Multi-criteria group decision-making method based on interdependent inputs of single-valued trapezoidal neutrosophic information

Abstract

Single-valued trapezoidal neutrosophic numbers (SVTNNs) are very useful tools for describing complex information, because they are able to maintain the completeness of the information and describe it accurately and comprehensively. This paper develops a method based on the single-valued trapezoidal neutrosophic normalized weighted Bonferroni mean (SVTNNWBM) operator to address multi-criteria group decision-making (MCGDM) problems. First, the limitations of existing operations for SVTNNs are discussed, after which improved operations are defined. Second, a new comparison method based on score function is proposed. Then, the entropy-weighted method is established in order to obtain objective expert weights, and the SVTNNWBM operator is proposed based on the new operations of SVTNNs. Furthermore, a single-valued trapezoidal neutrosophic MCGDM method is developed. Finally, a numerical example and comparison analysis are conducted to verify the practicality and effectiveness of the proposed approach.

Appendices

Appendix 1

Proof

In the following steps, Eq. (15) will be proved using mathematical induction on n.

• 1. The following equation must be proved first:

  $$\begin{aligned} {\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) &=\, \left\langle {\left[ {{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i1}^{p} a_{j1}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i2}^{p} a_{j2}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i3}^{p} a_{j3}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i4}^{p} a_{j4}^{q} } \right],} \right. \hfill \\ &\qquad \left( {\frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{p} \left( {T_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}},} \right. \hfill \\ &\qquad \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}, \hfill \\ &\qquad\left. {\left. { \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} +a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)}\right\rangle \hfill \\ \end{aligned}$$

  (26)

  • (a) Utilizing the operations for SVTNNs and mathematical induction on n, we have

    $$\tilde{a}_{i}^{p} = \left\langle {\left[ {a_{i1}^{p} ,a_{i2}^{p} ,a_{i3}^{p} ,a_{i4}^{p} } \right],\left( {\left( {T_{{\tilde{a}_{i} }} } \right)^{p} ,1 - \left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} ,1 - \left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} } \right)} \right\rangle ,$$
    $$\tilde{a}_{j}^{q} = \left\langle {\left[ {a_{j1}^{q} ,a_{j2}^{q} ,a_{j3}^{q} ,a_{j4}^{q} } \right],\left( {\left( {T_{{\tilde{a}_{j} }} } \right)^{q} ,1 - \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} ,1 - \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} } \right)} \right\rangle ,$$
    $$\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} = \left\langle {\left[ {a_{i1}^{p} a_{j1}^{q} ,a_{i2}^{p} a_{j2}^{q} ,a_{i3}^{p} a_{j3}^{q} ,a_{i4}^{p} a_{j4}^{q} } \right],\left( {\begin{array}{*{20}l} {\left( {T_{{\tilde{a}_{i} }} } \right)^{p} \left( {T_{{\tilde{a}_{j} }} } \right)^{q} ,1 - \left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} ,} \hfill \\ {1 - \left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} } \hfill \\ \end{array} } \right)} \right\rangle$$

    When n = 2, the following equation can be calculated:$\begin{aligned} {\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) &=\, \frac{{w_{1} w_{2} }}{{1 - w_{1} }}\left( {\tilde{a}_{1}^{p} \otimes \tilde{a}_{2}^{q} } \right) + \frac{{w_{2} w_{1} }}{{1 - w_{2} }}\left( {\tilde{a}_{2}^{p} \otimes \tilde{a}_{1}^{q} } \right) \hfill \\ &= \,\left\langle {\left[ {{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i1}^{p} a_{j1}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i2}^{p} a_{j2}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i3}^{p} a_{j3}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i4}^{p} a_{j4}^{q} } \right],} \right. \hfill \\ &\qquad \left( {\frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{p} \left( {T_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}},} \right. \hfill \\ &\qquad \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}, \hfill \\ &\qquad\left. {\left. { \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{2}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)} \right\rangle . \hfill \\ \end{aligned}$.In other words, when n = 2, Eq. (26) is true.

  • (b) Suppose that when n = k, Eq. (26) is true. That is,

    $$\begin{aligned} {\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) &=\, \left\langle {\left[ {{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i1}^{p} a_{j1}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i2}^{p} a_{j2}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i3}^{p} a_{j3}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i4}^{p} a_{j4}^{q} } \right],} \right. \hfill \\ &\qquad \left( {\frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{p} \left( {T_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}},} \right. \hfill \\ &\qquad \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}, \hfill \\ &\qquad\left. {\left. {\frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)} \right\rangle \hfill \\ \end{aligned}$$

    (27)

    Then, when n = k + 1, the following result can be obtained:

    $${\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left({\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) = {\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) + \mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{k + 1}^{q} } \right) + \mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {\tilde{a}_{k + 1}^{p} \otimes \tilde{a}_{j}^{q} } \right).$$

    (28)

    Next, the following equation must be proved:

    $$\begin{aligned} \mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{k + 1}^{q} } \right) = \left\langle {\left[ {\mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right),\mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {a_{i2}^{p} \otimes a_{k + 1,2}^{q} } \right),\mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {a_{i3}^{p} \otimes a_{k + 1,3}^{q} } \right),\mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {a_{i4}^{p} \otimes a_{k + 1,4}^{q} } \right)} \right],} \right. \hfill \\ \quad \left( {\frac{{\mathop \oplus \limits_{i = 1}^{k} \frac{1}{2}\frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{k + 1,3}^{q} - a_{i2}^{p} \otimes a_{k + 1,2}^{q} + a_{i4}^{p} \otimes a_{k + 1,4}^{q} - a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{P} \left( {T_{{\tilde{a}_{k + 1} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{k} \frac{1}{2}\frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{k + 1,3}^{q} - a_{i2}^{p} \otimes a_{k + 1,2}^{q} + a_{i4}^{p} \otimes a_{k + 1,4}^{q} - a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right]}},} \right. \hfill \\ \quad \frac{{\mathop \oplus \limits_{i = 1}^{k} \frac{1}{2}\frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{k + 1,3}^{q} - a_{i2}^{p} \otimes a_{k + 1,2}^{q} + a_{i4}^{p} \otimes a_{k + 1,4}^{q} - a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - I_{{\tilde{a}_{k + 1} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{k} \frac{1}{2}\frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{k + 1,3}^{q} - a_{i2}^{p} \otimes a_{k + 1,2}^{q} + a_{i4}^{p} \otimes a_{k + 1,4}^{q} - a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right]}}, \hfill \\ \quad \left. {\left. {\frac{{\mathop \oplus \limits_{i = 1}^{k} \frac{1}{2}\frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{k + 1,3}^{q} - a_{i2}^{p} \otimes a_{k + 1,2}^{q} + a_{i4}^{p} \otimes a_{k + 1,4}^{q} - a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - F_{{\tilde{a}_{k + 1} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{k} \frac{1}{2}\frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{k + 1,3}^{q} - a_{i2}^{p} \otimes a_{k + 1,2}^{q} + a_{i4}^{p} \otimes a_{k + 1,4}^{q} - a_{i1}^{p} \otimes a_{k + 1,1}^{q} } \right]}}} \right)} \right\rangle . \hfill \\ \end{aligned}$$

    (29)

    In the following steps, Eq. (29) will be proved using mathematical induction on k.

  • (i) When k = 2, the following result can be calculated:

    $$\begin{aligned} \mathop \oplus \limits_{i = 1}^{2} \frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{3}^{q} } \right) = \frac{{w_{1} w_{3} }}{{1 - w_{1} }}\left( {\tilde{a}_{1}^{p} \otimes \tilde{a}_{3}^{q} } \right) + \frac{{w_{2} w_{3} }}{{1 - w_{2} }}\left( {\tilde{a}_{2}^{p} \otimes \tilde{a}_{3}^{q} } \right) \hfill \\ = \left\langle {\left[ {\mathop \oplus \limits_{i = 1}^{2} \frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left( {a_{i1}^{p} \otimes a_{3,1}^{q} } \right),\mathop \oplus \limits_{i = 1}^{2} \frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left( {a_{i2}^{p} \otimes a_{3,2}^{q} } \right),\mathop \oplus \limits_{i = 1}^{2} \frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left( {a_{i3}^{p} \otimes a_{3,3}^{q} } \right),\mathop \oplus \limits_{i = 1}^{2} \frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left( {a_{i4}^{p} \otimes a_{3,4}^{q} } \right)} \right],} \right. \hfill \\ \left( {\frac{{\mathop \oplus \limits_{i = 1}^{2} \frac{1}{2}\frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{3,3}^{q} - a_{i2}^{p} \otimes a_{3,2}^{q} + a_{i4}^{p} \otimes a_{3,4}^{q} - a_{i1}^{p} \otimes a_{3,1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{P} \left( {T_{{\tilde{a}_{3} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{2} \frac{1}{2}\frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{3,3}^{q} - a_{i2}^{p} \otimes a_{3,2}^{q} + a_{i4}^{p} \otimes a_{3,4}^{q} - a_{i1}^{p} \otimes a_{3,1}^{q} } \right]}},} \right. \hfill \\ \frac{{\mathop \oplus \limits_{i = 1}^{2} \frac{1}{2}\frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{3,3}^{q} - a_{i2}^{p} \otimes a_{3,2}^{q} + a_{i4}^{p} \otimes a_{3,4}^{q} - a_{i1}^{p} \otimes a_{3,1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - I_{{\tilde{a}_{3} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{2} \frac{1}{2}\frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{3,3}^{q} - a_{i2}^{p} \otimes a_{3,2}^{q} + a_{i4}^{p} \otimes a_{3,4}^{q} - a_{i1}^{p} \otimes a_{3,1}^{q} } \right]}}, \hfill \\ \left. {\left. {\frac{{\mathop \oplus \limits_{i = 1}^{2} \frac{1}{2}\frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{3,3}^{q} - a_{i2}^{p} \otimes a_{3,2}^{q} + a_{i4}^{p} \otimes a_{3,4}^{q} - a_{i1}^{p} \otimes a_{3,1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - F_{{\tilde{a}_{3} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{2} \frac{1}{2}\frac{{w_{i} w_{3} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{3,3}^{q} - a_{i2}^{p} \otimes a_{3,2}^{q} + a_{i4}^{p} \otimes a_{3,4}^{q} - a_{i1}^{p} \otimes a_{3,1}^{q} } \right]}}} \right)} \right\rangle . \hfill \\ \end{aligned}$$

    That is, when k = 2, Eq. (29) is true.

  • (ii) Suppose that when k = l, Eq. (29) is true. That is,

    $$\begin{aligned} \mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{l + 1}^{q} } \right) = \left\langle {\left[ {\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left( {a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left( {a_{i2}^{p} \otimes a_{l + 1,2}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left( {a_{i3}^{p} \otimes a_{l + 1,3}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left( {a_{i4}^{p} \otimes a_{l + 1,4}^{q} } \right)} \right],} \right. \hfill \\ \quad \left( {\frac{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 1,3}^{q} - a_{i2}^{p} \otimes a_{l + 1,2}^{q} + a_{i4}^{p} \otimes a_{l + 1,4}^{q} - a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{P} \left( {T_{{\tilde{a}_{l + 1} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 1,3}^{q} - a_{i2}^{p} \otimes a_{l + 1,2}^{q} + a_{i4}^{p} \otimes a_{l + 1,4}^{q} - a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right]}},} \right. \hfill \\ \quad \frac{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 1,3}^{q} - a_{i2}^{p} \otimes a_{l + 1,2}^{q} + a_{i4}^{p} \otimes a_{l + 1,4}^{q} - a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - I_{{\tilde{a}_{l + 1} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 1,3}^{q} - a_{i2}^{p} \otimes a_{l + 1,2}^{q} + a_{i4}^{p} \otimes a_{l + 1,4}^{q} - a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right]}}, \hfill \\ \left. {\left. {\quad \frac{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 1,3}^{q} - a_{i2}^{p} \otimes a_{l + 1,2}^{q} + a_{i4}^{p} \otimes a_{l + 1,4}^{q} - a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - F_{{\tilde{a}_{l + 1} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 1,3}^{q} - a_{i2}^{p} \otimes a_{l + 1,2}^{q} + a_{i4}^{p} \otimes a_{l + 1,4}^{q} - a_{i1}^{p} \otimes a_{l + 1,1}^{q} } \right]}}} \right)} \right\rangle . \hfill \\ \end{aligned}$$

    Then, when k = l + 1, the following result can be calculated:

    $$\begin{aligned} \mathop \oplus \limits_{i = 1}^{l + 1} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{l + 2}^{q} } \right) = \mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{l + 2}^{q} } \right) + \frac{{w_{l + 1} w_{l + 2} }}{{1 - w_{l + 1} }}\left( {\tilde{a}_{l + 1}^{p} \otimes \tilde{a}_{l + 2}^{q} } \right) \hfill \\ = \left\langle {\left[ {\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i2}^{p} \otimes a_{l + 2,2}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i3}^{p} \otimes a_{l + 2,3}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i4}^{p} \otimes a_{l + 2,4}^{q} } \right)} \right],} \right. \hfill \\ \quad \left( {\frac{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{P} \left( {T_{{\tilde{a}_{l + 2} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{2 + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]}},} \right. \hfill \\ \quad \frac{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - I_{{\tilde{a}_{l + 2} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]}}, \hfill \\ \quad \left. {\left. {\frac{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - F_{{\tilde{a}_{l + 2} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]}}} \right)} \right\rangle \hfill \\ \oplus \left\langle {\left[ {\frac{{w_{l + 1} w_{l + 2} }}{{1 - w_{l + 1} }}a_{l + 1,1}^{p} a_{l + 2,1}^{q} ,\frac{{w_{l + 1} w_{l + 2} }}{{1 - w_{l + 1} }}a_{l + 1,2}^{p} a_{l + 2,2}^{q} ,\frac{{w_{l + 1} w_{l + 2} }}{{1 - w_{l + 1} }}a_{l + 1,3}^{p} a_{l + 2,3}^{q} ,\frac{{w_{l + 1} w_{l + 2} }}{{1 - w_{l + 1} }}a_{l + 1,4}^{p} a_{l + 2,4}^{q} } \right]} \right. \hfill \\ \left. {\quad \left( {\left( {T_{{\tilde{a}_{l + 1} }} } \right)^{p} \left( {T_{{\tilde{a}_{l + 2} }} } \right)^{q} ,1 - \left( {1 - I_{{\tilde{a}_{l + 1} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{l + 2} }} } \right)^{q} ,1 - \left( {1 - F_{{\tilde{a}_{l + 1} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{l + 2} }} } \right)^{q} } \right)} \right\rangle . \hfill \\ = \left\langle {\left[ {\mathop \oplus \limits_{i = 1}^{l + 1} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l + 1} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i2}^{p} \otimes a_{l + 2,2}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l + 1} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i3}^{p} \otimes a_{l + 2,3}^{q} } \right),\mathop \oplus \limits_{i = 1}^{l + 1} \frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left( {a_{i4}^{p} \otimes a_{l + 2,4}^{q} } \right)} \right],} \right. \hfill \\ \quad \left( {\frac{{\mathop \oplus \limits_{i = 1}^{l + 1} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{P} \left( {T_{{\tilde{a}_{l + 2} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l + 1} \frac{1}{2}\frac{{w_{i} w_{2 + 1} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]}},} \right. \hfill \\ \quad \frac{{\mathop \oplus \limits_{i = 1}^{l + 1} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - I_{{\tilde{a}_{l + 2} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l + 1} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]}}, \hfill \\ \quad \left. {\left. {\frac{{\mathop \oplus \limits_{i = 1}^{l + 1} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{P} \left( {1 - F_{{\tilde{a}_{l + 2} }} } \right)^{q} }}{{\mathop \oplus \limits_{i = 1}^{l + 1} \frac{1}{2}\frac{{w_{i} w_{l + 2} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} \otimes a_{l + 2,3}^{q} - a_{i2}^{p} \otimes a_{l + 2,2}^{q} + a_{i4}^{p} \otimes a_{l + 2,4}^{q} - a_{i1}^{p} \otimes a_{l + 2,1}^{q} } \right]}}} \right)} \right\rangle . \hfill \\ \end{aligned}$$

    That is, when k = l + 1, Eq. (29) is true.

  • (iii) So, for all k, Eq. (29) is true. The following equation can be proved in a similar fashion, and the proof is omitted here.

    $$\begin{aligned} \mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {\tilde{a}_{k + 1}^{p} \otimes \tilde{a}_{j}^{q} } \right) = \left\langle {\left[ {\mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right),\mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {a_{k + 1,2}^{p} \otimes a_{j2}^{q} } \right),\mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {a_{k + 1,3}^{p} \otimes a_{j3}^{q} } \right),\mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {a_{k + 1,4}^{p} \otimes a_{j4}^{q} } \right)} \right],} \right. \hfill \\ \quad \left( {\frac{{\mathop \oplus \limits_{j = 1}^{k} \frac{1}{2}\frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left[ {a_{k + 1,3}^{p} \otimes a_{j3}^{q} - a_{k + 1,2}^{p} \otimes a_{j2}^{q} + a_{k + 1,4}^{p} \otimes a_{j4}^{q} - a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right]\left( {T_{{\tilde{a}_{k + 1} }} } \right)^{P} \left( {T_{{\tilde{a}_{j} }} } \right)^{q} }}{{\mathop \oplus \limits_{j = 1}^{k} \frac{1}{2}\frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left[ {a_{k + 1,3}^{p} \otimes a_{j3}^{q} - a_{k + 1,2}^{p} \otimes a_{j2}^{q} + a_{k + 1,4}^{p} \otimes a_{j4}^{q} - a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right]}},} \right. \hfill \\ \quad \frac{{\mathop \oplus \limits_{j = 1}^{k} \frac{1}{2}\frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left[ {a_{k + 1,3}^{p} \otimes a_{j3}^{q} - a_{k + 1,2}^{p} \otimes a_{j2}^{q} + a_{k + 1,4}^{p} \otimes a_{j4}^{q} - a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{k + 1} }} } \right)^{P} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} }}{{\mathop \oplus \limits_{j = 1}^{k} \frac{1}{2}\frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left[ {a_{k + 1,3}^{p} \otimes a_{j3}^{q} - a_{k + 1,2}^{p} \otimes a_{j2}^{q} + a_{k + 1,4}^{p} \otimes a_{j4}^{q} - a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right]}}, \hfill \\ \left. {\left. {\quad \frac{{\mathop \oplus \limits_{j = 1}^{k} \frac{1}{2}\frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left[ {a_{k + 1,3}^{p} \otimes a_{j3}^{q} - a_{k + 1,2}^{p} \otimes a_{j2}^{q} + a_{k + 1,4}^{p} \otimes a_{j4}^{q} - a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{k + 1} }} } \right)^{P} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} }}{{\mathop \oplus \limits_{j = 1}^{k} \frac{1}{2}\frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left[ {a_{k + 1,3}^{p} \otimes a_{j3}^{q} - a_{k + 1,2}^{p} \otimes a_{j2}^{q} + a_{k + 1,4}^{p} \otimes a_{j4}^{q} - a_{k + 1,1}^{p} \otimes a_{j1}^{q} } \right]}}} \right)} \right\rangle . \hfill \\ \end{aligned}$$

    (30)

    Using Eqs. (27), (29) and (20), Eq. (28) can be transformed as follows:

    $$\begin{aligned} {\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) &=\, {\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{j}^{q} } \right) + \mathop \oplus \limits_{i = 1}^{k} \frac{{w_{i} w_{k + 1} }}{{1 - w_{i} }}\left( {\tilde{a}_{i}^{p} \otimes \tilde{a}_{k + 1}^{q} } \right) + \mathop \oplus \limits_{j = 1}^{k} \frac{{w_{k + 1} w_{j} }}{{1 - w_{k + 1} }}\left( {\tilde{a}_{k + 1}^{p} \otimes \tilde{a}_{j}^{q} } \right). \hfill \\ &= \,\left\langle {\left[ {{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i1}^{p} a_{j1}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i2}^{p} a_{j2}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i3}^{p} a_{j3}^{q} ,{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i4}^{p} a_{j4}^{q} } \right],} \right. \hfill \\ &\qquad \left( {\frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {T_{{\tilde{a}_{i} }} } \right)^{p} \left( {T_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}},} \right. \hfill \\ &\qquad \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}, \hfill \\ &\qquad\left. {\left. { \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{k + 1}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)}\right\rangle . \hfill \\ \end{aligned}$$

    Then, when n = k + 1, Eq. (26) is true. Therefore, Eq. (26) is true for all $n$.

• 2. Using the SVTNN operations and Eq. (26), Eq. (15) can be obtained. This completes the proof of Theorem 3.

Appendix 2

Proof

For an arbitrary i, there are a _i1 ≥ b _i1, a _i2 ≥ b _i2, a _i3 ≥ b _i3, a _i4 ≥ b _i4; therefore, it is easy to obtain the following inequalities:

$$a_{i1}^{p} \ge b_{i1}^{q} ,a_{i2}^{p} \ge b_{i2}^{q} ,a_{i3}^{p} \ge b_{i3}^{q} ,\;{\text{and}}\;a_{i4}^{p} \ge b_{i4}^{q} ,$$

then

$$\left( {{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{{w_{i} w_{j} }}{{1 - w_{i}}}a_{i1}^{p} a_{j1}^{q} } \right)^{{\frac{1}{p + q}}} \ge \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}b_{i1}^{p} b_{j1}^{q} }\right)^{{\frac{1}{p + q}}} ,\quad \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i2}^{p} a_{j2}^{q} }\right)^{{\frac{1}{p + q}}} \ge \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}b_{i2}^{p} b_{j2}^{q} }\right)^{{\frac{1}{p + q}}} ,\quad \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i3}^{p} a_{j3}^{q} }\right)^{{\frac{1}{p + q}}} \ge \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}b_{i3}^{p} b_{j3}^{q} }\right)^{{\frac{1}{p + q}}} ,\quad \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}a_{i4}^{p} a_{j4}^{q} }\right)^{{\frac{1}{p + q}}} \ge \left({{\mathop{\mathop{\oplus}\limits_{i,j = 1}}\limits_{ i \ne j}^{n}}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}b_{i4}^{p} b_{j4}^{q} }\right)^{{\frac{1}{p + q}}} .$$

The truth-membership, indeterminacy-membership and falsity-membership parts can be proved using mathematical induction on n.

$$\left( {\frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j}}}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p}a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} }\right]\left( {T_{{\tilde{a}_{i} }} } \right)^{p} \left({T_{{\tilde{a}_{j} }} } \right)^{q}}}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[{a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p}a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)^{{\frac{1}{p+ q}}} \ge \left( \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j =1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 -w_{i} }}\left[ {b_{i3}^{p} b_{j3}^{q} - b_{i2}^{p} b_{j2}^{q} +b_{i4}^{p} b_{j4}^{q} - b_{i1}^{p} b_{j1}^{q} } \right]\left({T_{{\tilde{b}_{i} }} } \right)^{p} \left( {T_{{\tilde{b}_{j} }} }\right)^{q}}}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[{b_{i3}^{p} b_{j3}^{q} - b_{i2}^{p} b_{j2}^{q} + b_{i4}^{p}b_{j4}^{q} - b_{i1}^{p} b_{j1}^{q} } \right]}} \right),$$

$$1 - \left( {1 - \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - I_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)^{{\frac{1}{p + q}}} \le 1 - \left( {1 - \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {b_{i3}^{p} b_{j3}^{q} - b_{i2}^{p} b_{j2}^{q} + b_{i4}^{p} b_{j4}^{q} - b_{i1}^{p} \beta_{j1}^{q} } \right]\left( {1 - I_{{\tilde{b}_{i} }} } \right)^{p} \left( {1 - I_{{\tilde{b}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {b_{i3}^{p} b_{j3}^{q} - b_{i2}^{p} b_{j2}^{q} + b_{i4}^{p} b_{j4}^{q} - b_{i1}^{p} \beta_{j1}^{q} } \right]}}} \right)^{{\frac{1}{p + q}}} ,\;{\text{and}}$$

$$1 - \left( {1 - \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]\left( {1 - F_{{\tilde{a}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{a}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {a_{i3}^{p} a_{j3}^{q} - a_{i2}^{p} a_{j2}^{q} + a_{i4}^{p} a_{j4}^{q} - a_{i1}^{p} a_{j1}^{q} } \right]}}} \right)^{{\frac{1}{p + q}}} \le 1 - \left( {1 - \frac{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {b_{i3}^{p} b_{j3}^{q} - b_{i2}^{p} b_{j2}^{q} + b_{i4}^{p} b_{j4}^{q} - b_{i1}^{p} b_{j1}^{q} } \right]\left( {1 - F_{{\tilde{b}_{i} }} } \right)^{p} \left( {1 - F_{{\tilde{b}_{j} }} } \right)^{q} }}{{{\mathop{\mathop{\oplus}\nolimits_{i,j = 1}}\limits_{ i \ne j}^{n}} \frac{1}{2}\frac{{w_{i} w_{j} }}{{1 - w_{i} }}\left[ {b_{i3}^{p} b_{j3}^{q} - b_{i2}^{p} b_{j2}^{q} + b_{i4}^{p} b_{j4}^{q} - b_{i1}^{p} b_{j1}^{q} } \right]}}} \right)^{{\frac{1}{p + q}}} .$$

Then, using the new comparison method in Sect. 3.2, Theorem 3 can be proved.