I have a facility location problem with a non-linear objective;

  • There are fixed costs $S_j$ to opening facility $j$
  • $Y_j$ is a binary, $1$ if facility $j$ is opened, $0$ otherwise
  • $D_j$ is the number of products that will be gathered at facility $j$
  • It is cheaper to assign more products to an open facility as fixed costs can be spread. Therefore, there is a negative slope of $-a\cdot D_j$ when a facility is open. Indicating that when more products are assigned to an open collection point, this will be deducted from the fixed cost.

This gives the objective function $$S_j \cdot Y_j - a \cdot D_j \cdot Y_j$$

How do I linearize this to create a linear programming problem?


1 Answer 1


One approach is to perform the usual linearization of a product of a bounded variable and a binary variable, by introducing a new variable to represent the product, along with additional linear constraints to enforce the desired relationships. A simpler approach is to replace $S_j Y_j - a D_j Y_j$ with $S_j Y_j - a D_j$ and enforce the logical implication $D_j > 0 \implies Y_j = 1$. Equivalently, you can enforce the contrapositive $$Y_j = 0 \implies D_j = 0$$ either directly as an indicator constraint or indirectly via linear big-M constraint $$D_j \le M_j Y_j,$$ where $M_j$ is a (small) upper bound on $D_j$ when $Y_j = 1$. For example, you can take $M_j$ to be the total number of products.

  • $\begingroup$ Thank you for you answer! However, I think there is still an issue, due to something I haven't explained yet. Facilities have internal supply; w_j. Therefore D_j = w_j + X_ij, where X_ij is a binary variable, 1 if customer i is assigned to facility j, 0 otherwise. Therefore, D_j is always >0, as each facility has w_j. With the latter method you explained, this would always lead to Y_j=1 $\endgroup$
    – user9867
    Commented Jun 22, 2022 at 12:54
  • $\begingroup$ @user9867 Your problem statement says that $D_j$ is "the number of products that will be gathered at facility $j$". So now you are saying that a positive amount will be gathered at facility $j$ ($D_j >0$ even if facility $j$ is closed ($Y_j=0$)? $\endgroup$
    – prubin
    Commented Jun 22, 2022 at 15:16
  • $\begingroup$ @user9867 Do you maybe mean instead that $D_j = w_j + \sum_i X_{ij}$? $\endgroup$
    – RobPratt
    Commented Jun 22, 2022 at 18:00
  • $\begingroup$ @RobPratt yes that is what I mean! It is possible to have products at a facility without opening a collection point, therefore D_j will always be positive $\endgroup$
    – user9867
    Commented Jun 23, 2022 at 6:47
  • $\begingroup$ Then I suggest replacing $D_j$ with $D_j-w_j$ in the constraints. $\endgroup$
    – RobPratt
    Commented Jun 23, 2022 at 12:18

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