Discussion
A three link problem can not be solved graphically, and most transportation network problems involve a large numbers of nodes and links. Eash et al., for instance, studied the road net on DuPage County where there were about 30,000 one-way links and 9,500 nodes. Because problems are large, an algorithm is needed to solve the assignment problem, and the Frank-Wolfe algorithm (modified a bit since first published) is used. Start with an all or nothing assignment, and then follow the rule developed by Frank-Wolfe to iterate toward the minimum value of the objective function. (The algorithm applies successive feasible solutions to achieve convergence to the optimal solution. It uses an efficient search procedure to move the calculation rapidly toward the optimal solution.) Travel times correspond to the dual variables in this programming problem.
It is interesting that the Frank-Wolfe algorithm was available in 1956. Its application was developed in 1968, and it took almost another two decades before the first equilibrium assignment algorithm was embedded in commonly used transportation planning software (Emme and Emme/2, developed by Florian and others in Montreal). We would not want to draw any general conclusion from the slow application observation, mainly because we can find counter examples about the pace and pattern of technique development. For example, the simplex method for the solution of linear programming problems was worked out and widely applied prior to the development of much of programming theory.
The problem statement and algorithm have general applications across civil engineering -– hydraulics, structures, and construction. (See Hendrickson and Janson 1984).
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