A note on time-indexed formulations for the resource-constrained project scheduling problem

The classical time-indexed 0–1 linear programming formulations for the resource-constrained project scheduling problem involve binary variables indicating whether an activity starts precisely at or before a given time period. In the literature, references to less classical “on/off” formulations, that involve binary variables indicating whether an activity is in progress during a time period, can also be found. These formulations were not compared to the classical ones in terms of linear programming (LP) relaxations. In this paper we show that the previously proposed on/off formulations are weaker than the classical formulations and we obtain a stronger on/off formulation via non singular transformations of the classical formulations. We also remark that additional time-indexed formulations, presented as appealing in the literature, are in fact either weaker or equivalent to the classical ones.