Lecture 3 Dr. Anna Nagurney John F. Smith Memorial Professor Isenberg School of Management University of Massachusetts Amherst, Massachusetts 01003 c 2009
Cost is a disutility - Cost is a function of travel time, probability of an accident, scenery of a link. Assume that all such factors can be lumped together into a disutility. Both economists and traffic engineers work on determining travel cost functions on the links. In particular, we consider travel cost functions of a user exercised via links of the network.
Modes of Transportation www.fredrikmedia.se Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 3
In the first generation model, travel cost of users was assumed constant (depends only on the characteristics of a link and can be determined a priori - known as uncongested networks. In the second generation model, the networks are congested, that is, the user s travel cost depends on the characteristics of the link, but also on the flow on that link.
The Simplest Model The Standard Model uncongested cost (time) Often one can substitute cost with time.
Bureau of Public Roads (BPR) Cost Function c a = c 0 a [ 1 + α ( fa t a )β ] where c a : travel time on link a f a : link flow on link a c 0 a : free flow travel time t α: practical capacity of link a α, β: model parameters (typically α = 0.15, β = 4)
Modes of Transportation, Marrakech, Morocco Copyright 2006 by of Ronald Correia
Suppose now that we have 2 classes of users that perceive cost in different way. More General Model c 1 a = c 1 a (f 1 a, f 2 a ) c 2 a = c 2 a (f 1 a, f 2 a ) Can generalize the 2-class cost structure to k classes or modes. But when you make the travel choice you choose paths, not links.
Path Cost Relationship to Link Costs Let C p denote the user s or personal travel cost along path p. C p = C p (f ) = a c a (f a )δ ap, where f is a vector and { 1, if link a is contained in path p; δ ap = 0, otherwise.
Example: Simplest - Linear c a (f a ) = g a f a + h a, g a, h a > 0 and constant. g a is the congestion factor. c a (f a ) = h a - is the uncongested term.
Network Example w 1 = (x 1, x 3 ) p 1 = (a, b), p 2 = (a, c) c a (f a ) = 10f a + 5 c b (f b ) = f b + 10 c c (f c ) = 5f c + 5 Suppose that the travel demand is d w1 = 10 and that F p1 = 5, F p2 = 5. What is C p1 =? What is C p2 =?
Another type of cost is the social or total cost. In the simplest case: ĉ a (f a ) = c a (f a ) f a and if c a is linear, then: ĉ a (f a ) = (g a f a + h a ) f a = g a f 2 a + h a f a. Hence, if the user cost function on a link is linear, then the total cost is quadratic.
Network Example w 1 = (x 1, x 2 ) c a (f a ) = 10f a + 5 ĉ a (f a ) = 10fa 2 + 5f a c b (f b ) = 4f b + 10 ĉ b (f b ) = 4fb 2 b Suppose now that p 1 = (a), p 2 = (b); d w1 = 20, and F p1 = 10, F p2 = 10. What are the user and total costs on the links a and b?
The Marginal Total Cost The marginal total cost ĉ a (f a ), where ĉ a (f a ) = c a (f a ) f a f a In the uncongested model, the marginal total cost is a constant. Hence, the marginal total cost in congested networks must be an increasing function of the link flows.
The Total Network Cost Different ways expressing it. S(f ) = a ĉ a (f a ) S(f ) = a c a (f a ) f a S(f, F ) = p C p (f ) F p
For more advanced formulations and associated theory, see Professor Nagurney s Fulbright Network Economics lectures. http://supernet.som.umass.edu/austria lectures/fulmain.html