T-norms and T-conorms hesitant fuzzy Einstein aggregation operator and its application to decision making

Abstract

In this paper, we define some Einstein operations on TrCHFS and develop three arithmetic averaging operators, that is TrCHFEWA operator, TrCHFEOWA operator, and TrCHFEHWA operator, for aggregating trapezoidal cubic fuzzy information. The TrCHFEHWA operator generalizes both the TrCHFEWA and TrCHFEOWA 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 TrCHFEHWA operator to multiple attribute decision making with trapezoidal cubic hesitant fuzzy information. Further, a numerical example is provided to illustrate the flexibility and accuracy of the proposed operators. Last, the proposed methods are compared with the existing methods to examine the best developing skill initiatives.

Appendices

Appendix 1: Proof of Proposition 1

(1) $A_{1} + A_{2} = A_{2} + A_{1} ;$

$$ \begin{aligned} & A_{1} + A_{2} = \mathop \cup \limits_{\begin{aligned} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } , \hfill \\ \varGamma_{1}^{ - } \in I_{1}^{ - } ,\,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } ,p_{2}^{ + } \in \varsigma_{2}^{ + } ,\,q_{1}^{ + } \in \chi_{1}^{ + } ,\,q_{2}^{ + } \in \chi_{2}^{ + } ,s_{1}^{ + } \in \psi_{1}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } , \hfill \\ r_{2}^{ + } \in \upsilon_{2}^{ + } ,\,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,\varGamma_{2}^{ + } \in I_{2}^{ + } ,p_{1} \in \varsigma_{1} ,\,p_{2} \in \varsigma_{2} ,\,q_{1} \in \chi_{1} ,q_{2} \in \chi_{2} ,\,s_{1} \in \psi_{1} ,\,s_{2} \in \psi_{2} , \hfill \\ r_{1} \in \upsilon_{1} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{1} \in I_{1} ,\,\varGamma_{2} \in I_{2} \hfill \\ \end{aligned} } \\ & \quad \langle \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ - } ,\,I_{{A_{2} }}^{ - } )[\tfrac{{p_{1}^{ - } (h) + p_{2}^{ - } (h)}}{{(1 + p_{1}^{ - } (h))(1 - p_{2}^{ - } (h))}},\,\tfrac{{q_{1}^{ - } (h) + q_{2}^{ - } (h)}}{{(1 + q_{1}^{ - } (h))(1 - q_{2}^{ - } (h))}},} \\ {\tfrac{{r_{1}^{ - } (h) + r_{2}^{ - } (h)}}{{(1 + r_{1}^{ - } (h))(1 - r_{2}^{ - } (h))}},\,\tfrac{{s_{1}^{ - } (h) + s_{2}^{ - } (h)}}{{(1 + s_{1}^{ - } (h))(1 - s_{2}^{ - } (h))}}],} \\ {\hbox{max} (I_{{A_{1} }}^{ + } ,\,I_{{A_{2} }}^{ + } )[\tfrac{{p_{1}^{ + } (h) + p_{2}^{ + } (h)}}{{(1 + p_{1}^{ + } (h))(1 - p_{2}^{ + } (h))}},\,\tfrac{{q_{1}^{ + } (h) + q_{2}^{ + } (h)}}{{(1 + q_{1}^{ + } (h))(1 - q_{2}^{ + } (h))}},} \\ {\tfrac{{r_{1}^{ + } (h) + r_{2}^{ + } (h)}}{{(1 + r_{1}^{ + } (h))(1 - r_{2}^{ + } (h))}},\,\tfrac{{s_{1}^{ + } (h) + s_{2}^{ + } (h)}}{{(1 + s_{1}^{ + } (h))(1 - s_{2}^{ + } (h))}}]} \\ \end{array} } \right], \\ \end{aligned} $$

$$ \left[ {\begin{array}{*{20}c} {\hbox{min} (I_{{A_{1} }} ,\,I_{{A_{2} }} )\left[ {\tfrac{{p_{1} (h)p_{2} (h)}}{{1 - ((1 - p_{1} (h))(1 - p_{2} (h)))}},\,\tfrac{{q_{1} (h)q_{2} (h)}}{{1 - ((1 - q_{1} (h))(1 - q_{2} (h)))}}} \right.,} \\ {\left. {\tfrac{{r_{1} (h)r_{2} (h)}}{{1 - ((1 - r_{1} (h))(1 - r_{2} (h)))}},\,\tfrac{{s_{1} (h)s_{2} (h)}}{{1 - ((1 - s_{1} (h))(1 - s_{2} (h)))}}} \right]} \\ \end{array} } \right] $$

$$ \begin{aligned} = \mathop \cup \limits_{\begin{aligned} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } , \hfill \\ \varGamma_{1}^{ - } \in I_{1}^{ - } ,\,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } ,p_{2}^{ + } \in \varsigma_{2}^{ + } ,\,q_{1}^{ + } \in \chi_{1}^{ + } ,\,q_{2}^{ + } \in \chi_{2}^{ + } , \hfill \\ s_{1}^{ + } \in \psi_{1}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,r_{2}^{ + } \in \upsilon_{2}^{ + } ,\,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,\varGamma_{2}^{ + } \in I_{2}^{ + } , \hfill \\ p_{1} \in \varsigma_{1} ,\,p_{2} \in \varsigma_{2} ,\,q_{1} \in \chi_{1} ,q_{2} \in \chi_{2} ,\,s_{1} \in \psi_{1} ,\,s_{2} \in \psi_{2} ,r_{1} \in \upsilon_{1} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{1} \in I_{1} ,\,\varGamma_{2} \in I_{2} \hfill \\ \end{aligned} } \hfill \\ \langle \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{2} }}^{ - } ,\,I_{{A_{1} }}^{ - } )\left[ {\tfrac{{p_{2}^{ - } (h) + p_{1}^{ - } (h)}}{{(1 + p_{2}^{ - } (h))(1 - p_{1}^{ - } (h))}},\,\tfrac{{q_{2}^{ - } (h) + q_{1}^{ - } (h)}}{{(1 + q_{2}^{ - } (h))(1 - q_{1}^{ - } (h))}}} \right.,} \\ {\left. {\tfrac{{r_{2}^{ - } (h) + r_{1}^{ - } (h)}}{{(1 + r_{2}^{ - } (h))(1 - r_{1}^{ - } (h))}},\,\tfrac{{s_{2}^{ - } (h) + s_{1}^{ - } (h)}}{{(1 + s_{2}^{ - } (h))(1 - s_{1}^{ - } (h))}}} \right],} \\ {\hbox{max} (I_{{A_{1} }}^{ + } ,\,I_{{A_{2} }}^{ + } )\left[ {\tfrac{{p_{2}^{ + } (h) + p_{1}^{ + } (h)}}{{(1 + p_{2}^{ + } (h))(1 - p_{1}^{ + } (h))}},\,\tfrac{{q_{2}^{ + } (h) + q_{1}^{ + } (h)}}{{(1 + q_{2}^{ + } (h))(1 - q_{1}^{ + } (h))}}} \right.,} \\ {\left. {\tfrac{{r_{2}^{ + } (h) + r_{1}^{ + } (h)}}{{(1 + r_{2}^{ + } (h))(1 - r_{1}^{ + } (h))}},\,\tfrac{{s_{2}^{ + } (h) + s_{1}^{ + } (h)}}{{(1 + s_{2}^{ + } (h))(1 - s_{1}^{ + } (h))}}} \right]} \\ \end{array} } \right], \hfill \\ \end{aligned} $$

$$ \left[ {\begin{array}{*{20}c} {\hbox{min} (I_{{A_{2} }} ,\,I_{{A_{1} }} )\left[ {\tfrac{{p_{2} (h)p_{1} (h)}}{{1 - ((1 - p_{2} (h))(1 - p_{1} (h)))}},\,\tfrac{{q_{2} (h)q_{1} (h)}}{{1 - ((1 - q_{2} (h))(1 - q_{1} (h)))}}} \right.,} \\ {\left. {\tfrac{{r_{2} (h)r_{1} (h)}}{{1 - ((1 - r_{2} (h))(1 - r_{1} (h)))}},\,\tfrac{{s_{2} (h)s_{1} (h)}}{{1 - ((1 - s_{2} (h))(1 - s_{1} (h)))}}} \right]} \\ \end{array} } \right] $$

$$ = A_{2} + A_{1} $$

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} ) = \mathop \cup \limits_{\begin{aligned} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } , \hfill \\ \varGamma_{1}^{ - } \in I_{1}^{ - } ,\,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } ,p_{2}^{ + } \in \varsigma_{2}^{ + } ,\,q_{1}^{ + } \in \chi_{1}^{ + } ,\,q_{2}^{ + } \in \chi_{2}^{ + } , \hfill \\ s_{1}^{ + } \in \psi_{1}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,r_{2}^{ + } \in \upsilon_{2}^{ + } ,\,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,\varGamma_{2}^{ + } \in I_{2}^{ + } , \hfill \\ p_{1} \in \varsigma_{1} ,\,p_{2} \in \varsigma_{2} ,\,q_{1} \in \chi_{1} ,q_{2} \in \chi_{2} ,\,s_{1} \in \psi_{1} ,\,s_{2} \in \psi_{2} ,r_{1} \in \upsilon_{1} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{1} \in I_{1} ,\,\varGamma_{2} \in I_{2} \hfill \\ \end{aligned} } \\ & \quad \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ - } ,\,I_{{A_{2} }}^{ - } )} \\ {\left[ {\tfrac{{[(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.,} \\ {\tfrac{{[(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 } }},} \\ {\tfrac{{[(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. {\tfrac{{[(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., \\ & \quad \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ + } ,\,I_{{A_{2} }}^{ + } )} \\ {\left[ {\tfrac{{[(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.,} \\ {\tfrac{{[(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 } }},} \\ {\tfrac{{[(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. {\tfrac{{[(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], \\ & \quad \left. {\left[ {\begin{array}{*{20}c} {\hbox{min} (I_{{A_{1} }} ,\,I_{{A_{2} }} )} \\ {\left[ {\tfrac{{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.,} \\ {\tfrac{{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 } }},} \\ {\tfrac{{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. {\tfrac{{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} = \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } \\ & \quad \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ - } )} \\ {\left[ {\tfrac{{[1 + p_{1}^{ - } (h)]^{\lambda } - [1 - p_{1}^{ - } (h)]^{\lambda } }}{{[1 + p_{1}^{ - } (h)]^{\lambda } + [1 - p_{1}^{ - } (h)]^{\lambda } }}} \right.,} \\ {\tfrac{{[1 + q_{1}^{ - } (h)]^{\lambda } - [1 - q_{1}^{ - } (h)]^{\lambda } }}{{[1 + q_{1}^{ - } (h)]^{\lambda } + [1 - q_{1}^{ - } (h)]^{\lambda } }},} \\ {\tfrac{{[1 + r_{1}^{ - } (h)]^{\lambda } - [1 - r_{1}^{ - } (h)]^{\lambda } }}{{[1 + r_{1}^{ - } (h)]^{\lambda } + [1 - r_{1}^{ - } (h)]^{\lambda } }},} \\ {\left. {\tfrac{{[1 + s_{1}^{ - } (h)]^{\lambda } - [1 - s_{1}^{ - } (h)]^{\lambda } }}{{[1 + s_{1}^{ - } (h)]^{\lambda } + [1 - s_{1}^{ - } (h)]^{\lambda } }}} \right]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ + } )} \\ {\left[ {\tfrac{{[1 + p_{1}^{ + } (h)]^{\lambda } - [1 - p_{1}^{ + } (h)]^{\lambda } }}{{[1 + p_{1}^{ + } (h)]^{\lambda } + [1 - p_{1}^{ + } (h)]^{\lambda } }}} \right.,} \\ {\tfrac{{[1 + q_{1}^{ + } (h)]^{\lambda } - [1 - q_{1}^{ + } (h)]^{\lambda } }}{{[1 + q_{1}^{ + } (h)]^{\lambda } + [1 - q_{1}^{ + } (h)]^{\lambda } }},} \\ {\tfrac{{[1 + r_{1}^{ + } (h)]^{\lambda } - [1 - r_{1}^{ + } (h)]^{\lambda } }}{{[1 + r_{1}^{ + } (h)]^{\lambda } + [1 - r_{1}^{ + } (h)]^{\lambda } }},} \\ {\left. {\tfrac{{[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}{*{20}c} {\hbox{min} (I_{{A_{1} }} )\left[ {\tfrac{{2[p_{1} (h)]^{\lambda } }}{{[(2 - p_{1} (h)]^{\lambda } + [p_{1} (h)]^{\lambda } }},\,\tfrac{{2[q_{1} (h)]^{\lambda } }}{{[(2 - q_{1} (h)]^{\lambda } + [q_{1} (h)]^{\lambda } }}} \right.,} \\ {\left. {\tfrac{{2[r_{A} (h)]^{\lambda } }}{{[(2 - r_{1} (h)]^{\lambda } + [r_{1} (h)]^{\lambda } }},\,\tfrac{{2[s_{1} (h)]^{\lambda } }}{{[(2 - s_{1} (h)]^{\lambda } + [s_{1} (h)]^{\lambda } }}} \right]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

$$ \begin{aligned} & \lambda A_{2} = \mathop \cup \limits_{\begin{subarray}{l} p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } ,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{2}^{ + } \in \varsigma_{2}^{ + } , \\ q_{2}^{ + } \in \chi_{2}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{2}^{ + } \in \upsilon_{2}^{ + } ,\varGamma_{2}^{ + } \in I_{2}^{ + } ,\,p_{2} \in \varsigma_{2} ,\,q_{2} \in \chi_{2} , \\ s_{2} \in \psi_{2} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{2} \in I_{2} \end{subarray} } \\ & \quad \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{2} }}^{ - } )} \\ {\left[ {\tfrac{{[1 + p_{2}^{ - } (h)]^{\lambda } - [1 - p_{2}^{ - } (h)]^{\lambda } }}{{[1 + p_{2}^{ - } (h)]^{\lambda } + [1 - p_{2}^{ - } (h)]^{\lambda } }}} \right.,} \\ {\tfrac{{[1 + q_{2}^{ - } (h)]^{\lambda } - [1 - q_{2}^{ - } (h)]^{\lambda } }}{{[1 + q_{2}^{ - } (h)]^{\lambda } + [1 - q_{2}^{ - } (h)]^{\lambda } }},} \\ {\tfrac{{[1 + r_{2}^{ - } (h)]^{\lambda } - [1 - r_{2}^{ - } (h)]^{\lambda } }}{{[1 + r_{2}^{ - } (h)]^{\lambda } + [1 - r_{2}^{ - } (h)]^{\lambda } }},} \\ {\left. {\tfrac{{[1 + s_{2}^{ - } (h)]^{\lambda } - [1 - s_{2}^{ - } (h)]^{\lambda } }}{{[1 + s_{2}^{ - } (h)]^{\lambda } + [1 - s_{2}^{ - } (h)]^{\lambda } }}} \right]} \\ \end{array} } \right],\,\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{2} }}^{ + } )} \\ {\left[ {\tfrac{{[1 + p_{2}^{ + } (h)]^{\lambda } - [1 - p_{2}^{ + } (h)]^{\lambda } }}{{[1 + p_{2}^{ + } (h)]^{\lambda } + [1 - p_{2}^{ + } (h)]^{\lambda } }}} \right.,} \\ {\tfrac{{[1 + q_{2}^{ + } (h)]^{\lambda } - [1 - q_{2}^{ + } (h)]^{\lambda } }}{{[1 + q_{2}^{ + } (h)]^{\lambda } + [1 - q_{2}^{ + } (h)]^{\lambda } }},} \\ {\tfrac{{[1 + r_{2}^{ + } (h)]^{\lambda } - [1 - r_{2}^{ + } (h)]^{\lambda } }}{{[1 + r_{2}^{ + } (h)]^{\lambda } + [1 - r_{2}^{ + } (h)]^{\lambda } }},} \\ {\left. {\tfrac{{[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}{*{20}c} {\hbox{min} (I_{{A_{2} }} )\left[ {\tfrac{{2[p_{2} (h)]^{\lambda } }}{{[(2 - p_{2} (h)]^{\lambda } + [p_{2} (h)]^{\lambda } }},\,\tfrac{{2[q_{2} (h)]^{\lambda } }}{{[(2 - q_{2} (h)]^{\lambda } + [q_{2} (h)]^{\lambda } }}} \right.,} \\ {\left. {\tfrac{{2[r_{2} (h)]^{\lambda } }}{{[(2 - r_{2} (h)]^{\lambda } + [r_{2} (h)]^{\lambda } }},\,\tfrac{{2[s_{2} (h)]^{\lambda } }}{{[(2 - s_{2} (h)]^{\lambda } + [s_{2} (h)]^{\lambda } }}} \right]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

$$ \begin{aligned} & \lambda A_{2} + \lambda A_{1} = \mathop \cup \limits_{\begin{aligned} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } , \hfill \\ \varGamma_{1}^{ - } \in I_{1}^{ - } ,\,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } ,p_{2}^{ + } \in \varsigma_{2}^{ + } ,\,q_{1}^{ + } \in \chi_{1}^{ + } ,\,q_{2}^{ + } \in \chi_{2}^{ + } ,s_{1}^{ + } \in \psi_{1}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } , \hfill \\ r_{2}^{ + } \in \upsilon_{2}^{ + } ,\,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,\varGamma_{2}^{ + } \in I_{2}^{ + } ,p_{1} \in \varsigma_{1} ,\,p_{2} \in \varsigma_{2} ,\,q_{1} \in \chi_{1} ,q_{2} \in \chi_{2} ,\,s_{1} \in \psi_{1} ,\,s_{2} \in \psi_{2} , \hfill \\ r_{1} \in \upsilon_{1} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{1} \in I_{1} ,\,\varGamma_{2} \in I_{2} \hfill \\ \end{aligned} } \\ & \quad \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{2} }}^{ - } ,\,I_{{A_{1} }}^{ - } )} \\ {\left[ {\tfrac{{[(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.,} \\ {\tfrac{{[(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 } }},} \\ {\tfrac{{[(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. {\tfrac{{[(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[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{2} }}^{ + } ,\,I_{{A_{1} }}^{ + } )} \\ {\left[ {\tfrac{{[(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.,} \\ {\tfrac{{[(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 } }},} \\ {\tfrac{{[(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. {\tfrac{{[(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], \\ & \quad \left. {\left[ {\begin{array}{*{20}c} {\hbox{min} (I_{{A_{2} }} ,\,I_{{A_{1} }} )} \\ {\left[ {\tfrac{{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.,} \\ {\tfrac{{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 } }},} \\ {\tfrac{{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. {\tfrac{{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 = \mathop \cup \limits_{\begin{subarray}{l} p_{A}^{ - } \in \varsigma_{a}^{ - } ,\,q_{A}^{ - } \in \chi_{A}^{ - } ,s_{A}^{ - } \in \psi_{A}^{ - } ,\,r_{A}^{ - } \in \upsilon_{A}^{ - } ,\varGamma_{A}^{ - } \in I_{a}^{ - } ,\,p_{A}^{ + } \in \varsigma_{A}^{ + } , \\ q_{A}^{ + } \in \chi_{A}^{ + } ,\,s_{A}^{ + } \in \psi_{A}^{ + } ,\,r_{A}^{ + } \in \upsilon_{A}^{ + } ,\varGamma_{A}^{ + } \in I_{A}^{ + } ,\,p_{A} \in \varsigma_{A} ,\,q_{A} \in \chi_{A} , \\ s_{A} \in \psi_{A} ,\,r_{A} \in \upsilon_{A} ,\,\varGamma_{A} \in I_{A} \end{subarray} } \\ & \quad \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{[1 + p_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - p_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + p_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - p_{A}^{ - } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}} \right.,} \\ {\left. {\tfrac{{[1 + r_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - r_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + r_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - r_{A}^{ - } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[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 \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{[1 + p_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - p_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + p_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - p_{A}^{ + } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }}} \right.,} \\ {\left. {\tfrac{{[1 + r_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - r_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + r_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - r_{A}^{ + } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[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], \\ & \quad \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{2[p_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - p_{A} (h)]^{{\lambda_{1} }} + [p_{A} (h)]^{{\lambda_{1} }} }},\,\tfrac{{2[q_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - q_{A} (h)]^{{\lambda_{1} }} + [q_{A} (h)]^{{\lambda_{1} }} }}} \right.,} \\ {\left. {\tfrac{{2[r_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - r_{A} (h)]^{{\lambda_{1} }} + [r_{A} (h)]^{{\lambda_{1} }} }},\,\tfrac{{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 = \mathop \cup \limits_{\begin{subarray}{l} p_{A}^{ - } \in \varsigma_{a}^{ - } ,\,q_{A}^{ - } \in \chi_{A}^{ - } ,s_{A}^{ - } \in \psi_{A}^{ - } ,\,r_{A}^{ - } \in \upsilon_{A}^{ - } ,\varGamma_{A}^{ - } \in I_{a}^{ - } ,\,p_{A}^{ + } \in \varsigma_{A}^{ + } , \\ q_{A}^{ + } \in \chi_{A}^{ + } ,\,s_{A}^{ + } \in \psi_{A}^{ + } ,\,r_{A}^{ + } \in \upsilon_{A}^{ + } ,\varGamma_{A}^{ + } \in I_{A}^{ + } ,\,p_{A} \in \varsigma_{A} ,\,q_{A} \in \chi_{A} , \\ s_{A} \in \psi_{A} ,\,r_{A} \in \upsilon_{A} ,\,\varGamma_{A} \in I_{A} \end{subarray} } \\ & \quad \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{[1 + p_{A}^{ - } (h)]^{{\lambda_{2} }} - [1 - p_{A}^{ - } (h)]^{{\lambda_{2} }} }}{{[1 + p_{A}^{ - } (h)]^{{\lambda_{2} }} + [1 - p_{A}^{ - } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}} \right.,} \\ {\left. {\tfrac{{[1 + r_{A}^{ - } (h)]^{{\lambda_{2} }} - [1 - r_{A}^{ - } (h)]^{{\lambda_{2} }} }}{{[1 + r_{A}^{ - } (h)]^{{\lambda_{2} }} + [1 - r_{A}^{ - } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[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., \\ & \quad \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{[1 + p_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - p_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + p_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - p_{A}^{ + } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[1 + q_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - q_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + q_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - q_{A}^{ + } (h)]^{{\lambda_{2} }} }}} \right.,} \\ {\left. {\tfrac{{[1 + r_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - r_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + r_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - r_{A}^{ + } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[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], \\ & \quad \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{2[p_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - p_{A} (h)]^{{\lambda_{2} }} + [p_{A} (h)]^{{\lambda_{2} }} }},\,\tfrac{{2[q_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - q_{A} (h)]^{{\lambda_{2} }} + [q_{A} (h)]^{{\lambda_{2} }} }}} \right.,} \\ {\left. {\tfrac{{2[r_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - r_{A} (h)]^{{\lambda_{2} }} + [r_{A} (h)]^{{\lambda_{2} }} }},\,\tfrac{{2[s_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - s_{A} (h)]^{{\lambda_{2} }} + [s_{A} (h)]^{{\lambda_{2} }} }}} \right]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

$$ \begin{aligned} & = \mathop \cup \limits_{\begin{subarray}{l} p_{A}^{ - } \in \varsigma_{a}^{ - } ,\,q_{A}^{ - } \in \chi_{A}^{ - } ,s_{A}^{ - } \in \psi_{A}^{ - } ,\,r_{A}^{ - } \in \upsilon_{A}^{ - } ,\varGamma_{A}^{ - } \in I_{a}^{ - } ,\,p_{A}^{ + } \in \varsigma_{A}^{ + } , \\ q_{A}^{ + } \in \chi_{A}^{ + } ,\,s_{A}^{ + } \in \psi_{A}^{ + } ,\,r_{A}^{ + } \in \upsilon_{A}^{ + } ,\varGamma_{A}^{ + } \in I_{A}^{ + } ,\,p_{A} \in \varsigma_{A} ,\,q_{A} \in \chi_{A} , \\ s_{A} \in \psi_{A} ,\,r_{A} \in \upsilon_{A} ,\,\varGamma_{A} \in I_{A} \end{subarray} } \\ & \quad \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{[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} }} }},\,\tfrac{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}} \right.,} \\ {\left. {\tfrac{{[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} }} }},\,\tfrac{{[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., \\ & \quad \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{[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} }} }},\,\tfrac{{[1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }}} \right.,} \\ {\left. {\tfrac{{[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} }} }},\,\tfrac{{[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], \\ & \quad \hbox{min} (I_{A} )\left. {\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{2[p_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - p_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [p_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }},\,\tfrac{{2[q_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - q_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [q_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}} \right.,} \\ {\left. {\tfrac{{2[r_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - r_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [r_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }},\,\tfrac{{2[s_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - s_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [s_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}} \right]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

$$ = (\lambda_{1} + \lambda_{2} )A. $$

Appendix 2: Proof of Theorem 1

Assume that $n = 1,$ TrCHFEWA $(A_{1} ,\,A_{2} , \ldots ,\,A_{n} ) = \mathop \oplus \limits_{j = 1}^{k} w_{1} A_{1}$

$$ \langle (\lambda (A_{1} + A_{2} ) = \lambda A_{2} + \lambda A_{1} $$

$$ \lambda (A_{1} + A_{2} ) = \mathop \cup \limits_{\begin{aligned} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } , \hfill \\ \varGamma_{1}^{ - } \in I_{1}^{ - } ,\,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } ,p_{2}^{ + } \in \varsigma_{2}^{ + } ,\,q_{1}^{ + } \in \chi_{1}^{ + } ,\,q_{2}^{ + } \in \chi_{2}^{ + } ,s_{1}^{ + } \in \psi_{1}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } , \hfill \\ r_{2}^{ + } \in \upsilon_{2}^{ + } ,\,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,\varGamma_{2}^{ + } \in I_{2}^{ + } ,p_{1} \in \varsigma_{1} ,\,p_{2} \in \varsigma_{2} ,\,q_{1} \in \chi_{1} ,q_{2} \in \chi_{2} ,\,s_{1} \in \psi_{1} ,\,s_{2} \in \psi_{2} , \hfill \\ r_{1} \in \upsilon_{1} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{1} \in I_{1} ,\,\varGamma_{2} \in I_{2} \hfill \\ \end{aligned} } $$

$$ \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ - } I_{{A_{2} }}^{ - } )} \\ {[\tfrac{{[(1 + p_{1}^{ - } (h))(1 - p_{1}^{ - } (h))]^{\lambda } [(1 + p_{2}^{ - } (h))(1 - p_{2}^{ - } (h))]^{\lambda } }}{{[(1 + p_{1}^{ - } (h))(1 - p_{1}^{ - } (h))]^{\lambda } [(1 + p_{2}^{ - } (h))(1 - p_{2}^{ - } (h))]^{\lambda } }},} \\ {\tfrac{{[(1 + q_{1}^{ - } (h))(1 - q_{1}^{ - } (h))]^{\lambda } [(1 + q_{2}^{ - } (h))(1 - q_{2}^{ - } (h))]^{\lambda } }}{{[(1 + q_{1}^{ - } (h))(1 - q_{1}^{ - } (h))]^{\lambda } [(1 + q_{2}^{ - } (h))(1 - q_{2}^{ - } (h))]^{\lambda } }},} \\ {\tfrac{{[(1 + r_{1}^{ - } (h))(1 - r_{1}^{ - } (h))]^{\lambda } [(1 + r_{2}^{ - } (h))(1 - r_{2}^{ - } (h))]^{\lambda } }}{{[(1 + r_{1}^{ - } (h))(1 - r_{1}^{ - } (h))]^{\lambda } [(1 + r_{2}^{ - } (h))(1 - r_{2}^{ - } (h))]^{\lambda } }},} \\ {\tfrac{{[(1 + s_{1}^{ - } (h))(1 - s_{1}^{ - } (h))]^{\lambda } [(1 + s_{2}^{ - } (h))(1 - s_{2}^{ - } (h))]^{\lambda } }}{{[(1 + s_{1}^{ - } (h))(1 - s_{1}^{ - } (h))]^{\lambda } [(1 + s_{2}^{ - } (h))(1 - s_{2}^{ - } (h))]^{\lambda } }}]} \\ \end{array} } \right],} \right. $$

$$ \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ + } I_{{A_{2} }}^{ + } )} \\ {\left[ {\tfrac{{[(1 + p_{1}^{ + } (h))(1 - p_{1}^{ + } (h))]^{\lambda } [(1 + p_{2}^{ + } (h))(1 - p_{2}^{ + } (h))]^{\lambda } }}{{[(1 + p_{1}^{ + } (h))(1 - p_{1}^{ + } (h))]^{\lambda } [(1 + p_{2}^{ + } (h))(1 - p_{2}^{ + } (h))]^{\lambda } }}} \right.,} \\ {\tfrac{{[(1 + q_{1}^{ + } (h))(1 - q_{1}^{ + } (h))]^{\lambda } [(1 + q_{2}^{ + } (h))(1 - q_{2}^{ + } (h))]^{\lambda } }}{{[(1 + q_{1}^{ + } (h))(1 - q_{1}^{ + } (h))]^{\lambda } [(1 + q_{2}^{ + } (h))(1 - q_{2}^{ + } (h))]^{\lambda } }},} \\ {\tfrac{{[(1 + r_{1}^{ + } (h))(1 - r_{1}^{ + } (h))]^{\lambda } [(1 + r_{2}^{ + } (h))(1 - r_{2}^{ + } (h))]^{\lambda } }}{{[(1 + r_{1}^{ + } (h))(1 - r_{1}^{ + } (h))]^{\lambda } [(1 + r_{2}^{ + } (h))(1 - r_{2}^{ + } (h))]^{\lambda } }},} \\ {\left. {\tfrac{{[(1 + s_{1}^{ + } (h))(1 - s_{1}^{ + } (h))]^{\lambda } [(1 + s_{2}^{ + } (h))(1 - s_{2}^{ + } (h))]^{\lambda } }}{{[(1 + s_{1}^{ + } (h))(1 - s_{1}^{ + } (h))]^{\lambda } [(1 + s_{2}^{ + } (h))(1 - s_{2}^{ + } (h))]^{\lambda } }}} \right]} \\ \end{array} } \right] $$

$$ \left. {\left[ {\begin{array}{*{20}c} {\hbox{min} (I_{{A_{1} }} I_{{A_{2} }} )} \\ {\left[ {\tfrac{{2[p_{1} (h)p_{2} (h)]^{\lambda } }}{{[(4 - 2p_{1} (h) - 2p_{2} (h) - p_{1} (h)p_{2} (h)]^{\lambda } + [p_{1} (h)p_{2} (h)]^{\lambda } }}} \right.,} \\ {\tfrac{{2[q_{1} (h)q_{2} (h)]^{\lambda } }}{{[(4 - 2q_{1} (h) - 2q_{2} (h) - q_{1} (h)q_{2} (h)]^{\lambda } + [q_{1} (h)q_{2} (h)]^{\lambda } }},} \\ {\tfrac{{2[r_{1} (h)r_{2} (h)]^{\lambda } }}{{[(4 - 2r_{1} (h) - 2r_{2} (h) - r_{1} (h)r_{2} (h)]^{\lambda } + [r_{1} (h)r_{2} (h)]^{\lambda } }},} \\ {\left. {\tfrac{{2[s_{1} (h)s_{2} (h)]^{\lambda } }}{{[(4 - 2s_{1} (h) - 2s_{2} (h) - s_{1} (h)s_{2} (h)]^{\lambda } + [s_{1} (h)s_{2} (h)]^{\lambda } }}} \right]} \\ \end{array} } \right]} \right\rangle $$

And we have

$$ \lambda A_{1} = \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\{ {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } )[\tfrac{{[(1 + p_{1}^{ - } (h))^{\lambda } - (1 - p_{1}^{ - } (h))^{\lambda } ]}}{{[(1 + p_{1}^{ - } (h))^{\lambda } + (1 - p_{1}^{ - } (h))^{\lambda } ]}},} \\ {\tfrac{{[(1 + q_{1}^{ - } (h))^{\lambda } - (1 - q_{1}^{ - } (h))^{\lambda } ]}}{{[(1 + q_{1}^{ - } (h))^{\lambda } + (1 - q_{1}^{ - } (h))^{\lambda } ]}},} \\ {\tfrac{{[(1 + r_{1}^{ - } (h))^{\lambda } - (1 - r_{1}^{ - } (h))^{\lambda } ]}}{{[(1 + r_{1}^{ - } (h))^{\lambda } + (1 - r_{1}^{ - } (h))^{\lambda } ]}},} \\ {\tfrac{{[(1 + s_{1}^{ - } (h))^{\lambda } - (1 - s_{1}^{ - } (h))^{\lambda } ]}}{{[(1 + s_{1}^{ - } (h))^{\lambda } + (1 - s_{1}^{ - } (h))^{\lambda } ]}}]} \\ {\hbox{max} (I_{A}^{ + } )[\tfrac{{[(1 + p_{1}^{ + } (h))^{\lambda } - (1 - p_{1}^{ + } (h))^{\lambda } ]}}{{[(1 + p_{1}^{ + } (h))^{\lambda } + (1 - p_{1}^{ + } (h))^{\lambda } ]}},} \\ {\tfrac{{[(1 + q_{1}^{ + } (h))^{\lambda } - (1 - q_{1}^{ + } (h))^{\lambda } ]}}{{[(1 + q_{1}^{ + } (h))^{\lambda } + (1 - q_{1}^{ + } (h))^{\lambda } ]}},} \\ {\tfrac{{[(1 + r_{1}^{ + } (h))^{\lambda } - (1 - r_{1}^{ + } (h))^{\lambda } ]}}{{[(1 + r_{1}^{ + } (h))^{\lambda } + (1 - r_{1}^{ + } (h))^{\lambda } ]}},} \\ {\tfrac{{[(1 + s_{1}^{ + } (h))^{\lambda } - (1 - s_{1}^{ + } (h))^{\lambda } ]}}{{[(1 + s_{1}^{ + } (h))^{\lambda } + (1 - s_{1}^{ + } (h))^{\lambda } ]}}]} \\ \end{array} } \right]} \right\}, $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\left[ {\tfrac{{2p_{1}^{\lambda } (h)}}{{[(2 - p_{1} (h)]^{\lambda } + [p_{1} (h)]^{\lambda } }}} \right.,} \\ {\tfrac{{2q_{1}^{\lambda } (h)}}{{[(2 - q_{1} (h)]^{\lambda } + [q_{1} (h)]^{\lambda } }},} \\ {\tfrac{{2r_{1}^{\lambda } (h)}}{{[(2 - r_{1} (h)]^{\lambda } + [r_{1} (h)]^{\lambda } }},} \\ {\left. {\tfrac{{2s_{1}^{\lambda } (h)}}{{[(2 - s_{1} (h)]^{\lambda } + [s_{1} (h)]^{\lambda } }}} \right]} \\ \end{array} } \right]} \right\rangle $$

$$ \lambda A_{2} = \mathop \cup \limits_{\begin{subarray}{l} p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } ,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{2}^{ + } \in \varsigma_{2}^{ + } , \\ q_{2}^{ + } \in \chi_{2}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{2}^{ + } \in \upsilon_{2}^{ + } ,\varGamma_{2}^{ + } \in I_{2}^{ + } ,\,p_{2} \in \varsigma_{2} ,\,q_{2} \in \chi_{2} , \\ s_{2} \in \psi_{2} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{2} \in I_{2} \end{subarray} } $$

$$ \begin{aligned} & \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } )\left[ {\tfrac{{[(1 + p_{2}^{ - } (h))^{\lambda } - (1 - p_{2}^{ - } (h))^{\lambda } ]}}{{[(1 + p_{2}^{ - } (h))^{\lambda } + (1 - p_{2}^{ - } (h))^{\lambda } ]}},\,\tfrac{{[(1 + q_{2}^{ - } (h))^{\lambda } - (1 - q_{2}^{ - } (h))^{\lambda } ]}}{{[(1 + q_{2}^{ - } (h))^{\lambda } + (1 - q_{2}^{ - } (h))^{\lambda } ]}}} \right.,} \\ {\left. {\tfrac{{[(1 + r_{2}^{ - } (h))^{\lambda } - (1 - r_{2}^{ - } (h))^{\lambda } ]}}{{[(1 + r_{2}^{ - } (h))^{\lambda } + (1 - r_{2}^{ - } (h))^{\lambda } ]}},\,\tfrac{{[(1 + s_{2}^{ - } (h))^{\lambda } - (1 - s_{2}^{ - } (h))^{\lambda } ]}}{{[(1 + s_{2}^{ - } (h))^{\lambda } + (1 - s_{2}^{ - } (h))^{\lambda } ]}}} \right],} \\ {\hbox{max} (I_{A}^{ + } )\left[ {\tfrac{{[(1 + p_{2}^{ + } (h))^{\lambda } - (1 - p_{2}^{ + } (h))^{\lambda } ]}}{{[(1 + p_{2}^{ + } (h))^{\lambda } + (1 - p_{2}^{ + } (h))^{\lambda } ]}},\,\tfrac{{[(1 + q_{2}^{ + } (h))^{\lambda } - (1 - q_{2}^{ + } (h))^{\lambda } ]}}{{[(1 + q_{2}^{ + } (h))^{\lambda } + (1 - q_{2}^{ + } (h))^{\lambda } ]}}} \right.,} \\ {\left. {\tfrac{{[(1 + r_{2}^{ + } (h))^{\lambda } - (1 - r_{2}^{ + } (h))^{\lambda } ]}}{{[(1 + r_{2}^{ + } (h))^{\lambda } + (1 - r_{2}^{ + } (h))^{\lambda } ]}},\,\tfrac{{[(1 + s_{2}^{ + } (h))^{\lambda } - (1 - s_{2}^{ + } (h))^{\lambda } ]}}{{[(1 + s_{2}^{ + } (h))^{\lambda } + (1 - s_{2}^{ + } (h))^{\lambda } ]}}} \right]} \\ \end{array} } \right]} \right., \\ & \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {[\tfrac{{2p_{2}^{\lambda } (h)}}{{[(2 - p_{2} (h)]^{\lambda } + [p_{2} (h)]^{\lambda } }},\,\tfrac{{2q_{2}^{\lambda } (h)}}{{[(2 - q_{2} (h)]^{\lambda } + [q_{2} (h)]^{\lambda } }},} \\ {\tfrac{{2r_{2}^{\lambda } (h)}}{{[(2 - r_{2} (h)]^{\lambda } + [r_{2} (h)]^{\lambda } }},\,\tfrac{{2s_{2}^{\lambda } (h)}}{{[(2 - s_{2} (h)]^{\lambda } + [s_{2} (h)]^{\lambda } }}]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

$$ \lambda A_{2} + \lambda A_{1} = \mathop \cup \limits_{\begin{aligned} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } , \hfill \\ \varGamma_{1}^{ - } \in I_{1}^{ - } ,\,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } ,p_{2}^{ + } \in \varsigma_{2}^{ + } ,\,q_{1}^{ + } \in \chi_{1}^{ + } ,\,q_{2}^{ + } \in \chi_{2}^{ + } ,s_{1}^{ + } \in \psi_{1}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } , \hfill \\ r_{2}^{ + } \in \upsilon_{2}^{ + } ,\,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,\varGamma_{2}^{ + } \in I_{2}^{ + } ,p_{1} \in \varsigma_{1} ,\,p_{2} \in \varsigma_{2} ,\,q_{1} \in \chi_{1} ,q_{2} \in \chi_{2} ,\,s_{1} \in \psi_{1} ,\,s_{2} \in \psi_{2} , \hfill \\ r_{1} \in \upsilon_{1} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{1} \in I_{1} ,\,\varGamma_{2} \in I_{2} \hfill \\ \end{aligned} } $$

$$ \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } )\left[ {\tfrac{{[(1 + p_{2}^{ - } (h))(1 - p_{2}^{ - } (h))]^{\lambda } [(1 + p_{1}^{ - } (h))(1 - p_{1}^{ - } (h))]^{\lambda } }}{{[(1 + p_{2}^{ - } (h))(1 - p_{2}^{ - } (h))]^{\lambda } [(1 + p_{1}^{ - } (h))(1 - p_{1}^{ - } (h))]^{\lambda } }}} \right.,} \\ {\tfrac{{[(1 + q_{2}^{ - } (h))(1 - q_{2}^{ - } (h))]^{\lambda } [(1 + q_{1}^{ - } (h))(1 - q_{1}^{ - } (h))]^{\lambda } }}{{[(1 + q_{2}^{ - } (h))(1 - q_{2}^{ - } (h))]^{\lambda } [(1 + q_{1}^{ - } (h))(1 - q_{1}^{ - } (h))]^{\lambda } }},} \\ {\tfrac{{[(1 + r_{2}^{ - } (h))(1 - r_{2}^{ - } (h))]^{\lambda } [(1 + r_{1}^{ - } (h))(1 - r_{1}^{ - } (h))]^{\lambda } }}{{[(1 + r_{2}^{ - } (h))(1 - r_{2}^{ - } (h))]^{\lambda } [(1 + r_{1}^{ - } (h))(1 - r_{1}^{ - } (h))]^{\lambda } }},} \\ {\left. {\tfrac{{[(1 + s_{2}^{ - } (h))(1 - s_{2}^{ - } (h))]^{\lambda } [(1 + s_{1}^{ - } (h))(1 - s_{1}^{ - } (h))]^{\lambda } }}{{[(1 + s_{2}^{ - } (h))(1 - s_{2}^{ - } (h))]^{\lambda } [(1 + s_{1}^{ - } (h))(1 - s_{1}^{ - } (h))]^{\lambda } }}} \right]} \\ \end{array} } \right], $$

$$ \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ + } )\left[ {\tfrac{{[(1 + p_{2}^{ + } (h))(1 - p_{2}^{ + } (h))]^{\lambda } [(1 + p_{1}^{ + } (h))(1 - p_{1}^{ + } (h))]^{\lambda } }}{{[(1 + p_{2}^{ + } (h))(1 - p_{2}^{ + } (h))]^{\lambda } [(1 + p_{1}^{ + } (h))(1 - p_{1}^{ + } (h))]^{\lambda } }}} \right.,} \\ {\tfrac{{[(1 + q_{2}^{ + } (h))(1 - q_{2}^{ + } (h))]^{\lambda } [(1 + q_{1}^{ + } (h))(1 - q_{1}^{ + } (h))]^{\lambda } }}{{[(1 + q_{2}^{ + } (h))(1 - q_{2}^{ + } (h))]^{\lambda } [(1 + q_{1}^{ + } (h))(1 - q_{1}^{ + } (h))]^{\lambda } }},} \\ {\tfrac{{[(1 + r_{2}^{ + } (h))(1 - r_{2}^{ + } (h))]^{\lambda } [(1 + r_{1}^{ + } (h))(1 - r_{1}^{ + } (h))]^{\lambda } }}{{[(1 + r_{2}^{ + } (h))(1 - r_{2}^{ + } (h))]^{\lambda } [(1 + r_{1}^{ + } (h))(1 - r_{1}^{ + } (h))]^{\lambda } }},} \\ {\left. {\tfrac{{[(1 + s_{2}^{ + } (h))(1 - s_{2}^{ + } (h))]^{\lambda } [(1 + s_{1}^{ + } (h))(1 - s_{1}^{ + } (h))]^{\lambda } }}{{[(1 + s_{2}^{ + } (h))(1 - s_{2}^{ + } (h))]^{\lambda } [(1 + s_{1}^{ + } (h))(1 - s_{1}^{ + } (h))]^{\lambda } }}} \right]} \\ \end{array} } \right], $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {[\tfrac{{2[p_{2} (h)p_{1} (h)]^{\lambda } }}{{[(4 - 2p_{2} (h) - 2p_{1} (h) - p_{2} (h)p_{1} (h)]^{\lambda } + [p_{2} (h)p_{1} (h)]^{\lambda } }},} \\ {\tfrac{{2[q_{2} (h)q_{1} (h)]^{\lambda } }}{{[(4 - 2q_{2} (h) - 2q_{1} (h) - q_{2} (h)q_{1} (h)]^{\lambda } + [q_{2} (h)q_{1} (h)]^{\lambda } }},} \\ {\tfrac{{2[r_{2} (h)r_{1} (h)]^{\lambda } }}{{[(4 - 2r_{2} (h) - 2r_{1} (h) - r_{2} (h)r_{1} (h)]^{\lambda } + [s_{2} (h)s_{1} (h)]^{\lambda } }},} \\ {\tfrac{{2[s_{2} (h)s_{1} (h)]^{\lambda } }}{{[(4 - 2s_{2} (h) - 2s_{1} (h) - s_{2} (h)s_{1} (h)]^{\lambda } + [s_{2} (h)s_{1} (h)]^{\lambda } }}]} \\ \end{array} } \right]} \right\rangle $$

so, we have $\lambda (A_{1} + A_{2} ) = \lambda A_{2} + \lambda A_{1} .$

$$ \lambda_{1} A + \lambda_{2} A = (\lambda_{1} + \lambda_{2} )A $$

$$ \lambda_{1} A = \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \begin{aligned} & \left\langle {\left\{ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } )[\tfrac{{[1 + p_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - p_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + p_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - p_{A}^{ - } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }},} \\ {\tfrac{{[1 + r_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - r_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + r_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - r_{A}^{ - } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[1 + s_{A}^{ - } (h)]^{{\lambda_{1} }} - [1 - s_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{[1 + s_{A}^{ - } (h)]^{{\lambda_{1} }} + [1 - s_{A}^{ - } (h)]^{{\lambda_{1} }} }}],} \\ {\hbox{max} (I_{A}^{ + } )[\tfrac{{[1 + p_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - p_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + p_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - p_{A}^{ + } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }},} \\ {\tfrac{{[1 + r_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - r_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + r_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - r_{A}^{ + } (h)]^{{\lambda_{1} }} }},\,\tfrac{{[1 + s_{A}^{ + } (h)]^{{\lambda_{1} }} - [1 - s_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{[1 + s_{A}^{ + } (h)]^{{\lambda_{1} }} + [1 - s_{A}^{ + } (h)]^{{\lambda_{1} }} }}],} \\ \end{array} } \right\}} \right. \\ & \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {[\tfrac{{2[p_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - p_{A} (h)]^{{\lambda_{1} }} + [p_{A} (h)]^{{\lambda_{1} }} }},\,\tfrac{{2[q_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - q_{A} (h)]^{{\lambda_{1} }} + [q_{A} (h)]^{{\lambda_{1} }} }},} \\ {\tfrac{{2[r_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - r_{A} (h)]^{{\lambda_{1} }} + [r_{A} (h)]^{{\lambda_{1} }} }},\,\tfrac{{2[s_{A} (h)]^{{\lambda_{1} }} }}{{[(2 - s_{A} (h)]^{{\lambda_{1} }} + [s_{A} (h)]^{{\lambda_{1} }} }}]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

And

$$ \lambda_{2} A = \mathop \cup \limits_{\begin{subarray}{l} p_{2}^{ - } \in \varsigma_{2}^{ - } ,\,q_{2}^{ - } \in \chi_{2}^{ - } ,s_{2}^{ - } \in \psi_{2}^{ - } ,\,r_{2}^{ - } \in \upsilon_{2}^{ - } ,\varGamma_{2}^{ - } \in I_{2}^{ - } ,\,p_{2}^{ + } \in \varsigma_{2}^{ + } , \\ q_{2}^{ + } \in \chi_{2}^{ + } ,\,s_{2}^{ + } \in \psi_{2}^{ + } ,\,r_{2}^{ + } \in \upsilon_{2}^{ + } ,\varGamma_{2}^{ + } \in I_{2}^{ + } ,\,p_{2} \in \varsigma_{2} ,\,q_{2} \in \chi_{2} , \\ s_{2} \in \psi_{2} ,\,r_{2} \in \upsilon_{2} ,\,\varGamma_{2} \in I_{2} \end{subarray} } $$

$$ \begin{aligned} & \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } )[\tfrac{{[1 + p_{A}^{ - } (h)]^{{\lambda_{2} }} - [1 - p_{A}^{ - } (h)]^{{\lambda_{2} }} }}{{[1 + p_{A}^{ - } (h)]^{{\lambda_{2} }} + [1 - p_{A}^{ - } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[1 + q_{A}^{ - } (h)]^{{\lambda_{2} }} - [1 - q_{A}^{ - } (h)]^{{\lambda_{2} }} }}{{[1 + q_{A}^{ - } (h)]^{{\lambda_{2} }} + [1 - q_{A}^{ - } (h)]^{{\lambda_{2} }} }},} \\ {\tfrac{{[1 + r_{A}^{ - } (h)]^{{\lambda_{2} }} - [1 - r_{A}^{ - } (h)]^{{\lambda_{2} }} }}{{[1 + r_{A}^{ - } (h)]^{{\lambda_{2} }} + [1 - r_{A}^{ - } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[1 + s_{A}^{ - } (h)]^{{\lambda_{2} }} - [1 - s_{A}^{ - } (h)]^{{\lambda_{2} }} }}{{[1 + s_{A}^{ - } (h)]^{{\lambda_{2} }} + [1 - s_{A}^{ - } (h)]^{{\lambda_{2} }} }}],} \\ {\hbox{max} (I_{A}^{ + } )[\tfrac{{[1 + p_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - p_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + p_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - p_{A}^{ + } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[1 + q_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - q_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + q_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - q_{A}^{ + } (h)]^{{\lambda_{2} }} }},} \\ {\tfrac{{[1 + r_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - r_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + r_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - r_{A}^{ + } (h)]^{{\lambda_{2} }} }},\,\tfrac{{[1 + s_{A}^{ + } (h)]^{{\lambda_{2} }} - [1 - s_{A}^{ + } (h)]^{{\lambda_{2} }} }}{{[1 + s_{A}^{ + } (h)]^{{\lambda_{2} }} + [1 - s_{A}^{ + } (h)]^{{\lambda_{2} }} }}]} \\ \end{array} ,} \right],} \right. \\ & \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {[\tfrac{{2[p_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - p_{A} (h)]^{{\lambda_{2} }} + [p_{A} (h)]^{{\lambda_{2} }} }},\,\tfrac{{2[q_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - q_{A} (h)]^{{\lambda_{2} }} + [q_{A} (h)]^{{\lambda_{2} }} }},} \\ {\tfrac{{2[r_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - r_{A} (h)]^{{\lambda_{2} }} + [r_{A} (h)]^{{\lambda_{2} }} }},\,\tfrac{{2[s_{A} (h)]^{{\lambda_{2} }} }}{{[(2 - s_{A} (h)]^{{\lambda_{2} }} + [s_{A} (h)]^{{\lambda_{2} }} }}]} \\ \end{array} } \right]} \right\rangle \\ \end{aligned} $$

$$ = \mathop \cup \limits_{\begin{aligned} p_{A}^{ - } \in \varsigma_{a}^{ - } ,\,q_{A}^{ - } \in \chi_{A}^{ - } , \hfill \\ s_{A}^{ - } \in \psi_{A}^{ - } ,\,r_{A}^{ - } \in \upsilon_{A}^{ - } , \hfill \\ \varGamma_{A}^{ - } \in I_{a}^{ - } ,\,p_{A}^{ + } \in \varsigma_{A}^{ + } , \hfill \\ q_{A}^{ + } \in \chi_{A}^{ + } ,\,s_{A}^{ + } \in \psi_{A}^{ + } ,\,r_{A}^{ + } \in \upsilon_{A}^{ + } , \hfill \\ \varGamma_{A}^{ + } \in I_{A}^{ + } ,\,p_{A} \in \varsigma_{A} ,\,q_{A} \in \chi_{A} , \hfill \\ s_{A} \in \psi_{A} ,\,r_{A} \in \upsilon_{A} ,\,\varGamma_{A} \in I_{A} \hfill \\ \end{aligned} } $$

$$ \left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } )} \\ {\left[ {\tfrac{{[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} }} }},\,\tfrac{{[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} }} }}} \right.,} \\ {\left. {\tfrac{{[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} }} }},\,\tfrac{{[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],} \\ {\hbox{max} (I_{A}^{ + } )} \\ {\left[ {\tfrac{{[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} }} }},\,\tfrac{{[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} }} }}} \right.,} \\ {\left. {\tfrac{{[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} }} }},\,\tfrac{{[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., $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {[\tfrac{{2[p_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - p_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [p_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }},\,\tfrac{{2[q_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - q_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [q_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }},} \\ {\tfrac{{2[r_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - r_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [r_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }},\,\tfrac{{2[s_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}{{[(2 - s_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} + [s_{A} (h)]^{{\lambda_{1} + \lambda_{2} }} }}]} \\ \end{array} } \right]} \right\rangle $$

$$ = \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}c} {\tfrac{{[[1 + p_{1}^{ - } (h)]^{{\varpi_{1} }} - [1 - p_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + p_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} + [1 - p_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }},\,\tfrac{{[1 + q_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} - [1 - q_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + q_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} + [1 - q_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }},} \\ {\tfrac{{[1 + r_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} - [1 - r_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + r_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} + [1 - r_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }},\,\tfrac{{[1 + s_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} - [1 - s_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + s_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} + [1 - s_{1}^{ - } (h)]^{{^{{\varpi_{1} }} }} }}} \\ \end{array} } \right]} \right.; $$

$$ \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}c} {\tfrac{{[1 + p_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} - [1 - p_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + p_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} + [1 - p_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }},\,\tfrac{{[1 + q_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} - [1 - q_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + q_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} + [1 - q_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }},} \\ {\tfrac{{[1 + r_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} - [1 - r_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + r_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} + [1 - r_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }},\,\tfrac{{[1 + s_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} - [1 - s_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }}{{[1 + s_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} + [1 - s_{1}^{ + } (h)]^{{^{{\varpi_{1} }} }} }}} \\ \end{array} } \right]; $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\tfrac{{2[p_{1} (h)]^{{^{{\varpi_{1} }} }} }}{{[(2 - p_{1} (h)]^{{^{{\varpi_{1} }} }} + [p_{1} (h)]^{{^{{\varpi_{1} }} }} }},\,\tfrac{{2[q_{1} (h)]^{{\varpi_{1} }} }}{{[(2 - q_{1} (h)]^{{^{{\varpi_{1} }} }} + [q_{1} (h)]^{{^{{\varpi_{1} }} }} }},} \\ {\tfrac{{2[r_{1} (h)]^{{^{{\varpi_{1} }} }} }}{{[(2 - r_{1} (h)]^{{^{{\varpi_{1} }} }} + [r_{1} (h)]^{{^{{\varpi_{1} }} }} }},\,\tfrac{{2[s_{1} (h)]^{{^{{\varpi_{1} }} }} }}{{[(2 - s_{1} (h)]^{{^{{\varpi_{1} }} }} + [s_{1} (h)]^{{^{{\varpi_{1} }} }} }}} \\ \end{array} } \right]} \right\rangle . $$

Assume that $n = k,$ TrCHFEWA $(A_{1} ,\,A_{2} , \ldots ,\,A_{n} ) = \mathop \oplus \limits_{j = 1}^{k} w_{j} A_{j}$

$$ \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}l} {\tfrac{{[\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ - } (h)]^{\varpi } - \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right];} \right. $$

$$ \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}l} {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right]; $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [p_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - p_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [p_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [q_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - q_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [q_{1} (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [r_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - r_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [r_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [s_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - s_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [s_{1} (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right]} \right\rangle . $$

Then when $n = k + 1,$ we have TrCHFEWA $(A_{1} ,\,A_{2} , \ldots ,\,A_{k + 1} ) = {\text{TrCHFEWA}}(A_{1} ,\,A_{2} , \ldots ,\,A_{k} ) \oplus A_{k + 1} )$

$$ \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}l} {\tfrac{{[\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ - } (h)]^{\varpi } - \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right];} \right. $$

$$ \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}l} {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right]; $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [p_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - p_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [p_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [q_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - q_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [q_{1} (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [r_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - r_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [r_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k} [s_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k} [(2 - s_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [s_{1} (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right]} \right\rangle \oplus_{k + 1} $$

$$ \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}l} {\tfrac{{[\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ - } (h)]^{\varpi } - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right];} \right. $$

$$ \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}l} {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right]; $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [p_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - p_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [p_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [q_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - q_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [q_{1} (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [r_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - r_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [r_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{k + 1} [s_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - s_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [s_{1} (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right]} \right\rangle $$

$$ = \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}l} {\tfrac{{[\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ - } (h)]^{\varpi } - \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ - } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ - } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right],} \right. $$

$$ \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}l} {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + p_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - p_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + q_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - q_{1}^{ + } (h)]^{{^{\varpi } }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + r_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - r_{1}^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [1 + s_{1}^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k} [1 - s_{1}^{ + } (h)]^{{^{\varpi } }} }}} \hfill \\ \end{array} } \right], $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [p_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - p_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [p_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [q_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - q_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [q_{1} (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [r_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - r_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [r_{1} (h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{k + 1} [s_{1} (h)]^{{^{\varpi } }} }}{{\mathop \prod \nolimits_{j = 1}^{k + 1} [(2 - s_{1} (h)]^{{^{\varpi } }} + \mathop \prod \nolimits_{j = 1}^{k + 1} [s_{1} (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right]} \right\rangle . $$

Especially, if $w = (\tfrac{1}{n},\,\tfrac{1}{n}, \ldots ,\,\tfrac{1}{n})^{T} ,$ then the TrCHFEWA operator is reduced to the trapezoidal cubic hesitant fuzzy Einstein weighing averaging operator, which is shown as follows:

$$ \mathop \cup \limits_{\begin{subarray}{l} p_{1}^{ - } \in \varsigma_{1}^{ - } ,\,q_{1}^{ - } \in \chi_{1}^{ - } ,s_{1}^{ - } \in \psi_{1}^{ - } ,\,r_{1}^{ - } \in \upsilon_{1}^{ - } ,\varGamma_{1}^{ - } \in I_{1}^{ - } ,\,p_{1}^{ + } \in \varsigma_{1}^{ + } , \\ q_{1}^{ + } \in \chi_{1}^{ + } ,\,s_{1}^{ + } \in \psi_{1}^{ + } ,\,r_{1}^{ + } \in \upsilon_{1}^{ + } ,\varGamma_{1}^{ + } \in I_{1}^{ + } ,\,p_{1} \in \varsigma_{1} ,\,q_{1} \in \chi_{1} , \\ s_{1} \in \psi_{1} ,\,r_{1} \in \upsilon_{1} ,\,\varGamma_{1} \in I_{1} \end{subarray} } $$

$$ \left\langle {\hbox{max} (I_{A}^{ - } )\left[ {\begin{array}{*{20}l} {\tfrac{{[\mathop \prod \nolimits_{j = 1}^{n} [1 + p_{1}^{ - } (h)]^{{\tfrac{1}{n}}} - \mathop \prod \nolimits_{j = 1}^{n} [1 - p_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + p_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - p_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + q_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - q_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + q_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - q_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + r_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - r_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + r_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - r_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + s_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - s_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \limits_{j = 1}^{n} [1 + s_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - s_{1}^{ - } (h)]^{{^{{\tfrac{1}{n}}} }} }}} \hfill \\ \end{array} } \right];} \right. $$

$$ \hbox{max} (I_{A}^{ + } )\left[ {\begin{array}{*{20}l} {\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + p_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - p_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + p_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - p_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + q_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - q_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + q_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - q_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }},} \hfill \\ {\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + r_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - r_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + r_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - r_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }},\,\tfrac{{\mathop \prod \nolimits_{j = 1}^{n} [1 + s_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} - \mathop \prod \nolimits_{j = 1}^{n} [1 - s_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [1 + s_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [1 - s_{1}^{ + } (h)]^{{^{{\tfrac{1}{n}}} }} }}} \hfill \\ \end{array} } \right]; $$

$$ \left. {\hbox{min} (I_{A} )\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{n} [p_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [(2 - p_{1} (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [p_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }},\,\tfrac{{2\mathop \prod \nolimits_{j = 1}^{n} [q_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [(2 - q_{1} (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [q_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }},} \\ {\tfrac{{2\mathop \prod \nolimits_{j = 1}^{n} [r_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [(2 - r_{1} (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [r_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} [s_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }}{{\mathop \prod \nolimits_{j = 1}^{n} [(2 - s_{1} (h)]^{{^{{\tfrac{1}{n}}} }} + \mathop \prod \nolimits_{j = 1}^{n} [s_{1} (h)]^{{^{{\tfrac{1}{n}}} }} }}} \\ \end{array} } \right]} \right\rangle . $$