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Changing assignment algorithms: the price of better convergence
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TRB Planning ApplicationsMay 2009, Houston,TX
Changing assignment algorithms: the price of better
convergence
Michael Florian and Shuguang He
INRO
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TRB Planning ApplicationsMay 2009, Houston,TX
Contents
The need for better convergenceNew assignment algorithm with pathsResults on some test problemsUniqueness considerationsIllustration of non unique resultsConclusions
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TRB Planning ApplicationsMay 2008, Houston,TX
The need for better convergence
- The linear approximation (Frank-Wolfe) algorithm is the most commonly used traffic assignment method
- It has the advantage of requiring small amount of RAM- It is easy to explain and is quite robust- It has the drawback of requiring a large number of
iterations to obtain a very fine solution- Any path analyses require the re-running of the
assignment to obtain the desired results or storing the paths (a very large number) or storing a very large number of paths for a limited number of iterations
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TRB Planning ApplicationsMay 2008, Houston,TX
The need for better convergence
- Certain applications require fine solutions in order to compare scenarios and carry out economic evaluation;
- The current generation of computers provide plenty of RAM and multiple processors;
- This opens up the possibility of implementing faster converging algorithms that require larger amounts of RAM; also it is possible to store and manipulate paths;
- It also opens up the possibility of parallel implementations of classical methods.
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TRB Planning ApplicationsMay 2008, Houston,TX
The need for better convergence
- The method that we chose to implement is an adaptation of the projected gradient method in the space of path flows;
- O-D pairs are considered sequentially with projected gradient descent directions;
- It provides finer solutions in much shorter time than that required by the linear approximation method;
- Path analyses can be carried out quickly and iterative equilibration algorithms can benefit from the information contained in a previous assignment (warm start)
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TRB Planning ApplicationsMay 2008, Houston,TX
New Equilibrium Traffic Assignment with Path Flows
1. Compute the average cost of all used paths (by O-D pair)
2. Reduce the flow of paths that have a larger cost than the average and
3. Increase the flow on paths that have a smaller cost than the average
4. Just keep the paths with positive flow5. Add a path if it is shorter than the current
equilibrated solution
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TRB Planning ApplicationsMay 2008, Houston,TX
Contents
The need for better convergenceNew assignment algorithm with pathsResults on some test problemsUniqueness considerationsIllustration of non unique resultsConclusions
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TRB Planning ApplicationsMay 2008, Houston,TX
New Equilibrium Traffic Assignment with Path Flows
- We compared the performance of the algorithm on several single and multi-class equilibrium assignments
- The convergence criterion used for these tests is a measure of Relative Gap (RelGap) for a current iteration:RelGap = Total travel time – Total travel time on
shortest pathTotal travel time
- Values of RelGap of the order of 10-5 or 10-6 are considered to be very good
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TRB Planning ApplicationsMay 2008, Houston,TX
Results on some test problems
- The platforms used for some of these tests are Dell desktop PC‘s based Intel processors at 2.4 to 3.00 GHz;
- Compared algorithms: 1 linear approximation method (Frank-Wolfe)
1000 iterations; 2 projected gradient algorithm.
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal Regional Planning Network
3 classes 1,425 zones13,491 nodes33,540 links
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal Regional Planning Network
3 classes 1,425 zones13,491 nodes33,540 links
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Performance Results
TRB Planning ApplicationsMay 2008, Houston,TX
Thanks to Pierre Tremblay, MTQ
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Using saved paths for new assignmnet
TRB Planning ApplicationsMay 2008, Houston,TX
Using 2006 assignment for 2015 assignment – about 10% increase in demand ( all to E-6)
2006 assignment - 22.71 min.2015 assignment with saved paths - 4.91 min
2015 assignment - 29.81 min
Demand Forecast by Mode - MTQ
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MAG Regional Planning Network
TRB Planning ApplicationsMay 2008, Houston,TX
21 modes 2041 centroids12938 regular nodes 39731 directional links 1896 turn table entries
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MAG Regional Planning Network
TRB Planning ApplicationsMay 2008, Houston,TX
21 modes 2041 centroids12938 regular nodes 39731 directional links 1896 turn table entries
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TRB Planning ApplicationsMay 2008, Houston,TX
3.12 Ghz – 8 processors at MAG – thanks to Vladimir Livshitz
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TRB Planning ApplicationsMay 2008, Houston,TX
RTA Sydney, Australia Test Network
4 modes 1155 centroids12893 regular nodes34551 directional links 8415 turn table entries
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TRB Planning ApplicationsMay 2008, Houston,TX
RTA Sydney, Australia Test Network
4 modes 1155 centroids12893 regular nodes34551 directional links 8415 turn table entries
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TRB Planning ApplicationsMay 2008, Houston,TX
Performance Results
Sydney Highway Assignment
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
0 10 20 30 40 50 60 70
minutes
rela
tiv
e g
ap
FW
PG
Thanks to Matthew Wilson, RTA
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TRB Planning ApplicationsMay 2008, Houston,TX
Portland Test Network
1,260 zones 8,794 nodes26,091 links 7,010 turns
4 classes of trafficSOVHOVHeavy Trucks Medium Trucks
2000 Base South Corridor
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TRB Planning ApplicationsMay 2008, Houston,TX
Portland Test Network
1,260 zones 8,794 nodes26,091 links 7,010 turns
4 classes of trafficSOVHOVHeavy Trucks Medium Trucks
2000 Base South Corridor
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TRB Planning ApplicationsMay 2008, Houston,TX
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
0 20 40 60 80 100 120 140 160
rela
tiv
e g
ap
minutes
Portland
F&W AssignmentProjected Gradient
Performance Results
Thanks to Metro Portland
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TRB Planning ApplicationsMay 2008, Houston,TX
SFCTA Test Network
4 classes of traffic 2266 centroids20490 regular nodes61615 directional links 9461 turns
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Performance Results
TRB Planning ApplicationsMay 2008, Houston,TX
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
0 100 200 300 400 500
rela
tiv
e g
ap
minutes
SFCTA PM Highway AssignmentIntel 3 GHz PC
Emme 521: FW
Emme 525: path
Thanks to Elisabeth Sall
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TRB Planning ApplicationsMay 2008, Houston,TX
Contents
The need for better convergenceNew assignment algorithm with pathsResults on some test problemsUniqueness considerationsIllustration of non unique resultsConclusions
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TRB Planning ApplicationsMay 2008, Houston,TX
Uniqueness considerations
- A little appreciated fact is that the equilibrium assignment does not guarantee unique paths or class flows;
- But, running the same code produces the same results so non uniqueness of certain results is a property that was not all that “visible” in practice;
- Non uniqueness is a very “elusive” property if one works with the same code.
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TRB Planning ApplicationsMay 2008, Houston,TX
Uniqueness considerations
- Different assignment algorithms produce slightly different class flows so results do change; the question is by how much;
- Regardless of the algorithm used, the only unique results are the total flows and the class impedances
- This remains true if one uses a slightly different implementation of the F&W algorithm so switching F&W implementations would change the results somewhat as well.
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TRB Planning ApplicationsMay 2008, Houston,TX
How different are the results?
- The results that may change are all related to the analysis of paths resulting from the assignment;
- These include; select link and generalized select link analyses, pure times vs. generalized cost, average tolls paid, sub-area traversal matrices, class flows,….. - Regardless of the algorithm used, the only unique results are
the total flows and the class impedances
- The implication is that in “feedback” model equilibration one should use schemes that average class impedances and not class volumes!
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TRB Planning ApplicationsMay 2008, Houston,TX
“feedback” equilibration and evaluation
- The averaging scheme used should rely on unique results: total link flows or class impedances should be used:
- This ensures compatibility with mode and destination choice models and near compatibility with results obtained when the assignment is carried out with the linear approximation method:
- Economic evaluation methods based on changes in accessibility times (impedances) will be nearly the same as those obtained with assignments done with the linear approximation method.
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TRB Planning ApplicationsMay 2008, Houston,TX
Contents
The need for better convergenceNew assignment algorithm with pathsResults on some test problemsUniqueness considerationsIllustration of non unique resultsConclusions
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TRB Planning ApplicationsMay 2008, Houston,TX
Chicago Test Network
1790 centroids11192 regular nodes39018 directional links
We carried out several select link assignments to see the differences in link flows
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TRB Planning ApplicationsMay 2008, Houston,TX
Select Link Flows
Projected gradient flows E-6
Linear aproximation flows E-4
Scale=75
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TRB Planning ApplicationsMay 2008, Houston,TX
Select Link Flow Differences
3 trips
Scale=1
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TRB Planning ApplicationsMay 2008, Houston,TX
Select Link Flows
Linear approximation flows E-4
Projected gradient flows E-6
Scale=75
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TRB Planning ApplicationsMay 2008, Houston,TX
Select Link Differences
1 trip
Scale=1
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal Regional Planning Network
3 classes 1,425 zones13,491 nodes33,540 links
We compared the class flows for the Montreal assignment:
Linear Approximation at 10^-4 relative gap
vs.
Projected Gradient at 10^-6 relative gap
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal network : Total Flows
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal network : Class 1 SOV
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal network: Class 2 Light trucks
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TRB Planning ApplicationsMay 2008, Houston,TX
Montreal network: Class 3 heavy trucks
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Seattle Regional Planning Model
TRB Planning ApplicationsMay 2008, Houston,TX
15 modes 30 transit vehicle types 1155 centroids 834 transit lines 5888 regular nodes 25856 transit line segments20633 directional links 16864 turn table entries
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Seattle Regional Planning Model
TRB Planning ApplicationsMay 2008, Houston,TX
15 modes 30 transit vehicle types 1155 centroids 834 transit lines 5888 regular nodes 25856 transit line segments20633 directional links 16864 turn table entries
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TRB Planning ApplicationsMay 2008, Houston,TX
Seattle Regional Planning Model
- These are the results of comparing the
results of the model equilibration after replacing the linear approximation algorithm with the projected gradient algorithm;
- The results were provided to us by PSRC staff.
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TRB Planning ApplicationsMay 2008, Houston,TX
PSRC Travel Model Documentation(for Version 1.0)
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TRB Planning ApplicationsMay 2008, Houston,TX
Comparison of Model Equilibration Results
20.4 hrs vs. 10 hrsResults carried out by PSRC planning staff and presented with the permission of PSRC
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TRB Planning ApplicationsMay 2008, Houston,TX
Comparison of Model Equilibration Results
20.4 hrs vs. 10 hrs of computing times (6 “feedback” loops on Intel 2.4 Ghz)
Result differences of the order of 0.2% to 0.5% .
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TRB Planning ApplicationsMay 2008, Houston,TX
Comparison of Model Equilibration
Results carried out by PSRC planning staff and
presented with the permission of PSRC
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TRB Planning ApplicationsMay 2008, Houston,TX
Comparison of Model Equilibration Results
20.4 hrs vs. 10 hrs of computing times
(6 “feedback” loops on Intel 2.4 Ghz)
Result differences of the order of 0.2%
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Total Flows Comparison
Sydney Users' Conference
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SOV Impedances
Sydney Users' Conference
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SOV Travel Time Distribution
Sydney Users' Conference
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TRB Planning ApplicationsMay 2008, Houston,TX
Contents
The need for better convergenceNew assignment algorithm with pathsResults on some test problemsUniqueness considerationsIllustration of non unique resultsConclusions
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TRB Planning ApplicationsMay 2008, Houston,TX
It is worth paying the “price” for faster convergence!