I have a specific scheduling problem I am interested in. I wish to schedule n meetings (hereafter 2 for simplicity) that maximizes the number of people able to attend, with j time slots and m people. In trying to format this as an optimization problem I came up with the following:
$$argmin_{x} \frac{1}{2} \sum^m (1 - \sum^j c_{l,i} x_i )(2 - \sum^j c_{l,i} x_i )$$ $$st \sum^j x_i = 2$$
The vector c represents the availability of each person l for the time slot i, and is 1 if available for that spot and 0 otherwise. X is the current assignment of meetings to slots, and should some to n (here 2). Therefore, $\sum^j c_{l,i} x_i$ represents the number of scheduled meetings that individual l will be able to attend. The polynomial in the cost ensures that the cost for individuals able to attend one or both of the meetings have cost 0 and those who can attend 0 meetings have cost 2.
My question is whether there is a more efficient representation of this problem, and if not is there a practical advice for implementing a solution to this optimization problem?
You can model the problem linearly. Let binary variable $y_l$ indicate whether person $l$ can attend. Then the problem is to maximize $\sum_{l=1}^m y_l$ subject to: \begin{align} \sum_{i=1}^j x_i &= n \\ y_l &\le \sum_{i=1}^j c_{l,i} x_i &&\text{for all $l$} \\ x_i &\in \{0,1\} &&\text{for all $i$}\\ y_l &\in \{0,1\} &&\text{for all $l$} \end{align}