The Evolution of Transport Planning

The Evolution of Transport Planning

On Proportionality and Uniqueness in Equilibrium Assignment Michael Florian Calin D. Morosan

Background Several bush-based algorithms for computing equilibrium assignments can obtain very finely converged flows Bar Gera (2002) Dial (2006) Gentile (2012) All require post processing to obtain unique path flows by maximizing the entropy of path flows TAPAS method (Bar-Gera, 2010) exhibits proportionality and hence uniqueness of path flows

Motivation Bush-based methods are not particularly efficient for solving multi-class network equilibrium models usually require cycling over the classes cannot be efficiently parallelized The motivation of this investigation is the need for a more efficient multi-class traffic assignment that can obtain unique path and class flows

Findings The bi-conjugate variant of the linear approximation method solves multi-class network equilibrium models efficiently The path and class flows exhibit proportionality and hence uniqueness as the relative gap decreases The linear approximation method is not as efficient but also shares the near uniqueness properties

Path Flows and Proportionality Two classes of traffic: 1->8 100 trips; 2->8 60 trips Total link flow indicated on the links 1 100 40 5 40 160 3 2 60 160 4 7 120 6 120 8

Path Flows and Proportionality These path flows are proportional 25/75=15/45=40/120=1/3 1 100 25;15 5 25;15 100;60 3 2 60 100;60 4 7 75;45 6 75;45 8

Path Flows and Proportionality These path flows are not proportional 40/60 is not equal to 0/60 1 100 40;0 5 40;0 100;60 3 2 60 100;60 4 7 60;60 6 60;60 8

Available Data for Comparison A two-class network equilibrium problem on the Chicago test network Three sets of origin-destination matrices cars and trucks three different levels of congestion Flows obtained with the TAPAS algorithm run to a relative gap of 10-12

The Chicago Test Network 1,790 zones 11,192 nodes 39,018 links two classes cars and trucks truck prohibition on 563 links

Algorithms Used in the Computations 1 A multi-threaded implementation of the classical linear approximation method, as implemented in Emme 4.1 Standard traffic assignment 2 A multi-threaded, conjugate direction linear approximation method* as implemented in Emme 4.1 SOLA traffic assignment The convergence of the second-order method is one order of magnitude better with reasonable computation times for relative gaps of 10-5 and 10-6 *As described in Mitradjieva, M. and Lindberg, P.O. (2013)

Car Flows Comparison TAPAS (10-12 relative gap) and SOLA (10-6 relative gap) have nearly identical car flows SOLA traffic assignment runs in 12.7 minutes on a 16core, 2.9 GHz Xeon processor (32 threads)

Truck Flows Comparison TAPAS (10-12 relative gap) and SOLA (10-6 relative gap) have nearly identical truck flows SOLA traffic assignment runs in 12.7 minutes on a 16core, 2.9 GHz Xeon processor (32 threads)

Paired Alternative Segments SOLA - Relative Gap 10-6 Ratio of flow on paired alternative segments is ~200/233 = 0.8584 200 233

Paired Alternative Segments SOLA Car flows O-D pairs contributing flow to each leg of the pair of alternative segments O-D demand are obtained with a selectlink analysis Slope is ~200/233 =.8584

Paired Alternative Segments SOLA Truck flows O-D pairs contributing flow to each leg of the pair of alternative segments O-D demand are obtained with a selectlink analysis Slope is ~200/233 =.8584

Comparison using a log-log scale A comparison of the flow differences between SOLA flows at different levels of convergence TAPAS flows at 10-12 relative gap

Car Flows Comparison: SOLA vs TAPAS SOLA 10-3 Relative Gap

Car Flows Comparison: SOLA vs TAPAS SOLA 10-5 Relative Gap

Car Flows Comparison: SOLA vs TAPAS SOLA 10-7 Relative Gap RMSE=0.89

Truck Flows Comparison: SOLA vs TAPAS SOLA 10-3 Relative Gap

Truck Flows Comparison: SOLA vs TAPAS SOLA 10-5 Relative Gap

Truck Flows Comparison: SOLA vs TAPAS SOLA 10-7 Relative Gap

Car Flows Comparison More congestion TAPAS (10-12 relative gap) and SOLA (10-6 relative gap) have nearly identical car flows SOLA traffic assignment runs in 17.7 minutes on a 16-core, 2.9 GHz Xeon processor (32 threads)

Truck Flows Comparison More congestion TAPAS (10-12 relative gap) and SOLA (10-6 relative gap) have nearly identical truck flows SOLA traffic assignment runs in 17.7 minutes on a 16-core, 2.9 GHz Xeon processor (32 threads)

Car Flows Comparison Even more congestion TAPAS (10-12 relative gap) and SOLA (10-6 relative gap) have nearly identical car flows SOLA traffic assignment runs in 32 minutes on a 16-core, 2.9 GHz Xeon processor (32 threads)

Truck Flows Comparison Even more congestion TAPAS (10-12 relative gap) and SOLA (10-6 relative gap) have nearly identical truck flows SOLA traffic assignment runs in 32 minutes on a 16-core, 2.9 GHz Xeon processor (32 threads)

Back to Linear Approximation Interesting to see how the linear approximation method performs Standard traffic assignment was run for 15,000 iterations to achieve a relative gap of 2.3*10-6 (not recommended for use in practice ) 3.5 hours with the linear approximation method on a 16-core, 2.9 GHz Xeon processor (32 threads)

Car Flows Comparison TAPAS vs Linear Approximation RMSE=1.35

Truck Flows Comparison TAPAS vs Linear Approximation

Conclusions Results obtained with the linear approximation method and the second-order method (SOLA) exhibit near class uniqueness and proportionality The SOLA traffic assignment, a multi-threaded biconjugate variant of the linear approximation method, provides an attractive and computationally efficient method for solving multi-class assignments to fine convergence