# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 1 Introduction  
Mathematical programming problems have different forms based on the nature of objective functions and constraints. The geometric programming problem, a typical form of mathematical optimization characterized by objective and constraint functions of a particular form, was introduced by [1]. Later, the advanced study in the domain of the GPP was performed by [2, 3]. Several engineering applications [4] have investigated the effectiveness and importance of the GPP. The GP optimization approach inevitably outperforms other existing techniques due to the objective function's relative magnitudes instead of the decision variables. Initially, the GP technique's basic working principle is based on finding the optimal solution of the objective function and then proceeding further to determine the optimal values of the design variables. This characteristic feature of the GPP is essential and fruitful in circumstances where the decision-makers are interested in first finding the optimal values of the objective function. Thus, the polynomial structure of the objectives and constraints leads the GPP towards the simpler convex solution space [2, 4]. GP optimization techniques can tackle this situation, and the computational activities are aborted to obtain the optimum design vectors. One of the GP techniques' most crucial advantages over others can be regarded as it mitigates the complex optimization problems into thedifferent piecewise linear algebraic equations. On the other hand, the GP approach deals only with the posynomial types of algebraic terms, meaning that it solely facilitates the objective function and the constraints with posynomial structures, which can be considered significant drawbacks. Many engineering optimization problems deal with the manufacturing and production processes of some products, machinery parts, and the raw equipment used in the final usable machines and products. They care about the structure, dimensions, quality, and specifications of the raw material parts that are very important to be transformed and converted into usable products. For example, the cofferdam, shaft, journal bearings, etc., are the raw parts that are the building block raw parts of the various products and machines. Hence, the mathematical models with the specifications of these raw parts are built up and further used in manufacturing and producing the final products. Sometimes, the perfect specifications cannot be achieved due to some vagueness or technical errors in the functioning machine, for which the experts/managers allow some marginal variations among the specifications and dimensions of such raw parts. Afterward, it can be managed or adjusted to some extent. In system reliability modeling, various parameters can be taken as varying between some specified intervals. This means that the parameters can be taken as uncertain, and using some specified tools, they can be converted into crisp ones. 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. Instead of taking the vague and random parameters, one can assume the continuous variations in the parameters' values can be tackled with the two-level mathematical programming techniques discussed in this paper. Additionally, the sensitivity and post-optimality of the obtained solution results are waived off due to the working procedure of the proposed approach. Hence, the proposed bonded approach for the GPP can be easily implemented on various non-engineering problems while dealing with varying parameters. The remaining part of the paper is summarized as follows: In Section 2, some relevant literature is discussed, while Section 3 presents the basic concepts and modeling of standard geometric programming problems along with the proposed bounded solution methods.