# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 3 Geometric Programming Problem: Basic Concepts
### Geometric Programming Problem under Varying Parameters  
In reality, optimization problems may contain uncertainty among the parameters that cannot be ignored. Due to the existence of uncertainty among parameters in the real world, many researchers have investigated the problem of decision-making in a fuzzy environment and management science. Different real-life problems inherently involve uncertainty in the parameters' values. In this case, the decision-makers are not able to provide fixed/exact values of the respective parameters. However, depending on some previous experience or knowledge, the decision-makers may furnish some estimated/most likely values of the parameters that lead to vagueness or ambiguousness. The inconsistent, inappropriate, inaccurate, indeterminate knowledge and lack of information result in vague and ambiguous situations. Thus, the parameters are not precise in such cases. Briefly, one can differentiate between stochastic and fuzzy techniques for tackling the uncertain parameters. Uncertainty arises due to randomness, which can be tackled by using stochastic techniques, whereas the fuzzy approaches can be applied when uncertainty arises due to vagueness.  
Various interactive and effective algorithms are investigated for solving the GPP when the RHS in the constraint is known exactly. However, many applications of geometric programming are engineering design problems in which some of the deterministic parameters in the RHS are defined in an estimated interval of actual values. There are also many cases when the RHS may not be depicted in a precise manner. For example, in the machining economics model, the tool life may fluctuate due to different machining operations and conditions. In this proposed GPP, uncertainty present in the data us varying between some specified intervals that differ from both types of the above-discussed uncertainties. The mathematical model of the GPP under varying parameters can be represented as follows (18):  
\[\begin{array}{ll}\textbf{Proposed Model}\\ \text{F}=\underset{x}{\text{Minimize}}&\sum_{k=1}^{l_{0}}c_{0k}\prod_{j=1}^{ m}x_  
the two-level mathematical programming problems into the single-level mathematical programming problem (26), which can be stated as follows:  
\[\begin{array}{ll}\textbf{Model B}\\ \overline{F}=\underset{y}{\text{Maximize}}&\prod_{k=1}^{l_{0}}\left(\frac{c_{0k} }{y_{0k}}\right)^{y_{0k}}\prod_{i=1}^{m}\prod_{k=1}^{l_{i}}\left(\frac{c_{ik}y _{i0}}{\underline{B}_{i}y_{ik}}\right)^{y_{ik}}\\ \text{s.\,x.}&\\ &\sum_{k=1}^{l_{0}}y_{0k}=1,\quad i=1,2,\ldots,m,\\ &\sum_{k=1}^{l_{0}}\alpha_{0k}y_{0k}+\sum_{i=1}^{m}\sum_{k=1}^{l}\beta_{ik}y_{ ik}=0,\;\;j=1,2,\ldots,n,\\ y_{ik}\geq 0,\;\;\forall\;\;i,k,\\ &\underline{B_{i}}\leq\bar{B_{i}}\leq\overline{B_{i}},\;\;\bar{B_{i}}\in[ \underline{B_{i}}\overline{B_{i}}],\;\;\forall\;\;i=1,2,\ldots,m.\end{array} \tag{26}\]  
Model B is a nonlinear constrained programming problem and can be solved by using several efficient methods. Thus, Model A and Model B provide the lower and upper bound to the geometric programming problem under varying parameters and calculate the objective value directly without violating the optimal range of the objective values where they should lie. A comprehensive study about the relationship between the globally optimal cost and the optimal dual value can be found in [4].