Indicator Variables for On/Off Activities

Many optimization problems involve activities that can be either on or off: a factory operates or stays idle, a facility is opened or not, a delivery route is used or not. To model these decisions, we introduce a binary indicator variable:

$$y = \begin{cases} 1 & \text{if the activity is on} \\ 0 & \text{if the activity is off} \end{cases}$$

If the activity has a continuous level $x \geq 0$ (units produced, hours run, MW generated, etc.), we must force $x = 0$ whenever $y = 0$. This is done with a big-M linking constraint:

$$0 \leq x \leq M\, y,$$

where $M$ is an upper bound on $x$ (typically the capacity). When $y = 0$, the constraint forces $x \leq 0$, so $x = 0$. When $y = 1$, the constraint allows $x$ to be anywhere from $0$ to $M$.

For instance, a plant with capacity $200$ units is modeled by

$$x \leq 200\, y, \quad x \geq 0, \quad y \in \{0,1\}.$$

If $y = 0$, then $x \leq 0$ forces $x = 0$. If $y = 1$, then $0 \leq x \leq 200$.