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.
Addison, J.D. and Heydecker, B.G. (1998), "System Optimal Assignment with Departure Time Choice", Griffiths, J.D. (Ed.) Mathematics in Transport Planning and Control, Emerald Group Publishing Limited, Bingley, pp. 145-154. https://doi.org/10.1108/9780585474182-014
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