A numerical comparison of three
heuristic methods for path reassignment
for dynamic user equilibrium
Ying-en Ge and Malachy Carey
16 September 2004
School of Management & Economics
Queen’s University Belfast
BT7 1NN
16 Sep 04
Transport Workshop at Queen's
1
Introduction
•
Dynamic traffic assignment (DTA)
1. Network loading,
with inflows/ assignment to spatial paths taken as given
compute new path travel times
2. Spatial path reassignment (based on travel-times from 1)
•
Three methods for path reassignment
–
–
–
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Pair-wise swapping method
Wu et al. (1998) method
Lo & Szeto (2002) method
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Pair-wise swapping method
• Step 1
At iteration n, for each time interval i, note the path with
n
current highest cost (travel time) i2
and path with
n
lowest cost i1
[ or variants of this, e.g. choose the same paths for
several time intervals, etc.]
• Step 2
For each time interval i, switch proportion sin of inflow
from higher cost to lower cost path
n
n
n
n
n
si = an (i 2  i1 ) (i1  i 2 )
where an is a chosen parameter (1 > an > 0)
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Wu et al. (1998) method
• VI formulation
• The solution of the VI formulation is obtained by
solving a series of quadratic programs below
min
f 

ip
1
 n n
n
n 2

(
f
)(
f

f
)

(
f

f
ip
ip
ip
ip ) 
 ip
2a


(1)
where a is a positive constant.
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Lo & Szeto (2002) method
Step 1. Compute gipn and vin for all i and p by:
gipn = max{0, fipn –b[ipn – (uipn –b(p fipn – di))]}
vin = uin -b(p gipn – di)
Step 2. Compute fipn and uin for all i and p
fipn+1 = fipn - tn gn (fipn - gipn)
uin+1 = uin - tn gn (uin - vin)
where tn = dn(1 -0.25bm-1), dn(0,2) such that
tn(0,1) and gn = r1/r2 with
r1 = ip(fipn - gipn)2 +b2i(gipn–gi)2
and
r2 = r1 + bi(p fipn – gipn)2
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Numerical experiments
• Scenario Settings
– 2-link network
– Network loading
– Travel demand
• Convergence measure
– Maximum absolute difference
• Numerical experiments
– Effects of parameters in three methods
– Convergence measure values over iterations, and
– Accuracy of numerical solutions
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Effects of the parameter an in the pair-wise swapping method
1.0
maximum absolute difference
an
1/(n+1)
1/sqrt(n+2)
1
0.8
0.6
Note: 1/(n+1) < 1/sqrt(n+2) < 1
0.4
0.2
0.0
0
30
60
90
120
150
iteration
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Change in convergence values over iteration
1.0
maximum absolute difference
from the path-paired method (an = 1)
from the Wu et al method (a = 2)
0.8
from the path-paired method (b = 0.50, tn = 1)
0.6
0.4
0.2
0.0
0
200
400
600
800
1000
iteration
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Minimum values of convergence measure
Convergence measure
Pair-wise swapping
method (an = 1)
Wu et al. method
(a = 2)
Lo & Szeto method
(b = 0.5 and tn =1.0)
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Maximum absolute difference
0.011116
0.058925
0.002874
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2.00
link
1
tolerance ( )
0.0625 (5%)
0.0125 (1%)
0.00625 (0.5%)
max
2
travel time
1.85
1.70
1.55
1.40
1.25
0
3
6
9
12
15
time horizon
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Stopping iterations when given tolerances for
maximum absolute difference are satisfied
tolerance for maximum
absolute difference
Pair-wise swapping method
(an = 1)
Wu et al. method (a = 2)
Lo & Szeto method
(b = 0.50 and tn = 1.00)
0.0625 0.0125 0.00625
(5%)
(1%) (0.5%)
17
112

290


305
630
783
Note: The percentages given in the round brackets after each tolerance
represent the proportion of a tolerance to the free-flow travel
time of the shorter of the two paths [1.25 minutes].
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Summary
• Preferred parameter values
• Not able to set an arbitrarily small tolerance
• Performance of three methods
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