How to linearize $f(x,y) = (ax+by)/(x+y)$?

Question

I have a problem which is mainly linear but it has a non-linear component. The objective function is

obj = Linear_term + $c*f(x,y)$ where,

$f(x,y) = (G_1 x_1 + G_2 x_2)/(x_1 + x_2)$.

The decision variables and parameters are as follows.

$0 < b_1 <1$ :: decision variable

$0 <b_2 <1$ :: decision variable

$c>1$ :: integer decision variable

$Q_1$ :: constant

$Q_2$ :: constant

$G_1$ :: constant

$G_2$ :: constant

$x_1 = Q_1 * b_1$

$x_2 = Q_2 * b_2$

My questions are:

How I can model $cf(x,y)$ in MIP? Please note it is also probable that more than two decision variables of $b$ appear in the last equation.

How do I break this fraction and model it in linear form?

Answer 1

Without the $c$ variable, you could do a Charnes-Cooper transformation, followed by a linearization of a product of a continuous and binary variable, as shown in my answer https://math.stackexchange.com/questions/3500493/doing-a-charnes-cooper-transformation-with-matrices-and-an-zero-one-constraint/3500608#3500608

If $c$ has a small enough upper bound, you can solve a separate problem for each value of $c$ and take the best.

Alternatively, you can introduce a variable $z$ to replace $c\cdot f(x,y)$ in the objective function and impose constraint $(x_1+x_2)z=c(G_1 x_1+G_2 x_2)$, which you can linearize by linearizing the resulting products of continuous and binary variables.