# Exploiting identical subproblems in column generation

As suggested, this is a new post. I have the following column generation model. With the following master problem $$MP$$: \begin{align} &\text{minimize} &\sum_t \sum_s \text{slack}_{ts} \\ &\text{subject to} &\sum_i \sum_r \text{motivation}_{its}^r \lambda_{ir} + \text{slack}_{ts} & = \text{demand}_{ts} &&\forall t,s &&\text(\text{\pi_{ts} free})\\ &&\sum_r \lambda_{ir} &= 1 &&\forall i &&\text(\text{\mu_i free})\\ &&\lambda_{ir} &\in\mathbb{Z}^+ &&\forall i,r\\ &&\text{slack}_{ts} &\ge 0 &&\forall t,s \end{align} And the individual subproblems $$SP(i)$$: \begin{align} &\text{minimize} &-\sum_{t,s} \pi_{ts} \text{motivation}_{its} - \mu_i \\ &\text{subject to} &-M(1-x_{its}) \le \text{motivation}_{its} - \text{mood}_{it} &\le M(1-x_{its}) &&\forall t,s \\ &&\text{motivation}_{its} &\le x_{its} && \forall t,s \\\ &&\alpha_{it} \sum_s x_{its} + \text{mood}_{it} &= 1 &&\forall t\\ &&\text{motivation}_{its} &\in[0,1] &&\forall t,s \\ &&\text{mood}_{it} &\in[0,1] &&\forall t \\ &&x_{its}&\in \{0,1\} &&\forall t,s\\ \end{align}

This assumes individual nurses, with different regulatory requirements or preferences f.e. However, if this assumption is relaxed, that every nurse is the same, how would this change the model formulation and why should the model be changed? For example, as done by Purnomo and Bard (2007), they translated the decision variable to be an integer, rather than being binary. Is the necessary as $$motivation^r_{its}$$ is continuous

EDIT: The new objective function looks like this. $$-\sum_{t,s}^{}\pi_{ts}motivation_{ts}-\min \mu_i$$

• @nflgreaternba If you use the suggested approach, the (single) subproblem objective function becomes $-\sum_{t,s} \pi_{ts} \text{motivation}_{ts}-\mu$, where $\mu$ is the dual variable for the convexity/cardinality constraint. You can think of the new $\lambda_r$ variable as the aggregation $\sum_i \lambda_{ir}$. Feb 12 at 4:56
• So the new objective function is not $-\sum_{t,s}^{}\pi_{ts}motivations_{ts}-\min\mu_i$ but rather $\min−\sum_{t,s}^{}\pi_{ts}motivation_{ts}-\mu$ (which is the same objective function as in the case of individual subproblems but dropping the index on \mu). Correct? Feb 13 at 10:05