Suppose that, the following non-linear objective arises (MAX/MIN): \begin{equation} \max\frac{\sum\limits_{j} a_{j} x_{j}}{\sum\limits_{j} b_{j} x_{j}} \end{equation}
1) Replace the expression $\dfrac{1}{\sum\limits_{j} b_{j} x_{j}}$ by a variable $t$.
2) Represent the products $x_{j} t$ by variables $w_{j}$. The objective now becomes:
\begin{equation} \max \sum_{j} a_{j} w_{j}. \end{equation} Introduce a constraint: \begin{equation} \sum_{j} b_{j} w_{j}=1 \end{equation} in order to satisfy condition 1.
Convert the original constraints of the form: \begin{align} \sum_{j} d_{j} x_{j} &\leqq e \\ \sum_{j} d_{j} w_{j}-e t &\leqq 0 \end{align} It must be pointed out that this transformation is only valid if the denominator $\sum\limits_{j} b_{j} x_{j}$ is always of the same sign and non-zero. If necessary (and it is valid), an extra constraint must be introduced to ensure this. If $\sum\limits_{j} b_{j} x_{j}$ always be negative the directions of the inequalities in the constraints above must, of course, be reversed.
Reference: Model Building in Mathematical Programming
