# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 4 Computational Study
### Application to System Reliability Optimization  
Assume system reliability having \(n\) components connected in series. Suppose \(r_{i}\ (i=1,2,\ldots,n)\) represents the individual reliability of the \(i\)-th component of the system. Similarly, \(R_{s}\ (r_{1},r_{2},\ldots,r_{n})\) is the reliability of the whole series system. Consequently, \(C_{s}\ (r_{1},r_{2},\ldots,r_{n})\)depicts the total cost of \(n\) components associated with the system reliability. It seldom happens that the system reliability is maximized when the cost of the associated system is exactly known; however, some varying cost may make it easier to execute the smooth function of the framework. The obtained lower and upper bounds on the cost objective function will ensure the variation in total cost associated with the system and help with allocating the budget for maintenance or renovation, etc. In the same manner, minimizing the system cost under the varying reliability of the whole system would be quite a worthwhile task. The minimization of the system cost without affecting the system reliability is much needed to ensure the longer performance of the components. Thus, we considered that the system reliability is varying between some specified intervals and bounded by upper and lower bounds. This situation is quite common due to uncertainty in the failure of any components. In real-life scenarios, the minimization of the total system cost by maintaining the system reliability would be a more prominent modeling texture of the reliability optimization problems (see [17; 29; 30]). Therefore, the mathematical model for the minimization of system cost under varying system reliability takes the form of the geometric programming problem and can be represented as follows (33):  
\[ \begin{array}{ll}\text{F}=\text{Minimize}&C_{s}\left(r_{1},r_{2},\ldots,r_{ n}\right)=+\sum_{i=1}^{n}C_{i}r_{i}^{a_{i}}\\ \text{s.\ x.}&\\ &\left(r_{1}\times r_{2}\times\cdots\times r_{n}\right)=\prod_{i=1}^{n}r_{i} \geq\tilde{R}_{s}\\ &\underline{R}_{s}\leq\tilde{R}_{s}\leq\overline{R}_{s}\\ &0\leq r_{i}\leq 1\quad\forall\ \ i=1,2,\ldots,n.\end{array}  \tag{33}\]  
where \(\alpha_{i}\) is the acceptable tolerance linked with the \(i\)-th component. We considered the three components connected in series, and the relevant data are summarized in Table 2.  
Since all the parameters are crisp except the system reliability, then Problems (23) and (26) can be utilized to obtain the upper and lower bounds of the objective values in Problem (33). According to the problems (23) and (26), the formulations of the upper and lower bounds for Problem (33) can be presented as follows (34):  
\[ \begin{array}{ll}\textbf{Model A}\\ \underline{F}=\text{Minimize}&\left(\frac{150}{y_{01}}\right)^{y_{01}}\left( \frac{210}{y_{02}}\right)^{y_{02}}\left(\frac{270}{y_{03}}\right)^{y_{03}} \left(\frac{1}{\overline{R}_{s}y_{11}}\right)^{y_{11}}\left(\frac{1}{\overline {R}_{s}y_{12}}\right)^{y_{12}}\left(\frac{1}{\overline{R}_{s}y_{13}}\right)^ {y_{13}}\\ \text{s.\ x.}&\\ &y_{01}+y_{02}+y_{03}=1,\\ &20y_{01}+15y_{02}+10y_{03}+y_{11}+y_{12}+y_{13}=0,\\ &y_{01}-y_{02}-y_{03}+y_{11}+y_{12}+y_{13}=0,\\ &y_{01},y_{02},y_{03},y_{11},y_{12},y_{13}\geq 0\\ &0.6358\leq\tilde{R}_{s}\leq 0.9776,\ \ \forall\ \ i=1,2,3.\end{array}  \tag{34}\]  
whereas the upper bound can be stated as follows (35): \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\boldsymbol{C_{1}}\) & \(\boldsymbol{C_{2}}\) & \(\boldsymbol{C_{3}}\) & \(\boldsymbol{\alpha_{1}}\) & \(\boldsymbol{\alpha_{2}}\) & \(\boldsymbol{\alpha_{3}}\) & \(\underline{R_{s}}\leq\overline{R_{s}}\leq\overline{R_{s}}\) \\ \hline
150 & 210 & 270 & 20 & 15 & 10 & \(0.6358\leq\tilde{R}_{s}\leq 0.9776\) \\ \hline \end{tabular}
\end{table} 
Table 2: Input data for the system reliability optimization problem. \[