## Appendix A Numerical Example for the Proposed FIS-Based System  
Equation (14) and (15), are used for the numerical example of the proposed FIS-based system. The multi-input signals for users U\({}_{1}\) and U\({}_{2}\) are (\(d_{1}\) = 60 m, \(SNR_{1}\) = 14 dB, and \(d_{f1}\) = 10 m) and (\(d_{2}\) = 180 m, \(SNR_{2}\) = 10 dB, and \(d_{f2}\) = 28 m), respectively. For user U\({}_{1}\) the fuzzy values are calculated by,  
\[T(s)=\begin{cases}h\left(\frac{s-e}{\alpha-e}\right)&\text{for}\quad\quad e \leq s\leq\alpha\\ h\left(\frac{s-f}{\alpha-f}\right)&\text{for}\quad\quad\alpha\leq s\leq f\\ 0&\text{otherwise}\end{cases} \tag{16}\]  
For \(d_{1}\) = 60 it touches two triangular MFs (L and M), refer to Figure 3. For L it uses second part of the equation. Here \(s\) = 60, \(h\) = 1, \(\alpha\) = 50, and \(f\) = 100. Then \(T_{L}(60)\) = 0.6666. For M it uses first part of the equation. Here \(s\) = 60, \(h\) = 1, \(\alpha\) = 100, and \(e\) = 50. Then \(T_{M}(60)\) = 0.2.  
For \(SNR_{1}\) = 14 dB again it touches the two MFs (L and M), refer to Figure 4. For L, \(s\) = 14, \(h\) = 1, \(\alpha\) = 7.5, \(e\) = 0, and \(f\) = 15, using the second part of. Then \(T_{L}(14)\) = 0.1333. For M, \(s\) = 14, \(h\) = 1, \(\alpha\) = 15, \(e\) = 7.5 and \(f\) = 22.5. Then \(T_{M}(14)\) = 0.8666.  
Similarly, for \(d_{f1}\) = 10 m the L MF has \(s\) = 10, \(h\) = 1, \(\alpha\) = 7.5, \(e\) = 0, and \(f\) = 15. Then \(T_{L}(10)\) = 0.6666. When \(d_{f1}\) is M then \(s\) = 14, \(h\) = 1, \(\alpha\) = 15, \(e\) = 7.5 and \(f\) = 22.5. Then \(T_{M}(10)\) = 0.3333.  
From Table 1 the possible rule combination for the multi-input variables are calculated through rule base stage which uses AND operation to create fuzzy outputs which are,  
* \(\Delta\mu(C_{1})\) = min (0.6666, 0.1333, 0.6666) = 0.1333, and \(\mu_{C_{1}}\) = 0.3332.
* \(\Delta\mu(C_{2})\) = min (0.6666, 0.1333, 0.3333), and \(\mu_{C_{2}}\) = 0.4998.
* \(\Delta\mu(C_{3})\) = min (0.6666, 0.8666, 0.6666) = 0.6666, and \(\mu_{C_{3}}\) = 0.3332.
* \(\Delta\mu(C_{4})\) = min (0.6666, 0.8666, 0.3333) = 0.3333, and \(\mu_{C_{4}}\) = 0.3332.
* \(\Delta\mu(C_{5})\) = min (0.2, 0.1333, 0.6666) = 0.1333, and \(\mu_{C_{5}}\) = 0.4998.
* \(\Delta\mu(C_{6})\) = min (0.2, 0.1333, 0.3333) = 0.1333, and \(\mu_{C_{6}}\) = 0.6664.
* \(\Delta\mu(C_{7})\) = min (0.2, 0.8666, 0.6666) = 0.2, and \(\mu_{C_{7}}\) = 0.3332.
* \(\Delta\mu(C_{8})\) = min (0.2, 0.8666, 0.3333) = 0.2, and \(\mu_{C_{1}}\) = 0.4998.  
Here \(\Delta\mu_{C_{i}}\) is the corresponding fuzzy outputs for rule \(C_{i}\) which is the FRBS rules combination as shown in Table 1. \(\mu_{C_{i}}\) is the centroid of respective MF in Figure 6. By putting the values in Equation (15) it gives the output power factor \(p_{1}^{{}^{\prime}}\) = 0.383.  
The same is repeated for user U\({}_{2}\) with multi-input signal (\(d_{2}\) = 180 m, \(SNR_{2}\) = 10 dB, and \(d_{f2}\) = 28 m). The output power factor is \(p_{2}^{{}^{\prime}}\) = 0.749. Both the power factors are normalized to obtain the actual power factors \(P_{1}\) and \(P_{2}\), respectively, using Equation (2). The normalized power factors are \(P_{1}\) = 0.34, and \(P_{2}\) = 0.66, respectively.