Choosing the Tightest Big-M Value

To switch an activity $x \geq 0$ on or off using a binary indicator $y \in \{0, 1\}$, we typically write the big-M constraint

$$x \leq M \cdot y.$$

If $y = 0$, the constraint becomes $x \leq 0$, forcing $x = 0$. If $y = 1$, it becomes $x \leq M$, allowing $x$ to take any value up to $M$.

For this formulation to be valid, $M$ must be at least as large as the largest value $x$ could attain in the feasible region — otherwise we cut off legitimate solutions.

But $M$ should not be unnecessarily large either. The tightest big-M is the smallest valid value: the actual maximum that $x$ can take when $y = 1$. Tighter values produce a stronger LP relaxation, which dramatically speeds up integer-programming solvers.

In the simplest case, when $x$ already has an explicit upper bound $x \leq U$, the tightest choice is

$$M = U.$$

For instance, if a generator produces $0 \leq x \leq 130$ MW when running, then

$$x \leq 130\, y$$

is the tightest big-M formulation of "$x = 0$ unless the generator is on."