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A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application to Multiattribute Group Decision Making

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Abstract

The purpose of this paper is to introduce some new Bonferroni mean operators under interval-valued 2-tuple linguistic environment. First, a class of new operational laws of interval-valued 2-tuple linguistic are proposed. Then, we put forward some new interval-valued 2-tuple linguistic Bonferroni mean (IV2TLBM) operators. Moreover, properties and special cases of new aggregation operators are investigated. The main characteristic of the IV2TLBM is that the interrelationship among the input arguments and the closed operations are taken into account. Finally, an approach to multiple attributes group decision making is presented, and a numerical example is given to illustrate the proposed method.

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Acknowledgments

The work was supported by National Natural Science Foundation of China (Nos. 71301001, 71371011, 11426033), Provincial Natural Science Research Project of Anhui Colleges (No.KJ2015A379), Higher School Specialized Research Fund for the Doctoral Program (No.20123401110001), Humanity and Social Science Youth Foundation of Ministry of Education (No. 13YJC630092), Anhui Provincial Philosophy and Social Science Planning Youth Foundation (No. AHSKQ2014D13), The Doctoral Scientific Research Foundation of Anhui University.

Author information

Authors and Affiliations

  1. School of Mathematical Science, Anhui University, Hefei, Anhui, China

    Xi Liu, Huayou Chen & Ligang Zhou

  2. Department of Foundation, Anhui Occupational College of City Management, Hefei, Anhui, China

    Xi Liu

  3. Signal and Image Processing Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, USA

    Ligang Zhou

  4. School of Economics, Anhui University, Hefei, Anhui, China

    Zhifu Tao

Authors
  1. Xi Liu
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  2. Zhifu Tao
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  3. Huayou Chen
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  4. Ligang Zhou
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Corresponding author

Correspondence to Huayou Chen.

Appendix

Appendix

The Proof of Theorem 3.1:

  1. (1)

    According to the Definition 3.1, we know that \(A \oplus B = \left[ \begin{aligned} \Delta \left\{ {\phi^{ - 1} \left[ {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))} \right]} \right\}, \hfill \\ \Delta \left\{ {\phi^{ - 1} \left[ {\phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))} \right]} \right\} \hfill \\ \end{aligned} \right]\). It is clear that \(\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right) \le \Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right),\,\,\Delta^{ - 1} \left( {s_{k} , \alpha_{k} } \right) \le \Delta^{ - 1} \left( {s_{l} , \alpha_{l} } \right),\) and \(\Delta^{ - 1} :S \times \left[ { - \frac{1}{2g},\frac{1}{2g}} \right) \to [0,1],\,\,\phi :\left[ {0, 1} \right] \to [0, + \infty )\) are strictly increasing function, such that

    $$\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} )) \le \phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))$$

    and

    $$\begin{aligned} \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{k} , \alpha_{k} } \right)} \right) \in [0, + \infty ), \hfill \\ \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{l} , \alpha_{l} } \right)} \right) \in [0, + \infty ). \hfill \\ \end{aligned}$$

    Noting that \(\phi^{ - 1} :[0, + \infty ) \to \left[ {0, 1} \right]\) is also a strictly increasing function, we have

    $$\phi^{ - 1} \left( {\phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{k} , \alpha_{k} } \right)} \right)} \right) \le \phi^{ - 1} \left( {\phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{l} , \alpha_{l} } \right)} \right)} \right)$$

    and

    $$\phi^{ - 1} \left( {\phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{k} , \alpha_{k} } \right)} \right)} \right), \phi^{ - 1} \left( {\phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{l} , \alpha_{l} } \right)} \right)} \right) \in \left[ {0, 1} \right].$$

    Thus \(\Delta (\phi^{ - 1} \left( {\phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{k} , \alpha_{k} } \right)} \right)} \right)) \le \Delta \left( {\phi^{ - 1} \left( {\phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right) + \phi \left( {\Delta^{ - 1} \left( {s_{l} , \alpha_{l} } \right)} \right)} \right)} \right)\) and

    $$\Delta \left( {\phi^{ - 1} \left[ {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))} \right]} \right),\Delta \left( {\phi^{ - 1} \left[ {\phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))} \right]} \right) \in S \times [ - \frac{1}{2g},\frac{1}{2g}).$$

    So, we can get

    $$\begin{aligned} &[(s_{i} ,\alpha_{i} ),(s_{j} ,\alpha_{j} )] \oplus [(s_{k} ,\alpha_{k} ),(s_{l} ,\alpha_{l} )] \hfill \\ &\quad= \left[ {\Delta \left( {\phi^{ - 1} \left[ {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))} \right]} \right),\Delta \left( {\phi^{ - 1} \left[ {\phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))} \right]} \right)} \right] \in \varOmega \hfill \\ \end{aligned}$$
  2. (2)

    The proof is similar to that (1), it is omitted here.

  3. (3)

    From Definition 3.1, we can get

    $$\lambda \odot A = \left[ {\Delta \left\{ {\phi^{ - 1} \left[ {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right]} \right\}, \Delta \left\{ {\phi^{ - 1} \left[ {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right]} \right\}} \right].$$

    Since \(\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right) \le \Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right),\phi :\left[ {0, 1} \right] \to [0, + \infty )\) is a strictly increasing function, then \(\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right) \le \lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right),{\text{ and}}\,\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right),\lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right) \in [0, + \infty )\) where λ ≥ 0. Besides, \(\phi^{ - 1} :[0, + \infty ) \to \left[ {0, 1} \right]\) is also a strictly increasing function, we have

    $$\phi^{ - 1} \left( {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right) \le \phi^{ - 1} \left( {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right),$$

    and \(\phi^{ - 1} \left( {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right), \phi^{ - 1} \left( {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right) \in \left[ {0, 1} \right].\) Obviously, \(\Delta (\phi^{ - 1} \left[ {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right]) \le \Delta \left( {\phi^{ - 1} \left[ {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right]} \right),\) and \(\Delta \left( {\phi^{ - 1} [\lambda \phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))]} \right),\Delta \left( {\phi^{ - 1} [\lambda \phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right) \in S \times \left[ { - \frac{1}{2g},\frac{1}{2g}} \right)\). Thus

    $$\lambda \odot A = \left[ {\Delta \left\{ {\phi^{ - 1} \left[ {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right]} \right\}, \Delta \left\{ {\phi^{ - 1} \left[ {\lambda \phi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right]} \right\}} \right] \in \varOmega .$$
  4. (4)

    For any interval-valued 2-tuple linguistic, we get \(\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right) \le \Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right).\). From Definition 3.1, we have

    $$A^{\lambda } = \left[ {\Delta \left( {\varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right]} \right),\Delta \left( {\varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right]} \right)} \right] .$$

In view of the function \(\varphi :\left[ {0, 1} \right] \to \left[ {0, + \infty } \right)\) is strictly decreasing function, we obtain

$$\lambda \varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) \ge \lambda \varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))$$

and \(\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right), \lambda \varphi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right) \in [0, + \infty )\)

Correspondingly, the inverse function \(\varphi^{ - 1} :\left[ {0, + \infty } \right) \to \left[ {0, 1} \right]\) is also strictly decreasing function, then

$$\varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right] \le \varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right],$$

and \(\varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right], \varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right] \in \left[ {0, 1} \right].\)

Thus, \(\Delta \left( {\varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} \right]} \right) \le \Delta \left( {\varphi^{ - 1} \left[ {\lambda \varphi \left( {\Delta^{ - 1} \left( {s_{j} , \alpha_{j} } \right)} \right)} \right]} \right)\) and\(\Delta \left( {\varphi^{ - 1} [\lambda \varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))]} \right),\Delta \left( {\varphi^{ - 1} [\lambda \varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right) \in S \times \left[ { - \frac{1}{2g},\frac{1}{2g}} \right)\).

Therefore, we have

$$A^{\lambda } = \left[ {\Delta \left\{ {\varphi^{ - 1} [\lambda \varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))]} \right\},\Delta \left\{ {\varphi^{ - 1} [\lambda \varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right\}} \right] \in \varOmega$$

Combining (1) with (4), we have that such operational laws are closed and the results of the operation are also interval-valued 2-tuple linguistic variables in Ω, which completes the proof. □

The Proof of Theorem 3.2:

(1) and (2) are easy to be verified, which is omitted;

(3)

$$\begin{aligned} &\lambda \odot (A \oplus B) \hfill \\ &\quad= \lambda \odot \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} [\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} [\phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} \left( {\lambda \phi \left[ {\Delta^{ - 1} \left( {\Delta \left[ {\phi^{ - 1} \left( {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))} \right)} \right]} \right)} \right]} \right)} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} \left( {\lambda \phi \left[ {\Delta^{ - 1} \left( {\Delta \left[ {\phi^{ - 1} \left( {\phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))} \right)} \right]} \right)} \right]} \right)} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} \left( {\lambda [\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]} \right)} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} \left( {\lambda [\phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]} \right)} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} [\lambda \phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \lambda \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} [\lambda \phi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \lambda \phi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= (\lambda \odot A) \oplus (\lambda \odot B) .\hfill \\ \end{aligned}$$

(4)

$$\begin{aligned} &(A \otimes B)^{\lambda } \hfill \\ &= \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} [\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} , \hfill \\ &\Delta \left\{ {\varphi^{ - 1} [\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} \left( {\lambda \varphi \left[ {\Delta^{ - 1} (\Delta (\varphi^{ - 1} [\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]))} \right]} \right)} \right\}, \hfill \\ &\Delta \left\{ {\varphi^{ - 1} \left( {\lambda \varphi \left[ {\Delta^{ - 1} (\Delta (\varphi^{ - 1} [\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]))} \right]} \right)} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} \left( {\lambda [\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]} \right)} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} , \hfill \\ &\Delta \left\{ {\varphi^{ - 1} \left( {\lambda [\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]} \right)} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad = \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} [\lambda \varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + \lambda \varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))]} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} , \hfill \\ &\Delta \left\{ {\varphi^{ - 1} [\lambda \varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} )) + \lambda \varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))]} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= A^{\lambda } \otimes B^{\lambda }. \hfill \\ \end{aligned}$$

Similarly, it is obtained that (5)–(8) hold, which completes the proof. □

The Proof of Theorem 3.3:

By using mathematical induction on n.

  1. (1)

    For n = 2, we have

    $$\begin{aligned} &[(s_{1} ,\alpha_{1} ),(s^{\prime}_{1} ,\alpha^{\prime}_{1} )] \oplus [(s_{2} ,\alpha_{2} ),(s^{\prime}_{2} ,\alpha^{\prime}_{2} )] \hfill \\ &\quad = \left[ {\Delta \left\{ {\phi^{ - 1} [\sum\limits_{i = 1}^{2} {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))} ]} \right\},\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{2} {\phi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} ))} } \right]} \right\}} \right] .\hfill \\ \end{aligned}$$

    When n = k − 1, k ∊ N+, (1) holds, that is

    $$\mathop \oplus \limits_{i = 1}^{k - 1} \left[ {\left( {s_{i} , \alpha_{i} } \right), (s_{i}^{\prime } , \alpha_{i}^{\prime } )} \right] = \left[ {\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k - 1} {\phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} } \right]} \right\},\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k - 1} {\phi \left( {\Delta^{ - 1} \left( {s_{i}^{\prime } , \alpha_{i}^{\prime } } \right)} \right)} } \right]} \right\}} \right],$$

    then

    $$\begin{aligned} &\mathop \oplus \limits_{i = 1}^{k} [(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )] \hfill \\ &\quad = \left[ {\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k - 1} {\phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} } \right]} \right\},\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k - 1} {\phi \left( {\Delta^{ - 1} \left( {s_{i}^{\prime } , \alpha_{i}^{\prime } } \right)} \right)} } \right]} \right\},} \right] \oplus \left[ {\left( {s_{k} , \alpha_{k} } \right), \left( {s_{k}^{\prime } , \alpha_{k}^{\prime } } \right)} \right] \hfill \\ &\quad = \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} \left[ {\phi \left( {\Delta^{ - 1} \left[ {\Delta \left( {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k - 1} {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))} } \right]} \right)} \right]} \right) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))} \right]} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} \left[ {\phi \left( {\Delta^{ - 1} \left[ {\Delta \left( {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k - 1} {\phi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime\prime}_{i} ))} } \right]} \right)} \right]} \right) + \phi (\Delta^{ - 1} (s^{\prime}_{k} ,\alpha^{\prime}_{k} ))} \right]} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad = \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} \left[ {\left( {\sum\limits_{i = 1}^{k - 1} {\phi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))} } \right) + \phi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))} \right]} \right\}{\kern 1pt} {\kern 1pt} , \hfill \\ &\Delta \left\{ {\phi^{ - 1} \left[ {\left( {\sum\limits_{i = 1}^{k - 1} {\phi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} ))} } \right) + \phi (\Delta^{ - 1} (s^{\prime}_{k} ,\alpha^{\prime}_{k} ))} \right]} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad = \left[ {\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k} {\phi \left( {\Delta^{ - 1} \left( {s_{i} , \alpha_{i} } \right)} \right)} } \right]} \right\},\Delta \left\{ {\phi^{ - 1} \left[ {\sum\limits_{i = 1}^{k} {\phi \left( {\Delta^{ - 1} \left( {s_{i}^{\prime } , \alpha_{i}^{\prime } } \right)} \right)} } \right]} \right\}} \right]. \hfill \\ \end{aligned}$$

    So (1) holds for n = k. Thus (1) holds for all n.

  2. (2)

    The proof is similar to (1), thus it is omitted. □

The Proof of Theorem 4.1:

Based on the Definition 3.1, we can get

$$\begin{aligned} &[(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )]^{p} \otimes [(s_{j} ,\alpha_{j} ),(s^{\prime}_{j} ,\alpha^{\prime}_{j} )]^{q} \hfill \\ &\quad= \left[ {\Delta \left\{ {\varphi^{ - 1} \left( {p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} ))} \right)} \right\},\left\{ {\varphi^{ - 1} \left( {p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} ))} \right)} \right\}} \right] \hfill \\ &\qquad{\kern 1pt} \otimes \left[ {\Delta \left\{ {\varphi^{ - 1} \left( {p\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))} \right)} \right\},\left\{ {\varphi^{ - 1} \left( {p\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))} \right)} \right\}} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} , \hfill \\ &\Delta \left\{ {\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))]} \right\} \hfill \\ \end{aligned} \right]. \hfill \\ \end{aligned}$$

It follows from Theorem 3.3 that

$$\begin{aligned} &\mathop \oplus \limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} ([(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )]^{p} \otimes [(s_{j} ,\alpha_{j} ),(s^{\prime}_{j} ,\alpha^{\prime}_{j} )]^{q} ) \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} \left( {\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \left( {\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))} \right]} \right)} } \right)} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} \left( {\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \left( {\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))} \right]} \right)} } \right)} \right\} \hfill \\ \end{aligned} \right] .\hfill \\ \end{aligned}$$

So, we can obtain that

$$\begin{aligned} &\left[ {\frac{1}{n(n - 1)} \odot \left( {\mathop \oplus \limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} ([(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )]^{p} \otimes [(s_{j} ,\alpha_{j} ),(s^{\prime}_{j} ,\alpha^{\prime}_{j} )]^{q} )} \right)} \right]^{{\frac{1}{p + q}}} \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\left[ {\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \left( {\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right)} } \right]} \right)} \right\}, \hfill \\ &\Delta \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\left[ {\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \left( {\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))]} \right)} } \right]} \right)} \right\} \hfill \\ \end{aligned} \right]^{{\frac{1}{p + q}}} \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left( {\Delta^{ - 1} \left[ {\Delta \left( {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\left( {\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \{ } \varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right)} \right)} \right)} \right]} \right)} \right)} \right\}, \hfill \\ &\Delta \left\{ {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left( {\Delta^{ - 1} \left[ {\Delta \left( {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\left( {\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \{ } \varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))]} \right)} \right)} \right)} \right]} \right)} \right)} \right\} \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left\{ {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left[ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \left( {\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right)} } \right)} \right]} \right)} \right\}, \hfill \\ &\Delta \left\{ {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left[ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi \left( {\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))]} \right)} } \right)} \right]} \right)} \right\} \hfill \\ \end{aligned} \right] .\hfill \\ \end{aligned}$$

Thus, the proof of Theorem 4.1 is completed. □

The Proof of Theorem 4.2:

  1. (1)

    By Theorem 4.1, it has

    $$\begin{aligned} &ATS - I 2TLBM^{p,q} (A_{ 1} , A_{ 2} , \ldots ,A_{n} ) \hfill \\ &= \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi} (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))]} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi} (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))]} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right]. \hfill \\ \end{aligned}$$

    Since \(A_{i} = \left[ {\left( {s_{i} , \alpha_{i} } \right), \left( {s_{i}^{\prime } , \alpha_{i}^{\prime } } \right)\left] { = } \right[\left( {s_{k} , \alpha_{k} } \right), \left( {s_{l} , \alpha_{l} } \right)} \right],\quad i = 1, 2, \ldots , n\), then

    $$\begin{aligned} &ATS - I2TLBM^{p,q} (A_{1} ,A_{2} , \ldots, A_{n} ) \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}n(n - 1)\phi (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} )) + q\varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))])} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}n(n - 1)\phi (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} )) + q\varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))])} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi (\varphi^{ - 1} [(p + q)\varphi (\Delta^{ - 1} (s_{k} ,\alpha_{k} ))])} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi (\varphi^{ - 1} [(p + q)\varphi (\Delta^{ - 1} (s_{l} ,\alpha_{l} ))])} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\ &\quad= \left[ {\left( {s_{k} , \alpha_{k} } \right), \left( {s_{l} , \alpha_{l} } \right)} \right]. \hfill \\ \end{aligned}$$
  2. (2)

    According to the Definition 2.5, we can get

    $$\begin{aligned} &S(A_{i} ) = \frac{{\Delta^{ - 1} (s_{i} ,\alpha_{i} ) + \Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )}}{2}; \hfill \\ &S(A^{ + } ) = \frac{{\Delta^{ - 1} (\mathop {\hbox{max} }\limits_{i} (s_{i} ,\alpha_{i} )) + \Delta^{ - 1} (\mathop {\hbox{max} }\limits_{i} (s^{\prime}_{i} ,\alpha^{\prime}_{i} ))}}{2} \hfill \\ &S(A^{ - } ) = \frac{{\Delta^{ - 1} (\mathop {\hbox{min} }\limits_{i} (s_{i} ,\alpha_{i} )) + \Delta^{ - 1} (\mathop {\hbox{min} }\limits_{i} (s^{\prime}_{i} ,\alpha^{\prime}_{i} ))}}{2}. \hfill \\ \end{aligned}$$

    Then, \(S\left( {A^{ - } } \right) \le S\left( {A_{i} } \right) \le S\left( {A^{ + } } \right)\) for all i. Since ATS-I2TLBM satisfies the idempotency, we have

    $$\begin{aligned} A^{ + } = ATS - I 2TLBM{^{p,q}} (A^{ + } , A^{ + } , \ldots , A^{ + } )\;{\text{and}} \hfill \\ A^{ - } = ATS - I 2TLBM{^{p,q}} (A^{ - } , A^{ - } , \ldots , A^{ - } ). \hfill \\ \end{aligned}$$

    Besides, φ is a strictly decreasing function and ϕ is a strictly increasing function, we obtain

    $$\begin{aligned} &A^{ - } = ATS - I2TLBM^{p,q} (A^{ - } ,A^{ - } , \ldots ,A^{ - } ) \hfill \\ &\quad= \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}n(n - 1)\phi (\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (\mathop {\hbox{min} }\limits_{i} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (\mathop {\hbox{min} }\limits_{i} (s_{i} ,\alpha_{i} )))} \right]} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}n(n - 1)\phi (\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (\mathop {\hbox{min} }\limits_{i} (s^{\prime}_{i} ,\alpha^{\prime})) + q\varphi (\Delta^{ - 1} (\mathop {\hbox{min} }\limits_{i} (s^{\prime}_{i} ,\alpha^{\prime})))} \right]} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\ &\quad \le \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi} (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))])} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} {\phi} (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))])} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\&\quad \le \left[ \begin{aligned}& \Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}n(n - 1)\phi (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (\mathop {\hbox{max} }\limits_{i} (s_{i} ,\alpha_{i} ))) + q\varphi (\Delta^{ - 1} (\mathop {\hbox{max} }\limits_{i} (s_{i} ,\alpha_{i} )))])} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}n(n - 1)\phi (\varphi^{ - 1} [p\varphi (\Delta^{ - 1} (\mathop {\hbox{max} }\limits_{i} (s^{\prime}_{i} ,\alpha^{\prime}_{i} ))) + q\varphi (\Delta^{ - 1} (\mathop {\hbox{max} }\limits_{i} (s^{\prime}_{i} ,\alpha^{\prime})))])} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\ &\quad= ATS - I2TLBM^{p,q} (A^{ + } , A^{ + } , \ldots , A^{ + } ) = A^{ + }. \hfill \\ \end{aligned}$$

    Thus, we have \(A^{ - } \le ATS - I 2TLBM^{p,q} (A_{ 1} , A_{ 2} , \ldots ,A_{n} ) \le A^{ + } .\)

  3. (3)

    According to the Definition 4.1, it has

    $$\begin{aligned} &ATS - I2TLBM^{p,q} (A_{1} ,A_{2} , \ldots ,A_{n} ) \hfill \\ &\quad = \left[ {\frac{1}{n(n - 1)} \odot \left( {\mathop \oplus \limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} ([(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )]^{p} \otimes [(s_{j} ,\alpha_{j} ),(s^{\prime}_{j} ,\alpha^{\prime}_{j} )]^{q} )} \right)} \right]^{{\frac{1}{p + q}}}. \hfill \\ \end{aligned}$$

    If \(A_{{i}}^{\prime } = \left[ {\left( {s_{{\sigma(i)}} , \alpha_{{\sigma(i)}} } \right), \left( {s_{{\sigma (i)}}^{\prime } , \alpha_{{\sigma (i)}}^{\prime } } \right)} \right]\) is any permutation of \(A_{i} = \left[ {\left( {s_{i} , \alpha_{i} } \right), \left( {s_{i}^{\prime } , \alpha_{i}^{\prime } } \right)} \right] \, (i = 1, 2, \ldots ,n)\), then for any two interval-valued linguistic 2-tuples Ai and Aj, we have \(k, l \in \{ 1, 2 ,\ldots ,n\} ,\) such that \(A_{i} = [(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )] = [(s_{\sigma (k)} ,\alpha_{\sigma (k)} ),(s^{\prime}_{\sigma (k)} ,\alpha^{\prime}_{\sigma (k)} )] = A^{\prime}_{k}\) and \(A_{j} = [(s_{j} ,\alpha_{j} ),(s^{\prime}_{j} ,\alpha^{\prime}_{j} )] = [(s_{\sigma (l)} ,\alpha_{\sigma (l)} ),(s^{\prime}_{\sigma (l)} ,\alpha^{\prime}_{\sigma (l)} )] = A^{\prime}_{l} ,\) then,

    $$\begin{aligned} &ATS - I2TLBM^{p,q} (A_{1} ,A_{2} , \ldots ,A_{n} ) \hfill \\ &\quad= \left[ {\frac{1}{n(n - 1)} \odot \left( {\mathop \oplus \limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} ([(s_{i} ,\alpha_{i} ),(s^{\prime}_{i} ,\alpha^{\prime}_{i} )]^{p} \otimes [(s_{j} ,\alpha_{j} ),(s^{\prime}_{j} ,\alpha^{\prime}_{j} )]^{q} )} \right)} \right]^{{\frac{1}{p + q}}} \hfill \\ &\quad = \left[ {\frac{1}{n(n - 1)} \odot \left( {\mathop \oplus \limits_{\begin{aligned} k \ne l \\ k,l = 1 \end{aligned} }^{n} ([(s_{\sigma (k)} ,\alpha_{\sigma (k)} ),(s^{\prime}_{\sigma (k)} ,\alpha^{\prime}_{\sigma (k)} )]^{p} \otimes [(s_{\sigma (l)} ,\alpha_{\sigma (l)} ),(s^{\prime}_{\sigma (l)} ,\alpha^{\prime}_{\sigma (l)} )]^{q} )} \right)} \right]^{{\frac{1}{p + q}}} \hfill \\ &\quad = ATS - I2TLBM^{p,q} (A_{1}^{\prime } , A_{2}^{\prime } , \ldots ,A_{n}^{\prime } ). \hfill \\ \end{aligned}$$
  4. (4)

    According to the concept of Archimedean t-norm and s-norm, we know φ is a strictly decreasing function and ϕ is a strictly increasing function. Applying Theorem 4.1, we obtain that

    $$\begin{aligned} &ATS - I2TLBM^{p,q} (A_{1} ,A_{2} , \ldots ,A_{n} ) \hfill \\ &\quad = \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} \phi \left( {\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (s_{i} ,\alpha_{i} )) + q\varphi (\Delta^{ - 1} (s_{j} ,\alpha_{j} ))} \right]} \right)} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} \phi \left( {\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (s^{\prime}_{i} ,\alpha^{\prime}_{i} )) + q\varphi (\Delta^{ - 1} (s^{\prime}_{j} ,\alpha^{\prime}_{j} ))} \right]} \right)} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\ &\quad \ge \left[ \begin{aligned} &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} \phi \left( {\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (s_{i}^{ * } ,\alpha_{i}^{ * } )) + q\varphi (\Delta^{ - 1} (s_{j}^{ * } ,\alpha_{j}^{ * } ))} \right]} \right)} \right)} \right\}} \right)} \right), \hfill \\ &\Delta \left( {\varphi^{ - 1} \left( {\frac{1}{p + q}\varphi \left\{ {\phi^{ - 1} \left( {\frac{1}{n(n - 1)}\sum\limits_{\begin{aligned} i \ne j \\ i,j = 1 \end{aligned} }^{n} \phi \left( {\varphi^{ - 1} \left[ {p\varphi (\Delta^{ - 1} (s_{i}^{ * \prime } ,\alpha_{i}^{ * \prime } )) + q\varphi (\Delta^{ - 1} (s_{j}^{ * \prime } ,\alpha_{j}^{ * \prime } ))} \right]} \right)} \right)} \right\}} \right)} \right) \hfill \\ \end{aligned} \right] \hfill \\ &\quad= ATS - I 2TLBM^{p,q} (A_{1}^{*} , A_{2}^{*} , \ldots ,A_{n}^{*} ). \hfill \\ \end{aligned}$$

    Thus, the proof of Theorem 4.2 has been finished. □

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Liu, X., Tao, Z., Chen, H. et al. A New Interval-valued 2-Tuple Linguistic Bonferroni Mean Operator and Its Application to Multiattribute Group Decision Making. Int. J. Fuzzy Syst. 19, 86–108 (2017). https://doi.org/10.1007/s40815-015-0130-4

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