Clarification on constraint formulation

Question

I have the following constraint and would like to know if it can be modelled like this.

$$\sum_{t\in T}\sum_{j=k}^{k+5}q_{itj}\geq 3\cdot\left( 1-\sum_{j=k}^{k+5}r_{ij} \right) \forall i\in I, k\geq K_i^{init}$$

I have the set $I$ (persons), $T$ (machine) and $K$ (day). $q_{itk}$ indicates whether the machine is worked on by the person on the day ($q_{itk}=1$). $r_{ik}$ indicates whether the person has worked for the last time on this day ($r_{ik}=1$ if yes). Now I want to make sure that as long as the person has not worked for the last time, they work at least 3 times within a 6-day window. In doing so, $r_{ik}$ can only have the value 1 once, so $\sum_{k\in K}r_{ik}=1 \forall i \in I$. In addition, each worker has a first working day $K_i^{init}$, so it should only apply to this and all subsequent days. Is the modelling correct?

Answer 1

I think it's very close, but the summation for $r$ needs to go back to $K_i^{init}$ otherwise you'll drop consideration of the fact that they may have worked for the last time prior to the beginning of the 6-day window.

For example, suppose you have a 16-day period. They satisfy the constraint over the first two 6-day periods and stop work on day 9. For the constraint with a start day of $k=10$, your current formulation would not sum over $r_{i,9}$, and would start re-enforcing the constraint.

I think this would be better: $$ \sum_{t \in T} \sum_{j=k}^{k+5} q_{itj} \geq 3\left(1-\sum_{j=K_i^{init}}^{k+5} r_{ij}\right) \quad \forall i \in I,\,k \geq K_i^{init}$$

The only other thing I'd recommend considering is whether $r_{ik}=1$ means that the constraint shouldn't apply over the period ending on on day $k$ or the period ending on day $k+1$. The way you stated that it meant they 'worked for the last time on this day' makes it sound like they are in fact working on day $k$ and that the constraint should apply to that period and then stop applying on the day after. But that's just a question of how you choose to define the variable and how you use it in the rest of the formulation.