Trapezoidal cubic fuzzy number Einstein hybrid weighted averaging operators and its application to decision making

Abstract

In this paper, we define some Einstein operations on trapezoidal cubic fuzzy set and develop three arithmetic averaging operators, that is trapezoidal cubic fuzzy Einstein weighted averaging (TrCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator and trapezoidal cubic fuzzy Einstein hybrid weighted averaging (TrCFEHWA) operator, for aggregating trapezoidal cubic fuzzy information. The TrCFEHWA operator generalizes both the TrCFEWA and TrCFEOWA operators. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply on the TrCFEHWA operator to multiple attribute decision making with trapezoidal cubic fuzzy information. Finally, a numerical example is providing to demonstrate the submission of the established approach.

Appendices

Appendix A: Proof of Proposition 1

• (1) $A_1 +A_2 =A_2 +A_1 ;$

  $$\begin{aligned} A_1 +A_2= & {} \left\langle \left[ \begin{array}{c} \max (I_{A_1 }^- , I_{A_2 }^- ) \\ \left[ \frac{p_1^- (h)+p_2^- (h)}{1+p_1^- (h)p_2^- (h)}, \right. \\ \frac{q_1^- (h)+q_2^- (h)}{1+q_1^- (h)q_2^- (h)}, \\ \frac{r_1^- (h)+r_2^- (h)}{1+r_1^- (h)r_2^- (h))}, \\ \left. \frac{s_1^- (h)+s_2^- (h)}{1+s_1^- (h)s_2^- (h))}\right] , \\ {\max (I_{A_1 }^+ , I_{A_2 }^+ )} \\ \left[ \frac{p_1^+ (h)+p_2^+ (h)}{1+p_1^+ (h)p_2^+ (h))}, \right. \\ {\frac{q_1^+ (h)+q_2^+ (h)}{1+q_1^+ (h)q_2^+ (h))},} \\ {\frac{r_1^+ (h)+r_2^+ (h)}{1+r_1^+ (h)r_2^+ (h))},} \\ \left. \frac{s_1^+ (h)+s_2^+ (h)}{1+s_1^+ (h)s_2^+ (h))}\right] , \\ \end{array} \right] , \right. \\&\left. \left[ {{\begin{array}{c} {\min (\mu _{A_1 } , \mu _{A_2 } )} \\ \left[ \frac{p_1 (h)\cdot p_2 (h)}{1+(\left( {1-p_1 (h)} \right) \left( {1-p_2 (h)} \right) )}, \right. \\ {\frac{q_1 (h)\cdot q_2 (h)}{1+(\left( {1-q_1 (h)} \right) \left( {1-q_2 (h)} \right) )},} \\ {\frac{r_1 (h)\cdot r_2 (h)}{1+(\left( {1-r_1 (h)} \right) \left( {1-r_2 (h)} \right) )},} \\ \left. \frac{s_1 (h)\cdot s_2 (h)}{1+(\left( {1-s_1 (h)} \right) \left( {1-s_2 (h)} \right) )}\right] \\ \end{array} }} \right] \right\rangle \\= & {} \left\langle \left[ {{\begin{array}{c} {\max (I_{A_2 }^- , I_{A_1 }^- )} \\ \left[ \frac{p_2^- (h)+p_1^- (h)}{1+p_2^- (h)p_1^- (h)},\right. \\ {\frac{q_2^- (h)+q_1^- (h)}{1+q_2^- (h)q_1^- (h)},} \\ {\frac{r_2^- (h)+r_1^- (h)}{1+r_2^- (h)r_1^- (h)},} \\ \left. \frac{s_2^- (h)+s_1^- (h)}{1+s_2^- (h)s_1^- (h)}\right] , \\ {\max (I_{A_2 }^+ , I_{A_1 }^+ )} \\ \left[ \frac{p_2^+ (h)+p_1^+ (h)}{1+p_2^+ (h)p_1^+ (h)},\right. \\ {\frac{q_2^+ (h)+q_1^+ (h)}{1+q_2^+ (h)q_1^+ (h)},} \\ {\frac{r_2^+ (h)+r_1^+ (h)}{1+r_2^+ (h)r_1^+ (h)},} \\ \left. \frac{s_2^+ (h)+s_1^+ (h)}{1+s_2^+ (h)s_1^+ (h)}\right] , \\ \end{array} }} \right] , \right. \\&\left. \left[ {{\begin{array}{c} {\min (\mu _{A_2 } , \mu _{A_1 } )} \\ \left[ \frac{p_2 (h)\cdot p_1 (h)}{1+(\left( {1-p_2 (h)} \right) \left( {1-p_1 (h)} \right) )},\right. \\ {\frac{q_2 (h)\cdot q_1 (h)}{1+(\left( {1-q_2 (h)} \right) \left( {1-q_1 (h)} \right) )},} \\ {\frac{r_2 (h).r_1 (h)}{1+(\left( {1-r_2 (h)} \right) \left( {1-r_1 (h)} \right) )},} \\ \left. \frac{s_2 (h)\cdot s_1 (h)}{1+(\left( {1-s_2 (h)} \right) \left( {1-s_1 (h)} \right) )}\right] \\ \end{array} }} \right] \right\rangle \\= & {} A_2 +A_1 \end{aligned}$$

  Hence $A_1 +A_2 =A_2 +A_1$.

• (2) $\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1$

  $$\begin{aligned}&\lambda (A_1 +A_2 )\\&\quad =\left\langle \left[ {{\begin{array}{c} {\max (I_{A_1 }^- I_{A_2 }^- )} \\ \left[ \frac{[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }}{[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }},\right. \\ {\frac{[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }}{[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }},} \\ {\frac{[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }}{[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }},} \\ \left. \frac{[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }}{[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }}\right] \\ \end{array} }} \right] , \right. \\&\qquad \left. \left[ {{\begin{array}{c} {\max (I_{A_1 }^+ I_{A_2 }^+ )} \\ \left[ \frac{[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }[(1+p_{p2}^+ (h))(1-p_2^+ (h))]^{\lambda }}{[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }},\right. \\ {\frac{[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }}{[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }},} \\ {\frac{[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }}{[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }},} \\ \left. \frac{[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }}{[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }}\right] \\ \end{array} }} \right] , \right. \\&\qquad \left. \left[ {{\begin{array}{c} {\min (\mu _{A_1 } \mu _{A_2 } );} \\ \left[ \frac{2[p_1 (h)p_2 (h)]^{\lambda }}{[(4-2p_1 (h)-2p_2 (h)-p_1 (h)p_2 (h)]^{\lambda }+[p_1 (h)p_2 (h)]^{\lambda }},\right. \\ {\frac{2[q_1 (h)q_2 (h)]^{\lambda }}{[(4-2q_1 (h)-2q_2 (h)-q_1 (h)q_2 (h)]^{\lambda }+[q_1 (h)q_2 (h)]^{\lambda }},} \\ {\frac{2[r_1 (h)r_2 (h)]^{\lambda }}{[(4-2r_1 (h)-2r_2 (h)-r_1 (h)r_2 (h)]^{\lambda }+[r_1 (h)r_2 (h)]^{\lambda }},} \\ \left. \frac{2[s_1 (h)s_2 (h)]^{\lambda }}{[(4-2s_1 (h)-2s_2 (h)-s_1 (h)s_2 (h)]^{\lambda }+[s_1 (h)s_2 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \end{aligned}$$

  and we have

  $$\begin{aligned}&\lambda A_1=\left\langle \left[ {{\begin{array}{c} \max (I_{A_1 }^- )\left[ \frac{[(1+p_1^- (h))^{\lambda }-(1-p_1^- (h))^{\lambda }]}{[(1+p_1^- (h))^{\lambda }+(1-p_1^- (h))^{\lambda }]},\right. \\ {\frac{[(1+q_1^- (h))^{\lambda }-(1-q_1^- (h))^{\lambda }]}{[(1+q_1^- (h))^{\lambda }+(1-q_1^- (h))^{\lambda }]},} \\ {\frac{[(1+r_1^- (h))^{\lambda }-(1-r_1^- (h))^{\lambda }]}{[(1+r_1^- (h))^{\lambda }+(1-r_1^- (h))^{\lambda }]},} \\ \left. \frac{[(1+s_1^- (h))^{\lambda }-(1-s_1^- (h))^{\lambda }]}{[(1+s_1^- (h))^{\lambda }+(1-s_1^- (h))^{\lambda }]}\right] \\ \max (I_{A_1 }^+ )\left[ \frac{[(1+p_1^+ (h))^{\lambda }-(1-p_1^+ (h))^{\lambda }]}{[(1+p_1^+ (h))^{\lambda }+(1-p_1^+ (h))^{\lambda }]},\right. \\ {\frac{[(1+q_1^+ (h))^{\lambda }-(1-q_1^+ (h))^{\lambda }]}{[(1+q_1^+ (h))^{\lambda }+(1-q_1^+ (h))^{\lambda }]},} \\ {\frac{[(1+r_1^+ (h))^{\lambda }-(1-r_1^+ (h))^{\lambda }]}{[(1+r_1^+ (h))^{\lambda }+(1-r_1^+ (h))^{\lambda }]},} \\ \left. \frac{[(1+s_1^+ (h))^{\lambda }-(1-s_1^+ (h))^{\lambda }]}{[(1+s_1^+ (h))^{\lambda }+(1-s_1^+ (h))^{\lambda }]}\right] \\ \end{array} }} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} \min (\mu _{A_1 } );\left[ \frac{2p_1^\lambda (h)}{[(2-p_1 (h)]^{\lambda }+[p_1 (h)]^{\lambda }},\right. \\ {\frac{2q_1^\lambda (h)}{[(2-q_1 (h)]^{\lambda }+[q_1 (h)]^{\lambda }},} \\ {\frac{2r_1^\lambda (h)}{[(2-r_1 (h)]^{\lambda }+[r_1 (h)]^{\lambda }},} \\ \left. \frac{2s_1^\lambda (h)}{[(2-s_1 (h)]^{\lambda }+[s_1 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \\&\lambda A_2 =\left\langle \left[ {{\begin{array}{c} \max (I_A^- )\left[ \frac{[(1+p_2^- (h))^{\lambda }-(1-p_2^- (h))^{\lambda }]}{[(1+p_2^- (h))^{\lambda }+(1-p_2^- (h))^{\lambda }]}, \right. \\ {\frac{[(1+q_2^- (h))^{\lambda }-(1-q_2^- (h))^{\lambda }]}{[(1+q_2^- (h))^{\lambda }+(1-q_2^- (h))^{\lambda }]},} \\ {\frac{[(1+r_2^- (h))^{\lambda }-(1-r_2^- (h))^{\lambda }]}{[(1+r_2^- (h))^{\lambda }+(1-r_2^- (h))^{\lambda }]},} \\ \left. \frac{[(1+s_2^- (h))^{\lambda }-(1-s_2^- (h))^{\lambda }]}{[(1+s_2^- (h))^{\lambda }+(1-s_2^- (h))^{\lambda }]}\right] ; \\ \max (I_A^- )\left[ \frac{[(1+p_2^+ (h))^{\lambda }-(1-p_2^+ (h))^{\lambda }]}{[(1+p_2^+ (h))^{\lambda }+(1-p_2^+ (h))^{\lambda }]},\right. \\ {\frac{[(1+q_2^+ (h))^{\lambda }-(1-q_2^+ (h))^{\lambda }]}{[(1+q_2^+ (h))^{\lambda }+(1-q_2^+ (h))^{\lambda }]},} \\ {\frac{[(1+r_2^+ (h))^{\lambda }-(1-r_2^+ (h))^{\lambda }]}{[(1+r_2^+ (h))^{\lambda }+(1-r_2^+ (h))^{\lambda }]},} \\ \left. \frac{[(1+s_2^+ (h))^{\lambda }-(1-s_2^+ (h))^{\lambda }]}{[(1+s_2^+ (h))^{\lambda }+(1-s_2^+ (h))^{\lambda }]}\right] \\ \end{array} }} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} \min (\mu _{A_2 } );\left[ \frac{2p_2^\lambda (h)}{[(2-p_2 (h)]^{\lambda }+[p_2 (h)]^{\lambda }},\right. \\ {\frac{2q_2^\lambda (h)}{[(2-q_2 (h)]^{\lambda }+[q_2 (h)]^{\lambda }},} \\ {\frac{2r_2^\lambda (h)}{[(2-r_2 (h)]^{\lambda }+[r_2 (h)]^{\lambda }},} \\ \left. \frac{2s_2^\lambda (h)}{[(2-s_2 (h)]^{\lambda }+[s_2 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \\&\lambda A_2 +\lambda A_1 \\&\quad =\left\langle \max (I_{A_2 }^- ,I_{A_1 }^- )\left[ {\begin{array}{c} \frac{[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }}{[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }}, \\ \frac{[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }}{[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }}, \\ \frac{[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }}{[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }}, \\ \frac{[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }}{[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }} \\ \end{array}} \right] \right. \\&\quad \max (I_{A_2 }^+ ,I_{A_1 }^+ )\left. \left[ {\begin{array}{c} \frac{[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }}{[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }}, \\ \frac{[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }}{[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }}, \\ \frac{[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }}{[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }}, \\ \frac{[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }}{[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }} \\ \end{array}} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} {\min (\mu _{A_2 } \mu _{A_1 } );} \\ \left[ \frac{2[p_2 (h)p_1 (h)]^{\lambda }}{[(4-2p_2 (h)-2p_1 (h)-p_2 (h)p_1 (h)]^{\lambda }+[p_2 (h)p_1 (h)]^{\lambda }},\right. \\ {\frac{2[q_2 (h)q_1 (h)]^{\lambda }}{[(4-2q_2 (h)-2q_1 (h)-q_2 (h)q_1 (h)]^{\lambda }+[q_2 (h)q_1 (h)]^{\lambda }},} \\ {\frac{2[r_2 (h)r_1 (h)]^{\lambda }}{[(4-2r_2 (h)-2r_1 (h)-r_2 (h)r_1 (h)]^{\lambda }+[r_2 (h)r_1 (h)]^{\lambda }},} \\ \left. \frac{2[s_2 (h)s_1 (h)]^{\lambda }}{[(4-2s_2 (h)-2s_1 (h)-s_2 (h)s_1 (h)]^{\lambda }+[s_2 (h)s_1 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \end{aligned}$$

  so, we have $\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1.$

• (3) $\lambda _1 A+\lambda _2 A=(\lambda _1 +\lambda _2 )A$

  $$\begin{aligned} \lambda _1 A= & {} \left\langle \left[ {{\begin{array}{c} \max (I_A^- ),\left[ \frac{[1+p_A^- (h)]^{\lambda _1 }-[1-p_A^- (h)]^{\lambda _1 }}{[1+p_A^- (h)]^{\lambda _1 }+[1-p_A^- (h)]^{\lambda _1 }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda _1 }-[1-q_A^- (h)]^{\lambda _1 }}{[1+q_A^- (h)]^{\lambda _1 }+[1-q_A^- (h)]^{\lambda _1 }},} \\ {\frac{[1+r_A^- (h)]^{\lambda _1 }-[1-r_A^- (h)]^{\lambda _1 }}{[1+r_A^- (h)]^{\lambda _1 }+[1-r_A^- (h)]^{\lambda _1 }},} \\ {\frac{[1+s_A^- (h)]^{\lambda _1 }-[1-s_A^- (h)]^{\lambda _1 }}{[1+s_A^- (h)]^{\lambda _1 }+[1-s_A^- (h)]^{\lambda _1 }}} \\ \max (I_A^+ ),\left[ \frac{[1+p_A^+ (h)]^{\lambda _1 }-[1-p_A^+ (h)]^{\lambda _1 }}{[1+p_A^+ (h)]^{\lambda _1 }+[1-p_A^+ (h)]^{\lambda _1 }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda _1 }-[1-q_A^+ (h)]^{\lambda _1 }}{[1+q_A^+ (h)]^{\lambda _1 }+[1-q_A^+ (h)]^{\lambda _1 }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda _1 }-[1-r_A^+ (h)]^{\lambda _1 }}{[1+r_A^+ (h)]^{\lambda _1 }+[1-r_A^+ (h)]^{\lambda _1 }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda _1 }-[1-s_A^+ (h)]^{\lambda _1 }}{[1+s_A^+ (h)]^{\lambda _1 }+[1-s_A^+ (h)]^{\lambda _1 }}\right] \\ \end{array} }} \right] \right. \\&\left. \left[ {{\begin{array}{c} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda _1 }}{[(2-p_A (h)]^{\lambda _1 }+[p_A (h)]^{\lambda _1 }},\right. \\ {\frac{2[q_A (h)]^{\lambda _1 }}{[(2-q_A (h)]^{\lambda _1 }+[q_A (h)]^{\lambda _1 }},} \\ {\frac{2[r_A (h)]^{\lambda _1 }}{[(2-r_A (h)]^{\lambda _1 }+[r_A (h)]^{\lambda _1 }},} \\ \left. \frac{2[s_A (h)]^{\lambda _1 }}{[(2-s_A (h)]^{\lambda _1 }+[s_A (h)]^{\lambda _1 }}\right] \\ \end{array} }} \right] \right\rangle \end{aligned}$$

  and

  $$\begin{aligned} \lambda _2 A= & {} \left\langle \left[ {{\begin{array}{c} \max (I_A^- ),\left[ \frac{[1+p_A^- (h)]^{\lambda _2 }-[1-p_A^- (h)]^{\lambda _2 }}{[1+p_A^- (h)]^{\lambda _2 }+[1-p_A^- (h)]^{\lambda _2 }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda _2 }-[1-q_A^- (h)]^{\lambda _2 }}{[1+q_A^- (h)]^{\lambda _2 }+[1-q_A^- (h)]^{\lambda _2 }},} \\ {\frac{[1+r_A^- (h)]^{\lambda _2 }-[1-r_A^- (h)]^{\lambda _2 }}{[1+r_A^- (h)]^{\lambda _2 }+[1-r_A^- (h)]^{\lambda _2 }},} \\ {\frac{[1+s_A^- (h)]^{\lambda _2 }-[1-s_A^- (h)]^{\lambda _2 }}{[1+s_A^- (h)]^{\lambda _2 }+[1-s_A^- (h)]^{\lambda _2 }}} \\ \max (I_A^+ ),\left[ \frac{[1+p_A^+ (h)]^{\lambda _2 }-[1-p_A^+ (h)]^{\lambda _2 }}{[1+p_A^+ (h)]^{\lambda _2 }+[1-p_A^+ (h)]^{\lambda _2 }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda _2 }-[1-q_A^+ (h)]^{\lambda _2 }}{[1+q_A^+ (h)]^{\lambda _2 }+[1-q_A^+ (h)]^{\lambda _2 }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda _2 }-[1-r_A^+ (h)]^{\lambda _2 }}{[1+r_A^+ (h)]^{\lambda _2 }+[1-r_A^+ (h)]^{\lambda _2 }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda _2 }-[1-s_A^+ (h)]^{\lambda _2 }}{[1+s_A^+ (h)]^{\lambda _2 }+[1-s_A^+ (h)]^{\lambda _2 }}\right] \\ \end{array} }} \right] \right. \\&\left. \left[ {{\begin{array}{c} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda _2 }}{[(2-p_A (h)]^{\lambda _2 }+[p_A (h)]^{\lambda _2 }},\right. \\ {\frac{2[q_A (h)]^{\lambda _2 }}{[(2-q_A (h)]^{\lambda _2 }+[q_A (h)]^{\lambda _2 }},} \\ {\frac{2[r_A (h)]^{\lambda _2 }}{[(2-r_A (h)]^{\lambda _2 }+[r_A (h)]^{\lambda _2 }},} \\ \left. \frac{2[s_A (h)]^{\lambda _2 }}{[(2-s_A (h)]^{\lambda _2 }+[s_A (h)]^{\lambda _2 }}\right] \\ \end{array} }} \right] \right\rangle \\= & {} \left\langle \left[ {{\begin{array}{c} \max (I_A^- ),\left[ \frac{[1+p_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-p_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+p_A^- (h)]^{\lambda _1 +\lambda _2 }+[1-p_A^- (h)]^{\lambda _1 +\lambda _2 }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-q_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+q_A^- (h)]^{\lambda _1 +\lambda _2 }+[1-q_A^- (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{[1+r_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-r_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+r_A^- (h)]^{\lambda _1 +\lambda _2 }+[1-r_A^- (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{[1+s_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-s_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+s_A^- (h)]^{\lambda _2 }+[1-s_A^- (h)]^{\lambda _1 +\lambda _2 }}} \\ \max (I_A^+ ),\left[ \frac{[1+p_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-p_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+p_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-p_A^+ (h)]^{\lambda _1 +\lambda _2 }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-q_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+q_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-q_A^+ (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-r_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+r_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-r_A^+ (h)]^{\lambda _1 +\lambda _2 }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-s_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+s_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-s_A^+ (h)]^{\lambda _1 +\lambda _2 }}\right] \\ \end{array} }} \right] \right. \\&\left. \left[ {{\begin{array}{c} {\min (\mu _A )} \\ \left[ \frac{2[p_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-p_A (h)]^{\lambda _1 +\lambda _2 }+[p_A (h)]^{\lambda _1 +\lambda _2 }},\right. \\ {\frac{2[q_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-q_A (h)]^{\lambda _1 +\lambda _2 }+[q_A (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{2[r_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-r_A (h)]^{\lambda _1 +\lambda _2 }+[r_A (h)]^{\lambda _1 +\lambda _2 }},} \\ \left. \frac{2[s_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-s_A (h)]^{\lambda _1 +\lambda _2 }+[s_A (h)]^{\lambda _1 +\lambda _2 }}\right] \\ \end{array} }} \right] \right\rangle \\= & {} (\lambda _1 +\lambda _2 )A. \end{aligned}$$

Appendix B: Proof of Theorem 1

Assume that $n=1,$ TrCFEWA $(A_1 , A_2 ,..., A_n )=\mathop {\oplus }\nolimits _{j=1}^k w_1 A_1$

$$\begin{aligned}&\langle \max (\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1 \\&\lambda (A_1 +A_2 )\\&\quad =\left\langle \left[ {{\begin{array}{c} {\max (I_{A_1 }^- I_{A_2 }^- )} \\ \left[ \frac{[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }}{[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }},\right. \\ {\frac{[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }}{[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }},} \\ {\frac{[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }}{[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }},} \\ \left. \frac{[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }}{[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }}\right] \\ \end{array} }} \right] , \right. \\&\qquad \left. \left[ {{\begin{array}{c} {\max (I_{A_1 }^+ I_{A_2 }^+ )} \\ \left[ \frac{[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }}{[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }},\right. \\ {\frac{[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }}{[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }},} \\ {\frac{[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }}{[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }},} \\ \left. \frac{[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }}{[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }}\right] \\ \end{array} }} \right] , \right. \\&\qquad \left. \left[ {{\begin{array}{c} {\min (\mu _{A_1 } \mu _{A_2 } );} \\ \left[ \frac{2[p_1 (h)p_2 (h)]^{\lambda }}{[(4-2p_1 (h)-2p_2 (h)-p_1 (h)p_2 (h)]^{\lambda }+[p_1 (h)p_2 (h)]^{\lambda }},\right. \\ {\frac{2[q_1 (h)q_2 (h)]^{\lambda }}{[(4-2q_1 (h)-2q_2 (h)-q_1 (h)q_2 (h)]^{\lambda }+[q_1 (h)q_2 (h)]^{\lambda }},} \\ {\frac{2[r_1 (h)r_2 (h)]^{\lambda }}{[(4-2r_1 (h)-2r_2 (h)-r_1 (h)r_2 (h)]^{\lambda }+[r_1 (h)r_2 (h)]^{\lambda }},} \\ \left. \frac{2[s_1 (h)s_2 (h)]^{\lambda }}{[(4-2s_1 (h)-2s_2 (h)-s_1 (h)s_2 (h)]^{\lambda }+[s_1 (h)s_2 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \end{aligned}$$

and we have

$$\begin{aligned}&\lambda A_1 =\left\langle \left[ {{\begin{array}{c} \max (I_{A_1 }^- )\left[ \frac{[(1+p_1^- (h))^{\lambda }-(1-p_1^- (h))^{\lambda }]}{[(1+p_1^- (h))^{\lambda }+(1-p_1^- (h))^{\lambda }]},\right. \\ {\frac{[(1+q_1^- (h))^{\lambda }-(1-q_1^- (h))^{\lambda }]}{[(1+q_1^- (h))^{\lambda }+(1-q_1^- (h))^{\lambda }]},} \\ {\frac{[(1+r_1^- (h))^{\lambda }-(1-r_1^- (h))^{\lambda }]}{[(1+r_1^- (h))^{\lambda }+(1-r_1^- (h))^{\lambda }]},} \\ \left. \frac{[(1+s_1^- (h))^{\lambda }-(1-s_1^- (h))^{\lambda }]}{[(1+s_1^- (h))^{\lambda }+(1-s_1^- (h))^{\lambda }]}\right] \\ \max (I_{A_1 }^+ )\left[ \frac{[(1+p_1^+ (h))^{\lambda }-(1-p_1^+ (h))^{\lambda }]}{[(1+p_1^+ (h))^{\lambda }+(1-p_1^+ (h))^{\lambda }]},\right. \\ {\frac{[(1+q_1^+ (h))^{\lambda }-(1-q_1^+ (h))^{\lambda }]}{[(1+q_1^+ (h))^{\lambda }+(1-q_1^+ (h))^{\lambda }]},} \\ {\frac{[(1+r_1^+ (h))^{\lambda }-(1-r_1^+ (h))^{\lambda }]}{[(1+r_1^+ (h))^{\lambda }+(1-r_1^+ (h))^{\lambda }]},} \\ \left. \frac{[(1+s_1^+ (h))^{\lambda }-(1-s_1^+ (h))^{\lambda }]}{[(1+s_1^+ (h))^{\lambda }+(1-s_1^+ (h))^{\lambda }]}\right] \\ \end{array} }} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} \min (\mu _{A_1 } );\left[ \frac{2p_1^\lambda (h)}{[(2-p_1 (h)]^{\lambda }+[p_1 (h)]^{\lambda }},\right. \\ {\frac{2q_1^\lambda (h)}{[(2-q_1 (h)]^{\lambda }+[q_1 (h)]^{\lambda }},} \\ {\frac{2r_1^\lambda (h)}{[(2-r_1 (h)]^{\lambda }+[r_1 (h)]^{\lambda }},} \\ \left. \frac{2s_1^\lambda (h)}{[(2-s_1 (h)]^{\lambda }+[s_1 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \\&\lambda A_2 =\left\langle \left[ {{\begin{array}{c} \max (I_A^- )\left[ \frac{[(1+p_2^- (h))^{\lambda }-(1-p_2^- (h))^{\lambda }]}{[(1+p_2^- (h))^{\lambda }+(1-p_2^- (h))^{\lambda }]},\right. \\ {\frac{[(1+q_2^- (h))^{\lambda }-(1-q_2^- (h))^{\lambda }]}{[(1+q_2^- (h))^{\lambda }+(1-q_2^- (h))^{\lambda }]},} \\ {\frac{[(1+r_2^- (h))^{\lambda }-(1-r_2^- (h))^{\lambda }]}{[(1+r_2^- (h))^{\lambda }+(1-r_2^- (h))^{\lambda }]},} \\ \left. \frac{[(1+s_2^- (h))^{\lambda }-(1-s_2^- (h))^{\lambda }]}{[(1+s_2^- (h))^{\lambda }+(1-s_2^- (h))^{\lambda }]}\right] , \\ \max (I_A^- )\left[ \frac{[(1+p_2^+ (h))^{\lambda }-(1-p_2^+ (h))^{\lambda }]}{[(1+p_2^+ (h))^{\lambda }+(1-p_2^+ (h))^{\lambda }]},\right. \\ {\frac{[(1+q_2^+ (h))^{\lambda }-(1-q_2^+ (h))^{\lambda }]}{[(1+q_2^+ (h))^{\lambda }+(1-q_2^+ (h))^{\lambda }]},} \\ {\frac{[(1+r_2^+ (h))^{\lambda }-(1-r_2^+ (h))^{\lambda }]}{[(1+r_2^+ (h))^{\lambda }+(1-r_2^+ (h))^{\lambda }]},} \\ \left. \frac{[(1+s_2^+ (h))^{\lambda }-(1-s_2^+ (h))^{\lambda }]}{[(1+s_2^+ (h))^{\lambda }+(1-s_2^+ (h))^{\lambda }]}\right] \\ \end{array} }} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} \min (\mu _{A_2 } );\left[ \frac{2p_2^\lambda (h)}{[(2-p_2 (h)]^{\lambda }+[p_2 (h)]^{\lambda }},\right. \\ {\frac{2q_2^\lambda (h)}{[(2-q_2 (h)]^{\lambda }+[q_2 (h)]^{\lambda }},} \\ {\frac{2r_2^\lambda (h)}{[(2-r_2 (h)]^{\lambda }+[r_2 (h)]^{\lambda }},} \\ \left. \frac{2s_2^\lambda (h)}{[(2-s_2 (h)]^{\lambda }+[s_2 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \\&\lambda A_2 +\lambda A_1 \\&\quad =\left\langle \left[ {{\begin{array}{c} {\max (I_{A_2 }^- I_{A_1 }^- )} \\ \left[ \frac{[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }}{[(1+p_2^- (h))(1-p_2^- (h))]^{\lambda }[(1+p_1^- (h))(1-p_1^- (h))]^{\lambda }},\right. \\ {\frac{[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }}{[(1+q_2^- (h))(1-q_2^- (h))]^{\lambda }[(1+q_1^- (h))(1-q_1^- (h))]^{\lambda }},} \\ {\frac{[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }}{[(1+r_2^- (h))(1-r_2^- (h))]^{\lambda }[(1+r_1^- (h))(1-r_1^- (h))]^{\lambda }},} \\ \left. \frac{[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }}{[(1+s_2^- (h))(1-s_2^- (h))]^{\lambda }[(1+s_1^- (h))(1-s_1^- (h))]^{\lambda }}\right] \\ \end{array} }} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} {\max (I_{A_2 }^- I_{A_1 }^- )} \\ \left[ \frac{[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }}{[(1+p_2^+ (h))(1-p_2^+ (h))]^{\lambda }[(1+p_1^+ (h))(1-p_1^+ (h))]^{\lambda }},\right. \\ {\frac{[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }}{[(1+q_2^+ (h))(1-q_2^+ (h))]^{\lambda }[(1+q_1^+ (h))(1-q_1^+ (h))]^{\lambda }},} \\ {\frac{[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }}{[(1+r_2^+ (h))(1-r_2^+ (h))]^{\lambda }[(1+r_1^+ (h))(1-r_1^+ (h))]^{\lambda }},} \\ \left. \frac{[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }}{[(1+s_2^+ (h))(1-s_2^+ (h))]^{\lambda }[(1+s_1^+ (h))(1-s_1^+ (h))]^{\lambda }}\right] \\ \end{array} }} \right] , \right. \\&\quad \left. \left[ {{\begin{array}{c} {\min (\mu _{A_2 } \mu _{A_1 } );} \\ \left[ \frac{2[p_2 (h)p_1 (h)]^{\lambda }}{[(4-2p_2 (h)-2p_1 (h)-p_2 (h)p_1 (h)]^{\lambda }+[p_2 (h)p_1 (h)]^{\lambda }},\right. \\ {\frac{2[q_2 (h)q_1 (h)]^{\lambda }}{[(4-2q_2 (h)-2q_1 (h)-q_2 (h)q_1 (h)]^{\lambda }+[q_2 (h)q_1 (h)]^{\lambda }},} \\ {\frac{2[r_2 (h)r_1 (h)]^{\lambda }}{[(4-2r_2 (h)-2r_1 (h)-r_2 (h)r_1 (h)]^{\lambda }+[s_2 (h)s_1 (h)]^{\lambda }},} \\ \left. \frac{2[s_2 (h)s_1 (h)]^{\lambda }}{[(4-2s_2 (h)-2s_1 (h)-s_2 (h)s_1 (h)]^{\lambda }+[s_2 (h)s_1 (h)]^{\lambda }}\right] \\ \end{array} }} \right] \right\rangle \end{aligned}$$

so, we have $\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1$.

$$\begin{aligned}&\lambda _1 A+\lambda _2 A=(\lambda _1 +\lambda _2 )A \\&\quad \lambda _1 A=\left\langle \left[ {{\begin{array}{c} \max (I_A^- ),\left[ \frac{[1+p_A^- (h)]^{\lambda _1 }-[1-p_A^- (h)]^{\lambda _1 }}{[1+p_A^- (h)]^{\lambda _1 }+[1-p_A^- (h)]^{\lambda _1 }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda _1 }-[1-q_A^- (h)]^{\lambda _1 }}{[1+q_A^- (h)]^{\lambda _1 }+[1-q_A^- (h)]^{\lambda _1 }},} \\ {\frac{[1+r_A^- (h)]^{\lambda _1 }-[1-r_A^- (h)]^{\lambda _1 }}{[1+r_A^- (h)]^{\lambda _1 }+[1-r_A^- (h)]^{\lambda _1 }},} \\ \frac{[1+s_A^- (h)]^{\lambda _1 }-[1-s_A^- (h)]^{\lambda _1 }}{[1+s_A^- (h)]^{\lambda _1 }+[1-s_A^- (h)]^{\lambda _1 }} \\ \max (I_A^+ ),\left[ \frac{[1+p_A^+ (h)]^{\lambda _1 }-[1-p_A^+ (h)]^{\lambda _1 }}{[1+p_A^+ (h)]^{\lambda _1 }+[1-p_A^+ (h)]^{\lambda _1 }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda _1 }-[1-q_A^+ (h)]^{\lambda _1 }}{[1+q_A^+ (h)]^{\lambda _1 }+[1-q_A^+ (h)]^{\lambda _1 }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda _1 }-[1-r_A^+ (h)]^{\lambda _1 }}{[1+r_A^+ (h)]^{\lambda _1 }+[1-r_A^+ (h)]^{\lambda _1 }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda _1 }-[1-s_A^+ (h)]^{\lambda _1 }}{[1+s_A^+ (h)]^{\lambda _1 }+[1-s_A^+ (h)]^{\lambda _1 }}\right] \\ \end{array} }} \right] \right. \\&\quad \left. \left[ {{\begin{array}{c} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda _1 }}{[(2-p_A (h)]^{\lambda _1 }+[p_A (h)]^{\lambda _1 }},\right. \\ {\frac{2[q_A (h)]^{\lambda _1 }}{[(2-q_A (h)]^{\lambda _1 }+[q_A (h)]^{\lambda _1 }},} \\ {\frac{2[r_A (h)]^{\lambda _1 }}{[(2-r_A (h)]^{\lambda _1 }+[r_A (h)]^{\lambda _1 }},} \\ \left. \frac{2[s_A (h)]^{\lambda _1 }}{[(2-s_A (h)]^{\lambda _1 }+[s_A (h)]^{\lambda _1 }}\right] \\ \end{array} }} \right] \right\rangle \\ \end{aligned}$$

and

$$\begin{aligned} \lambda _2 A= & {} \left\langle \left[ {{\begin{array}{c} \max (I_A^- ),\left[ \frac{[1+p_A^- (h)]^{\lambda _2 }-[1-p_A^- (h)]^{\lambda _2 }}{[1+p_A^- (h)]^{\lambda _2 }+[1-p_A^- (h)]^{\lambda _2 }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda _2 }-[1-q_A^- (h)]^{\lambda _2 }}{[1+q_A^- (h)]^{\lambda _2 }+[1-q_A^- (h)]^{\lambda _2 }},} \\ {\frac{[1+r_A^- (h)]^{\lambda _2 }-[1-r_A^- (h)]^{\lambda _2 }}{[1+r_A^- (h)]^{\lambda _2 }+[1-r_A^- (h)]^{\lambda _2 }},} \\ {\frac{[1+s_A^- (h)]^{\lambda _2 }-[1-s_A^- (h)]^{\lambda _2 }}{[1+s_A^- (h)]^{\lambda _2 }+[1-s_A^- (h)]^{\lambda _2 }}} \\ \max (I_A^+ ),\left[ \frac{[1+p_A^+ (h)]^{\lambda _2 }-[1-p_A^+ (h)]^{\lambda _2 }}{[1+p_A^+ (h)]^{\lambda _2 }+[1-p_A^+ (h)]^{\lambda _2 }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda _2 }-[1-q_A^+ (h)]^{\lambda _2 }}{[1+q_A^+ (h)]^{\lambda _2 }+[1-q_A^+ (h)]^{\lambda _2 }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda _2 }-[1-r_A^+ (h)]^{\lambda _2 }}{[1+r_A^+ (h)]^{\lambda _2 }+[1-r_A^+ (h)]^{\lambda _2 }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda _2 }-[1-s_A^+ (h)]^{\lambda _2 }}{[1+s_A^+ (h)]^{\lambda _2 }+[1-s_A^+ (h)]^{\lambda _2 }}\right] \\ \end{array} }} \right] \right. \\&\left. \left[ {{\begin{array}{c} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda _2 }}{[(2-p_A (h)]^{\lambda _2 }+[p_A (h)]^{\lambda _2 }},\right. \\ {\frac{2[q_A (h)]^{\lambda _2 }}{[(2-q_A (h)]^{\lambda _2 }+[q_A (h)]^{\lambda _2 }},} \\ {\frac{2[r_A (h)]^{\lambda _2 }}{[(2-r_A (h)]^{\lambda _2 }+[r_A (h)]^{\lambda _2 }},} \\ \left. \frac{2[s_A (h)]^{\lambda _2 }}{[(2-s_A (h)]^{\lambda _2 }+[s_A (h)]^{\lambda _2 }}\right] \\ \end{array} }} \right] \right\rangle \\= & {} \left\langle \left[ {{\begin{array}{c} \max (I_A^- ),\left[ \frac{[1+p_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-p_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+p_A^- (h)]^{\lambda _1 +\lambda _2 }+[1-p_A^- (h)]^{\lambda _1 +\lambda _2 }},\right. \\ {\frac{[1+q_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-q_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+q_A^- (h)]^{\lambda _1 +\lambda _2 }+[1-q_A^- (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{[1+r_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-r_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+r_A^- (h)]^{\lambda _1 +\lambda _2 }+[1-r_A^- (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{[1+s_A^- (h)]^{\lambda _1 +\lambda _2 }-[1-s_A^- (h)]^{\lambda _1 +\lambda _2 }}{[1+s_A^- (h)]^{\lambda _2 }+[1-s_A^- (h)]^{\lambda _1 +\lambda _2 }}} \\ \max (I_A^+ ),\left[ \frac{[1+p_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-p_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+p_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-p_A^+ (h)]^{\lambda _1 +\lambda _2 }},\right. \\ {\frac{[1+q_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-q_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+q_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-q_A^+ (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{[1+r_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-r_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+r_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-r_A^+ (h)]^{\lambda _1 +\lambda _2 }},} \\ \left. \frac{[1+s_A^+ (h)]^{\lambda _1 +\lambda _2 }-[1-s_A^+ (h)]^{\lambda _1 +\lambda _2 }}{[1+s_A^+ (h)]^{\lambda _1 +\lambda _2 }+[1-s_A^+ (h)]^{\lambda _1 +\lambda _2 }}\right] \\ \end{array} }} \right] \right. \\&\left. \left[ {{\begin{array}{c} \min (\mu _A )\left[ \frac{2[p_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-p_A (h)]^{\lambda _1 +\lambda _2 }+[p_A (h)]^{\lambda _1 +\lambda _2 }},\right. \\ {\frac{2[q_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-q_A (h)]^{\lambda _1 +\lambda _2 }+[q_A (h)]^{\lambda _1 +\lambda _2 }},} \\ {\frac{2[r_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-r_A (h)]^{\lambda _1 +\lambda _2 }+[r_A (h)]^{\lambda _1 +\lambda _2 }},} \\ \left. \frac{2[s_A (h)]^{\lambda _1 +\lambda _2 }}{[(2-s_A (h)]^{\lambda _1 +\lambda _2 }+[s_A (h)]^{\lambda _1 +\lambda _2 }}\right] \\ \end{array} }} \right] \right. \rangle \\= & {} \max (I_{A_1 }^- )\left[ {{\begin{array}{c} {\frac{[[1+p_1^- (h)]^{\varpi _1 }-[1-p_1^- (h)]^{^{\varpi _1 }}}{[1+p_1^- (h)]^{^{\varpi _1 }}+[1-p_1^- (h)]^{^{\varpi _1 }}},} \\ {\frac{[1+q_1^- (h)]^{^{\varpi _1 }}-[1-q_1^- (h)]^{^{\varpi _1 }}}{[1+q_1^- (h)]^{^{\varpi _1 }}+[1-q_1^- (h)]^{^{\varpi _1 }}},} \\ {\frac{[1+r_1^- (h)]^{^{\varpi _1 }}-[1-r_1^- (h)]^{^{\varpi _1 }}}{[1+r_1^- (h)]^{^{\varpi _1 }}+[1-r_1^- (h)]^{^{\varpi _1 }}},} \\ {\frac{[1+s_1^- (h)]^{^{\varpi _1 }}-[1-s_1^- (h)]^{^{\varpi _1 }}}{[1+s_1^- (h)]^{^{\varpi _1 }}+[1-s_1^- (h)]^{^{\varpi _1 }}}} \\ \end{array} }} \right] ; \\&\max (I_{A_1 }^+ )\left[ {{\begin{array}{c} {\frac{[1+p_1^+ (h)]^{^{\varpi _1 }}-[1-p_1^+ (h)]^{^{\varpi _1 }}}{[1+p_1^+ (h)]^{^{\varpi _1 }}+[1-p_1^+ (h)]^{^{\varpi _1 }}},} \\ {\frac{[1+q_1^+ (h)]^{^{\varpi _1 }}-[1-q_1^+ (h)]^{^{\varpi _1 }}}{[1+q_1^+ (h)]^{^{\varpi _1 }}+[1-q_1^+ (h)]^{^{\varpi _1 }}},} \\ {\frac{[1+r_1^+ (h)]^{^{\varpi _1 }}-[1-r_1^+ (h)]^{^{\varpi _1 }}}{[1+r_1^+ (h)]^{^{\varpi _1 }}+[1-r_1^+ (h)]^{^{\varpi _1 }}},} \\ {\frac{[1+s_1^+ (h)]^{^{\varpi _1 }}-[1-s_1^+ (h)]^{^{\varpi _1 }}}{[1+s_1^+ (h)]^{^{\varpi _1 }}+[1-s_1^+ (h)]^{^{\varpi _1 }}}} \\ \end{array} }} \right] ; \\&\min (\mu _{A_1 } )\left[ {{\begin{array}{c} {\frac{2[p_1 (h)]^{^{\varpi _1 }}}{[(2-p_1 (h)]^{^{\varpi _1 }}+[p_1 (h)]^{^{\varpi _1 }}},} \\ {\frac{2[q_1 (h)]^{\varpi _1 }}{[(2-q_1 (h)]^{^{\varpi _1 }}+[q_1 (h)]^{^{\varpi _1 }}},} \\ {\frac{2[r_1 (h)]^{^{\varpi _1 }}}{[(2-r_1 (h)]^{^{\varpi _1 }}+[r_1 (h)]^{^{\varpi _1 }}},} \\ {\frac{2[s_1 (h)]^{^{\varpi _1 }}}{[(2-s_1 (h)]^{^{\varpi }}+[s_1 (h)]^{^{\varpi _1 }}}} \\ \end{array} }} \right] . \end{aligned}$$

Assume that $n=k,$ TrCFEWA $(A_1 , A_2 ,..., A_n )=\mathop {\oplus }\nolimits _{j=1}^k w_j A_j$

$$\begin{aligned}&\langle \max (I_A^- )\left[ \begin{array}{c} \frac{[\mathop {\prod }\nolimits _{j=1}^k [1+p_1^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^k [1-p_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+p_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-p_1^- (h)]^{^{\varpi }}}, \\ {\frac{\mathop {\prod }\nolimits _{j=1}^k [1+q_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-q_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+q_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-q_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^k [1+r_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-r_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+r_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-r_1^- (h)]^{^{\varpi }}},} \\ \frac{\mathop {\prod }\nolimits _{j=1}^k [1+s_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-s_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+s_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-s_1^- (h)]^{^{\varpi }}} \\ \end{array}\right] , \\&\max (I_A^+ )\left[ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^k [1+p_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-p_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+p_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-p_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^k [1+q_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-q_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+q_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-q_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^k [1+r_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-r_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+r_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-r_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^k [1+s_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-s_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [1+s_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-s_1^+ (h)]^{^{\varpi }}}} \\ \end{array} }} \right] ; \\&\min (\mu _A )\left[ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^k [p_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-p_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [p_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^k [q_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-q_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [q_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^k [r_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-r_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [r_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^k [s_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-s_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [s_1 (h)]^{^{\varpi }}}} \\ \end{array} }} \right] . \end{aligned}$$

Then when $n=k+1$, we have

TrCFEWA $(A_1 , A_2 ,..., A_{k+1} )=$ TrCFEWA $(A_1 , A_2 ,..., A_k )\oplus A_{k+1} )$

$$\begin{aligned}&\langle \max (I_A^- )\left[ {\begin{array}{c} \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+p_1^- (h)]^{\varpi }-} \prod _{j=1}^k {[1-p_1^- (h)]^{\varpi }} }{\prod _{j=1}^k {[1+p_1^- (h)]^{\varpi }+} \prod _{j=1}^k {[1-p_1^- (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+q_1^- (h)]^{\varpi }-} \prod _{j=1}^k {[1-q_1^- (h)]^{\varpi }} }{\prod _{j=1}^k {[1+q_1^- (h)]^{\varpi }+} \prod _{j=1}^k {[1-q_1^- (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+r_1^- (h)]^{\varpi }-} \prod _{j=1}^k {[1-r_1^- (h)]^{\varpi }} }{\prod _{j=1}^k {[1+r_1^- (h)]^{\varpi }+} \prod _{j=1}^k {[1-r_1^- (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+s_1^- (h)]^{\varpi }-} \prod _{j=1}^k {[1-s_1^- (h)]^{\varpi }} }{\prod _{j=1}^k {[1+s_1^- (h)]^{\varpi }+} \prod _{j=1}^k {[1-s_1^- (h)]^{\varpi }} } \\ \end{array}} \right] ; \\&\langle \max (I_A^+ )\left[ {\begin{array}{c} \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+p_1^+ (h)]^{\varpi }-} \prod _{j=1}^k {[1-p_1^+ (h)]^{\varpi }} }{\prod _{j=1}^k {[1+p_1^+ (h)]^{\varpi }+} \prod _{j=1}^k {[1-p_1^+ (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+q_1^+ (h)]^{\varpi }-} \prod _{j=1}^k {[1-q_1^+ (h)]^{\varpi }} }{\prod _{j=1}^k {[1+q_1^+ (h)]^{\varpi }+} \prod _{j=1}^k {[1-q_1^+ (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+r_1^+ (h)]^{\varpi }-} \prod _{j=1}^k {[1-r_1^+ (h)]^{\varpi }} }{\prod _{j=1}^k {[1+r_1^+ (h)]^{\varpi }+} \prod _{j=1}^k {[1-r_1^+ (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^k {[1+s_1^+ (h)]^{\varpi }-} \prod _{j=1}^k {[1-s_1^+ (h)]^{\varpi }} }{\prod _{j=1}^k {[1+s_1^+ (h)]^{\varpi }+} \prod _{j=1}^k {[1-s_1^+ (h)]^{\varpi }} } \\ \end{array}} \right] ; \\&\min (\mu _A )\left[ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^k [p_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-p_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [p_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^k [q_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-q_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [q_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^k [r_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-r_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [r_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^k [s_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^k [(2-s_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [s_1 (h)]^{^{\varpi }}}} \\ \end{array} }} \right] \oplus _{k+1} \\&\langle \max (I_A^- )\left[ {\begin{array}{c} \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+p_1^- (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-p_1^- (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+p_1^- (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-p_1^- (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+q_1^- (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-q_1^- (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+q_1^- (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-q_1^- (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+r_1^- (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-r_1^- (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+r_1^- (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-r_1^- (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+s_1^- (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-s_1^- (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+s_1^- (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-s_1^- (h)]^{\varpi }} } \\ \end{array}} \right] ; \\&\langle \max (I_A^+ )\left[ {\begin{array}{c} \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+p_1^+ (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-p_1^+ (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+p_1^+ (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-p_1^+ (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+q_1^+ (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-q_1^+ (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+q_1^+ (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-q_1^+ (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+r_1^+ (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-r_1^+ (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+r_1^+ (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-r_1^+ (h)]^{\varpi }} }, \\ \frac{\mathop {\prod }\nolimits _{j=1}^{k+1} {[1+s_1^+ (h)]^{\varpi }-} \prod _{j=1}^{k+1} {[1-s_1^+ (h)]^{\varpi }} }{\prod _{j=1}^{k+1} {[1+s_1^+ (h)]^{\varpi }+} \prod _{j=1}^{k+1} {[1-s_1^+ (h)]^{\varpi }} } \\ \end{array}} \right] ; \\&\min (\mu _A )\left[ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [p_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-p_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [p_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [q_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-q_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [q_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [r_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-r_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [r_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [s_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-s_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [s_1 (h)]^{^{\varpi }}}} \\ \end{array} }} \right] \\= & {} \max (I_A^- )\left[ {{\begin{array}{c} {\frac{[\mathop {\prod }\nolimits _{j=1}^{k+1} [1+p_1^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^k [1-p_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+p_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-p_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+q_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-q_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+q_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-q_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+r_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-r_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+r_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-r_1^- (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+s_1^- (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-s_1^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+s_1^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-s_1^- (h)]^{^{\varpi }}}} \\ \end{array} }} \right] , \\&\max (I_A^+ )\left[ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+p_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-p_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+p_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-p_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+q_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-q_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+q_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-q_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+r_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-r_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+r_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-r_1^+ (h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+s_1^+ (h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^k [1-s_1^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [1+s_1^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^k [1-s_1^+ (h)]^{^{\varpi }}}} \\ \end{array} }} \right] , \\&\min (\mu _A )\left[ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [p_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-p_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [p_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [q_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-q_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [q_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [r_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-r_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [r_1 (h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^{k+1} [s_1 (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^{k+1} [(2-s_1 (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^{k+1} [s_1 (h)]^{^{\varpi }}}} \\ \end{array} }} \right] . \end{aligned}$$

In particular, if $w=(\frac{1}{n}, \frac{1}{n},...., \frac{1}{n})^{T},$ then the TrCFEWA operator is reduced to the trapezoidal cubic fuzzy Einstein weighing averaging operator, which is shown as follows:

$$\begin{aligned}&\langle \max (I_A^- )\left[ {{\begin{array}{c} {\frac{[\mathop {\prod }\nolimits _{j=1}^n [1+p_1^- (h)]^{\frac{1}{n}}-\mathop {\prod }\nolimits _{j=1}^n [1-p_1^- (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_1^- (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-p_1^- (h)]^{^{\frac{1}{n}}}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^- (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-q_1^- (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^- (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-q_1^- (h)]^{^{\frac{1}{n}}}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^- (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-r_1^- (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^- (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-r_1^- (h)]^{^{\frac{1}{n}}}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^- (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-s_1^- (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^- (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-s_1^- (h)]^{^{\frac{1}{n}}}}} \\ \end{array} }} \right] ; \\&\max (I_A^+ )\left[ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_1^+ (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-p_1^+ (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_1^+ (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-p_1^+ (h)]^{^{\frac{1}{n}}}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^+ (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-q_1^+ (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_1^+ (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-q_1^+ (h)]^{^{\frac{1}{n}}}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^+ (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-r_1^+ (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_1^+ (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-r_1^+ (h)]^{^{\frac{1}{n}}}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^+ (h)]^{^{\frac{1}{n}}}-\mathop {\prod }\nolimits _{j=1}^n [1-s_1^+ (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_1^+ (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [1-s_1^+ (h)]^{^{\frac{1}{n}}}}} \\ \end{array} }} \right] ; \\&\min (\mu _A )\left[ {{\begin{array}{l} {\frac{2\mathop {\prod }\nolimits _{j=1}^n [p_1 (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_1 (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [p_1 (h)]^{^{\frac{1}{n}}}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [q_1 (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_1 (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [q_1 (h)]^{^{\frac{1}{n}}}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [r_1 (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_1 (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [r_1 (h)]^{^{\frac{1}{n}}}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [s_1 (h)]^{^{\frac{1}{n}}}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_1 (h)]^{^{\frac{1}{n}}}+\mathop {\prod }\nolimits _{j=1}^n [s_1 (h)]^{^{\frac{1}{n}}}}} \\ \end{array} }} \right] . \end{aligned}$$

Appendix C: Proof of Proposition 2

• (1) (Idempotency) Since $A_j =A$ are equal to

  $$\begin{aligned} \left\{ {{\begin{array}{l} {\langle [p^{-}(h), q^{-}(h), r^{-}(h), s^{-}(h)],(I_A^- )} \\ {[p^{+}(h), q^{+}(h), r^{+}(h), s^{+}(h)],(I_A^+ )} \\ {[p(h), q(h), r(h), s(h)],(\mu _A )\rangle |h\in H} \\ \end{array} }} \right\} \end{aligned}$$

  for $(j=1, 2,..., n),$ then TrCFEWA

  $$\begin{aligned}&(A_1 , A_2 ,..., A_n )\\&=\langle \max [I_A^- ] \left[ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_j^- (h)]^{\varpi _j }-\mathop {\prod }\nolimits _{j=1}^n [1-p_j^- (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_j^- (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_j^- (h)]^{^{\varpi _j }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_j^- (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-q_j^- (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_j^- (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_j^- (h)]^{^{\varpi _j }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_j^- (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-r_j^- (h)]^{\varpi _j }}{\mathop {\prod }\nolimits _{j=1}^n [1+r_j^- (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_j^- (h)]^{\varpi _j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_j^- (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-s_j^- (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_j^- (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_j^- (h)]^{^{\varpi _j }}}} \\ \end{array} }} \right] ; \\&\max [(I_A^+ ]\left[ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_j^+ (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-p_j^+ (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_j^+ (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_j^+ (h)]^{^{\varpi _j }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_j^+ (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-q_j^+ (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_j^+ (h)]^{^{\varpi _j }}+\mathop {\prod }\limits _{j=1}^n [1-q_j^+ (h)]^{^{\varpi _j }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_j^+ (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-r_j^+ (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_j^+ (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_j^+ (h)]^{^{\varpi _j }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_j^+ (h)]^{^{\varpi _j }}-\mathop {\prod }\nolimits _{j=1}^n [1-s_j^+ (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_j^+ (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_j^+ (h)]^{^{\varpi _j }}}} \\ \end{array} }} \right] ; \\&\min [(\mu _A ]\left[ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^n [p_j (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_j (h)]^{\varpi _j }+\mathop {\prod }\nolimits _{j=1}^n [p_j (h)]^{^{\varpi _j }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [q_j (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_j (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [q_j (h)]^{^{\varpi _j }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [r_j (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_j (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [r_j (h)]^{^{\varpi _j }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [s_j (h)]^{^{\varpi _j }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_j (h)]^{^{\varpi _j }}+\mathop {\prod }\nolimits _{j=1}^n [s_j (h)]^{^{\varpi _j }}}} \\ \end{array} }} \right] \\&=\langle \max [(I_A^- ]\left[ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p^{-}(h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p^{-}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p^{-}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p^{-}(h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q^{-}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-q^{-}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q^{-}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q^{-}(h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r^{-}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-r^{-}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r^{-}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r^{-}(h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s^{-}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-s^{-}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s^{-}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s^{-}(h)]^{^{\varpi }}}} \\ \end{array} }} \right] ; \\&\max [(I_A^+ )]\left[ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p^{+}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-p^{+}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p^{+}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p^{+}(h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+q^{+}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-q^{+}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q^{+}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q^{+}(h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+r^{+}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-r^{+}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r^{+}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r^{+}(h)]^{^{\varpi }}},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n [1+s^{+}(h)]^{^{\varpi }}-\mathop {\prod }\nolimits _{j=1}^n [1-s^{+}(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s^{+}(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s^{+}(h)]^{^{\varpi }}}} \\ \end{array} }} \right] ; \\&\min [(\mu _A )]\left[ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^n [p(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p(h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [p(h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [q(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [q(h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [r(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [r(h)]^{^{\varpi }}},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n [s(h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s(h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [s(h)]^{^{\varpi }}}} \\ \end{array} }} \right] \\&=\langle [I_A^- ]\left[ {{\begin{array}{c} {\frac{[1+p^{-}(h)]^{\varpi }-[1-p^{-}(h)]^{^{\varpi }}}{[1+p^{-}(h)]^{^{\varpi }}+[1-p^{-}(h)]^{^{\varpi }}},} \\ {\frac{[1+q^{-}(h)]^{^{\varpi }}-[1-q^{-}(h)]^{^{\varpi }}}{[1+q^{-}(h)]^{^{\varpi }}+[1-q^{-}(h)]^{^{\varpi }}},} \\ {\frac{[1+r^{-}(h)]^{^{\varpi }}-[1-r^{-}(h)]^{^{\varpi }}}{[1+r^{-}(h)]^{^{\varpi }}+[1-r^{-}(h)]^{^{\varpi }}},} \\ {\frac{[1+s^{-}(h)]^{^{\varpi }}-[1-s^{-}(h)]^{^{\varpi }}}{[1+s^{-}(h)]^{^{\varpi }}+[1-s^{-}(h)]^{^{\varpi }}}} \\ \end{array} }} \right] ; \\&{[}I_A^+ ]\left[ {{\begin{array}{c} {\frac{[1+p^{+}(h)]^{^{\varpi }}-[1-p^{+}(h)]^{^{\varpi }}}{[1+p^{+}(h)]^{^{\varpi }}+[1-p^{+}(h)]^{^{\varpi }}},} \\ {\frac{[1+q^{+}(h)]^{^{\varpi }}-[1-q^{+}(h)]^{^{\varpi }}}{[1+q^{+}(h)]^{^{\varpi }}+[1-q^{+}(h)]^{^{\varpi }}},} \\ {\frac{[1+r^{+}(h)]^{^{\varpi }}-[1-r^{+}(h)]^{^{\varpi }}}{[1+r^{+}(h)]^{^{\varpi }}+[1-r^{+}(h)]^{^{\varpi }}},} \\ {\frac{[1+s^{+}(h)]^{^{\varpi }}-[1-s^{+}(h)]^{^{\varpi }}}{[1+s^{+}(h)]^{^{\varpi }}+[1-s^{+}(h)]^{^{\varpi }}}} \\ \end{array} }} \right] ; \\&{[}(\mu _A )]\left[ {{\begin{array}{c} {\frac{2[p(h)]^{^{\varpi }}}{[(2-p(h)]^{\mu }+[p(h)]^{^{\varpi }}},} \\ {\frac{2[q(h)]^{^{\varpi }}}{[(2-q(h)]^{^{\varpi }}+[q(h)]^{^{\varpi }}},} \\ {\frac{2[r(h)]^{^{\varpi }}}{[(2-r(h)]^{^{\varpi }}+[r(h)]^{^{\varpi }}},} \\ {\frac{2[s(h)]^{^{\varpi }}}{[(2-s(h)]^{^{\varpi }}+[s(h)]^{^{\varpi }}}} \\ \end{array} }} \right] = \\&\left\{ {{\begin{array}{l} {\langle [p^{-}(h), q^{-}(h), r^{-}(h), s^{-}(h)],(I_A^- )} \\ {[p^{+}(h), q^{+}(h), r^{+}(h), s^{+}(h)],(I_{A^{+}} )} \\ {[p(h), q(h), r(h), s(h)],(\mu _A )\rangle |h\in H} \\ \end{array} }} \right\} =A \end{aligned}$$

  TrCFEWA $(A_1 , A_2 ,..., A_n )=A.$ The proof is completed.

• (2) (Boundary): Let $f(x)=\frac{(1-x)}{(1+x)} \quad x\in [0, 1]$; then $\frac{-2}{(1-x)^{2}}<0$; that is, f(x) is a decreasing function. Since $p_{\min }^- \le p_j^- \le p_{\max }^- ,$ then for all j, we have $f(p_{\min }^- )\le f(p_j^- )\le f(p_{\max }^- );$ that is $\frac{1-p_{\max }^- }{1+p_{\max }^- }\le \frac{1-p_j^- }{1+p_j^- }\le \frac{1-p_{\min }^- }{1+p_{\min }^- }$. Let $w=(w_1 , w_2 ,..., w_n )$ be the weight vector of $(A_1 , A_2 ,..., A_n )$, such that $w_j \in [0, 1]$ and $\mathop {\sum }\nolimits _{j=1}^n w_j =1.$ Then, for all $w_j \in [0, 1]$, we have $\left( {\frac{1-p_{\max }^- }{1+p_{\max }^- }} \right) ^{w_j }\le \left( {\frac{1-p_j^- }{1+p_j^- }} \right) ^{w_j }\le \left( {\frac{1-p_{\min }^- }{1+p_{\min }^- }} \right) ^{w_j }$. Thus

  $$\begin{aligned}&\mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_{\max }^- }{1+p_{\max }^- }} \right) ^{w_j }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_j^- }{1+p_j^- }} \right) ^{w_j }\\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_{\min }^- }{1+p_{\min }^- }} \right) ^{w_j } \\&\quad \Leftrightarrow \frac{1-p_{\max }^- }{1+p_{\max }^- }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_j^- }{1+p_j^- }} \right) ^{w_j }\le \frac{1-p_{\min }^- }{1+p_{\min }^- } \\&\quad \Leftrightarrow \frac{2}{1+p_{\max }^- }\le 1+\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_j^- }{1+p_j^- }} \right) ^{w_j }\le \frac{2}{1+p_{\min }^- } \\&\quad \Leftrightarrow \frac{1+p_{\min }^- }{2}\le \frac{1}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{1-p_j^- }{1+p_j^- }} \right) } \right) ^{w_j }}\le \frac{1+p_{\max }^- }{2}\\&\quad \Leftrightarrow 1+p_{\min }^- \le \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{1-p_j^- }{1+p_j^- }} \right) } \right) ^{w_j }}\le 1+p_{\max }^-\\&\quad \Leftrightarrow p_{\min }^- \le \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{1-p_j^- }{1+p_j^- }} \right) } \right) ^{w_j }}-1\le p_{\max }^-\\ \end{aligned}$$

  that is

  $$\begin{aligned} p_{\min }^- \le \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^- } \right) ^{w_j }}\le p_{\max }^- \\ \end{aligned}$$

  Similarly, we have

  $$\begin{aligned} {[}q_{\min }^-\le & {} \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^- } \right) ^{w_j }}\le q_{\max }^- , \\ r_{\min }^-\le & {} \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^- } \right) ^{w_j }}\le r_{\max }^- , \\ s_{\min }^-\le & {} \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^- } \right) ^{w_j }}\le s_{\max }^- ], \\&\max [I_j^- ]A_j^- \\= & {} \max [I_j^- ]\left\{ {{\begin{array}{l} {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^- } \right) ^{w_j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^- } \right) ^{w_j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^- } \right) ^{w_j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^- } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^- } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^- } \right) ^{w_j }}} \\ \end{array} }} \right\} ; \end{aligned}$$

  Let $g(y)=\frac{(1-y)}{(1+y)} \quad y\in [0, 1]$; then $\frac{-2}{(1-y)^{2}}<0;$that is, g(y) is a decreasing function. Since $p_{\min }^+ \le p_j^+ \le p_{\max }^+ ,$ then for all j, we have $g(p_{\min }^+ )\le g(p_j^+ )\le g(p_{\max }^+ );$ that is $\frac{1-p_{\max }^+ }{1+p_{\max }^+ }\le \frac{1-p_j^+ }{1+p_j^+ }\le \frac{1-p_{\min }^+ }{1+p_{\min }^+ }$. Let $w=(w_1 , w_2 ,..., w_n )$ be the weight vector of $(A_1 , A_2 ,..., A_n )$, such that $w_j \in [0, 1]$ and $\mathop {\sum }\nolimits _{j=1}^n w_j =1$. Then, for all $w_j \in [0, 1]$, we have $\left( {\frac{1-p_{\max }^+ }{1+p_{\max }^+ }} \right) ^{w_j }\le \left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) ^{w_j }\le \left( {\frac{1-p_{\min }^+ }{1+p_{\min }^+ }} \right) ^{w_j }$. Thus

  $$\begin{aligned}&\mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_{\max }^+ }{1+p_{\max }^+ }} \right) ^{w_j }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) ^{w_j }\\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_{\min }^+ }{1+p_{\min }^+ }} \right) ^{w_j } \\&\quad \Leftrightarrow \frac{1-p_{\max }^+ }{1+p_{\max }^+ }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) ^{w_j }\le \frac{1-p_{\min }^+ }{1+p_{\min }^+ } \\&\quad \Leftrightarrow \frac{2}{1+p_{\max }^+ }\le 1+\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) ^{w_j }\le \frac{2}{1+p_{\min }^+ }\\&\quad \Leftrightarrow \frac{1+p_{\min }^+ }{2}\le \frac{1}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) } \right) ^{w_j }}\le \frac{1+p_{\max }^+ }{2}\\&\quad \Leftrightarrow 1+p_{\min }^+ \le \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) } \right) ^{w_j }}\le 1+p_{\max }^+\\&\quad \Leftrightarrow p_{\min }^+ \le \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{1-p_j^+ }{1+p_j^+ }} \right) } \right) ^{w_j }}-1\le p_{\max }^+\\ \end{aligned}$$

  that is

  $$\begin{aligned} p_{\min }^+ \le \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^+ } \right) ^{w_j }}\le p_{\max }^+ \end{aligned}$$

  Similarly, we have

  $$\begin{aligned} {[}q_{\min }^+\le & {} \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^+ } \right) ^{w_j }}\le q_{\max }^+ , \\ r_{\min }^+\le & {} \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^+ } \right) ^{w_j }}\le r_{\max }^+ , \\ s_{\min }^+\le & {} \frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^+ } \right) ^{w_j }}\le s_{\max }^+ ], \\&\max [I_j^+ ]A_j^+ \\= & {} \max [I_j^+ ]\left\{ {{\begin{array}{c} {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+p_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-p_j^+ } \right) ^{w_j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+q_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-q_j^+ } \right) ^{w_j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+r_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-r_j^+ } \right) ^{w_j }},} \\ {\frac{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^+ } \right) ^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {1+s_j^+ } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {1-s_j^+ } \right) ^{w_j }}} \\ \end{array} }} \right\} \end{aligned}$$

  and Let $h(z)=\frac{(2-z)}{z}$, $z\in [0,1]$; then $\frac{-2}{z^{2}}<0$; that is, h(z) is a decreasing function. Since $p_{\min } \le p_j \le p_{\max } ,$ then for all j, we have $h(p_{\min } )\le h(p_j )\le h(p_{\max } );$ that is $\frac{2-p_{\max } }{h}\le \frac{2-p_j }{h}\le \frac{2-p_{\min } }{h}$. Let $w=(w_1 , w_2 ,..., w_n )$ be the weight vector of $(A_1 , A_2 ,..., A_n )$, such that $w_j \in [0, 1]$ and $\mathop {\sum }\nolimits _{j=1}^n w_j =1.$ Then, for all $w_j \in [0, 1]$, we have

  $$\begin{aligned} \left( {\frac{2-p_{\max } }{p_{\max } }} \right) ^{w_j }\le \left( {\frac{2-p_j }{p_j }} \right) ^{w_j }\le \left( {\frac{2-p_{\min } }{p_{\min } }} \right) ^{w_j }. \end{aligned}$$

  Thus

  $$\begin{aligned}&\mathop {\prod }\nolimits _{j=1}^n \left( {\frac{2-p_{\max } }{p_{\max }^- }} \right) ^{w_j }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{2-p_j }{p_j }} \right) ^{w_j }\\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{2-p_{\min } }{p_{\min } }} \right) ^{w_j } \\&\quad \Leftrightarrow \frac{2-p_{\max } }{p_{\max } }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{2-p_j }{p_j }} \right) ^{w_j }\le \frac{2-p_{\min } }{p_{\min } } \\&\quad \Leftrightarrow \frac{2}{p_{\max } }\le \mathop {\prod }\nolimits _{j=1}^n \left( {\frac{2-p_j }{p_j }} \right) ^{w_j }+1\le \frac{2}{p_{\min } } \\&\quad \Leftrightarrow \frac{p_{\min } }{2}\le \frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{2-p_j }{p_j }} \right) } \right) ^{w_j }+1}\le \frac{p_{\max } }{2}\\&\quad \Leftrightarrow p_{\min } \le \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left( {\left( {\frac{2-p_j }{p_j }} \right) } \right) ^{w_j }}\le p_{\max } \\ \end{aligned}$$

  that is

  $$\begin{aligned} p_{\min } \le \frac{2\mathop {\prod }\nolimits _{j=1}^n (p_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+p_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {p_j } \right) ^{w_j }}\le p_{\max } \end{aligned}$$

  Similarly, we have

  $$\begin{aligned} q_{\min }\le & {} \frac{2\mathop {\prod }\nolimits _{j=1}^n (q_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+q_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {q_j } \right) ^{w_j }}\le q_{\max } , \\ r_{\min }\le & {} \frac{2\mathop {\prod }\nolimits _{j=1}^n (r_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+r_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {r_j } \right) ^{w_j }}\le r_{\max } , \\ s_{\min }\le & {} \frac{2\mathop {\prod }\nolimits _{j=1}^n (s_j )^{w_j }}{\mathop {\prod }\limits _{j=1}^n \left( {2+s_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {s_j } \right) ^{w_j }}\le s_{\max } \\ \end{aligned}$$

  That is $\min [\mu _j ]A_j =$

  $$\begin{aligned} \min [\mu _j ]\left\{ {{\begin{array}{c} {\frac{2\mathop {\prod }\nolimits _{j=1}^n (p_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+p_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {p_j } \right) ^{w_j }},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n (q_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+q_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {q_j } \right) ^{w_j }},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n (r_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+r_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {r_j } \right) ^{w_j }},} \\ {\frac{2\mathop {\prod }\nolimits _{j=1}^n (s_j )^{w_j }}{\mathop {\prod }\nolimits _{j=1}^n \left( {2+s_j } \right) ^{w_j }-\mathop {\prod }\nolimits _{j=1}^n \left( {s_j } \right) ^{w_j }}} \\ \end{array} }} \right\} . \end{aligned}$$

  The proof is completed

• (3) (Monotonicity)

  Since

  $$\begin{aligned} \{p_A^- (h)\le & {} p_B^- (h),\\ q_A^- (h)\le & {} q_B^- (h), r_A^- (h)\le r_B^- (h), \\ s_A^- (h)\le & {} s_B^- (h)],(I_A^- )\le (I_B^- )\}; \\ \{p_A^+ (h)\le & {} p_B^+ (h), q_A^+ (h)\le q_B^+ (h), \\ r_A^+ (h)\le & {} r_B^+ (h), s_A^+ (h)\le s_B^+ (h)], \\ (I_A^+ )\le & {} (I_B^+ )\} \end{aligned}$$

  and

  $$\begin{aligned} \{p_A (h)\le & {} p_B (h), q_A (h)\le q_B (h), \\ r_A (h)\le & {} r_B (h), s_A (h)\le s_B (h)], \\ (\mu _A )\le & {} (\mu _B )\} \end{aligned}$$

  Since

  $$\begin{aligned} \frac{1+p_A^- (h)}{1+p_A^- (h)}\le & {} \frac{1+p_B^- (h)}{1+p_B^- (h)}, \\ \frac{1+q_A^- (h)}{1+q_A^- (h)}\le & {} \frac{1+q_B^- (h)}{1+q_B^- (h)}, \\ \frac{1+r_A^- (h)}{1+r_A^- (h)}\le & {} \frac{1+r_B^- (h)}{1+r_B^- (h)}, \\ \frac{1+s_A^- (h)}{1+s_A^- (h)}\le & {} \frac{1+s_B^- (h)}{1+s_B^- (h)}, \\ \max \{(I_A^- )\le & {} (I_B^- )\} \\ \frac{1+p_A^+ (h)}{1+p_A^+ (h)}\le & {} \frac{1+p_B^+ (h)}{1+p_B^+ (h)}, \\ \frac{1+q_A^+ (h)}{1+q_A^+ (h)}\le & {} \frac{1+q_B^+ (h)}{1+q_B^+ (h)}, \\ \frac{1+r_A^+ (h)}{1+r_A^+ (h)}\le & {} \frac{1+r_B^+ (h)}{1+r_B^+ (h)}, \\ \frac{1+s_A^+ (h)}{1+s_A^+ (h)}\le & {} \frac{1+s_B^+ (h)}{1+s_B^+ (h)}, \end{aligned}$$
  $$\begin{aligned}&\max \{(I_A^+ )\le (I_B^+ )\}\\&\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_A^- (h)}{1+p_A^- (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_B^- (h)}{1+p_B^- (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_A^- (h)}{1+q_A^- (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_B^- (h)}{1+q_B^- (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_A^- (h)}{1+r_A^- (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_B^- (h)}{1+r_B^- (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_A^- (h)}{1+s_A^- (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_B^- (h)}{1+s_B^- (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n (I_A^- )\le \mathop {\prod }\nolimits _{j=1}^n (I_B^- ); \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_A^+ (h)}{1+p_A^+ (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_B^+ (h)}{1+p_B^+ (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_A^+ (h)}{1+q_A^+ (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_B^+ (h)}{1+q_B^+ (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_A^+ (h)}{1+r_A^+ (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_B^+ (h)}{1+r_B^+ (h)}} \right\} ^{w_j } \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_A^+ (h)}{1+s_A^+ (h)}} \right\} ^{w_j } \\&\quad \le \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_B^+ (h)}{1+s_B^+ (h)}} \right\} ^{w_j }, \\&\quad \mathop {\prod }\nolimits _{j=1}^n (I_A^+ )\le \mathop {\prod }\nolimits _{j=1}^n (I_B^+ ); \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_A^- (h)}{1+p_A^- (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_B^- (h)}{1+p_B^- (h)}} \right\} ^{w_j }}, \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_A^- (h)}{1+q_A^- (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_B^- (h)}{1+q_B^- (h)}} \right\} ^{w_j }}, \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_A^- (h)}{1+r_A^- (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_B^- (h)}{1+r_B^- (h)}} \right\} ^{w_j }}, \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_A^- (h)}{1+s_A^- (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_B^- (h)}{1+s_B^- (h)}} \right\} ^{w_j }}, \\&\quad \mathop {\prod }\nolimits _{j=1}^n (I_A^- )\le \mathop {\prod }\nolimits _{j=1}^n (I_B^- ); \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_A^+ (h)}{1+p_A^+ (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+p_B^+ (h)}{1+p_B^+ (h)}} \right\} ^{w_j }}, \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_A^+ (h)}{1+q_A^+ (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+q_B^+ (h)}{1+q_B^+ (h)}} \right\} ^{w_j }}, \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_A^+ (h)}{1+r_A^+ (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+r_B^+ (h)}{1+r_B^+ (h)}} \right\} ^{w_j }}, \\&\quad \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_A^+ (h)}{1+s_A^+ (h)}} \right\} ^{w_j }} \\&\quad \ge \frac{2}{1+\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{1+s_B^+ (h)}{1+s_B^+ (h)}} \right\} ^{w_j }}, \\&\quad \mathop {\prod }\nolimits _{j=1}^n (I_A^+ )\le \mathop {\prod }\nolimits _{j=1}^n (I_B^+ ). \end{aligned}$$

  We have

  $$\begin{aligned}&\frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_A^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p_A^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_A^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_A^- (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_B^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p_B^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_B^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_B^- (h)]^{^{\varpi }}}, \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_A^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-q_A^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_A^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_A^- (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_B^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-q_B^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_B^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_B^- (h)]^{^{\varpi }}}, \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_A^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-r_A^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_A^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_A^- (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_B^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-r_B^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_B^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_B^- (h)]^{^{\varpi }}}, \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_A^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-s_A^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_A^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_A^- (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_B^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-s_B^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_B^- (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_B^- (h)]^{^{\varpi }}}, \\&\quad \max \{(I_A^- )\ge (I_B^- )\} \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_A^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p_A^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_A^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_A^+ (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+p_B^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-p_B^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+p_B^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-p_B^+ (h)]^{^{\varpi }}}, \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_A^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-q_A^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_A^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_A^+ (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+q_B^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-q_B^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+q_B^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-q_B^+ (h)]^{^{\varpi }}}, \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_A^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-r_A^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_A^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_A^+ (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+r_B^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-r_B^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+r_B^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-r_B^+ (h)]^{^{\varpi }}}, \\&\quad \frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_A^+ (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-s_A^+ (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_A^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_A^+ (h)]^{^{\varpi }}} \\&\quad \ge \frac{\mathop {\prod }\nolimits _{j=1}^n [1+s_B^- (h)]^{\varpi }-\mathop {\prod }\nolimits _{j=1}^n [1-s_B^- (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [1+s_B^+ (h)]^{^{\varpi }}+\mathop {\prod }\nolimits _{j=1}^n [1-s_B^+ (h)]^{^{\varpi }}}, \\&\quad \max \{(I_A^+ )\ge (I_B^+ )\} \end{aligned}$$

  Since

  $$\begin{aligned} \frac{2-p_B (h)}{p_B (h)}\ge & {} \frac{2-p_A (h)}{p_A (h)}, \\ \frac{2-q_B (h)}{q_B (h)}\ge & {} \frac{2-q_A (h)}{q_A (h)}, \\ \frac{2-r_B (h)}{r_B (h)}\ge & {} \frac{2-r_A (h)}{r_A (h)}, \\ \frac{2-s_B (h)}{s_B (h)}\ge & {} \frac{2-s_A (h)}{s_A (h)}, \min \{(\mu _A )\ge (\mu _B )\} \end{aligned}$$

  then $\left\{ {\frac{2-p_B (h)}{p_B (h)}} \right\} ^{w_j }\ge \left\{ {\frac{2-p_A (h)}{p_A (h)}} \right\} ^{w_j }$

  $$\begin{aligned}&\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-p_B (h)}{p_B (h)}} \right\} ^{w_j }\ge \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-p_A (h)}{p_A (h)}} \right\} ^{w_j } \\&\quad \frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-p_B (h)}{p_B (h)}} \right\} ^{w_j }+1}\frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-p_A (h)}{p_A (h)}} \right\} ^{w_j }+1}; \\&\quad \left\{ {\frac{2-q_B (h)}{q_B (h)}} \right\} ^{w_j }\ge \\&\quad \left\{ {\frac{2-q_A (h)}{q_A (h)}} \right\} ^{w_j }\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-q_B (h)}{q_B (h)}} \right\} ^{w_j }\ge \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-q_A (h)}{q_A (h)}} \right\} ^{w_j } \\&\quad \frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-q_B (h)}{q_B (h)}} \right\} ^{w_j }+1}\frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-q_A (h)}{q_A (h)}} \right\} ^{w_j }+1}; \\&\quad \left\{ {\frac{2-r_B (h)}{r_B (h)}} \right\} ^{w_j }\ge \\&\quad \left\{ {\frac{2-r_A (h)}{r_A (h)}} \right\} ^{w_j }\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-r_B (h)}{r_B (h)}} \right\} ^{w_j }\ge \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-r_A (h)}{r_A (h)}} \right\} ^{w_j } \\&\quad \frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-r_B (h)}{r_B (h)}} \right\} ^{w_j }+1}\frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-r_A (h)}{r_A (h)}} \right\} ^{w_j }+1}; \\&\quad \left\{ {\frac{2-s_B (h)}{s_B (h)}} \right\} ^{w_j }\ge \\&\quad \left\{ {\frac{2-s_A (h)}{s_A (h)}} \right\} ^{w_j }\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-s_B (h)}{s_B (h)}} \right\} ^{w_j }\ge \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-s_A (h)}{s_A (h)}} \right\} ^{w_j } \\&\quad \frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-s_B (h)}{s_B (h)}} \right\} ^{w_j }+1}\frac{1}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-s_A (h)}{s_A (h)}} \right\} ^{w_j }+1}, \\&\quad \min \{(\mu _A )\ge (\mu _B )\} \\&\quad \mathop {\prod }\nolimits _{j=1}^n \left\{ {(\mu _A )} \right\} ^{w_j }\ge \mathop {\prod }\nolimits _{j=1}^n \left\{ {(\mu _B )} \right\} ^{w_j }. \\&\quad \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-p_B (h)}{p_B (h)}} \right\} ^{w_j }+1}\le \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-p_B (h)}{p_B (h)}} \right\} ^{w_j }+1}; \\&\quad \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-q_B (h)}{q_B (h)}} \right\} ^{w_j }+1}\le \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-q_B (h)}{q_B (h)}} \right\} ^{w_j }+1}, \\&\quad \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-r_B (h)}{r_B (h)}} \right\} ^{w_j }+1}\le \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-r_B (h)}{r_B (h)}} \right\} ^{w_j }+1}, \\&\quad \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-s_B (h)}{s_B (h)}} \right\} ^{w_j }+1}\le \frac{2}{\mathop {\prod }\nolimits _{j=1}^n \left\{ {\frac{2-s_B (h)}{s_B (h)}} \right\} ^{w_j }+1}. \\&\quad \min [\mu _A ]\le \min [\mu _B ]; \\&\quad \frac{2\mathop {\prod }\nolimits _{j=1}^n [p_B (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_B (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [p_B (h)]^{^{\varpi }}} \\&\quad \le \frac{2\mathop {\prod }\nolimits _{j=1}^n [p_A (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-p_A (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [p_A (h)]^{^{\varpi }}}, \\&\quad \frac{2\mathop {\prod }\nolimits _{j=1}^n [q_B (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_B (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [q_B (h)]^{^{\varpi }}} \\&\quad \le \frac{2\mathop {\prod }\nolimits _{j=1}^n [q_A (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-q_A (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [q_A (h)]^{^{\varpi }}}, \\&\quad \frac{2\mathop {\prod }\nolimits _{j=1}^n [r_B (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_B (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [r_B (h)]^{^{\varpi }}} \\&\quad \le \frac{2\mathop {\prod }\nolimits _{j=1}^n [r_A (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-r_A (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [r_A (h)]^{^{\varpi }}}, \\&\quad \frac{2\mathop {\prod }\nolimits _{j=1}^n [s_B (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_B (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [s_B (h)]^{^{\varpi }}} \\&\quad \le \frac{2\mathop {\prod }\nolimits _{j=1}^n [s_A (h)]^{^{\varpi }}}{\mathop {\prod }\nolimits _{j=1}^n [(2-s_A (h)]^{\mu }+\mathop {\prod }\nolimits _{j=1}^n [s_A (h)]^{^{\varpi }}}, \\ \end{aligned}$$

  We can get TrCFEWA $(A_1 , A_2 ,..., A_n )\le$ TrCFEWA $(B_1 , B_2 ,..., B_n ),$ which complete the proof . $\square$