# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 4 Computational Study
#### Model B  
\[ \begin{array}{ll}\overline{F}=\underset{y}{\text{Maximize}}&\left(\frac{2}{y_{01 }}\right)^{y_{01}}\left(\frac{4}{y_{02}}\right)^{y_{02}}\left(\frac{2y_{10}}{ \bar{B}_{1}y_{11}}\right)^{y_{11}}\left(\frac{1.6y_{10}}{\bar{B}_{1}y_{12}} \right)^{y_{12}}\left(\frac{1.9y_{20}}{\bar{B}_{2}y_{21}}\right)^{y_{21}}\left( \frac{3.1y_{20}}{\bar{B}_{2}y_{22}}\right)^{y_{21}}\\ \text{s. x.}&y_{01}+y_{02}=1,\\ &y_{01}+y_{02}-2y_{11}+y_{12}+2y_{21}+2.2y_{22}=0,\\ &y_{01}-y_{02}-y_{11}+1.4y_{21}=0,\\ &-y_{01}+y_{02}+2y_{11}+y_{12}+y_{21}=0,\\ &y_{01}-y_{02}+y_{11}+2y_{12}+y_{21}+y_{22}=0,\\ &y_{01},y_{02},y_{11},y_{21},y_{12},y_{22}\geq 0\\ &3\leq\bar{B}_{1}\leq 3.2,\ \ 2\leq\bar{B}_{2}\leq 2.4,\ \ \forall\ i=1,2,\ldots,m.\end{array}  \tag{29}\]  
Thus, Problems (28) and (29) are the required upper and lower bounds for the geometric programming problem (27). Upon solving the problem at zero degree of difficulty, the upper and lower bounds for the objective functions are obtained as \(\overline{F}=8.5429\) and \(\underline{F}=4.9271\), respectively. However, the objective values at \(\bar{B}_{1}=3.2\) and \(\bar{B}_{2}=2.4\) are found to be \(\bar{F}=5.6212\). Therefore, the obtain objective function lies between the range of the upper and lower bounds, which shows that it is justified to reduce the objective values at the maximum RHS under variations. **Example 2** ([4]). : _This illustration belongs to a design problem of a journal bearing. The texture of the journal bearing is an inverse problem, where the eccentricity ratio and attitude angle are obtained for a defined load and speed. The engineers may not have experience in modeling the structure of this new type of journal bearing. The volume of steel, the thickness of the intermediate layer and nickel barrier, and the dimension of the plated overlay of the journal bearing are assumed to be unknown. Thus, the values of these parameters have been depicted between some specified closed intervals and taken in the form of lower and upper bounds, respectively. Hence, the varying solution outcomes will also come by ensuring the optimal objectives between corresponding intervals. Figure 2 represents the structure of the journal bearing used in this example. Hence, some parameters of the model are approximately known and are estimated by the engineers. Suppose that \(x_{1}\) is the radial clearance, \(x_{2}\) the fluid force, \(x_{3}\) the diameter, \(x_{4}\) the rotation speed, and \(x_{5}\) the length-to-diameter ratio._  
Figure 2: Illustrative figure of a journal bearing. The following mathematical programming formulation can depict the design problem as a geometric programming problem (30):  
\[ \begin{array}{ll}\begin{array}{ll}\text{F}=\text{Minimize}&0.5x_{1}^{2}x_{2}x_{4 }x_{5}+1.1x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}\\ \text{s. x.}&\\ &8.4x_{1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}x_{5}\leq\tilde{B}_{1}\\ &0.5x_{2}x_{3}+x_{1}x_{4}^{-1}x_{5}^{-1}+1.6x_{3}x_{4}\leq\tilde{B}_{2}\\ &x_{1},x_{2},x_{3},x_{4},x_{5}\geq 0.\end{array}\end{array}  \tag{30}\]  
where \(\tilde{B}_{1}\in(4,4.2)\) and \(\tilde{B}_{2}\in(0,1)\) are the varying parameters. Since all the parameters are crisp except the RHS, then Problems (23) and (26) can be utilized to obtain the upper and lower bound of the objective values in Problem (30). According to the problems (23) and (26), the formulations of the upper and lower bounds for Problem (30) can be presented as follows (31):  
**Model A**  
\[