Crew allocation in airline operations

Crew allocation problems in the airline industry involve assigning crews to airplanes. Such problems are modeled with boolean decision variables x_{ij}, with x_{ij} = 1 if the i-th crew is assigned to the j-th plane, and 0 otherwise. Other constraints may be added; for example, to express that exactly one crew is to be assigned to each plane, we write the constraint

 forall : i ~:~ sum_{j} x_{ij} = 1 .