CVEN 4402 Week 7 Workshop
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
CVEN 4402 Week 7 Workshop
Convergence and UE with Elastic Demand
Objective. Learn about different formulations for the traffic assignment problem and an approach to
defining convergence.
Convergence
An important question when using our traffic assignment models is when do we stop? If we stop the model too soon, our link flow solutions may not have reached equilibrium. However, there is a computational cost associated with running more iterations (which is especially a concern for large networks or applications that involve running the model many times). Furthermore, how do we even know when the network has reached equilibrium (meaning the model has converged)?
Question 1.
Given the following network and information about the current iteration, determine whether the user equilibrium model has converged. Assume that q14 = 6.
50+x
34
Figure 1. Network for questions 1-2.
Scenario A.
Link# |
Flow |
1 |
6 |
2 |
0 |
3 |
3.8337 |
4 |
2.1663 |
5 |
3.8337 |
Scenario B.
Link# |
Flow |
1 |
4.005 |
2 |
1.995 |
3 |
2.015 |
4 |
1.990 |
5 |
4.010 |
User Equilibrium with Elastic Demand
The focus of this week’s tutorial will be on the user equilibrium model in which we consider demand to be elastic, i.e., the demand for travel (# of trips) changes with the cost of travel (travel time). We will focus on a simple example first to understand the formulation and terminology, then solve a bigger example using the definition of equilibrium, and finally solve a larger instance using MSA.
First we will introduce the formulation, and the linearized subproblem that we will solve as part of the MSA algorithm. We will not focus on how to derive these at the moment, as it goes beyond the scope of the class.
Formulation:
xa qrs
min z(x, q) = ∑ ∫ ta(业) d业 − ∑ ∫ Drs(−)1 (业) d业
a 0 rs 0
s.t.
fkrs = qrs
fkrs ≥ 0
qrs ≥ 0
The first term in the objective function is the same as in the regular UE problem. The second term, which has a minus sign, represents the elastic demand term. We don’t need to dwell on the meaning on this formulation, but the intuition behind it is that the objective function seeks to find values of x (link flows)
and q (path flows) such that the travel times are consistent for both the link travel cost function ta(x) and the inverse demand function Drs(−)1 (q).
The demand function tells us, for a specific travel time, what will be the demand; the inverse demand function Drs(−)1 (q) tells us for a given level of demand, what will be the travel time that induced it.
Question 2.
Using the same network in Figure 1, but assume a demand function of the following form:
q13 = 10 − ( − 50)
q14 = 20 − ( − 50)
Perform four iterations of MSA to find the user equilibrium with variable demand flows.
2022-09-08