For all $i \in I,j\in J$ and $k\in K$, define variables $x_{ij}, z_{ijk}\in\{0,1\}$, $y_{ij}\geq 0$ and constants $c_j, e_{ijk}, d_j, f_j >0$.

We have the following constraint $$\sum_{j\in J_1}c_j\frac{x_{ij}}{\sum_{k\in K}e_{ijk}z_{ijk}} + \sum_{j\in J_2}c_jf_j x_{ij} = \sum_{j\in J_1}d_j\frac{y_{ij}}{\sum_{k\in K}e_{ijk}z_{ijk}} + \sum_{j\in J_2}d_jf_j y_{ij} \quad \forall i\in I,$$

where $J_1, J_2 \subset J$.

We also know that $$\sum_{k\in K}e_{ijk}z_{ijk}=1 \quad \forall i\in I,j\in J.$$

Now is there a way to linearize these two components? $$\frac{x_{ij}}{\sum_{k\in K}e_{ijk}z_{ijk}}, \quad \frac{y_{ij}}{\sum_{k\in K}e_{ijk}z_{ijk}}$$

I saw some interesting answers here, here and here but I'm having some problems using them for our problem.


1 Answer 1


The two denominators are equal to $1$, so just omit the denominators, yielding a linear constraint.

  • $\begingroup$ Thank you. I didn't notice that! Is there a way in general to linearize division of two variables? $\endgroup$
    – Vitamin Z
    Jan 1, 2022 at 15:36
  • $\begingroup$ Here is a Taylor series approximation for a function of two variables math.stackexchange.com/questions/69610/… $\endgroup$
    – Daniel
    Jan 2, 2022 at 3:25

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