Characteristics of Routes in a Road Traffic Assignment

Transcription

1 Characteristics of Routes in a Road Traffic Assignment by David Boyce Northwestern University, Evanston, IL Hillel Bar-Gera Ben-Gurion University of the Negev, Israel at the PTV Vision Users Group Meeting Philadelphia, May 15-16, 2008

2 Questions about routes and link flows in a road traffic assignment What are the basic assumptions and properties of a road traffic assignment? How many routes are typically found in the solutions of an assignment model, and how are they related to the level of congestion? What are the properties of the route flows, and how do they affect route flow-based analyses, including microsimulation? What effects do approximate solutions have on the route and link flow results?

3 Outline of this talk Review of principles of route choice model solved by a road traffic assignment method; Properties of route counts for three OD matrices for the Chicago region; Properties of route flow for a typical origin; Conversion of continuous route flows to discrete vehicle trajectories for VISSIM; Some questions about approximation solutions for future study.

4 What is road traffic assignment? Traditionally, a procedure for loading an OD matrix onto the links of a road network; A model of route choice behavior over a road network, describing an equilibrium of users route choices and OD travel times; A mathematical model solved by an iterative solution procedure, or algorithm; A critical step in the Sequential Procedure to determine OD travel times, which influence OD and mode choices through feedback.

5 Traffic assignment assumptions - 1 A substantial period, compared to the duration of trips, during which network congestion is relatively constant (high or low), such as a period of one or two hours. Such models are static, and lack dynamic attributes. An input OD matrix of the continuous flow of vehicles per hour from origins to destinations. A road network description of nodes, links and link travel time - flow functions that increase indefinitely as flow increases without limit.

6 100 Link 1: Travel Time vs. Flow Link 1 Travel Time (minutes) Link ,000 1,500 2,000 2,500 3,000 3,500 4,000 Link 1 Flow (vehicles/hour)

7 Traffic assignment assumptions - 2 The travel time function for each link is typically defined on its own flow, ignoring effects of flows of opposing or conflicting links; however, the link s nominal capacity may reflect the effect of intersecting links. Link capacity is not a strict upper limit on flow. Drivers have perfect information about travel times (deterministic), or perfect information plus a random perception error (a limited stochastic case); models with truly stochastic travel times are much more complex.

8 User-equilibrium (UE) principles Deterministic: For each origin-destination pair of zones, all used routes have equal travel times, and no unused route has a lower travel time. (Wardrop-Beckmann) Stochastic: For each origin-destination zone pair, all used routes have equal perceived travel times, and no unused route has a lower perceived travel time.

9 Basic two link problem - A graphical analysis - Consider two one-way links with fixed inflow and outflow in vehicles per hour; Determine the flows on each link by graphically equating the link times to solve for the user-equilibrium flows; This graphical solution can be related to a mathematically formulated problem.

10 d Link 1 d (vehicles/hour) Node A Link 2 Two-Link Example Node B

11 Link Travel Times vs. Flows Link 1 80 Link 1 Travel Time (minutes) Link 2 Equilibrium Travel Time = Link 2 Travel Time (minutes) User Equilibrium Flows = (1,522; 2,478) ,000 1,500 2,000 2,500 3,000 3,500 4,000 0 Link 1 Flow (vehicles/hour) Link 2 Flow = Link 1 Flow

12 Properties of deterministic UE models Link flows are uniquely determined; Route flows of OD pairs served by multiple routes are not uniquely determined due to swapping possibilities; Routes, or sequences of links, are uniquely determined, since link costs are unique functions of link flows; Multi-class link flows are also not unique; Solution attributes may depend greatly on the precision of algorithm convergence.

13 Measuring solution effectiveness Only two-link problems can be solved graphically or algebraically; generally, others must be solved by some iterative method. To monitor the improvement in the solution as it approaches the user equilibrium (optimum), a measure of the solution s quality is needed. One measure is the objective function value being minimized, the sum of the area under the link travel time functions; however, this is not a good measure since it changes little as the solution approaches the equilibrium.

14 Measuring the solution (con d) A better measure is Total Excess Cost (Gap), where cost, or impedance, is a weighted sum of travel time, travel distance, tolls, etc. For each OD pair, the OD Excess Cost is the difference between each route s time and the time of the shortest route, weighted by the route s flow, and summed over all routes. Total Excess Cost is the sum of OD Excess Cost over all OD pairs, Often the TEC or Gap is standardized by the value of the objective function, called the Relative Gap, to facilitate comparisons across models or networks.

15 The definition of Total Excess Cost suggests the cost and flow of every used route is needed for its computation. In fact, the TEC is found for any solution to the problem by computing an all-or-nothing assignment on the link costs from that given solution. Using the all-or-nothing flows and the given link costs, compute the total cost and call it the Minimum Cost. Total Excess Cost = Cost of solution minus Minimum Cost

16 Monitoring the convergence Total Excess Cost, or Gap, approaches zero as the assignment solution improves; The process of the solution approaching zero Total Excess Cost is called Convergence; Some practitioners use the term Closure, implying the true equilibrium solution was reached. For most problems, the true equilibrium is not actually reached; So, we need criteria for judging when the solution has adequately converged.

17 Convergence based on Excess Cost When to stop the solution procedure, and accept the solution as having satisfactorily converged, depends on: How precise an answer is desired? Differences among scenarios should reflect the scenarios, not error in convergence; Are clock time and computing resources available? At what point are the link flows stable? Stability is a matter of judgment.

18 Computational studies Three OD matrices were prepared, and their precisely converged routes were computed; These matrices were assigned with VISUM to study its solution properties and implications for exporting route flows to VISSIM; The trip distribution function is: d c pqm pqm = = A p time B q pqm exp + ( β c ) 0.15 length min mile pqm pqm 0.05 toll min $ ( min) = ( min) + ( mile) + () $ + pqm

19 Only the value of the cost sensitivity parameter β was varied among the three matrices: high cost sensitivity resulting in shorter trips medium cost sensitivity low cost sensitivity resulting in longer trips The models were solved for the Chicago regional network (1790 zones; 39,000 links) using equilibrium travel costs; Histograms of the OD flows were plotted to understand their basic properties.

20 Fig. 1 Zone Map of the Chicago Region 1777

21 Fig. 2 Roadway Network of the Chicago Region

22 2,000,000 1,500,000 Frequency Distribution of InterZonal OD Cell Values CS 05 CS 10 CS 20 Number of Cells 1,000, , E E E E E E E E E E E E+02 Category Value - Lower Limit

23 10,000,000 Logarithmic Frequency Distribution of Interzonal OD Cell Values CS 05 1,000,000 CS ,000 CS 20 Number of Cells 10,000 1, E E E E E E E E E E E E+02 Category Value - Lower Limit

24 Number of OD Pairs versus Number of Routes per OD Pair: CS E+07 Cumulative Proportion of OD Pairs 1.E+00 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 1.E+06 1.E-01 1.E+05 1.E-02 Number of OD Pairs 1.E+04 1.E+03 1.E-03 1.E-04 1.E+02 96% of OD pairs 1.E-05 1.E+01 1,920 routes 1.E-06 1.E ,000 10,000 Number of Routes per OD Pair 1.E-07

25 Number of OD Pairs versus Number of Routes per OD Pair: CS ,000,000 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 1,000,000 3,003,534 OD pairs VISUM solution 100,000 Origin-based algorithm solution Number of OD Pairs 10,000 1,000 Total No. of VISUM routes ,000 10,000 Number of Routes per OD Pair 1,920 routes

26 Number of OD Pairs versus Number of Routes per OD Pair: CS Cumulative Proportion of OD Pairs 1.E+07 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 1.E+00 1.E+06 1.E-01 1.E+05 1.E-02 Number of OD Pairs 1.E+04 1.E+03 1.E-03 1.E-04 1.E+02 92% of OD pairs 21,360 routes 1.E-05 1.E+01 1.E-06 1.E ,000 10, ,000 1.E-07 Number of Routes per OD Pair

27 Number of OD Pairs versus Number of Routes per OD Pair: CS ,000,000 3,166,736 OD pairs 1,000,000 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs Number of OD Pairs 100,000 10,000 1,000 VISUM solution Origin-based algorithm solution Total No. of VISUM routes ,360 routes ,000 10, ,000 Number of Routes per OD Pair

28 Number of OD Pairs versus Number of Routes per OD Pair: CS E+07 Cumulative Proportion of OD Pairs 1.E+00 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 1.E+06 1.E-01 1.E+05 1.E-02 Number of OD Pairs 1.E+04 1.E+03 1.E+02 78% of OD pairs 459,264 routes 1.E-03 1.E-04 1.E+01 1.E-05 1.E ,000 10, ,000 1,000,000 1.E-06 Number of Routes per OD Pair

29 10,000,000 1,000,000 Number of OD Pairs versus Number of Routes per OD Pair: CS OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 3,168,379 OD pairs VISUM solution 100,000 Origin-based algorithm solution Number of OD Pairs 10,000 1, Total No. of VISUM routes 459,264 routes ,000 10, ,000 1,000,000 Number of Routes per OD Pair

30 10,000,000 Number of OD Pairs by Cost Sensitivity cs 05 1,000,000 cs ,000 cs 20 Number of OD Pairs 10,000 1, Number of Routes per OD Pair

31 Conclusions so far Number of routes in the solution set is very large, and increases with congestion; Number of OD pairs with only one route is much larger than expected from a UE viewpoint; some routes are very long; Number of routes in the VISUM solution set is much smaller than we found in a more highly converged solution. The relationship between convergence and the number of routes is subtle and requires further study.

32 What about route flows? As noted, route flows for OD pairs with multiple routes are not unique, An additional assumption is needed to determine most likely route flows, which are consistent across scenarios; Consistent route flows may be important for select link and select zone analyses, based on route flows; For these analyses, having a more complete route set is also likely to be important. Research in progress is comparing the properties of consistent route flows to an arbitrary solution.

33 10,000 1, Number of Routes vs. OD Flow for Origin Zones 34 and 628 Origin Zone 34 Origin Zone 628 Dest. 33-1,728 Dest. 32-1,920 Dest Dest E+03 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-06 1.E-05 1.E-04 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 OD Flow (vph) 1.E-12 Number of Routes

34 1.E+01 1,600 1,400 1,200 1, Frequency Distribution of Route Flows from Zone 34 with Multiple Routes 1.E-25 1.E-24 1.E-23 1.E-22 1.E-21 1.E-20 1.E-19 1.E-18 1.E-17 1.E-16 1.E-15 1.E-14 1.E-13 1.E-12 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 Categories of Route Flow (vph) 1.E-27 1.E-26 Number of Routes

35 Implications for route flows for VISSIM Realizing that route flow solutions contain very many small flows, possibly over multiple routes, raises questions about how to export these flows to VISSIM for micro-simulation; A natural answer would appear to be Monte Carlo sampling from the route flow solution; Each route would have a chance of being selected proportional to its share of the total flow in the OD matrix. Studies are needed to assess whether multiple samples would be required.

36 Reducing solution time by rounding Analysis of the many small OD flows in the OD matrices, and their effect on solution times, raised the question of rounding, either to integers or larger real-valued flows; This question was studied by rounding the matrices to integers by a method that preserved the total flow in each matrix, as shown in the next slides.

37 1,400,000 Total Interzonal Flow for Original and Rounded Trip Matrices 1,350,000 1,300,000 cs 20 decimal cs 10 decimal cs 05 decimal cs 20 integer cs 10 integer cs 05 integer Vehicle Trips per Hour 1,250,000 1,200,000 1,150,000 1,100,000 1,050,000 1,000,000 Trip Matrix

38 10,000,000 1,000,000 Frequency Distribution of Decimal and Integer InterZonal OD Cell Values cs 05 - decimal cs 10 - decimal cs 20 - decimal 100,000 cs 05 - integer Number of Cells 10,000 1,000 cs 10 - integer cs 20 - integer to <1 1 to <10 10 to < to <1,000 1,000 to Max Category Value

39 10,000,000 Number of OD Pairs vs. Number of Routes by Cost Sensitivity and Treatment cs 05 - decimal Number of OD Pairs 1,000, ,000 10,000 1, cs 10 - decimal cs 20 - decimal cs 05 - integer cs 10 - integer cs 20 - integer Number of Routes per OD Pair

40 10,000,000 Number of OD Pairs vs. Number of Routes by Cost Sensitivity and Treatment cs 05 - decimal cs 10 - decimal 1,000,000 cs 20 - decimal cs 05 - integer Number of OD Pairs 100,000 10,000 cs 10 - integer cs 20 - integer 1, Number of Routes per OD Pair

41 Number of OD Pairs versus Number of Routes per OD Pair: CS ,000,000 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 3,003,534 OD pairs 1,000, ,635 OD pairs VISUM integer solution 100,000 VISUM decimal solution Number of OD Pairs 10,000 1,000 Origin-based algorithm solution ,000 10,000 Number of Routes per OD Pair 1,920 routes

42 Number of OD Pairs versus Number of Routes per OD Pair: CS ,000,000 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 1,000,000 3,166,736 OD pairs 545,452 OD pairs VISUM integer solution 100,000 VISUM decimal solution Number of OD Pairs 10,000 1,000 Origin-based algorithm solution Total No. of VISUM integer routes Total No. of VISUM decimal routes ,360 routes ,000 10, ,000 Number of Routes per OD Pair

43 Number of OD Pairs versus Number of Routes per OD Pair: CS ,000,000 OBA: 3,168,206 positive-valued interzonal OD flows plus 34,104 zero pairs 1,000, ,000 3,168,379 OD pairs 640,210 OD pairs VISUM integer solution VISUM decimal solution Origin-based algorithm solution Number of OD Pairs 10,000 1,000 Total No. of VISUM integer routes Total No. of VISUM decimal routes ,264 routes ,000 10, ,000 1,000,000 Number of Routes per OD Pair

44 Implications for routes To assess whether the number of routes found in our solutions were at all plausible, we mapped the links used in each OD matrix by the zone pair with the largest number of routes. For comparison, the routes used in the other two matrices were mapped also. These maps of used links are shown next. Our interpretation is that the routes shown are quite plausible.

45 OD pair CS ,536 routes 52.9 min vph CS routes 48.1 min vph CS ,920 routes 44.4 min vph

46 OD pair CS ,048 routes 96.7 min vph CS ,268 routes 93.3 min vph CS routes 90.3 min vph

47 OD pair CS ,264 routes min vph CS routes min vph CS routes 98.7 min vph

48 Questions for future study From these explorations, we suggest there are two types of questions that may lead to findings helpful to practitioners and software developers in the future: What are the basic properties of routes sets and route flows in congested road networks, and how do they vary by network configuration, coding practice, congestion level, etc.? Assuming an improved understanding of network properties, how should models be efficiently solved in practice, and what level of detail is required in the solutions?