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.
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References
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Atanassov KT (1994) New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst 61:137–142
Beliakov G, Pradera A, Calvo T (2007) Aggregation functions: a guide for practitioners. Springer, Heidelberg, Berlin, NewYork
Bustince H, Burillo P (1996) Structures on intuitionistic fuzzy relations. Fuzzy Sets Syst 78:293–303
Deschrijver G, Kerre EE (2003) On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst 133:227–235
Deschrijver G, Kerre EE (2007) On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision. Inf Sci 177:1860–1866
Fahmi A, Abdullah S, Amin F, Siddiqui N, Ali A (2017) Aggregation operators on triangular cubic fuzzy numbers and its application to multi-criteria decision making problems. J Intell Fuzzy Syst 33:3323–3337
Fahmi A, Abdullah S, Amin F, Ali A (2017) Precursor Selection for Sol-Gel Synthesis of Titanium Carbide Nanopowders by a New Cubic Fuzzy Multi-Attribute Group Decision-Making Model. Journal of Intelligent Systems. from. https://doi.org/10.1515/jisys-2017-0083
Fahmi A, Abdullah S, Amin F, Ali A (2018) Weighted average rating (war) method for solving group decision making problem using triangular cubic fuzzy hybrid aggregation (Tcfha). Punjab Univ J Math 50:23–34
Grabisch M, Marichal JL, Mesiar R, Pap E (2009) Aggregation functions. Cambridge University Press, Cambridge
Jun YB, Kim CS, Yang Ki O (2011) Annals of fuzzy mathematics and informatics. Cubic Sets 4:83–98
Liu HW, Wang GJ (2007) Multi-criteria decision-making methods based on intuitionistic fuzzy sets. Eur J Operat Res 179:220–233
Liu Y, Wu J, Liang C (2017) Some Einstein aggregating operators for trapezoidal intuitionistic fuzzy MAGDM and application in investment evolution. J Intell Fuzzy Syst 32:63–74
Turksen IB (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20:191–210
Wang W, Liu X (2013) Some operations over Atanassov’s intuitionistic fuzzy sets based on Einstein t-norm and t-conorm. Int J Uncertain Fuzziness Knowl Based Syst 21:263–276
Wang J, Zhang Z (2009) Multi-criteria decision-making method with incomplete certain information based on intuitionistic fuzzy number. Control Decis 24:226–230
Wu J, Cao QW (2013) Same families of geometric aggregation operators with intuitionistic trapezoidal fuzzy numbers. Appl Math Model 37:318–327
Wu J, Liu Y (2013) An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers. Comput Ind Eng 66:311–324
Xu Z, Cai X (2010) Recent advances in intuitionistic fuzzy information aggregation. Fuzzy Optim Decis Mak 9:359–381
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Zadeh LA (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 1:28–44
Zhang X, Jin F, Liu P (2013) A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number. Appl Math Model 37:3467–3477
Zhang W, Li X, Ju Y (2014) Some aggregation operators based on Einstein operations under interval-valued dual hesitant fuzzy setting and their application. Math Probl Eng. https://doi.org/10.1155/2014/958927
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Appendices
Appendix A: Proof of Proposition 1
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(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 \(A_1 +A_2 =A_2 +A_1\).
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(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 \(\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1.\)
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(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 \((A_1 , A_2 ,..., A_n )=\mathop {\oplus }\nolimits _{j=1}^k w_1 A_1 \)
and we have
so, we have \(\lambda (A_1 +A_2 )=\lambda A_2 +\lambda A_1\).
and
Assume that \(n=k,\) TrCFEWA \((A_1 , A_2 ,..., A_n )=\mathop {\oplus }\nolimits _{j=1}^k w_j A_j \)
Then when \(n=k+1\), we have
TrCFEWA \((A_1 , A_2 ,..., A_{k+1} )=\) TrCFEWA \((A_1 , A_2 ,..., A_k )\oplus A_{k+1} )\)
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:
Appendix C: Proof of Proposition 2
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(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.
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(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 . \(\square \)
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Fahmi, A., Abdullah, S., Amin, F. et al. Trapezoidal cubic fuzzy number Einstein hybrid weighted averaging operators and its application to decision making. Soft Comput 23, 5753–5783 (2019). https://doi.org/10.1007/s00500-018-3242-6
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DOI: https://doi.org/10.1007/s00500-018-3242-6