# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 5 Conclusions  
The geometric programming problem is an integrated part of mathematical programming and has real-life applications in many engineering problems such as gravel-box design, bar-truss region texture, system reliability optimization, etc. The concept of varying parameters under the objective functions is discussed with the aim that uncertainty is critically involved and affects engineering problems' formulations directly. The proposed research work is developed and introduces an interval-based solution approach to finding the upper and lower limits on the objective function of the varying parameters. The outer- and inner-level geometric programming problem is transformed into a single-level mathematical programming problem. The outcomes are summarized in the numerical illustrations and observed in the precise interval where they should exist. The system reliability optimization problem also provides evidence of the discussed problem's successful implementation and dynamic solution results. The minimum system cost is obtained at the utmost system reliability, which also falls into the lower and upper bounds of the system costs.  
The scope of usual sensitivity analysis is not further required due to the flexible nature of the proposed solution method. The propounded approach allows the abrupt  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Proposed Method** & **Traditional Methods** \\ \hline (1) Deals with the varying parameters & No scope for dealing with such parameters \\ (2) No need to consider the fuzzy or random & It may require the fuzzy or random parameters \\ parameters while dealing with uncertainty & while dealing with uncertainty \\ (3) No need to obtain the crisp or deterministic & It requires the crisp or deterministic version of \\ version of uncertain models & the uncertain models \\ (4) Sensitivity or post-optimal analysis is not & Sensitivity or post-optimal analysis can be \\ required for the obtained solutions & performed separately \\ (5) Less computational complexity in terms of & Comparatively involves more computational \\ mathematical calculations & complexity in terms of mathematical \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison between the proposed method and traditional methods.  
fluctuations among the parameters between given intervals for which bounds over the objective functions are directly obtained. It also makes the computational algorithm easier than other methods by ignoring the uncertain parameters such as fuzzy, stochastic, and other uncertain forms that yield a solution procedure that is comparatively more complex. The developed approach may be extended for future research to stochastic programming, bi-level or multilevel programming, and various engineering problems with real-life applications.  
Conceptualization, A.E.S., F.A. and S.A.; methodology, M.A. and S.A.; formal analysis, M.A. and F.A.; writing--original draft preparation, M.A., S.A. and A.E.S.; writing--review and editing, S.A., A.E.S. and F.A.; project administration, S.A. and I.A.; funding acquisition, S.A. All authors have read and agreed to the published version of the manuscript.  
The authors extend their appreciation to King Saud University for funding this work through the Researchers Supporting Project (RSP-2021/387), King Saud University, Riyadh, Saudi Arabia.  
Not applicable.  
The authors are very thankful to the Anonymous Reviewers and Editors for their insightful comments, which made the manuscript clearer and more readable. The authors extend their appreciation to King Saud University for funding this work through the Researchers Supporting Project (RSP-2021/387), King Saud University, Riyadh, Saudi Arabia.  
The authors declare no conflict of interest.