# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 3 Geometric Programming Problem: Basic Concepts
### Geometric Programming Problems
: The expression for \(F_{0}(x)\) can be written as  
\[F_{0}(x)=\sum_{i=1}^{n}\sum_{k=1}^{T_{0}}y_{ik}\left(\frac{c_{0k}\prod_{j=1}^{ m}x_{ij}^{a_{0 kj}}}{y_{ik}}\right) \tag{5}\]We can apply the weighted \(A.M.\geq G.M.\) inequality to this new expression for \(F_{0}(x)\) and obtain  
\[\left(\frac{c_{0 ik}\prod_{j=1}^{m}x_{ij}^{\alpha_{0 ikj}}}{\sum_{i=1}^{n}\sum_{k= 1}^{T_{0}}y_{ik}}\right)^{\sum_{i=1}^{n}\sum_{k=1}^{T_{0}}\delta_{k}}\geq\prod_{ i=1}^{n}\prod_{k=1}^{T_{0}}\left(\frac{c_{0 ik}\prod_{j=1}^{m}x_{ij}^{\alpha_{0 ikj}} }{y_{ik}}\right)^{\delta_{k}}\]  
or  
\[\left(\frac{F_{0}(x)}{n}\right)^{n}\geq\prod_{i=1}^{n}\left(\frac{C_{0 ik}}{y_{ ik}}\right)^{y_{ik}}\prod_{j=1}^{m}x_{ij}^{\sum_{k=1}^{T_{0}}\alpha_{0 ikj}y_{ ik}}\]  
using normality condition  
\[=\prod_{i=1}^{n}\prod_{k=1}^{T_{0}}\left(\frac{C_{0 ik}}{y_{ik}}\right)^{\delta _{ik}}\prod_{j=1}^{m}x_{ij}^{\sum_{k=1}^{T_{0}}\alpha_{0 ikj}y_{ik}} \tag{6}\]  
Again, \(F_{r}(x)\) can be written as  
\[g_{r}(x)=\sum_{i=1}^{n}\sum_{T_{r-1}+1}^{T_{r}}y_{ik}\left(\frac{c_{rik}\prod_{ j=1}^{m}x_{ij}^{\alpha_{rij}}}{y_{ik}}\right) \tag{7}\]  
Applying the weighted \(A.M.\geq G.M.\) inequality in (7), we have  
\[\left(\frac{F_{r}(x)}{\sum_{i=1}^{n}\delta_{ik}}\right)^{\sum_{i=1}^{n}y_{ik}} \geq\prod_{i=1}^{n}\prod_{k=T_{r-1}+1}^{T_{r}}\left(\frac{c_{rik}\prod_{j=1}^{ m}x_{ij}^{\alpha_{rikj}}y_{ik}}\right)^{y_{ik}}\]  
and  
\[(F_{r}(x))^{\sum_{i=1}^{n}y_{ik}}\geq\prod_{i=1}^{n}\prod_{k=T_{r-1}+1}^{T_{r }}\left(\frac{C_{rik}}{y_{ik}}\right)^{y_{ik}}\prod_{j=1}^{m}x_{ij}^{\sum_{k= T_{r-1}+1}^{T_{r}}\alpha_{rik}y_{ik}}\left(\sum_{s=1}^{n}y_{sk}\right)^{y_{ik}}\]  
\((r=1,2,\ldots,l).\) Using \(1\geq(F_{r}(x))^{\sum_{i=1}^{n}y_{ik}}(r=1,2,\ldots,l)\) (since \(F_{r}(x)\leq 1(r=1,2,\ldots,l)\)),  
\[1\geq\prod_{i=1}^{n}\prod_{k=T_{r-1}+1}^{T_{r}}\left(\frac{C_{rik}}{\delta_{ ik}}\right)^{y_{ik}}\left(\sum_{s=1}^{n}y_{sk}\right)^{y_{ik}}\prod_{j=1}^{m}x_{ ij}^{\sum_{k=T_{r-1}+1}^{T_{r}}\alpha_{rik}y_{ik}} \tag{8}\]  
Multiplying (6) and (8), we have  
\[\left(\frac{F_{0}(x)}{n}\right)^{n}\geq\prod_{i=1}^{n}\prod_{k=T_{r-1}+1}^{T_{ r}}\left(\frac{C_{0 ik}}{y_{ik}}\right)^{y_{ik}}\left(\sum_{s=1}^{n}y_{sk} \right)^{y_{ik}}\prod_{j=1}^{m}x_{ij}^{\sum_{k=1}^{T_{0}}\alpha_{0 ikj}\delta_{ ik}}\prod_{j=1}^{m}x_{ij}^{\sum_{k=1}^{T_{0}}\alpha_{0 ikj}y_{ik}+\sum_{j=1}^{l} \sum_{k=T_{r-1}+1}^{T_{r}}\alpha_{rij}y_{ik}} \tag{9}\]  
\((r=0,1,2,\ldots,l)\). Using orthogonality conditions, the inequality (9) becomes  
\[\left(\frac{F_{0}(x)}{n}\right)^{n}\geq\prod_{i=1}^{n}\prod_{k=1}^{T_{r}}\left( \frac{C_{ik}}{y_{ik}}\right)^{\delta_{ik}}\left(\sum_{s=1}^{n}y_{sk}\right)^{y_ {ik}},(r=0,1,2,\ldots,l)\]  
i.e., \(F_{0}(x)\geq n\sqrt[n]{F(y)}\). This completes the proof. Theorem 4.2: _Suppose that the constraint PGP (2) is super-consistent and that \(x^{*}\) is a solution for GP. Then, the corresponding DP (4) is consistent and has a solution \(\delta^{*}\) that satisfies_  
\[\left(\frac{c_{0 ik}\prod_{j=1}^{m}x_{ij}^{\alpha_{0 ikj}}}{\sum_{i=1}^{n}\sum_{k= 1}^{T_{0}}y_{ik}}\right)^{\sum_{i=1}^{n}\sum_{k=1}^{T_{0}}\delta_{k}}\geq\prod_{ i=1}^{n}\prod_{k=1}^{T_{0}}\left(\frac{c_{0 ik}\prod_{j=1}^{m}x_{ij}^{\alpha_{0 ikj}} }{y_{ik}}\right)^{\delta_{k}}\]0_and_  
\[y^{*}_{ik}=\left\{ \begin{array}{ll}\frac{u_{ik}(x^{*})}{\delta(x^{*})},&(i=1,2, \ldots,n;k=1,2,\ldots,T_{0})\\ \lambda_{ir}(y^{*})u_{ik}(x^{*})&(i=1,2,\ldots,n;k=T_{r-1}+1,2,\ldots,T_{r};r=1,2,\ldots,l)\end{array} \right.\]  
Proof. : Since GP is super-consistent, so is the associated CGP. Furthermore, since GP has a solution \(x^{*}=(x^{*}_{i1},x^{*}_{i2},\ldots,x^{*}_{ij})\), the associated GP has a solution \(p^{*}=(p^{*}_{i1},p^{*}_{i2},\ldots,p^{*}_{ij})\) given by \(p^{*}_{ij}=\ln x^{*}_{ij}\).