CVEN 4402 Week 9 Workshop
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CVEN 4402 Week 9 Workshop
System Optimal Assignment and Tolling
Objective. Learn about System Optimal Assignment
System Optimal (SO) Assignment
Up to this point we have focused on user equilibrium assignment, a model which attempts to characterize the behavior of users in a road network. System Optimal (SO) Assignment, on the other hand, focuses on finding the assignment of users to routes that minimizes total system travel time. While SO is not helpful in characterizing actual road conditions, it does help us develop an idea of the best possible scenario in a traffic network. Furthermore, the methods used to solve this problem extend to more general problems which focus not on describing (like UE), but rather prescribing optimal solutions.
Consider a transport network G = (N, A) . The formulation for the SO Assignment problem is given below:
min TSTT = min xata(xa)
s.t.
xa = ∑ ∑ ∑ fkrs 6a(r)k(s) ∀a ∈ A
r∈N s∈N k∈ΠTS
∑ fkrs = qrs ∀ T, s ∈ N
k ∈ΠTS
fkrs ≥ 0 ∀ T, s ∈ N, ∀ k ∈ Πrs
In contrast to the UE formulation, the formulation for the SO problem is relatively intuitive: in each link, there are xa units each with a travel time of ta(xa) . Therefore, the total travel time by all units on that link is xata(xa), and we want to minimize the total time travelled by all units on all links. The constraints are the same as that of the UE problem.
Question 1
Consider the 2-node, 2-link network shown below with a demand from node 1 to 2 of 20 units. Determine the User Equilibrium conditions and the System Optimal conditions. Will a tolling system improve the network performance? If so, what toll would be suitable? (Assume a value of time of $1/unit of time.)
Question 2
Consider the network shown below with demand from node 1 to 4 of 6 units . Perform 3 iterations of
MSA for SO flow assignment.
10x12 2 50+x24
50+x13 3 10x34
2022-09-08