Formulate a constraint set to identify the number of overlapping operations

Question

Let assume we have operation $r \in R$, machine $n \in N$ and day $t \in T$. $S_{r}$ is the service duration of operation $r$. Note that $S_{r}$ can be defined based on time slot (e.g., $S_{1}$ can be 2 time slots). In this problem, a machine can handle multiple operations at the same time.

$y_{rnt}$ is a binary variable indicating whether operation $r$ is allocated to machine $n$ on day $t$. $a_{rt}$ is a non-negative variable indicating the start time of serving operation $r$ on day $t$.

Given the above assumptions, how can I formulate a constraint set to restrict the number of overlapping operations on a machine to a predefined value of $MR$?

For more clarity, let us consider an example. For instance, we have 8 operations and one machine. If $S_{1}=3$, we should identify what operations overlap with operation $r=1$ during its service. Therefore, we should check three-time slots. The below example shows the overlapping operations with operation $r=1$, where each number represents an operation and each row represents overlapping operations at each time slot.

1, 2, 3, 4

1, 4, 5, 6

1, 5, 6, 7, 8

Let me know if you need further explanation. Thanks.

Answer 1

This seems to be a job shop scheduling problem. I am not expert in this problem, but in the job shop scheduling problem, the preventions of overlapping operations is one of the main constraints. You may use such a constraint and instead of setting the RHS of the inequality to zero (which states that overlapping is not allowed), you set your upper bound to the number of allowable overlapping operations.