changing assignment algorithms: the price of better convergence

TRB Planning ApplicationsMay 2009, Houston,TX

Changing assignment algorithms: the price of better

convergence

Michael Florian and Shuguang He

INRO


Contents

The need for better convergenceNew assignment algorithm with pathsResults on some test problemsUniqueness considerationsIllustration of non unique resultsConclusions


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



- 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.



- 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)


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


Contents



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


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.


Montreal Regional Planning Network

3 classes 1,425 zones13,491 nodes33,540 links

Performance Results


Thanks to Pierre Tremblay, MTQ

Using saved paths for new assignmnet


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

MAG Regional Planning Network


21 modes 2041 centroids12938 regular nodes 39731 directional links 1896 turn table entries


3.12 Ghz – 8 processors at MAG – thanks to Vladimir Livshitz


RTA Sydney, Australia Test Network

4 modes 1155 centroids12893 regular nodes34551 directional links 8415 turn table entries


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


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


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


SFCTA Test Network

4 classes of traffic 2266 centroids20490 regular nodes61615 directional links 9461 turns

Performance Results


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


Contents



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.


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.


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!


“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.


Contents



Chicago Test Network

1790 centroids11192 regular nodes39018 directional links

We carried out several select link assignments to see the differences in link flows


Select Link Flows

Projected gradient flows E-6

Linear aproximation flows E-4

Scale=75


Select Link Flow Differences

3 trips

Scale=1


Select Link Flows

Linear approximation flows E-4

Projected gradient flows E-6

Scale=75


Select Link Differences

1 trip

Scale=1


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


Montreal network : Total Flows


Montreal network : Class 1 SOV


Montreal network: Class 2 Light trucks


Montreal network: Class 3 heavy trucks

Seattle Regional Planning Model


15 modes 30 transit vehicle types 1155 centroids 834 transit lines 5888 regular nodes 25856 transit line segments20633 directional links 16864 turn table entries


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.


PSRC Travel Model Documentation(for Version 1.0)


Comparison of Model Equilibration Results

20.4 hrs vs. 10 hrsResults carried out by PSRC planning staff and presented with the permission of PSRC



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% .


Comparison of Model Equilibration

Results carried out by PSRC planning staff and

presented with the permission of PSRC



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%

Total Flows Comparison

Sydney Users' Conference

SOV Impedances


SOV Travel Time Distribution



Contents



It is worth paying the “price” for faster convergence!

changing assignment algorithms: the price of better convergence

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