In a scheduling model for maintenance of power generators the objective is to smooth the reserves. The authors use a quadratic objective to model and the results over a year look like:
MIQP’s still solve significantly slower than MIP models, so I was looking for a linear approximation. The first thing that comes to mind is to minimize the max. This gives:
The problem here is we get a few weeks with really low reserves. An alternative is to minimize the difference between the max and the min. This looks like:
This actually looks quite similar to the first picture, and we have achieved our goal: make the model linear.