Linguistic Dual Hesitant Fuzzy Preference Relations and Their Application in Group Decision-Making

Abstract

Intuitionistic fuzzy preference relations (IFPRs) can not only deal with the uncertainty and vagueness of decision makers’ judgments but also describe the information from the aspects of preferred and non-preferred, respectively. Thus, it has proved to be an efficient tool for solving group decision-making (GDM) problems. However, considering the expression of linguistic information of the current extended IFPRs is limited by a single linguistic term, it is still imperfect to deal with GDM problems. Hence, this paper first proposes a novel extended IFPRs as linguistic dual hesitant fuzzy preference relations (LDHFPRs) by utilizing a set of ordered linguistic terms to describe the preferred and non-preferred evaluation information, which recognize the uncertainty and hesitance of each decision maker and conform to real-life decision-making situations better. Subsequently, we construct the conditions of additive consistency and develop a maximum consistency linear programming model to cope with the problem of inconsistent LDHFPRs. Furthermore, a novel consensus reaching process which pays more attention to the minority but important individual opinions is established. Finally, a real-world application is utilized to demonstrate the effectiveness of the proposed method, and a comparison with the existing related works is presented to show the advantages and innovation of the proposed GDM method.

Appendices

Appendix A

Property 5

(the robust property)

Proof. (1) The sufficiency.□

Suppose the $R_{s} = (d_{s_ ij} )_{n \times n}$ is additively consistent, and for each triple of ($R_{s} = (d_{s_ ij} )_{n \times n}$), let $\delta (i) = k,$$\delta (k) = j$,$\delta (j) = i$, then we have

$$E\left( {d_{{s\delta (i)\delta (k)}} } \right) \oplus E\left( {d_{{s\delta (k)\delta (j)}} } \right) = E\left( {d_{{skj}} } \right) \oplus E\left( {d_{{sji}} } \right)$$

$$E\left({d_{s\delta(i)\delta(j)}}\right)\oplus (\tau,\tau)=E\left({d_{ski}}\right)\oplus (\tau,\tau)$$

Following property 4, if $R_{s} = (d_{s_ ij} )_{n \times n}$ is additively consistent, we have ${(}\sum\nolimits_{l = 1}^{{\# h_{sij} }} {\lambda_{ij}^{l} } p_{\alpha ij}^{l} s_{\alpha ij}^{l} ,\sum\nolimits_{{l^{\prime} { = 1}}}^{{\# g_{sij} }} {\lambda_{ij}^{{l^{\prime} }} p_{\beta ij}^{{l^{\prime} }} s_{\beta ij}^{{l^{\prime} }} } ) \oplus (\sum\nolimits_{l = 1}^{{\# h_{sij} }} {(1 - \lambda_{ij}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {\xi_{ji}^{{l^{\prime} }} p_{\alpha ij}^{l} s_{2\tau } s_{\alpha ji}^{{l^{\prime} }} } ,\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {(1 - \lambda_{ij}^{{l^{\prime} }} )} \sum\nolimits_{l = 1}^{{\# h_{sij} }} {\xi_{ji}^{l} p_{\beta ij}^{{l^{\prime} }} s_{2\tau } s_{\beta ji}^{l} } )) \oplus (s_{\tau } ,s_{\tau } )=$${(}\sum\nolimits_{l = 1}^{{\# h_{sik} }} {\lambda_{ik}^{l} } p_{\alpha ik}^{l} s_{\alpha ik}^{l} ,$$\sum\nolimits_{{l^{\prime} { = 1}}}^{{\# g_{sik} }} {\lambda_{ik}^{{l^{\prime} }} p_{\beta ik}^{{l^{\prime} }} s_{\beta ik}^{{l^{\prime} }} } )$$\oplus (\sum\nolimits_{l = 1}^{{\# h_{sik} }} {(1 - \lambda_{ik}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {\xi_{ki}^{{l^{\prime} }} p_{\alpha ik}^{l} s_{2\tau } s_{\alpha ki}^{{l^{\prime} }} } ,\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {(1 - \lambda_{ik}^{{l^{\prime} }} )} \sum\nolimits_{l = 1}^{{\# h_{sik} }} {\xi_{ki}^{l} p_{\beta ik}^{{l^{\prime} }} s_{2\tau } s_{\beta ki}^{l} } )) \oplus$${(}\sum\nolimits_{l = 1}^{{\# h_{skj} }} {\lambda_{kj}^{l} } p_{\alpha kj}^{l} s_{\alpha kj}^{l} ,\sum\nolimits_{{l^{\prime} { = 1}}}^{{\# g_{skj} }} {\lambda_{kj}^{{l^{\prime} }} p_{\beta kj}^{{l^{\prime} }} s_{\beta kj}^{{l^{\prime} }} } ) \oplus (\sum\nolimits_{l = 1}^{{\# h_{skj} }} {(1 - \lambda_{kj}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {\xi_{jk}^{{l^{\prime} }} p_{\alpha kj}^{l} s_{2\tau } s_{\alpha jk}^{{l^{\prime} }} } ,$$\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {(1 - \lambda_{kj}^{{l^{\prime} }} )} \sum\nolimits_{l = 1}^{{\# h_{skj} }} {\xi_{jk}^{l} p_{\beta kj}^{{l^{\prime} }} s_{2\tau } s_{\beta jk}^{l} } ))$, namely, $\begin{aligned} \left\{ \begin{aligned} \sum\nolimits_{l = 1}^{{\# h_{sij} }} {\lambda_{ij}^{l} } p_{\alpha ij}^{l} s_{\alpha ij}^{l} \oplus \sum\nolimits_{l = 1}^{{\# h_{sij} }} {(1 - \lambda_{ij}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {\xi_{ji}^{{l^{\prime} }} p_{\alpha ij}^{l} s_{2\tau } s_{\alpha ji}^{{l^{\prime} }} } \oplus s_{\tau } { = (}\sum\nolimits_{l = 1}^{{\# h_{sik} }} {\lambda_{ik}^{l} } p_{\alpha ik}^{l} s_{\alpha ik}^{l} \oplus \sum\nolimits_{l = 1}^{{\# h_{sik} }} {(1 - \lambda_{ik}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {\xi_{ki}^{{l^{\prime} }} p_{\alpha ik}^{l} s_{2\tau } s_{\alpha ki}^{{l^{\prime} }} } ) \oplus (\sum\nolimits_{l = 1}^{{\# h_{skj} }} {\lambda_{kj}^{l} } p_{\alpha kj}^{l} s_{\alpha kj}^{l} \oplus \sum\nolimits_{l = 1}^{{\# h_{skj} }} {(1 - \lambda_{kj}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {\xi_{jk}^{{l^{\prime} }} p_{\alpha kj}^{l} s_{2\tau } s_{\alpha jk}^{{l^{\prime} }} } ) \hfill \\ \sum\nolimits_{{l^{\prime} { = 1}}}^{{\# g_{sij} }} {\lambda_{ij}^{{l^{\prime} }} p_{\beta ij}^{{l^{\prime} }} s_{\beta ij}^{{l^{\prime} }} } \oplus \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {(1 - \lambda_{ij}^{{l^{\prime} }} )} \sum\nolimits_{l = 1}^{{\# h_{sij} }} {\xi_{ji}^{l} p_{\beta ij}^{{l^{\prime} }} s_{2\tau } s_{\beta ji}^{l} } \oplus s_{\tau } = (\sum\nolimits_{{l^{\prime} { = 1}}}^{{\# g_{sik} }} {\lambda_{ik}^{{l^{\prime} }} p_{\beta ik}^{{l^{\prime} }} s_{\beta ik}^{{l^{\prime} }} \oplus \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {(1 - \lambda_{ik}^{{l^{\prime} }} )} \sum\nolimits_{l = 1}^{{\# h_{sik} }} {\xi_{ki}^{l} p_{\beta ik}^{{l^{\prime} }} s_{2\tau } s_{\beta ki}^{l} } ) \oplus (\sum\nolimits_{{l^{\prime} { = 1}}}^{{\# g_{skj} }} {\lambda_{kj}^{{l^{\prime} }} p_{\beta kj}^{{l^{\prime} }} s_{\beta kj}^{{l^{\prime} }} } \oplus \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {(1 - \lambda_{kj}^{{l^{\prime} }} )} \sum\nolimits_{l = 1}^{{\# h_{skj} }} {\xi_{jk}^{l} p_{\beta kj}^{{l^{\prime} }} s_{2\tau } s_{\beta jk}^{l} } )} \hfill \\ \end{aligned} \right. \hfill \\ \hfill \\ \end{aligned}$

Since, $\begin{aligned} \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {\lambda _{{ij}}^{l} } p_{{\alpha ij}}^{l} s_{{\alpha ij}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(1 - \lambda _{{ij}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{2\tau }} s_{{\alpha ji}}^{{l^{\prime} }} } \oplus s_{\tau } = (\sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {\lambda _{{ik}}^{l} } p_{{\alpha ik}}^{l} s_{{\alpha ik}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(1 - \lambda _{{ik}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{2\tau }} s_{{\alpha ki}}^{{l^{\prime} }} } ) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } ) \hfill \\ \Rightarrow \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(((} \lambda _{{ik}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{2\tau }} s_{{\alpha ki}}^{{l^{\prime} }} ) - \lambda _{{ik}}^{l} p_{{\alpha ik}}^{l} s_{{\alpha ik}}^{l} )} \oplus s_{\tau } = \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(((\lambda _{{ij}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{2\tau }} s_{{\alpha ji}}^{{l^{\prime} }} )} } - \lambda _{{ij}}^{l} p_{{\alpha ij}}^{l} s_{{\alpha ij}}^{l} ) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } ) \hfill \\ \Rightarrow \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(((} \lambda _{{ik}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} (s_{{2\tau }} - s_{{\alpha ki}}^{{l^{\prime} }} )) - \lambda _{{ik}}^{l} p_{{\alpha ik}}^{l} s_{{\alpha ik}}^{l} )} \oplus s_{\tau } = \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(((\lambda _{{ij}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} (s_{{2\tau }} - s_{{\alpha ji}}^{{l^{\prime} }} } } )) - \lambda _{{ij}}^{l} p_{{\alpha ij}}^{l} s_{{\alpha ij}}^{l} ) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } ) \hfill \\ \Rightarrow \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(((} \lambda _{{ik}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{2\tau }} ) - ((\lambda _{{ik}}^{l} - 1)\sum\nolimits_{{l^{\prime} }}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{\alpha ki}}^{{l^{\prime} }} )} - \lambda _{{ik}}^{l} p_{{\alpha ik}}^{l} s_{{\alpha ik}}^{l} )} \oplus s_{\tau } = \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(((\lambda _{{ij}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{2\tau }} } } ) - ((\lambda _{{ij}}^{l} - 1)\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{\alpha ji}}^{{l^{\prime} }} )} - \lambda _{{ij}}^{l} p_{{\alpha ij}}^{l} s_{{\alpha ij}}^{l} ) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } ) \hfill \\ \Rightarrow \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(((1 - \lambda _{{ik}}^{l} )\sum\nolimits_{{l^{\prime} }}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{\alpha ki}}^{{l^{\prime} }} ) \oplus } (} \lambda _{{ik}}^{l} ((\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{2\tau }} ) - p_{{\alpha ik}}^{l} s_{{\alpha ik}}^{l} ))} - \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{2\tau }} )} \oplus s_{\tau } = \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(((1 - \lambda _{{ij}}^{l} )\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{\alpha ji}}^{{l^{\prime} }} ) \oplus } \lambda _{{ij}}^{l} ((\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{2\tau }} ) - p_{{\alpha ij}}^{l} s_{{\alpha ij}}^{l} } } )) - \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{2\tau }} } ) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } ) \hfill \\ \Rightarrow \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(((1 - \lambda _{{ik}}^{l} )\sum\nolimits_{{l^{\prime} }}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{\alpha ki}}^{{l^{\prime} }} ) \oplus } (} \lambda _{{ik}}^{l} ((\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{2\tau }} ) - p_{{\alpha ik}}^{l} s_{{\alpha ik}}^{l} ))} - s_{{2\tau }} \oplus s_{\tau } = \sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(1 - \lambda _{{ij}}^{l} )\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{\alpha ji}}^{{l^{\prime} }} \oplus } \lambda _{{ij}}^{l} ((\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{2\tau }} ) - p_{{\alpha ij}}^{l} s_{{\alpha ij}}^{l} } } )) - s_{{2\tau }} \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } ) \hfill \\ \Rightarrow (\sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(1 - \lambda _{{ik}}^{l} )\sum\nolimits_{{l^{\prime} }}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{\alpha ki}}^{{l^{\prime} }} ) \oplus } \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {\lambda _{{ik}}^{l} p_{{\alpha ik}}^{l} (s_{{2\tau }} - s_{{\alpha ik}}^{l} )) \oplus s_{\tau } = } } (\sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(1 - \lambda _{{ij}}^{l} )\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{\alpha ji}}^{{l^{\prime} }} \oplus } } \sum\nolimits_{{l^{\prime} }}^{{\# g_{{sij}} }} {\lambda _{{ij}}^{l} p_{{\alpha ij}}^{l} (s_{{2\tau }} - s_{{\alpha ij}}^{l} )) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } } ) \hfill \\ \Rightarrow (\sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {(1 - \lambda _{{ik}}^{l} )\sum\nolimits_{{l^{\prime} }}^{{\# g_{{sik}} }} {\xi _{{ki}}^{{l^{\prime} }} p_{{\alpha ik}}^{l} s_{{\alpha ki}}^{{l^{\prime} }} ) \oplus } \sum\nolimits_{{l = 1}}^{{\# h_{{sik}} }} {\lambda _{{ik}}^{l} p_{{\alpha ik}}^{l} s_{{2\tau }} s_{{\alpha ik}}^{l} ) \oplus s_{\tau } = } } (\sum\nolimits_{{l = 1}}^{{\# h_{{sij}} }} {(1 - \lambda _{{ij}}^{l} )\sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{sij}} }} {\xi _{{ji}}^{{l^{\prime} }} p_{{\alpha ij}}^{l} s_{{\alpha ji}}^{{l^{\prime} }} \oplus } } \sum\nolimits_{{l^{\prime} }}^{{\# g_{{sij}} }} {\lambda _{{ij}}^{l} p_{{\alpha ij}}^{l} s_{{2\tau }} s_{{\alpha ij}}^{l} ) \oplus (\sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {\lambda _{{kj}}^{l} } p_{{\alpha kj}}^{l} s_{{\alpha kj}}^{l} \oplus \sum\nolimits_{{l = 1}}^{{\# h_{{skj}} }} {(1 - \lambda _{{kj}}^{l} )} \sum\nolimits_{{l^{\prime} = 1}}^{{\# g_{{skj}} }} {\xi _{{jk}}^{{l^{\prime} }} p_{{\alpha kj}}^{l} s_{{2\tau }} s_{{\alpha jk}}^{{l^{\prime} }} } } ) \hfill \\ \end{aligned}$ Thus, we have

$\left\{ \begin{aligned} \sum\limits_{l = 1}^{{\# h_{sik} }} {(1 - \lambda_{ik}^{l} )\sum\limits_{{l^{\prime} }}^{{\# g_{sik} }} {\xi_{ki}^{{l^{\prime} }} p_{\alpha ik}^{l} s_{\alpha ki}^{{l^{\prime} }} \oplus } \sum\limits_{l = 1}^{{\# h_{sik} }} {\lambda_{ik}^{l} p_{\alpha ik}^{l} s_{2\tau } s_{\alpha ik}^{l} \oplus \tau } } = (\sum\limits_{l = 1}^{{\# h_{sij} }} {(1 - \lambda_{ij}^{l} )\sum\limits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {\xi_{ji}^{{l^{\prime} }} p_{\alpha ij}^{l} s_{\alpha ji}^{{l^{\prime} }} \oplus } } \sum\limits_{{l^{\prime} }}^{{\# g_{sij} }} {\lambda_{ij}^{l} p_{\alpha ij}^{l} s_{2\tau } s_{\alpha ij}^{l} ) \oplus (\sum\limits_{l = 1}^{{\# h_{skj} }} {\lambda_{kj}^{l} } p_{\alpha kj}^{l} s_{\alpha kj}^{l} \oplus \sum\limits_{l = 1}^{{\# h_{skj} }} {(1 - \lambda_{kj}^{l} )} \sum\limits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {\xi_{jk}^{{l^{\prime} }} p_{\alpha kj}^{l} s_{2\tau } s_{\alpha jk}^{{l^{\prime} }} } )} \hfill \\ \sum\limits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {(1 - \lambda_{ik}^{{l^{\prime} }} )} \sum\limits_{l = 1}^{{\# h_{sik} }} {\xi_{ki}^{l} p_{\beta ik}^{{l^{\prime} }} s_{\beta ki}^{l} \oplus } \sum\limits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {\lambda_{ik}^{{l^{\prime} }} } p_{\beta ik}^{{l^{\prime} }} s_{2\tau } s_{\beta ik}^{l} \oplus \tau = (\sum\limits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {(1 - \lambda_{ij}^{{l^{\prime} }} )} \sum\limits_{l = 1}^{{\# h_{sij} }} {\xi_{ji}^{l} p_{\beta ij}^{{l^{\prime} }} s_{2\tau } s_{\beta ji}^{l} } \oplus \sum\limits_{{l^{\prime} { = 1}}}^{{\# g_{sij} }} {\lambda_{ij}^{{l^{\prime} }} p_{\beta ij}^{{l^{\prime} }} s_{2\tau } s_{\beta ij}^{{l^{\prime} }} ) \oplus (\sum\limits_{{l^{\prime} { = 1}}}^{{\# g_{skj} }} {\lambda_{kj}^{{l^{\prime} }} p_{\beta kj}^{{l^{\prime} }} s_{\beta kj}^{{l^{\prime} }} } \oplus \sum\limits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {(1 - \lambda_{kj}^{{l^{\prime} }} )} \sum\limits_{l = 1}^{{\# h_{skj} }} {\xi_{jk}^{l} p_{\beta kj}^{{l^{\prime} }} s_{2\tau } s_{\beta jk}^{l} } )} \hfill \\ \end{aligned} \right.$, Namely, $\left\{ \begin{aligned} \sum\limits_{l = 1}^{{\# h_{sik} }} {\lambda_{ki}^{l} \sum\limits_{{l^{\prime} }}^{{\# g_{sik} }} {\xi_{ki}^{{l^{\prime} }} p_{\alpha ik}^{l} s_{\alpha ki}^{{l^{\prime} }} \oplus } \sum\limits_{l = 1}^{{\# h_{sik} }} {(1 - \lambda_{ki}^{l} )p_{\alpha ik}^{l} s_{2\tau } s_{\alpha ik}^{l} \oplus \tau } } = (\sum\limits_{l = 1}^{{\# h_{sij} }} {\lambda_{ji}^{l} \sum\limits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {\xi_{ji}^{{l^{\prime} }} p_{\alpha ij}^{l} s_{\alpha ji}^{{l^{\prime} }} \oplus } } \sum\limits_{{l^{\prime} }}^{{\# g_{sij} }} {(1 - \lambda_{ji}^{l} )p_{\alpha ij}^{l} s_{2\tau } s_{\alpha ij}^{l} ) \oplus (\sum\limits_{l = 1}^{{\# h_{skj} }} {\lambda_{kj}^{l} } p_{\alpha kj}^{l} s_{\alpha kj}^{l} \oplus \sum\limits_{l = 1}^{{\# h_{skj} }} {(1 - \lambda_{kj}^{l} )} \sum\limits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {\xi_{jk}^{{l^{\prime} }} p_{\alpha kj}^{l} s_{2\tau } s_{\alpha jk}^{{l^{\prime} }} } )} \hfill \\ \sum\limits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {\lambda_{ki}^{{l^{\prime} }} } \sum\limits_{l = 1}^{{\# h_{sik} }} {\xi_{ki}^{l} p_{\beta ik}^{{l^{\prime} }} s_{\beta ki}^{l} \oplus } \sum\limits_{{l^{\prime} = 1}}^{{\# g_{sik} }} {(1 - \lambda_{ki}^{{l^{\prime} }} } )p_{\beta ik}^{{l^{\prime} }} s_{2\tau } s_{\beta ik}^{l} \oplus \tau = (\sum\limits_{{l^{\prime} = 1}}^{{\# g_{sij} }} {\lambda_{ji}^{{l^{\prime} }} } \sum\limits_{l = 1}^{{\# h_{sij} }} {\xi_{ji}^{l} p_{\beta ij}^{{l^{\prime} }} s_{2\tau } s_{\beta ji}^{l} } \oplus \sum\limits_{{l^{\prime} { = 1}}}^{{\# g_{sij} }} {(1 - \lambda_{ji}^{l} )p_{\beta ij}^{{l^{\prime} }} s_{2\tau } s_{\beta ij}^{{l^{\prime} }} ) \oplus (\sum\limits_{{l^{\prime} { = 1}}}^{{\# g_{skj} }} {\lambda_{kj}^{{l^{\prime} }} p_{\beta kj}^{{l^{\prime} }} s_{\beta kj}^{{l^{\prime} }} } \oplus \sum\limits_{{l^{\prime} = 1}}^{{\# g_{skj} }} {(1 - \lambda_{kj}^{{l^{\prime} }} )} \sum\limits_{l = 1}^{{\# h_{skj} }} {\xi_{jk}^{l} p_{\beta kj}^{{l^{\prime} }} s_{2\tau } s_{\beta jk}^{l} } )} \hfill \\ \end{aligned} \right.$ In which,$\lambda_{ki}^{l} ,\lambda_{ki}^{{l^{\prime} }} ,\lambda_{ji}^{l} ,\lambda_{ji}^{{l^{\prime} }}$ be the 0–1 indicator variable with $\lambda_{ij}^{l} + \lambda_{ji}^{l} = 1,\lambda_{ik}^{{l^{\prime} }} + \lambda_{ki}^{{l^{\prime} }} = 1$.

Then, we can further obtain

$E(d_{ski} ) \oplus (\tau ,\tau ) = E(d_{skj} ) \oplus E(d_{s\delta ji} )$.

Thus, we have $E(d_{s\delta (i)\delta (j)} ) \oplus (s_{\tau } ,s_{\tau } ) = E(d_{s\delta (i)\delta (k)} ) \oplus E(d_{s\delta (k)\delta (j)} )$ held.

(2) The necessity.

Similar to the above procedure, the necessity can be easily derived, the detailed proof process is omitted here.

Appendix B

Notation List

Notation List
IFPRs	Intuitionistic fuzzy preference relations
GDM	Group decision making
LDHFPRs	Linguistic dual hesitant fuzzy preference relations
DM	Decision makers
FPRs	Fuzzy preference relations
MPRs	Multiplicative preference relations
LPRs	Linguistic preference relations
AHP	Analytic hierarchy processes
LIFVs LIFPRs LVIFPRs MLIFPRs PLDHFPRs RCLIFPRs RCLIFV IFSs LDHFS LDHFV ELDHFV PLDHFV RCLDHFV PCLDHFPRs CRP	Linguistic intuitionistic fuzzy variables Linguistic intuitionistic fuzzy preference relations Linguistic-valued intuitionistic fuzzy preference relations Multiplicative linguistic intuitionistic fuzzy preference relations Probability linguistic dual hesitant fuzzy preference relations Reverse complementary linguistic intuitionistic fuzzy preference relations Reverse complementary linguistic intuitionistic fuzzy variables Intuitionistic fuzzy sets Linguistic dual hesitant fuzzy set Linguistic dual hesitant fuzzy value Equivalent linguistic dual hesitant fuzzy value Possibility linguistic dual hesitant fuzzy value Reverse complementary linguistic dual hesitant fuzzy value Reverse complementary linguistic dual hesitant fuzzy preference relations Consensus reaching process