Lecture-11: Freight Assignment

Transcription

1 Lecture-11: Freight Assignment 1 F R E I G H T T R A V E L D E M A N D M O D E L I N G C I V L / D E P A R T M E N T O F C I V I L E N G I N E E R I N G U N I V E R S I T Y O F M E M P H I S 11/21/2014

2 Outline 1. Traffic Assignment-General concept 2. Application in Freight 3. Operational freight traffic assignment 4. Case Studies 2

3 Overview The procedure used to obtain expected traffic volume on the network is known as trip assignment. 3

4 Given Assignment Definition A graph representation of the urban transportation network The associated link performance functions, and An origin-destination matrix Find 4 the flow and the associated travel time on each of the network links. This problem is known as that of traffic assignment as the objective is to assign the O-D matrix onto the network.

5 Route Travel Time Capacity Link Performance Functions Mathematical Relationship Between Traffic Flow and Travel Time 5 Linear Relationship Non-Linear Relationship Free-Flow Travel Time Traffic Flow Traffic Flow

6 Link Performance Functions A steady-state link performance function is a positive, increasing, and convex curve. 6 Typical link performance functions do not consider queued vehicles in the traffic stream

7 Travel time on link Travel time on link , Link Performance Functions ,000 1,500 2,000 2,500 Volume in Vehicles/hour ,000 1,500 2,000 2, Volume in Vehicles/hour V=volume, C=capacity, t 0 =free flow travel time V t t0 1 C

8 Assignment Methods All-or-nothing (AON) User equilibrium (UE) System optimum (SO) Stochastic user equilibrium (SUE) Multiple user class assignment Stochastic multiple user class assignment 8

9 Classification of Traffic Assignment 9 User Class Capacity Restraint No Stochastic Effects Included Yes Single user class Multiple user class No All-or-nothing Yes UE SUE No A-O-N with multiple user classes Yes UE multiple user classes SUE Dial's Stochastic Dial's Stochastic

10 All or Nothing Assignment Simplest assignment methods 10 Does not consider any congestion effects Absence of congestion suggests that link costs are fixed All drivers are assigned to one route and no driver is assigned to other (less attractive routes)

11 All or Nothing Assignment 11 n min z ( y) subject to k g rs k rs g q r, s k rs a 0 k, r, s t y n a a minimizing the total travel time over a network with fixed travel time the auxiliary flow variable for path k connecting O-D pair r-s

12 Definition of Equilibria To solve the traffic assignment problem, it is required that the rule by which motorists choose a route be specified. 12 It is reasonable to assume that every motorist will try to minimize his or her own travel time when traveling form origin to destination. A stable condition is reached only when no traveler can improve his/her travel time by unilaterally changing routes.

13 Equilibrium UE definition implies that 13 motorists have full information (choice set and travel times), motorists consistently make the correct route choice decision all motorists are identical in their behavior These assumptions can be partially relaxed in the context of route choice under information provision. distinction between the travel time that individuals perceive and the actual travel time This definition characterizes the stochastic-user-equilibrium (SUE) condition.

14 Criterion-1 Wardrop s Equillibrium 14 The journey times on all routes actually uses are equal and less than those which would be experienced by a single vehicle on any unused route Criterion-2 User equilibrium conditions traffic arranges itself in such a way that no individual trip maker can reduce his/her path costs by switching routes.

15 A simple example of UE Link 1 15 O D q OD = x1 +x2 Link 2 t t2(x2) t1(x1) x2 x1 x

16 Operational UE 16 Path 1 Cost Path 2 Cost Trips on Path 1 Trips on Path 2 Operational definition of UE: For each O-D pair, at user equilibrium, the travel time on all used paths is equal, and (also) less than or equal to the travel time that would be experienced by a single vehicle on any unused path.

17 Example UE (40 min) (30 (40 min) (65 min) User Equilibrium is reached when no traveler can improve his travel time by unilaterally changing routes.

18 Formulating the Assignment Problem NOTATIONS Network G (N,A) 18 N is set of consecutively numbered nodes A is a set of consecutively numbered arcs (links) R denote the set of origin centroids (which are the nodes at which flows are generated) S denote the set of destination centroids (which are the nodes at which flows terminate) q rs is the trip rate between origin r and destination s during the period of analysis x a and t a, represent the flow and travel time, respectively, on link a

19 NOTATIONS Formulating the Assignment Problem t t ( x ) a a a f rs k 19 : where t. represents the relationship between flow and travel time for link a a : represents flow on path k connecting origin r and destination s such that path k K rs rs c : is travel time on path k is the sum of the travel time on the links comprising this path. k rs c t k K, r R, s S where rs k a rs a k rs k 1 if link is part of path k connecting O-D pair r s 0 otherwise Using the same indicator variable, the link flow can be expressed as a function of the path flow, that is x f rs rs a k a, k a r s

20 User Equilibrium Formulation 20 min z(x) = t ( ) subject to f k rs k x a rs f q r, s k rs 0 k, r, s rs rs a k a, k a r s a 0 a d x f a flow conservation constraint Non negativity constraint δ rs a,k = 1 if link a is on path k between o d pair rs 0 otherwise definitional constraints

21 Significance: Significance of User Equilibrium 21 Reasonable assumption for representation of human behavior In order to asses the network performance for given demands UE conditions are assumed Limitations: Assumption that each user minimizes travel time implies each user has perfect information on all conditions and routes Individuals are assumed to behave identically

22 UE Note Demand for travel depends on the activity pattern, and hence not uniform over time and space. 22 However transportation planners analyze networks only for certain periods of the day morning peaks, evening peaks etc. depending on objective of analysis => O-D flows are considered constant for such analysis (steady-state) -> static assignment Flow is present simultaneously on all links of a path (static conditions)

23 System Optimal Assignment min z(x) = xt ( t ( ) xd) subject to to k k f f rs rs k k a a a a a rs rs f f qq rr, s, s k k rs rs x a 00 kk, r, r, s, s x f a rs rs a k a, k a r s 0 23 Total system travel time flow conservation constraint Non negativity constraint Definitional constraint δ rs a,k = 1 if link a is on path k between o d pair rs 0 otherwise

24 SO Properties The SO formulation is subject to the same set of constraints as the UE problem and differs only in its objective function 24 The SO flow pattern does not generally represent an equilibrium solution in congested networks Consequently, the SO flow pattern is not an appropriate descriptive model of actual user behavior

25 Significance of System Optimal In many transportation system analysis problem it is useful to know the best performance possible for the network and OD demand 2.This is useful for control action (pricing, tolling) as well as to compare alternative solution strategies 3.Solution procedures for SO are virtually identical to those for UE

26 Solving UE 26

27 Solving UE 27 Substitute x2= 12-x1 Differentiate w.r.t x1 and equate to zero x1= 5.8, x2 = 6.2.

28 Solving SO Let us consider the same example 28 For SO

29 Substitute x2= 12-x1 Solving SO 29 Differentiate the equation and set it to zero x1 = 5.3,x2= 6.7

30 Comparison of Methods 30 Type t1 t2 x1 x2 UE Z(x) SO Z(x) AON UE SO

31 Stochastic Methods 31 Emphasize the variability in driver perception of cost Need to consider second best routes No perfect information about network characteristics Different travel costs perception Eliminates zero volume links Requires large number of iterations and hence a longer run time See more in Modeling Transport by Ortuzar and Williumsen, Chapter 10.

32 Older approach Multiclass traffic assignment Pce conversion Preload the usual class first The Modern approach Each user class is assigned simultaneously. See a worked out example in Nagurney (2000) Available in TransCAD, Cube, VISUM 32 Nagurney, A. (2000). A Multiclass, Multicriteria Traffic Network Equilibrium Model. Mathematical and Computer Modelling 32 (2000)

33 Truck counts By vehicle class By facility type By time of day Screen lines Cordon lines Model Validation 33 Develop RMSE, R 2 or other goodness-of-fit measures

34 Screenlines Example 34 Share of counts per screenline > 75% 50% - 75% < 50%

35 Simulation All vehicles all day 140,000 Model Validation Example , ,000 R² = ,000 60,000 40,000 20, ,000 40,000 60,000 80, , , ,000 Count

36 Interstate Freeways/ Expressways Other Principal Arterial Minor Arterial Collector Local Daily VMT Millions VMT based comparison Observed Modeled

37 %RMSE Volume class comparison % 120% 100% 80% 60% 40% 20% 0% <5,000 5,000-10,000 10,000-20,000 20,000-30,000 30,000-40,000 40,000-50,000 50,000-60,000 60,000-70,000 70,000-80,000 80,000-90,000 90, ,000 >100,000 Volume Class

38 State of Practice 38 QRFM based Source: Freight demand modeling: Tools for public sector decision making

39 FAF State of Practice (sample only) AON (older versions) Freight only SUE (TransCAD based) Florida Multiclass UE Indiana QRFM Iowa AON Maryland Multiclass UE Oregon Multiclass SUE Ohio Multi UE Wisconsin Multiclass UE 39