Modeling Setup Times and Precedences

In a scheduling model, a precedence constraint requires one task to finish before another begins. Given interval variables for tasks $A$ and $B,$ the requirement that $B$ starts only after $A$ ends is expressed as the linear inequality

$$\texttt{model.Add(b\_start >= a\_end)}.$$

We write this as $A \prec B,$ read "$A$ precedes $B$."

For example, suppose two tasks have been declared with start, duration, and end variables. Task $A$ has duration $5$ and task $B$ has duration $3.$ Adding the single line

$$\texttt{model.Add(b\_start >= a\_end)}$$

forces the solver to choose values such that $b\_start \geq a\_end.$ One feasible assignment is

$$a\_start = 0,\quad a\_end = 5,\quad b\_start = 5,\quad b\_end = 8.$$

The constraint is a pure inequality between integer variables; no Boolean machinery is needed when the order of the two tasks is fixed in advance.