1. Transportation Planning Asian Institute of Technology
Traffic Assignment
Transportation Planning
Asian Institute of Technology

2. Contents Basic Concepts Traffic Assignment Methods
All-or-Nothing Assignment
Stochastic Methods
Congested Assignment

3. Basic Concepts Introduction
The equilibrium point between demand and supply defines the price at which goods will be exchanged.
Supply side is made up of road (transport) network, S(L,C)
Cost C is a function of distance, free-flow speed, capacity, and speed-flow relationship
Demand side is made up of the number of trips by O-D pair and mode under a given level of service.

4. Basic Concepts Introduction
In private transport mode, a main element defining LOS is travel time.
In public transport mode, addition consideration should be given to routes, capacity, frequency and other random characteristics.
At a higher level resulting flow may affect choices of mode, destination and time of day for travel. (System Equilibrium)

5. Basic Concepts
Definitions and Notation Tijr = number of trips from i and j via route r Va = the flow on link a C(Va) = the cost-flow relationship for link a c(Va) = the actual cost for a particular level of flow Va c(Va) = 0 is free-flow cost cijr = cost of traveling from i and j via route r dijra = 1 if link a is on path r from i to j; 0 otherwise superscript n shows iteration #, and superscript * indicates optimum value.

6. Basic Concepts Speed-Flow and Cost-Flow Curve Speed, S (km/h)
Travel Time (minutes/km)
Vmax
Vmax
Flow, V
Flow, V

7. Basic Concepts Speed-Flow and Cost-Flow Relationship
The cost on a link a is a function of all the flows V in the network.
Ca = Ca({V})
Considering long links to simplify the problem (separable):
Ca = Ca(Va)
The function should be
Realistic - Allowing overloading
Non-decreasing - Easy to transfer
Continuous and differentiable

8. Basic Concepts Total operating cost = VaCa(Va)
Marginal cost contributed by the marginal addition of a vehicle to the stream
(10.3)
BPR Function
(10.6)
UK Department of Transport divides speed-flow into three regions: free-flow, moderate flow, and over-capacity.

9. Traffic Assignment Methods
Introduction
Primary objectives
To obtain aggregate network measures
To estimate zone-to-zone travel costs
To obtain reasonable link flow & identify congested links
Congestion will be studied using peak-hour trip matrices

10. Traffic Assignment Methods
Route Choice
Most common approximation considers only time and monetary costs (proportional to travel distance)
Different drivers may choose different routes when traveling between the same two points
Different individual perceptions
Congestion effects affect shorter route first and attract more traffic to initially less attractive routes

11. Traffic Assignment Methods
Example Demand 3,500 from A to B Capacity (through)= 1,000 vph Capacity (bypass) = 3,000 vph A B
Stochastic Effects? No
Yes No
All-or-nothing
pure stochastic
Capacity Restraint?
Wardop’s equilibrium
Stochastic user equilibrium
Yes

12. Traffic Assignment Methods
Tree Building
Identifies a set of routes which might be considered attractive to drivers.
All dA = ∞ except dS = 0
set L = loose-end containing all nodes
Set origin S as current node A then
Check each link (A, B). If dA + dA,B < dB then dB = dA + dA,B
Remove A from L then select another node until L is empty.

13. All-or-Nothing Assignment
Example A 5 6 2 10 4 8 4 3 6 4 C 3 4 8 10 5 2 3 2 D B
A-C = 400 B-C = 300
A-D = 200 B-D = 100

14. All-or-Nothing Assignment
Example A 5 6 2 10 4 8 4 3 6 4 C 3 4 8 10 5 2 3 2 D B

15. All-or-Nothing Assignment
Example A
600
400
400
200
400
200 C
200
200 D B
200

16. All-or-Nothing Assignment
Example A
600
400
400
200
400
300
500 C
300
300
200
100
300 D B
300
Does not reflect real situation
Act like desired lines

17. Stochastic Methods Varied perceptions of costs and other attributes.
Second-best routes must be identified. The number of second-best routes may be very large.
Simulation-based method
Proportion-based method

18. Stochastic Method Simulation-based Method
Relies on Monte Carlo Simulation
Distribution of perceived costs are assumed independent.
Drivers choose the route that minimizes their perceived route costs.
Proportion of drivers
Mean link cost

19. Stochastic Method Simulation-based Method
Split population traveling between each OD pair into N segments. (Tij/N trips)
n = 0
n = n+1
For each O-D compute perceived cost for each link using random numbers to sample link cost. Build a minimum path from i to j and assign Tij/N
If n = N stop, otherwise go to 2.

20. Congested Assignment Wardrop’s Equilibrium
No user can improve the trip travel time (cost) by unilaterally change changing the routes
Traffic arranges itself in congested networks such that all used routes between O-D pair have equal and minimum costs while all unused routes have greater of equal costs.

21. Congested Assignment Example Time Ct = 10 + 0.02Vt Cb = 15 + 0.005Vb
Town
Bypass
Time
Time
Flow, V
Flow, V
Ct = Vt
Cb = Vb
Vb + Vt = V; Vb = 0.8V – 200 = 1,400

22. Congested Assignment Wardrop’s Equilibrium Second Principle:
under social equilibrium conditions, traffic should be arranged in congested networks in such a way that the average or total travel cost is minimized.
This become known as social equilibrium or system optimum.

23. Congested Assignment Wardrop’s Equilibrium Complexity City suburb
Route 1:
x1 = 3 km
v1f = 45 km/hr
t1f = (3x45)/60 = 4 min
t1 = 4 + (q1/1000)2 min
Route 2:
x2 = 6 km
v2f = 60 km/hr
t2f = (6x60)/60 = 6 min
t2 = 6 + 2(q2/500) min

24. Congested Assignment
Wardrop’s Equilibrium Complexity

25. Congested Assignment Incremental Assignment
Traffic does not all come at the same time, but gradually spread over a given time period.
Incremental loading traffic into the network is more realistic.
Easy to program and solve.
Represents a build up of traffic congestion during a short peak period.

26. Congested Assignment Incremental Assignment
Select an initial set of current link costs. Set Va = 0 and select fraction pn such that (Typical value 0.4, 0.3, 0.2, 0.1)
Build the set of minimum cost trees using current costs. Set n = n + 1
Load Tn = pnT (all-or-nothing) with a set of auxiliary flow Fa. Calculate flow on each link as:
Calculate a new set of current link cost based on the flows Van
Continue until all fractions of T is assigned.

27. Congested Assignment Example
Consider again the problem of the two routes, town centre and bypass, of previous example. We split the demand of 2000 trips into four increments of 0.4, 0.3, 0.2 and 0.1 of this demand, i.e. 800, 600, 400 and 200 trips. At each increment we calculate the new travel costs using equations
The following table summarizes the results of this algorithm

28. Congested Assignment Example V = 2000 Ct = 10 + 0.02Vt
Cb = Vb N
Increment
Flow Town
Cost Town
Flow Bypass
Cost Bypass 10 15 1
800 26 2
600 18 3
400
1000 20 4
200
1200 21

29. Congested Assignment Method of Successive Averages (MSA)
Iterative algorithm developed to overcome the problem of overloading on low- capacity links.
Heuristics approach.
Provide close solution to real equilibrium.

30. Congested Assignment Method of Successive Averages (MSA)
Initialize all Va = 0 and set n = 0
Build a set of minimum cost tree with the current costs; set n = n + 1
Load the whole matrix T (all-or-nothing) then obtain auxiliary flow Fa.
Calculated the current flows
where 0 ≤ f ≤ 1
Calculate a new set of current link costs based on flows Van. Stop when converged.

31. Congested Assignment Method of Successive Averages (MSA) Iteration f
Flow town
Cost town
Flow bypass
Cost bypass 1 F
2000 Vn 50 15 2
1/2
1000 40 20 3
1/3
667
23.3
1333
21.7 4
1/4
500
1500
22.5 5
1/5
800 26
1200 21

32. Congested Assignment Method of Successive Averages (MSA) Iteration f
Flow town
Cost town
Flow bypass
Cost bypass 6 F
2000 Vn
1/6
667
23.3
1333
21.7 7
1/7
572
21.4
1428
22.1 8
1/8
750 25
1250
21.2 9
1/9 10
1/10
600 22
1400

33. Assignment #9
Providing two origins, A and B, and two destinations, L and M, with the following trip interchanges
A − L = 600 A − M = 400
B − L = 300 B − M = 400
Find the shortest path and the link volumes by all-or-nothing assignment 5 11 13 2 A C F I L 15 9 12 7 7 9 9 11 7 3 B D G J 12 11 7 9 9 7 3 2 E H K M

34. Assignment #9
Consider the following network where there are 100 vehicles per hour travelling from A to D and 500 from B to D. The travel time versus flow relationships are depicted in the figure in minutes and the flow q in vehicles per hour.
Use an incremental loading technique with fractions 40, 30, 20 and 10% of the total demand to obtain an approximation to equilibrium assignment.
t = q
t = q A C D
t = q
t = q
t = q A C