# Problems with Big-M Constraint

I have the following constraints for my roster optimisation problem: \begin{align} &(1-r_{i,t})\le \sum_{j=t-\chi}^{t-1}sc_{i,j}\quad &\forall i\in I, t\in \{1+\chi,\ldots,T\} \end{align} \begin{align} &M\cdot (1-r_{i,t})\ge \sum_{j=t-\chi}^{t-1}sc_{i,j}\quad &\forall i\in I, t\in \{1+\chi,\dots,T\} \end{align} \begin{align} &r_{i,t}=0\quad &\forall i\in I, t\in \{1,\ldots,\chi\} \end{align} It is intended to ensure that the binary variable $$r_{it}$$ assumes the value one if the binary variable $$sc_{i,t}$$ was always zero in the previous $$\chi$$ days (there was no shift change in the last $$\chi$$ days). Otherwise, $$r_{i,t}=0$$ should apply if there was such a shift change. The variable $$r_{i,t}$$ is then used in the model to enable work performance to be restored. Now to my question / problem.

1. Is it possible to improve / summarise the three constraints?
2. Main problem: The variable $$r_{i,t}$$ can also take the value of 1 in this modelling if, for example, there was no change on the first three days ($$t=1-3$$). However, I actually want to model that this possible assignment of $$r_{i,t}$$ is only possible from the first shift change in the planning horizon. In other words, only once $$sc_{i,t}=1$$ has applied.

Example 1: $$\chi=2$$ applies. $$sc_{i,t}=[0,0,0,1,1,0,0,0,0]$$, then it currently follows that$$r_{i,t}= [0,0,1,1,0,0,0,0,1]$$.

Example 2 (as it should be): $$\chi=2$$ applies. $$sc_{i,t}=[0,0,0,0,1,1,0,0,0]$$, then it should follow: $$r_{i,t}= [0,0,0,0,0,0,0,0,1]$$.

1. You are enforcing: \begin{align} \lnot sc_{t-\chi}\land\dots\land \lnot sc_{t-1} \implies r_t &\equiv \lnot (\lnot sc_{t-\chi}\land\dots\land \lnot sc_{t-1}) \lor r_t \\ &\equiv (sc_{t-\chi}\lor\dots\lor sc_{t-1}) \lor r_t \\ &\equiv sc_{t-\chi}+\dots+sc_{t-1}+r_t \ge 1 \\ &\equiv 1-r_t \le sc_{t-\chi}+\dots+sc_{t-1} \end{align} which is your first constraint.
You are also enforcing: \begin{align} sc_{t-\chi}\lor\dots\lor sc_{t-1} \implies \lnot r_t &\equiv \lnot ( sc_{t-\chi}\lor\dots\lor sc_{t-1}) \lor \lnot r_t \\ &\equiv (\lnot sc_{t-\chi}\land\dots\land \lnot sc_{t-1}) \lor \lnot r_t \\ &\equiv \land_{k=t-\chi}^{t-1} (\lnot sc_k \lor \lnot r_t) \\ &\equiv 1-sc_{k}+1-r_t \ge 1 &&\forall k=t-\chi,\dots,t-1\\ &\equiv sc_k \le 1-r_t &&\forall k=t-\chi,\dots,t-1\\ \end{align} This is the disaggregated version of your second constraint (tighter), with $$M=\chi$$.
1. If my understanding is correct (?), you want to enforce: $$(\lnot sc_{t-\chi}\land\dots\land \lnot sc_{t-1}) \land sc_t \implies r_t$$ Using the same methodology: \begin{align} \lnot sc_{t-\chi}\land\dots\land \lnot sc_{t-1} \land sc_t \implies r_t &\equiv \lnot (\lnot sc_{t-\chi}\land\dots\land \lnot sc_{t-1} \land sc_t) \lor r_t \\ &\equiv (sc_{t-\chi}\lor\dots\lor sc_{t-1} \lor \lnot sc_t) \lor r_t \\ &\equiv sc_{t-\chi}+\dots+sc_{t-1}+(1-sc_t)+r_t \ge 1 \\ &\equiv sc_t \le sc_{t-\chi}+\dots+sc_{t-1}+r_t \end{align}