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T-spherical fuzzy TODIM method for multi-criteria group decision-making problem with incomplete weight information

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Abstract

In this paper, we investigate the multi-criteria group decision-making (MCGDM) problems with incomplete weight information under T-spherical fuzzy environment. Firstly, motivated by the idea of the intuitionistic fuzzy interaction method, we propose some operation laws of T-spherical fuzzy numbers (T-SFNs), as well as some T-spherical fuzzy interaction aggregation operators, such as the T-spherical fuzzy weighted averaging interaction (T-SFWAI) operator, the T-spherical fuzzy weighted geometric interaction (T-SFWGI) operator, the T-spherical fuzzy ordered weighted interaction aggregation operators and the generalized T-spherical fuzzy interaction aggregation operators. Then, some desirable properties of the proposed T-SFWAI and T-SFWGI operators and some special cases of the generalized T-spherical fuzzy interaction aggregation operators are discussed in detail. Secondly, for the situations where the information about the weights of criteria is partly known or completely unknown, we establish two optimization models to determine the weights of criteria based on the maximizing deviation method and Lagrange function method, respectively. Thirdly, the traditional TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) approach is extended to solve MCGDM problem under the T-spherical fuzzy environment by defining the distance between T-SFNs, score function and accuracy function of T-spherical fuzzy number. Finally, a numerical example is given to illustrate the application of the extended TODIM approach, and further the sensitivity analysis and comparison analysis are carried out to demonstrate the influence of parameter on the final result and the effectiveness of the extended method.

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Acknowledgements

This research is supported by Natural Science Foundation of China (71873015), Program for New Century Excellent Talents in University of China (NCET-13-0037) and Humanities and Social Sciences Foundation of Ministry of Education of China (14YJA630019).

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Correspondence to Yanbing Ju.

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Appendices

Appendix A: Proof of Theorem 1

According to Definition 7, we can easily get (1)–(4) of Theorem 1, so the proofs of (1)–(4) are omitted here. In what follows, we just give the proofs of (5) and (6) of Theorem 1.

(5) According to the operation laws in Definition 7, we have

$$ {\kern 1pt} {\kern 1pt} {\kern 1pt} (a_{1} )^{{\lambda_{1} }} = \left\langle {((1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{\lambda_{1} }} - (1 - \nu_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{{\lambda_{1} }} )^{1/q} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ((1 - \nu_{1}^{q} )^{{\lambda_{1} }} - (1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{\lambda_{1} }} )^{1/q} ,{\kern 1pt} {\kern 1pt} (1 - (1 - \nu_{1}^{q} )^{{\lambda_{1} }} )^{1/q} } \right\rangle . $$
$$ {\kern 1pt} {\kern 1pt} {\kern 1pt} (a_{1} )^{{\lambda_{2} }} = \left\langle {((1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{\lambda_{2} }} - (1 - \nu_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{{\lambda_{2} }} )^{1/q} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ((1 - \nu_{1}^{q} )^{{\lambda_{2} }} - (1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{\lambda_{2} }} )^{1/q} ,{\kern 1pt} (1 - (1 - \nu_{1}^{q} )^{{\lambda_{2} }} )^{1/q} } \right\rangle . $$

and

$$ {\kern 1pt} {\kern 1pt} {\kern 1pt} (a_{1} )^{{(\lambda_{1} + \lambda_{2} )}} = \left\langle {\left( {(1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} - (1 - \nu_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} } \right)^{1/q} ,{\kern 1pt} {\kern 1pt} \left( {(1 - \nu_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} - (1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} } \right)^{1/q} ,{\kern 1pt} \left( {1 - (1 - \nu_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} } \right)^{1/q} } \right\rangle . $$

By Definition 7, we can further calculate the aggregation value of \(a_{1}^{{\lambda_{1} }}\) and \(a_{1}^{{\lambda_{2} }}\):

$$ a_{1}^{{\lambda_{1} }} \otimes a_{1}^{{\lambda_{2} }} = \left\langle {\left( {(1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} - (1 - \nu_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} } \right)^{1/q} ,{\kern 1pt} \left( {(1 - \nu_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} - (1 - \nu_{1}^{q} - \eta_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} } \right)^{1/q} ,{\kern 1pt} \left( {1 - (1 - \nu_{1}^{q} )^{{(\lambda_{1} + \lambda_{2} )}} } \right)^{1/q} } \right\rangle . $$

Therefore, the expression \(a_{1}^{{\lambda_{1} }} \otimes a_{1}^{{\lambda_{2} }} = {\kern 1pt} {\kern 1pt} (a_{1} )^{{(\lambda_{1} + \lambda_{2} )}}\) holds.

(6) According to the operation laws in Definition 7, we have

$$ {\kern 1pt} (a_{1} )^{\lambda } = \left\langle {\left( {(1 - \nu_{1}^{q} - \eta_{1}^{q} )^{\lambda } - (1 - \nu_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{\lambda } } \right)^{1/q} ,{\kern 1pt} \left( {(1 - \nu_{1}^{q} )^{\lambda } - (1 - \nu_{1}^{q} - \eta_{1}^{q} )^{\lambda } } \right)^{1/q} ,\left( {1 - (1 - \nu_{1}^{q} )^{\lambda } } \right)^{1/q} } \right\rangle , $$
$$ {\kern 1pt} (a_{2} )^{\lambda } = \left\langle {\left( {(1 - \nu_{2}^{q} - \eta_{2}^{q} )^{\lambda } - (1 - \nu_{2}^{q} - \eta_{2}^{q} - \mu_{2}^{q} )^{\lambda } } \right)^{1/q} ,\left( {(1 - \nu_{2}^{q} )^{\lambda } - (1 - \nu_{2}^{q} - \eta_{2}^{q} )^{\lambda } } \right)^{1/q} ,{\kern 1pt} \left( {1 - (1 - \nu_{2}^{q} )^{\lambda } } \right)^{1/q} } \right\rangle , $$

and

$$ \begin{aligned} a_{1} \otimes a_{2} & = \langle \left( {(1 - v_{1}^{q} - \eta_{1}^{q} )(1 - v_{2}^{q} - \eta_{2}^{q} ) - (1 - v_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )(1 - v_{2}^{q} - \eta_{2}^{q} - \mu_{2}^{q} )} \right)^{1/q} , \\ & \,\,\,\,\left( {(1 - v_{1}^{q} )(1 - v_{2}^{q} ) - (1 - v_{1}^{q} - \eta_{1}^{q} )(1 - v_{2}^{q} - \eta_{2}^{q} )} \right)^{1/q} , \\ & \,\,\,\,\left( {1 - (1 - v_{1}^{q} )(1 - v_{2}^{q} )} \right)^{1/q} \rangle . \\ \end{aligned} $$

Further, we have the following two expressions by Definition 7:

$$ \begin{gathered} (a_{1} \otimes a_{2} )^{\lambda } = < ((1 - v_{1}^{q} - \eta_{1}^{q} )^{\lambda } (1 - v_{2}^{q} - \eta_{2}^{q} )^{\lambda } - (1 - v_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{\lambda } (1 - v_{2}^{q} - \eta_{2}^{q} - \mu_{2}^{q} )^{\lambda } )^{1/q} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ((1 - v_{1}^{q} )^{\lambda } (1 - v_{2}^{q} )^{\lambda } - (1 - v_{1}^{q} - \eta_{1}^{q} )^{\lambda } (1 - v_{2}^{q} - \eta_{2}^{q} )^{\lambda } )^{1/q} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1 - (1 - v_{1}^{q} )^{\lambda } (1 - v_{2}^{q} )^{\lambda } )^{1/q} > . \hfill \\ \end{gathered} $$

and

$$ \begin{gathered} a_{1}^{\lambda } \otimes a_{2}^{\lambda } = < ((1 - v_{1}^{q} - \eta_{1}^{q} )^{\lambda } (1 - v_{2}^{q} - \eta_{2}^{q} )^{\lambda } - (1 - v_{1}^{q} - \eta_{1}^{q} - \mu_{1}^{q} )^{\lambda } (1 - v_{2}^{q} - \eta_{2}^{q} - \mu_{2}^{q} )^{\lambda } )^{1/q} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ((1 - v_{1}^{q} )^{\lambda } (1 - v_{2}^{q} )^{\lambda } - (1 - v_{1}^{q} - \eta_{1}^{q} )^{\lambda } (1 - v_{2}^{q} - \eta_{2}^{q} )^{\lambda } )^{1/q} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1 - (1 - v_{1}^{q} )^{\lambda } (1 - v_{2}^{q} )^{\lambda } )^{1/q} > . \hfill \\ \end{gathered} $$

therefore, the expression \((a_{1} \otimes a_{2} )^{\lambda } = a_{1}^{\lambda } \otimes a_{2}^{\lambda }\) holds.

Appendix B: Proof of Theorem 2

The proof of this theorem consists of the following two parts.

(1) Firstly, we prove that the aggregation value by the T-SFWAI operator is still a T-SFN.

Let \(\mu_{*} = (1 - \prod\nolimits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } )^{1/q}\), \(\eta_{*} = (\prod\nolimits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} - } \prod\nolimits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } )^{1/q}\) and \(v_{*} = (\prod\nolimits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } - \prod\nolimits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } )^{1/q}\).

Then, we can prove that Eq. (7) satisfies the following two conditions:

(a) \(0 \le \mu_{*}^{q} + \eta_{*}^{q} + v_{*}^{q} \le 1\) for \(q \ge 1\);

(b) \(0 \le \mu_{*}^{q} \le 1\), \(0 \le \eta_{*}^{q} \le 1\) and \(0 \le v_{*}^{q} \le 1\).

We prove that condition (a) holds first.

$$ \begin{aligned} & \mu_{*}^{q} + \eta_{*}^{q} + v_{*}^{q} = 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } + \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } + \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } \\ & \quad = 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } . \\ \end{aligned} $$

According to Definition 2, we have \(0 \le \mu_{j}^{q} + \eta_{j}^{q} + v_{j}^{q} \le 1\) for \(q \ge 1\); then, we can derive the following inequality:

$$ \begin{aligned} 0 & \le 1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} \le 1 \Rightarrow 0 \le (1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} \le 1 \Rightarrow 0 \le \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } \le 1 \Rightarrow \\ 0 & \le 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } \le 1.i.e.,0 \le \mu_{*}^{q} + \eta_{*}^{q} + v_{*}^{q} \le 1. \\ \end{aligned} $$

So we know that condition (a) holds.

According to Definition 2, we have

$$ \begin{gathered} \mu_{j}^{q} \in [0,1] \Rightarrow 1 - \mu_{j}^{q} \in [0,1] \Rightarrow (1 - \mu_{j}^{q} )^{{w_{j} }} \in [0,1] \Rightarrow \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} \in [0,1] \Rightarrow 1 - } \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} \in [0,1]} \hfill \\ \Rightarrow \left( {1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} \in [0,1],.i.e.,0 \le \mu_{*} \le 1. \hfill \\ \end{gathered} $$

Similarly, we can get \(0 \le \eta_{*} \le 1\) and \(0 \le v_{*} \le 1\). Then, we prove that condition (b) holds.

Therefore, the aggregation value by the T-SFWAI operator is still a T-SFN.

(2) Secondly, we prove that Eq. (7) holds by using mathematical induction method on positive integer n.

(a) When \(n = 1\), we have

$$ w_{1} a_{1} = \left\langle {(1 - (1 - \mu_{1}^{q} )^{{w_{1} }} )^{1/q} ,((1 - \mu_{1}^{q} )^{{w_{1} }} - (1 - \mu_{1}^{q} - \eta_{1}^{q} )^{{w_{1} }} )^{1/q} ,((1 - \mu_{1}^{q} - \eta_{1}^{q} )^{{w_{1} }} - (1 - \mu_{1}^{q} - \eta_{1}^{q} - v_{j}^{q} )^{{w_{1} }} )^{1/q} } \right\rangle $$

Thus, Eq. (7) holds for \(n = 1\).

(b) Suppose that Eq. (7) holds for \(n = k\), that is,

$$ \begin{aligned} {\text{T } - \text{ SFWAI}}(a_{1} ,a_{2} ,...,a_{k} ) & = \left\langle {\left( {1 - \prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} ,\left( {\prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} ,} \right. \\ {\kern 1pt} & \quad \left. {\left( {\prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} } \right\rangle . \\ \end{aligned} $$

Then, when \(n = k + 1\), by inductive assumption and Definition 7, we have

$$ \begin{aligned} & {\text{T } - \text{ SFWAI}}(a_{1} ,a_{2} ,...,a_{k} ,a_{k + 1} ) \\ & = {\text{T } - \text{ SFWAI}}(a_{1} ,a_{2} ,...,a_{k} ) \oplus (w_{k + 1} a_{k + 1} ) \\ & = \left\langle {\left( {1 - \prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} ,\left( {\prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} ,} \right. \\ & \quad \left. {{\kern 1pt} \left( {\prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} } \right\rangle \\ & \quad \oplus \left\langle {\left( {1 - (1 - \mu_{k + 1}^{q} )^{{w_{k + 1} }} } \right)^{1/q} ,\left( {(1 - \mu_{k + 1}^{q} )^{{w_{k + 1} }} - (1 - \mu_{k + 1}^{q} - \eta_{k + 1}^{q} )^{{w_{k + 1} }} } \right)^{1/q} ,} \right.\left. {\left( {(1 - \mu_{k + 1}^{q} - \eta_{k + 1}^{q} )^{{w_{k + 1} }} - (1 - \mu_{k + 1}^{q} - \eta_{k + 1}^{q} - v_{k + 1}^{q} )^{{w_{k + 1} }} } \right)^{1/q} } \right\rangle \\ & = \left\langle {\left( {1 - \prod\limits_{j = 1}^{k + 1} {(1 - \mu_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} ,\left( {\prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{k + 1} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} ,} \right. \\ &\left. { \quad \left( {\prod\limits_{j = 1}^{k} {(1 - \mu_{j}^{q} - \eta_{j}^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{k + 1} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } } \right)^{1/q} } \right\rangle . \\ \end{aligned} $$

Therefore, we can deduce that Eq. (7) holds for the positive integer \(n = k + 1\). According to the above two steps, we know that Eq. (7) holds for any positive integer, which proves the correctness of Theorem 2.

Appendix C: Proof of Theorem 3

If all \(a_{j} = \left\langle {\mu_{j} ,\eta_{j} ,v_{j} } \right\rangle\) (j = 1,2,…,n) are equal and \(a_{j} = < \mu ,\eta ,v > = a\) for all j = 1,2,…,n, then we have

$$ \begin{aligned} & {\text{T } - \text{ SFWAI}}(a_{1} ,a_{2} , \ldots ,a_{n} ) \\ & = \left\langle {\left( {1 - \prod\limits_{j = 1}^{n} {(1 - \mu^{q} )^{{w_{j} }} } } \right)^{1/q} ,} \right.\left( {\prod\limits_{j = 1}^{n} {(1 - \mu^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{n} {(1 - \mu^{q} - \eta^{q} )^{{w_{j} }} } } \right)^{1/q} , \\ &\left. {{\kern 1pt} \quad {\kern 1pt} {\kern 1pt} \left( {\prod\limits_{j = 1}^{n} {(1 - \mu^{q} - \eta^{q} )^{{w_{j} }} } - \prod\limits_{j = 1}^{n} {(1 - \mu^{q} - \eta^{q} - v^{q} )^{{w_{j} }} } } \right)^{1/q} } \right\rangle \\ & = \left\langle {\left( {1 - (1 - \mu^{q} )^{{\sum\limits_{j = 1}^{n} {w_{j} } }} } \right)^{1/q} ,} \right.\left( {(1 - \mu^{q} )^{{\sum\limits_{j = 1}^{n} {w_{j} } }} - (1 - \mu^{q} - \eta^{q} )^{{\sum\limits_{j = 1}^{n} {w_{j} } }} } \right)^{1/q} , \\ &\left. { \quad {\kern 1pt} \left( {(1 - \mu^{q} - \eta^{q} )^{{\sum\limits_{j = 1}^{n} {w_{j} } }} - (1 - \mu^{q} - \eta^{q} - v^{q} )^{{\sum\limits_{j = 1}^{n} {w_{j} } }} } \right)^{1/q} } \right\rangle \\ & = \left\langle {\left( {1 - (1 - \mu^{q} )} \right)^{1/q} ,\left( {(1 - \mu^{q} ) - (1 - \mu^{q} - \eta^{q} )} \right)^{1/q} ,{\kern 1pt} \left( {(1 - \mu^{q} - \eta^{q} ) - (1 - \mu^{q} - \eta^{q} - v^{q} )} \right)^{1/q} } \right\rangle \\ & \quad = \left\langle {\mu ,\eta ,v} \right\rangle . \\ \end{aligned} $$

Appendix D: Proof of theorem 4

Denote the aggregation results of \(a_{j}\) and \(\tilde{a}_{j}\) (j = 1,2,…,n) by \(a\) and \(\tilde{a}\), respectively, i.e., \(a = T - SFWAI(a_{1} ,a_{2} , \ldots ,a_{n} )\) and \(\tilde{a} = T - SFWAI(\tilde{a}_{1} ,\tilde{a}_{2} , \ldots ,\tilde{a}_{n} )\). Then, according to the definition of the score function in Definition 3, we have

$$ S(a) = \frac{1}{2}\left( {2 - 2\prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } + \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } } \right) $$

and

$$ S(\tilde{a}) = \frac{1}{2}\left( {2 - 2\prod\limits_{j = 1}^{n} {(1 - \tilde{\mu }_{j}^{q} )^{{w_{j} }} } + \prod\limits_{j = 1}^{n} {(1 - \tilde{\mu }_{j}^{q} - \tilde{\eta }_{j}^{q} - \tilde{v}_{j}^{q} )^{{w_{j} }} } } \right) $$

Since \(\mu_{j} \ge \tilde{\mu }_{j}\), we have

$$ 1 - \mu_{j}^{q} \le 1 - \tilde{\mu }_{j}^{q} \Rightarrow (1 - \mu_{j}^{q} )^{{w_{j} }} \le (1 - \tilde{\mu }_{j}^{q} )^{{w_{j} }} \Rightarrow \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } \le \prod\limits_{j = 1}^{n} {(1 - \tilde{\mu }_{j}^{q} )^{{w_{j} }} } \Rightarrow 1 - \prod\limits_{j = 1}^{n} {(1 - \tilde{\mu }_{j}^{q} )^{{w_{j} }} } \le 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} )^{{w_{j} }} } . $$

Similarly, since \( \mu_{j} + \eta_{j} + v_{j} \le \tilde{\mu }_{j} + \tilde{\eta }_{j} + \tilde{v}_{j} \)

$$ \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{q} - \eta_{j}^{q} - v_{j}^{q} )^{{w_{j} }} } \ge \prod\limits_{j = 1}^{n} {(1 - \tilde{\mu }_{j}^{q} - \tilde{\eta }_{j}^{q} - \tilde{v}_{j}^{q} )^{{w_{j} }} } . $$

Therefore, we have \(S(a) \ge S(\tilde{a})\), which means that the monotonicity of the T-SFWAI operator holds.

Appendix E: Proof of theorem 8

Since \(\tilde{a}_{j} = \left\langle {\tilde{\mu }_{j} ,\tilde{\eta }_{j} ,\tilde{v}_{j} } \right\rangle\) (j = 1,2,…,n) is a permutation of \(a_{j} = \left\langle {\mu_{j} ,\eta_{j} ,v_{j} } \right\rangle\) (j = 1,2,…,n), we have \(\tilde{a}_{\delta (j)} = a_{\delta (j)}\) (j = 1,2,…,n). Then, according to Definition 10, we know that the expression \(T - SFOWAI(\tilde{a}_{1} ,\tilde{a}_{2} ,...,\tilde{a}_{n} ) = T - SFOWAI(a_{1} ,a_{2} ,...,a_{n} )\) holds.

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Ju, Y., Liang, Y., Luo, C. et al. T-spherical fuzzy TODIM method for multi-criteria group decision-making problem with incomplete weight information. Soft Comput 25, 2981–3001 (2021). https://doi.org/10.1007/s00500-020-05357-x

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