I am currently setting up a nurse rostering model. In it, I have several regulatory constraints, such as limits on working days and so on. Apparently these are considered hard constraints. However, I am struggling to determine if my other constraints are considered hard or soft.
This is the model so far: $$\text{minimize} \sum_t \sum_s slack_{ts} $$ $$\sum_{i}^{}motivation_{its}+slack_{ts}=demand_{ts}\forall t,s\quad \quad\quad \quad\quad \quad (1)$$ $$\sum_s x_{its}\le 1\quad \forall i, t\quad \quad\quad \quad\quad \quad (2)$$ $$ \sum_{j=t}^{t+4}\sum_{s}^{}x_{ijs}\le 4\quad\forall i,t\in \{1,\ldots,D - 4\}\quad \quad\quad \quad\quad \quad (3)$$ $$mood_{it} + M\cdot (1-x_{its}) \geq motivation_{its} \geq mood_{it} - M\cdot (1-x_{its})\quad \forall i,t,s\quad \quad\quad \quad\quad \quad (4)$$ $$motivation_{its} \leq x_{its}\quad \forall i,t,s\quad \quad\quad \quad\quad \quad (5)$$ $$mood_{it}=1-\alpha_{it}\cdot \sum_s x_{its}\quad \forall i,t\\ \alpha_{it}\sim U(0,1)\quad \forall i,t\quad \quad\quad \quad\quad \quad (6)$$ $$x_{its}\in \{0,1\}\quad \quad\quad \quad\quad \quad (7)$$ $$mood_{it}, motivation_{its}\in[0,1]\quad \quad\quad \quad\quad \quad (8)$$
Currently, i would refer to constraint (2) and (3) as hard constraints, but what are the other, i.e. (4), (5) and (6)?
