where ai} > 0 is the minimum time interval which must elapse between an ordered pair of events of types i and j whether or not there are intermediate events. (5) The last requirement proves to be crucial in railway applications of the kind considered below, and marks the main point of divergence from standard transportation network theory. In its absence, the present problem reduces to a variant of the Travelling Salesman Problem (Reference 1) by regarding events as towns and equating time with distance. We are tacitly assuming in (4) and (5) above that real events have zero duration. In an application when this is not the case, the theory can still be applied by denning theoretical events as, say, the start of corresponding real events, provided that a constraint of the form (5) holds for the starting times. The basic scheduling problem can now be stated in the following form: 'Find a schedule for the events