Add second "constraint" to model a binary variable

Question

in my model I have the binary variable $f_{ij}$ which pushes a time-dependent $j$ integer variable $D_{ij}$ to zero if $f_{ij}$ takes the value 1 and keeps the integer number if $f_{ij}$ equals 0. Yes, I have linearised it accordingly. The variable $f_{ij}$ is currently coded like this. It takes the value 1 if the additional binary variable $z_{ij}$ was always zero in the last $\alpha$ days.

\begin{align} &(1-f_{ij})\le\sum_{t=j-\alpha}^{j-1}z_{it}&\quad\forall i\in I, j\in \{1+\alpha,\ldots,J\} \\ &M\cdot (1-f_{ij})\ge \sum_{t=j-\alpha}^{j-1}z_{it}&\quad\forall i\in I, j\in \{1+\alpha,\dots,J\} \end{align}

In addition to zero, I want to introduce that the variable $f_{ij}$ can also take the value 1 if the third binary variable $x_{ij}$ was zero in each of the last $\beta$ days. How can I additionally model this?

Answer 1

Your second constraint is an aggregation of $$f_{ij}\le 1-z_{it}$$ for $t\in\{j-\alpha,\dots,j-1\}$, which enforces the logical implication $$f_{ij}\implies \lnot z_{it}.$$ To instead enforce $$f_{ij}\implies (\lnot z_{it}\lor \lnot x_{it}),$$ impose $$f_{ij}\le (1-z_{it})+(1-x_{it}).$$