# Fuzzy-Based Control System for Solar-Powered Bulk Service Queueing Model with Vacation
## 3 Materials and Methods
### Parametric Nonlinear Program (P-NLP)  
In a solar-powered queuing system, uncertainties in energy availability and server reliability due to weather fluctuations and renewable intermittency necessitate the use of fuzzy modeling. To handle these uncertainties, Parametric Nonlinear Programming (P-NLP) is used to derive membership functions for key performance metrics such as energy availability, service delay, and system reliability. These membership functions are often complex and non-analytic, making direct derivation impractical. Therefore, the \(\alpha\)-cut approach is applied. For each \(\alpha\)-level \(\alpha\)\(\in\) [0, 1], the fuzzy objective is decomposed into an interval defined by its lower and upper bounds, which are obtained by solving two crisp nonlinear programs. The P-NLP formulations for determining the lower and upper bounds of the \(\alpha\)-cuts of \(p_{m}\), \(m\) = 1(1)5 are defined as follows:  
\[p_{m}(\alpha)=\big{\{}x_{m}\in X/\xi_{p_{m}}(x_{m})\geq\alpha\big{\}},m=1,2,3,4.\]  
The \(\alpha\)-cuts of metrics \(T_{i}\{P_{1}=EL_{s},P_{2}=EW_{s},P_{3}=TP\}\) are obtained using MFs \(\xi_{p_{m}}(z)\) of \(P_{j}\), where \(j\) = 1, 2, 3. The \(\alpha\)-cut of \(p_{m}\), \(m\) = 1(1)5 is a crisp interval and can be obtained in the following form:  
\[p_{m}(\alpha)=[(x_{m})_{\alpha}^{LB},(x_{m})_{\alpha}^{UB}]=[\min_{m\in p}\{ \xi_{p_{m}}(x_{m})\geq\alpha\},\max\{\xi_{p_{m}}(x_{m})\geq\alpha\}] \tag{1}\]  
Thus, the fuzzified model FM\(/\)FM\(/\)1 for the dual orbit imperfect service system reduces to an all-crisp M\(/\)M\(/\)1 dual orbit imperfect service model with different \(\alpha\)-cut sets \(\{p_{m}(\alpha)/0<\alpha\leq 1,m=1,2,3,4\}\). To derive \(P\), at least one of the following four situations holds such that \(\xi_{p_{j}}(z)=\alpha\)  
\[\text{I.}\ (\xi_{p_{1}}(x_{1})=\alpha,\xi_{p_{2}}(x_{2})>\alpha,\xi_{p_{3}}(x_{3})> \alpha,\xi_{p_{4}}(x_{4})>\alpha) \tag{2}\]  
\[\text{II.}\ (\xi_{p_{1}}(x_{1})>\alpha,\xi_{p_{2}}(x_{2})=\alpha,\xi_{p_{3}}(x_{3})> \alpha,\xi_{p_{4}}(x_{4})>\alpha) \tag{3}\]  
\[\text{III.}\ (\xi_{p_{1}}(x_{1})>\alpha,\xi_{p_{2}}(x_{2})>\alpha,\xi_{p_{3}}(x_{3})= \alpha,\xi_{p_{4}}(x_{4})>\alpha) \tag{4}\]  
\[\text{IV.}\ (\xi_{p_{1}}(x_{1})>\alpha,\xi_{p_{2}}(x_{2})>\alpha,\xi_{p_{3}}(x_{3})> \alpha,\xi_{p_{4}}(x_{4})=\alpha) \tag{5}\]  
The shape function of \(\xi_{p}(z)\) can be obtained from the above four situations through the lower bound (LB) and upper bound (UB). \[\text{Now, }\left[P\right]_{\alpha}^{LB}=\min_{x_{m}\in X_{m}/x_{m}>0}P_{j} \text{and }\left[P_{j}\right]_{\alpha}^{UB}=\min_{x_{m}\in X_{m}/x_{m}>0}P_{j}, \tag{6}\]where \(P_{1}=EL_{s}\), \(P_{2}=EW_{s}\), \(P_{3}=TP\). From Equation (5), one gets \([(x_{m})_{a_{1}}^{LB},((x_{m})_{a_{1}}^{UB}]\subseteq[(x_{m})_{a_{2}}^{LB},(x_{m })_{a_{2}}^{UB}],0<\alpha_{2}<\alpha_{1}\leq 1\). To obtain the membership \(\xi_{P_{j}}(z)\), it is sufficient to determine the shape function of \(\xi_{P_{j}}(z)\) which is similar to \(LB[P_{j}]_{a}^{LB}\) and \(UB[P_{j}]_{a}^{UB},j=1,2,3\). \[\text{Now, }[P_{j}]_{a}^{LB}=\min_{\{x_{k}\in X_{K}/x_{k}>0\}}P_{j}\text{and }[P_{j}]_{a}^{UB}=\max_{\{x_{k}\in X_{K}/x_{k}>0\}}P_{j}\text{ s.t }x_{k}\in[(x_{k})_{a_{1}}^{LB},((x_{k})_{a_{1}}^{UB}], \tag{7}\]  
At least one \(y_{k}\)(k = 1, 2, 3, 4) must attain the boundaries of the crisp intervals given in Equation (7). Here, \([P_{j}]_{a}^{LB}\) is increasing and \([P_{j}]_{a}^{UB}\) is decreasing with respect to \"\(\alpha\)\"; thus, \(MF\xi_{P_{j}}(z)\) can be determined from (7). If both \([P_{j}]_{a}^{LB}\) and \([P_{j}]_{a}^{UB}\) are invertible with respect to \"\(\alpha\)\", then \(L(x)=\left([P_{j}]_{a}^{LB}\right)^{-1}\), \(R(x)=\left([P_{j}]_{a}^{UB}\right)^{-1}\), where \(P_{1}=EL_{s}\), \(P_{2}=EW_{s}\), \(P_{3}=TP\). The left and right shape functions are given by  
\[\xi_{P_{j}}(x)=\left\{