$$Z_{MP} = \min \sum_{d=1}^{\delta} \sum_{k=1}^{K} [c^k Q_{dk} + \theta_{dk}]$$ $$\sum_{i=1}^{N} X_{idk} \leq Q_{dk}, \quad \forall d \in \mathbb{D}, k \in \mathbb{K},$$ $$Q_{dk} \geq Q_{dk'}, \quad \forall k, k' \in \mathbb{K}, k' > k, d \in \mathbb{D},$$ $$\sum_{d=1}^{\delta} \sum_{k=1}^{K} X_{idk} = 1 \hspace{1in} \forall i \in \mathbb{N}$$ $$\text{Combinatorial benders cuts}$$ $$X_{idk} \in \{0,1\}, Q_{dk} \in \{0,1\}, \theta_{dk} \geq 0$$ Where Q determines whether OR is opened or not, while X is the assignment variable. Solving this problem gives me as set of surgeries for each $$\mathbb{N}_{dk}$$ for which subproblem is solved. The assignment in the master problem changes the structure of my subproblem. That is, it is possible to have 2 surgeries assigned to (day1, OR1) in one iteration while it could be 4 in the next iteration. The cut added is of the form where $$Z^{SP}_{dk}$$ is cost of subproblem for (d,k). $$\theta_{dk} \geq Z^{SP}_{dk} - (Z^{SP}_{dk} - L) (\sum_{i: X^*_{idk} = 0} X_{idk} + \sum_{i: X^*_{idk}=1} (1-X_{idk}) )$$
1. Using this form, I add a cut for each (day = d,OR=k) tuple, if $$\theta_{dk} < Z^{SP}_{dk}$$ This cut essentially ensures that the solution is not repeated or the solution that is found is at least >= L (Lower Bound). The subproblem has the cost of idle time, overtime, and a cost of starting surgery $$i$$ in a particular timeslot $$t$$ i.e. $$C_{idtk}$$. Also must be noted that adding surgery for (day p, OR q) another (day r,OR s) may reduce the cost of the (r,s) depending on the comparison of idle time cost vs $$C_{it}$$ parameter. The convergence is too slow and some additional strengthening cuts do not work which are discussed Benders Decomposition cuts for MILP problem with further separable subproblems How can this be improved.