Fundamentals of Transportation/Route Choice/Background
Some additional topics in Route Choice:
Integrating travel choices 
The urban transportation planning model evolved as a set of steps to be followed and models evolved for use in each step. Sometimes there were steps within steps, as was the case for the first statement of the Lowry model. In some cases, it has been noted that steps can be integrated. More generally, the steps abstract from decisions that may be made simultaneously and it would be desirable to better replicate that in the analysis.
Disaggregate demand models were first developed to treat the mode choice problem. That problem assumes that one has decided to take a trip, where that trip will go, and at what time the trip will be made. They have been used to treat the implied broader context. Typically, a nested model will be developed, say, starting with the probability of a trip being made, then examining the choice among places, and then mode choice. The time of travel is a bit harder to treat.
Wilson’s doubly constrained entropy model has been the point of departure for efforts at the aggregate level. That model contains the constraint
QijSij = C
where the Sij are the travel costs, Qij refers to travel demand, and C is a resource constraint to be sized when fitting the model with data. Instead of using that form of the constraint, the monotonically increasing resistance function used in traffic assignment can be used. The result determines zone-to-zone movements and assigns traffic to networks, and that makes much sense from the way one would imagine the system works. Zone-to-zone traffic depends on the resistance occasioned by congestion.
Alternatively, the link resistance function may be included in the objective function (and the total cost function eliminated from the constraints).
A generalized disaggregate choice approach has evolved as has a generalized aggregate approach. The large question is that of the relations between them. When we use a macro model, we would like to know the disaggregate behavior it represents. If we are doing a micro analysis, we would like to know the aggregate implications of the analysis.
Wilson derives a gravity-like model with weighted parameters that say something about the attractiveness of origins and destinations. Without too much math we can write probability of choice statements based on attractiveness, and these take a form similar to some varieties of disaggregate demand models.
Integrating travel demand with route assignment 
It has long been recognized that travel demand is influenced by network supply. The example of a new bridge opening where none was before inducing additional traffic has been noted for centuries. Much research has gone into developing methods for allowing the forecasting system to directly account for this phenomenon. Evans (1974) published a doctoral dissertation on a mathematically rigorous combination of the gravity distribution model with the equilibrium assignment model. The earliest citation of this integration is the work of Irwin and Von Cube, as related by Florian et al. (1975), who comment on the work of Evans:
"The work of Evans resembles somewhat the algorithms developed by Irwin and Von Cube (“Capacity Restraint in Multi-Travel Mode Assignment Programs” H.R.B. Bulletin 347 (1962)) for a transportation study of Toronto, Canada. Their work allows for feedback between congested assignment and trip distribution, although they apply sequential procedures. Starting from an initial solution of the distribution problem, the interzonal trips are assigned to the initial shortest routes. For successive iterations, new shortest routes are computed, and their lengths are used as access times for input the distribution model. The new interzonal flows are then assigned in some proportion to the routes already found. The procedure is stopped when the interzonal times for successive iteration are quasi-equal."
Florian et al. proposed a somewhat different method for solving the combined distribution assignment, applying directly the Frank-Wolfe algorithm. Boyce et al. (1988) summarize the research on Network Equilibrium Problems, including the assignment with elastic demand.
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).