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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?

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    $\begingroup$ The inequalities are linear when you get rid of the fractions, so yes, it is a MIP. What is the objective? It depends on you. You didn't provide one. If you just want to solve the feasibility problem the objective may as well be 0. $\endgroup$ – Michal Adamaszek Mar 29 at 8:11
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Is this problem solvable as an integer linear programming one?

Yes.

Are the inequalities the constraints?

The inequalities and the equation. You might need to split the equation into two inequalities.

What is the objective function?

Pick anything.

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