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?