I have the unknowns $w,x,y,z$ that are all in $\mathbb{N}$ and $\gt0$.
The known parameters $\alpha,\beta,\gamma,\delta$ are all in $\mathbb{N}$ and $\gt0$ too.
Given $\alpha,\beta,\gamma,\delta$, I need to find $w,x,y,z$ and these inequalities must be satisfied:
$\frac{w}{x+w}\leq\frac{1}{\alpha}$
$\frac{y}{y+z}\leq\frac{1}{\beta}$
$\frac{w+z}{x+y}\leq\frac{1}{\gamma}$
And also the following must be satisfied:
$w+x+y+z=\delta$
Is this problem solvable as an integer linear programming one?
Are the inequalities the constraints?
What is the objective function?
Yes.
The inequalities and the equation. You might need to split the equation into two inequalities.
Pick anything.