As a task, I want to produce three products $x,y,z$ in different quantities $a,b,c>0$ respectively.
It is not always possible to produce the full amount of each product, because of a lack of resources.
Now I want to set an objective keeping the percentage between $a^*/a$, $b^*/b$ and $c^*/c$ as similar as possible with $a^*,b^*,c^*$ being the decision variables for the actual amount of produced numbers.
Moreover the following attributes have to hold:
- $a^* \leq a,b^*\leq b,c^*\leq c$
- $a^* + b^* + c^* \leq |\rm resources|$.
My thoughts on this:
- I thought that by maximizing the sum of $a^*/a$, $b^*/b$ and $c^*/c$ would have the wanted effect which is not the case.
- A way to do so would be to write an objective in the following way: $$\min\ \left|\frac{a^*}{a}-\frac{b^*}{b}\right| + \left|\frac{b^*}{b}-\frac{c^*}{c}\right| + \left|\frac{a^*}{a}-\frac{c^*}{c}\right| $$ which is not linear and perhaps not the best way to formulate.
- Is there a better objective function without having to linearize the upper part?