# Is this problem solvable with positive integer linear programming?

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?

• 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. – Michal Adamaszek Mar 29 at 8:11