## 4 q-ROF Aczel-Alsina Aggregation Operators  
This portion elaborates on some q-ROF average AOs utilizing AA operations. We discuss q-ROFAAWA, q-ROFAAWG, q-ROFAAOWA, q-ROFAAOWG, q-ROFAAHA, and q-ROFAAHG operators in detail. Definition 9: Let \(\tau_{i}=(\alpha_{i},\ \beta_{i})\) be some q-ROFNs and \(\zeta_{i}=(\zeta_{1},\ \zeta_{2},\ \ldots,\ \zeta_{n})^{T}\) be the weight vector (WV) of \(\tau_{i}\), having condition \(\zeta_{i}\geq 0\) and \(\sum\limits_{i=1}^{n}\zeta_{i}=1\). Then, the q-ROFAAWA operator is the function: \(M^{*n}\to M^{*}\), defined as:  
\[q-ROFAAWA\ (\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})=\mathop{\oplus}\limits_{i=1}^{ n}(\zeta_{i}\tau_{i})=\zeta_{1}\tau_{1}\oplus\zeta_{2}\tau_{2}\oplus\ldots \oplus\zeta_{n}\tau_{n}\]  
By using the AA operations on q-ROFNs, we derive the following theorem. Theorem 4.1: _Let \(\tau_{i}=(\alpha_{i},\ \beta_{i})\) denote some q-ROFNs. Then aggregated values of \(\tau_{i}^{\prime}\)s utilizing the q-ROFAAWA AOs is also known as a q-ROFN and given by:_  
\[=\left(\sqrt[q]{1-e^{-(\sum\limits_{i=1}^{n}\left((\zeta_{i}(-\ln \left(1-(\alpha_{\tau_{1}})^{q}\right))^{N})^{\frac{1}{N}}\right))}}\right.  
**Proof.** Since \(\tau_{i}=(\alpha_{i},\ \beta_{i});\ (i=1,\ 2,\ \ldots,\ n).\)  
\[q-ROFAWA\ (\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})=\underset{i=1}{\overset{k}{ \oplus}}(\alpha_{i},\ \beta_{i})\] \[=\left(\sqrt[q]{1-e^{-\left(\underset{i=1}{\overset{n}{\sum}}\ (\zeta_{i}((-\ln\left(1-(\alpha_{\tau_{i}}\right)^{q}))^{N}\right)^{\frac{1}{ N}})\right)}}e^{-\left(\underset{i=1}{\overset{n}{\sum}}\ (((\zeta_{i}(-\ln\left(\beta_{\tau_{i}}\right)^{N})^{N})^{\frac{1}{ N}}))\right)}\right)\] \[=\left(\sqrt[q]{1-e^{-\left((-\ln\left(1-(\alpha_{\tau_{i}}\right)^ {q})\right)^{N}\right)^{\frac{1}{N}}}}e^{-\left((-\ln\left(\beta_{\tau_{i}} \right)^{N})^{N}\right)^{\frac{1}{N}}}\right)\] \[=\left(\sqrt[q]{1-e^{\ln\left(1-\alpha\right)}},e^{\ln\left(\beta \right)}\right)=(\alpha,\ \beta)=\tau\]  
\(\square\)  
**Theorem 4. ** (_Boundedness_) _Let \(\tau_{i}=(\alpha,\ \beta)\) be an accumulation of \(q\)-ROFNs. Let \(\tau^{-}=\min(\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})\) and \(\tau^{+}=\max(\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})\). Then_  
\[\tau^{-}\leq q-ROFAWA\ (\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})\leq\tau^{+}.\]  
**Proof. ** Let \(\tau_{i}=(\alpha,\ \beta)\) be an accumulation of \(q\)-ROFNs. Let \(\tau^{-}=\min(\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})=(\alpha_{\tau}^{-},\ \beta_{ \tau}^{-})\) and \(\tau^{+}=\max(\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})=(\alpha_{\tau}^{+},\ \beta_{ \tau}^{+})\). We obtain \(\alpha_{\tau}^{-}=\min(\alpha_{i})\), \(\beta_{\tau}^{-}=\max(\beta_{i})\), \(\alpha_{\tau}^{+}=\max(\alpha_{i})\), \(\ \beta_{\tau}^{+}=\min(\beta_{i})\). Hence, we obtained subsequent inequality as given below:  
\[\sqrt[q]{1-e^{-\left(\underset{i=1}{\overset{n}{\sum}}\ ((\zeta_{i}((-\ln\left(1-( \alpha_{\tau}^{-}\tau_{i}\right)^{q})))^{N})\right)^{\frac{1}{N}})}}\leq\sqrt[q]{1-e^{- \left(\underset{i=1}{\overset{n}{\sum}}\ ((\zeta_{i}(-\ln\left(1-(\alpha_{\tau}^{+}\tau_{i}\right)^{q}))^{N})^{\frac{1} {N}})\right)}}\] \[\leq e^{-\left(\underset{i=1}{\overset{n}{\sum}}\ ((\zeta_{i}(-\ln\left(\beta_{\tau}^{-}\right)^{N})^{N}) ^{\frac{1}{N}}\right)}\leq e^{-\left(\underset{i=1}{\overset{n}{\sum}}\ ((\zeta_{i}(-\ln\beta_{\tau})^{N})^{\frac{1}{N}}\right)} \leq e^{-\left(\underset{i=1}{\overset{n}{\sum}}\ ((\zeta_{i}(-\ln\left(\beta_{\tau}^{+}\right)^{N})^{ \frac{1}{N}})\right.}\right)\] Therefore, \[\tau^{-}\leq q-ROFAWA\ (\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})\leq\tau^{+}\]  
\(\square\)  
**Theorem 5. ** (_Monotonicity_) _Let \(\tau_{i}=(\alpha_{i},\ \beta_{i})\) and \(\tau_{i}^{\prime}=\left(\alpha_{i}^{\prime},\ \beta_{i}^{\prime}\right)\ (i=1,\ 2,\ \ldots,\ n)\) be two \(q\)-ROFSs such that \(\tau_{i}\leq\tau_{i}^{\prime}\) i.e., \(\alpha_{i}\leq\alpha_{i}^{\prime}\) and \(\beta_{i}\geq\ \beta_{i}^{\prime}\ \forall i\). Then_  
\[q-ROFAWA\ (\tau_{1},\ \tau_{2},\ \ldots,\tau_{n})\leq q-ROFAWA\ (\tau_{1}^{\prime},\ \tau_{2}^{ \prime},\ \ldots,\tau_{n}^{\prime})\]  
**Proof.