Lecture 3 Cost Structure

Transcription

1 Lecture 3 Dr. Anna Nagurney John F. Smith Memorial Professor Isenberg School of Management University of Massachusetts Amherst, Massachusetts c 2009

2 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.

3 Modes of Transportation Dr. Anna Nagurney FOMGT 341 Transportation and Logistics - Lecture 3

4 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.

5 The Simplest Model The Standard Model uncongested cost (time) Often one can substitute cost with time.

6 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)

7 Modes of Transportation, Marrakech, Morocco Copyright 2006 by of Ronald Correia

8 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.

9 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.

10 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.

11 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 =?

12 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.

13 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?

14 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.

15 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

16 For more advanced formulations and associated theory, see Professor Nagurney s Fulbright Network Economics lectures. lectures/fulmain.html