# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 4 Computational Study
### Analyses of Computational Complexity and Discussions  
This proposed bounded solution method captures the behavior of varying parameters and provides the interval-based solution of the objective function. Most often, uncertain parameters exist in any form, such as they may take the form of randomness, fuzziness, and any other aspects of uncertainty. The uncertainty among parameters arises due to vagueness being able to be dealt with by using fuzzy approaches, whereas the stochastic technique is applied when the uncertainty involves randomness among the parameters. More precisely, contrary to other uncertain optimization approaches, the developed approach adheres to comparatively less computational complexity in the sense of mathematical computation (e.g., some mathematical calculations are used to derive the crisp or deterministic version of fuzzy or random parameters), and there is no scope for obtaining the deterministic version of the problem for such varying geometric programming problems. The beauty of the proposed method can be highlighted by the fact that sensitivity analyses (post-optimality analysis) do not need to be performed because the continuous variations among the parametric values directly produce the range of optimal objective functions from the interval parameters. Thus, the propounded solution approach can be the most prominent and efficient decision-maker while dealing with uncertain parameters other than the fuzzy or stochastic form. The generalization of the conventional geometric programming problem of constant parameters is highlighted for interval parameters. The most prominent and extensive idea is to determine the lower and upper bounds of the range by applying the two-level mathematical programming technique to geometric programming problems. With the aid of a strong duality theorem, the two-level geometric programming problems are converted into a pair of one-level geometric programming problems to implement the computational study. When all the varying parameters degenerate to constant parameters, the two-level geometric programming problem turns into the conventional geometric programming problem. In general, in interval geometric programming problems, it may probably happen that the problem is infeasible for some specified range of varying parameters. Thus, our proposed methods are free from infeasibility and ignore those complexities due to infeasible values. The proposed method obtains the lower and upper bounds of the feasible solutions directly. In addition, the suggested method does not examine the range of values that results in infeasibility. Furthermore, developing two-level geometric programming problems can determine the lower and upper bounds of the objective values. However, mathematical programming is nonlinear in the case of geometric programming problems, which may be very typical for solving large-scale problems. The comparative study is presented in Table 3.  
The presented work can be described as an empirical case research work by filling the various gaps [8; 9; 17; 19; 29] such as instant variation among parameters, two-level mathematical programming, duality theory in the GPP, and the automatic post-optimal analysis metric. In system reliability modeling, various parameters can be taken as varying between some specified intervals. For example, the cofferdam, shaft, journal bearings, etc., are the raw parts that are the building block materials of the various products and machines. This means that the parameters can be taken as uncertain, and using some specified tools, they can be converted into a crisp one. In the literature, the concept of fuzzy and random parameters is available, which deals with the vagueness and randomness in the parameters. However, we have provided an opportunity to define the parameters under the continuous variations bounded by upper and lower limits.