1 Introduction
1.1 Research Motivation
1.2 Contributions

1. Some new operational laws are presented in order to fair treatment of belongingness and nonbelongingness grades.

2. Four new operators, namely qROF improved power weighted averaging and geometric (qROFIPWA and qROFIPWG, resp.) operators, qROF improved power weighted averaging and geometric MSM (qROFIPWAMSM and qROFIPWGMSM, resp.) operators are developed.

3. A novel DM approach is developed based on the proposed operators. This proposed approach can resolve the limitations of Liu et al. (2020).

4. To show the efficiency of the proposed methodology, a personnel selection problem is considered under qROF setting.

5. A detailed comparative investigation is demonstrated to validate the superiority of the proposed model.
2 Preliminaries
2.1 qRung Orthopair Fuzzy Sets (qROFSs)
Definition 1 (Yager, 2017).
Definition 2 (Liu and Wang, 2018).
Definition 3 (Liu and Wang, 2018).
Definition 4 (Liu and Wang, 2018).
Definition 5 (Liu and Wang, 2018).
(3)
\[\begin{aligned}{}(\mathrm{i})& \hspace{2.5pt}{\Theta _{1}}\otimes {\Theta _{2}}=\big\langle \sqrt[q]{1\big(1{\Delta _{1}^{q}}\big)\big(1{\Delta _{2}^{q}}\big)},{\nabla _{1}}{\nabla _{2}}\big\rangle ,\end{aligned}\](4)
\[\begin{aligned}{}(\mathrm{ii})& \hspace{2.5pt}{\Theta _{1}}\otimes {\Theta _{2}}=\big\langle {\Delta _{1}}{\Delta _{2}},\sqrt[q]{\big(1\big(1{\nabla _{1}^{q}}\big)\big(1{\nabla _{2}^{q}}\big)\big)}\big\rangle ,\end{aligned}\]Definition 6.
2.2 Power Averaging Operator (PAO)
Definition 7 (Yager, 2001).
(8)
\[ PA({b_{1}},{b_{2}},\dots ,{b_{n}})=\frac{{\textstyle\textstyle\sum _{i=1}^{n}}(1+\psi ({b_{i}})){b_{i}}}{{\textstyle\textstyle\sum _{i=1}^{n}}(1+\psi ({b_{i}}))},\]3 qROF Improved Power Weighted Operators
3.1 New Operations Between qROFNs
Definition 8.
(9)
\[\begin{aligned}{}(\mathrm{i})& \hspace{2.5pt}{\Theta _{1}}\tilde{\oplus }{\Theta _{2}}=\left\langle \sqrt[q]{1{\prod \limits_{r=1}^{2}}\big(1{\Delta _{r}^{q}}\big)},\sqrt[q]{{\prod \limits_{r=1}^{2}}\big(1{\Delta _{r}^{q}}\big){\prod \limits_{r=1}^{2}}\big(1{\Delta _{r}^{q}}{\nabla _{r}^{q}}\big)}\right\rangle ;\end{aligned}\](10)
\[\begin{aligned}{}(\mathrm{ii})& \hspace{2.5pt}{\Theta _{1}}\tilde{\otimes }{\Theta _{2}}=\left\langle \sqrt[q]{{\prod \limits_{r=1}^{2}}\big(1{\nabla _{r}^{q}}\big){\prod \limits_{r=1}^{2}}\big(1{\Delta _{r}^{q}}{\nabla _{r}^{q}}\big)},\sqrt[q]{1{\prod \limits_{r=1}^{2}}\big(1{\nabla _{r}^{q}}\big)}\right\rangle ;\end{aligned}\]Example 1.
Example 2.
Example 3.
Example 4.
Theorem 1.

(i) ${\Theta _{1}}\tilde{\oplus }\hspace{0.1667em}{\Theta _{2}}={\Theta _{2}}\tilde{\oplus }{\Theta _{1}}$;

(ii) ${\Theta _{1}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{2}}={\Theta _{2}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{1}}$;

(iii) $\lambda ({\Theta _{1}}\tilde{\oplus }{\Theta _{2}})=\lambda {\Theta _{1}}\tilde{\oplus }\hspace{0.1667em}\lambda {\Theta _{2}}$;

(iv) ${({\Theta _{1}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{2}})^{\lambda }}={\Theta _{1}^{\lambda }}\tilde{\otimes }\hspace{0.1667em}{\Theta _{2}^{\lambda }}$;

(v) $({\lambda _{1}}+{\lambda _{2}}){\Theta _{1}}={\lambda _{1}}{\Theta _{1}}\tilde{\oplus }\hspace{0.1667em}{\lambda _{2}}{\Theta _{1}}$;

(vi) ${\Theta _{1}^{{\lambda _{1}}+{\lambda _{2}}}}={\Theta _{1}^{{\lambda _{1}}}}\tilde{\otimes }\hspace{0.1667em}{\Theta _{1}^{{\lambda _{2}}}}$.
3.2 qROF Improved Power Weighted Averaging Operators
Definition 9.
Theorem 2.
(15)
\[\begin{aligned}{}& \text{qROFIPWA}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\left\langle {\Bigg(1{\prod \limits_{r=1}^{n}}{\big(1{\Delta _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}},{\Bigg({\prod \limits_{r=1}^{n}}{\big(1{\Delta _{r}^{q}}\big)^{{\Omega _{r}}}}{\prod \limits_{r=1}^{n}}{\big(1{\Delta _{r}^{q}}{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}}\right\rangle .\end{aligned}\]Theorem 3 (Idempotency).
Theorem 4 (Boundedness).
Theorem 5 (Monotonicity).
Definition 10.
(16)
\[ {\text{qROFIPWAMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})={\Bigg(\frac{1}{{^{n}}{c_{p}}}\hspace{0.1667em}\hspace{0.1667em}\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\bigoplus }\Bigg(\hspace{0.1667em}{\underset{j=1}{\overset{p}{\bigotimes }}}(n{\Omega _{{t_{j}}}}{\Theta _{{t_{j}}}})\Bigg)\Bigg)^{\frac{1}{p}}},\]Theorem 6.
(17)
\[\begin{aligned}{}& {\text{qROFIPWAMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\Bigg\langle \Bigg(\Bigg(1\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1{\prod \limits_{j=1}^{p}}\big(1{\big(1{({\Delta _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{r}}}\\ {} & \hspace{2em}{\Bigg({\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg)^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}},\\ {} & \hspace{2em}\Bigg(1\Bigg(1\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1{\prod \limits_{j=1}^{p}}\big(1{\big(1{({\Delta _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}}\Bigg\rangle .\end{aligned}\]Theorem 7 (Idempotency).
Theorem 8 (Boundedness).
Theorem 9 (Monotonicity).
3.3 qROF Improved Power Weighted Geometric Operators
Theorem 10.
(19)
\[\begin{aligned}{}& \text{qROFIPWG}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\left\langle {\Bigg({\prod \limits_{r=1}^{n}}{\big(1{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}{\prod \limits_{r=1}^{n}}{\big(1{\Delta _{r}^{q}}{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}},\right.\left.{\Bigg(1{\prod \limits_{r=1}^{n}}{\big(1{\nabla _{r}^{q}}\big)^{{\Omega _{r}}}}\Bigg)^{\frac{1}{q}}}\right\rangle .\end{aligned}\]Theorem 11.
Theorem 12.
Theorem 13.
Definition 12.
(20)
\[ {\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})=\frac{1}{p}{\Big(\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\widetilde{\bigotimes }}\Big({\underset{j=1}{\overset{p}{\widetilde{\bigoplus }}}}{\Theta _{{t_{j}}}^{{\Omega _{{t_{j}}}}}}\Big)\Big)^{\frac{1}{{^{n}}{c_{p}}}}},\]Theorem 14.
(21)
\[\begin{aligned}{}& {\text{qROFIPWGMSM}^{(p)}}({\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{n}})\\ {} & \hspace{1em}=\Bigg\langle \Bigg(1\Bigg(1\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1{\prod \limits_{j=1}^{p}}\big(1{\big(1{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}},\\ {} & \hspace{2em}\Bigg(\Bigg(1\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg(1{\prod \limits_{j=1}^{p}}\big(1{\big(1{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\\ {} & \hspace{2em}+{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\big)+{\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg){^{\frac{1}{{^{n}}{c_{p}}}}}\\ {} & \hspace{2em}+{\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg){^{\frac{1}{p}}}\\ {} & \hspace{2em}{\Bigg({\Bigg(\prod \limits_{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}\Bigg({\prod \limits_{j=1}^{p}}{\big(1{({\Delta _{{t_{j}}}})^{q}}{({\nabla _{{t_{j}}}})^{q}}\big)^{n{\Omega _{{t_{j}}}}}}\Bigg)\Bigg)^{\frac{1}{{^{n}}{c_{p}}}}}\Bigg)^{\frac{1}{p}}}\Bigg){^{\frac{1}{q}}}\Bigg\rangle .\end{aligned}\]Theorem 15.
Theorem 16.
Theorem 17 (Monotonicity).
4 Group Decision Making Methodology
(22)
\[ {\tilde{\Theta }_{ij}^{(d)}}=\left\{\begin{array}{l@{\hskip4.0pt}l}\langle {\Delta _{ij}^{(d)}},{\nabla _{ij}^{(d)}}\rangle \hspace{1em}& \text{if}\hspace{2.5pt}{C_{j}}\hspace{2.5pt}\text{is of benefittype},\\ {} \langle {\nabla _{ij}^{(d)}},{\Delta _{ij}^{(d)}}\rangle \hspace{1em}& \text{if}\hspace{2.5pt}{C_{j}}\hspace{2.5pt}\text{is of costtype}.\end{array}\right.\](23)
\[ \textit{Supp}\big({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}}\big)=1\textit{Dist}\big({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}}\big)\big(d,s=1(1)l;d\ne s\big),\](24)
\[ \psi \big({\tilde{\Theta }_{ij}^{(d)}}\big)={\sum \limits_{s=1,s\ne d}^{l}}\textit{Supp}\big({\tilde{\Theta }_{ij}^{(d)}},{\tilde{\Theta }_{ij}^{(s)}}\big)\big(i=1(1)m;j=1(1)n;d=1(1)l\big).\](25)
\[ {\Omega _{ij}^{(d)}}=\frac{{\eta _{d}}(1+\psi ({\tilde{\Theta }_{ij}^{(d)}}))}{{\textstyle\textstyle\sum _{d=1}^{l}}{\eta _{d}}(1+\psi ({\tilde{\Theta }_{ij}^{(d)}}))}.\](26)
\[\begin{aligned}{}& \text{qROFIPWA}\big({\tilde{\Theta }_{ij}^{(1)}},{\tilde{\Theta }_{ij}^{(2)}},\dots ,{\tilde{\Theta }_{ij}^{(l)}}\big)\\ {} & \hspace{1em}=\Bigg\langle {\Bigg(1{\prod \limits_{d=1}^{l}}{\big(1{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}},\\ {} & \hspace{2em}{\Bigg({\prod \limits_{d=1}^{l}}{\big(1{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}{\prod \limits_{d=1}^{l}}{\big(1{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}}{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}}\Bigg\rangle ,\end{aligned}\](27)
\[\begin{aligned}{}& \text{qROFIPWG}\big({\tilde{\Theta }_{ij}^{(1)}},{\tilde{\Theta }_{ij}^{(2)}},\dots ,{\tilde{\Theta }_{ij}^{(l)}}\big)\\ {} & \hspace{1em}=\Bigg\langle {\Bigg({\prod \limits_{d=1}^{l}}{\big(1{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}{\prod \limits_{d=1}^{l}}{\big(1{\big({\tilde{\Delta }_{ij}^{(d)}}\big)^{q}},{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}},\\ {} & \hspace{2em}{\Bigg(1{\prod \limits_{d=1}^{l}}{\big(1{\big({\tilde{\nabla }_{ij}^{(d)}}\big)^{q}}\big)^{{\Omega _{ij}^{(d)}}}}\Bigg)^{\frac{1}{q}}}\Bigg\rangle .\end{aligned}\](28)
\[ \textit{Supp}({\Theta _{ij}},{\Theta _{iy}})=1\textit{Dist}({\Theta _{ij}},{\Theta _{iy}})\big(j,y=1(1)n;j\ne y\big),\](29)
\[ \psi ({\Theta _{ij}})={\sum \limits_{y=1,y\ne j}^{n}}\textit{Supp}({\Theta _{ij}},{\Theta _{iy}})\big(i=1(1)m;j=1(1)n\big).\](30)
\[ {\Omega _{ij}}=\frac{{\varpi _{j}}(1+\psi ({\Theta _{ij}}))}{{\textstyle\textstyle\sum _{j=1}^{n}}{\varpi _{j}}(1+\psi ({\Theta _{ij}}))}\big(i=1(1)m;j=1(1)n\big).\](31)
\[\begin{aligned}{}{\Theta _{i}}& ={\text{qROFIPWAMSM}^{(p)}}({\Theta _{i1}},{\Theta _{i2}},\dots ,{\Theta _{in}})\\ {} & ={\Bigg(\frac{1}{{^{n}}{c_{p}}}\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\bigoplus }\Bigg({\underset{j=1}{\overset{p}{\bigotimes }}}(n{\Omega _{i{t_{j}}}}{\Theta _{i{t_{j}}}})\Bigg)\Bigg)^{\frac{1}{p}}},\end{aligned}\](32)
\[\begin{aligned}{}{\Theta _{i}}& ={\text{qROFIPWGMSM}^{(p)}}({\Theta _{i1}},{\Theta _{i2}},\dots ,{\Theta _{in}})\\ {} & =\frac{1}{p}{\Big(\underset{1\leqslant {t_{1}}<{t_{2}}<\cdots <{t_{p}}\leqslant n}{\widetilde{\bigotimes }}\Big({\underset{j=1}{\overset{\hspace{0.1667em}p}{\widetilde{\bigoplus }}}}{\Theta _{i{t_{j}}}^{n{\Omega _{i{t_{j}}}}}}\Big)\Big)^{\frac{1}{{^{n}}{c_{p}}}}}.\end{aligned}\]5 Application of the Proposed Methodology
5.1 Problem Description
5.2 Problem Solution
Table 1
Expert  Alternative  ${L_{1}}$  ${L_{2}}$  ${L_{3}}$  ${L_{4}}$ 
${D_{1}}$  ${X_{1}}$  $\langle 0.2,0.6\rangle $  $\langle 0.4,0.6\rangle $  $\langle 0.4,0.3\rangle $  $\langle 0.5,0.6\rangle $ 
${X_{2}}$  $\langle 0.6,0.5\rangle $  $\langle 0.6,0.5\rangle $  $\langle 0.5,0.4\rangle $  $\langle 0.5,0.2\rangle $  
${X_{3}}$  $\langle 0.8,0.2\rangle $  $\langle 0.5,0.2\rangle $  $\langle 0.5,0.3\rangle $  $\langle 0.4,0.3\rangle $  
${X_{4}}$  $\langle 0.5,0.6\rangle $  $\langle 0.3,0.5\rangle $  $\langle 0.5,0.2\rangle $  $\langle 0.5,0.2\rangle $  
${X_{5}}$  $\langle 0.5,0.3\rangle $  $\langle 0.4,0.7\rangle $  $\langle 0.6,0.4\rangle $  $\langle 0.6,0.6\rangle $  
${X_{1}}$  $\langle 0.4,0.6\rangle $  $\langle 0.2,0.2\rangle $  $\langle 0.5,0.4\rangle $  $\langle 0.4,0.6\rangle $  
${D_{2}}$  ${X_{2}}$  $\langle 0.5,0.1\rangle $  $\langle 0.6,0.4\rangle $  $\langle 0.5,0.5\rangle $  $\langle 0.4,0.3\rangle $ 
${X_{3}}$  $\langle 0.7,0.3\rangle $  $\langle 0.4,0.2\rangle $  $\langle 0.4,0.1\rangle $  $\langle 0.5,0.4\rangle $  
${X_{4}}$  $\langle 0.5,0.4\rangle $  $\langle 0.5,0.7\rangle $  $\langle 0.5,0.6\rangle $  $\langle 0.3,0.8\rangle $  
${X_{5}}$  $\langle 0.6,0.4\rangle $  $\langle 0.3,0.3\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.4,0.2\rangle $  
${X_{1}}$  $\langle 0.7,0.7\rangle $  $\langle 0.5,0.4\rangle $  $\langle 0.2,0.4\rangle $  $\langle 0.4,0.6\rangle $  
${D_{3}}$  ${X_{2}}$  $\langle 0.4,0.2\rangle $  $\langle 0.5,0.4\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.5,0.1\rangle $ 
${X_{3}}$  $\langle 0.5,0.3\rangle $  $\langle 0.4,0.2\rangle $  $\langle 0.4,0.3\rangle $  $\langle 0.6,0.4\rangle $  
${X_{4}}$  $\langle 0.3,0.5\rangle $  $\langle 0.5,0.4\rangle $  $\langle 0.5,0.2\rangle $  $\langle 0.8,0.2\rangle $  
${X_{5}}$  $\langle 0.4,0.6\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.4,0.4\rangle $  $\langle 0.6,0.1\rangle $ 
Table 2
Alternative  ${L_{1}}$  ${L_{2}}$  ${L_{3}}$  ${L_{4}}$ 
${X_{1}}$  $\langle 0.457546,0.727680\rangle $  $\langle 0.377018,0.456022\rangle $  $\langle 0.411592,0.370298\rangle $  $\langle 0.438196,0.600809\rangle $ 
${X_{2}}$  $\langle 0.516059,0.345696\rangle $  $\langle 0.578450,0.442128\rangle $  $\langle 0.528034,0.422266\rangle $  $\langle 0.464399,0.226244\rangle $ 
${X_{3}}$  $\langle 0.710191,0.264441\rangle $  $\langle 0.438196,0.200154\rangle $  $\langle 0.438588,0.246261\rangle $  $\langle 0.500551,0.375744\rangle $ 
${X_{4}}$  $\langle 0.461582,0.510394\rangle $  $\langle 0.443096,0.594864\rangle $  $\langle 0.5,0.425597\rangle $  $\langle 0.578171,0.513749\rangle $ 
${X_{5}}$  $\langle 0.526362,0.432299\rangle $  $\langle 0.437910,0.502261\rangle $  $\langle 0.562724,0.362961\rangle $  $\langle 0.534898,0.427268\rangle $ 
5.3 Effects of the Parameter ‘p’ on Ranking Orders
Table 3
Parameter  Score value  Ranking order 
$p=1$  ${V_{1}}=0.1497$, ${V_{2}}=0.1332$, ${V_{3}}=0.1849$, ${V_{4}}=0.0222$, ${V_{5}}=0.1092$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
$p=2$  ${V_{1}}=0.1202$, ${V_{2}}=0.1362$, ${V_{3}}=0.1927$, ${V_{4}}=0.0404$, ${V_{5}}=0.1162$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
$p=3$  ${V_{1}}=0.1040$, ${V_{2}}=0.1388$, ${V_{3}}=0.2002$, ${V_{4}}=0.0579$, ${V_{5}}=0.1242$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
$p=4$  ${V_{1}}=0.0641$, ${V_{2}}=0.0307$, ${V_{3}}=0.0484$, ${V_{4}}=0.0109$, ${V_{5}}=0.0199$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
Table 4
Parameter  Score value  Ranking order 
$p=1$  ${V_{1}}=0.0173$, ${V_{2}}=0.0043$, ${V_{3}}=0.0014$, ${V_{4}}=0.0020$, ${V_{5}}=0.0017$  ${X_{1}}\succ {X_{2}}>{X_{4}}>{X_{3}}>{X_{5}}$ 
$p=2$  ${V_{1}}=0.0788$, ${V_{2}}=0.1435$, ${V_{3}}=0.1823$, ${V_{4}}=0.0452$, ${V_{5}}=0.1155$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
$p=3$  ${V_{1}}=0.1772$, ${V_{2}}=0.0349$, ${V_{3}}=0.1132$, ${V_{4}}=0.1631$, ${V_{5}}=0.0176$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
$p=4$  ${V_{1}}=0.1280$, ${V_{2}}=0.0609$, ${V_{3}}=0.0883$, ${V_{4}}=0.0983$, ${V_{5}}=0.0538$  ${X_{3}}\succ {X_{2}}>{X_{5}}>{X_{4}}>{X_{2}}$ 
5.4 Comparative Analysis with Existing Methods
Table 5
Method  Score value  Ranking order 
Jana et al. (2019b) with qROFDWA operator  ${V_{1}}=0.0212$, ${V_{2}}=0.1885$, ${V_{3}}=0.2743$, ${V_{4}}=0.0522$, ${V_{5}}=0.1233$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Wei et al. (2018) with qROFGWHM operator  ${V_{1}}=0.2942$, ${V_{2}}=0.0703$, ${V_{3}}=0.0272$, ${V_{4}}=0.1913$, ${V_{5}}=0.1324$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Wei et al. (2018) with qROFGWGHM operator  ${V_{1}}=0.0884$, ${V_{2}}=0.3162$, ${V_{3}}=0.3317$, ${V_{4}}=0.2660$, ${V_{5}}=0.3294$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Liu and Liu (2018) with qROFWBM operator  ${V_{1}}=0.7144$, ${V_{2}}=0.5922$, ${V_{3}}=0.5780$, ${V_{4}}=0.6416$, ${V_{5}}=6163$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Yang and Pang (2020) with qROFWBMDA operator  ${V_{1}}=0.4993$, ${V_{2}}=0.2204$, ${V_{3}}=0.0846$, ${V_{4}}=0.4227$, ${V_{5}}=0.2986$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Liu and Wang (2018) with qROFWA operator  ${V_{1}}=0.0919$, ${V_{2}}=0.1568$, ${V_{3}}=0.2051$, ${V_{4}}=0.0309$, ${V_{5}}=0.1130$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Garg and Chen (2020) with qROFWNA operator  ${V_{1}}=0.5931$, ${V_{2}}=0.7337$, ${V_{3}}=0.7525$, ${V_{4}}=0.6783$, ${V_{5}}=0.7411$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Liu et al. (2020) with qROFPWMSM operator  ${V_{1}}=0.1154$, ${V_{2}}=0.1143$, ${V_{3}}=0.1656$, ${V_{4}}=0.0348$, ${V_{5}}=0.0934$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Proposed method with qROFIPWA operator and qROFIPWAMSM operator  ${V_{1}}=0.1202$, ${V_{2}}=0.1362$, ${V_{3}}=0.1927$, ${V_{4}}=0.0404$, ${V_{5}}=0.1162$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Proposed method with qROFIPWG operator and qROFIPWGMSM operator  ${V_{1}}=0.0788$, ${V_{2}}=0.1435$, ${V_{3}}=0.1823$, ${V_{4}}=0.0452$, ${V_{5}}=0.1156$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
5.5 Comparative Analysis Based on Biasness of Experts
Table 6
Method  Score value  Ranking order 
Jana et al. (2019b) with qROFDWA operator  ${V_{1}}=0.0343$, ${V_{2}}=0.2170$, ${V_{3}}=0.2042$, ${V_{4}}=0.0133$, ${V_{5}}=0.1445$  ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{1}}\succ {X_{4}}$ 
Wei et al. (2018) with qROFGWHM operator  ${V_{1}}=0.2538$, ${V_{2}}=0.0407$, ${V_{3}}=0.0642$, ${V_{4}}=0.2449$, ${V_{5}}=0.1066$  ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Wei et al. (2018) with qROFGWGHM operator  ${V_{1}}=0.1324$, ${V_{2}}=0.3172$, ${V_{3}}=0.3312$, ${V_{4}}=0.2084$, ${V_{5}}=0.3526$  ${X_{5}}\succ {X_{3}}\succ {X_{2}}\succ {X_{4}}\succ {X_{1}}$ 
Liu and Liu (2018) with qROFWBM operator  ${V_{1}}=0.7144$, ${V_{2}}=0.5845$, ${V_{3}}=0.5640$, ${V_{4}}=0.6416$, ${V_{5}}=6163$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Yang and Pang (2020) with qROFWBMDA operator  ${V_{1}}=0.4570$, ${V_{2}}=0.1966$, ${V_{3}}=0.0703$, ${V_{4}}=0.44442$, ${V_{5}}=0.2833$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Liu and Wang (2018) with qROFWA operator  ${V_{1}}=0.0405$, ${V_{2}}=0.1789$, ${V_{3}}=0.1774$, ${V_{4}}=0.0122$, ${V_{5}}=0.1350$  ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Garg and Chen (2020) with qROFWNA operator  ${V_{1}}=0.5930$, ${V_{2}}=0.7395$, ${V_{3}}=0.7113$, ${V_{4}}=0.6782$, ${V_{5}}=0.7410$  ${X_{5}}\succ {X_{3}}\succ {X_{2}}\succ {X_{4}}\succ {X_{1}}$ 
Liu et al. (2020) with qROFPWMSM operator  ${V_{1}}=0.1154$, ${V_{2}}=0.1161$, ${V_{3}}=0.1556$, ${V_{4}}=0.0348$, ${V_{5}}=0.0934$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Proposed method with qROFIPWA operator and qROFIPWAMSM operator  ${V_{1}}=0.1202$, ${V_{2}}=0.1422$, ${V_{3}}=0.1645$, ${V_{4}}=0.0404$, ${V_{5}}=0.1162$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Proposed method with qROFIPWG operator and qROFIPWGMSM operator  ${V_{1}}=0.0788$, ${V_{2}}=0.1523$, ${V_{3}}=0.1562$, ${V_{4}}=0.0452$, ${V_{5}}=0.1155$  ${X_{3}}\succ {X_{2}}\succ {X_{5}}\succ {X_{4}}\succ {X_{1}}$ 
Table 7
Expert  Alternative  ${L_{1}}$  ${L_{2}}$  ${L_{3}}$  ${L_{4}}$ 
${D_{1}}$  X_{1}  $\langle 0.5,0\rangle $  $\langle 0.4,0.4\rangle $  $\langle 0.4,0.3\rangle $  $\langle 0.5,0\rangle $ 
${X_{2}}$  $\langle 0.7,0.2\rangle $  $\langle 0.51,0\rangle $  $\langle 0.5,0.1\rangle $  $\langle 0.6,0.2\rangle $  
${X_{3}}$  $\langle 0.3,0.2\rangle $  $\langle 0.38,0.4\rangle $  $\langle 0.6,0\rangle $  $\langle 0.4,0.3\rangle $  
${X_{4}}$  $\langle 0.5,0\rangle $  $\langle 0.502,0.4\rangle $  $\langle 0.5,0.3\rangle $  $\langle 0.5,0\rangle $  
${X_{5}}$  $\langle 0.6,0.1\rangle $  $\langle 0.3,0\rangle $  $\langle 0.6,0.2\rangle $  $\langle 0.6,0.3\rangle $  
${X_{1}}$  $\langle 0.5,0.3\rangle $  $\langle 0.5,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.6,0.3\rangle $  
${D_{2}}$  ${X_{2}}$  $\langle 0.6,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.206,0.2\rangle $  $\langle 0.4,0\rangle $ 
${X_{3}}$  $\langle 0.4,0.3\rangle $  $\langle 0.7,0\rangle $  $\langle 0.35,0.2\rangle $  $\langle 0.6,0.4\rangle $  
${X_{4}}$  $\langle 0.45,0.4\rangle $  $\langle 0.5,0.2\rangle $  $\langle 0.6,0\rangle $  $\langle 0.5,0.2\rangle $  
${X_{5}}$  $\langle 0.4,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.4,0.4\rangle $  $\langle 0.6,0\rangle $  
${X_{1}}$  $\langle 0.5,0.3\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.5,0\rangle $  $\langle 0.6,0.3\rangle $  
${D_{3}}$  ${X_{2}}$  $\langle 0.6,0.1\rangle $  $\langle 0.49,0.2\rangle $  $\langle 0.7,0\rangle $  $\langle 0.2,0.2\rangle $ 
${X_{3}}$  $\langle 0.7,0\rangle $  $\langle 0.6,0.2\rangle $  $\langle 0.71,0.4\rangle $  $\langle 0.4,0\rangle $  
${X_{4}}$  $\langle 0.5,0.3\rangle $  $\langle 0.7,0\rangle $  $\langle 0.5,0.2\rangle $  $\langle 0.6,0.1\rangle $  
${X_{5}}$  $\langle 0.62,0.3\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.3,0\rangle $  $\langle 0.4,0.2\rangle $ 
Table 8
Method  Score value  Ranking order 
Liu and Liu (2018) with qROFWBM operator  ${V_{1}}=0.019$, ${V_{2}}=0.019$, ${V_{3}}=0.019$, ${V_{4}}=0.019$, ${V_{5}}=0.019$  ${X_{1}}={X_{2}}={X_{3}}={X_{4}}={X_{5}}$ 
Yang and Pang (2020) with qROFWBMDA operator  ${V_{1}}=0.0749$, ${V_{2}}=0.0459$, ${V_{3}}=0.0633$, ${V_{4}}=0.0262$, ${V_{5}}=0.0352$  ${X_{2}}\succ {X_{4}}\succ {X_{5}}\succ {X_{3}}\succ {X_{1}}$ 
Liu et al. (2020) with qROFPWMSM operator  ${V_{1}}=0.264$, ${V_{2}}=0.264$, ${V_{3}}=0.264$, ${V_{4}}=0.264$, ${V_{5}}=0.264$  ${X_{1}}={X_{2}}={X_{3}}={X_{4}}={X_{5}}$ 
Proposed method with qROFIPWA operator and qROFIPWAMSM operator  ${V_{1}}=0.2204$, ${V_{2}}=0.2458$, ${V_{3}}=0.2010$, ${V_{4}}=0.2327$, ${V_{5}}=0.2278$  ${X_{2}}\succ {X_{4}}\succ {X_{5}}\succ {X_{1}}\succ {X_{3}}$ 
Proposed method with qROFIPWG operator and qROFIPWGMSM operator  ${V_{1}}=0.2165$, ${V_{2}}=0.2463$, ${V_{3}}=0.2031$, ${V_{4}}=0.2297$, ${V_{5}}=0.2257$  ${X_{2}}\succ {X_{4}}\succ {X_{5}}\succ {X_{1}}\succ {X_{3}}$ 
Table 9
Expert  Alternative  ${L_{1}}$  ${L_{2}}$  ${L_{3}}$  ${L_{4}}$ 
${D_{1}}$  X_{1}  $\langle 1,0\rangle $  $\langle 0.4,0.4\rangle $  $\langle 0.4,0.3\rangle $  $\langle 1,0\rangle $ 
${X_{2}}$  $\langle 0.7,0.2\rangle $  $\langle 0.1,0\rangle $  $\langle 0.5,0.1\rangle $  $\langle 0.6,0.2\rangle $  
${X_{3}}$  $\langle 1,0\rangle $  $\langle 0.5,0.4\rangle $  $\langle 1,0\rangle $  $\langle 0.4,0.3\rangle $  
${X_{4}}$  $\langle 0.5,0.3\rangle $  $\langle 0.5,0.4\rangle $  $\langle 0.5,0.3\rangle $  $\langle 1,0\rangle $  
${X_{5}}$  $\langle 0.6,0.1\rangle $  $\langle 1,0\rangle $  $\langle 0.6,0.2\rangle $  $\langle 0.6,0.3\rangle $  
${X_{1}}$  $\langle 0.5,0.3\rangle $  $\langle 1,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.6,0.3\rangle $  
${D_{2}}$  ${X_{2}}$  $\langle 1,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 1,0\rangle $  $\langle 0.4,0.5\rangle $ 
${X_{3}}$  $\langle 0.4,0.3\rangle $  $\langle 0.7,0.2\rangle $  $\langle 0.4,0.2\rangle $  $\langle 1,0\rangle $  
${X_{4}}$  $\langle 0.5,0.4\rangle $  $\langle 1,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.5,0.4\rangle $  
${X_{5}}$  $\langle 1,0\rangle $  $\langle 0.6,0.3\rangle $  $\langle 1,0\rangle $  $\langle 0.6,0.4\rangle $  
${X_{1}}$  $\langle 0.5,0.3\rangle $  $\langle 0.6,0.3\rangle $  $\langle 1,0\rangle $  $\langle 0.6,0.3\rangle $  
${D_{3}}$  ${X_{2}}$  $\langle 1,0\rangle $  $\langle 0.5,0.2\rangle $  $\langle 0.7,0.2\rangle $  $\langle 1,0\rangle $ 
${X_{3}}$  $\langle 0.7,0.2\rangle $  $\langle 1,0\rangle $  $\langle 0.8,0.2\rangle $  $\langle 0.4,0.5\rangle $  
${X_{4}}$  $\langle 1,0\rangle $  $\langle 0.7,0.1\rangle $  $\langle 1,0\rangle $  $\langle 0.6,0.1\rangle $  
${X_{5}}$  $\langle 0.6,0.3\rangle $  $\langle 0.6,0.3\rangle $  $\langle 0.3,0.5\rangle $  $\langle 1,0\rangle $ 
Table 10
Method  Score value  Ranking order 
Yang and Pang (2020) with qROFWBMDA operator  ${V_{1}}=0.6133$, ${V_{2}}=0.8372$, ${V_{3}}=0.6602$, ${V_{4}}=0.5890$, ${V_{5}}=0.6265$  ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{1}}\succ {X_{4}}$ 
Liu et al. (2020) with qROFPWMSM operator  ${V_{1}}={V_{2}}={V_{3}}={V_{4}}={V_{5}}=1$.  ${X_{1}}={X_{2}}={X_{3}}={X_{4}}={X_{5}}$ 
Proposed method with qROFIPWG operator and qROFIPWGMSM operator  ${V_{1}}=0.5623$, ${V_{2}}=0.7078$, ${V_{3}}=0.6156$, ${V_{4}}=0.5533$, ${V_{5}}=0.5863$  ${X_{2}}\succ {X_{3}}\succ {X_{5}}\succ {X_{1}}\succ {X_{4}}$ 
Table 11
Alternative  ${L_{1}}$  ${L_{2}}$  ${L_{3}}$  ${L_{4}}$ 
${X_{1}}$  $\langle 0.5,0\rangle $  $\langle 0.4,0.4\rangle $  $\langle 0.4,0.3\rangle $  $\langle 0.5,0.6\rangle $ 
${X_{2}}$  $\langle 0.7,0.2\rangle $  $\langle 0.5,0\rangle $  $\langle 0.5,0.1\rangle $  $\langle 0.6,0.4\rangle $ 
${X_{3}}$  $\langle 0.3,0.6\rangle $  $\langle 0.3,0.4\rangle $  $\langle 0.6,0\rangle $  $\langle 0.4,0.3\rangle $ 
${X_{4}}$  $\langle 0.5,0\rangle $  $\langle 0.7,0.5\rangle $  $\langle 0.5,0.3\rangle $  $\langle 0.5,0.8\rangle $ 
${X_{5}}$  $\langle 0.6,0.5\rangle $  $\langle 0.3,0\rangle $  $\langle 0.6,0.4\rangle $  $\langle 0.6,0.3\rangle $ 
Table 12
Method  Score value  Ranking order 
Yang and Pang (2020) with qROFWBMDA operator  Cannot be determined due to division by zero  Cannot be generated 
Proposed method with qROFIPWA operator and qROFIPWAMSM operator  ${V_{1}}=0.0443$, ${V_{2}}=0.2818$, ${V_{3}}=0.0709$, ${V_{4}}=0.0453$, ${V_{5}}=0.2666$  ${X_{2}}\succ {X_{5}}\succ {X_{3}}\succ {X_{4}}\succ {X_{1}}$ 
Proposed method with qROFIPWG operator and qROFIPWGMSM operator  ${V_{1}}=0.0014$, ${V_{2}}=0.2545$, ${V_{3}}=0.0589$, ${V_{4}}=0.0926$, ${V_{5}}=0.1826$  ${X_{2}}\succ {X_{5}}\succ {X_{3}}\succ {X_{1}}\succ {X_{4}}$ 