Abstract
We develop a new compatibility for the interval fuzzy preference relations based on the continuous ordered weighted averaging (COWA) operator and use it to determine the weights of experts in group decision making (GDM). We define some concepts of the compatibility degree and the compatibility index for the two interval fuzzy preference relations based on the COWA operator. We study some desirable properties of the compatibility index and investigate the relationship between the each expert’s interval fuzzy preference relation and the synthetic interval fuzzy preference relation. The prominent characteristic of the compatibility index based on the COWA operator is that it can deal with the compatibility of all the arguments by using a controlled parameter considering the attitude of decision maker rather than the compatibility of the simply two points in intervals. To determine the experts’ weights in the GDM with the interval fuzzy preference relations, we propose an optimal model based on the criterion of minimizing the compatibility index. In the end, we give a numerical example to develop the new approach to GDM with interval fuzzy preference relations.
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Acknowledgments
The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions with regard to this paper. The work was supported by National Natural Science Foundation of China (No. 71071002, 71371011; 71301001), Higher School Specialized Research Fund for the Doctoral Program (No. 20123401110001), The Scientific Research Foundation of the Returned Overseas Chinese Scholars, Anhui Provincial Natural Science Foundation (No. 1308085QG127), Provincial Natural Science Research Project of Anhui Colleges (No. KJ2012A026), Humanity and Social Science Youth foundation of Ministry of Education (13YJC630092), Humanities and Social Science Research Project of Department of Education of Anhui Province (No. SK2013B041).
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Appendix
Appendix
Proof of Theorem 1
-
(1)
By Definition 7, we have \(C_\lambda (a,b)=\left| {F_Q (a)-F_Q (b)} \right| \ge 0\).
-
(2)
By Definition 7, we obtain \(C_\lambda (a,a)=\left| {F_Q (a)-F_Q (a)} \right| \) \(=0\).
-
(3)
By Definition 7, we get
$$\begin{aligned} C_\lambda (a,b)&= \left| {F_Q (a)-F_Q (b)} \right| \\&= \left| {F_Q (b)-F_Q (a)} \right| =C_\lambda (b,a). \end{aligned}$$ -
(4)
If \(C_\lambda (a,b)=0\) and \(C_\lambda (b,c)=0\), then by Definition 7, we have \(F_Q (a)=F_Q (b)\) and \(F_Q (b)=F_Q (c)\). Thus, \(F_Q (a)=F_Q (c)\). That is,
$$\begin{aligned} C_\lambda (a,c)=\vert F_Q (a)-F_Q (c)\vert =0. \end{aligned}$$ -
(5)
By Definition 7, we get that
$$\begin{aligned} C_\lambda (a,c)&=\vert F_Q (a)-F_Q (c)\vert \\&=\vert (F_Q (a)-F_Q (b))-(F_Q (b)-F_Q (c))\vert \\&\le \vert F_Q (a)-F_Q (b)\vert +\vert (F_Q (b)-F_Q (c))\vert \\&=C_\lambda (a,b)+C_\lambda (b,c). \end{aligned}$$
This completes the proof of Theorem 1. \(\square \)
Proof of Theorem 2
By Eq. (8), we have
and
Then \(C_\lambda (a(-s,-t),a(0,0))=C_\lambda (a(s,t),a(0,0))\).
Similarly, by Eq. (8), we get
and
Thus, \(C_\lambda (a(s,-t),a(0,0))=C_\lambda (a(-s,t),a(0,0))\). This completes the proof of Theorem 2. \(\square \)
Proof of Theorem 3
Necessity. If \(a\) and \(b\) are perfectly compatible, then by Eq. (8), for all attitudinal character \(\lambda \in [0,1]\), we have \(C_\lambda (a,b)=0\), i.e.,
Let \(\lambda =1\), then \(a^U=b^U\). And let \(\lambda =0\), then \(a^L=b^L\). Thus, \(a=b\).
Sufficiency. If \(a=b\), then \(a^U=b^U\) and \(a^L=b^L\). Thus, for all attitudinal character \(\lambda \), we have \(\lambda (a^U-b^U)+(1-\lambda )(a^L-b^L)=0\), which means \(C_\lambda (a,b)=0\). Therefore, \(a\) and \(b\) are perfectly compatible. This completes the proof of Theorem 3. \(\square \)
Proof of Theorem 5
-
(1)
Since \(C_\lambda (a_{ij} ,b_{ij} )\ge 0\) for all \(i,j=1,2,\ldots ,n\), then by Eq. (8), we obtain that
$$\begin{aligned} C_\lambda (A,B)=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,b_{ij} )} } \ge 0. \end{aligned}$$ -
(2)
Since \(C_\lambda (a_{ij} ,a_{ij} )=0\) for all \(i,j=1,2,\ldots ,n\), then by Eq. (8), we have
$$\begin{aligned} C_\lambda (A,A)=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,a_{ij} )} } =0. \end{aligned}$$ -
(3)
Since \(C_\lambda (a_{ij} ,b_{ij} )=C_\lambda (b_{ij} ,a_{ij} )\) for all \(i,j=1,2,\ldots ,n\), then by Eq. (8), we get that
$$\begin{aligned}&\!\!C_\lambda (A,B)=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,b_{ij} )} } \\&\quad =\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (b_{ij} ,a_{ij} )} } =C_\lambda (B,A). \end{aligned}$$ -
(4)
If \(C_\lambda (A,B)=0\) and \(C_\lambda (B,D)=0\), then by Eq. (8), we have that
$$\begin{aligned} C_\lambda (A,B)=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,b_{ij} )} } =0, \end{aligned}$$and
$$\begin{aligned} C_\lambda (B,D)=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (b_{ij} ,d_{ij} )} } =0. \end{aligned}$$Thus, \(C_\lambda (a_{ij} ,b_{ij} )=0\) and \(C_\lambda (b_{ij} ,d_{ij} )=0\), for all \(i,j=1,2,\ldots ,n\). Then by Theorem 1, we get \(C_\lambda (a_{ij} ,d_{ij} )=0\) for all \(i,j=1,2,\ldots ,n\). Therefore,
$$\begin{aligned} C_\lambda (A,D)=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,d_{ij} )} } =0. \end{aligned}$$ -
(5)
By Eq. (8) and Theorem 1, we obtain
$$\begin{aligned} C_\lambda (A,D)&=\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,d_{ij} )} } \\&\le \sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (a_{ij} ,b_{ij} )} } +\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {C_\lambda (b_{ij} ,d_{ij} )} }\\&=C_\lambda (A,B)+C_\lambda (B,D). \end{aligned}$$
This completes the proof of Theorem 5. \(\square \)
Proof of Theorem 6
Since \(A=(a_{ij} )_{n\times n} \in \Omega _n \), where \(a_{ij} =[a_{ij}^L ,a_{ij}^U ]\). By Definition 3, we have
By Eq. (12), we obtain \(\bar{a}_{ij} =[\bar{a}^L_{ij} ,\bar{a}^U_{ij} ]=1-a_{ij} =[1-a_{ij}^U ,1-a_{ij}^L ]\), i.e.,
Similarly, we can get
Thus,
and
From Definition 3, we can see that \(\bar{A}=(\bar{a}_{ij} )_{n\times n} \in \Omega _n \), which completes the proof of Theorem 6. \(\square \)
Proof of Theorem 7
Since \(\bar{A}=(\bar{a}_{ij} )_{n\times n} \) and \(\bar{B}=(\bar{b}_{ij} )_{n\times n} \) are complementary matrices of \(A\) and \(B\), respectively, we have
and
Then by Definition 10, we get
This completes the proof of Theorem 7. \(\square \)
Proof of Theorem 10
By Definition 14, we have that
That is, \(\tilde{a}_{ij} ^L{=}\sum \nolimits _{k=1}^m {\omega ^{(k)}(a_{ij} ^{(k)})^L} \) and \(\tilde{a}_{ij} ^U{=}\sum \nolimits _{k=1}^m {\omega ^{(k)}(a_{ij} ^{(k)})^U} \), where \({\varvec{\omega }}=(\omega ^{(1)},\omega ^{(2)},\ldots , \ \omega ^{(n)})\) is the weighting vector of \(m\) experts, which satisfies that \(\omega ^{(k)}\ge 0\) for all \(j=1,2,\ldots ,m\) and \(\sum \nolimits _{k=1}^m {\omega ^{(k)}} =1\).
Since \(CI_\lambda (A^{(k)},B)\le \alpha \), we get
Thus, according to Definition 12, we obtain that
This completes the proof of Theorem 10. \(\square \)
Proof of Theorem 11
Since for a given attitudinal character \(\lambda \), \(\tilde{A}\) and \(B\) are not perfectly compatible, there exists \(i_0 ,j_0 \in \{1,2,\ldots ,n\}\), which satisfy
Thus,
By Eq. (19), we get \(e_{k_1 k_2 } =e_{k_2 k_1 } \), \(\forall k_1 ,k_2 =1,2,\ldots ,m\). Therefore, \(E=(e_{k_1 k_2 } )_{m\times m} \) is a definite matrix. Thus, it is also a nonsingular matrix. We can construct the Lagrange function corresponding to the model of Eq. (23):
where \(\mu \) is called the Lagrange multiplier.
Taking the partial derivatives of equation mentioned above with respect to \(\omega \) and \(\mu \), and setting them to be equal to 0, we obtain that
By solving these equations, we get
With the fact that \(\frac{\partial ^2L(\omega ,\mu )}{\partial \omega ^2}=2E\), which means that \(f(\omega )\) is a strictly convex function, \(\omega ^*\) is the unique optimal solution to Eq. (23). This completes the proof of Theorem 11. \(\square \)
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Zhou, L., He, Y., Chen, H. et al. Compatibility of interval fuzzy preference relations with the COWA operator and its application to group decision making. Soft Comput 18, 2283–2295 (2014). https://doi.org/10.1007/s00500-013-1201-9
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DOI: https://doi.org/10.1007/s00500-013-1201-9