In a fixed cost problem, what does the binary variable indicate?

In a fixed cost problem, the binary variable is fundamentally used to signify the activation or deactivation of a particular decision or activity. When the binary variable takes a value of 1, it indicates that the activity is being undertaken, meaning that the costs or investments associated with it are incurred. Conversely, a value of 0 indicates that the activity is not being undertaken, and thus, those costs are not incurred.

This is crucial in modeling scenarios where decisions are of an all-or-nothing nature, such as in facility location problems, production decisions, or project initiation, where the fixed costs only apply if the project goes ahead. Therefore, the binary variable effectively encapsulates the decision-making process that leads to the allocation of resources and capital, directly linking it to fixed costs.