TRIP DISTRIBUTION MODELLING

Transcription

1 TRIP DISTRIBUTION MODELLING

2 Trip Production / Attraction How many trips are produced and attracted in each zone?

3 Trip Distribution How many trips are made between each zone pair?

4 TRIP MATRIX Destinations Origins j...n Σ 1 T11 T12 T13 T1j T1n O1 2 T21 T22 T23 T2j T2n O2 : i Ti1 Ti2 Ti3 Tij Tin Oi : n Tn1 Tn2 Tn3 Tnj Tnn On Σ D1 D2 D3 Dj Dn ΣTij = T O i = Σ T ij j D j = Σ T ij i Origin (trip generation) constraint Destination (trip attraction) constraint

5 Trip Distribution Given a location, where do people go to satisfy demand for an activity type? Trip Distribution Trip interchanges increase with decrease impedance (distance, travel time, travel cost) between zones. Trip interchanges increase with increased zone attractiveness (square footage of retail or population for example).

6 Trip Distribution Models Growth factor models: When a base O-D matrix is given, future matrix is estimated. - Uniform factor - Singly constrained factors - Doubly constrained factors Gravity models.

7 GROWTH FACTORS Uniform factor T ij = k t ij t: Present trips T: Future trips k: Uniform factor Singly constrained growth factor T ij = O i /o i t ij = A i t ij A i : Origin specific factors O i : Target year generations o i : Base year generations Σ T ij D j i Destination constraints are not satisfied

8 GROWTH FACTORS Singly constrained growth factor T ij = D j /d j t ij = B j t ij Σ T ij O i j Average factor T ij = ½ (O i /o i + D j /d j ) t ij = ½ (A i + B j ) t ij Σ T ij O i j B j : Destination specific factors D j : Target year attractions d j : Base year attractions Origin constraints are not satisfied Σ T ij D j i Non of the trip-end constraints are satisfied

9 GROWTH FACTORS Doubly constrained growth factor T ij = k i k j t ij k i :Origin specific factors k j :Destination specific factors O i = Σ T ij D j = Σ T ij k i and k j are calculated so as both constraints are satisfied (Furness, 1965)

10 FURNESS ALGORTIHM (1965) T ij = A i B j O i /o i D j /d j t ij t ij = Base year trips from zone i to zone j T ij = Target year trips from zone i to zone j o i = Base year trips generated from zone i d j = Base year trips attracted to zone j O i = Target year trips generated from zone i D j = Target year trips attracted to zone j A i and B j = Balancing factors

11 FURNESS ALGORTIHM (1965) O i = Σ T ij Origin constraints j =1,..,n (1) j D j = Σ T ij Destination constraints i =1,...,n (2) i Set B j = 1 for all j and calculate A i from (1) A i = o i / ( Σ D j /d j t ij ) j Replace A i in (2) and calculate B j from (2) B j = d j / ( Σ A i O i /o i t ij ) i A i = o i / ( Σ B j D j /d j t ij ) j

12 FURNESS ALGORTIHM (1965) Test convergence If A i (k+1) - A i (k) = є If B j (k+1) - B j (k) = є stop. Otherwise, continue iteration. k: Iteration number. This method produces solutions within 3 to 5% of the target values in a few iterations. It may require correcting trip-end estimates produced by the trip generation model.

13 Disadvantages of Growth Factor Models New spatial developments for the study area cannot be accommodated. Changes in transport costs due to improvements cannot be taken into account. The method depends heavily on the reliability of the data in the individual cells of the base year matrix. There are instances when problems can occur with the iteration process. Only reasonable for short-term planning horizons.

14 Gravity Model Originally generated from an anology with Newton s gravitational law. Tij = α Pi Pj / (d ij ) 2 (Casey, 1955) T ij = Trips from zone i to zone j α= Propotionality factor Pi = Population of zone i Pj = Population of zone j dij = Distance between zone i and zone j

15 Gravity Model The model was further generalised by assuming that the effect of distance or seperation could be modelled better by a decreasing function (deterrence function). T ij = k i k j O i D j f (c ij ) T ij = Trips from zone i to zone j D j = Trip attraction in zone j O i = Trip production in zone i f (c ij ) = Deterrence function of travel impedance between zone i and zone j k i, k j = Balanging factors

16 Gravity Model Fij : Deterrence function (Friction factors) Fij can be a function of distance, time or cost. Cij : Travel impedance.

17 Different Deterrence Functions Source: J.G. Ortuzar, L.G. Willumsen, Modelling Transport, 4th Edt., Wiley, 2011

18 Travel Impedance Generalised cost of travel Cij = a1 tij + a2 Fij + a3 Pj + δ tij : Travel time Fij : Fare (Travel cost) Pj : Parking cost δ : Modal penalty that represents all other attributes not included in the generalised cost (e.g., comfort, safety, convenience)

19 TRIP LENGTH DISTRIBUTION (%) Travel time (min)

20 TLD Checks

21 Gravity Model

22 A Simplified Travel Matrix

23 CALIBRATION OF GRAVITY MODEL BASE YEAR OBSERVED TRIP MATRIX tij ZONE oi dj BASE YEAR TRAVEL TIMES cij (min) ZONE

24 CALIBRATION OF GRAVITY MODEL Phase 1: Step 1. Construct Base Year (Observed) Trip Length Frequency Distribution (TLD)

25 CALIBRATION OF GRAVITY MODEL -β cij Step 2. Choose some form of f (cij), e.g. f (cij) = e tij = Ai Bj oi dj f (cij) Step 3. Set Ai = 1 and Bj = 1 for all i and j. Step 4. Calculate the initial fij (0) = tij/(oi x dj) from the base year trip matrix. ln fij (0) = - β cij Iteration No. = (1) Step 5. Estimate β from the linear regression analysis. -β cij Step 6. Calculate fij (1) = e Step 7. Set Bj = 1 for all j. Σ tij = oi Origin constraints, for all i. j

26 CALIBRATION OF GRAVITY MODEL Step 8. From the origin constraints, calculate Ai = 1 / Σ dj f ij (1) Step 9. Calculate tij (1) = oi x dj x fij (1) / ( Σ dj x fij (1) ) j Note that the origin constraints of the tij (1) trip matrix are satisfied, but the destination constraints are not.

27 CALIBRATION OF GRAVITY MODEL Convergence Test Step 10. Construct resulting Trip Length Frequency Distribution TLD (1). Compare TLD (1) with the Base Year (Observed) TLD. If converged, Go To Phase 2. Iteration No. = (2) If not converged, recalculate fij (2) for the Iteration 2. fij (2) = fij (1) x TLD / TLD (1) Go To Step 5.

28 ITERATION 1 Initial fij (0) = tij / (oi x dj) fij (0) Regression 1 3 0, , , , , ,00119 ITERATION 1 cij fij (0) ln (fij 0) fij (1) = exp (-β cij) 15 0, , , , , , , , , , , , , , , , , ,15E-05 β (1) = 0,1863 fij (1) , , , , , , fij (1) x dj , , , , , , , ,962 tij (1) = oi x fij 1( fij (1) * dj) ZONE oi dj y = -0,1863x

29 CONVERGENCE TEST 1 cij OBSERVED Iteration (1) (min) TRIPS TLD TRIPS TLD (1) TLD / TLD (1) ,00 0 0, , ,266 0, , ,034 5, , ,589 0, , ,105 2, ,10 6 0,006 17, , ,00 0,70 0,60 0,50 0,40 0,30 0,20 0,10 0,00 0,589 0,35 0,266 0,20 0,25 0,10 0,105 0,10 0,00 0,00 0,034 0,

30 ITERATION 2 cij fij (1) fij (2) = fij (1) TLD / TLD (1) ln (fij 2) fij (3) = exp (-β cij) Regression , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,00000 β (2) = 0,1597-6,00000 y = -0,1597x -7,00000 fij (3) , , , , , , fij (3) x dj ,429 16,932 27,338 47, ,616 0,148 1,003 3,767 tij (2) = oi x fij ( fij (3) * dj) ZONE oi dj , ,00000

31 CONVERGENCE TEST 2 cij OBSERVED Iteration (2) (min) TRIPS TLD TRIPS TLD (2) TLD / TLD (2) ,00 0 0, , ,278 0, , ,022 9, , ,486 0, , ,186 1, , ,027 3, , ,00 0,50 0,40 0,30 0,20 0,10 0,00 0,486 0,35 0,278 0,25 0,20 0,186 0,10 0,10 0,00 0,022 0,

32 ITERATION 3 ITERATION 3 cij fij (3) fij (4) = fij (3) TLD / TLD (2) ln (fij 4) fij (5) = exp (-β cij) Regression , , , , , , , , , , , , , , , , , , , , , , , y = -0,1458x 53 0, , , , ,00000 β (3) = 0,1458-8,00000 fij (3) , , , , , , ,00000 fij (3) x dj ,060 21,745 33,676 60, ,255 0,308 1,774 6,337 tij (3) = oi x fij (5) x dj / ( ( fij (5)* dj) ZONE oi dj

33 CONVERGENCE TEST 3 cij OBSERVED Iteration (3) (min) TRIPS TLD TRIPS TLD (3) TLD / TLD (3) ,00 0 0, , ,275 0, , ,025 7, , ,470 0, , ,196 1, , ,034 2, , ,00 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 0,470 0,35 0,275 0,25 0,20 0,196 0,10 0,10 0,025 0,034 0,

34 ITERATION N cij fij (N) = exp (-β cij) 15 0, , , , , , β (N) = 0,0274 fij (N) , , , , , , fij (N) x dj , , , , , , , ,752 tij (N) = oi x( fij (N) * dj) ZONE oi dj

35 CONVERGENCE TEST 3 cij OBSERVED Iteration (N) (min) TRIPS TLD TRIPS TLD (N) TLD / TLD (N) ,00 0 0, , ,220 0, , ,080 2, , ,285 1, , ,242 1, , ,174 0, , ,00 0,40 0,35 0,35 0,30 0,25 0,20 0,15 0,10 0,10 0,220 0,20 0,080 0,285 0,250,242 0,174 0,10 TLD TLD (N) 0,05 0,00 0,

36 PHASE 2 Calculation of Mi and Nj by Furness Algorithm: [tij] Base Year = Mi x Nj x [tij] Estimated Mi and Nj: Balancing Factors Σ tij dj Destination constraints, for all j. i Σ [tij] Base Year = oi Origin constraints, for all i. j Mi = oi / Σ Nj x [tij] Estimated Σ [tij] Base Year = dj Destination constraints, for all j. i Nj = dj / Σ Mi x [tij] Estimated Mi and Nj can be calculated by Furness Algorithm.

37 PHASE 2 Nj1 1,508 0,537 1,020 Mi1 Nj*tij2 120,834 56, , ,234 1, ,166 93, , ,979 0,910 Mi1*tij2 81, , , , , , , , ,527 Nj2 1,614 0,565 0,891 Mi2 Nj*tij2 129,329 59, , ,855 1, ,340 98, , ,764 0,906 SYNTHETIC TRAVEL MATRIX tij Model ZONE oi dj K Factors ZONE ,499 0,815 0, ,841 1,125 1,282

38 K Factors -β cij tij = Kij Ai Bj oi dj e Some practical studies have used K factors in an attemp to improve the calibration of the model. The best advise: DO NOT USE THEM!

39 Gravity Model Calibration Homework 1 O/D Trip Matrix Target Year Zone no Generation Target Year Attraction tij (min)

40 NUMERICAL EXAMPLE 1 X1 X2 Residents Workers Cars Number of Zone per HH per HH per HH Population Employment HH ,000 2,000 2, ,000 12,000 8, ,000 6,000 6,250 60,000 20,000 17,083 Avg. HH size 3.51 Observed Daily Travel Matrix Zone Generation 1 5, , ,000 1,880 34, ,000 16,000 22,000 40,000 Attraction 7,770 48,200 24,030 80,000 Base-year data Calculated

41 Model 1: Regression Analysis X1: Workers per HH X2: Cars per HH Y:Trips per HH per day Y = X X2 (t) ( t > 1.96 at 95 % confidence level) Generations Zone (Trips/HH/Day) (Trips/Day) Model/Observed , , , Total 83,

42 Model 2: Category Analysis Trips/HH/Day Number of Cars Number of wo or less Generations Zone (Trips/HH/Day) (Trips/Day) Model/Obs , , , , Model 2 fits better with the observed number of trips produced from all of zones. Therefore, Model 2 is better than Model 1. Base-year data Calculated

43 NUMERICAL EXAMPLE 2: Model 2 Results Of Example 1: Zone Generation 1??? 6,000 2??? 33,333 3??? 38,750 Attraction 7,808 46,850 23,425 78,083 We will use a gravity model to estimate tij =? Gravity Model tij = Ai Bj oi dj f(cij) f(cij) = exp (-β*cij) β= 0.60 β is calibrated so as model TLD is reproduced as closely as possible. oi : Trips generated at zone i. dj : Trips attracted to zone j. cij : Travel time between zone i and zone j (min.) β : Calibration coefficient.

44 TRIP LENGTH DISTRIBUTION Trip Length Distribution (TLD) Frequency Observed Model Travel Time (min) Base Year Travel Time Matrix (cij Matrix) (minutes) Zone

45 ESTIMATED BASE-YEAR TRAVEL MATRIX Zone Generation Ai 1 5, , ,475 2,188 33, ,049 15,371 21,459 38, Attraction 7,876 47,039 23,795 78,710 Bj Ai and Bj are calibrated so as both constraints are satisfied (Furness algorithm). K Factors Zone Kij= Estimated tij / Observed tij tij = Kij Ai Bj oi dj f(cij)

46 10 YEARS LATER X1 X2 Residents Workers Cars Zone per HH per HH per HH Population Employment ,000 3, ,000 14, ,000 8,000 70,000 25,000 Travel Time Matrix (cij Matrix, minutes) Zone Calculate travel matrix in 10 years by using the model calibrated with base-year data!

47 DESIRE LINES in ISTANBUL (2006)

48 DESIRE LINES in ISTANBUL (2023) Source: IMP, JICA (2007) 48

49 Trip Distribution Checks Powerful method for adjusting traffic volumes is in the distribution process. Mean Trip Length Shortening or increasing the average trip lengths will in turn raise or lower traffic volumes. Trip Length Distribution

50 Travel Time Distribution (Istanbul, 2006)

51 Travel Time Distribution by Trip Purpose (Istanbul, 2006)

52 Travel Time Distribution of HBW Trips

53 Travel Time Distribution of HBS Trips

54 Travel Time Distribution of HBO Trips

55 Travel Time Distribution of NHB Trips

56 Average Travel Time (Istanbul, 2006)

57 Average Travel Time (Istanbul, )

58 IUAP (2006) Trip Distribution Model -β tij = Ai Bj oi dj cij Cij : Travel time β = β1 + β2 xij xij = 1, If i and j are on the opposite sides of the Bosphorus. xij = 0, Otherwise. β1 β2 HBW HBS HBO NHB

59 ACCESSIBILITY Accessibility (or just Access) is the ability to reach desired goods, services, activities and destinations (together called opportunities). A highway or transit improvement can increase the services and jobs accessible from a neighborhood. Access is the ultimate goal of most transportation, excepting the small portion of travel in which movement is an end in itself, (e.g., cruising, historic train rides, horseback riding, jogging).

60 ACCESSIBILITY Accessibility reflects both mobility (people s ability to travel) and land use patterns (the location of activities). This perspective gives greater consideration to nonmotorized modes and accessible land use patterns. Accessibility tends to be optimized with multi-modal transportation and more compact, mixed-use, walkable communities, which reduces the amount of travel required to reach destinations.

61 ACCESS AND MOBILITY

62 CONTINOUS ACCESSIBILITY MEASURES These indicators combine both the attractiveness of the service or opportunity under consideration such as the number of jobs, retail floor space and the disbenefit or disutility associated with travel. Hansen/gravity measure Relative Hansen/gravity measure Simple utility or logsum measure

63 HANSEN GRAVITY MEASURE Hansen/gravitymeasure of the relevant facilities or services (e.g. jobs, schools, hospitals, food shops, etc.) k, with respect to the origin zone location i for the population segment s.

64 HANSEN GRAVITY MEASURE

65 RELATIVE HANSEN GRAVITY MEASURE Relative Hansen/gravity measure of the relevant facilities or services (e.g. jobs, schools, hospitals, food shops, etc.) k, with respect to the origin zone location i for the population segment s.

66 RELATIVE HANSEN GRAVITY MEASURE Relative Hansen/gravity measure of the relevant facilities or services (e.g. jobs, schools, hospitals, food shops, etc.) k, with respect to the origin zone location i for the population segment s.

67 SIMPLE UTILITY/LOGSUM MEASURE This provides an indication of the "expected maximumutility" for the individual/population segment s resident in origin zone location i of the available choice set encompassing a set of opportunities available in the Base case O with differing levels of (dis)utility.

68 SIMPLE UTILITY/LOGSUM MEASURE

69 LOGSUM MEASURE USED FOR BURSA A j = Ln i P i E j e -ß t ij A j : attractiveness of zone j P i : population in zone i. E j : job places (opportunities) in zone j. t ij : travel time from i to j. ß : calibration coefficient of the deterrence function in the gravity model (ß = )

70 PRESENT LOGSUM MEASURES (2010)

71 FUTURE LOGSUM MEASURES (2030)

72 CHANGES IN LOGSUM MEASURES ( )