# Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach
## 3 Geometric Programming Problem: Basic Concepts
### Geometric Programming Problems  
Geometric programming problems fall under a class of nonlinear programming problems characterized by objective and constraint functions in a special form. The texture of GPP is quite different from other mathematical programming problems and depends on the characterization of decision variables in its product form. Thus, the modeling structure of different engineering problems inevitably adheres to the form of the GPP while optimizing the real-life problems. It is introduced for the solution of the algebraic nonlinear programming problems under the linear or nonlinear constraints, used to solve dynamic optimization problems. The useful impact in the area can be realized by its enormous application in integrated circuit design, manufacturing system design, and project management. Therefore, the standard form of GPP formulations can be represented as follows (1):  
\[ \begin{array}{ll}\text{F}=\underset{x}{\text{Minimize}}&\sum_{k=1}^{l_{0}}c_ {0k}\prod_{j=1}^{n}x_{j}^{a_{0 kj}}\\ \text{s.\ x.}&\sum_{k=1}^{l_{i}}c_{ik}\prod_{j=1}^{n}x_{j}^{\beta_{ikj}}\leq 1,\quad i=1,2,\ldots,m,\\ &x_{j}\geq 0,\quad j=1,2,\ldots,n.\end{array}  \tag{1}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Monomial** & **Polynomial** & **Posynomial** \\ \hline (1) Deals with a single term & Having one or more term & Having one or more term \\ \hline (2) Sum of monomials is not a monomial & Sum of polynomials is a polynomial & Sum of posynomials is a posynomial \\ \hline (3) Subtraction of monomials is not a monomial & Subtraction of polynomials is a polynomial & Subtraction of posynomials is not a \\ \hline (4) Multiplication of monomials is a monomial & Multiplication of polynomials is a polynomial \\ \hline (5) Division of a monomial by other & Division of a polynomial by other monomials is a polynomial & Division of a posynomial by other monomials is a posynomial \\ \hline (6) The mathematical expression of a & The mathematical expression of a polynomial & The mathematical expression of a posynomial \\ monomial is \(F(x)=cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}\), \(c>0\) & is \(F(x)=\sum_{i=1}^{n}c_{i}x_{1}^{a_{1i}}x_{2}^{a_{2i}}\cdots x_{n}^{a_{ni}}\), \(c_{i}>0\) \\ \hline (7) Example: 0.84, \(9a^{3}b^{-9}\), \(17\sqrt{c/a}\) & Example: 0.43, \(9a^{3}b^{-9}\), \(-5c/a\), \(6a-8c\) & Example: 0.59, \(9a^{3}b^{-9}\), \(17\sqrt{c/a}+a^{7}b^{4}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison between monomial, polynomial, and posynomial. where \(l_{0}\) is the number of terms present in the objective function, while the inequality constraints include \(l_{i}\) terms for \(i=1,2,\ldots,m\). Geometric programming problems have a strong duality theorem, and hence, geometric programming problems with enormously nonlinear constraints can be depicted correspondingly as one with only linear constraints. Moreover, if the primal problem is in the form of a posynomial, then a global solution of a minimization-type problem can be determined by solving its dual maximization-type problem. The dual problem contains the desirable characteristics of being linearly constrained and with an objective function having wholesome features. This leads towards the development of the most promising solution methods for the geometric programming problems. Assume that we replace the right-hand-side term (RHS) of the constraints in the GPP (1). Then, the modified GPP can be given as follows (2):  
\[ \begin{array}{ll}\textbf{Primal}&\\ \text{F}=\underset{x}{\text{Minimize}}&\sum_{k=1}^{l_{0}}c_{0k}\prod_{j=1}^{ m}x_{j}^{a_{0 kj}}=F_{0}(x)(say)\\ \text{s.\ x.}&\\ &\sum_{k=1}^{l_{i}}c_{ik}\prod_{j=1}^{n}x_{j}^{\beta_{kj}}\leq B_{i},\quad i=1,2,\ldots,m,\\ &x_{j}\geq 0,\quad\ j=1,2,\ldots,n.\end{array}  \tag{2}\]  
where \(B_{i},\ \forall\ i=1,2,\ldots,m\) are non-negative numbers. If \(B_{i}=1,\ \forall\ i\), then this modified geometric programming problem (2) is the standard geometric programming problem (1).