I got an ILP Model where $c_i$ represents the starting time for a visit$_i$. $c_i$ is already constraint by a number of constraints, one is $c_i > 0$.
I have now outside of my model 0 or multiple desired intervals for the visit$_i$, lets call them $dStart_{ik}$ and $dEnd_{ik}$ where k represents the k-th desired interval. i started by adding two variables per k: $desiredStart_{ik}$ and $desiredEnd_{ik}$.
I added the constraints
$desiredStart_{ik} >= dStart_{ik}$
$desiredStart_{ik} >= c_{i}$
$desiredEnd_{ik} <= dEnd_{ik}$
$desiredEnd_{ik} <= c_{i+1}$
$desiredDuration_{ik} <= desiredEnd_{ik} - desiredStart_{ik}$
with the last constraint, i get the problem if the intervals do not overlap, the $desiredDuration_{ik}$ gets negative. So I added a "helper" Variable $desiredHelper_{ik}$ to it so it looks like this:
$desiredDuration_{ik} <= desiredEnd_{ik} - desiredStart_{ik} + desiredHelper_{ik}$
The general problem of my model is a minimsationProblem, so I added these two terms to the target function:
$+100000 \sum_{i=0}^N \sum_{k=0}^N desiredHelper_{ik} -\sum_{i=0}^N \sum_{k=0}^N desiredDuration_{ik} $
With this formulation I see a problem, it favours solutions where the desired interval is closer by the actual solution. Is there a better formulation? how could I avoid the $desiredHelper_{ik}$ variable?
