Hesitant Fuzzy 2-Dimension Linguistic Term Set and its Application to Multiple Attribute Group Decision Making

Abstract

Hesitant fuzzy linguistic term set (HFLTS) is a powerful tool for solving the situations in which a decision maker hesitates among several consecutive linguistic terms in providing his or her preference to an alternative. To reflect the different importance degrees or weights of all possible linguistic terms in a HFLTS, an extension of HFLTS called probabilistic linguistic term set (PLTS) is proposed through adding probabilities. Note that for a PLTS, the importance information of all possible linguistic terms is described by crisp numbers. However, in practical applications, especially under uncertain environment, it may be difficult for decision makers to provide the importance information by crisp numbers. To accurately preserve the complete evaluation information provided by decision makers, motivated by the idea of 2-dimension linguistic variables, this paper proposes the concept of hesitant fuzzy 2-dimension linguistic term set, which includes not only possible linguistic terms expressing the evaluation value to an object, but also the importance degree of each linguistic term denoted by a linguistic term. Firstly, the operations and comparison laws between hesitant fuzzy 2-dimension linguistic elements (HF2DLEs) are defined. Then, some generalized aggregation operators are proposed for aggregating HF2DLEs, such as generalized hesitant fuzzy 2-dimension linguistic weighted average (G-HF2DLWA) operator, generalized hesitant fuzzy 2-dimension linguistic ordered weighted average (G-HF2DLOWA) operator and generalized hesitant fuzzy 2-dimension linguistic hybrid weighted average (G-HF2DLHWA) operator. Furthermore, some desirable properties and special cases of these operators are discussed. Based on the G-HF2DLOWA and G-HF2DLWA operators, an approach to multiple attribute group decision making is developed under hesitant fuzzy 2-dimension linguistic environment. Finally, a numerical example is given to verify the practicality and effectiveness of the proposed method.

Appendix

1.1 The Proof of Theorem 1

Proof (1) For n = 1, Eq. (17) is hold obviously.

(2) For n = 2, since

$$\hat{h}_{{S_{1} }}^{\lambda } = \bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} }} {\left\{ {\left( {f^{ - 1} (f(\dot{s}_{{a_{1} }} )^{\lambda } ),\ddot{s}_{{b_{1} }} } \right)} \right\}},$$

then

$$w_{1} \hat{h}_{{S_{1} }}^{\lambda } = \bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} }} {\left\{ {\left( {f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{1} }} )^{\lambda } )^{{w_{1} }} ),\ddot{s}_{{b_{1} }} } \right)} \right\}}.$$

Similarly,

$$w_{2} \hat{h}_{{S_{2} }}^{\lambda } = \bigcup\nolimits_{{(\dot{s}_{{a_{2} }} ,\ddot{s}_{{b_{2} }} ) \in \hat{h}_{{S_{2} }} }} {\left\{ {\left( {f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{2} }} )^{\lambda } )^{{w_{2} }} ),\ddot{s}_{{b_{2} }} } \right)} \right\}}.$$

Then,

$$\begin{aligned} w_{1} \hat{h}_{{S_{1} }}^{\lambda } + w_{2} \hat{h}_{{S_{2} }}^{\lambda } \\ & = \bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} ,(\dot{s}_{{a_{2} }} ,\ddot{s}_{{b_{2} }} ) \in \hat{h}_{{S_{2} }} }} {\left\{ {\left( {f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{1} }} )^{\lambda } )^{{w_{1} }} ) + f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{2} }} )^{\lambda } )^{{w_{2} }} )} \right.} \quad \right.- f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{1} }} )^{\lambda } )^{{w_{1} }} )} \\ & \left. {\qquad \left. {f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{2} }} )^{\lambda } )^{{w_{2} }} ),\;\;\hbox{min} (\ddot{s}_{{b_{1} }} ,\ddot{s}_{{b_{2} }} )} \right)} \right\} \\ {\kern 1pt} & = \bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} ,(\dot{s}_{{a_{2} }} ,\ddot{s}_{{b_{2} }} ) \in \hat{h}_{{S_{2} }} }} {\left\{ {\left( {f^{ - 1} (1 - (1 - f(\dot{s}_{{a_{1} }} )^{\lambda } )^{{w_{1} }} (1 - f(\dot{s}_{{a_{2} }} )^{\lambda } )^{{w_{2} }} ),\;\;\hbox{min} (\ddot{s}_{{b_{1} }} ,\ddot{s}_{{b_{2} }} )} \right)} \right\}} \\ & = \bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} ,(\dot{s}_{{a_{2} }} ,\ddot{s}_{{b_{2} }} ) \in \hat{h}_{{S_{2} }} }} {\left\{ {\left( {f^{ - 1} \left( {1 - \prod\limits_{i = 1}^{2} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right),\;\;\hbox{min} (\ddot{s}_{{b_{1} }} ,\ddot{s}_{{b_{2} }} )} \right)} \right\}} . \\ \end{aligned}$$

Based on Eq. (13), we obtain

$$\begin{aligned} {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} ) & = (w_{1} \hat{h}_{{S_{1} }}^{\lambda } + w_{2} \hat{h}_{{S_{2} }}^{\lambda } )^{1/\lambda } \\ & = \left\{ {\bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} ,(\dot{s}_{{a_{2} }} ,\ddot{s}_{{b_{2} }} ) \in \hat{h}_{{S_{2} }} }} {\left\{ {\left( {f^{ - 1} \left( {1 - \prod\limits_{i = 1}^{2} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right),\hbox{min} (\ddot{s}_{{b_{1} }} ,\ddot{s}_{{b_{2} }} )} \right)} \right\}} } \right\}^{1/\lambda } \\ & = \bigcup\nolimits_{{(\dot{s}_{{a_{1} }} ,\ddot{s}_{{b_{1} }} ) \in \hat{h}_{{S_{1} }} ,(\dot{s}_{{a_{2} }} ,\ddot{s}_{{b_{2} }} ) \in \hat{h}_{{S_{2} }} }} {\left\{ {\left( {f^{ - 1} \left( {\left( {1 - \prod\limits_{i = 1}^{2} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } } \right),\hbox{min} (\ddot{s}_{{b_{1} }} ,\ddot{s}_{{b_{2} }} )} \right)} \right\}} . \\ \end{aligned}$$

Therefore, when n = 2, Eq. (17) is hold.

(3) If Eq. (17) holds for n = k, that is

$${\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{k} }} ) = \bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{ - 1} \left( {\left( {1 - \prod\limits_{i = 1}^{k} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } } \right),\mathop {\hbox{min} }\limits_{1 \le i \le k} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} ,$$

i.e.,

$$\sum\limits_{i = 1}^{k} {w_{i} \hat{h}_{{S_{i} }}^{\lambda } } = \bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{ - 1} \left( {1 - \prod\limits_{i = 1}^{k} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right),\mathop {\hbox{min} }\limits_{1 \le i \le k} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} .$$

Then, when n = k+1, by Eqs. (10), (12) and (13), we have

$$\begin{aligned} {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{k + 1} }} ) = \left[ {\left( {\sum\limits_{i = 1}^{k} {w_{i} \hat{h}_{{S_{i} }}^{\lambda } } } \right) + w_{k + 1} \hat{h}_{{S_{k + 1} }}^{\lambda } } \right]^{1/\lambda } \hfill \\ = \left[ {\bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{ - 1} \left( {\left( {1 - \prod\limits_{i = 1}^{k} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right) + (1 - (1 - f(\dot{s}_{{a_{k + 1} }} )^{\lambda } )^{{w_{k + 1} }} )} \right. - } \right.} \right.} \left( {1 - \prod\limits_{i = 1}^{k} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)} \right. \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \quad \left. {\left. {\left. {(1 - (1 - f(\dot{s}_{{a_{k + 1} }} )^{\lambda } )^{{w_{k + 1} }} )),\mathop {\hbox{min} }\limits_{1 \le i \le k + 1} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} \right]^{1/\lambda } \hfill \\ = \bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{ - 1} \left( {\left( {1 - \left( {\prod\limits_{i = 1}^{k} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)(1 - f(\dot{s}_{{a_{k + 1} }} )^{\lambda } )^{{w_{k + 1} }} } \right)^{1/\lambda } } \right),\mathop {\hbox{min} }\limits_{1 \le i \le k + 1} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} \hfill \\ = \bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{ - 1} \left( {\left( {1 - \prod\limits_{i = 1}^{k + 1} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } } \right),\mathop {\hbox{min} }\limits_{1 \le i \le k + 1} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} . \hfill \\ \end{aligned}$$

i.e., Equation (17) holds for n = k+1. Thus, we can obtain that Eq. (17) holds for all n, which completes the proof of Theorem 1.

1.2 The Proof of Case (2) of the G-HF2DLWA Operator

Proof $\mathop { \lim }\limits_{\lambda \to 0} {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ) = \bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{ - 1} \left( {\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } } \right),\mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} .$

First, let us consider $\mathop {\lim }\limits_{\lambda \to 0} f^{ - 1} \left( {\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } } \right)$. Based on the L’ Hopital’s rule, we can prove that

$$\begin{aligned} \mathop {\lim }\limits_{\lambda \to 0} f^{ - 1} \left( {\left( {1-\prod\limits_{i=1}^{n} {(1-f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } } \right) = f^{-1} \left( {\mathop {\lim }\limits_{\lambda \to 0} e^{{\frac{{\ln \left( {1-\prod\nolimits_ {i=1}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)}}{\lambda }}} }\ \ \right) \hfill \\ = f^{-1} \left( {e^{{ - \mathop {\lim }\limits_{\lambda \to 0}\ \left( {\prod\limits_{i = 1}^{n} {(1-f(\dot{s}_{{a_{i} }} )^{\lambda }\ )^{{w_{i} }} } } \right)\left( {\sum\limits_{k=1}^{n} {w_{k} \frac{ - 1}{{1 - f(\dot{s}_{{a_{k} }} )^{\lambda } }}f(\dot{s}_{{a_{k} }} )^{\lambda } \ln (f(\dot{s}_{{a_{k} }} ))} } \right)}} } \right) \hfill\\ = f^{-1} \left( {e^{{\mathop {\lim }\limits_{\lambda \to 0}\ \sum\limits_{k=1}^{n} {w_{k} \frac{{\prod\limits_{i = 1}^{n} {(1-f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } }}{{1-f(\dot{s}_{{a_{k} }} )^{\lambda } }}\ln (f(\dot{s}_{{a_{k} }} ))} }} } \right)= f^{-1} \left( {e^{{\sum\limits_{k=1}^{n} {w_{k} \left( {\prod\limits_{i = 1}^{n} {\left( {\mathop {\lim }\limits_{\lambda \to 0} \frac{{1 - f(\dot{s}_{{a_{i} }} )^{\lambda } }}{{1-f(\dot{s}_{{a_{k} }} )^{\lambda } }}}\ \right)^{{w_{i} }} } } \right)\ln (f(\dot{s}_{{a_{k} }} ))} }} } \right) \hfill \\ = f^{-1} \left( {e^{{\sum\limits_{k=1}^{n} {w_{k} \left( {\prod\limits_{i = 1}^{n} {\left( {\frac{{\ln (f(\dot{s}_{{a_{i} }} ))}}{{\ln f(\dot{s}_{{a_{k} }} )}}}\ \right)^{{w_{i} }} } } \right)\ln (f(\dot{s}_{{a_{k} }} ))} }} } \right)= f^{-1} \left( {e^{{\sum\limits_{k=1}^{n} {w_{k} \left( {\prod\limits_{i = 1}^{n} {\left( {\ln (f(\dot{s}_{{a_{i} }} ))} \right)^{{w_{i} }} } } \right)} }} } \right) \hfill \\ = f^{-1} \left( {e^{{\prod\limits_{i=1}^{n} {\left( {\ln (f(\dot{s}_{{a_{i} }} ))} \right)^{{w_{i} }} } }} } \right)= f^{-1} \left( {\prod\limits_{i=1}^{n} {f(\dot{s}_{{a_{i} }} )^{{w_{i} }} } } \right). \hfill \\ \end{aligned}$$

Therefore, the proof of Case (2) is completed.

1.3 The Proof of Theorem 2

Proof Since $\hat{h}_{{S_{i} }} = \{ (\dot{s}_{a} ,\ddot{s}_{b} )\}$ for all i (i = 1, 2,…,n), then

$$\begin{aligned} {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ) & = \bigcup\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{{ - 1}} \left( {\left( {1 - \prod\limits_{{i = 1}}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{{1/\lambda }} } \right),\mathop {\min }\limits_{{1 \le i \le n}} (\ddot{s}_{{b_{i} }} )} \right)} \right\}} \\ {\kern 1pt} & = \bigcup\nolimits_{{(\dot{s}_{a} ,\ddot{s}_{b} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{{ - 1}} \left( {\left( {1 - \prod\limits_{{i = 1}}^{n} {(1 - f(\dot{s}_{a} )^{\lambda } )^{{w_{i} }} } } \right)^{{1/\lambda }} } \right),\mathop {\min }\limits_{{1 \le i \le n}} (\ddot{s}_{b} )} \right)} \right\}} \\ {\kern 1pt} & = \bigcup\nolimits_{{(\dot{s}_{a} ,\ddot{s}_{b} ) \in \hat{h}_{{S_{i} }} }} {\left\{ {\left( {f^{{ - 1}} (f(\dot{s}_{a} )),\mathop {\min }\limits_{{1 \le i \le n}} (\ddot{s}_{b} )} \right)} \right\}} \\ {\kern 1pt} & = \{ (\dot{s}_{a} ,\ddot{s}_{b} )\} . \\ \end{aligned}$$

1.4 The Proof of Theorem 3

Proof Since $\dot{s}_{{a_{i} }} \le \dot{s}_{{c_{i} }}$ and $\ddot{s}_{{b_{i} }} \le \ddot{s}_{{d_{i} }}$, for all i (i = 1, 2,…,n), we have

$$\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } \le \left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{c_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } ,\quad \mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{b_{i} }} ) \le \mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{d_{i} }} ).$$

Then,

$$\begin{aligned} \frac{1}{{l_{1} }}\sum\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left( {\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } \times f(\mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{b_{i} }} ))} \right)} \hfill \\ \le \frac{1}{{l_{2} }}\sum\nolimits_{{(\dot{s}_{{c_{i} }} ,\ddot{s}_{{d_{i} }} ) \in \hat{h}_{{S_{i} }}^{'} }} {\left( {\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{c_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } \times f(\mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{d_{i} }} ))} \right)} , \hfill \\ \end{aligned}$$

i.e.,

$$E ( {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} )) \le E({\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} )),$$

where l ₁ and l ₂ are the numbers of 2DLVs in ${\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} )$ and ${\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} )$, respectively.

• 1 If $E ( {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} )) < E({\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ))$, according to the comparison laws of HF2DLEs, we can obtain ${\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ) < {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ).$

• 2 If $E ( {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} )){ = }E({\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ))$, since $\dot{s}_{{a_{i} }} \le \dot{s}_{{c_{i} }}$ and $\ddot{s}_{{b_{i} }} \le \ddot{s}_{{d_{i} }}$, for all i (i = 1, 2,…,n), then $\dot{s}_{{a_{i} }} { = }\dot{s}_{{c_{i} }}$ and $\ddot{s}_{{b_{i} }} { = }\ddot{s}_{{d_{i} }}$, i = 1, 2,…, n,

and thus,

$$\begin{aligned} \frac{1}{{l_{1} }}\sum\nolimits_{{(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} }} {\left( {\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{a_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } \times f(\mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{b_{i} }} )) - E ( {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ))} \right)^{ 2} } \hfill \\ { = }\frac{1}{{l_{2} }}\sum\nolimits_{{(\dot{s}_{{c_{i} }} ,\ddot{s}_{{d_{i} }} ) \in \hat{h}_{{S_{i} }}^{'} }} {\left( {\left( {1 - \prod\limits_{i = 1}^{n} {(1 - f(\dot{s}_{{c_{i} }} )^{\lambda } )^{{w_{i} }} } } \right)^{1/\lambda } \times f(\mathop {\hbox{min} }\limits_{1 \le i \le n} (\ddot{s}_{{d_{i} }} )){ - }E({\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ))} \right)^{ 2} } , \hfill \\ \end{aligned}$$

i.e.,

$$V ( {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} )){ = }V({\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} )).$$

Therefore, according to the comparison laws of HF2DLEs, we can obtain

$${\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ){\text{ = G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ).$$

To sum up, if $\dot{s}_{{a_{i} }} \le \dot{s}_{{c_{i} }}$ and $\ddot{s}_{{b_{i} }} \le \ddot{s}_{{d_{i} }}$, for all i (i = 1, 2,…,n), then

$${\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ) \le {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ),$$

which completes the proof of Theorem 3.

1.5 The Proof of Theorem 4

Proof Since $\dot{s}_{{a^{ - } }} = \mathop {\hbox{min} }\limits_{1 \le i \le n} \{ \dot{s}_{{a_{i} }} |(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} \}$, $\ddot{s}_{{b^{ - } }} = \mathop {\hbox{min} }\limits_{1 \le i \le n} \{ \ddot{s}_{{b_{i} }} |(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} \}$, $\dot{s}_{{a^{ + } }} = \mathop {\hbox{max} }\limits_{1 \le i \le n} \{ \dot{s}_{{a_{i} }} |(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} \}$ and $\ddot{s}_{{b^{ + } }} = \mathop {\hbox{max} }\limits_{1 \le i \le n} \{ \ddot{s}_{{b_{i} }} |(\dot{s}_{{a_{i} }} ,\ddot{s}_{{b_{i} }} ) \in \hat{h}_{{S_{i} }} \}$, we have $\dot{s}_{{a^{ - } }} \le \dot{s}_{{a_{i} }} \le \dot{s}_{{a^{ + } }}$, $\ddot{s}_{{b^{ - } }} \le \ddot{s}_{{b_{i} }} \le \ddot{s}_{{b^{ + } }}$, i = 1, 2,…,n.

Therefore, according to the monotonicity property of the G-HF2DLWA operator, we can obtain

$$\{ (\dot{s}_{{a^{ - } }} ,\ddot{s}_{{b^{ - } }} )\} \le {\text{G-HF2DLWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ) \le \{ (\dot{s}_{{a^{ + } }} ,\ddot{s}_{{b^{ + } }} )\} ,$$

which completes the proof of Theorem 4.

1.6 The Proof of Theorem 5

Proof Since ω = (ω ₁, ω ₂, …, ω _n )^T is a position weight vector, $(\hat{h}^{\prime}_{{S_{1} }} ,\hat{h}^{\prime}_{{S_{2} }} , \ldots ,\hat{h}^{\prime}_{{S_{n} }} )$ is any permutation of $(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} )$, we have $\hat{h}_{{S_{\sigma (i)} }}^{{}} = \hat{h}^{\prime}_{{S_{\sigma (i)} }}$, then

$$\omega_{i} \hat{h}_{{S_{\sigma (i)} }}^{\lambda } = \omega_{i} \left( {\hat{h}^{\prime}_{{S_{\sigma (i)} }} } \right)^{\lambda } ,i=1,2,...,n.$$

Thus,

$$\left( {\sum\limits_{i = 1}^{n} {\omega_{i} \hat{h}_{{S_{\sigma (i)} }}^{\lambda } } } \right)^{1/\lambda } = \left( {\sum\limits_{i = 1}^{n} {\omega_{i} (\hat{h}_{{S_{\sigma (i)} }}^{'} )^{\lambda } } } \right)^{1/\lambda } ,$$

i.e.,

$${\text{G-HF2DLOWA}}(\hat{h}_{{S_{1} }} ,\hat{h}_{{S_{2} }} , \ldots ,\hat{h}_{{S_{n} }} ) = {\text{G-HF2DLOWA}}(\hat{h}_{{S_{1} }}^{'} ,\hat{h}_{{S_{2} }}^{'} , \ldots ,\hat{h}_{{S_{n} }}^{'} ),$$

which completes the proof of Theorem 9.