The ‘‘traffic-assignment’’ model concerns minimizing travel time over a large network, where traffic enters the network at a number of origins and must flow to a number of different destinations. We can consider this model as a multicommodity-flow problem by defining a commodity as the traffic that must flow between a particular origin– destination pair. As an alternative to the usual node–arc formulation, consider chain flows. A chain is merely a directed path through a network from an origin to a destination. In particular, let

For example, the network in Fig. shows the arc flows of one of the commodities, those vehicles entering node 1 and flowing to node 5. The chains connecting the origin–destination pair 1–5 can be used to express the flow in this network as:

Frequently an upper bound ui is imposed upon the total flow on each arc i. These restrictions are modeled as:

 

The summation indices jk correspond to chains joining the kth origin–destination pair. The total number of arcs is I and the total number of origin–destination pairs is K. The requirement that certain levels of traffic must flow between origin–destination pairs can be formulated as follows:

Finally, suppose that the travel time over any arc is t_i , so that, if x_i units travel over arc i, the total travel time on arc on arc i is t_i xi .

a) Complete the ‘‘arc–chain’’ formulation of the traffic-assignment problem by specifying an objective function that minimizes total travel time on the network. [Hint. Define the travel time over a chain, using the ai j data.]

b) In reality, generating all the chains of a network is very difficult computationally. Suppose enough chains have been generated to determine a basic feasible solution to the linear program formulated in part (a). Show how to compute the reduced cost of the next chain to enter the basis from those generated thus far.

c) Now consider the chains not yet generated. In order for the current solution to be optimal, the minimum reduced costs of these chains must be nonnegative. How would you find the chain with the minimum reduced cost for each ‘‘commodity’’? [Hint. The reduced costs are, in general,

where π_i and u_k are the shadow prices associated with the capacity restriction on the ith constraint and flow requirement between the kth origin–destination pair. What is the sign of πi ?] d) Give an economic interpretation of πi . In the reduced cost of part (c), do the values of πi depend on which commodity flows over arc i?

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In addition, define 4 = {1 1 10 if arc i is in chain j, which connects origin-destination pair k, otherwise. == Flow over chain j between origin-destination pair k. In addition, define 4 = {1 1 10 if arc i is in chain j, which connects origin-destination pair k, otherwise. == Flow over chain j between origin-destination pair k.