Appendix 1: Proof of theorem 3.1
Obviously, rules (1) and (2) are correct. For rule (3), we have
$$\lambda (sd_{1} \oplus sd_{2} ) = \lambda \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} ,\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\alpha _{1} + \alpha _{2} }} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {} } \right.} \left. {\{ \{ [\mu _{1}^{L} + \mu _{2}^{L} - \mu _{1}^{L} \mu _{2}^{L} ,\mu _{1}^{U} + \mu _{2}^{U} - \mu _{1}^{U} \mu _{2}^{U} ]\} ,\{ [\nu _{1}^{L} \nu _{2}^{L} ,\nu _{1}^{U} \nu _{2}^{U} ]\} \} } \right) = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} ,\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\lambda (\alpha _{1} + \alpha _{2} )}} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {} } \right.} \left. {\left. {\{ \{ [1 - (1 - (\mu _{1}^{L} + \mu _{2}^{L} - \mu _{1}^{L} \mu _{2}^{L} ))^{\lambda } ,1 - (1 - (\mu _{1}^{U} + \mu _{2}^{U} - \mu _{1}^{U} \mu _{2}^{U} ))^{\lambda } ]\} ,\{ [(\nu _{1}^{L} \nu _{2}^{L} )^{\lambda } ,(\nu _{1}^{U} \nu _{2}^{U} )^{\lambda } ]\} \} } \right\rangle } \right\}, \lambda sd_{1} = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} }} {\left( {s_{{\lambda \alpha _{1} }} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} }} {\{ \{ [1 - (1 - \mu _{1}^{L} )^{\lambda } ,1 - (1 - \mu _{1}^{U} )^{\lambda } ]\} ,\{ [(\nu _{1}^{L} )^{\lambda } ,(\nu _{1}^{U} )^{\lambda } ]\} \} } } \right)} \lambda sd_{2} = \bigcup\nolimits_{{\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\lambda \alpha _{2} }} ,\bigcup\nolimits_{{[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {\{ \{ [1 - (1 - \mu _{2}^{L} )^{\lambda } ,1 - (1 - \mu _{2}^{U} )^{\lambda } ]\} ,\{ [(\nu _{2}^{L} )^{\lambda } ,(\nu _{2}^{U} )^{\lambda } ]\} \} } } \right)} \lambda sd_{1} \oplus \lambda sd_{2} = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} ,\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\lambda (\alpha _{1} + \alpha _{2} )}} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {} } \right.} \left. {\{ \{ [1 - (1 - (\mu _{1}^{L} + \mu _{2}^{L} - \mu _{1}^{L} \mu _{2}^{L} ))^{\lambda } ,1 - (1 - (\mu _{1}^{U} + \mu _{2}^{U} - \mu _{1}^{U} \mu _{2}^{U} ))^{\lambda } ]\} ,\{ [(\nu _{1}^{L} \nu _{2}^{L} )^{\lambda } ,(\nu _{1}^{U} \nu _{2}^{U} )^{\lambda } ]\} \} } \right) = \lambda (sd_{1} \oplus sd_{2} );$$
For rule (4),
$$sd_{1} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd}} {\left( {s_{{\alpha _{1}^{\lambda } }} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} }} {\{ \{ [(\mu _{1}^{L} )^{\lambda } ,(\mu _{1}^{U} )^{\lambda } ]\} ,\{ [1 - (1 - \nu _{1}^{L} )^{\lambda } ,1 - (1 - \nu _{1}^{U} )^{\lambda } ]\} \} } } \right)} , sd_{2} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\alpha _{2}^{\lambda } }} ,\bigcup\nolimits_{{[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {\{ \{ [(\mu _{2}^{L} )^{\lambda } ,(\mu _{2}^{U} )^{\lambda } ]\} ,\{ [1 - (1 - \nu _{2}^{L} )^{\lambda } ,1 - (1 - \nu _{2}^{U} )^{\lambda } ]\} \} } } \right)} , sd_{1} \otimes sd_{2} = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} ,\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\alpha _{1} \alpha _{2} }} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {} } \right.} \left. {\{ \{ [\mu _{1}^{L} \mu _{2}^{L} ,\mu _{1}^{U} \mu _{2}^{U} ]\} ,\{ [\nu _{1}^{L} + \nu _{2}^{L} - \nu _{1}^{L} \nu _{2}^{L} ,\nu _{1}^{U} + \nu _{2}^{U} - \nu _{1}^{U} \nu _{2}^{U} ]\} \} } \right), sd_{1} ^{\lambda } \otimes sd_{2} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} ,\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{(\alpha _{1} \alpha _{2} )^{\lambda } }} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {} } \right.} \left. {\{ \{ [(\mu _{1}^{L} \mu _{2}^{L} )^{\lambda } ,(\mu _{1}^{U} \mu _{2}^{U} )^{\lambda } ]\} ,\{ [1 - (1 - (\nu _{1}^{L} + \nu _{2}^{L} - \nu _{1}^{L} \nu _{2}^{L} ))^{\lambda } ,1 - (1 - (\nu _{1}^{U} + \nu _{2}^{U} - \nu _{1}^{U} \nu _{2}^{U} ))^{\lambda } ]\} \} } \right) = (sd_{1} \otimes sd_{2} )^{\lambda } ;$$
For rule (5),
$$\begin{aligned} & \lambda_{1} sd = \bigcup\limits_{{(s_{\alpha } ,\tilde{h},\tilde{g}) \in sd}} {\left( {s_{{\lambda_{1} \alpha }} ,\bigcup\limits_{{[\mu^{L} ,\mu^{U} ] \in \tilde{h},[\nu^{L} ,\nu^{U} ] \in \tilde{g}}} {\{ \{ [1 - (1 - \mu^{L} )^{{\lambda_{1} }} ,1 - (1 - \mu^{U} )^{{\lambda_{1} }} ]\} ,\{ [(\nu^{L} )^{{\lambda_{1} }} ,(\nu^{U} )^{{\lambda_{1} }} ]\} \} } } \right)} , \\ & \lambda_{2} sd = \bigcup\limits_{{(s_{\alpha } ,\tilde{h},\tilde{g}) \in sd}} {\left( {s_{{\lambda_{2} \alpha }} ,\bigcup\limits_{{[\mu^{L} ,\mu^{U} ] \in \tilde{h},[\nu^{L} ,\nu^{U} ] \in \tilde{g}}} {\{ \{ [1 - (1 - \mu^{L} )^{{\lambda_{2} }} ,1 - (1 - \mu^{U} )^{{\lambda_{2} }} ]\} ,\{ [(\nu^{L} )^{{\lambda_{2} }} ,(\nu^{U} )^{{\lambda_{2} }} ]\} \} } } \right)} , \\ & \lambda_{1} sd + \lambda_{2} sd = \bigcup\limits_{{(s_{\alpha } ,\tilde{h},\tilde{g}) \in sd}} {\left( {s_{{(\lambda_{1} + \lambda_{2} )\alpha }} ,} \right.} \\ & \quad \left. {\bigcup\limits_{{[\mu^{L} ,\mu^{U} ] \in \tilde{h},[\nu^{L} ,\nu^{U} ] \in \tilde{g}}} {\{ \{ [1 - (1 - \mu^{L} )^{{\lambda_{1} + \lambda_{2} }} ,1 - (1 - \mu^{U} )^{{\lambda_{1} + \lambda_{2} }} ]\} ,\{ [(\nu^{L} )^{{\lambda_{1} + \lambda_{2} }} ,(\nu^{U} )^{{\lambda_{1} + \lambda_{2} }} ]\} \} } } \right) \\ & \quad = (\lambda_{1} + \lambda_{2} )sd; \\ \end{aligned}$$
For rule (6)
$$sd^{{\lambda _{1} }} = \bigcup\nolimits_{{\left( {s_{\alpha } ,\tilde{h},\tilde{g}} \right) \in sd}} {\left( {s_{{\alpha ^{{\lambda _{1} }} }} ,\bigcup\nolimits_{{[\mu ^{L} ,\mu ^{U} ] \in \tilde{h},[\nu ^{L} ,\nu ^{U} ] \in \tilde{g}}} {\{ \{ [(\mu ^{L} )^{{\lambda _{1} }} ,(\mu ^{U} )^{{\lambda _{1} }} ]\} ,\{ [1 - (1 - \nu ^{L} )^{{\lambda _{1} }} ,1 - (1 - \nu ^{U} )^{{\lambda _{1} }} ]\} \} } } \right)} , sd^{{\lambda _{2} }} = \bigcup\nolimits_{{\left( {s_{\alpha } ,\tilde{h},\tilde{g}} \right) \in sd}} {\left( {s_{{\alpha ^{{\lambda _{2} }} }} ,\bigcup\nolimits_{{[\mu ^{L} ,\mu ^{U} ] \in \tilde{h},[\nu ^{L} ,\nu ^{U} ] \in \tilde{g}}} {\{ \{ [(\mu ^{L} )^{{\lambda _{2} }} ,(\mu ^{U} )^{{\lambda _{2} }} ]\} ,\{ [1 - (1 - \nu ^{L} )^{{\lambda _{2} }} ,1 - (1 - \nu ^{U} )^{{\lambda _{2} }} ]\} \} } } \right)} , sd^{{\lambda _{1} }} \otimes sd^{{\lambda _{2} }} = \bigcup\nolimits_{{\left( {s_{\alpha } ,\tilde{h},\tilde{g}} \right) \in sd}} {\left( {s_{{\alpha ^{{\lambda _{1} + \lambda _{2} }} }} ,} \right.} \left. {\bigcup\nolimits_{{[\mu ^{L} ,\mu ^{U} ] \in \tilde{h},[\nu ^{L} ,\nu ^{U} ] \in \tilde{g}}} {\{ \{ [(\mu ^{L} )^{{\lambda _{1} + \lambda _{2} }} ,(\mu ^{U} )^{{\lambda _{1} + \lambda _{2} }} ]\} ,\{ [1 - (1 - \nu ^{L} )^{{\lambda _{1} + \lambda _{2} }} ,1 - (1 - \nu ^{U} )^{{\lambda _{1} + \lambda _{2} }} ]\} \} } } \right) = sd^{{\lambda _{1} + \lambda _{2} }}$$
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Appendix 2: Proof of Theorem 4.1
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1.
When \(n = 1,\) obviously, it is right.
$$WGIVDHFLPA(sd) = \left( {s_{\alpha } ,\bigcup\limits_{{[\mu^{L} ,\mu^{U} ] \in \tilde{h},[\nu^{L} ,\nu^{U} ] \in \tilde{g}}} {\{ \{ [\mu_{j}^{L} ,\mu_{j}^{U} ]\} ,\{ [\nu_{j}^{L} ,\nu_{j}^{U} ]\} \} } } \right).$$
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2.
When \(n = 2,\)
$$sd_{1} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} }} {\left( {s_{{\alpha _{1} ^{\lambda } }} ,\bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} }} {\left\{ {\left\{ {\left[ {(\mu _{1}^{L} )^{\lambda } ,(\mu _{1}^{U} )^{\lambda } } \right]} \right\},\left\{ {\left[ {1 - (1 - \nu _{1}^{L} )^{\lambda } ,1 - (1 - \nu _{1}^{U} )^{\lambda } } \right]} \right\}} \right\}} } \right)} , \frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}sd_{1} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} }} {\left( {s_{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}\alpha _{1} ^{\lambda } }} ,} \right.} \bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} }} {\left\{ {\left\{ {\left[ {1 - (1 - (\mu _{1}^{L} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} ,1 - (1 - (\mu _{1}^{U} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right.,} \left. {\left. {\left\{ {\left[ {(1 - (1 - \nu _{1}^{L} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} ,(1 - (1 - \nu _{1}^{U} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right\}} \right), sd_{2} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\alpha _{2} ^{\lambda } }} ,\bigcup\nolimits_{{[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {\left\{ {\left\{ {\left[ {(\mu _{2}^{L} )^{\lambda } ,(\mu _{2}^{U} )^{\lambda } } \right]} \right\},\left\{ {\left[ {1 - (1 - \nu _{2}^{L} )^{\lambda } ,1 - (1 - \nu _{2}^{U} )^{\lambda } } \right]} \right\}} \right\}} } \right)} , \frac{{\omega _{2} (1 + T(sd_{2} )}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}sd_{2} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}\alpha _{2} ^{\lambda } }} ,} \right.} \bigcup\nolimits_{{[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {\left\{ {\left\{ {\left[ {1 - (1 - (\mu _{2}^{L} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} ,1 - (1 - (\mu _{2}^{U} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right.,} \left. {\left. {\left\{ {\left[ {(1 - (1 - \nu _{2}^{L} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} ,(1 - (1 - \nu _{2}^{U} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right\}} \right), \frac{{\omega _{1} (1 + T(sd_{1} )}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}sd_{1} ^{\lambda } + \frac{{\omega _{2} (1 + T(sd_{2} )}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}sd_{2} ^{\lambda } = \bigcup\nolimits_{{\left( {s_{{\alpha _{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} } \right) \in sd_{1} ,\left( {s_{{\alpha _{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} } \right) \in sd_{2} }} {\left( {s_{{\sum\limits_{{j = 1}}^{2} {\frac{{\omega _{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}(\alpha _{j} )^{\lambda } } }} ,} \right.} \bigcup\nolimits_{{[\mu _{1}^{L} ,\mu _{1}^{U} ] \in \tilde{h}_{1} ,[\mu _{2}^{L} ,\mu _{2}^{U} ] \in \tilde{h}_{2} ,[\nu _{1}^{L} ,\nu _{1}^{U} ] \in \tilde{g}_{1} ,[\nu _{2}^{L} ,\nu _{2}^{U} ] \in \tilde{g}_{2} }} {\left\{ {\left\{ {\left[ {1 - (1 - (\mu _{1}^{L} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} (1 - (\mu _{2}^{L} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} ,} \right.} \right.} \right.} \, \left. {\left. {1 - (1 - (\mu _{1}^{U} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} (1 - (\mu _{2}^{U} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}, \left\{ {\left[ {(1 - (1 - \nu _{1}^{L} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} (1 - (1 - \nu _{2}^{L} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right.} \right., \left. {\left. {\left. {\left. {(1 - (1 - \nu _{1}^{U} )^{\lambda } )^{{\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} (1 - (1 - \nu _{2}^{U} )^{\lambda } )^{{\frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right\}} \right), \left( {\frac{{\omega _{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}sd_{1} ^{\lambda } + \frac{{\omega _{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}sd_{2} ^{\lambda } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} = \bigcup\nolimits_{{\left( {s_{{\alpha _{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} } \right) \in sd_{j} }} {\left( {s_{{\left( {\sum\limits_{{j = 1}}^{2} {\frac{{\omega _{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{{i = 1}}^{2} {\omega _{i} (1 + T(sd_{i} ))} }}(\alpha _{j} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} }} ,} \right.} \cup _{{[\mu _{j}^{L} ,\mu _{j}^{U} ] \in \tilde{h}_{j} ,[\nu _{j}^{L} ,\nu _{j}^{U} ] \in \tilde{g}_{j} }} \left\{ {\left\{ {\left[ {\left( {1 - \prod\limits_{{j = 1}}^{2} {(1 - (\mu _{j}^{L} )^{\lambda } )^{{\frac{{\omega _{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{{i = 1}}^{n} {\omega _{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} ,\left( {1 - \prod\limits_{{j = 1}}^{2} {(1 - (\mu _{j}^{U} )^{\lambda } )^{{\frac{{\omega _{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{{i = 1}}^{n} {\omega _{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} } \right]} \right\},} \right. \left. {\left. {\left\{ {\left[ {1 - \left( {1 - \prod\limits_{{j = 1}}^{2} {(1 - (1 - \nu _{j}^{L} )^{\lambda } )^{{\frac{{\omega _{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{{i = 1}}^{n} {\omega _{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} ,1 - \left( {1 - \prod\limits_{{j = 1}}^{2} {(1 - (1 - \nu _{j}^{U} )^{\lambda } )^{{\frac{{\omega _{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{{i = 1}}^{n} {\omega _{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} } \right]} \right\}} \right\}} \right).$$
So when \(n = 2,\) Theorem 4.1 also is right.
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3.
Suppose when \(n = k,\) Theorem 4.1 is right, then we have
$$\begin{aligned} & WGIVDHFLPA_{\omega ,\lambda } (sd_{1} ,sd_{2} ,\ldots ,sd_{k} ) = \bigcup\limits_{{(s_{{\alpha_{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} ) \in sd_{j} }} {\left( {s_{{\left( {\sum\nolimits_{j = 1}^{k} {\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}(\alpha_{j} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} }} ,} \right.} \\ & \quad \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} ,[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\left\{ {\left\{ {\left[ {\left( {1 - \prod\limits_{j = 1}^{k} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,\left( {1 - \prod\limits_{j = 1}^{k} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\},} \right.} \\ &\left. {\left. { \quad \quad \left\{ {\left[ {1 - \left( {1 - \prod\limits_{j = 1}^{k} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,1 - \left( {1 - \prod\limits_{j = 1}^{k} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\}} \right\}} \right), \\ & \mathop \oplus \limits_{j = 1}^{k} \omega_{j} sd_{j}^{\lambda } = \bigcup\limits_{{(s_{{\alpha_{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} ) \in sd_{j} }} {\left( {s_{{\sum\nolimits_{j = 1}^{k} {\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{k} {\omega_{i} (1 + T(sd_{i} ))} }}(\alpha_{j} )^{\lambda } } }} ,} \right.} \\ & \quad \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} ,[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\left\{ {\left\{ {\left[ {1 - \prod\limits_{j = 1}^{k} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } ,1 - \prod\limits_{j = 1}^{k} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right]} \right\},} \right.} \\ &\left. {\left. { \quad \quad \left\{ {\left[ {\prod\limits_{j = 1}^{k} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } ,\prod\limits_{j = 1}^{k} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right]} \right\}} \right\}} \right). \\ \end{aligned}$$
Then when \(n = k + 1,\)
$$\begin{aligned} & WGIVDHFLPA_{\omega ,\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{k + 1} ) \\ & \quad = \left( {\left( {\mathop \oplus \limits_{j = 1}^{k} \frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}sd_{j}^{\lambda } } \right) \oplus \frac{{\omega_{k + 1} (1 + T(sd_{k + 1} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}sd_{k + 1}^{\lambda } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = \bigcup\limits_{{(s_{{\alpha_{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} ) \in sd_{j} }} {\left( {s_{{\left( {\sum\nolimits_{j = 1}^{k + 1} {\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}(\alpha_{j} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} }} ,} \right.} \\ & \quad \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} ,[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\left\{ {\left\{ {\left[ {\left( {1 - \prod\limits_{j = 1}^{k + 1} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,\left( {1 - \prod\limits_{j = 1}^{k + 1} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\},} \right.} \\ &\left. {\left. { \quad \quad \left\{ {\left[ {1 - \left( {1 - \prod\limits_{j = 1}^{k + 1} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,1 - \left( {1 - \prod\limits_{j = 1}^{k + 1} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\}} \right\}} \right). \\ \end{aligned}$$
So, when \(n = k + 1,\) Theorem 4.1 is right too.
According to steps (1), (2), (3), we can conclude that Theorem 4.1 is right for all n.□
Appendix 3: Proof of Theorem 4.4
1. When \(n = 1,\) obviously, it is right.
2. When \(n = 2,\)
$$\begin{aligned} & (\lambda sd_{1} )^{{\frac{{(\omega_{1} (1 + T(sd_{1} )))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} = \bigcup\limits_{{(s_{{\alpha_{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} ) \in sd_{1} }} {\left( {s_{{\left( {\lambda \alpha_{1} } \right)^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} }} ,} \right.} \\ & \quad \bigcup\limits_{{[\mu_{1}^{L} ,\mu_{1}^{U} ] \in \tilde{h}_{1} ,[\nu_{1}^{L} ,\nu_{1}^{U} ] \in \tilde{g}_{1} }} {\left\{ {\left\{ {\left[ {(1 - (1 - \mu_{1}^{L} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} ,(1 - (1 - \mu_{1}^{U} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\},} \right.} \\ & \left. {\left. { \quad \quad \left\{ {\left[ {1 - (1 - (\nu_{1}^{L} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} ,1 - (1 - (\nu_{1}^{U} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right\}} \right), \\ & (\lambda sd_{2} )^{{\frac{{(\omega_{2} (1 + T(sd_{2} )))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} = \bigcup\limits_{{(s_{{\alpha_{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} ) \in sd_{2} }} {\left( {s_{{\left( {\lambda \alpha_{2} } \right)^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} }} ,} \right.} \\ & \quad \bigcup\limits_{{[\mu_{2}^{L} ,\mu_{2}^{U} ] \in \tilde{h}_{2} ,[\nu_{2}^{L} ,\nu_{2}^{U} ] \in \tilde{g}_{2} }} {\left\{ {\left\{ {\left[ {(1 - (1 - \mu_{2}^{L} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} ,(1 - (1 - \mu_{2}^{U} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\},} \right.} \\ & \left. {\left. { \quad \quad \left\{ {\left[ {1 - (1 - (\nu_{2}^{L} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} ,1 - (1 - (\nu_{2}^{U} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right\}} \right), \\ & (\lambda sd_{1} )^{{\omega_{1} }} \otimes (\lambda sd_{2} )^{{\omega_{2} }} = \bigcup\limits_{{(s_{{\alpha_{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} ) \in sd_{1} ,(s_{{\alpha_{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} ) \in sd_{2} }} {} \\ & \quad \quad \left( {s_{{(\lambda \alpha_{1} )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (\lambda \alpha_{2} )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} }} ,} \right.\bigcup\limits_{{[\mu_{1}^{L} ,\mu_{1}^{U} ] \in \tilde{h}_{1} ,[\mu_{2}^{L} ,\mu_{2}^{U} ] \in \tilde{h}_{2} ,[\nu_{1}^{L} ,\nu_{1}^{U} ] \in \tilde{g}_{1} ,[\nu_{2}^{L} ,\nu_{2}^{U} ] \in \tilde{g}_{2} }} {} \\ & \quad \quad \left\{ {\left\{ {\left[ {(1 - (1 - \mu_{1}^{L} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (1 - \mu_{2}^{L} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} ,(1 - (1 - \mu_{1}^{U} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (1 - \mu_{2}^{U} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\},} \right. \\ & \left. {\left. { \quad \quad \left\{ {\left[ {1 - (1 - (\nu_{1}^{L} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (\nu_{2}^{L} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} ,1 - (1 - (\nu_{1}^{U} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (\nu_{2}^{U} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right]} \right\}} \right\}} \right), \\ & \frac{1}{\lambda }((\lambda sd_{1} )^{{\omega_{1} }} \otimes (\lambda sd_{2} )^{{\omega_{2} }} ) = \bigcup\limits_{{\langle \tilde{s}_{{\vartheta_{1} }} ,\tilde{h}_{1} ,\tilde{g}_{1} \rangle \in sd_{1} ,\langle \tilde{s}_{{\vartheta_{2} }} ,\tilde{h}_{2} ,\tilde{g}_{2} \rangle \in sd_{2} }} {} \\ & \quad \quad \left( {s_{{\frac{1}{\lambda }\left( {(\lambda \alpha_{1} )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (\lambda \alpha_{2} )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right)}} } \right.,\bigcup\limits_{{[\mu_{1}^{L} ,\mu_{1}^{U} ] \in \tilde{h}_{1} ,[\mu_{2}^{L} ,\mu_{2}^{U} ] \in \tilde{h}_{2} ,[\nu_{1}^{L} ,\nu_{1}^{U} ] \in \tilde{g}_{1} ,[\nu_{2}^{L} ,\nu_{2}^{U} ] \in \tilde{g}_{2} }} {} \\ & \quad \quad \left\{ {\left\{ {\left[ {1 - \left( {1 - (1 - (1 - \mu_{1}^{L} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (1 - \mu_{2}^{L} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right)} \right.} \right.^{{\frac{1}{\lambda }}} ,} \right. \\ & \left. {\left. { \quad \quad 1 - \left( {1 - (1 - (1 - \mu_{1}^{U} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (1 - \mu_{2}^{U} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right)^{{\frac{1}{\lambda }}} } \right]} \right\}, \\ & \quad \quad \left\{ {\left[ {\left( {1 - (1 - (\nu_{1}^{L} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (\nu_{2}^{L} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right)^{{\frac{1}{\lambda }}} ,} \right.} \right. \\ & \quad \quad \left. {\left. {\left. {\left. {\left( {1 - (1 - (\nu_{1}^{U} )^{\lambda } )^{{\frac{{\omega_{1} (1 + T(sd_{1} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} (1 - (\nu_{2}^{U} )^{\lambda } )^{{\frac{{\omega_{2} (1 + T(sd_{2} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } \right)^{{\frac{1}{\lambda }}} } \right]} \right\}} \right\}} \right). \\ \end{aligned}$$
So when \(n = 2,\) Theorem 4.4 also is right.
3. Suppose when \(n = k,\) Theorem 4.4 is right, then we have
$$\begin{aligned} & WGIVDHFLPGA_{\omega ,\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{k} ) = \bigcup\limits_{{(s_{{\alpha_{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} ) \in sd_{j} }} {} \\ & \quad \quad \left( {s_{{\frac{1}{\lambda }\prod\nolimits_{j = 1}^{k} {(\lambda \alpha_{j} )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } }} ,} \right.\bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} ,[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {} \\ & \quad \quad \left\{ {\left\{ {\left[ {1 - \left( {1 - \prod\limits_{j = 1}^{k} {(1 - (1 - \mu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,1 - \left( {1 - \prod\limits_{j = 1}^{k} {(1 - (1 - \mu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\},} \right. \\& \left. {\left. { \quad \quad \left\{ {\left[ {\left( {1 - \prod\limits_{j = 1}^{k} {(1 - (\nu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,\left( {1 - \prod\limits_{j = 1}^{k} {(1 - (\nu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\}} \right\}} \right), \\ & \quad \mathop \otimes \limits_{j = 1}^{k} (\lambda sd_{j} )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} = \bigcup\limits_{{(s_{{\alpha_{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} ) \in sd_{j} }} {\left( {s_{{\prod\nolimits_{j = 1}^{k} {(\lambda \alpha_{j} )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } }} ,} \right.} \\ & \quad \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} ,[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\left\{ {\left\{ {\left[ {\prod\limits_{j = 1}^{k} {(1 - (1 - \mu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } ,\prod\limits_{j = 1}^{k} {(1 - (1 - \mu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right]} \right\},} \right.} \\& \left. {\left. { \quad \quad \left\{ {\left[ {1 - \prod\limits_{j = 1}^{k} {(1 - (\nu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } ,1 - \prod\limits_{j = 1}^{k} {(1 - (\nu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{2} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right]} \right\}} \right\}} \right), \\ \end{aligned}$$
then when \(n = k + 1,\)
$$\begin{aligned} & WGIVDHFLPGA_{\omega ,\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{k + 1} ) = \frac{1}{\lambda }\left( {\left( {\mathop \otimes \limits_{j = 1}^{k} (\lambda sd_{j} )^{{\omega_{j} }} } \right) \otimes (\lambda sd_{k + 1} )^{{\omega_{k + 1} }} } \right) \\ & \quad = \bigcup\limits_{{(s_{{\alpha_{j} }} ,\tilde{h}_{j} ,\tilde{g}_{j} ) \in sd_{j} }} {\left( {s_{{\frac{1}{\lambda }\prod\nolimits_{j = 1}^{n} {(\lambda \alpha_{j} )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } }} ,} \right.} \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} ,[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {} \\ & \quad \quad \left\{ {\left\{ {\left[ {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \mu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \mu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\}} \right., \\ & \quad \quad \left. {\left. {\left\{ {\left[ {\left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\nu_{j}^{L} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,\left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\nu_{j}^{U} )^{\lambda } )^{{\frac{{\omega_{j} (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {\omega_{i} (1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right]} \right\}} \right\}} \right). \\ \end{aligned}$$
So, when \(n = k + 1,\) Theorem 4.4 is right too.
According to steps (1), (2), (3), we can conclude that Theorem 4.4 is right for all n.□
Appendix 4: Proof of Theorem 4.7
1. Assume that \((sd_{1}^{*} ,sd_{2}^{*} , \ldots ,sd_{n}^{*} )\) is any permutation of \((sd_{1} ,sd_{2} , \ldots ,sd_{n} ),\) then for each \(sd_{j} ,\) there exists one and only one \(sd_{k}^{*}\) such that \(sd_{j} = sd_{k}^{*}\) and vice versa. And, \(T(sd_{j} ) = T(sd_{k}^{*} ).\) Thus, based on Theorems 4.2 and 4.5, we have
$$\begin{aligned} & GIVDHFLPA_{\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{n} ) = \left( {\frac{{ \oplus_{j = 1}^{n} (1 + T(sd_{j} ))sd_{j}^{\lambda } }}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} = \left( {\frac{{ \oplus_{k = 1}^{n} (1 + T(sd_{k}^{*} ))sd_{k}^{*\lambda } }}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = GIVDHFLPA_{\lambda } (sd_{1}^{*} ,sd_{2}^{*} , \ldots ,sd_{n}^{*} ). \\ & GIVDHFLPGA_{\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{n} ) = \frac{1}{\lambda }\left( {\mathop \otimes \limits_{j = 1}^{n} (\lambda sd_{j} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } \right) \\ & \quad = \frac{1}{\lambda }\left( {\mathop \otimes \limits_{k = 1}^{n} (\lambda sd_{k}^{*} )^{{\frac{{1 + T(sd_{k}^{*} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } \right) = GIVDHFLPGA_{\lambda } (sd_{1}^{*} ,sd_{2}^{*} , \ldots ,sd_{n}^{*} ). \\ \end{aligned}$$
2. Since \(sd_{j} = sd\) for all \(j = 1,2, \ldots ,n,\) thus
$$\begin{aligned} & GIVDHFLPA_{\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{n} ) = \bigcup\limits_{{(s_{\alpha } ,\tilde{h},\tilde{g}) \in sd}} {\left( {s_{\alpha } ,\bigcup\limits_{{[\mu^{L} ,\mu^{U} ] \in \tilde{h},[\nu^{L} ,\nu^{U} ] \in \tilde{g}}} {\left\{ {\{ [\mu^{L} ,\mu^{U} ]\} ,\{ [\nu^{L} ,\nu^{U} ]\} } \right\}} } \right)} \\ & \quad = sd = GIVDHFLPGA_{\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{n} ). \\ \end{aligned}$$
3. Let \(sd^{ - } = (s_{\alpha }^{ - } ,\tilde{h}^{ - } ,\tilde{g}^{ - } ),sd^{ + } = \langle s_{\alpha }^{ + } ,\tilde{h}^{ + } ,\tilde{g}^{ + } \rangle ,\) where \(s_{\alpha }^{ - } = \min_{j} (s_{{\alpha_{j} }} ),s_{\alpha }^{ + } = \max_{j} (s_{{\alpha_{j} }} ),\)
$$\begin{aligned} \tilde{h}^{ - } & = \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} }} {\{ [\mu^{L - } ,\mu^{U - } ]\} } = \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} }} {\left\{ {\left[ {\mathop {\hbox{min} }\limits_{1 \le j \le n} \mu_{j}^{L} ,\mathop {\hbox{min} }\limits_{1 \le j \le n} \mu_{j}^{U} } \right]} \right\}} , \\ \tilde{h}^{ + } & = \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} }} {\{ [\mu^{L + } ,\mu^{U + } ]\} } = \bigcup\limits_{{[\mu_{j}^{L} ,\mu_{j}^{U} ] \in \tilde{h}_{j} }} {\left\{ {\left[ {\mathop {\hbox{max} }\limits_{1 \le j \le n} \mu_{j}^{L} ,\mathop {\hbox{max} }\limits_{1 \le j \le n} \mu_{j}^{U} } \right]} \right\}} , \\ \tilde{g}^{ - } & = \bigcup\limits_{{[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\{ [\nu^{L - } ,\nu^{U - } ]\} } = \bigcup\limits_{{[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\left\{ {\left[ {\mathop {\hbox{max} }\limits_{1 \le j \le n} \nu_{j}^{L} ,\mathop {\hbox{max} }\limits_{1 \le j \le n} \nu_{j}^{U} } \right]} \right\}} , \\ \tilde{g}^{ + } & = \bigcup\limits_{{[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\{ [\nu^{L + } ,\nu^{U + } ]\} } = \bigcup\limits_{{[\nu_{j}^{L} ,\nu_{j}^{U} ] \in \tilde{g}_{j} }} {\left\{ {\left[ {\mathop {\hbox{min} }\limits_{1 \le j \le n} \nu_{j}^{L} ,\mathop {\hbox{min} }\limits_{1 \le j \le n} \nu_{j}^{U} } \right]} \right\}} , \\ \end{aligned}$$
for all \(j = 1,2, \ldots ,n,\) we have
$$\begin{aligned} & \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{L + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{U + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \ge \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \ge \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{L - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{U - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} , \\ \end{aligned}$$
and mean while
$$\begin{aligned} & \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{L - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) + \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{U - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) \\ & \quad \ge \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) + \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) \\ & \quad \ge \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{L + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) + \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{U + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right), \\ \end{aligned}$$
so we have
$$\begin{aligned} & \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \quad - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) \\ & \quad \le \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{L + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{U + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \quad - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{L + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{U + } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right). \\ \end{aligned}$$
And
$$\begin{aligned} & \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \quad - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) \\ & \quad \ge \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{L - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} + \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu^{U - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \quad - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{L - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right) - \left( {1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu^{U - } )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} } \right). \\ \end{aligned}$$
Obviously, \(max_{j} (s_{{\alpha_{j} }} ) \ge s_{{\left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} (\alpha_{j} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} }} \ge \min_{j} (s_{{\alpha_{j} }} ),\) then according to Definitions 3.3, 3.5 and Theorem 4.2, we have\(sd^{ + } \ge GIVDHFLPA_{\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{n} ) \ge sd^{ - } .\)
Similarly, according to Definitions 3.3, 3.5 and Theorem 4.5, we have
$$sd^{ + } \ge GIVDHFLPGA_{\lambda } (sd_{1} ,sd_{2} , \ldots ,sd_{n} ) \ge sd^{ - } ,$$
which completes the proof.□
Appendix 5: Proof of Theorem 4.8
Based on Lemma 4.1, we have
$$\begin{aligned} \prod\limits_{j = 1}^{n} {\alpha_{j}^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } & \le \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\alpha_{j} } , \\ \prod\limits_{j = 1}^{n} {(\mu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } & \le \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\mu_{j}^{L} } = 1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \mu_{j}^{L} )} \\ & \le 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } , \\ \prod\limits_{j = 1}^{n} {(\mu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } & \le \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\mu_{j}^{U} } = 1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \mu_{j}^{U} )} \\ & \le 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } , \\ \prod\limits_{j = 1}^{n} {(\nu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } & \le \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\nu_{j}^{L} } = 1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \nu_{j}^{L} )} \\ & \le 1 - \prod\limits_{j = 1}^{n} {(1 - \nu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } , \\ \prod\limits_{j = 1}^{n} {(\nu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } & \le \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\nu_{j}^{U} } = 1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \nu_{j}^{U} )} \\ & \le 1 - \prod\limits_{j = 1}^{n} {(1 - \nu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } . \\ \end{aligned}$$
By Definitions 3.3, 3.4 and 3.5, we have \(\otimes_{j = 1}^{n} (sd_{j}^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} ) \le \oplus_{j = 1}^{n} \left( {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}sd_{j} } \right),\) which completes the proof of Theorem 4.8. □
Appendix 6: Proof of Theorem 4.9
Based on Lemma 1, we have
$$\begin{aligned} & \prod\limits_{j = 1}^{n} {\alpha_{j}^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } = \left( {\prod\limits_{j = 1}^{n} {(\alpha_{j}^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\alpha_{j}^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} , \\ & \prod\limits_{j = 1}^{n} {(\mu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } = \left( {\prod\limits_{j = 1}^{n} {((\mu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \le \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(\mu_{j}^{L} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} = \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (\mu_{j}^{L} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad \le \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} , \\ & \prod\limits_{j = 1}^{n} {(\mu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } = \left( {\prod\limits_{j = 1}^{n} {((\mu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(\mu_{j}^{U} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (\mu_{j}^{U} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\mu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} , \\ & \quad \quad 1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (1 - \nu_{j}^{L} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = 1 - \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \nu_{j}^{L} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {\prod\limits_{j = 1}^{n} {(1 - \nu_{j}^{L} )^{{\frac{{\lambda (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} = 1 - \prod\limits_{j = 1}^{n} {(1 - \nu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } , \\ & \quad \quad 1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \nu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (1 - \nu_{j}^{U} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = 1 - \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \nu_{j}^{U} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {\prod\limits_{j = 1}^{n} {(1 - \nu_{j}^{U} )^{{\frac{{\lambda (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} = 1 - \prod\limits_{j = 1}^{n} {(1 - \nu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } . \\ \end{aligned}$$
By Theorems 4.3 and 4.6, we have \(\otimes_{j = 1}^{n} (sd_{j}^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} ) \le \left( { \oplus_{j = 1}^{n} \left( {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}sd_{j}^{\lambda } } \right)} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} ,\) which completes the proof of Theorem 4.9. □
Appendix 7: Proof of Theorem 4.10
$$\begin{aligned} & \frac{1}{\lambda }\prod\nolimits_{j = 1}^{n} {(\lambda \alpha_{j} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } \le \frac{1}{\lambda }\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(\lambda \alpha_{j} )} = \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}\alpha_{j} } , \\ & 1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \mu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (1 - \mu_{j}^{L} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = 1 - \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \mu_{j}^{L} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {\prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{L} )^{{\frac{{\lambda (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } , \\ & 1 - \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (1 - \mu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (1 - \mu_{j}^{U} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = 1 - \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - \mu_{j}^{U} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le 1 - \left( {\prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{U} )^{{\frac{{\lambda (1 + T(sd_{j} ))}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} = 1 - \prod\limits_{j = 1}^{n} {(1 - \mu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } , \\ & \prod\limits_{j = 1}^{n} {(\nu_{j}^{L} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } = \left( {\prod\limits_{j = 1}^{n} {((\nu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(\nu_{j}^{L} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (\nu_{j}^{L} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\nu_{j}^{L} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} , \\ & \prod\limits_{j = 1}^{n} {(\nu_{j}^{U} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } = \left( {\prod\limits_{j = 1}^{n} {((\nu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {\sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(\nu_{j}^{U} )^{\lambda } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \\ & \quad = \left( {1 - \sum\limits_{j = 1}^{n} {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}(1 - (\nu_{j}^{U} )^{\lambda } )} } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} \le \left( {1 - \prod\limits_{j = 1}^{n} {(1 - (\nu_{j}^{U} )^{\lambda } )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-0pt} \lambda }}} . \\ \end{aligned}$$
Thus, we have \(\frac{1}{\lambda }\left( { \otimes_{j = 1}^{n} ((\lambda sd_{j} )^{{\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}}} )} \right) \le \oplus_{j = 1}^{n} \left( {\frac{{1 + T(sd_{j} )}}{{\sum\nolimits_{i = 1}^{n} {(1 + T(sd_{i} ))} }}sd_{j} } \right),\) which completes the proof of Theorem 4.10. □
