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David Boyce and Jun Xie Northwestern University, Evanston, Illinois

Assigning User Class Link and Route Flows Uniquely to Urban Road Networks ______________________________________________________________________________. David Boyce and Jun Xie Northwestern University, Evanston, Illinois Tongji University, Shanghai, China

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David Boyce and Jun Xie Northwestern University, Evanston, Illinois

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  1. Assigning User Class Link and Route Flows Uniquely to Urban Road Networks______________________________________________________________________________ David Boyce and Jun Xie Northwestern University, Evanston, Illinois Tongji University, Shanghai, China __________________________________________________________________________________ Research in collaboration with Hillel Bar-Gera and Yu (Marco) Nie ___________________________________________________________________________________ 14th TRB Transportation Planning Applications Conference May 8, 2013

  2. Objectives of this talk Review the status of solving the traffic assignment problem (TAP); Describe the condition of proportionality to solve for unique class link flows and route flows; Analyze class link flows on the Chicago regional network with regard to: order of the assignment of the trip matrices; flows over a pair of links and alternative segments; Explore the prospects for the validation of proportionality in a real network. 2

  3. What is the Traffic Assignment Problem? • Traditionally, a method for “loading” origin-destination (O-D) flows onto a road network; • A simple behavioral model of route choice in a congested urban road network; • A method for determining link flows and travel times and costs for: • computing origin-destination-mode flows; • computing emissions from cars and trucks; • Inputs for analyses using route flows and multi-class link flows.

  4. Brief History of TAP 1952 Wardrop proposed criteria for route choice 1956 Beckmann formulated the user-equilibrium route choice problem with variable demand 1958 Algorithms for the shortest route problem 1958-75 Heuristic methods for assigning traffic to road networks 1973 Solution of Beckmann’s formulation with the link-based linear approximation method 1978 First link-based practitioner code 1990~ Research on quick-precision methods

  5. Quick-precision methods 1994 Gradient Projection (Jayakrishnan et al.) 2002 Origin-Based Assignment (Bar-Gera) 2006 Algorithm B (Dial) 2009 Projected Gradient (Florian et al.) 2010 TAPAS (Bar-Gera) Applications of TAPAS (Bar-Gera et al.)

  6. Who needs route flows or user class link flows? • Solution of TAP finds unique total link flows, but not route flows or user class links flows; • However, transportation planners require: • Link flows by user class for project evaluation; • Route flows for select link analyses to find the source of link flows by origin and destination; • This talk examines the idea of applying a condition of proportionality to determine user class link and route flows uniquely.

  7. Multiple user-equilibrium route flow solutions Car 2 100 40 40 160 1 4 D 120 60 120 Truck 3

  8. How to choose a unique route flow solution? C 100 2 40 40 160 4 D 1 120 120 60 T 3 Proportionality Condition The same proportions apply to all travelers facing a choice between a pair of alternative segments (PAS). Consider the pair of segments [1,2,4] and [1,3,4]. First segment’s proportion is 40/(40+120)=1/4.

  9. How to choose a unique route flow solution? Proportionality Condition Same proportions apply to all travelers facing a choice between a pair of alternative segments (PAS). Consider the pair of segments [1,2,4] and [1,3,4]. First segment proportion is 40/(40+120)=1/4. 25 C 2 100 40 40 160 4 D 1 75 120 120 60 T 3 For trips by Car to D, the proportion is 25/(25+75)=1/4.

  10. How to choose a unique route flow solution? Proportionality Condition The same proportions apply to all travelers facing a choice between a pair of alternative segments (PAS). Consider the pair of segments [1,2,4] and [1,3,4]. First segment’s proportion is 40/(40+120)=1/4. 15 C 2 100 40 40 160 4 D 1 120 120 45 60 T 3 For trips by Truck to D the proportion is 15/(15+45)=1/4.

  11. Basis for the proportionality condition • Reasons: • Simple, reasonable, consistent, stable, and useful. • Proportionality is testable, but not yet verified. • Are there any other practical suggestions? Implications: Any user-equilibrium (UE) route that can be used, keeping the same total link flows, should be used Traffic Assignment by Paired Alternative Segments (TAPAS), as well as two commercial software systems, now implement the condition of proportionality.

  12. Chicago regional zone system and road network

  13. Construction of trip matrices for our experiments • Person trips by mode are directly related to the number of origins and destinations and inversely related to UE travel time, distance, tolls and transit fares: • This function results in many small OD flows << 1; • Cost sensitivity parameter β = 0.20 determines mode and trip length with a moderate level of endogenous congestion; • Truck trips are exogenous, and represented in car equivalent units.

  14. Solution procedure with VISUM-LUCE • The car and truck trip matrices were assigned to UE travel time routes as precisely as possible assuming that both consider only travel time in choosing their routes (Relative Gap = 1E-8). • Class link and route flows were adjusted for proportionality with a post-processing procedure. • The result is two link flow arrays for cars and trucks: • one solution for class link flows without proportionality • uniquely determined class link flow solution with proportionality

  15. Segments 1 and 2 connecting node 8032 to node 10344 North Avenue

  16. In performing the assignment, we noticed an unexpected result in the class flows without proportionality: cars are assigned mainly on segment 1, and trucks mainly on segment 2. The order of the initial assignment was cars followed by trucks.

  17. By reversing the assignment order of the trip matrices, the class flows without proportionality were changed dramatically.

  18. North Avenue PAS: Car and truck OD flows with and without proportionality Car: With: Segt 1 - 249; Segt 2 - 114 vph Without: 338; 25 vph Truck: With: Segt 1 - 145; Segt 2 - 66 vph Without: 55; 155 vph

  19. North Avenue EB select link analysis: car and truck OD flows Car: With - 1,052 vph; Without - 1,171 vph Truck: With - 696 vph; Without - 577 vph

  20. Car and truck link flows: without proportionality Order of initial assignment: truck/car versus car/truck Car flows on 30,147 links Root mean square difference = 33.2 Truck flows on 29,810 links Root mean square difference = 33.4

  21. Car and truck link flows: with proportionality Order of initial assignment: truck/car versus car/truck Car flows on 30,149 links Root mean square difference = 4.5 Truck flows on 29,831 links Root mean square difference = 4.5

  22. Two Links and a Pair of Alternative Segments Left – Edens Exp′way link and segment Right – Cicero Avenue link and segment Among 7,266 pairs of alternative segments: • expressway link lies in 101 segments; • arterial link lies in 45 segments; • both lie in 11 pairs of alternative segments.

  23. Flows on two links without and with proportionality Flows on two segments with proportionality (TAPAS)

  24. Difference in CAR link flows: with minus without proportionality versus car link flow with proportionality

  25. Difference in TRUCK link flows: with minus without proportionality versus truck link flow withproportionality

  26. Histogram of 30,365 used links by car link flow differences

  27. Histogram of 30,365 used links by car link flow differences

  28. Conclusions so far • Differences in car and truck flows with and without proportionality are primarily less than +/–100 vph; • Differences mainly occur on link flows < 2,000 vph; • Adjustments can occur only on pairs of segments; • From related research, we know that links found in pairs of segments are primarily links with flows under 2,000 vph: if links are ranked ordered by flow, links with flow < 2,000 vph comprise 77% of links in PASs; • Additional studies are required to reach more specific conclusions.

  29. Progress on validation of proportionality • The static traffic assignment model is very simple, but it has proven to be useful for plan evaluation. “Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.” George Box and Norman Draper, EmpiricalModel-Building, Wiley, 1987, p. 74. • How to validate the condition of proportionality? • locate where alternative routes occur; • identify a high volume pair of segments; • observe segment flows over many days to identify whether the pattern is stable.

  30. Test for proportionality of observed segment flows

  31. Conclusions • The charts show substantial differences with respect to class link flows by order of assignment, and with and without proportionality; • The proportionality condition can be applied to determine class link flows and route flows uniquely. Without proportionality, these flows are not comparable among scenarios. • Initial validation tests of proportionality appear promising; next we need to examine whether these patterns repeat daily or weekly.

  32. Number of O-D pairs without and with proportionality by order of assignment

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