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This paper investigates the temporal inflow profile that minimises the total cost of travel for a single route. The problem is formulated to consider the case in which the total demand to be serviced is fixed. The approach used here is a direct calculation of the first order variation of total system cost with respect to variations in the inflow profile. Two traffic models are considered; the bottleneck with deterministic queue and the kinematic wave model. For the bottleneck model a known solution is recovered. The wave model proves more difficult and after eliminating the possibility of a smooth inflow profile the restricted case of constant inflow is solved. As the space of possible profiles is finite dimensional in this case, the standard techniques of calculus apply. We establish a pair of equations that are satisfied simultaneously by the optimal inflow and time of first departure.

Time of concentration (T_c) is one of the main inputs in rainfall–runoff model which depends on catchment length, slope, soil properties and surface cover. Factor such as floodplain also has a significant contribution on the flood wave travel time. Floodplain which influences the flow and the travelling time is not possible to be calculated using common T_c formulae. One approach to handle this complex behaviour is to deploy the hydrodynamic model as part of the rainfall–runoff model. This chapter explains the application of hydrodynamic approach to determine T_c for large catchment with the effect of floodplain. A hydrodynamic river model for Sg Relai was developed as part of the rainfall–runoff model covering 460 km² catchment area. It includes channels covering 90 km long which is extended to the floodplain based on the digital terrain model. The simulation results show that once the flood water spill to the floodplain, the channel travelling time is delayed by several hours. The delay of the travel time increases as the rainfall intensity increase which demonstrates that hydrodynamic modelling with the integration of floodplain is capable to compute the variation of T_c.