how to express an intertemporal implication constraint in integer linear programming (ILP)?
Base yourself on the fact that $x_{d,i} \land x_{d',j} $ happens iff $ x_{d,i} + x_{d',j} =2  $. 

$\forall (d,d') \in D^2, \forall (i,j) \in T^2, i&lt;j, \sum\limits_{d'' \in D \\ i&lt;j&lt;k} x_{d'',k} \leq \sum\limits_{i&lt;j&lt;k} a_k + (j-i)(2-x_{d,i} - x_{d',j}) $

$\forall (d,d') \in D^2, \forall (i,j) \in T^2, i&lt;j, \sum\limits_{d'' \in D \\ i&lt;j&lt;k} x_{d'',k} \geq \sum\limits_{i&lt;j&lt;k} a_k + (j-i)(x_{d,i} + x_{d',j} - 2) $

Check these equations in two cases : $ x_{d,i} + x_{d',j} =2 $ and $ x_{d,i} + x_{d',j} &lt; 2  $

Depending on your $a_k$ and problem, you may get a better multiplicative constant than $(j-i)$. If you can, do it, otherwise the continuous relaxation is going to suck.