The paper considers an existing model for optimizing time-varying flows on a congested network, and extends it so as to improve its accuracy while reducing computation. The model has been developed in a series of articles and has a variety of applications. Solving the model is a significant problem, since it is a non-linear program and can be very large for even small or medium size applications. Hitherto the model has been solved by taking a piecewise linear approximation and solving this as a linear program, perhaps taking advantage of the staircase structure of the constraints. However, this staircase is not available in the extended model. Further, the existing solution methods do not take advantage of the network structure of the problem. Here the paper shows that a piecewise linear version of the model can be stated as a pure processing network (PPN); this allows algorithms for PPNs to be used to solve the model. It also proposes a penalty (or barrier) function solution method. This reduces the problem to successively reoptimizing a program equivalent to the well-known static traffic assignment model. The latter is a convex cost, linearly constrained, uncapacitated network flow problem. The above solution methods apply both to the basic model and the extended model.
