Compute time between tasks

Question

I am trying to solve an optimization problem in which there is a set of tasks, $S$, where $s_i$ and $e_i$ are the starting and ending time of task $i \in S$. Each task $i $ must be done within its own time window $[a_i, b_i]$: \begin{equation} a_i \le s_i, \forall i \in S \end{equation} \begin{equation} e_i \le b_i, \forall i \in S \end{equation}

I would like to compute the time between all the pairs of tasks. That is, $s_j-e_i$ (if $i$ is scheduled before $j$) or $s_i-e_j$ (if $j$ is scheduled before $i$).

The objective is to minimize the times between all the pairs of tasks. How can I define it?

Answer 1

What you could do is the following: Introduce variable $t_{ij}$ that is the time between tasks $i$ and $j$. This variable is only defined for once per pair $ij$, therefore the $j >i$. Then, you can write:

\begin{align} \ & \min \sum_{i \in S}\sum_{j \in S, j>i} t_{ij} \; \\ \end{align}

and adding the following two constraints:

\begin{align} & s_j - e_i \leq t_{ij} & \forall i,j \in S, j>i \tag1 \\ & s_i - e_j \leq t_{ij} & \forall i,j \in S, j>i \tag2 \\ \end{align}