NA Operator-Based Interval-Valued q-Rung Orthopair Fuzzy PSI-COPRAS Group Decision-Making Method

Abstract

Interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs) have been widely used to address uncertain or imprecision data in practical decision-making. One contribution of this study is to develop a new score function for comparing the magnitude of IVq-ROFNs. We construct the variance value-based index for analyzing different score functions and the results show that the developed score function well improves the fluctuation of IVq-ROFNs and has much stronger ability to distinguish IVq-ROFNs. The second contribution is that interval-valued q-rung orthopair fuzzy neutral operational law is investigated and a new interval-valued q-rung orthopair fuzzy weighted average neutral aggregating (IVq-ROFWANA) operator is proposed. We conduct a detail comparison analysis with the existing operators and show that the calculated results obtained by them are consistent. The third contribution is that a new interval-valued q-rung orthopair fuzzy preference selection index and complex proportional assessment (PSI-COPRAS) method is developed. On the one hand, we propose the similarity-proximity degree based-weight determination method for identifying the experts’ weights and modify the classical PSI method to identify weights of attributes under IVq-ROFNs environments. On the other hand, we introduce interval-valued q-rung orthopair fuzzy COPRAS to rank alternatives. Finally, the proposed method is implemented in warning risk evaluation of hypertension in residents to demonstrate its applicability. Comparing with the classical TOPSIS, MOORA and MABAC, the biggest advantage of the developed method is that that it can not only fully consider the influences of the maximum and minimum attribute indices but also allow both the weights of attributes and experts to be completely unknown in advance.

Appendices

Appendix 1

1.1 Proof of Theorem 3.1

Proof

According to definition of the developed score function ${\text{S}}\left( a \right)$, we take the partial derivative of $v_{a}^{ - }$ and get:

$\frac{\partial S\left( a \right)}{{\partial v_{a}^{ - } }} = - \frac{1}{3} \times \left[ \begin{gathered} \frac{1}{3} \times \left( {q \times \left( {(v_{a}^{ - } )^{q - 1} + \left( {\frac{{v_{a}^{ - } + v_{a}^{ + } }}{2}} \right)^{q - 1} \times \frac{1}{2}} \right)} \right) + \frac{1}{2}\left( {q \times (v_{a}^{ - } )^{q - 1} } \right) \hfill \\ + \frac{\pi }{2} \times \sin \left( {\left( {\frac{{((v_{a}^{ - } )^{q} + (v_{a}^{ + } )^{q} )}}{2}} \right)\frac{\pi }{2}} \right) \times \frac{{q \times (v_{a}^{ - } )^{q - 1} }}{2} \hfill \\ \end{gathered} \right]$.

It is easy to see that $\frac{\partial S\left( a \right)}{{\partial v_{a}^{ - } }} < 0$ when $u_{a}^{ - } ,u_{a}^{ + } ,v_{a}^{ + }$ remain unchanged.

Analogously, we take the partial derivative of $v_{a}^{ + }$ and get:

$\frac{\partial S\left( a \right)}{{\partial v_{a}^{ + } }} = - \frac{1}{3} \times \left[ \begin{gathered} \frac{1}{3} \times \left( {q \times \left( {(v_{a}^{ + } )^{q - 1} + \left( {\frac{{v_{a}^{ - } + v_{a}^{ + } }}{2}} \right)^{q - 1} \times \frac{1}{2}} \right)} \right) + \frac{1}{2}\left( {q \times (v_{a}^{ + } )^{q - 1} } \right) \hfill \\ + \frac{\pi }{2} \times \sin \left( {\left( {\frac{{((v_{a}^{ - } )^{q} + (v_{a}^{ + } )^{q} )}}{2}} \right)\frac{\pi }{2}} \right) \times \frac{{q \times (v_{a}^{ + } )^{q - 1} }}{2} \hfill \\ \end{gathered} \right]$.

It is easy to see that $\frac{\partial S\left( a \right)}{{\partial v_{a}^{ + } }} < 0$ when $u_{a}^{ - } ,u_{a}^{ + } ,v_{a}^{ - }$ remain unchanged. That is to say, the developed score function s (a) is monotonically decreasing with respect to $v_{a}^{ - }$ and $v_{a}^{ + }$.

On the other hand, we take the partial derivative of $u_{a}^{ - }$ and get:

$$\frac{{\partial S\left( a \right)}}{{\partial u_{a}^{ - } }} = \frac{1}{3} \times {\text{ }}\left[ {\begin{array}{*{20}c} {\frac{1}{3} \times \left( {q \times \left( {(u_{a}^{ - } )^{{q - 1}} + \left( {\frac{{u_{a}^{ - } + u_{a}^{ + } }}{2}} \right)^{{q - 1}} \times \frac{1}{2}} \right)} \right) + \frac{1}{2}\left( {q \times (u_{a}^{ - } )^{{q - 1}} } \right)} \\ { + \frac{\pi }{2} \times \sin \left( {\left( {\frac{{((u_{a}^{ - } )^{q} + (u_{a}^{{ + ^{q} }} )}}{2}} \right)\frac{\pi }{2}} \right) \times \frac{{q \times (u_{a}^{ - } )^{{q - 1}} }}{2}} \\ \end{array} } \right].$$

It is easy to see that $\frac{\partial S\left( a \right)}{{\partial u_{a}^{ - } }} > 0$ when $u_{a}^{ + } ,v_{a}^{ - } ,v_{a}^{ + }$ remain unchanged.

We take the partial derivative of $u_{a}^{ + }$ and get:

$$\frac{\partial S\left( a \right)}{{\partial u_{a}^{ + } }} = \frac{1}{3} \times \left[ \begin{gathered} \frac{1}{3} \times \left( {q \times \left( {(u_{a}^{ + } )^{q - 1} + \left( {\frac{{u_{a}^{ - } + u_{a}^{ + } }}{2}} \right)^{q - 1} \times \frac{1}{2}} \right)} \right) + \frac{1}{2}\left( {q \times (u_{a}^{ + } )^{q - 1} } \right) \hfill \\ + \frac{\pi }{2} \times \sin \left( {\left( {\frac{{((u_{a}^{ - } )^{q} + (u_{a}^{ + } )^{q} )}}{2}} \right)\frac{\pi }{2}} \right) \times \frac{{q \times (u_{a}^{ + } )^{q - 1} }}{2} \hfill \\ \end{gathered} \right]$$

It is easy to see that $\frac{\partial S\left( a \right)}{{\partial u_{a}^{ + } }} > 0$ when $u_{a}^{ - } ,v_{a}^{ - } ,v_{a}^{ + }$ remain unchanged. That is to say, the developed score function $S\left( a \right)$ is monotonically increasing with respect to $u_{a}^{ - }$ and $u_{a}^{ + }$.

Therefore, we can conclude ${\text{ S}}\left( {a_{1} } \right)$ > ${\text{S}}\left( {a_{2} } \right)$ if they satisfy $u_{{a_{1} }}^{ - } > u_{{a_{2} }}^{ - }$, $u_{{a_{1} }}^{ + } > u_{{a_{2} }}^{ + }$, $v_{{a_{1} }}^{ - } < v_{{a_{2} }}^{ - }$ and $v_{{a_{1} }}^{ + } < v_{{a_{2} }}^{ + }$. Obviously, the proof of the property (4) is completed. Meanwhile, it can be seen from the proof (4) that the score value will be larger when the membership value u is larger and the non-membership value v is smaller. Thus, the score value of $a_{{{\text{max}}}} = \left( {\left[ {1,1\left] , \right[0,0} \right]} \right)$ is the maximum, and the calculated value is ${\text{S}}\left( {a_{{{\text{max}}}} } \right)_{ } = 1$; while the score values of $a_{{{\text{min}}}} = \left( {\left[ {0,0} \right],\left[ {1,1} \right]} \right)$ is the minimum, and it is calculated as ${\text{S}}\left( {a_{{{\text{min}}}} } \right)_{ } = - 1$. Therefore, the value range of score function is $- 1 \le S\left( a \right) \le 1$. That is to say, the proofs of the properties (1–3) are also completed.

1.2 Proof of Definition 4.2

By Eq. (16) and when $k = 2$, it can be obtained:

$$\begin{gathered} {\text{PS}}\left( {2\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }}} \right)} \right) \hfill \\ = {\text{PS}}\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }},{\text{PS}}\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }}} \right)} \right) \hfill \\ = {\text{PS}}\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }},\sqrt[{\text{q}}]{{1 - \left( {\pi _{a}^{ + } } \right)^{{\text{q}}} }}} \right) \hfill \\ = \sqrt[{\text{q}}]{{1 - \left( {1 - \left( {u_{a}^{ - } } \right)^{{\text{q}}} - \left( {v_{a}^{ - } } \right)^{{\text{q}}} } \right)\left( {1 - 1 + \left( {\pi _{a}^{ + } } \right)^{{\text{q}}} } \right)}} \hfill \\ = \sqrt[{\text{q}}]{{1 - \left( {\left( {\pi _{a}^{ + } } \right)^{{\text{q}}} } \right)^{2} }} \hfill \\ \end{gathered}$$

We suppose that the following equation holds for any $n = k$, namely,

$$PS\left( {n\left( {\sqrt[q]{{\left( {u_{a}^{ - } } \right)^{q} + \left( {v_{a}^{ - } } \right)^{q} }}} \right)} \right) = \sqrt[q]{{1 - \left( {\left( {\pi_{a}^{ + } } \right)^{q} } \right)^{n} }}$$

Thus, when $n = k + 1$, it can be obtained

$$\begin{gathered} {\text{PS}}\left( {\left( {n + 1} \right)\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }}} \right)} \right) \hfill \\ = {\text{PS}}\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }},{\text{PS}}\left( {{\text{n}}\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }}} \right)} \right) \hfill \\ = {\text{PS}}\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }},\sqrt[{\text{q}}]{{1 - \left( {\left( {\pi _{a}^{ + } } \right)^{{\text{q}}} } \right)^{{\text{n}}} }}} \right) \hfill \\ = \sqrt[{\text{q}}]{{1 - \left( {1 - \left( {u_{a}^{ - } } \right)^{{\text{q}}} - \left( {v_{a}^{ - } } \right)^{{\text{q}}} } \right)\left( {\left( {\pi _{a}^{ + } } \right)^{{\text{q}}} } \right)^{{\text{n}}} }} \hfill \\ = \sqrt[{\text{q}}]{{1 - \left( {\left( {\pi _{a}^{ + } } \right)^{{\text{q}}} } \right)^{{{\text{n}} + 1}} }} \hfill \\ \end{gathered}$$

That is to say, Eq. (16) holds when $n = k + 1$. Thus, ${\text{PS}}\left( {k\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {v_{a}^{ - } } \right)^{{\text{q}}} }}} \right)} \right) =$ $\sqrt[{\text{q}}]{{1 - (\left( {{\uppi }_{a}^{ + } } \right)^{{\text{q}}} )^{k} }}$. Similarity, we can also prove that ${\text{PS}}\left( {k\left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ + } } \right)^{{\text{q}}} + \left( {v_{a}^{ + } } \right)^{{\text{q}}} }}} \right)} \right) = \sqrt[{\text{q}}]{{1 - (\left( {{\uppi }_{a}^{ - } } \right)^{{\text{q}}} )^{k} }}.$

1.3 Proof of Definition 4.3

For the case of $k = 2$, it can be obtained:

$$\begin{gathered} {\text{MCS}}\left( {2a} \right) = \left({MCS^-}\left( {2a} \right),{MCS^+}\left( {2a} \right)\right)= \left( {\sqrt[{\text{q}}]{{\left( {u_{a}^{ - } } \right)^{{\text{q}}} + \left( {u_{a}^{ - } } \right)^{{\text{q}}} }},\sqrt[{\text{q}}]{{\left( {u_{a}^{ + } } \right)^{{\text{q}}} + \left( {u_{a}^{ + } } \right)^{{\text{q}}} }}} \right) \hfill \\ = \left( {\sqrt[{\text{q}}]{{2\left( {u_{a}^{ - } } \right)^{{\text{q}}} }},\sqrt[{\text{q}}]{{2\left( {u_{a}^{ + } } \right)^{{\text{q}}} }}} \right) = \sqrt[{\text{q}}]{2}\left( {u_{a}^{ - } ,u_{a}^{ + } } \right) \hfill \\ \end{gathered}$$

Thus, Eq. (20) holds when $k = 2$.

We suppose that the following equation holds for any $n = k$, namely,

$MCS\left( {na} \right) = \left( {\sqrt[q]{n}u_{a}^{ - } ,\sqrt[q]{n}u_{a}^{ + } } \right)$.

When $n = k + 1$, it can be obtained:

$$\begin{gathered} MCS\left( {\left( {n + 1} \right)a} \right) = MCS\left( {na,a} \right) = \left( {\sqrt[q]{{n\left( {u_{a}^{ - } } \right)^{q} + \left( {u_{a}^{ - } } \right)^{q} }},\sqrt[q]{{n\left( {u_{a}^{ + } } \right)^{q} + \left( {u_{a}^{ + } } \right)^{q} }}} \right) \hfill \\ = \left( {\sqrt[q]{{n + 1}}u_{a}^{ - } ,\sqrt[q]{{n + 1}}u_{a}^{ + } } \right) \hfill \\ \end{gathered}$$

That is to say, Eq. (20) holds when ${ } n = k + 1$. the proof process of Eq. (20) is completed. Analogously, we can also prove that Eq. (21) holds, namely, $NCS\left( {ka} \right) = \left( {\sqrt[q]{k}v_{a}^{ - } ,\sqrt[q]{k}v_{a}^{ + } } \right)$.

1.4 Proof of Theorem 4.1

Proof. (1) According to Definition 4.1, it is easy to obtain:

$${a}_{1}{\Theta }_{NA}{a}_{2}=\left(\begin{array}{c}\left[\begin{array}{c} \sqrt[q]{\frac{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({u}_{{a}_{2}}^{-}\right)}^{q}}{\left({\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}+{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{1}}^{+}\right)}^{q}{\left({\pi }_{{a}_{2}}^{+}\right)}^{q}\right),}\\ \sqrt[q]{\frac{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({u}_{2}^{+}\right)}^{q}}{\left({\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{1}}^{-}\right)}^{q}{\left({\pi }_{{a}_{2}}^{-}\right)}^{q}\right),}\end{array}\right],\\ \left[\begin{array}{c} \sqrt[q]{\frac{{\left({v}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}}{\left({\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}+{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{1}}^{+}\right)}^{q}{\left({\pi }_{{a}_{2}}^{+}\right)}^{q}\right),}\\ \sqrt[q]{\frac{{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}{\left({\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{1}}^{-}\right)}^{q}{\left({\pi }_{{a}_{2}}^{-}\right)}^{q}\right),}\end{array}\right]\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{c} \sqrt[q]{\frac{{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({u}_{{a}_{1}}^{-}\right)}^{q}}{\left({\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}+{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{2}}^{+}\right)}^{q}{\left({\pi }_{{a}_{1}}^{+}\right)}^{q}\right),}\\ \sqrt[q]{\frac{{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({u}_{{a}_{1}}^{+}\right)}^{q}}{\left({\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}+{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{2}}^{-}\right)}^{q}{\left({\pi }_{{a}_{1}}^{-}\right)}^{q}\right),}\end{array}\right],\\ \left[\begin{array}{c} \sqrt[q]{\frac{{\left({v}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}}{\left({\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}+{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{1}}^{+}\right)}^{q}{\left({\pi }_{{a}_{2}}^{+}\right)}^{q}\right),}\\ \sqrt[q]{\frac{{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}{\left({\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}+{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}\right)}\times \left(1-{\left({\pi }_{{a}_{2}}^{-}\right)}^{q}{\left({\pi }_{{a}_{1}}^{-}\right)}^{q}\right),}\end{array}\right]\end{array}\right)={a}_{2}{\Theta }_{NA}{a}_{1}$$

(2) By Formulas (15) and (22), it is easy to obtain:

$$k{a}_{1}{\Theta }_{NA}{ka}_{2}=\left(\begin{array}{c}\left[\begin{array}{c}\begin{array}{c}\sqrt[q]{\frac{{{MCS}^{-}\left(k{a}_{1},{ka}_{2}\right)}^{q}}{{{MCS}^{-}\left(k{a}_{1},{ka}_{2}\right)}^{q}+{{NCS}^{-}\left(k{a}_{1},{ka}_{2}\right)}^{q}}}\times \\ PS\left(PS\left(k\sqrt[q]{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}}\right),PS\left(k\sqrt[q]{{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}}\right)\right),\\ \end{array}\\ \sqrt[q]{\frac{{{MCS}^{+}\left(k{a}_{1},{ka}_{2}\right)}^{q}}{{{MCS}^{+}\left(k{a}_{1},{ka}_{2}\right)}^{q}+{{NCS}^{+}\left(k{a}_{1},{ka}_{2}\right)}^{q}}}\times \\ PS\left(PS\left(k\sqrt[q]{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}}\right),PS\left(k\sqrt[q]{{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}\right)\right)\end{array}\right],\\ \left[\begin{array}{c}\begin{array}{c}\sqrt[q]{\frac{{{NCS}^{-}\left(k{a}_{1},{ka}_{2}\right)}^{q}}{{{MCS}^{-}\left(k{a}_{1},{ka}_{2}\right)}^{q}+{{NCS}^{-}\left(k{a}_{1},{ka}_{2}\right)}^{q}}}\times \\ PS\left(PS\left(k\sqrt[q]{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}}\right),PS\left(k\sqrt[q]{{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}}\right)\right),\\ \end{array}\\ \sqrt[q]{\frac{{{NCS}^{+}\left(k{a}_{1},{ka}_{2}\right)}^{q}}{{{MCS}^{+}\left(k{a}_{1},{ka}_{2}\right)}^{q}+{{NCS}^{+}\left(k{a}_{1},{ka}_{2}\right)}^{q}}}\times \\ PS\left(PS\left(k\sqrt[q]{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}}\right),PS\left(k\sqrt[q]{{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}\right)\right)\end{array}\right]\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{k\left({{\left({u}_{{a}_{1}}^{-}\right)}^{q}+\left({u}_{{a}_{2}}^{-}\right)}^{q}\right)}{k\left({{\left({u}_{{a}_{1}}^{-}\right)}^{q}+\left({u}_{{a}_{2}}^{-}\right)}^{q}\right)+k\left({{\left({v}_{{a}_{1}}^{-}\right)}^{q}+\left({v}_{{a}_{2}}^{-}\right)}^{q}\right)}\times \left(1-{\left({{\pi }_{{a}_{1}}^{+}}^{q}\right)}^{k}{\left({{\pi }_{{a}_{2}}^{+}}^{q}\right)}^{k}\right)},\\ \sqrt[q]{\frac{k\left({{\left({u}_{{a}_{1}}^{+}\right)}^{q}+\left({u}_{{a}_{2}}^{+}\right)}^{q}\right)}{k\left({{\left({u}_{{a}_{1}}^{+}\right)}^{q}+\left({u}_{{a}_{2}}^{+}\right)}^{q}\right)+k\left({{\left({v}_{{a}_{1}}^{+}\right)}^{q}+\left({v}_{{a}_{2}}^{+}\right)}^{q}\right)}\times \left(1-{\left({{\pi }_{{a}_{1}}^{-}}^{q}\right)}^{k}{\left({{\pi }_{{a}_{2}}^{-}}^{q}\right)}^{k}\right)}\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{k\left({{\left({v}_{{a}_{1}}^{-}\right)}^{q}+\left({v}_{{a}_{2}}^{-}\right)}^{q}\right)}{k\left({{\left({u}_{{a}_{1}}^{-}\right)}^{q}+\left({u}_{{a}_{2}}^{-}\right)}^{q}\right)+k\left({{\left({v}_{{a}_{1}}^{-}\right)}^{q}+\left({v}_{{a}_{2}}^{-}\right)}^{q}\right)}\times \left(1-{\left({{\pi }_{{a}_{1}}^{+}}^{q}\right)}^{k}{\left({{\pi }_{{a}_{2}}^{+}}^{q}\right)}^{k}\right)},\\ \sqrt[q]{\frac{k\left({{\left({v}_{{a}_{1}}^{+}\right)}^{q}+\left({v}_{{a}_{2}}^{+}\right)}^{q}\right)}{k\left({{\left({u}_{{a}_{1}}^{+}\right)}^{q}+\left({u}_{{a}_{2}}^{+}\right)}^{q}\right)+k\left({{\left({v}_{{a}_{1}}^{+}\right)}^{q}+\left({v}_{{a}_{2}}^{+}\right)}^{q}\right)}\times \left(1-{\left({{\pi }_{{a}_{1}}^{-}}^{q}\right)}^{k}{\left({{\pi }_{{a}_{2}}^{-}}^{q}\right)}^{k}\right)}\end{array}\right]\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({u}_{{a}_{2}}^{-}\right)}^{q}}{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}}}PS\left(k\left(\sqrt[q]{1-{\left({\pi }_{{a}_{1}}^{+}\right)}^{q}({{\pi }_{{a}_{2}}^{+})}^{q}}\right)\right),\\ \sqrt[q]{\frac{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({u}_{{a}_{2}}^{+}\right)}^{q}}{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}}PS\left(k\left(\sqrt[q]{1-{\left({\pi }_{{a}_{1}}^{-}\right)}^{q}({{\pi }_{{a}_{2}}^{-})}^{q}}\right)\right)\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{{\left({v}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}}{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({u}_{{a}_{2}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{2}}^{-}\right)}^{q}}}PS\left(k\left(\sqrt[q]{1-{\left({\pi }_{{a}_{1}}^{+}\right)}^{q}({{\pi }_{{a}_{2}}^{+})}^{q}}\right)\right),\\ \sqrt[q]{\frac{{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({u}_{{a}_{2}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{2}}^{+}\right)}^{q}}}PS\left(k\left(\sqrt[q]{1-{\left({\pi }_{{a}_{1}}^{-}\right)}^{q}({{\pi }_{{a}_{2}}^{-})}^{q}}\right)\right)\end{array}\right]\end{array}\right)=k({a}_{1}{\Theta }_{NA}{a}_{2})$$

(3) For any two real number $k_{1} ,k_{2}$, using Eq. (22) we can obtain.

$${k}_{1}a=\left(\begin{array}{c}\left[\sqrt[q]{\frac{{{(u}_{a}^{-})}^{q}}{{{(u}_{a}^{-})}^{q}+{{(v}_{a}^{-})}^{q}}}\times \left(1-{({({\pi }_{a}^{+})}^{q})}^{{k}_{1}}\right),\sqrt[q]{\frac{{{(u}_{a}^{+})}^{q}}{{{(u}_{a}^{+})}^{q}+{{(v}_{a}^{+})}^{q}}}\times \left(1-{({({\pi }_{a}^{-})}^{q})}^{{k}_{1}}\right)\right],\\ \left[\sqrt[q]{\frac{{{(v}_{a}^{-})}^{q}}{{{(u}_{a}^{-})}^{q}+{{(v}_{a}^{-})}^{q}}}\times \left(1-{({({\pi }_{a}^{+})}^{q})}^{{k}_{1}}\right),\sqrt[q]{\frac{{{(v}_{a}^{+})}^{q}}{{{(u}_{a}^{+})}^{q}+{{(v}_{a}^{+})}^{q}}}\times \left(1-{({({\pi }_{a}^{-})}^{q})}^{{k}_{1}}\right)\right]\end{array}\right),$$

$$k_{2} a = \left( {\begin{array}{*{20}c} {\left[ {\sqrt[q]{{\frac{{(u_{a}^{ - } )^{q} }}{{(u_{a}^{ - } )^{q} + (v_{a}^{ - } )^{q} }}}} \times \left( {1 - ((\pi _{a}^{ + } )^{q} )^{{k_{2} }} } \right),\sqrt[q]{{\frac{{(u_{a}^{ + } )^{q} }}{{(u_{a}^{ + } )^{q} + (v_{a}^{ + } )^{q} }}}} \times \left( {1 - ((\pi _{a}^{ - } )^{q} )^{{k_{2} }} } \right)} \right],} \\ {\left[ {\sqrt[q]{{\frac{{(v_{a}^{ - } )^{q} }}{{(u_{a}^{ - } )^{q} + (v_{a}^{ - } )^{q} }}}} \times \left( {1 - ((\pi _{a}^{ + } )^{q} )^{{k_{2} }} } \right),\sqrt[q]{{\frac{{(v_{a}^{ + } )^{q} }}{{(u_{a}^{ + } )^{q} + (v_{a}^{ + } )^{q} }}}} \times \left( {1 - ((\pi _{a}^{ - } )^{q} )^{{k_{2} }} } \right)} \right]} \\ \end{array} } \right).$$

Using Eqs. (15) and (22) we can obtain

$$\left( {k_{1} a} \right)\Theta_{NA} \left( {k_{2} a} \right) =\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{{{MCS}^{-}({k}_{1}a,{k}_{2}a)}^{q}}{{{MCS}^{-}({k}_{1}a,{k}_{2}a)}^{q}+{{NCS}^{-}({k}_{1}a,{k}_{2}a)}^{q}}}\times PS\left(PS({k}_{1}\sqrt[q]{{({u}_{a}^{-})}^{q}+{({v}_{a}^{-})}^{q}}),PS({k}_{2}\sqrt[q]{{({u}_{a}^{-})}^{q}+{({v}_{a}^{-})}^{q}})\right),\\ \sqrt[q]{\frac{{{MCS}^{+}({k}_{1}a,{k}_{2}a)}^{q}}{{{MCS}^{+}({k}_{1}a,{k}_{2}a)}^{q}+{{NCS}^{+}({k}_{1}a,{k}_{2}a)}^{q}}}\times PS\left(PS({k}_{1}\sqrt[q]{{({u}_{a}^{+})}^{q}+{({v}_{a}^{+})}^{q}}),PS({k}_{2}\sqrt[q]{{({u}_{a}^{+})}^{q}+{({v}_{a}^{+})}^{q}})\right)\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{{{NCS}^{-}({k}_{1}a,{k}_{2}a)}^{q}}{{{MCS}^{-}({k}_{1}a,{k}_{2}a)}^{q}+{{NCS}^{-}({k}_{1}a,{k}_{2}a)}^{q}}}\times PS\left(PS({k}_{1}\sqrt[q]{{({u}_{a}^{-})}^{q}+{({v}_{a}^{-})}^{q}}),PS({k}_{2}\sqrt[q]{{({u}_{a}^{-})}^{q}+{({v}_{a}^{-})}^{q}})\right),\\ \sqrt[q]{\frac{{{NCS}^{+}({k}_{1}a,{k}_{2}a)}^{q}}{{{MCS}^{+}({k}_{1}a,{k}_{2}a)}^{q}+{{NCS}^{+}({k}_{1}a,{k}_{2}a)}^{q}}}\times PS\left(PS({k}_{1}\sqrt[q]{{({u}_{a}^{+})}^{q}+{({v}_{a}^{+})}^{q}}),PS({k}_{2}\sqrt[q]{{({u}_{a}^{+})}^{q}+{({v}_{a}^{+})}^{q}})\right)\end{array}\right]\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{({k}_{1}+{k}_{2}){({u}_{a}^{-})}^{q}}{({k}_{1}+{k}_{2}){({u}_{a}^{-})}^{q}+({k}_{1}+{k}_{2}){({v}_{a}^{-})}^{q}}}\times PS\left(\sqrt[q]{1-{({({\pi }_{a}^{+})}^{q})}^{{k}_{1}}},\sqrt[q]{1-{({({\pi }_{a}^{+})}^{q})}^{{k}_{2}}}\right),\\ \sqrt[q]{\frac{({k}_{1}+{k}_{2}){({u}_{a}^{+})}^{q}}{({k}_{1}+{k}_{2}){({u}_{a}^{+})}^{q}+({k}_{1}+{k}_{2}){({v}_{a}^{+})}^{q}}}\times PS\left(\sqrt[q]{1-{({({\pi }_{a}^{-})}^{q})}^{{k}_{1}}},\sqrt[q]{1-{({({\pi }_{a}^{-})}^{q})}^{{k}_{2}}}\right)\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{({k}_{1}+{k}_{2}){({v}_{a}^{-})}^{q}}{({k}_{1}+{k}_{2}){({u}_{a}^{-})}^{q}+({k}_{1}+{k}_{2}){({v}_{a}^{-})}^{q}}}\times PS\left(\sqrt[q]{1-{({({\pi }_{a}^{+})}^{q})}^{{k}_{1}}},\sqrt[q]{1-{({({\pi }_{a}^{+})}^{q})}^{{k}_{2}}}\right),\\ \sqrt[q]{\frac{({k}_{1}+{k}_{2}){({v}_{a}^{+})}^{q}}{({k}_{1}+{k}_{2}){({u}_{a}^{+})}^{q}+({k}_{1}+{k}_{2}){({v}_{a}^{+})}^{q}}}\times PS\left(\sqrt[q]{1-{({({\pi }_{a}^{-})}^{q})}^{{k}_{1}}},\sqrt[q]{1-{({({\pi }_{a}^{-})}^{q})}^{{k}_{2}}}\right)\end{array}\right]\end{array}\right)=\left(\begin{array}{c}\left[\sqrt[q]{\frac{{({u}_{a}^{-})}^{q}}{{({u}_{a}^{-})}^{q}+{({v}_{a}^{-})}^{q}}\times \left(1-{({({\pi }_{a}^{+})}^{q})}^{({k}_{1}+{k}_{2})}\right)},\sqrt[q]{\frac{{({u}_{a}^{+})}^{q}}{{({u}_{a}^{+})}^{q}+{({v}_{a}^{+})}^{q}}\times \left(1-{({({\pi }_{a}^{-})}^{q})}^{({k}_{1}+{k}_{2})}\right)}\right],\\ \left[\sqrt[q]{\frac{{({v}_{a}^{-})}^{q}}{{({u}_{a}^{-})}^{q}+{({v}_{a}^{-})}^{q}}\times \left(1-{({({\pi }_{a}^{+})}^{q})}^{({k}_{1}+{k}_{2})}\right)},\sqrt[q]{\frac{{({v}_{a}^{+})}^{q}}{{({u}_{a}^{+})}^{q}+{({v}_{a}^{+})}^{q}}\times \left(1-{({({\pi }_{a}^{-})}^{q})}^{({k}_{1}+{k}_{2})}\right)}\right]\end{array}\right)=({k}_{1}+{k}_{2})a$$

Thus, the proof of Theorem 4.1 is completed.

1.5 Proof of Definition 4.5

Proof

When $n = 1$, we have:

$$IVq - ROFWANA\left( {a_{1} } \right) ={\lambda }_{1}{a}_{1}=\left(\begin{array}{c}\left[\sqrt[q]{\frac{{\left({u}_{{a}_{1}}^{-}\right)}^{q}}{{\left({u}_{{a}_{1}}^{-}\right)}^{q}+{\left({v}_{{a}_{1}}^{-}\right)}^{q}}\times \left(1-{\left({\left({\pi }_{{a}_{1}}^{+}\right)}^{q}\right)}^{{\lambda }_{1}}\right)},\sqrt[q]{\frac{{\left({u}_{{a}_{1}}^{+}\right)}^{q}}{{\left({u}_{{a}_{1}}^{+}\right)}^{q}+{\left({v}_{{a}_{1}}^{+}\right)}^{q}}\times \left(1-{\left({\left({\pi }_{{a}_{1}}^{-}\right)}^{q}\right)}^{{\lambda }_{1}}\right)}\right],\\ \left[\sqrt[q]{\frac{{({v}_{{a}_{1}}^{-})}^{q}}{{({u}_{{a}_{1}}^{-})}^{q}+{({v}_{{a}_{1}}^{-})}^{q}}\times \left(1-{({({\pi }_{{a}_{1}}^{+})}^{q})}^{{\lambda }_{1}}\right)},\sqrt[q]{\frac{{({v}_{{a}_{1}}^{+})}^{q}}{{({u}_{{a}_{1}}^{+})}^{q}+{({v}_{{a}_{1}}^{+})}^{q}}\times \left(1-{({({\pi }_{{a}_{1}}^{-})}^{q})}^{{\lambda }_{1}}\right)}\right]\end{array}\right)$$

Suppose Eq. (14) holds when $n = k$, that is to say:

$$IVq - ROFWANA\left( {a_{1} ,a_{2} , \ldots ,a_{k} } \right) =\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{\sum_{i=1}^{k}{\lambda }_{i}{\left({u}_{{a}_{i}}^{-}\right)}^{q}}{\sum_{i=1}^{k}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{-}\right)}^{q}+\left({v}_{{a}_{i}}^{-}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}\right)},\\ \sqrt[q]{\frac{\sum_{i=1}^{k}{\lambda }_{i}{\left({u}_{{a}_{i}}^{+}\right)}^{q}}{\sum_{i=1}^{k}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{+}\right)}^{q}+\left({v}_{{a}_{i}}^{+}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}\right)}\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{\sum_{i=1}^{k}{\lambda }_{i}{\left({v}_{{a}_{i}}^{-}\right)}^{q}}{\sum_{i=1}^{k}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{-}\right)}^{q}+\left({v}_{{a}_{i}}^{-}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}\right)},\\ \sqrt[q]{\frac{\sum_{i=1}^{k}{\lambda }_{i}{\left({v}_{{a}_{i}}^{+}\right)}^{q}}{\sum_{i=1}^{k}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{+}\right)}^{q}+\left({v}_{{a}_{i}}^{+}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}\right)}\end{array}\right]\end{array}\right)$$

If $n = k + 1$, we have:

$$IVq - ROFWANA\left( {a_{1} ,a_{2} ,a_{3} , \ldots ,a_{k} ,a_{k + 1} } \right)$$

$$= IVq - ROFWANA\left( {a_{1} ,a_{2} ,a_{3} , \ldots ,a_{k} } \right) \Theta_{NA} \left( {\lambda_{k + 1} a_{k + 1} } \right) =\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{{{MCS}^{-}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}}{\left(\begin{array}{c}{{MCS}^{-}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\\ {{+NCS}^{-}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\end{array}\right)}}\times PS\left(\begin{array}{c}\sqrt[q]{1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}},\\ \sqrt[q]{1-{\left({\left({\pi }_{{a}_{k+1}}^{+}\right)}^{q}\right)}^{{\lambda }_{k+1}}}\end{array}\right),\\ \sqrt[q]{\frac{{{MCS}^{+}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}}{\left(\begin{array}{c}{{MCS}^{+}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\\ {{+NCS}^{+}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\end{array}\right)}}\times PS\left(\begin{array}{c}\sqrt[q]{1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}},\\ \sqrt[q]{1-{\left({\left({\pi }_{{a}_{k+1}}^{-}\right)}^{q}\right)}^{{\lambda }_{k+1}}}\end{array}\right)\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{{{NCS}^{-}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}}{\left(\begin{array}{c}{{MCS}^{-}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\\ {{+NCS}^{-}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\end{array}\right)}}\times PS\left(\begin{array}{c}\sqrt[q]{1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}},\\ \sqrt[q]{1-{\left({\left({\pi }_{{a}_{k+1}}^{+}\right)}^{q}\right)}^{{\lambda }_{k+1}}}\end{array}\right),\\ \sqrt[q]{\frac{{{NCS}^{+}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}}{\left(\begin{array}{c}{{MCS}^{+}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\\ {{+NCS}^{+}\left(IVq-ROFWANA\left({a}_{1},{a}_{2},{a}_{3}\dots {a}_{k}\right),{\lambda }_{k+1}{a}_{k+1}\right)}^{q}\end{array}\right)}}\times PS\left(\begin{array}{c}\sqrt[q]{1-\prod_{i=1}^{k}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}},\\ \sqrt[q]{1-{\left({\left({\pi }_{{a}_{k+1}}^{-}\right)}^{q}\right)}^{{\lambda }_{k+1}}}\end{array}\right)\end{array}\right]\end{array}\right)$$

According to formula (14), we get:

$${\text{MCS}}\left( {IVq - ROFWANA\left( {a_{1} ,a_{2} ,a_{3} ,\ldots,a_{k} } \right),\left( {\lambda_{k + 1} a_{k + 1} } \right)} \right)^{{\text{q}}} = MCS\left( {IVq - ROFWANA\left( {a_{1} ,a_{2} ,a_{3} ,\ldots,a_{k} } \right)} \right)^{q} + MCS\left( {\lambda_{k + 1} a_{k + 1} } \right) = \left( {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{k}}} \lambda_{{\text{i}}} \left( {u_{{a_{i} }}^{ - } } \right)^{{\text{q}}} + \lambda_{k + 1} \left( {u_{{a_{k + 1} }}^{ - } } \right)^{{\text{q}}} ,\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{k}}} \lambda_{{\text{i}}} \left( {u_{{a_{i} }}^{ + } } \right)^{{\text{q}}} + \lambda_{k + 1} \left( {u_{{a_{k + 1} }}^{ + } } \right)^{{\text{q}}} } \right)$$

Similarly, it is easy to obtain:

$${\text{NCS}}\left( {{\text{IVq}} - {\text{ROFWANA}}\left( {a_{1} ,a_{2} ,a_{3} ,\ldots,a_{k} } \right),\left( {\lambda_{k + 1} a_{k + 1} } \right)} \right)^{{\text{q}}} = {\text{NCS}}\left( {{\text{IVq}} - {\text{ROFWANA}}\left( {a_{1} ,a_{2} ,a_{3} ,\ldots,a_{k} } \right)} \right)^{{\text{q}}} + {\text{NCS}}\left( {\lambda_{k + 1} a_{k + 1} } \right) = \left( {\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{k}}} \lambda_{{\text{i}}} \left( {v_{{a_{i} }}^{ - } } \right)^{{\text{q}}} + \lambda_{k + 1} \left( {v_{{a_{k + 1} }}^{ - } } \right)^{{\text{q}}} ,\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{k}}} \lambda_{{\text{i}}} \left( {v_{{a_{i} }}^{ + } } \right)^{{\text{q}}} + \lambda_{k + 1} \left( {v_{{a_{k + 1} }}^{ + } } \right)^{{\text{q}}} } \right)$$

According to formula (14), we get:

$$IVq - ROFWANA\left( {a_{1} ,a_{2} ,a_{3} ,\ldots,a_{k + 1} } \right) =\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{\sum_{i=1}^{k+1}{\lambda }_{i}{\left({u}_{{a}_{i}}^{-}\right)}^{q}}{\sum_{i=1}^{k+1}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{-}\right)}^{q}+\left({v}_{{a}_{i}}^{-}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k+1}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}\right)},\\ \sqrt[q]{\frac{\sum_{i=1}^{k+1}{\lambda }_{i}{\left({u}_{{a}_{i}}^{+}\right)}^{q}}{\sum_{i=1}^{k+1}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{+}\right)}^{q}+\left({v}_{{a}_{i}}^{+}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k+1}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}\right)}\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{\sum_{i=1}^{k+1}{\lambda }_{i}{\left({v}_{{a}_{i}}^{-}\right)}^{q}}{\sum_{i=1}^{k+1}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{-}\right)}^{q}+\left({v}_{{a}_{i}}^{-}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k+1}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}\right)},\\ \sqrt[q]{\frac{\sum_{i=1}^{k+1}{\lambda }_{i}{\left({v}_{{a}_{i}}^{+}\right)}^{q}}{\sum_{i=1}^{k+1}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{+}\right)}^{q}+\left({v}_{{a}_{i}}^{+}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{k+1}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}\right)}\end{array}\right]\end{array}\right)$$

Therefore, the expression (23) holds for n = k + 1. According to the mathematical induction, the proof of Eq. (23) is completed.

1.6 Proof of Property 4.1

Proof $IVq - ROFWANA\left( {a_{1} ,a_{2} ,a_{3} ,\ldots,a_{n} } \right)$$=\left(\begin{array}{c}\left[\begin{array}{c}\sqrt[q]{\frac{\sum_{i=1}^{n}{\lambda }_{i}{\left({u}_{{a}_{i}}^{-}\right)}^{q}}{\sum_{i=1}^{k}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{-}\right)}^{q}+\left({v}_{{a}_{i}}^{-}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{n}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}\right)},\\ \sqrt[q]{\frac{\sum_{i=1}^{n}{\lambda }_{i}{\left({u}_{{a}_{i}}^{+}\right)}^{q}}{\sum_{i=1}^{n}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{+}\right)}^{q}+\left({v}_{{a}_{i}}^{+}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{n}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}\right)}\end{array}\right],\\ \left[\begin{array}{c}\sqrt[q]{\frac{\sum_{i=1}^{n}{\lambda }_{i}{\left({v}_{{a}_{i}}^{-}\right)}^{q}}{\sum_{i=1}^{k}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{-}\right)}^{q}+\left({v}_{{a}_{i}}^{-}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{n}{\left({\left({\pi }_{{a}_{i}}^{+}\right)}^{q}\right)}^{{\lambda }_{i}}\right)},\\ \sqrt[q]{\frac{\sum_{i=1}^{n}{\lambda }_{i}{\left({v}_{{a}_{i}}^{+}\right)}^{q}}{\sum_{i=1}^{n}{\lambda }_{i}\left({{\left({u}_{{a}_{i}}^{+}\right)}^{q}+\left({v}_{{a}_{i}}^{+}\right)}^{q}\right)}\times \left(1-\prod_{i=1}^{n}{\left({\left({\pi }_{{a}_{i}}^{-}\right)}^{q}\right)}^{{\lambda }_{i}}\right)}\end{array}\right]\end{array}\right)$

$$=\left(\begin{array}{c}\left[\sqrt[\mathrm{q}]{\frac{{({u}_{a}^{-})}^{\mathrm{q}}}{{({u}_{a}^{-})}^{\mathrm{q}}+{({v}_{a}^{-})}^{\mathrm{q}}}\times \left(1-{({({\uppi }_{a}^{+})}^{\mathrm{q}})}^{\sum_{\mathrm{i}=1}^{\mathrm{n}}{\lambda }_{\mathrm{i}}}\right)},\sqrt[\mathrm{q}]{\frac{{({u}_{a}^{+})}^{\mathrm{q}}}{{({u}_{a}^{+})}^{\mathrm{q}}+{({v}_{a}^{+})}^{\mathrm{q}}}\times \left(1-{({({\uppi }_{a}^{-})}^{\mathrm{q}})}^{\sum_{\mathrm{i}=1}^{\mathrm{n}}{\lambda }_{\mathrm{i}}}\right)}\right],\\ \left[\sqrt[\mathrm{q}]{\frac{{({v}_{a}^{-})}^{\mathrm{q}}}{{({u}_{a}^{-})}^{\mathrm{q}}+{({v}_{a}^{-})}^{\mathrm{q}}}\times \left(1-{({({\uppi }_{a}^{+})}^{\mathrm{q}})}^{\sum_{\mathrm{i}=1}^{\mathrm{n}}{\lambda }_{\mathrm{i}}}\right)},\sqrt[\mathrm{q}]{\frac{{({v}_{a}^{+})}^{\mathrm{q}}}{{({u}_{a}^{+})}^{\mathrm{q}}+{({v}_{a}^{+})}^{\mathrm{q}}}\times \left(1-{({({\uppi }_{a}^{-})}^{\mathrm{q}})}^{\sum_{\mathrm{i}=1}^{\mathrm{n}}{\lambda }_{\mathrm{i}}}\right)}\right]\end{array}\right)=\left(\begin{array}{c}\left[\sqrt[\mathrm{q}]{\frac{{({u}_{a}^{-})}^{\mathrm{q}}}{{({u}_{a}^{-})}^{\mathrm{q}}+{({v}_{a}^{-})}^{\mathrm{q}}}\times \left({({u}_{a}^{-})}^{\mathrm{q}}+{({v}_{a}^{-})}^{\mathrm{q}}\right)},\sqrt[\mathrm{q}]{\frac{{({u}_{a}^{+})}^{\mathrm{q}}}{{({u}_{a}^{+})}^{\mathrm{q}}+{({v}_{a}^{+})}^{\mathrm{q}}}\times \left({({u}_{a}^{+})}^{\mathrm{q}}+{({v}_{a}^{+})}^{\mathrm{q}}\right)}\right],\\ \left[\sqrt[\mathrm{q}]{\frac{{({v}_{a}^{-})}^{\mathrm{q}}}{{({u}_{a}^{-})}^{\mathrm{q}}+{({v}_{a}^{-})}^{\mathrm{q}}}\times \left({({u}_{a}^{-})}^{\mathrm{q}}+{({v}_{a}^{-})}^{\mathrm{q}}\right)},\sqrt[\mathrm{q}]{\frac{{({v}_{a}^{+})}^{\mathrm{q}}}{{({u}_{a}^{+})}^{\mathrm{q}}+{({v}_{a}^{+})}^{\mathrm{q}}}\times \left({({u}_{a}^{+})}^{\mathrm{q}}+{({v}_{a}^{+})}^{\mathrm{q}}\right)}\right]\end{array}\right)=\left(\left[{{u}^{-}}_{a},{{u}^{+}}_{a}\right],\left[{{v}^{-}}_{a},{{v}^{+}}_{a}\right]\right)=a$$

Appendix 2

See Tables 13, 14, 15, 16 and Fig. 7

Table 13 Interval-valued q-rung orthopair fuzzy decision matrix

Table 14 Sensitive analysis of the parameter q

Table 15 The ranking values and the ranking orders of schemes obtained by different operators with case of different values of q

Table 16 The final ranking values and the ranking orders of schemes obtained by different decision-making methods with case of different values of q

Fig. 7

GDM Flow based on the developed PSI-COPRAS method