New work of trapezoidal cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging operator and decision making

Abstract

In this paper, we define some Einstein operations on trapezoidal cubic linguistic uncertain fuzzy numbers and develop two arithmetic averaging operators, that is, trapezoidal cubic linguistic uncertain fuzzy Einstein weighted averaging operator and trapezoidal cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging (TrCLUFEHWA) operator, for aggregating trapezoidal cubic linguistic uncertain fuzzy information. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply the TrCLUFEHWA operator to multiple-attribute decision making with trapezoidal cubic linguistic uncertain fuzzy information. Finally, a numerical example is provided to demonstrate the submission of the established approach.

Appendices

Appendix A: Proof of Proposition 1

• (1)
  $$ A_{1} + A_{2} = A_{2} + A_{1} ; $$
  $$ A_{1} + A_{2} = {[}s_{{\theta_{1} + \theta_{2} }} ,\,s_{{t_{1} + t_{2} }} ],\,\left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ - } ,\,I_{{A_{2} }}^{ - } )} \\ {{[}\tfrac{{p_{1}^{ - } (h) + p_{2}^{ - } (h)}}{{1 + p_{1}^{ - } (h)p_{2}^{ - } (h)}},\,\tfrac{{q_{1}^{ - } (h) + q_{2}^{ - } (h)}}{{1 + q_{1}^{ - } (h)q_{2}^{ - } (h)}},} \\ {\tfrac{{r_{1}^{ - } (h) + r_{2}^{ - } (h)}}{{1 + r_{1}^{ - } (h)r_{2}^{ - } (h))}},\,\tfrac{{s_{1}^{ - } (h) + s_{2}^{ - } (h)}}{{1 + s_{1}^{ - } (h)s_{2}^{ - } (h))}}],} \\ {\hbox{max} (I_{{A_{1} }}^{ + } ,\,I_{{A_{2} }}^{ + } ){[}\tfrac{{p_{1}^{ + } (h) + p_{2}^{ + } (h)}}{{1 + p_{1}^{ + } (h)p_{2}^{ + } (h))}},\,\tfrac{{q_{1}^{ + } (h) + q_{2}^{ + } (h)}}{{1 + q_{1}^{ + } (h)q_{2}^{ + } (h))}},} \\ {\tfrac{{r_{1}^{ + } (h) + r_{2}^{ + } (h)}}{{1 + r_{1}^{ + } (h)r_{2}^{ + } (h))}},\,\tfrac{{s_{1}^{ + } (h) + s_{2}^{ + } (h)}}{{1 + s_{1}^{ + } (h)s_{2}^{ + } (h))}}],} \\ \end{array} } \right],} \right. $$
  $$ \left. {\left[ {\begin{array}{*{20}c} {\hbox{min} (\mu_{{A_{1} }} ,\,\mu_{{A_{2} }} ){[}\tfrac{{p_{1} (h) \cdot p_{2} (h)}}{{1 + (\left( {1 - p_{1} (h)} \right)\,\left( {1 - p_{2} (h)} \right))}},\,\tfrac{{q_{1} (h) \cdot q_{2} (h)}}{{1 + (\left( {1 - q_{1} (h)} \right)\,\left( {1 - q_{2} (h)} \right))}},} \\ {\tfrac{{r_{1} (h) \cdot r_{2} (h)}}{{1 + (\left( {1 - r_{1} (h)} \right)\,\left( {1 - r_{2} (h)} \right))}},\,\tfrac{{s_{1} (h) \cdot s_{2} (h)}}{{1 + (\left( {1 - s_{1} (h)} \right)\,\left( {1 - s_{2} (h)} \right))}}]} \\ \end{array} } \right]} \right\rangle , $$
  $$ = {[}s_{{\theta_{2} + \theta_{1} }} ,\,s_{{t_{2} + t_{1} }} ],\,\left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{2} }}^{ - } ,\,I_{{A_{1} }}^{ - } )} \\ {{[}\tfrac{{p_{2}^{ - } (h) + p_{1}^{ - } (h)}}{{1 + p_{2}^{ - } (h)p_{1}^{ - } (h)}},\,\tfrac{{q_{2}^{ - } (h) + q_{1}^{ - } (h)}}{{1 + q_{2}^{ - } (h)q_{1}^{ - } (h)}},} \\ {\tfrac{{r_{2}^{ - } (h) + r_{1}^{ - } (h)}}{{1 + r_{2}^{ - } (h)r_{1}^{ - } (h)}},\,\tfrac{{s_{2}^{ - } (h) + s_{1}^{ - } (h)}}{{1 + s_{2}^{ - } (h)s_{1}^{ - } (h)}}],} \\ {\hbox{max} (I_{{A_{2} }}^{ + } ,\,I_{{A_{1} }}^{ + } ){[}\tfrac{{p_{2}^{ + } (h) + p_{1}^{ + } (h)}}{{1 + p_{2}^{ + } (h)p_{1}^{ + } (h)}},\,\tfrac{{q_{2}^{ + } (h) + q_{1}^{ + } (h)}}{{1 + q_{2}^{ + } (h)q_{1}^{ + } (h)}},} \\ {\tfrac{{r_{2}^{ + } (h) + r_{1}^{ + } (h)}}{{1 + r_{2}^{ + } (h)r_{1}^{ + } (h)}},\,\tfrac{{s_{2}^{ + } (h) + s_{1}^{ + } (h)}}{{1 + s_{2}^{ + } (h)s_{1}^{ + } (h)}}],} \\ \end{array} } \right],} \right. $$
  $$ \left. {\left[ {\begin{array}{*{20}c} {\hbox{min} (\mu_{{A_{2} }} ,\,\mu_{{A_{1} }} ){[}\tfrac{{p_{2} (h) \cdot p_{1} (h)}}{{1 + (\left( {1 - p_{2} (h)} \right)\,\left( {1 - p_{1} (h)} \right))}},\,\tfrac{{q_{2} (h) \cdot q_{1} (h)}}{{1 + (\left( {1 - q_{2} (h)} \right)\,\left( {1 - q_{1} (h)} \right))}},} \\ {\tfrac{{r_{2} (h) \cdot r_{1} (h)}}{{1 + (\left( {1 - r_{2} (h)} \right)\,\left( {1 - r_{1} (h)} \right))}},\,\tfrac{{s_{2} (h) \cdot s_{1} (h)}}{{1 + (\left( {1 - s_{2} (h)} \right)\,\left( {1 - s_{1} (h)} \right))}}]} \\ \end{array} } \right]} \right\rangle , $$
  $$ = A_{2} + A_{1} . $$

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

• (2)
  $$ \lambda (A_{1} + A_{2} ) = \lambda A_{2} + \lambda A_{1} $$
  $$ \lambda (A_{1} + A_{2} ) = {[}s_{{\theta_{A}^{\lambda } }} ,\,s_{{t_{A}^{\lambda } }} ],\,\left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ - } I_{{A_{2} }}^{ - } )} \\ {\left[ {\tfrac{{{[}(1 + p_{1}^{ - } (h))(1 - p_{1}^{ - } (h))]^{\lambda } {[}(1 + p_{2}^{ - } (h))(1 - p_{2}^{ - } (h))]^{\lambda } }}{{{[}(1 + p_{1}^{ - } (h))(1 - p_{1}^{ - } (h))]^{\lambda } {[}(1 + p_{2}^{ - } (h))(1 - p_{2}^{ - } (h))]^{\lambda } }},} \right.} \\ {\tfrac{{{[}(1 + q_{1}^{ - } (h))(1 - q_{1}^{ - } (h))]^{\lambda } {[}(1 + q_{2}^{ - } (h))(1 - q_{2}^{ - } (h))]^{\lambda } }}{{{[}(1 + q_{1}^{ - } (h))(1 - q_{1}^{ - } (h))]^{\lambda } {[}(1 + q_{2}^{ - } (h))(1 - q_{2}^{ - } (h))]^{\lambda } }},} \\ {\tfrac{{{[}(1 + r_{1}^{ - } (h))(1 - r_{1}^{ - } (h))]^{\lambda } {[}(1 + r_{2}^{ - } (h))(1 - r_{2}^{ - } (h))]^{\lambda } }}{{{[}(1 + r_{1}^{ - } (h))(1 - r_{1}^{ - } (h))]^{\lambda } {[}(1 + r_{2}^{ - } (h))(1 - r_{2}^{ - } (h))]^{\lambda } }},} \\ {\left. {\tfrac{{{[}(1 + s_{1}^{ - } (h))(1 - s_{1}^{ - } (h))]^{\lambda } {[}(1 + s_{2}^{ - } (h))(1 - s_{2}^{ - } (h))]^{\lambda } }}{{{[}(1 + s_{1}^{ - } (h))(1 - s_{1}^{ - } (h))]^{\lambda } {[}(1 + s_{2}^{ - } (h))(1 - s_{2}^{ - } (h))]^{\lambda } }}} \right]} \\ \end{array} } \right],} \right. $$
  $$ \left[ {\begin{array}{*{20}c} {\hbox{max} (I_{{A_{1} }}^{ + } I_{{A_{2} }}^{ + } )} \\ {\left[ {\tfrac{{{[}(1 + p_{1}^{ + } (h))(1 - p_{1}^{ + } (h))]^{\lambda } {[}(1 + p_{p2}^{ + } (h))(1 - p_{2}^{ + } (h))]^{\lambda } }}{{{[}(1 + p_{1}^{ + } (h))(1 - p_{1}^{ + } (h))]^{\lambda } {[}(1 + p_{2}^{ + } (h))(1 - p_{2}^{ + } (h))]^{\lambda } }},} \right.} \\ {\tfrac{{{[}(1 + q_{1}^{ + } (h))(1 - q_{1}^{ + } (h))]^{\lambda } {[}(1 + q_{2}^{ + } (h))(1 - q_{2}^{ + } (h))]^{\lambda } }}{{{[}(1 + q_{1}^{ + } (h))(1 - q_{1}^{ + } (h))]^{\lambda } {[}(1 + q_{2}^{ + } (h))(1 - q_{2}^{ + } (h))]^{\lambda } }},} \\ {\tfrac{{{[}(1 + r_{1}^{ + } (h))(1 - r_{1}^{ + } (h))]^{\lambda } {[}(1 + r_{2}^{ + } (h))(1 - r_{2}^{ + } (h))]^{\lambda } }}{{{[}(1 + r_{1}^{ + } (h))(1 - r_{1}^{ + } (h))]^{\lambda } {[}(1 + r_{2}^{ + } (h))(1 - r_{2}^{ + } (h))]^{\lambda } }},} \\ {\left. {\tfrac{{{[}(1 + s_{1}^{ + } (h))(1 - s_{1}^{ + } (h))]^{\lambda } {[}(1 + s_{2}^{ + } (h))(1 - s_{2}^{ + } (h))]^{\lambda } }}{{{[}(1 + s_{1}^{ + } (h))(1 - s_{1}^{ + } (h))]^{\lambda } {[}(1 + s_{2}^{ + } (h))(1 - s_{2}^{ + } (h))]^{\lambda } }}} \right]} \\ \end{array} } \right], $$
  $$ \left. {,\,\left[ {\begin{array}{*{20}c} {\hbox{min} (\mu_{{A_{1} }} \mu_{{A_{2} }} );} \\ {{[}\tfrac{{2{[}p_{1} (h)p_{2} (h)]^{\lambda } }}{{{[}(4 - 2p_{1} (h) - 2p_{2} (h) - p_{1} (h)p_{2} (h)]^{\lambda } + {[}p_{1} (h)p_{2} (h)]^{\lambda } }},} \\ {\tfrac{{2{[}q_{1} (h)q_{2} (h)]^{\lambda } }}{{{[}(4 - 2q_{1} (h) - 2q_{2} (h) - q_{1} (h)q_{2} (h)]^{\lambda } + {[}q_{1} (h)q_{2} (h)]^{\lambda } }},} \\ {\tfrac{{2{[}r_{1} (h)r_{2} (h)]^{\lambda } }}{{{[}(4 - 2r_{1} (h) - 2r_{2} (h) - r_{1} (h)r_{2} (h)]^{\lambda } + {[}r_{1} (h)r_{2} (h)]^{\lambda } }},} \\ {\tfrac{{2{[}s_{1} (h)s_{2} (h)]^{\lambda } }}{{{[}(4 - 2s_{1} (h) - 2s_{2} (h) - s_{1} (h)s_{2} (h)]^{\lambda } + {[}s_{1} (h)s_{2} (h)]^{\lambda } }}]} \\ \end{array} } \right]} \right\rangle , $$

  and we have

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

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

• (3)
  $$ \lambda_{1} A + \lambda_{2} A = (\lambda_{1} + \lambda_{2} )A $$
  $$ \lambda_{1} A = {[}s_{{\theta_{A}^{\lambda } }} ,\,s_{{t_{A}^{\lambda } }} ],\,\left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } ),\left[ {\tfrac{{{[}1 + p_{A}^{ - } (h)]^{{\lambda_{1} }} - {[}1 - p_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{{[}1 + p_{A}^{ - } (h)]^{{\lambda_{1} }} + {[}1 - p_{A}^{ - } (h)]^{{\lambda_{1} }} }},} \right.} \\ {\tfrac{{{[}1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} - {[}1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{{[}1 + q_{A}^{ - } (h)]^{{\lambda_{1} }} + {[}1 - q_{A}^{ - } (h)]^{{\lambda_{1} }} }},} \\ {\tfrac{{{[}1 + r_{A}^{ - } (h)]^{{\lambda_{1} }} - {[}1 - r_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{{[}1 + r_{A}^{ - } (h)]^{{\lambda_{1} }} + {[}1 - r_{A}^{ - } (h)]^{{\lambda_{1} }} }},} \\ {\tfrac{{{[}1 + s_{A}^{ - } (h)]^{{\lambda_{1} }} - {[}1 - s_{A}^{ - } (h)]^{{\lambda_{1} }} }}{{{[}1 + s_{A}^{ - } (h)]^{{\lambda_{1} }} + {[}1 - s_{A}^{ - } (h)]^{{\lambda_{1} }} }}} \\ {\hbox{max} (I_{A}^{ + } ),\left[ {\tfrac{{{[}1 + p_{A}^{ + } (h)]^{{\lambda_{1} }} - {[}1 - p_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{{[}1 + p_{A}^{ + } (h)]^{{\lambda_{1} }} + {[}1 - p_{A}^{ + } (h)]^{{\lambda_{1} }} }},} \right.} \\ {\tfrac{{{[}1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} - {[}1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{{[}1 + q_{A}^{ + } (h)]^{{\lambda_{1} }} + {[}1 - q_{A}^{ + } (h)]^{{\lambda_{1} }} }},} \\ {\tfrac{{{[}1 + r_{A}^{ + } (h)]^{{\lambda_{1} }} - {[}1 - r_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{{[}1 + r_{A}^{ + } (h)]^{{\lambda_{1} }} + {[}1 - r_{A}^{ + } (h)]^{{\lambda_{1} }} }},} \\ {\left. {\tfrac{{{[}1 + s_{A}^{ + } (h)]^{{\lambda_{1} }} - {[}1 - s_{A}^{ + } (h)]^{{\lambda_{1} }} }}{{{[}1 + s_{A}^{ + } (h)]^{{\lambda_{1} }} + {[}1 - s_{A}^{ + } (h)]^{{\lambda_{1} }} }}} \right]} \\ \end{array} ,} \right]} \right., $$
  $$ \left. {\left[ {\begin{array}{*{20}c} {\hbox{min} (\mu_{A} )\left[ {\tfrac{{2{[}p_{A} (h)]^{{\lambda_{1} }} }}{{{[}(2 - p_{A} (h)]^{{\lambda_{1} }} + {[}p_{A} (h)]^{{\lambda_{1} }} }},} \right.} \\ {\tfrac{{2{[}q_{A} (h)]^{{\lambda_{1} }} }}{{{[}(2 - q_{A} (h)]^{{\lambda_{1} }} + {[}q_{A} (h)]^{{\lambda_{1} }} }},\,\tfrac{{2{[}r_{A} (h)]^{{\lambda_{1} }} }}{{{[}(2 - r_{A} (h)]^{{\lambda_{1} }} + {[}r_{A} (h)]^{{\lambda_{1} }} }},} \\ {\left. {\tfrac{{2{[}s_{A} (h)]^{{\lambda_{1} }} }}{{{[}(2 - s_{A} (h)]^{{\lambda_{1} }} + {[}s_{A} (h)]^{{\lambda_{1} }} }}} \right]} \\ \end{array} } \right]} \right\rangle , $$

  and

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

Appendix B: Proof of Theorem 1

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

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

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

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

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

and we have

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

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

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

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

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

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

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

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

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

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

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

and

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

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

$$ = {[}s_{{\theta_{A}^{{\lambda_{1} + \lambda_{2} }} }} ,\,s_{{t_{A}^{{\lambda_{1} + \lambda_{2} }} }} ],\,\left\langle {\left[ {\begin{array}{*{20}c} {\hbox{max} (I_{A}^{ - } ),} \\ {{[}\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} }} }},} \\ {\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_{2} }} + {[}1 - s_{A}^{ - } (h)]^{{\lambda_{1} + \lambda_{2} }} }}} \\ {\hbox{max} (I_{A}^{ + } ),} \\ {{[}\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} }} }},} \\ {\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} }} }}]} \\ \end{array} } \right],} \right. $$

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

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

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

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

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

$$ {[}s_{{\theta_{A}^{\omega } }} ,\,s_{{t_{A}^{\omega } }} ],\,\langle \hbox{max} (I_{A}^{ - } ), $$

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

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

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

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

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

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

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

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

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

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

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

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

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

In particular, if $w = (\tfrac{1}{n},\,\tfrac{1}{n}, \ldots ,\,\tfrac{1}{n})^{T} ,$ then the TrCLUFEWA operator is reduced to the trapezoidal cubic linguistic uncertain fuzzy Einstein weighing averaging operator, which is given as follows:

$$ {[}s_{{\theta_{A}^{\omega } }} ,\,s_{{t_{A}^{\omega } }} ],\,\langle \hbox{max} (I_{A}^{ - } ), $$

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

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

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

Appendix C: Proof of Proposition 2

• (1) (Idempotency)

  Since $A_{j} = A$ are equal to $\left\{ {\begin{array}{*{20}c} {{[}s_{\theta } ,\,s_{t} ],\,\langle {[}p^{ - } (h),\,q^{ - } (h),\,r^{ - } (h),\,s^{ - } (h)],(I_{A}^{ - } )} \\ {{[}p^{ + } (h),\,q^{ + } (h),\,r^{ + } (h),\,s^{ + } (h)],(I_{A}^{ + } )} \\ {{[}p(h),\,q(h),\,r(h),\,s(h)],(\mu_{A} )\rangle |h \in H} \\ \end{array} } \right\}$ for $(j = 1,\,2, \ldots ,\,n),$ TrCLUFEWA $(A_{1} ,\,A_{2} , \ldots ,\,A_{n} ) = {[}s_{{\theta_{A}^{\omega } }} ,\,s_{{t_{A}^{\omega } }} ]\langle \hbox{max} {[}I_{A}^{ - } ]$

  $$ \left[ {\begin{array}{*{20}c} {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p_{j}^{ - } (h)]^{{\varpi_{j} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - p_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - p_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - q_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - q_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} }},} \\ {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - r_{j}^{ - } (h)]^{{\varpi_{j} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - r_{j}^{ - } (h)]^{{\varpi_{j} }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - s_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - s_{j}^{ - } (h)]^{{^{{\varpi_{j} }} }} }}} \\ \end{array} } \right], $$
  $$ \hbox{max} {[}(I_{A}^{ + } ]\left[ {\begin{array}{*{20}c} {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - p_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - p_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - q_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - q_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }},} \\ {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - r_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - r_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - s_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - s_{j}^{ + } (h)]^{{^{{\varpi_{j} }} }} }}} \\ \end{array} } \right], $$
  $$ \hbox{min} {[}(\mu_{A} ]\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}p_{j} (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - p_{j} (h)]^{{\varpi_{j} }} + \mathop \prod \limits_{j = 1}^{n} {[}p_{j} (h)]^{{^{{\varpi_{j} }} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}q_{j} (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - q_{j} (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}q_{j} (h)]^{{^{{\varpi_{j} }} }} }},} \\ {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}r_{j} (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - r_{j} (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}r_{j} (h)]^{{^{{\varpi_{j} }} }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}s_{j} (h)]^{{^{{\varpi_{j} }} }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - s_{j} (h)]^{{^{{\varpi_{j} }} }} + \mathop \prod \limits_{j = 1}^{n} {[}s_{j} (h)]^{{^{{\varpi_{j} }} }} }}} \\ \end{array} } \right], $$
  $$ = {[}s_{{\theta_{A}^{\omega } }} ,\,s_{{t_{A}^{\omega } }} ],\,\langle \hbox{max} {[}(I_{A}^{ - } ]\left[ {\begin{array}{*{20}c} {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p^{ - } (h)]^{\varpi } - \mathop \prod \limits_{j = 1}^{n} {[}1 - p^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - p^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - q^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - q^{ - } (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - r^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - r^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s^{ - } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - s^{ - } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s^{ - } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - s^{ - } (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right]; $$
  $$ \hbox{max} {[}(I_{A}^{ + } )]\left[ {\begin{array}{*{20}c} {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - p^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + p^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - p^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - q^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + q^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - q^{ + } (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - r^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + r^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - r^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s^{ + } (h)]^{{^{\varpi } }} - \mathop \prod \limits_{j = 1}^{n} {[}1 - s^{ + } (h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}1 + s^{ + } (h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}1 - s^{ + } (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right], $$
  $$ \hbox{min} {[}(\mu_{A} )]\left[ {\begin{array}{*{20}c} {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}p(h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - p(h)]^{\mu } + \mathop \prod \limits_{j = 1}^{n} {[}p(h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}q(h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - q(h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}q(h)]^{{^{\varpi } }} }},} \\ {\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}r(h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - r(h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}r(h)]^{{^{\varpi } }} }},\,\tfrac{{2\mathop \prod \limits_{j = 1}^{n} {[}s(h)]^{{^{\varpi } }} }}{{\mathop \prod \limits_{j = 1}^{n} {[}(2 - s(h)]^{{^{\varpi } }} + \mathop \prod \limits_{j = 1}^{n} {[}s(h)]^{{^{\varpi } }} }}} \\ \end{array} } \right], $$
  $$ = {[}s_{{\theta_{A}^{\omega } }} ,\,s_{{t_{A}^{\omega } }} ],\,\langle {[}I_{A}^{ - } ]\left[ {\begin{array}{*{20}c} {\tfrac{{{[}1 + p^{ - } (h)]^{\varpi } - {[}1 - p^{ - } (h)]^{{^{\varpi } }} }}{{{[}1 + p^{ - } (h)]^{{^{\varpi } }} + {[}1 - p^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{{[}1 + q^{ - } (h)]^{{^{\varpi } }} - {[}1 - q^{ - } (h)]^{{^{\varpi } }} }}{{{[}1 + q^{ - } (h)]^{{^{\varpi } }} + {[}1 - q^{ - } (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{{[}1 + r^{ - } (h)]^{{^{\varpi } }} - {[}1 - r^{ - } (h)]^{{^{\varpi } }} }}{{{[}1 + r^{ - } (h)]^{{^{\varpi } }} + {[}1 - r^{ - } (h)]^{{^{\varpi } }} }},\,\tfrac{{{[}1 + s^{ - } (h)]^{{^{\varpi } }} - {[}1 - s^{ - } (h)]^{{^{\varpi } }} }}{{{[}1 + s^{ - } (h)]^{{^{\varpi } }} + {[}1 - s^{ - } (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right], $$
  $$ {[}I_{A}^{ + } ]\left[ {\begin{array}{*{20}c} {\tfrac{{{[}1 + p^{ + } (h)]^{{^{\varpi } }} - {[}1 - p^{ + } (h)]^{{^{\varpi } }} }}{{{[}1 + p^{ + } (h)]^{{^{\varpi } }} + {[}1 - p^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{{[}1 + q^{ + } (h)]^{{^{\varpi } }} - {[}1 - q^{ + } (h)]^{{^{\varpi } }} }}{{{[}1 + q^{ + } (h)]^{{^{\varpi } }} + {[}1 - q^{ + } (h)]^{{^{\varpi } }} }},} \\ {\tfrac{{{[}1 + r^{ + } (h)]^{{^{\varpi } }} - {[}1 - r^{ + } (h)]^{{^{\varpi } }} }}{{{[}1 + r^{ + } (h)]^{{^{\varpi } }} + {[}1 - r^{ + } (h)]^{{^{\varpi } }} }},\,\tfrac{{{[}1 + s^{ + } (h)]^{{^{\varpi } }} - {[}1 - s^{ + } (h)]^{{^{\varpi } }} }}{{{[}1 + s^{ + } (h)]^{{^{\varpi } }} + {[}1 - s^{ + } (h)]^{{^{\varpi } }} }}} \\ \end{array} } \right], $$
  $$ {[}(\mu_{A} )]\left[ {\begin{array}{*{20}c} {\tfrac{{2{[}p(h)]^{{^{\varpi } }} }}{{{[}(2 - p(h)]^{\mu } + {[}p(h)]^{{^{\varpi } }} }},\,\tfrac{{2{[}q(h)]^{{^{\varpi } }} }}{{{[}(2 - q(h)]^{{^{\varpi } }} + {[}q(h)]^{{^{\varpi } }} }},} \\ {\tfrac{{2{[}r(h)]^{{^{\varpi } }} }}{{{[}(2 - r(h)]^{{^{\varpi } }} + {[}r(h)]^{{^{\varpi } }} }},\,\tfrac{{2{[}s(h)]^{{^{\varpi } }} }}{{{[}(2 - s(h)]^{{^{\varpi } }} + {[}s(h)]^{{^{\varpi } }} }}} \\ \end{array} } \right] = $$
  $$ \left\{ {\begin{array}{*{20}c} {{[}s_{\theta } ,\,s_{t} ],\,\langle {[}p^{ - } (h),\,q^{ - } (h),\,r^{ - } (h),\,s^{ - } (h)],(I_{A}^{ - } )} \\ {{[}p^{ + } (h),\,q^{ + } (h),\,r^{ + } (h),\,s^{ + } (h)],(I_{{A^{ + } }} )} \\ {{[}p(h),\,q(h),\,r(h),\,s(h)],(\mu_{A} )\rangle |h \in H} \\ \end{array} } \right\} = A. $$

  TrCLUFEWA $(A_{1} ,\,A_{2} , \ldots ,\,A_{n} ) = A.$

The proof is completed.