## 3 The Generalized p-Norm Knowledge Measure of q-ROFS
### The Generalized p-Norm Knowledge-Based Score Function of q-ROFSs  
To solve similar unreasonable comparison result, inspired by [PERSON]'s method [40], we present a notion of knowledge measure for q-ROFs to compare any two q-ROFSs, which have the characteristic of portraying information conveyed by the MD and NMD, as well as considering the inherent fuzziness of q-ROFSs. Definition 7: Let \(Y\) be a finite universal set, a q-ROFN \(\overline{A}\) is defined by the form of \(\left\langle u_{\overline{A}}(y),v_{\overline{A}}(y)\right\rangle.\) The generalized knowledge measure of \(\overline{A}\) is consist of a normalized sum of its p-norm distance from the reference q-ROFS \(Q=\left\langle y,0,0\right\rangle\) and p-norm variation between its MD and NMD. The normalized p-norm distance from the reference \(Q=\left\langle y,0,0\right\rangle\) can be denoted by \(\left(\left(u_{\overline{A}}^{q}(y)-0\right)^{\!p}+\left(v_{\overline{A}}^{q} (y)^{q}-0\right)^{\!p}+\left(1-\pi_{\overline{A}}^{q}(y)\right)^{\!p}\right)^{ \!\frac{1}{p}},\) and the p-norm variation between its MD and NMD can be expressed as \(\left|\left(u_{\overline{A}}^{q}(y)\right)^{\!p}-\left(v_{\overline{A}}^{q}(y) \right)^{\!p}\right|^{\!\frac{1}{p}}\). Then, the generalized p-norm knowledge measure of \(\overline{A}\) is listed as follows \(\left(p=1,2,\ldots,n\right)\):  
\[K_{\!F}\!\left(\overline{A}\right)=\frac{1}{2^{\frac{1}{p}}+1}\left[\left( \left(u_{\overline{A}}^{q}(y)\right)^{\!p}+\left(v_{\overline{A}}^{q}(y) \right)^{\!p}+\left(u_{\overline{A}}^{q}(y)+v_{\overline{A}}^{q}(y)\right)^{ \!p}\right)^{\!\frac{1}{p}}+\left|\left(u_{\overline{A}}^{q}(y)\right)^{\!p} -\left(v_{\overline{A}}^{q}(y)\right)^{\!p}\right|^{\!\frac{1}{p}}\right]. \tag{12}\]  
Based on the Definition 7, \(K_{\!F}\!\left(\overline{A}\right)\) meets the following properties:  
Theorem 1 (Monotonicity): Let \(Y\) be a finite universal set, a q-ROFN \(\overline{A}\) is defined by the form of \(\left\langle u_{\overline{A}}(y),v_{\overline{A}}(y)\right\rangle\). Then, the et al.easure \(K_{\!F}\!\left(\overline{A}\right)\) is a monotone function. Proof:  
(1) Firstly, that \(K_{\!F}\!\left(\overline{A}\right)\) is strictly monotonic with respect to the \(u_{\overline{A}}\) is proven by  
**Proof of Theorem 2 (2):** Suppose \(p=2\), \(q=3\), when \(u_{\overline{A}}(y)=1,v_{\overline{A}}(y)=0\), then  
\[K_{F}\Big{(}\overline{A}\Big{)}=\frac{1}{2^{\frac{1}{p}}+1}\left[\left(\left(u_{ \overline{A}}^{q}(y)\right)^{p}+\left(v_{\overline{A}}^{q}(y)\right)^{p}+ \left(u_{\overline{A}}^{q}(y)+v_{\overline{A}}^{q}(y)\right)^{p}\right)^{ \frac{1}{p}}+\left|\left(u_{\overline{A}}^{q}(y)\right)^{p}-\left(v_{\overline {A}}^{q}(y)\right)^{p}\right|^{\frac{1}{p}}\right]\]  
\[=\frac{1}{2^{\frac{1}{2}}+1}\left[\left(1^{2}+0+1^{2}\right)^{\frac{1}{2}}+ \left|1^{2}-0\right|^{\frac{1}{2}}\right]=1.\]  
Similarly, when \(u_{\overline{A}}(y)=0,v_{\overline{A}}(y)=1\), then \(K_{F}(\overline{A})=1\). The rest of properties can be easily proven, so are omitted here. Based on the above analysis, the novel score function of a q-ROFN is defined in the following. **Definition 8**. : _Suppose \(\overline{A}\in q-ROFS\) be a q-ROFN in a finite universal set \(Y=\{y_{1},y_{2},\ldots,y_{n}\}\), for \(C\gg 0\) and \(0<\delta\ll 1\), then the generalized p-norm knowledge-based score function of \(\overline{A}\) is depicted as:_  
\[Sco_{F}(\overline{A})= \begin{cases}\frac{e^{C(\frac{q}{A}(y)-v_{\overline{A}} ^{q}(y))}-1}{e^{C(\frac{q}{A}(y)-v_{\overline{A}}^{q}(y))}+1}\bigg{|}K_{F}( \overline{A}),&\text{if }u_{\overline{A}}(y)\
eq v_{\overline{A}}(y)\\ \delta K_{F}(\overline{A}),&\text{if }u_{\overline{A}}(y)=v_{\overline{A}}(y) \end{cases}  \tag{13}\]  
_in which the coefficient \(C\) is regarded as the sigmoidal shape of the defined score function around the points of \(u_{\overline{A}}=v_{\overline{A}}\).