In an all-integer linear program, what requirement is imposed on the variables?

In an all-integer linear program, the primary requirement is that all decision variables must take only integer values. Integer constraints are often necessary in situations where the variables represent discrete quantities, such as the number of items produced, the number of workers assigned to a task, or the number of vehicles used in a transport scenario. This restriction prevents the variables from assuming fractional values, which would not be feasible or meaningful in many practical applications.

For example, if a variable represents the number of chairs to manufacture, it makes sense only when that variable is an integer. Allowing fractional values would imply the production of a fraction of a chair, which is not possible in reality. Therefore, the characterization of the variables needing to take on only integer values aligns perfectly with the definitions and goals of an all-integer linear program.

The other options, such as requiring continuous values or allowing both integers and fractions, contradict the foundational principle of integer programming, which emphasizes the need for whole number solutions. While binary variables, which are either 0 or 1, can be a subset of integer variables, not all integer programs are limited to binary constraints. Thus, the specific requirement for all-integer linear programs is clearly defined by the necessity for integer values exclusively.