## 3 The Generalized p-Norm Knowledge Measure of q-ROFS
### Analysis of the Existing Comparative Methods of q-ROFSs  
As is well known, the common comparative methods for any two q-ROFNs are generally based on the defined function ([PERSON] et al. (2019); [PERSON] et al. (2019)). To mention it, an unreasonable phenomenon has occurred in the existing comparative methods of q-ROFSs, which may result in some undesirable results. Then, the detailed analysis can be shown as follows. (a) [PERSON] and [PERSON]'s score function ([PERSON] et al. (2019)). **Definition 4** ([PERSON] et al. (2019)). : _Let \(\overline{A}=\left(u_{\overline{A}},v_{\overline{A}}\right)\) and \(\overline{B}=\left(u_{\overline{B}},v_{\overline{B}}\right)\) are two \(q\)-ROFNs, then the score function \(S(\overline{A})\) and the accuracy function \(H(\overline{A})\) can be expressed as:_  
\[S\big{(}\overline{A}\big{)}=u_{\overline{A}}^{q}-v_{\overline{A}}^{q} \tag{7}\]\[H\big{(}\overline{A}\big{)}=u_{\overline{A}}^{q}+v_{\overline{A}}^{q} \tag{8}\]  
According to Definition 4, the comparative method is depicted as follows: if \(S(\overline{A})\succ S(\overline{B})\), then \(\overline{A}\succ\overline{B}\); otherwise, if \(S(\overline{A})=S(\overline{B})\), then the result can depend on the values of accuracy function. In detail, if \(H(\overline{A})\succ H(\overline{B})\), then \(\overline{A}\succ\overline{B}\); if \(H(\overline{A})=H(\overline{B})\), then \(\overline{A}=\overline{B}\). **Remark 1**. : _It can easily be observed that the above comparison method may result in unreasonable results, that is, when the value of \(q\) changes, the comparison results are completely reversed, which is counterintuitive. In addition, the above comparative method ignores the inherent uncertainties and vagueness of \(q\)-ROFSs, as well as the influence of abstention, all these may also lead to the undesirable phenomenon and weaker precision._  
**Example 1**. : _Suppose \(q=1.5\), take \(\overline{A}=(0.6,0.3)\) and \(\overline{B}=(0.5,0.1)\) for example, based on Equation (7), the obtained score function values are \(S(\overline{A})=0.3004\) and \(S(\overline{B})=0.3219\), then \(\overline{A}\prec\overline{B}\). However, when \(q\) is assigned to the value of \(3\), then the obtained results are \(S(\overline{A})=0.1890\) and \(S(\overline{B})=0.1240\), respectively, that is, \(\overline{A}\succ\overline{B}\). Therefore, the comparison method based on Definition 4 is unreasonable._  
(b)[PERSON] et al. 's score function [38]. **Definition 5** ([38]). : _Let \(\overline{A}=\big{(}u_{\overline{A}},v_{\overline{A}}\big{)}\) be a \(q\)-ROFN, and then the score function and the accuracy function represented in [38] can be as:_  
\[S\big{(}\overline{A}\big{)}=\frac{1}{2}(1+u_{\overline{A}}^{q}-v_{\overline{A }}^{q}) \tag{9}\]  
\[H\big{(}\overline{A}\big{)}=u_{\overline{A}}^{q}+v_{\overline{A}}^{q} \tag{10}\]  
In addition, [PERSON] et al. 's comparative method for any two q-ROFNs is the same as [36]. **Remark 2**. : _Although the above defined score function is different from [PERSON] and [PERSON]'s [36], the same problem occurs in this method, that is, with the transformation of the value of \(q\), thecomparison results change._  
(c)[PERSON] et al. 's score function [39]. **Definition 6** ([39]). : _For a \(q\)-ROFN \(\overline{A}=\big{(}u_{\overline{A}},v_{\overline{A}}\big{)}\), the score function is:_  
\[S\big{(}\overline{A}\big{)}=u_{\overline{A}}^{q}-v_{\overline{A}}^{q}+\Bigg{(} \frac{e^{u_{\overline{A}}^{q}-v_{\overline{A}}^{q}}}{e^{u_{\overline{A}}^{q} -v_{\overline{A}}^{q}}+1}-\frac{1}{2}\Bigg{)}\pi^{q}. \tag{11}\]  
**Remark 3**. : _It is shown that the improved score function described in Equation (11) provides unreasonable results in cases of \(u_{\overline{A}}=v_{\overline{A}}\). Regardless, whatever the value of \(q\) is, it always gives value 0._  
**Example 2**. : _Suppose \(\overline{A}=(0.6,0.6)\) and \(\overline{B}=(0.3,0.3)\), if we utilize the [PERSON] et al.