8), the following Compatibility Indexes can be worked out:
CIðRð1Þ ; PÞ ¼ 0:870;
C g Q ðRðlÞ Þ; g Q ðPÞ
CðRð3Þ ; PÞ ¼ 0:775:
CIðRð2Þ ; PÞ ¼ 0:702;
CIðRð3Þ ; PÞ ¼ 1:290:
Then, the Compatibility Indexes and the multiplicative preference relations are reordered as follows:
l¼1
CIðRðrð1ÞÞ ; PÞ ¼ CIðRð3Þ ; PÞ;
Thus
¼ CIðRð2Þ ; PÞ;
m 1 X Cðg Q ðR Þ; g Q ðPÞÞ 6 Cðg Q ðRðlÞ Þ; g Q ðPÞÞ m l¼1 _
CIðRðrð2ÞÞ ; PÞ ¼ CIðRð1Þ ; PÞ;
CIðRðrð3ÞÞ ; PÞ
Rðrð1ÞÞ ¼ Rð3Þ ; Rðrð2ÞÞ ¼ Rð1Þ ; Rðrð3ÞÞ
ð2Þ
¼R : Using Eq. (12) with Q ðrÞ ¼ r 2=3 , we get the following weights:
which completes the proof of the theorem. h
crð1Þ ¼ 0:58; crð2Þ ¼ 0:24; crð3Þ ¼ 0:18: Corollary 3. If the individual expected multiplicative preference rela. . ; g Q ðRðmÞ Þ are of acceptable compatibility, tions g Q ðRð1Þ Þ; g Q ðRð2Þ Þ; ._ then the expected CJM g Q ðR Þ is also acceptable compatibility, that is to say, _
Cðg Q ðRðlÞ Þ; g Q ðPÞÞ 6 s; 8 l ¼ 1; . . . ; m ) Cðg Q ðR Þ; g Q ðPÞÞ 6 s; where
ð20Þ
s is the threshold for acceptable compatibility.
_
Then, CJM R is calculated as: _
R ¼ ðRðrð1ÞÞ Þcrð1Þ ðRðrð2ÞÞ Þcrð2Þ ðRðrð3ÞÞ Þcrð3Þ 2 ½1:00;1:00 ½6:00;7:00 ½0:37;0:68 6 6 ½0:14;0:17 ½1:00;1:00 ½6:00;7:56 6 ¼6 6 ½1:47;2:70 ½0:13;0:17 ½1:00;1:00 6 4 ½0:15;0:17 ½0:14;0:17 ½0:13;0:16
_
ðlÞ
Cðg Q ðR Þ; g Q ðPÞÞ 6 Maxl¼1;...;m fCðg Q ðR Þ; g Q ðPÞÞg:
ð21Þ
4. Illustrative example Let us consider the example used by Wu et al. (2009). In this example, there are four DMs D ¼ fd1 ; d2 ; d3 ; d4 g. Suppose that the interval multiplicative preference relation P is given by a leading decision maker, and the interval multiplicative preference relations Rð1Þ ; Rð2Þ ; Rð3Þ are given by the other DMs, respectively. They are listed as follows: 3 2 ½1;1 ½6;7 ½1=7;1=5 ½7;8 ½1=6;1=5 7 6 ½1;1 ½6;7 ½6;7 ½1=7;1=5 7 6 ½1=7;1=6 7 6 ð1Þ R ¼6 ½1=7;1=6 ½1;1 ½6;7 ½7;8 7 7 6 ½5;7 7 6 ½1;1 ½6;7 5 4 ½1=8;1=7 ½1=7;1=6 ½1=7;1=6 ½5;6 ½5;7 ½1=8;1=7 ½1=7;1=6 ½1;1 3 2 ½1;1 ½6;7 ½1=2;1 ½7;8 ½1=2;1 7 6 ½1;1 ½6;7 ½6;7 ½1=7;1=6 7 6 ½1=7;1=6 7 6 7 Rð2Þ ¼ 6 ½1;2 ½1=7;1=6 ½1;1 ½5;7 ½7;8 7 6 7 6 ½1;1 ½6;7 5 4 ½1=8;1=7 ½1=7;1=6 ½1=7;1=5 ½1;2 ½6;7 ½1=8;1=7 ½1=7;1=6 ½1;1 3 2 ½1;1 ½6;7 ½1=2;1 ½5;6 ½1=8;1=7 7 6 ½1;1 ½6;8 ½6;7 ½1=7;1=6 7 6 ½1=7;1=6 7 6 ð3Þ 6 R ¼ 6 ½1;2 ½1=8;1=6 ½1;1 ½7;8 ½6;7 7 7 7 6 ½1;1 ½7;8 5 4 ½1=6;1=5 ½1=7;1=6 ½1=8;1=7 ½7;8 ½6;7 ½1=7;1=6 ½1=8;1=7 ½1;1 3 2 ½1;1 ½6;7 ½1=3;3=5 ½25=4;29=4 ½1=5;3=10 7 6 ½1;1 ½6;29=4 ½6;7 ½1=7;1=6 7 6 ½1=7;1=6 7 6 ½4=29;1=6 ½1;1 ½6;29=4 ½33=5;61=8 7 P¼6 7 6 ½5=3;3 7 6 ½1;1 ½25=4;29=4 5 4 ½4=25;4=29 ½1=7;1=6 ½4=29;1=6 ½10=3;5 ½6;7 ½8=61;5=33 ½4=29;4=25 ½1;1 According to Eq. (7), we can calculate the Compatibility Degree CðRðlÞ ; PÞ; l ¼ 1; 2; 3; 4:
3
7 ½6:00;7:00 ½0:14;0:17 7 7 ½6:35;7:56 ½6:40;7:40 7 7 7 ½1:00;1:00 ½6:56;7:56 5 ½4:55;5:82 ½5:74;7:00 ½0:14;0:16 ½0:13;0:15 ½1:00;1:00
_
Corollary 4. The Compatibility Degree between g Q ðR Þ and g Q ðPÞ is no more greater than the maximum of the Compatibility Degree between g Q ðRðlÞ Þ and g Q ðPÞ, which holds that
½5:76;6:77 ½0:17;0:22
According to Eq. (7) we can obtain that _
CðR ; PÞ ¼ 0:125 < ð1:150 þ 1:425 þ 0:775Þ=3 ¼ 1:117 which verifies the conclusion of Theorem 5. pffiffiffi Assume that the BUM function Q ðyÞ ¼ y, then we can obtain
Z 0
1
dQ ðyÞ 1 y dy ¼ dy 2
Z 0
1
y1=2 dy ¼
1 3
Thus, from Eq. (2), we construct the expected multiplicative preference relation matrix g Q ðRð1Þ Þ; g Q ðRð2Þ Þ; g Q ðRð3Þ Þ; g Q ðPÞ as follows:
3 1:0000 6:6494 0:1788 7:6517 0:1882 7 6 6 0:1504 1:0000 6:6494 6:6494 0:1788 7 7 6 7 g Q ðRð1Þ Þ ¼ 6 6 5:5928 0:1504 1:0000 6:6494 7:6517 7; 7 6 4 0:1307 0:1504 0:1504 1:0000 6:6494 5 2
5:3135 5:5928 0:1307 0:1504 1:0000 3 1:0000 6:6494 0:7937 7:6517 0:7937 7 6 6 0:1504 1:0000 6:6494 6:6494 0:1583 7 7 6 ð2Þ 7 g Q ðR Þ ¼ 6 6 1:2599 0:1504 1:0000 6:2573 7:6517 7; 7 6 4 0:1307 0:1504 0:1598 1:0000 6:6494 5 1:2599 6:3171 0:1307 0:1504 1:0000 3 2 1:0000 6:6494 0:7937 5:6462 0:1366 7 6 6 0:1504 1:0000 7:2685 6:6494 0:1583 7 7 6 ð3Þ 7 g Q ðR Þ ¼ 6 6 1:2599 0:1376 1:0000 7:6517 6:6494 7; 7 6 4 0:1771 0:1504 0:1307 1:0000 7:6517 5 7:3206 6:3171 0:1504 0:1307 1:0000 3 2 1:0000 6:6494 0:4829 6:9145 0:2733 7 6 6 0:1504 1:0000 6:8497 6:6494 0:1649 7 7 6 6 g Q ðPÞ ¼ 6 2:0708 0:1460 1:0000 6:8283 7:3019 7 7; 7 6 4 0:1446 0:1504 0:1464 1:0000 6:9680 5 3:6590 6:0643 0:1370 0:1435 1:0000 3 2 1:0000 6:6494 0:5550 6:4150 0:2025 7 6 6 0:1504 1:0000 7:0017 6:6494 0:1630 7 _ 7 6 7 gQ ðR Þ ¼ 6 6 1:8018 0:1428 1:0000 7:1351 7:0533 7: 7 6 4 0:1559 0:1504 0:1402 1:0000 7:2135 5 4:9383 6:1350 0:1418 0:1386 1:0000 2
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
According to Eq. (7), we can calculate the Compatibility Degree CðRðlÞ ; PÞ; l ¼ 1; 2; 3; 4:
Cðg Q ðRð1Þ Þ; g Q ðPÞÞ ¼ 1:225; Cðg Q ðRð2Þ Þ; g Q ðPÞÞ ¼ 1:525; _
Cðg Q ðRð3Þ Þ; g Q ðPÞÞ ¼ 0:825; Cðg Q ðR Þ; g Q ðPÞÞ ¼ 0:125: Thus, we can obtain that _
Cðg Q ðR Þ; g Q ðPÞÞ ¼ 0:125 < ð1:225 þ 1:525 þ 0:825Þ=3 ¼ 1:191 This confirms the conclusion of Theorem 6. From Theorems 3, 5, Corollary 1, and the numerical results, we can draw the following conclusions: _
_
(1) The CJM R ¼ ð r ij Þnn of all interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ is an interval multiplicative preference relation. Furthermore, if all Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are con_ sistent, then R is also consistent. (2) If an interval multiplicative preference relation R and each of interval multiplicative preference relation Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the interval multiplica_ _ tive preference relation R and the CJM R ¼ ð r ij Þnn of ð1Þ ð2Þ ðmÞ R ; R ; . . . ; R are of acceptable compatibility. (3) If an interval multiplicative preference relation R and each of interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the collective judgement matrix (CJM) of Rð1Þ ; Rð2Þ ; . . . ; RðmÞ can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. 5. Concluding remarks This study defined the concept of Compatibility Degree and Compatibility Index for two interval multiplicative preference relations. Then, we presented the CI-ICOWG operator, which aggregates interval multiplicative preference relations by the Compatibility Index (CI) value. The main novelty of the CIICOWG operator is that it can improve group Compatibility Degree. If the leading decision maker’s interval multiplicative preference relation P and each of interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the CJM is of acceptable compatibility. Furthermore, the CI-ICOWG operator guarantees that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. Acknowledgements The authors are very grateful to three anonymous referees for their valuable comments and suggestions. This work was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y6080215.
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Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Some properties of the induced continuous ordered weighted geometric operators in group decision making q Jian Wu *, Qing-wei Cao, Jun-lin Zhang School of Business Administration, Zhejiang Normal University, Jinhua 321004, PR China
a r t i c l e
i n f o
Article history: Received 29 August 2009 Received in revised form 10 January 2010 Accepted 18 March 2010 Available online 21 March 2010 Keywords: Group decision making Interval multiplicative preference relation Compatibility ICOWG operator
a b s t r a c t In Computers and Industrial Engineering 56 (2009) 1545–1552, an induced continuous ordered weighted geometric (ICOWG) operator is presented to deal with group decision making (GDM) problems with interval multiplicative preference relations. But, we still do not know whether the ICOWG operator can improve the consensus among a group of decision makers. The aim of this paper is to study some desired properties of the ICOWG operator in GDM problems. Firstly, the concept of Compatibility Degree and Compatibility Index (CI) is defined. We then present the Compatibility Index induced COWG (CI-ICOWG) operator to aggregate interval multiplicative preference relations, which induces the order of argument values based on the Compatibility Index of decision makers (DMs). The main novelty of the CI-ICOWG operator is that it aggregates individual preference relation in such a way that more importance is placed on the most compatibility one. Thus, the CI-ICOWG operator can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. Additionally, if the leading decision maker’s interval multiplicative preference relation P and each of interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then P and the collective judgement matrix (CJM) of Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility. Finally, an illustrative numerical example is used to verify the developed approaches. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In the process of group decision-making, decision maker (DM) usually needs to compare a finite set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g with respect to a single criterion, and constructs preference relations. In general, the preference relations take the form of multiplicative preference relation or fuzzy preference relation, whose elements estimate the dominance of one alternative over another and take the form of exact numerical values. However, many decision making processes take place in an environment, where the information is not precisely known. Thus, the DMs may have vague knowledge about the preference degrees of one alternative over another, and cannot estimate their preference with an exact numerical value, but with an interval number (Wu, Li, & Li, 2009). After constructing preference relations, the DMs then need to aggregate the preference relations into a collective one and rank the given alternatives. However, there usually arise situations of conflict among preference relations of DMs. Therefore, the consensus reaching process is necessary to obtain a final solution (Choudhury, Shankar, & Tiwari, 2006; Herrera-Viedma, Herreq
This manuscript was processed by Area Editor Imed Kacem. * Corresponding author. Tel.: +86 579 82298567. E-mail address: [email protected] (J. Wu).
0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.03.005
ra, & Chiclana, 2002a, Herrera-Viedma, Chiclana, & Herrera, 2002b, 2005, 2007). So, one key issue in the theory of group judgments aggregation is how to choose the aggregation operator, which will affect the group consensus (Dong, Xu, & Li, 2008a, 2008b; Wang & Fan, 2007; Xu, 2000, 2005, 2009). Yager (1988) provided a family of averaging operator called the ordered weighted averaging (OWA) operators, which allows the implementation the concept of fuzzy majority and has been extended extensively in the resolution process of different problems (Chiclana, Herrera, & Herrera-Viedma, 2003; Dubois, Fortemps, & Pirlot, 2001; Herrera, Herrera-Viedma, & Verdegay, 1997; Torra, 1997; Wang & Parkan, 2007; Wang, Luo, & Liu, 2007; Wu, Liang, & Huang, 2007). Recently, Wu et al. (2009) presented the induced continuous ordered weighted geometric (ICOWG) operator to aggregate interval multiplicative preference relations in group decision making (GDM) problems. In particular, we presented the reliability induced COWG (R-ICOWG) operator and the relative consensus degree induced COWG (RCD-ICOWG) operator. The main novelty of the R-ICOWG and RCD-ICOWG operators is that they permit the aggregation of DMs’ preference in such a way that more importance is given to the most consensus one. However, the knowledge on the use of the ICOWG operator as aggregation method is still quite limited. For example, few studies have been focused on the question that whether the ICOWG operator can improve the group Compatibility Degree.
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
This work focuses on resolving the question that how to measure and improve the consensus among a group of decision makers. To do so, we define the concept of Compatibility Degree for two interval multiplicative linguistic preference relations, and give a Compatibility Index. The Compatibility Index (CI) value measures the level of consensus between individual decision maker and the leading decision maker (Xu, 2004). In order to exert the effect of the leading decision maker, it is often required that the group opinion should be close to the one of the leading decision maker. Thus, the more compatibility in the information provided by the DM, the higher the weighting value should be placed on that information. With this Compatibility Index, we present a CI induced COWG (CI-ICOWG) operator to aggregate the interval multiplicative preference relations. The main novelty of this operator is that it aggregates individual preference relations in such a way that more importance is placed on the most compatibility one. Thus, the collective judgement matrix (CJM) by utilizing of the CI-ICOWG operator can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees (see the results of the illustrative example). Additionally, the leading decision maker’s interval multiplicative preference relation P and the CJM of Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, if P and Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility (see Corollary 1). It is noteworthy that the CI-ICOWG operator can be used for aggregating the interval multiplicative preference relations, thus these results are of great importance both theoretically and practically. In order to do that, this work is set out as follows. Section 2 gives a review of the IOWG, COWG and ICOWG operator. In Section 3, we define the concept of Compatibility Degree and Compatibility Index for two interval multiplicative preference relations. Then, the CI-ICOWG operator is presented and its desired properties are studied. Section 4 provides an illustrative example. Finally, in Section 5 we draw our conclusions.
Definition 3 (Wu et al., 2009). Suppose ½ai ; bi ði ¼ 1; 2; . . . ; nÞ is a set of interval numbers. An induced continuous ordered weighted geometric (ICOWG) operator is a mapping: Xþ ! Rþ , which has associated with it an exponential weighting vector w ¼ ðw1 ; Pn w2 ; . . . ; wn ÞT , such that wi 2 ½0; 1 and i¼1 wi ¼ 1, and is defined to aggregate the set of second arguments of list of n two tuples ðhu1 ; ½a1 ; b1 i; . . . ; hun ; ½an ; bn iÞ according to the following expression:
Uðhu1 ; ½a1 ; b1 i; . . . ; hun ; ½an ; bn iÞ ¼ IOWGðhu1 ; g Q ð½a1 ; b1 Þi; . . . ; hun ; g Q ð½an ; bn ÞiÞ ¼ IOWGðhurð1Þ ; g Q ð½arð1Þ ; brð1Þ Þi; . . . ; hurðnÞ ; g Q ð½arðnÞ ; brðnÞ ÞiÞ ¼
n Y
ðg Q ð½arðiÞ ; brðiÞ ÞÞwi
with r : f1; . . . ; ng ! f1; . . . ; ng being a permutation such that urðiÞ P urðiþ1Þ ; 8 i ¼ 1; . . . ; n 1; hurðiÞ ; g Q ½arðiÞ ; brðiÞ i is the two tuples with urðiÞ the ith highest value in the set fu1 ; . . . ; un g, and g Q ½arðiÞ ; brðiÞ is calculated by COWG operator (2). Definition 4 (Yager and Xu, 2006). If a DM provides his/her preference information on x according to the interval multiplicative prefe ¼ ða ~ij Þnn where erence relation A
~ij ¼ ½aij ; aþij ; a
~ji ¼ ½aji ; aþji ; a
aji aþij ¼ aþji aij ¼ 1;
P aji > 0;
aþii ¼ aii ¼ 1;
for all i; j ¼ 1; 2; . . . ; n:
~ij Þ ¼ g Q ð½aij ; aþij Þ ¼ aþij g Q ða
We start this section by making a generalization of the concepts of IOWG, COWG and ICOWG operators.
~ij Þ ¼ g Q ða
UGW ðhu1 ; a1 i; . . . ; hun ; an iÞ ¼
n a
ðbj Þwi
where bj is the jth largest element of the collection of the aggregated objects a1 ; a2 ; . . . ; an . Definition 2 (Yager and Xu, 2006). A continuous ordered weighted geometric averaging (COWG) operator is a mapping g : Xþ ! Rþ which has associated with it a basic unit-interval monotonic (BUM) function: Q : ½0; 1 ! ½0; 1 and is monotonic with the properties: (1) Q ð0Þ ¼ 0; (2) Q ð1Þ ¼ 1; and (3) Q ðxÞ P Q ðyÞ if x P y, such that
g Q ð½a; bÞ ¼ b: þ
0
b
1 ~ji Þ g Q ða
aij
!R 1 dQ ðyÞy dy 0
dy
;
aþij
for all i 6 j;
ð5Þ
where Q is a basic unit-interval monotonic (BUM) function. e¼ Definition 5. If the interval multiplicative preference relation A ~ij Þnn is consistent, where ða
aij ¼ aik aþkj ;
aþij ¼ aþik akj ;
for all i 6 k 6 j
¼ 1; 2; . . . ; n:
ð6Þ
3. The CI-ICOWG and its properties 3.1. The definition of Compatibility Degree and Compatibility Index Let Gn be a set of all n n interval multiplicative preference relations. ~ Þ e ¼ ða e ¼ ðb ~ij Þnn 2 Gn and B Definition 6. Let A ij nn 2 Gn , then the e e Compatibility Degree of A and B is defined as follows:
e BÞ e ¼ Cð A; ;
ð4Þ
ð1Þ
i¼1
R a 1 dQdyðyÞy dy
aþji
e ¼ ðg ða ~ij ÞÞnn the expected multiplicative preferThen we call g Q ð AÞ Q e where g ða ~ij Þ is the expected vaence relation corresponding to A, Q ~ij of the alternative xi over xj , obtained lue of preference degree a by COWG operator (2):
2. Preliminaries: IOWG, COWG and ICOWG operators
Definition 1 (Yager and Filev, 1999). An IOWG operator of dimenn sion n is a mapping, UGW : Rþ ! Rþ , to which a set of weights or a weighting vector is associated, W ¼ ðw1 ; w2 ; . . . ; wn ÞT ; wj 2 ½0; 1 Pn and j¼1 wj ¼ 1, and it is defined to aggregate the set of second arguments of list of two tuples fhu1 ; a1 i; . . . ; hun ; an ig, given on the basis of a positive ratio scale, according to the following expression:
ð3Þ
i¼1
ð2Þ
where X is the set of closed intervals, in which the lower limits of all closed intervals are positive, Rþ is the set of positive real numbers, and [a, b] is an closed interval in Xþ .
n X n 1X þ aij bji þ aþij bji n2 : 2 i¼1 j¼1
ð7Þ
e BÞ, e the greater the CompatObviously, the smaller the value of Cð A; e ibility Degree of the interval multiplicative preference relations A e and B.
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
e ¼ ðaij Þ e Definition 7. Let A nn 2 Gn and B ¼ ðbij Þnn 2 Gn , then
e BÞ e ¼ CIð A;
1 e BÞ e Cð A;
aij ¼ dij ð8Þ
Theorem 1. Let A ¼ ðaij Þnn 2 Gn and B ¼ ðbij Þnn 2 Gn , then e BÞ e P 0; (1) Cð A; e e and B e ¼ 0 if and only if A e are perfectly compatible. (2) Cð A; BÞ Proof (1) Since
" # n X n n X n n X X 1X þ þ þ þ þ a b þ aij bji ¼ aij bji þ aji bij þ aii bii 2 i¼1 j¼1 ij ji i>j j¼1 i¼1 1 nðn 1Þ n2 P 2þn ¼ 2 2 2 and
" # n X n n X n n X 1X 1 X þ þ þ þ a b ¼ a b þ aji bij þ aii bii 2 i¼1 j¼1 ij ji 2 i>j j¼1 ij ji i¼1 1 nðn 1Þ n2 P 2þn ¼ : 2 2 2 Then, we have n X n X þ aij bji þ aþij bji n2 P 0: i¼1
and thus, we can obtain
e DÞ e AÞ e ¼ 0: e ¼ Cð A; Cð A;
e and B. e is called the Compatibility Index of A e e The greater the value of CIð A; BÞ, the greater the Compatibility e and B. e Degree of the interval multiplicative preference relations A
e BÞ e ¼1 Cð A; 2
for all i; j 2 N
j¼1
e BÞ e ¼ 0, then a bþ ¼ a bþ and aþ b ¼ aþ b (2) Necessity. If Cð A; ij ji ji ij ij ji ji ij for all i; j 2 N. Thus, we can obtain aij ¼ bij for all i; j 2 N. e and B e are perfectly compatible. Therefore, A e and B e are perfectly compatible, then aij ¼ bij (3) Sufficiency. If A þ þ for all i; j 2 N. Thus, we have a ij bji ¼ 1 and aij bji ¼ 1 for all e e i; j 2 N. Therefore, Cð A; BÞ ¼ 0. h
3.2. The CI-ICOWG operator to aggregate interval multiplicative preference relations In a homogeneous GDM problem, the DMs have equal importance. However, in this situation, each expert always can have a Compatibility Index (CI) value associated with them, which measures the level of consensus between individual preference relations and the leading decision maker’s preference relation. Thus, the more compatibility is the information provided by the expert, the higher the weighting value should be placed on that information. In this section, we present the CI-ICOWG operator to aggregate interval multiplicative preferences. Then, we analyze the reciprocity, consistent and compatibility properties of the CJM, which is obtained by using of the CI-ICOWG operator. Definition 8. If a set of DMs D ¼ fd1 ; d2 ; . . . ; dm g provides preference about a set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g by means of interval multiplicative preference relations fRð1Þ ; Rð2Þ ; . . . ; RðmÞ g; RðlÞ 2 R, then a CI-IOWG operator is an IOWG operator in which its order-inducing values is the set of Compatibility Index values fCIð1Þ ; . . . ; CIðlÞ ; . . . ; CIðmÞ g. Definition 9. If Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are the interval multiplicative preference relations provided by m DMs, then the CJM _ _ R ¼ ð r ij Þnn is expressed as follows: _
R ¼ CI-IOWGðhCIð1Þ ; Rð1Þ i; hCIð2Þ ; Rð2Þ i; . . . ; hCIðmÞ ; RðmÞ iÞ ¼ CI-IOWGðhCIðrð1ÞÞ ; Rðrð1ÞÞ i; hCIðrð2ÞÞ ; Rðrð2ÞÞ i; . . . ; hCIðrðmÞÞ ; RðrðmÞÞ iÞ ¼ ðRðrð1ÞÞ Þcrð1Þ ðRðrð2ÞÞ Þcrð2Þ . . . ðRðrðmÞÞ ÞcrðmÞ ð9Þ
_ r ij
¼
ðrð1ÞÞ ðr ij Þcrð1Þ
ðrð2ÞÞ ðr ij Þcrð2Þ
ðrðmÞÞ crðmÞ ðr ij Þ
m Y ðrðlÞÞ ¼ ðr ij ÞcrðlÞ ;
ð10Þ
l¼1
e B; e D e 2 Gn , then we have Theorem 2. Let A; e AÞ e ¼ 0, (1) Reflexivity: Cð A; e e e e AÞ, (2) Symmetry: Cð A; BÞ ¼ Cð B; e BÞ e DÞ e ¼ 0 and Cð B; e DÞ e ¼ 0, then Cð A; e ¼ 0. (3) Transitivity: If Cð A;
Proof n X n 1X ða aþ þ aþij aji Þ n2 ¼ 0 2 i¼1 j¼1 ij ji
(1)
e AÞ e ¼ Cð A;
(2)
n X n X þ e BÞ e e ¼ 1 e AÞ Cð A; ða b þ aþij bji Þ ¼ Cð B; 2n2 i¼1 j¼1 ij ji
(3) Since
e BÞ e ¼ 0 and Cð B; e DÞ e ¼ 0; Cð A; from Theorem 1, we have
aij ¼ bij then
and bij ¼ dij
for all i; j 2 N
where ðrð1Þ; rð2Þ; . . . ; rðnÞÞ is a permutation of (1, 2, . . . , n) such that CIðrðl1ÞÞ P CIðrðlÞÞ and crðl1Þ P crðlÞ for all l ¼ 2; . . . ; m; hCIðrðlÞÞ ; RðrðlÞÞ i is the two tuple with CIðrðlÞÞ the lth biggest value in the set fCIð1Þ ; . . . ; CIðmÞ g; c ¼ ðcrð1Þ ; crð2Þ ; . . . ; crðmÞ ÞT is a weighting vector, P such that m l¼1 crðlÞ ¼ 1; crðlÞ 2 ½0; 1. Theorem 3. Let Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be interval multiplicative preferðlÞ ence _relations provided by m DMs, where RðlÞ ¼ ðrij Þnn , then their _ CJM R ¼ ð r ij Þnn is also an interval multiplicative preference_relation. Furthermore, if all Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are consistent, then R is also consistent. Proof (i) Since Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are interval multiplicative preference relations, we have
~rijrðlÞ ¼ ½r rij ðlÞ ; r rij ðlÞþ ; rðlÞþ
¼ r ij
rðlÞ
r ji
¼ 1; . . . ; m
~rjirðlÞ ¼ ½r rji ðlÞ ; ¼ 1;
rðlÞþ
r ij
rðlÞþ
rji
;
rðlÞ
P 0;
P r ij
rðlÞ
r ij
rðlÞþ
rji
for all i; j 2 N; l
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
Pl
Thus m m Y Y _ _ ðrðlÞÞþ crðlÞ ðrðlÞÞ crðlÞ r þij ¼ ðr ij Þ P ðr ij Þ ¼ r ij P 0; l¼1 _ r ij
l¼1 m Y
_þ
r ji ¼
ðrðlÞÞ
r ji
ðrðlÞÞþ
r ji
ðr ij
ðrðlÞÞþ crðlÞ Þ
¼ 1;
ðrðlÞÞ crðlÞ Þ
¼ 1:
l¼1 _þ r ij
m Y
_
r ji ¼
ðr ij
l¼1 _
_
Hence, R ¼ ð r ij Þnn is also an interval multiplicative preference relation. (ii) Since all Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are consistent, we can obtain rðlÞ rðlÞ rðlÞþ rðlÞþ rðlÞþ rðlÞ ¼ r ik r kj ; rij ¼ rik r kj for all l ¼ 1; 2; . . . ; rij m i 6 k 6 j ¼ 1; 2; . . . ; n. Then, we have _
r ij ¼ ¼
m m crðlÞ Y crðlÞ _ Y _ ðrðlÞÞ ðrðlÞÞþ r ik r kj ¼ r ik r þkj ; l¼1
and _þ
r ij ¼
m m crðlÞ Y crðlÞ Y ðrðlÞÞþ ðrðlÞÞþ ðrðlÞÞ r ij ¼ r ik r kj l¼1
¼
ðrðlÞÞþ
r ik
crðlÞ
l¼1
m Y
ðrðlÞÞ
r kj
crðlÞ
_
SðrðlÞÞ Sðrðl 1ÞÞ Q SðrðnÞÞ SðrðnÞÞ
ðrðlÞÞ
in
Pl
ðrðkÞÞ , and k¼1 CI ðrðlÞÞ ðrðlÞÞ ðrij ; CI ; CIðrðlÞÞ Þ is ðrðnÞÞ
where SðrðlÞÞ ¼
ð12Þ
r is the permutation such that the lth largest value in the set
CIðrð1ÞÞ P CIðrð2ÞÞ P P CIðrðnÞÞ P 0 ) crð1Þ P crð2Þ P crðnÞ P 0
l¼1
m Y
g. fCIðrð1ÞÞ ; . . . ; CI 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 compatibility is reduced, and therefore the above is obtained if the linguistic quantifier Q verifiers that the most the compatibility of an expert the higher the weighting value of that expert in the aggregation, i.e.:
l¼1
l¼1
crðlÞ ¼ Q
CI
m m crðlÞ Y crðlÞ Y ðrðlÞÞ ðrðlÞÞ ðrðlÞÞþ r ij ¼ r ik r kj l¼1
where SðlÞ ¼ k¼1 urðkÞ , and r is the permutation such that urðlÞ in rðlÞ ðpij ; urðlÞ ; v rðlÞ Þ is the lth largest value in the set fv 1 ; . . . ; v n g. Q is a function: ½0; 1 ! ½0; 1 such that Q ð0Þ ¼ 0; Q ð1Þ ¼ 1 and if x > y then Q ðxÞ P Q ðyÞ (Yager & Kacprzyk, 1997). In our case, we propose to use the Compatibility Index values associated with each one of the DMs, both as a weight associated to the argument and as the order inducing values ui ¼ v i ¼ CIðiÞ . Thus, the ordering of the preference values is first induced by the ordering of the DMs from most to least compatibility one, and the weights of the CI-IOWG operator is obtained by applying Eq. (11), which reduces to
_
¼ r þik r kj :
l¼1
_
Hence, R is also consistent, which completes the proof of the theorem. h Definition 10. Let Rð1Þ ; . . . ; RðlÞ ; . . . ; RðmÞ be the interval multiplica-
Theorem 4. Assuming the parameterized family of regular increasing monotone (RIM) quantifiers Q ðrÞ ¼ r a ; a P 0. If a 2 ½0; 1 and P SðrðlÞÞ ¼ lk¼1 CIðrðkÞÞ , then crðlÞ P crðlþ1Þ , for all l ¼ 1; 2; . . . ; m.
tive preference relations provided by m DMs, and g Q ðR Þ;
Proof. If a 2 ½0; 1, then the function Q ðrÞ ¼ ra is concave and we can have
g Q ðRð2Þ Þ; . . . ; g Q ðRðmÞ Þ be the corresponding expected multiplicative
QðT l Þ Q ðT l1 Þ P Q ðT lþ1 Þ Q ðT l Þ
ð1Þ
ðlÞ
preference relations, where g Q ðRðlÞ Þ ¼ g Q ðrij Þnn ; l ¼ 1; 2; . . . ; m, then their collective expected multiplicative preference relation _
_
g Q ðR Þ ¼ g Q ð r ij Þnn is expressed as follows: _
g Q ðR Þ ¼ CI-ICOWGðhCIð1Þ ; g Q ðRð1Þ Þi; . . . ; hCIðmÞ ; g Q ðRðmÞ ÞiÞ
ðlÞÞ Suppose T l ¼ SðSðrrðnÞÞ and SðrðlÞÞ ¼
Pl
k¼1 CI
ðrðkÞÞ
, then
SðrðlÞÞ Sðrðl 1ÞÞ crðlÞ ¼ Q Q ¼ Q ðT l Þ Q ðT l1 Þ SðrðnÞÞ SðrðnÞÞ and
¼ CI-IOWGðhCIðrð1ÞÞ ; g Q ðRðrð1ÞÞ Þi; . . . ; hCIðrðmÞÞ ; g Q ðRðrðmÞÞ ÞiÞ
crðlþ1Þ ¼ Q
¼ ðg Q ðRðrð1ÞÞ ÞÞcrð1Þ ðg Q ðRðrð2ÞÞ ÞÞcrð2Þ . . . ðg Q ðRðrðmÞÞ ÞÞcrðmÞ
Sðrðl þ 1ÞÞ SðrðlÞÞ Q ¼ QðT lþ1 Þ QðT l Þ SðrðnÞÞ SðrðnÞÞ
Thus, we can obtain _
g Q ð r ij Þ ¼
m Y
crðlÞ P crðlþ1Þ
rðlÞ
ðg Q ðr ij ÞÞcrðlÞ :
l¼1
which completes the proof of the theorem. h
Before implementing this operator to aggregate individual preference relations, we need to calculate its associated weights. Yager (2003) proposed a procedure to determine the weighting vector associated with an IOWA operator. In this case, each comment in ðlÞ ðlÞ the aggregation consists of a triple ðpij ; ul ; v l Þ, where pij is the argument value to aggregate, ul is the importance weight value associðlÞ ated to pij , and v l is the order inducing value. Thus, the aggregation is ð1Þ
ðnÞ
IOWAQ ðpij ; . . . ; pij Þ ¼
n X
rðlÞ
wl pij ;
l¼1
with
SðlÞ Sðl 1Þ wl ¼ Q Q SðnÞ SðnÞ
ð11Þ
Lemma 1. Xu, 2000Let xi > 0; bi > 0; i 2 N and n Y i¼1
b
xi i 6
n X
Pn
i¼1 bi
¼ 0, then
bi xi
ð13Þ
i¼1
Lemma 2. Liu and Chen, 2004 For ordered vector x ¼ ðx1 ; x2 ; . . . ; xn Þ; x1 P x2 P P xn , and weights w ¼ ðw1 ; w2 ; . . . ; wn Þ, then vector a1 ; a2 ; . . . an ; ai P 0 ði ¼ 1; 2; . . . ; nÞ.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 If w1 6 w2 6 6 wn , then
ð14Þ
104
J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
a1 w1 x1 þ a2 w2 x2 þ þ an wn xn a1 w1 þ a2 w2 þ þ an wn 6 ða1 x1 þ a2 x2 þ þ an xn Þ a1 þ a2 þ þ an
ð15Þ
with equality if and only if x1 ¼ x2 ¼ ¼ xn
_
CðRðlÞ ; PÞ 6 s; 8 l ¼ 1; . . . ; m ) CðR ; PÞ 6 s;
Definition 11. Suppose Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be the interval multiplicative preference relations provided by m DMs when comparing n alternatives, and P be the interval multiplicative preference relation given by a leading decision _maker. Let CðRðlÞ ; PÞ be Compatibil_ ity Degree of RðlÞ and P, and CðR ; PÞ be Compatibility Degree of R and P. From Theorem 1, it can be obtained that RðlÞ is perfectly compatible with P if CðRðlÞ ; PÞ ¼ 0. The smaller the value of CðRðlÞ ; PÞ, the more Compatibility Degree and consensus degree of RðlÞ and P. Theorem 5. Suppose Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be the interval multiplicative preference relations provided by m DMs when comparing n alternatives. Using the CJM as the aggregation procedure, it holds that: _
CðR ; PÞ 6
ð16Þ
n X n Y m m Y 1 X rðlÞ ðrij ÞcrðlÞ ðpji ÞcrðlÞ 2 n i¼1 j¼1 l¼1 l¼1
1 2n2
_
Corollary 2. The Compatibility Degree between R and P is no more greater than the maximum of the Compatibility Degree between RðlÞ and P, which holds that _
CðR ; PÞ 6 Maxl¼1;...;m fCðRðlÞ ; PÞg:
n X
n X
m Y
i¼1
j¼1
l¼1
ðrðlÞÞ þ crðlÞ pji Þ
ðr ij
6
l¼1
m X
relation of the leading decision maker, then we let Cðg Q ðRðlÞ Þ; g Q ðPÞÞ
rðlÞ þ crðlÞ
ðrij
pji Þ
þ
m Y
be Compatibility Degree of g Q ðRðlÞ Þ and g Q ðPÞ, and Cðg Q ðR ; PÞÞbe _
Compatibility Degree of g Q ðR Þ and g Q ðPÞ.
_
Cðg Q ðR ; PÞÞ 6
!
1 C g Q ðRðlÞ Þ; g Q ðPÞ m
pji Þ
Proof. By Eq. (7), we have _
Cðg Q ðR Þ; g Q ðPÞÞ ¼
crðlÞ ðrðijrðlÞÞ pþji Þ
l¼1
ðrðlÞÞþ crðlÞ pji Þ
ðr ij
l¼1
6
m X
Then n X n m m X X 1 X CðR ; PÞ 6 2 c ðrðrðlÞÞ pþji Þ þ crðlÞ ðrijðrðlÞÞþ pji Þ 2n i¼1 j¼1 l¼1 rðlÞ ij l¼1 ! m n X n X 1 X ðrðlÞÞ ðrðlÞÞþ þ crðlÞ ðr pji þ r ij pji Þ ¼ 2n2 i¼1 j¼1 ij l¼1 _
m X
¼
n X n Y m m Y 1 X rðlÞ ðg Q ðr ij ÞÞcrðlÞ ðg Q ðpji ÞÞcrðlÞ 2 n i¼1 j¼1 l¼1 l¼1
¼
n X n Y m crðlÞ 1 X rðlÞ g ðr Þg Q ðpji Þ n2 i¼1 j¼1 l¼1 Q ij
!
From Lemma 1, it can be obtained that
l¼1
l¼1
l¼1
rðlþ1Þ
Since CðR ; PÞ 6 CðR ; PÞ and crðlÞ P crðlþ1Þ Then, from Lemma 2, we have
crðlÞ CðRðrðlÞÞ ; PÞ 6
l¼1
n X n Y m 1 X rðlÞ crðlÞ g ðr Þ g Q ðpji Þ n2 i¼1 j¼1 l¼1 Q ij
m m crðlÞ X Y rðlÞ g Q ðr ij Þg Q ðpji Þ 6 crðlÞ g Q ðrrij ðlÞ Þg Q ðpji Þ
crðlÞ CðRrðlÞ ; PÞ
rðlÞ
m X
n X n _ 1 X g ð r ij Þ g Q ðpji Þ 2 n i¼1 j¼1 Q
¼
crðlÞ ðrðijrðlÞÞþ pji Þ
l¼1
¼
ð19Þ
rðlÞþ crðlÞ
ðr ij
l¼1
and m Y
ð18Þ
Definition 12. Suppose g Q ðRð1Þ Þ; g Q ðRð2Þ Þ; . . . ; g Q ðRðmÞ Þ be the corresponding expected multiplicative preference relations of m DMs, and g Q ðPÞ be the corresponding expected multiplicative preference
From Lemma 1, it can be obtained that m Y
s is the threshold for acceptable compatibility.
Theorem 6. Suppose Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be the interval multiplicative preference relations provided by m DMs when comparing n alternatives. Using the CI-ICOWG operator as the aggregation procedure, it holds that:
n X n n X n Y m _ 1 X 1 X rðlÞ CðR ; PÞ ¼ 2 r ij pji ¼ 2 ðr ÞcrðlÞ pji n i¼1 j¼1 n i¼1 j¼1 l¼1 ij _
¼
where
ð17Þ
_
m 1 X CðRðlÞ ; PÞ m l¼1
Proof. By Eq. (7), we have
¼
Corollary 1. If the individual interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the CJM _ R is also acceptable compatibility, that is to say,
m X l¼1
crðlÞ
m 1 X CðRðrðlÞÞ ; PÞ m l¼1
Then
!
m m 1 X 1 X CðRðrðlÞÞ ; PÞ ¼ CðRðlÞ ; PÞ ¼ m l¼1 m l¼1
_ n X n X m 1 X Cðg Q R ; g Q ðPÞÞ 6 2 c g rrðlÞ g Q pji n i¼1 j¼1 l¼1 rðlÞ Q ij ¼ crðlÞ
m X l¼1
n X n 1 X rðlÞ g Q ðpji Þ g r Q ij n2 i¼1 j¼1
!
Thus m 1 X CðR ; PÞ 6 CðRðlÞ ; PÞ m l¼1 _
which completes the proof of the theorem. h
¼ crðlÞ Cðg Q ðRrðlÞ Þ; g Q ðPÞÞ Since Cðg Q ðRrðlÞ Þ; g Q ðPÞÞ 6 Cðg Q ðRrðlþ1Þ Þ; g Q ðPÞÞ and crðlÞ P crðlþ1Þ Then, from Lemma 2, we have
105
J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106 m X
crðlÞ Cðg Q ðRrðlÞ Þ; g Q ðPÞÞ 6
l¼1
m X
crðlÞ
l¼1
m 1 X C g Q ðRrðlÞ Þ; g Q ðPÞ m l¼1
!
m 1 X C g Q ðRrðlÞ Þ; g Q ðPÞ ¼ m l¼1
¼
1 m
m X
CðRð1Þ ; PÞ ¼ 1:150;
CðRð2Þ ; PÞ ¼ 1:425;
According to Eq. (8), the following Compatibility Indexes can be worked out:
CIðRð1Þ ; PÞ ¼ 0:870;
C g Q ðRðlÞ Þ; g Q ðPÞ
CðRð3Þ ; PÞ ¼ 0:775:
CIðRð2Þ ; PÞ ¼ 0:702;
CIðRð3Þ ; PÞ ¼ 1:290:
Then, the Compatibility Indexes and the multiplicative preference relations are reordered as follows:
l¼1
CIðRðrð1ÞÞ ; PÞ ¼ CIðRð3Þ ; PÞ;
Thus
¼ CIðRð2Þ ; PÞ;
m 1 X Cðg Q ðR Þ; g Q ðPÞÞ 6 Cðg Q ðRðlÞ Þ; g Q ðPÞÞ m l¼1 _
CIðRðrð2ÞÞ ; PÞ ¼ CIðRð1Þ ; PÞ;
CIðRðrð3ÞÞ ; PÞ
Rðrð1ÞÞ ¼ Rð3Þ ; Rðrð2ÞÞ ¼ Rð1Þ ; Rðrð3ÞÞ
ð2Þ
¼R : Using Eq. (12) with Q ðrÞ ¼ r 2=3 , we get the following weights:
which completes the proof of the theorem. h
crð1Þ ¼ 0:58; crð2Þ ¼ 0:24; crð3Þ ¼ 0:18: Corollary 3. If the individual expected multiplicative preference rela. . ; g Q ðRðmÞ Þ are of acceptable compatibility, tions g Q ðRð1Þ Þ; g Q ðRð2Þ Þ; ._ then the expected CJM g Q ðR Þ is also acceptable compatibility, that is to say, _
Cðg Q ðRðlÞ Þ; g Q ðPÞÞ 6 s; 8 l ¼ 1; . . . ; m ) Cðg Q ðR Þ; g Q ðPÞÞ 6 s; where
ð20Þ
s is the threshold for acceptable compatibility.
_
Then, CJM R is calculated as: _
R ¼ ðRðrð1ÞÞ Þcrð1Þ ðRðrð2ÞÞ Þcrð2Þ ðRðrð3ÞÞ Þcrð3Þ 2 ½1:00;1:00 ½6:00;7:00 ½0:37;0:68 6 6 ½0:14;0:17 ½1:00;1:00 ½6:00;7:56 6 ¼6 6 ½1:47;2:70 ½0:13;0:17 ½1:00;1:00 6 4 ½0:15;0:17 ½0:14;0:17 ½0:13;0:16
_
ðlÞ
Cðg Q ðR Þ; g Q ðPÞÞ 6 Maxl¼1;...;m fCðg Q ðR Þ; g Q ðPÞÞg:
ð21Þ
4. Illustrative example Let us consider the example used by Wu et al. (2009). In this example, there are four DMs D ¼ fd1 ; d2 ; d3 ; d4 g. Suppose that the interval multiplicative preference relation P is given by a leading decision maker, and the interval multiplicative preference relations Rð1Þ ; Rð2Þ ; Rð3Þ are given by the other DMs, respectively. They are listed as follows: 3 2 ½1;1 ½6;7 ½1=7;1=5 ½7;8 ½1=6;1=5 7 6 ½1;1 ½6;7 ½6;7 ½1=7;1=5 7 6 ½1=7;1=6 7 6 ð1Þ R ¼6 ½1=7;1=6 ½1;1 ½6;7 ½7;8 7 7 6 ½5;7 7 6 ½1;1 ½6;7 5 4 ½1=8;1=7 ½1=7;1=6 ½1=7;1=6 ½5;6 ½5;7 ½1=8;1=7 ½1=7;1=6 ½1;1 3 2 ½1;1 ½6;7 ½1=2;1 ½7;8 ½1=2;1 7 6 ½1;1 ½6;7 ½6;7 ½1=7;1=6 7 6 ½1=7;1=6 7 6 7 Rð2Þ ¼ 6 ½1;2 ½1=7;1=6 ½1;1 ½5;7 ½7;8 7 6 7 6 ½1;1 ½6;7 5 4 ½1=8;1=7 ½1=7;1=6 ½1=7;1=5 ½1;2 ½6;7 ½1=8;1=7 ½1=7;1=6 ½1;1 3 2 ½1;1 ½6;7 ½1=2;1 ½5;6 ½1=8;1=7 7 6 ½1;1 ½6;8 ½6;7 ½1=7;1=6 7 6 ½1=7;1=6 7 6 ð3Þ 6 R ¼ 6 ½1;2 ½1=8;1=6 ½1;1 ½7;8 ½6;7 7 7 7 6 ½1;1 ½7;8 5 4 ½1=6;1=5 ½1=7;1=6 ½1=8;1=7 ½7;8 ½6;7 ½1=7;1=6 ½1=8;1=7 ½1;1 3 2 ½1;1 ½6;7 ½1=3;3=5 ½25=4;29=4 ½1=5;3=10 7 6 ½1;1 ½6;29=4 ½6;7 ½1=7;1=6 7 6 ½1=7;1=6 7 6 ½4=29;1=6 ½1;1 ½6;29=4 ½33=5;61=8 7 P¼6 7 6 ½5=3;3 7 6 ½1;1 ½25=4;29=4 5 4 ½4=25;4=29 ½1=7;1=6 ½4=29;1=6 ½10=3;5 ½6;7 ½8=61;5=33 ½4=29;4=25 ½1;1 According to Eq. (7), we can calculate the Compatibility Degree CðRðlÞ ; PÞ; l ¼ 1; 2; 3; 4:
3
7 ½6:00;7:00 ½0:14;0:17 7 7 ½6:35;7:56 ½6:40;7:40 7 7 7 ½1:00;1:00 ½6:56;7:56 5 ½4:55;5:82 ½5:74;7:00 ½0:14;0:16 ½0:13;0:15 ½1:00;1:00
_
Corollary 4. The Compatibility Degree between g Q ðR Þ and g Q ðPÞ is no more greater than the maximum of the Compatibility Degree between g Q ðRðlÞ Þ and g Q ðPÞ, which holds that
½5:76;6:77 ½0:17;0:22
According to Eq. (7) we can obtain that _
CðR ; PÞ ¼ 0:125 < ð1:150 þ 1:425 þ 0:775Þ=3 ¼ 1:117 which verifies the conclusion of Theorem 5. pffiffiffi Assume that the BUM function Q ðyÞ ¼ y, then we can obtain
Z 0
1
dQ ðyÞ 1 y dy ¼ dy 2
Z 0
1
y1=2 dy ¼
1 3
Thus, from Eq. (2), we construct the expected multiplicative preference relation matrix g Q ðRð1Þ Þ; g Q ðRð2Þ Þ; g Q ðRð3Þ Þ; g Q ðPÞ as follows:
3 1:0000 6:6494 0:1788 7:6517 0:1882 7 6 6 0:1504 1:0000 6:6494 6:6494 0:1788 7 7 6 7 g Q ðRð1Þ Þ ¼ 6 6 5:5928 0:1504 1:0000 6:6494 7:6517 7; 7 6 4 0:1307 0:1504 0:1504 1:0000 6:6494 5 2
5:3135 5:5928 0:1307 0:1504 1:0000 3 1:0000 6:6494 0:7937 7:6517 0:7937 7 6 6 0:1504 1:0000 6:6494 6:6494 0:1583 7 7 6 ð2Þ 7 g Q ðR Þ ¼ 6 6 1:2599 0:1504 1:0000 6:2573 7:6517 7; 7 6 4 0:1307 0:1504 0:1598 1:0000 6:6494 5 1:2599 6:3171 0:1307 0:1504 1:0000 3 2 1:0000 6:6494 0:7937 5:6462 0:1366 7 6 6 0:1504 1:0000 7:2685 6:6494 0:1583 7 7 6 ð3Þ 7 g Q ðR Þ ¼ 6 6 1:2599 0:1376 1:0000 7:6517 6:6494 7; 7 6 4 0:1771 0:1504 0:1307 1:0000 7:6517 5 7:3206 6:3171 0:1504 0:1307 1:0000 3 2 1:0000 6:6494 0:4829 6:9145 0:2733 7 6 6 0:1504 1:0000 6:8497 6:6494 0:1649 7 7 6 6 g Q ðPÞ ¼ 6 2:0708 0:1460 1:0000 6:8283 7:3019 7 7; 7 6 4 0:1446 0:1504 0:1464 1:0000 6:9680 5 3:6590 6:0643 0:1370 0:1435 1:0000 3 2 1:0000 6:6494 0:5550 6:4150 0:2025 7 6 6 0:1504 1:0000 7:0017 6:6494 0:1630 7 _ 7 6 7 gQ ðR Þ ¼ 6 6 1:8018 0:1428 1:0000 7:1351 7:0533 7: 7 6 4 0:1559 0:1504 0:1402 1:0000 7:2135 5 4:9383 6:1350 0:1418 0:1386 1:0000 2
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J. Wu et al. / Computers & Industrial Engineering 59 (2010) 100–106
According to Eq. (7), we can calculate the Compatibility Degree CðRðlÞ ; PÞ; l ¼ 1; 2; 3; 4:
Cðg Q ðRð1Þ Þ; g Q ðPÞÞ ¼ 1:225; Cðg Q ðRð2Þ Þ; g Q ðPÞÞ ¼ 1:525; _
Cðg Q ðRð3Þ Þ; g Q ðPÞÞ ¼ 0:825; Cðg Q ðR Þ; g Q ðPÞÞ ¼ 0:125: Thus, we can obtain that _
Cðg Q ðR Þ; g Q ðPÞÞ ¼ 0:125 < ð1:225 þ 1:525 þ 0:825Þ=3 ¼ 1:191 This confirms the conclusion of Theorem 6. From Theorems 3, 5, Corollary 1, and the numerical results, we can draw the following conclusions: _
_
(1) The CJM R ¼ ð r ij Þnn of all interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ is an interval multiplicative preference relation. Furthermore, if all Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are con_ sistent, then R is also consistent. (2) If an interval multiplicative preference relation R and each of interval multiplicative preference relation Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the interval multiplica_ _ tive preference relation R and the CJM R ¼ ð r ij Þnn of ð1Þ ð2Þ ðmÞ R ; R ; . . . ; R are of acceptable compatibility. (3) If an interval multiplicative preference relation R and each of interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the collective judgement matrix (CJM) of Rð1Þ ; Rð2Þ ; . . . ; RðmÞ can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. 5. Concluding remarks This study defined the concept of Compatibility Degree and Compatibility Index for two interval multiplicative preference relations. Then, we presented the CI-ICOWG operator, which aggregates interval multiplicative preference relations by the Compatibility Index (CI) value. The main novelty of the CIICOWG operator is that it can improve group Compatibility Degree. If the leading decision maker’s interval multiplicative preference relation P and each of interval multiplicative preference relations Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are of acceptable compatibility, then the CJM is of acceptable compatibility. Furthermore, the CI-ICOWG operator guarantees that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. Acknowledgements The authors are very grateful to three anonymous referees for their valuable comments and suggestions. This work was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y6080215.
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