Multiple-attribute decision-making based on picture fuzzy Archimedean power Maclaurin symmetric mean operators

Abstract

In this paper, a novel multiple-attribute decision-making method based on a set of Archimedean power Maclaurin symmetric mean operators of picture fuzzy numbers is proposed. The Maclaurin symmetric mean operator, power average operator, and operational rules based on Archimedean T-norm and T-conorm are introduced into picture fuzzy environment to construct the aggregation operators. The formal definitions of the aggregation operators are presented. Their general and specific expressions are established. The properties and special cases of the aggregation operators are, respectively, explored and discussed. Using the presented aggregation operators, a method for solving the multiple-attribute decision-making problems based on picture fuzzy numbers is designed. The method is illustrated through example and experiments and validated by comparisons. The results of the comparisons show that the proposed method is feasible and effective that can provide the generality and flexibility in aggregation of values of attributes and consideration of interactions among attributes and the capability to lower the negative effect of biased attribute values on the result of aggregation.

Appendixes

1.1 A. Proof of Theorem 1

Proof

• 1 The following equations are successively derived from the operational rules in Eqs. (1), (2), (3), and (4)

  $$\begin{aligned}(n\omega_{{i_{h} }} )\alpha_{{i_{h} }} & = \left\langle {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right), \, \varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} )} \right), \, \varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\nu_{{i_{h} }} )} \right)} \right\rangle , \\ \mathop \otimes \limits_{h = 1}^{k} \left( {(n\omega_{{i_{h} }} )\alpha_{{i_{h} }} } \right) & = \left\langle {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right), \, \psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} )} \right)} \right)} } \right), \, \psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\nu_{{i_{h} }} )} \right)} \right)} } \right)} \right\rangle , \\ \mathop \oplus \limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} \mathop \otimes \limits_{h = 1}^{k} \left( {(n\omega_{{i_{h} }} )\alpha_{{i_{h} }} } \right) & = \left\langle {\psi^{ - 1} \left( {\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right),} \right. \\ & \quad \varphi^{ - 1} \left( {\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right), \\ & \left. { \quad \varphi^{ - 1} \left( {\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\nu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right\rangle , \\ \frac{1}{{C_{n}^{k} }}\mathop \oplus \limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} \mathop \otimes \limits_{h = 1}^{k} \left( {(n\omega_{{i_{h} }} )\alpha_{{i_{h} }} } \right) & = \left\langle {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right., \\ & \quad \varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right), \\ & \left. { \quad \varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\nu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right\rangle , \\ \left( {\frac{1}{{C_{n}^{k} }}\mathop \oplus \limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} \mathop \otimes \limits_{h = 1}^{k} \left( {(n\omega_{{i_{h} }} )\alpha_{{i_{h} }} } \right)} \right)^{1/k} & = \left\langle {\varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right),} \right. \\ & \quad \psi^{ - 1} \left( {\frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right), \\ & \quad \left. {\psi^{ - 1} \left( {\frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\varphi (\nu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right)} \right\rangle . \\ \end{aligned}$$

• 2 The proof of “PFAPMSM^(k)(α₁, α₂, ..., α_n) is a PFN” is equivalent to the proof of “0 ≤ μ ≤ 1, 0 ≤ η ≤ 1, 0 ≤ ν ≤ 1, and 0 ≤ μ + η + ν ≤ 1”. The following is the proof of “0 ≤ μ ≤ 1”:

  • (i) According to the conditions 0 ≤ μ_ih ≤ 1, φ(t) and φ⁻¹(t) are monotonically decreasing, and ψ(t) and ψ⁻¹(t) are monotonically increasing, the following inequalities are successively derived

    $$\begin{aligned} (n\omega_{{i_{h} }} )\psi (0) & \le (n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} ) \le (n\omega_{{i_{h} }} )\psi (1), \\ \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (0)} \right) & \le \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right) \le \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (1)} \right), \\ \sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (0)} \right)} \right)} & \ge \sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} \ge \sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (1)} \right)} \right)} , \\ \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (0)} \right)} \right)} } \right) & \le \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right) \le \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (1)} \right)} \right)} } \right), \\ \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (0)} \right)} \right)} } \right)} \right)} & \le \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} \\ & \quad \le\,\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (1)} \right)} \right)} } \right)} \right)} . \\ \end{aligned}$$

    Since

    $$\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\sum\limits_{h = 1}^{k} {(n\omega_{{i_{h} }} ) = n\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\left( {\omega_{{i_{1} }} + \omega_{{i_{2} }} + \ldots + \omega_{{i_{k} }} } \right)} } } = n\frac{1}{n} = 1.$$

    The following inequalities are successively obtained

    $$\begin{aligned} \psi \left( {\varphi^{ - 1} \left( {k\varphi (0)} \right)} \right) \le \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} \le \psi \left( {\varphi^{ - 1} \left( {k\varphi (1)} \right)} \right), \hfill \\ \varphi^{ - 1} \left( {k\varphi (0)} \right) \le \psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right) \le \varphi^{ - 1} \left( {k\varphi (1)} \right), \hfill \\ \varphi (0) \ge \frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right) \ge \varphi (1), \hfill \\ 0 \le \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) \le 1. \hfill \\ \end{aligned}$$

    That is 0 ≤ μ ≤ 1. “0 ≤ η ≤ 1” and “0 ≤ ν ≤ 1” can be proved in a similar way.

  • (ii) The following is the proof of “0 ≤ μ + η + ν ≤ 1”:

    Since 0 ≤ μ ≤ 1, 0 ≤ η ≤ 1, and 0 ≤ ν ≤ 1, it is obtained that 0 ≤ μ + η + ν ≤ 3. According to the definition of a PFN in Def. 1, it is further obtained that μ_ih + η_ih + ν_ih ≤ 1 and μ_ih ≤ 1 – (η_ih + ν_ih).

    According to the conditions φ(t) and φ⁻¹(t) are monotonically decreasing, ψ(t) and ψ⁻¹(t) are monotonically increasing, ψ(1 − t) = φ(t), ψ⁻¹(t) = 1 − φ⁻¹(t), φ(1 − t) = ψ(t), and φ⁻¹(t) = 1 − ψ⁻¹(t), the following inequalities are successively derived

    $$\begin{aligned} (n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} ) \le (n\omega_{{i_{h} }} )\psi \left( {1 - (\eta_{{i_{h} }} + \nu_{{i_{h} }} )} \right), \hfill \\ (n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} ) \le (n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} + \nu_{{i_{h} }} ), \hfill \\ \varphi (\eta_{{i_{h} }} + \nu_{{i_{h} }} ) \le 2\varphi (\eta_{{i_{h} }} + \nu_{{i_{h} }} ) \le \varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} ), \hfill \\ (n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} ) \le (n\omega_{{i_{h} }} )\varphi (\eta_{{i_{h} }} + \nu_{{i_{h} }} ) \le (n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right), \hfill \\ \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right) \le \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right), \hfill \\ \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right) \le 1 - \varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right), \hfill \\ \sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} \ge \sum\limits_{h = 1}^{k} {\varphi \left( {1 - \varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} , \hfill \\ \sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} \ge \sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} , \hfill \\ \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right) \le \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right), \hfill \\ \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right) \le 1 - \psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right), \hfill \\ \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} \le \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {1 - \psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} , \hfill \\ \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} \le \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} , \hfill \\ \psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right) \le \psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right), \hfill \\ \psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right) \hfill \\ \le 1 - \varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right), \hfill \\ \frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right) \hfill \\ \ge \frac{1}{k}\varphi \left( {1 - \varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right)} \right), \hfill \\ \frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right) \hfill \\ \ge \frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right)} \right), \hfill \\ \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) \hfill \\ \le \varphi^{ - 1} \left( {\frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right)} \right)} \right), \hfill \\ \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) \hfill \\ \le 1 - \psi^{ - 1} \left( {\frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} ) + \varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) \hfill \\ \le 1 - \left( \begin{aligned} \psi^{ - 1} \left( {\frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\eta_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) + \hfill \\ \psi^{ - 1} \left( {\frac{1}{k}\psi \left( {\varphi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\varphi \left( {\psi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\psi \left( {\varphi^{ - 1} \left( {(n\omega_{{i_{h} }} )\left( {\varphi (\nu_{{i_{h} }} )} \right)} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) \hfill \\ \end{aligned} \right). \hfill \\ \end{aligned}$$

    That is, μ + η + ν ≤ 1.

    Since 0 ≤ μ + η + ν ≤ 3 and μ + η + ν ≤ 1 have been proved, 0 ≤ μ + η + ν ≤ 1 can be obtained.

1.2 B. Proof of Theorem 2

Proof

Since α_i = α = < μ_α, η_α, ν_α > for all i = 1, 2, ..., n, d(α_i, α_j) = 0 for all j = 1, 2, ..., n and j ≠ i. Based on the expression of T(α_p) in Def. 9, it is obtained that

$$n\omega_{{i_{h} }} = n{{\left( {\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\sum\limits_{h = 1}^{k} {\left( {1 + T(\alpha_{{i_{h} }} )} \right)} } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\sum\limits_{h = 1}^{k} {\left( {1 + T(\alpha_{{i_{h} }} )} \right)} } } \right)} {\sum\limits_{j = 1}^{n} {\left( {1 + T(\alpha_{j} )} \right)} }}} \right. \kern-0pt} {\sum\limits_{j = 1}^{n} {\left( {1 + T(\alpha_{j} )} \right)} }} = n{{\left( {1 + (n - 1)} \right)} \mathord{\left/ {\vphantom {{\left( {1 + (n - 1)} \right)} {\left( {n\left( {1 + (n - 1)} \right)} \right)}}} \right. \kern-0pt} {\left( {n\left( {1 + (n - 1)} \right)} \right)}} = 1.$$

Based on this and Theorem 1, it can be obtained that

$$\mu = \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} } \right)} \right)} } \right)} \right)} \right).$$

Since μ_i = μ_α, then

$$\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} = \sum\limits_{h = 1}^{k} {\varphi (\mu_{\alpha } )} = k\varphi (\mu_{\alpha } )\;{\text{and }}\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} } \right) = \varphi^{ - 1} \left( {k\varphi (\mu_{\alpha } )} \right).$$

The following equations are successively obtained

$$\begin{aligned} \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} } \right)} \right)} = \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {k\varphi (\mu_{\alpha } )} \right)} \right)} = \psi \left( {\varphi^{ - 1} \left( {k\varphi (\mu_{\alpha } )} \right)} \right), \hfill \\ \psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} } \right)} \right)} } \right) = \psi^{ - 1} \left( {\psi \left( {\varphi^{ - 1} \left( {k\varphi (\mu_{\alpha } )} \right)} \right)} \right) = \varphi^{ - 1} \left( {k\varphi (\mu_{\alpha } )} \right), \hfill \\ \frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} } \right)} \right)} } \right)} \right) = \frac{1}{k}\varphi \left( {\varphi^{ - 1} \left( {k\varphi (\mu_{\alpha } )} \right)} \right) = \varphi (\mu_{\alpha } ), \hfill \\ \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi (\mu_{{i_{h} }} )} } \right)} \right)} } \right)} \right)} \right) = \varphi^{ - 1} \left( {\varphi (\mu_{\alpha } )} \right) = \mu_{\alpha } , \hfill \\ \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) = \mu_{\alpha } . \hfill \\ \end{aligned}$$

This is μ = μ_α. Similarly, it can be proved that η = η_α and ν = ν_α. Thus, PFAPMSM^(k)(α₁, α₂, ..., α_n) = < μ_α, η_α, ν_α > .

1.3 C. Proof of Theorem 3

Proof

It can be derived from the definition of the PFAPMSM operator that

$$\left( {\frac{1}{{C_{n}^{k} }}\mathop \oplus \limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} \mathop \otimes \limits_{h = 1}^{k} \left( {\frac{{n\left( {1 + T(\beta_{{i_{h} }} )} \right)}}{{\sum\limits_{j = 1}^{n} {\left( {1 + T(\beta_{j} )} \right)} }}\beta_{{i_{h} }} } \right)} \right)^{1/k} = \left( {\frac{1}{{C_{n}^{k} }}\mathop \oplus \limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} \mathop \otimes \limits_{h = 1}^{k} \left( {\frac{{n\left( {1 + T(\alpha_{{i_{h} }} )} \right)}}{{\sum\limits_{j = 1}^{n} {\left( {1 + T(\alpha_{j} )} \right)} }}\alpha_{{i_{h} }} } \right)} \right)^{1/k} .$$

According to Def. 9, it is obtained that PFAPMSM^(k)(β₁, β₂, ..., β_n) = PFAPMSM^(k)(α₁, α₂, ..., α_n).

1.4 D. Proof of Theorem 4

Proof

Based on Theorem 2, it can be obtained that PFAPMSM^(k)(α⁻, α⁻, ..., α⁻) = αα⁻, PFAPMSM^(k)(α⁺, α⁺, ..., α⁺) = α⁺, and nξ_ih = 1 for both PFAPMSM^(k)(α⁻, α⁻, ..., α⁻) and PFAPMSM^(k)(α⁺, α⁺, ..., α⁺). According to the conditions μ⁻ ≤ μ_ih ≤ μ⁺, φ(t) and φ⁻¹(t) are monotonically decreasing, and ψ(t) and ψ⁻¹(t) are monotonically increasing, the following inequalities are successively derived

$$\begin{aligned} \psi (\mu^{ - } ) \le (n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} ) \le \psi (\mu^{ + } ), \hfill \\ \mu^{ - } \le \psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right) \le \mu^{ + } , \hfill \\ k\varphi (\mu^{ - } ) \ge \sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} \ge k\varphi (\mu^{ + } ), \hfill \\ \varphi^{ - 1} \left( {k\varphi (\mu^{ - } )} \right) \le \varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right) \le \varphi^{ - 1} \left( {k\varphi (\mu^{ + } )} \right), \hfill \\ \psi \left( {\varphi^{ - 1} \left( {k\varphi (\mu^{ - } )} \right)} \right) \le \frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} \le \psi \left( {\varphi^{ - 1} \left( {k\varphi (\mu^{ + } )} \right)} \right), \hfill \\ \varphi^{ - 1} \left( {k\varphi (\mu^{ - } )} \right) \le \psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right) \le \varphi^{ - 1} \left( {k\varphi (\mu^{ + } )} \right), \hfill \\ \varphi (\mu^{ - } ) \ge \frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right) \ge \varphi (\mu^{ + } ), \hfill \\ \mu^{ - } \le \varphi^{ - 1} \left( {\frac{1}{k}\varphi \left( {\psi^{ - 1} \left( {\frac{1}{{C_{n}^{k} }}\sum\limits_{{1 \le i_{1} < \ldots < i_{k} \le n}} {\psi \left( {\varphi^{ - 1} \left( {\sum\limits_{h = 1}^{k} {\varphi \left( {\psi^{ - 1} \left( {(n\omega_{{i_{h} }} )\psi (\mu_{{i_{h} }} )} \right)} \right)} } \right)} \right)} } \right)} \right)} \right) \le \mu^{ + } . \hfill \\ \end{aligned}$$

That is μ⁻ ≤ μ ≤ μ⁺. Similarly, it can be proved that η⁻ ≥ η ≥ η⁺ and ν⁻ ≥ ν ≥ ν⁺.

According to Def. 2, it is obtained that

PFAPMSM^(k)(α⁻, α⁻, ..., α⁻) ≤ PFAPMSM^(k)(α₁, α₂, ..., α_n) ≤ PFAPMSM^(k)(α⁺, α⁺, ..., α⁺)

Therefore, α⁻ ≤ PFAPMSM^(k)(α₁, α₂, ..., α_n) ≤ α⁺.