Modeling Yes/No Decisions with Binary Variables

A binary variable is a decision variable restricted to the two values $0$ and $1$:

$$x \in \{0, 1\}.$$

Binary variables model yes/no decisions. We set $x=1$ when the action is taken and $x=0$ when it is not. For instance, if a firm is deciding whether to launch a new product, we let

$$x = \begin{cases} 1 & \text{if the product is launched,} \\ 0 & \text{otherwise.} \end{cases}$$

Multiplying $x$ by a cost or revenue figure produces a quantity that activates only when the decision is "yes." If launching the product costs \80{,}000$and yields$$140{,}000$ in revenue, then

• Total launch cost: $80{,}000\, x$
• Total launch revenue: $140{,}000\, x$
• Net profit: $140{,}000\, x - 80{,}000\, x = 60{,}000\, x.$

When $x=1$ the profit is \60{,}000;$when$x=0$every term collapses to$0.$ This trick -- multiplying a constant by a binary -- is the foundation of every integer-programming formulation.