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A Framework of Group Decision Making with Hesitant Fuzzy Preference Relations Based on Multiplicative Consistency

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Abstract

The purpose of this paper is to develop a general framework of group decision making with hesitant fuzzy preference relations based on the multiplicative consistency. First, we define a consistency index to measure whether or not a hesitant fuzzy preference relation is of acceptably multiplicative consistency. A consistency improving process is developed to improve the consistency level of a hesitant fuzzy preference relation with unacceptably multiplicative consistency until it is acceptably multiplicative. Then, we present a group consensus index to measure the deviation measure between the individual hesitant fuzzy preference relations and their collective hesitant fuzzy preference relation. A consensus reaching process is developed to help the decision makers improve the consensus level among hesitant fuzzy preference relations until they meet a predefined consensus level. Subsequently, we establish a complete framework of group decision making with hesitant fuzzy preference relations. In this framework, individual multiplicative consistency and group consensus are simultaneously highlighted and the acceptably multiplicative consistency of each hesitant fuzzy preference relation is still maintained in the achievement of the predefined consensus level. Finally, an illustrative numerical example is given to verify the effectiveness and practicality of the developed method.

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Acknowledgments

The author thanks the anonymous referees for their valuable suggestions in improving this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 61375075), the Natural Science Foundation of Hebei Province of China (Grant No. F2012201020), and the Scientific Research Project of Department of Education of Hebei Province of China (Grant No. QN2016235).

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  1. College of Mathematics and Information Science, Hebei University, Baoding, 071002, Hebei, China

    Zhiming Zhang

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  1. Zhiming Zhang
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Correspondence to Zhiming Zhang.

Appendix

Appendix

The proof of Theorem 3.1

$$\begin{aligned} {\kern 1pt} {\text{CI}}\left( {H^{{\left( {t + 1} \right)}} } \right) & = d\left( {H^{{\left( {t + 1} \right)}} ,\tilde{H}^{{\left( {t + 1} \right)}} } \right) \le d\left( {H^{{\left( {t + 1} \right)}} ,\tilde{H}^{\left( t \right)} } \right) \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left[ \begin{aligned} \ln \left( {\left( {h_{ij}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ - \ln \left( {\left( {\tilde{h}_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) + \ln \left( {\left( {\tilde{h}_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ \end{aligned} \right]^{2} } } } \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left[ \begin{aligned} \left( {1 - \delta } \right)\ln \left( {\left( {h_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) + \delta \ln \left( {\left( {\tilde{h}_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ - \left( {1 - \delta } \right)\ln \left( {\left( {h_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) - \delta \ln \left( {\left( {\tilde{h}_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ - \ln \left( {\left( {\tilde{h}_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) + \ln \left( {\left( {\tilde{h}_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ \end{aligned} \right]}^{2} } } \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left[ \begin{aligned} \left( {1 - \delta } \right)\ln \left( {\left( {h_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) - \left( {1 - \delta } \right)\ln \left( {\left( {h_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ - \left( {1 - \delta } \right)\ln \left( {\left( {\tilde{h}_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) + \left( {1 - \delta } \right)\ln \left( {\left( {\tilde{h}_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ \end{aligned} \right]}^{2} } } \\ & = \frac{{2\left( {1 - \delta } \right)^{2} }}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left[ \begin{aligned} \ln \left( {\left( {h_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ - \ln \left( {\left( {\tilde{h}_{ij}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) + \ln \left( {\left( {\tilde{h}_{ji}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) \hfill \\ \end{aligned} \right]}^{2} } } \\ & = \left( {1 - \delta } \right)^{2} d\left( {H^{\left( t \right)} ,\tilde{H}^{\left( t \right)} } \right) \le {\text{CI}}\left( {H^{\left( t \right)} } \right) \\ \end{aligned}$$

which completes the proof. Moreover, \({\text{CI}}\left( {H^{\left( t \right)} } \right) \ge 0\), for each t. Thus, the sequence \(\left\{ {{\text{CI}}\left( {H^{\left( t \right)} } \right)} \right\}\) is monotonically decreasing and has lower bounds.

The proof of Theorem 3.2

(1) According to Eq. (11), for all \(i,j = 1,2, \ldots ,n\), we have

$$\begin{aligned} h_{{ij,{\text{c}}}}^{\sigma \left( s \right)} + h_{{ji,{\text{c}}}}^{\sigma \left( s \right)} & = \frac{{\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } }}{{\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } + \prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } }} \\ & \quad + \;\frac{{\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } }}{{\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } + \prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } }} = 1 \\ \end{aligned}$$
$$\begin{aligned} h_{{ii,{\text{c}}}}^{\sigma \left( s \right)} & = \frac{{\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ii,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } }}{{\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ii,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } + \prod\limits_{k = 1}^{m} {\left( {1 - \bar{h}_{ii,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } }} \\ & = 0.5 \\ \end{aligned}$$

According to Definition 2.4, H c is a HFPR.

(2) Let \(\tilde{H}_{\text{c}} = \left( {\tilde{h}_{{ij,{\text{c}}}} } \right)_{n \times n} = \left( {\left\{ {\left. {\tilde{h}_{{ij,{\text{c}}}}^{\sigma \left( s \right)} } \right|s = 1,2, \ldots ,l} \right\}} \right)_{n \times n}\) be the multiplicative consistent HFPR of \(H_{\text{c}} = \left( {h_{{ij,{\text{c}}}} } \right)_{n \times n}\) obtained by Eq. (6), i.e., \(\tilde{h}_{{ij,{\text{c}}}}^{\sigma \left( s \right)} = \frac{{\sqrt[n]{{\prod\limits_{k = 1}^{n} {h_{{ik,{\text{c}}}}^{\sigma \left( s \right)} \cdot h_{{kj,{\text{c}}}}^{\sigma \left( s \right)} } }}}}{{\sqrt[n]{{\prod\limits_{k = 1}^{n} {h_{{ik,{\text{c}}}}^{\sigma \left( s \right)} \cdot h_{{kj,{\text{c}}}}^{\sigma \left( s \right)} } }} + \sqrt[n]{{\prod\limits_{k = 1}^{n} {\left( {1 - h_{{ik,{\text{c}}}}^{\sigma \left( s \right)} } \right)\left( {1 - h_{{kj,{\text{c}}}}^{\sigma \left( s \right)} } \right)} }}}}\).

Let \(\varepsilon_{ij,k}^{\sigma \left( s \right)} = \ln \left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right) - \ln \left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right) - \ln \left( {\tilde{h}_{ij,k}^{\sigma \left( s \right)} } \right) + \ln \left( {\tilde{h}_{ji,k}^{\sigma \left( s \right)} } \right)\), then we have

$$\begin{aligned} {\text{CI}}\left( {H_{\text{c}} } \right) = d\left( {H_{\text{c}} ,\tilde{H}_{\text{c}} } \right) & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\ln \left( {h_{{ij,{\text{c}}}}^{\sigma \left( s \right)} } \right) - \ln \left( {h_{{ji,{\text{c}}}}^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \;\ln \left( {\tilde{h}_{{ij,{\text{c}}}}^{\sigma \left( s \right)} } \right) + \ln \left( {\tilde{h}_{{ji,{\text{c}}}}^{\sigma \left( s \right)} } \right)} \right)^{2} \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\ln \left( {\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } } \right) - \ln \left( {\prod\limits_{k = 1}^{m} {\left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right)^{{\lambda_{k} }} } } \right)} \right.} } } \\ & \quad \left. { - \;\ln \left( {\sqrt[n]{{\prod\limits_{t = 1}^{n} {h_{{it,{\text{c}}}}^{\sigma \left( s \right)} \cdot h_{{tj,{\text{c}}}}^{\sigma \left( s \right)} } }}} \right) + \ln \left( {\sqrt[n]{{\prod\limits_{t = 1}^{n} {h_{{jt,{\text{c}}}}^{\sigma \left( s \right)} \cdot h_{{ti,{\text{c}}}}^{\sigma \left( s \right)} } }}} \right)} \right)^{2} \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right) - \sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \;\sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\sqrt[n]{{\prod\limits_{t = 1}^{n} {\bar{h}_{it,k}^{\sigma \left( s \right)} \cdot \bar{h}_{tj,k}^{\sigma \left( s \right)} } }}} \right) + \sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\sqrt[n]{{\prod\limits_{t = 1}^{n} {\bar{h}_{jt,k}^{\sigma \left( s \right)} \cdot \bar{h}_{ti,k}^{\sigma \left( s \right)} } }}} \right)} \right)^{2} \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\bar{h}_{ij,k}^{\sigma \left( s \right)} } \right) - \sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\bar{h}_{ji,k}^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \;\sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\tilde{h}_{ij,k}^{\sigma \left( s \right)} } \right) + \sum\limits_{k = 1}^{m} {\lambda_{k} } \ln \left( {\tilde{h}_{ji,k}^{\sigma \left( s \right)} } \right)} \right)^{2} \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\sum\limits_{k = 1}^{m} {\lambda_{k} } \varepsilon_{ij,k}^{\sigma \left( s \right)} } \right)^{2} } } } \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\sum\limits_{k = 1}^{m} {\lambda_{k}^{2} } \left( {\varepsilon_{ij,k}^{\sigma \left( s \right)} } \right)^{2} + 2\sum\limits_{p = 1}^{m - 1} {\sum\limits_{q = p + 1}^{m} {\lambda_{p} \lambda_{q} \varepsilon_{ij,p}^{\sigma \left( s \right)} \varepsilon_{ij,q}^{\sigma \left( s \right)} } } } \right)} } } \\ & = \sum\limits_{k = 1}^{m} {\lambda_{k}^{2} } \left[ {\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\varepsilon_{ij,k}^{\sigma \left( s \right)} } \right)^{2} } } } } \right] \\ & \quad + \;2\sum\limits_{p = 1}^{m - 1} {\sum\limits_{q = p + 1}^{m} {\lambda_{p} \lambda_{q} \left[ {\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\varepsilon_{ij,p}^{\sigma \left( s \right)} \varepsilon_{ij,q}^{\sigma \left( s \right)} } } } } \right]} } \\ & \le \sum\limits_{k = 1}^{m} {\lambda_{k}^{2} } \left[ {\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\varepsilon_{ij,k}^{\sigma \left( s \right)} } \right)^{2} } } } } \right] \\ & \quad + \;2\sum\limits_{p = 1}^{m - 1} {\sum\limits_{q = p + 1}^{m} {\lambda_{p} \lambda_{q} \hbox{max} \left\{ {\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\varepsilon_{ij,p}^{\sigma \left( s \right)} } \right)^{2} } } } ,\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\varepsilon_{ij,q}^{\sigma \left( s \right)} } \right)^{2} } } } } \right\}} } \\ & \le \left( {\sum\limits_{k = 1}^{m} {\lambda_{k} } } \right)^{2} \mathop {\hbox{max} }\limits_{1 \le k \le m} \left\{ {\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\varepsilon_{ij,k}^{\sigma \left( s \right)} } \right)^{2} } } } } \right\} \\ & = \mathop {\hbox{max} }\limits_{1 \le k \le m} \left\{ {\frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\varepsilon_{ij,k}^{\sigma \left( s \right)} } \right)^{2} } } } } \right\} = \mathop {\hbox{max} }\limits_{1 \le k \le m} \left\{ {{\text{CI}}\left( {H_{k} } \right)} \right\} \\ \end{aligned}$$

(3) can be directly derived from (2).

The proof of Theorem 4.1

(1) According to Eqs. (11), (12) and (14), for all \(k = 1,2, \ldots ,m\), we have

$$\begin{aligned} {\text{GCI}}\left( {H_{k}^{{\left( {t + 1} \right)}} } \right) & = d\left( {H_{k}^{{\left( {t + 1} \right)}} ,H_{\text{c}}^{{\left( {t + 1} \right)}} } \right) \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\ln \left( {\left( {h_{ij,k}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji,k}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \;\ln \left( {\left( {h_{{ij,{\text{c}}}}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) + \ln \left( {\left( {h_{{ji,{\text{c}}}}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right)} \right)^{2} \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\ln \left( {\left( {h_{ij,k}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji,k}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \;\sum\limits_{p = 1}^{m} {\lambda_{p} } \ln \left( {\left( {h_{ij,p}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) + \sum\limits_{p = 1}^{m} {\lambda_{p} } \ln \left( {\left( {h_{ji,p}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right)} \right)^{2} \\ & = \frac{2}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\sum\limits_{p = 1}^{m} {\lambda_{p} \left( {\ln \left( {\left( {h_{ij,k}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji,k}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right)} \right.} } \right.} } } \\ & \quad \left. {\left. { - \ln \left( {\left( {h_{ij,p}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right) + \ln \left( {\left( {h_{ji,p}^{{\left( {t + 1} \right)}} } \right)^{\sigma \left( s \right)} } \right)} \right)} \right)^{2} \\ & = \frac{{2\left( {1 - \eta } \right)}}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\ln \left( {\left( {h_{ij,k}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji,k}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \sum\limits_{p = 1}^{m} {\lambda_{p} } \left( {\ln \left( {\left( {h_{ij,p}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right)} \right) + \sum\limits_{p = 1}^{m} {\lambda_{p} } \left( {\ln \left( {\left( {h_{ji,p}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right)} \right)} \right)^{2} \\ & = \frac{{2\left( {1 - \eta } \right)}}{{n\left( {n - 1} \right)l}}\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\sum\limits_{s = 1}^{l} {\left( {\ln \left( {\left( {h_{ij,k}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) - \ln \left( {\left( {h_{ji,k}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right)} \right.} } } \\ & \quad \left. { - \ln \left( {\left( {h_{ij,c}^{\left( t \right)} } \right)^{\sigma \left( s \right)} } \right) + \ln \left( {\left( {h_{ji,c}^{\left( t \right)} } \right)} \right)} \right)^{2} \\ & = \left( {1 - \eta } \right){\text{GCI}}\left( {H_{k}^{\left( t \right)} } \right) < {\text{GCI}}\left( {H_{k}^{\left( t \right)} } \right) \\ \end{aligned}$$

(2) Let \(\left\{ {H_{k}^{{\left( {t + 1} \right)}} = \left( {h_{ij,k}^{{\left( {t + 1} \right)}} } \right)_{n \times n} } \right\}\) (\(k = 1,2, \ldots ,m\)) be the HFPR sequence in the iteration t+1 derived by Eq. (14). From Eq. (14), we know that \(H_{k}^{{\left( {t + 1} \right)}}\) is the combination of \(H_{k}^{\left( t \right)}\) and \(H_{\text{c}}^{\left( t \right)}\). According to Theorem 3.2, we have \({\text{CI}}\left( {H_{k}^{{\left( {t + 1} \right)}} } \right) \le \hbox{max} \left\{ {{\text{CI}}\left( {H_{k}^{\left( t \right)} } \right),{\text{CI}}\left( {H_{\text{c}}^{\left( t \right)} } \right)} \right\} \le \mathop {\hbox{max} }\limits_{1 \le k \le m} \left\{ {{\text{CI}}\left( {H_{k}^{\left( t \right)} } \right)} \right\}\). Consequently, \(\mathop {\hbox{max} }\limits_{1 \le k \le m} \left\{ {{\text{CI}}\left( {H_{k}^{{\left( {t + 1} \right)}} } \right)} \right\} \le \mathop {\hbox{max} }\limits_{1 \le k \le m} \left\{ {{\text{CI}}\left( {H_{k}^{\left( t \right)} } \right)} \right\}\).

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Zhang, Z. A Framework of Group Decision Making with Hesitant Fuzzy Preference Relations Based on Multiplicative Consistency. Int. J. Fuzzy Syst. 19, 982–996 (2017). https://doi.org/10.1007/s40815-016-0219-4

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