Trip Distribution Modeling Milos N. Mladenovic Assistant Professor Department of Built Environment

Transcription

1 Trip Distribution Modeling Milos N. Mladenovic Assistant Professor Department of Built Environment

2 Course Outline Forecasting overview and data management Trip generation modeling Trip distribution modeling Mode split modeling Traffic assignment modeling Network theory Activity-based modeling Modeling practices 2

3 Outline Overview Modeling strategy and process Gravity-based model Synthetic OD estimation 3

4 Four-Step Travel Modeling 4

5 Overall (Matrix Manipulation) Process 5

6 Overall (Matrix Manipulation) Process 6

7 From Trip Generation to Trip Distribution 7

8 Trip Distribution Objective of trip generation modeling is to compute the number of trips by destination Determines Tij for Design Year Trips are classified into categories (e.g., work, shopping, social) Work trips sometimes have only mode option (no destination choice involved) 8

9 Trip Distribution Input and Output 9

10 Trip Distribution Input and Output 10

11 Key Concepts Internal, External-internal, and External-external trips are three distinct trip types since they exhibit different characteristics in relation to the study area Internal trips use gravity model External-internal trips can also use gravity model, but since the trip length characteristics of these trips are different from internal trips, a separate gravity model analysis is done For the External-external trips, travel surveys are used for the base-year trip distribution and a growth factor technique called FRATAR is used for the forecast trip distribution 11

12 Key Concepts Some persons making external-internal trips (e.g., persons who commute into the study area for work) also make internal trips during the day. Typically these internal trips are not picked up as part of the external station survey; consequently, these trips are not modeled. For many urban areas, such travel is a modest part of the total travel and the loss of these trips is not significant. For other urban areas, tourist areas in particular, internal travel by persons who commute into the study area is significant. Where this is the case, a separate internal trip purpose called external-local is used. The characteristics of these trips are obtained through an expanded set of questions administered as part of the external station survey. The participants are asked to answer questions about their internal trips in addition to their externalinternal trips. A separate trip generation and gravity model analysis is made for this internal trip purpose called external-local. 12

13 Inputs and Outputs Inputs for external-internal trips Productions for each external station Trip length frequency distribution Travel time impedance matrix Scaled NHB attractions by zone Inputs for external-external trips Ps and As for each external station O & D trip table (base-year) Outputs 24 hr external-internal trip table (P & A) 24 hr external-internal (O & D) (same outputs for external-external trips) 13

14 Inputs and Outputs Inputs for external-external trips purpose Productions for each external station zone from the external station travel survey Trip length frequency distribution obtained from analysis of the external station survey trips Zone to zone travel impedance matrix obtained from skimming the network Scaled NHB attractions by internal zones from trip generation Inputs (base year) for external-external trips purpose include Productions and attractions for each external station zone from the external station travel survey A base-year trip table in origin-destination format obtained from analysis of the external station survey trips 14

15 Trip Distribution Example 15

16 Trip Distribution Example 16

17 Trip Distribution Modeling Origins The first type of computerized trip distribution model was based on an analogy to Newton s law of gravity 3 F G 12 M M d

18 Gravity Model Interpretation M1 might represent the mass of trips available at, say, a residential area M2 the mass or attractiveness of a shopping area d12 the distance between the two areas (the actual distance, not as crow flies ) F12 the number of trips between the two areas These interpretations would imply through the gravity model that the greater the size or attractiveness of the two areas (masses) and the less the distance between them, the more would be the number of inter-area trips This was found to resemble many real world situations 18

19 Gravity Model Interpretation When the effect of several competing attraction areas (i.e., multiple masses) was taken into account, the gravity trip distribution model became T ij P i j ij b b b b A d A d... A d... A d 1 i1 2 i2 A d b j ij n in 19

20 Gravity Model Interpretation What this formula states in essence is that the percentage of the Pi trips produced by zone I allocated to destination zone j is dependent upon both the attractiveness (Aj) of zone j and travel time to that zone relative to the same features of all other attracting zones. Thus a zone in which a new shopping center is built (increased Aj) or to which a new transportation facility is constructed (decreased dij), increases its relative pull on the Pi trip productions and subsequently draws a greater proportion of these productions to itself. 20

21 Trip Distribution Generalization Trips between Zone 3 and TO zone = Trips produced in Zone 3 x Attractiveness of TO zone Attractiveness of all zones Zone 3 Productions:

22 Gravity Model Modification T ij P i n A j1 j A F j ij F K ij ij K ij where F ij is the travel time factor K ij is a specific zone-to-zone adjustment factor for taking into account the effect on travel patterns of defined social or economic linkages not otherwise incorporated in the gravity model formulation (somewhat equivalent to the g s in the Newton s gravity model) 22

23 Gravity Model Modification T ij P i n A j1 j A F j ij F K ij ij K ij One major change to be noted here is that it is no longer required to have a travel time function of the form b Instead, other functional forms can be used F at ij F ate ct d bt 2 F a bt 1 23

24 Gravity Model Modification All these factors express the general idea of a drop in F as t increases, but one may lead to more reliable trip distribution estimates than the others under certain circumstances. The second major change is, of course, in the inclusion of the Kij factors, these being added because of some unusual differences in travel distributions noted in some cities and attributed to the social and economic makeup of the zones. For example, zones with higher income have a greater tendency to make longer work trips than their lower income counterparts. Basically it stems from the inability of the Fij factors above to replicate the base year trip interchanges among zones. 24

25 Trip Distribution Calibration First step is usually establishing the minimum paths from each zone centroid to all others to obtain the current zone-to-zone travel times Calibration for trip distribution is the determination of the values of travel time factors (Fij) and zone-to-zone adjustment factors (Kij) which will produce the zone-to-zone trip tables of the base year from the trip ends (productions and attractions) which are observed in the base year These factors, Fij and Kij are then assumed constant over time, and by applying them to the trip ends computed for the forecasted year, the future trip-interchanges from zone-to-zone can be computed 25

26 Trip Distribution Constraints All trips from i go somewhere Destination j can receive trips from everywhere The total from all origins i and all destinations j must be equal Oi either come from counted data or from estimated Ti created in Trip Generation stage, or both In general, Tii need not be 0 i j j ij P A, T P, i T ij A j i 26

27 Trip Distribution Constraints Trip distribution for Tii is not a constant function, thus we need to compute values for each cell using a cost function accounting for actual conditions We know some total or at least minimum values, for some of the Pi and Aj because we have some partial information based on actual traffic counts, thus we won t let the Trip Generation or Trip Distribution procedures contradict these So the equation used for Tij must include both Pi and Aj and the cost function This combined function likely to give best fit over a wider range of trip lengths 27

28 Trip Distribution Example 28

29 Trip Distribution Example 29

30 Trip Distribution Tij ( Ai Pi)( B j Aj ) f ( cij) Where: The A i and B j are balancing factors that allow the totals to match without constraint violation f f f ( c ij ) c ij ( c ij ) e ( c ij ) c n n ij c ij e c ij Generalized cost function - Power - Exponential - Combined 30

31 Highway Link Performance Curve 31

32 PT Link Performance Curve 32

33 Determining Fij Factors 1. Assume all Kij= Assume initial values for travel time factors Fij (usually 1.0) at 5- minute intervals or other time intervals of your choice 3. Group the zones within the predetermined time intervals 4. Use these assumed travel time factors to distribute the trip ends among zones using the gravity model 5. Check the number of trips obtained by the gravity model vs. those observed in the base year data for each of the time interval zone groups 6. Adjust the travel time factors for each time interval as follows: F ij adjusted F ij old Total Total trips observed from data trips calculated from Gravity Model 33

34 Determining Fij Factors 7. Stop calibrating for Fij factors when the ratio for trips observed to trips calculated is close to 1.0 (in other words, when no improvements in Fij factors are realized from the old ones) 8. Use the obtained travel time factors to forecast the future trip interchanges using the gravity model 34

35 Gravity Model Example Assume all Kijs = 1.0 and group the zones within 5-min time intervals First Trial: Assume Fij = 1.0 and calculate Tij s from the Gravity Model and compare with the observed values A j The model now has the following form: Tij Pi A j 35

36 Gravity Model Example 1 st Iteration T 144, T T 183, T T , T T 99, T T

37 Gravity Model Example 1 st Iteration 37

38 Gravity Model Example 2 nd Iteration Using the observed trip ends (P i and A j ) and the F ij factors from the first trial, distribute the trips using the gravity model T '' ij P i A F j j ' ij A F ' ij 38

39 Gravity Model Example 2 nd Iteration T T T T T T T T T

40 Gravity Model Example 2 nd Iteration 40

41 Gravity Model Example 3 rd Iteration Similar to second trial, use the gravity model to distribute the observed trip ends utilizing the second trial factors " F ij 41

42 Gravity Model Example 3 rd Iteration T T T T T T T T T

43 Gravity Model Example 3 rd Iteration Since the calculated trips are within 0.95 to 1.05 of the observed trips for the same travel time intervals, the iteration process is terminated and the third trial travel time factors are adopted for future forecasts 43

44 Gravity Model - Checking Fij Factors Before their adoption is made final, the travel time factors should be checked if they approximate some reasonably regular decay with travel time, that is, if they are plotted against travel time, they should produce a graph similar to the one shown in figure However, badly behaved travel time factors will create considerable difficulty in balancing the zonal trip interchanges with the observed ones 44

45 Gravity Model - Row and Column Factoring The third trial trip interchange matrix: The matrix shows that the calculated trip productions (P i ) are equal to the observed values. However, this is not true for the trip attractions A j s. The trial and error procedure used to determine the F ij factors does not guarantee that the zonal trip attractions estimated by the model will be equal to those found in the observed data. So, it is common to use a successive column and row factoring process in which the zonal trip attractions and productions are re-estimated to make them equal to their observed values. 45

46 Gravity Model - Row and Column Factoring 46

47 Gravity Model - Row and Column Factoring 47

48 Gravity Model - Row and Column Factoring The final matrix is the one where the Pi s and Aj s are equal to their observed counterparts 48

49 Gravity Model Determining Kij Factors Determine the Kij factors from the final travel balance matrix using the following relationship K ij T T ij ij (observed) (calculated) 49

50 Gravity Model Design Year Forecast After the gravity model shown in equation is fully calibrated, the planner is in a position to forecast design year trip interchange 50

51 Gravity Model Design Year Forecast Using the Fij and Kij factors obtained in the base year gravity model calibration of the 3-zone example, the design year zonal trip interchanges are determined using the gravity model relationship z A j F ij K ij Z for production Zone #1 = 600(.852)(1.042)+1100(1.692)(1.159)+950(.714)(.658) Z for production Zone #2 = 600(1.692)(1.067)+1100(1.692)(.789)+950(.714)(1.135) Z for production Zone #3 = 600(.714)(.759)+1100(.714)(.938)+950(.852)(1.156) 51

52 Gravity Model Design Year Forecast Design year zonal trip interchanges T ij = P i A j F ij K ij Z i 52

53 Gravity Model Design Year Forecast T T T , , , T T T , , , T , T , T ,

54 GM - Row and Column Factoring 54

55 GM - Row and Column Factoring 55

56 GM - Row and Column Factoring 56

57 GM Final Trip Interchange Matrix 57

58 Synthetic OD Estimation Synthetic O-D estimation is a generalized formulation of the trip distribution model, where O-D demand is estimated from link flow counts Synthetic O-D generation is subject to information underspecification Inputs Link flows based on traffic counts Seed matrix based on trip distribution Paths for OD pairs Formulation Maximize the likelihood of a solution Constrained by satisfying the link counts 58

59 Maximum Likelihood Estimation Requires to define a measure of likelihood of each matrix There are two approaches to establish the likelihood of a matrix: One treats the trip as the base unit of observation Other treats the volume count as the base unit of observation Entropy is the number of permutations of trips that give rise to a trip table Entropy maximization and information minimization techniques have been used to solve a number of transport problems Combination or unordered arrangement 59

60 Maximum Likelihood Estimation Combination or unordered arrangement involves Selection of a subset of objects from a set without regard to order If repetitions are not permitted, the number of distinct combinations selecting r objects out of n is: The number of ways in which a person can select 5 apples from 20 apples with no replication 20!/(15!*5!) = 15,504 60

61 Maximum Likelihood Estimation Number of ways in which T trips can be divided into groups of Tij without replication 61

62 QueensOD A model for estimating origin-destination traffic demands based on observed link traffic flows, observed link turning movement counts, link travel times and, potentially, additional information on drivers' route choices. 62

63 QueensOD Example Link 5 carries lowest volume Constraint on O-D demands T24 and T14 Maximum value for either demand is 30 Link 1 carries second lowest volume Maximum value for T13 or T14 is 40 Total of 100 trips 63

64 QueensOD Example First feasible solution T13 = 40 T14 = Volume on link 1 T13 = 40-40=0 T23 = Volume on link 4 T13 = 70-40=30 T24 computed as: Volume on link 2 T23 = = 30 Volume on link 5 T14 = 30-0 = 30 64

65 QueensOD Example Second Feasible solution T13 = x T14 = 40 - x T23 = 70 - x T24 = x x 40 Starting solution T13 = 15 T14 = 25 T23 = 55 T24 = 5 65

66 QueensOD Example Solution is x that maximizes the expression on the right Trial and error solution x = 28 T13=28, T14=12, T23=42, T24=18 The above solution asks the question: how many different ways can the observed trip matrix be created from a fixed number of trips? An alternative solution asks the question: given a certain number of total link traffic counts, in how many ways can the observed link flows be created? This approach leads to a somewhat more complicated formulation. 66

67 Practical Considerations in Trip Distribution Zones with high and low productions and attractions Sparse matrices with zero values Future year matrix Observed and estimated trip lengths Home-based work trip distribution Intra-zonal trips 67

68 GM Advantages It accounts for competition of trips between land uses by using trip attractions vs. trip productions It is sensitive to changes in travel time between zones It recognizes trip purposes as affecting zonal interchanges It is easy to understand and therefore easy to apply in particular areas 68

69 GM Disadvantages It is very unlikely that the travel time factors by trip purpose would remain constant throughout the urban area to the design period The changing nature of travel times between zones with time of day makes questionable the use of single value of travel time factors The model tends to overestimate near trips and underestimate far trips It only shows an approximate agreement with field data and considerable playing with Kij factors is needed to balance the total number of productions and attractions 69