# How to model a binary variable?

I am trying to find a constraint for the following relationship, but am failing a bit at it right now. I want to find a linear constraint that does the following. The binary variable $$switch_{ot}$$ is to be introduced. It should always take the value 1, if the variable $$lamp_{ot}$$ was $$=0$$ in every of the periods $$t-\tau$$ to $$t-1$$. Otherwise $$switch_{ot}=0$$ shall be valid. How do I model something like this?

• Do you mean “if the variable $lamp_{ot’}=0$ for one $t-\tau\leq t’\leq t-1$” or “if the variables $lamp_{ot’}=0$ for all $t-\tau\leq t’\leq t-1$”
– Sune
May 25, 2023 at 17:14
• Woopsie. I meant for all. May 25, 2023 at 17:20
• Are the "lamp" variables binary? If not, what are their domains?
– prubin
May 25, 2023 at 17:56
• Yes they are. Both are binary May 25, 2023 at 17:57

In other words, you want to linearize this relationship: $$\text{switch}_{ot} \iff \bigwedge_{u=t-\tau}^{t-1} \lnot \text{lamp}_{ou}$$

Or as a product: $$\text{switch}_{ot} = \prod_{u=t-\tau}^{t-1} (1 - \text{lamp}_{ou})$$

Because $$\text{lamp}_{ou}$$ is binary, so is its "complement" $$1 - \text{lamp}_{ou}$$, and the usual linearization of a product of binary variables yields: \begin{align} \text{switch}_{ot} &\le 1 - \text{lamp}_{ou} &&\text{for u \in \{t-\tau, \dots, t-1\}} \tag1\label1\\ \text{switch}_{ot} &\ge \sum_{u=t-\tau}^{t-1} (1 - \text{lamp}_{ou}) - \tau + 1 \tag2\label2 \end{align}

Of course, \eqref{2} can be simplified as $$\text{switch}_{ot} \ge 1 - \sum_{u=t-\tau}^{t-1} \text{lamp}_{ou}$$

The answer of Rob is great. Another way to model this with fewer constraints needed is the following:

\begin{alignat}1 (1- \text{switch}_{ot}) &\le \sum_{u=t-\tau}^{t-1} \text{lamp}_{ou} \tag{1}\label{1A}\\ M \cdot (1- \text{switch}_{ot}) &\ge \sum_{u=t-\tau}^{t-1} \text{lamp}_{ou} \tag{2} \label{2A} \end{alignat}

Where $$M$$ is a large enough number. In your case this might be $$\tau$$. If all $$\text{lamp}_{ou}$$ variables are $$0$$, equation \eqref{1A} makes sure that the switch variable is 1. If any $$\text{lamp}_{ou}$$ is greater $$0$$, \eqref{2A} states that the switch variable must be 0.

• This is also correct. Your (1) is my (2), and your (2) is an aggregation of my (1). Yes, $M=\tau$ is valid. May 25, 2023 at 19:16
• You are right. Calling this a "different way" is maybe not the most correct statement ;) May 25, 2023 at 19:22