1. Trip Distribution Meeghat Habibian Transportation Demand Analysis

2. Demand models of trip distribution Destination choice models
Outline
Introduction
Hitchkock Model
Entropy Model
Gravity Model
Demand models of trip distribution
Destination choice models
Intervening Opportunity Model
Transportation Demand Analysis - Trip Distribution

3. Transportation Demand Analysis
Trip Distribution
Introduction

4. Second Step in Urban Transportation Modeling System
Socio-economic Forecasts
(Population, Employment, …)
Trip Generation
Trip Distribution
Transportation Demand Analysis - Trip Distribution
Mode Split
Trip Assignment

5. Situation Four-step model: Given Ti, Aj  Tij= f(Ti,Aj, …ij)
Direct approach:
Tij= f(Ti,Aj)=f(g(popi),h(popj))=Ψ(popi, popj)
More general:
Tij= f( …i, …j, …ij)
Example:
Tij= f(popi, popj, Number of callsij)
Transportation Demand Analysis - Trip Distribution

6. Definitions N = Number of zones in the urban area
i = Subscript, used to denote origin zones
j = Subscript, used to denote destination zones
𝑂 𝑖 = Number of trips originating in zone i
𝐷 𝑗 = Number of trips destined for zone j
𝑇 𝑖𝑗 = Number of trips (“flow”) from origin zone i to destination zone j
Transportation Demand Analysis - Trip Distribution

7. General Equations T=Total Trips= 𝑖 𝑂 𝑖 = 𝑗 𝐷 𝑗
Logical constraints which any feasible trip matrix must satisfy:
𝑗 𝑇 𝑖𝑗 = 𝑂 𝑖 𝑖=1,…,𝑛 [1]
𝑖 𝑇 𝑖𝑗 = 𝐷 𝑗 𝑗=1,…,𝑛 [2]
Transportation Demand Analysis - Trip Distribution

8. Elementary Trip Distribution Models
Growth factor models
Uniform growth factor model
Average growth factor model
Simple average
Fratar model
Positive and Negative points?
Transportation Demand Analysis - Trip Distribution

9. Transportation Demand Analysis
Trip Distribution
The Hitchkock Model

10. Hitchcock Model A minimum-cost flow problem over the network
𝑀𝑖𝑛 𝑧 𝑥 = 𝑖=1 𝐼 𝑗=1 𝐽 𝑐 𝑖𝑗 𝑋 𝑖𝑗
Such to:
𝑗 𝑥 𝑖𝑗 = 𝑂 𝑖 ∀𝑖=1,2,…,𝐼
Transportation Demand Analysis - Trip Distribution
𝑖 𝑥 𝑖𝑗 = 𝐷 𝑗 ∀𝑗=1,2,…,𝐽
𝑥 𝑖𝑗 ≥ ∀ 𝑖,𝑗

11. Hitchcock Model
𝑐 𝑖𝑗 and 𝑥 𝑖𝑗 : (fixed) cost (per unit of flow) and the flow, respectively, on the link leading from node i to node j
Oi : Total flow supplied by node i
Dj : Total flow required at node j.
Assume further:
∑ Oi = ∑ Dj
Transportation Demand Analysis - Trip Distribution
At the optimal solution:
Maximum number of links carrying flow equals minimum number of links that can connect I supply nodes to J demand nodes, that is (I+J-1)

12. Transportation Demand Analysis
Trip Distribution
The Entropy Model

13. Entropy Model
All of the states can be occurred in trip distribution matrix:
𝑇 𝑡 𝑇− 𝑡 11 𝑡 𝑇− 𝑡 11 − 𝑡 12 𝑡 13 …
T = All of the trips
It can be simplified as:
Transportation Demand Analysis - Trip Distribution
𝑇! 𝑖𝑗 𝑡 𝑖𝑗 !

14. Entropy Model
This model finds the highest probability of trip distribution matrix with respect to all constraints, so the objective function is:
𝑀𝑎𝑥 𝑇! 𝑖𝑗 𝑡 𝑖𝑗 !
And it equals to:
Transportation Demand Analysis - Trip Distribution
𝑀𝑎𝑥 𝑇! 𝑖𝑗 𝑡 𝑖𝑗 ! = ln 𝑇!− 𝑖𝑗 ln 𝑡 𝑖𝑗 !
Ln𝑇! Is constant and equation can be simplified as:
𝑀𝑎𝑥 − 𝑖𝑗 ln 𝑡 𝑖𝑗 !

15. Entropy Model The problem is changed to a minimization problem:
min 𝑖𝑗 ln 𝑡 𝑖𝑗 !
(Stirling Approximation log n!≈nlogn−n)
Therefore:
min 𝑖𝑗 (𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗 − 𝑡 𝑖𝑗 )
Transportation Demand Analysis - Trip Distribution
𝑖𝑗 𝑡 𝑖𝑗 =𝑇 is constant again and equation can be simplified as:
min 𝑖𝑗 𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗

16. Entropy Model with respect to just one constraint
max (− 𝑖 𝑗 𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗 )
𝑠.𝑡. 𝑖 𝑗 𝑡 𝑖𝑗 =𝑇
𝑡 𝑖𝑗 ≥0
Transportation Demand Analysis - Trip Distribution

17. Entropy Model
If 𝜆 is a dual variable, with Lagrange multiplier solution :
Transportation Demand Analysis - Trip Distribution

18. Example
20000 trips estimated in peak hour for an area contains 5 origins and 5 destinations, make the trip distribution matrix through entropy approach.
With the information above, the most likely matrix is:
×5 =800
Transportation Demand Analysis - Trip Distribution

19. Transportation Demand Analysis
Trip Distribution
The Gravity Model

20. Concept m2 r m1
Transportation Demand Analysis - Trip Distribution

21. Analogy T ij ≅ O i i=1,…,N T ij ≅ D j j=1,…,N
T ij ≅𝑔( 1 c ij )= f ij i, j = 1,…,N
Therefore:
𝑇 𝑖𝑗 = 𝑘 𝑖𝑗 𝑂 𝑖 𝐷 𝑗 .𝑔( 1 c ij )= 𝑘 𝑖 𝑘 𝑗 𝑂 𝑖 𝐷 𝑗 . 𝑔( 1 c ij )
kij: constant of proportionality for pair i-j
ki: constant of proportionality for zone i (simplification)
c ij :Impedance (e.g., time, distance, cost) between i and j
f ij = Impedance function; 𝜕 f ij 𝜕 c ij <0
Transportation Demand Analysis - Trip Distribution

22. Proportional constants
T ij =kikj O i D j f ij
Remember:
𝑗 𝑇 𝑖𝑗 = 𝑂 𝑖 𝑖=1,…,𝑛
𝑘𝑖= 1 𝑗 𝑘𝑗 𝐷 𝑗 𝑓 𝑖𝑗
𝑖 𝑇 𝑖𝑗 = 𝐷 𝑗 𝑗=1,…,𝑛
𝑘𝑗= 1 𝑖 𝑘𝑖 𝑂 𝑖 𝑓 𝑖𝑗
Transportation Demand Analysis - Trip Distribution

23. Formulation Observation trips  Many Tij=0 For simplification:
Assume: kj=1
Therefore:
𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗
Note:
Constraint on destinations is relaxed:
𝑘𝑗= 1 𝑖 𝑘𝑖 𝑂 𝑖 𝑓 𝑖𝑗 is not in place!
Transportation Demand Analysis - Trip Distribution

24. Example
Determine the trip distribution matrix for following four cities assuming fij= tij-2
Travel-time Matrix (minutes)
Time 1 2 3 4 7 35 45 40 5 20 12 8
Transportation Demand Analysis - Trip Distribution
Trip Generation data
City 1 2 3 4
Product
4724
901
193
108
Attract
4909
774
174 69

25. Example Travel-time Matrix (minutes) Calculate fij= tij-2 Time 1 2 3 47 35 45 40 5 20 12 8
Transportation Demand Analysis - Trip Distribution
Calculate fij= tij-2
*0.001 1 2 3 4
20.408
0.816
0.494
0.625
40.000
2.500
6.944
15.625

26. estimated by trip generation /observed
Example
Trip distribution matrix
The sum of attracted trips are not satisfied!
Reason: The attraction constraint has been ignored: 1 2 3 4
4688 30
4724
101
777 11 12
901 19 15
151 8
193 20 10 66
108 Aj
4820
842
176 88
estimated by trip generation /observed
4909
774
174 69
Transportation Demand Analysis - Trip Distribution Nj
1.019
0.919
0.989
0.784

27. Row - Column Factor Correction Method
Concept
to proportionally adjust the trip matrix until it approximately (e.g., ±5%) matches the forecast year row and column sums
An iterative procedure is required to balance rows and columns Let:
O i n = j T ij n for the n th iteration
T ij 0 = base year O−D flow
O i new = forecast year row sum
Similar definitions for D j n 𝑎𝑛𝑑 𝐷 𝑗 𝑛𝑒𝑤
Transportation Demand Analysis - Trip Distribution

28. Row - Column Factor Correction Method
𝑘=1 Nj
𝑇 𝑖𝑗 𝑘 = 𝑇 𝑖𝑗 𝑘−1 ( 𝐷 𝑗 𝑛𝑒𝑤 𝐷 𝑗 𝑘−1 )
k=k+1
Yes
𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒? No
k=k+1
Transportation Demand Analysis - Trip Distribution Mi
𝑇 𝑖𝑗 𝑘 = 𝑇 𝑖𝑗 𝑘−1 ( 𝑂 𝑖 𝑛𝑒𝑤 𝑂 𝑖 𝑘−1 )
𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒? No
Yes
Yes
Stop

29. Row-Column correction example
Transportation Demand Analysis - Trip Distribution

30. Row-Column correction example
Transportation Demand Analysis - Trip Distribution

31. Row-Column correction example
Transportation Demand Analysis - Trip Distribution

32. Row-Column correction example
Transportation Demand Analysis - Trip Distribution

33. Impedance Function
Common impedance functions (transportation system effect):
Hyperbolic:
Exponential:
𝑓 𝑖𝑗 = 𝑐 𝑖𝑗 𝜃 𝜃<0
𝑓 𝑖𝑗 =𝜃1𝑒𝑥𝑝 𝜃2 𝑐 𝑖𝑗 𝜃1>0, 𝜃2<0
Transportation Demand Analysis - Trip Distribution
θ, θ1, θ2= parameters which must be estimated from observed data

34. Hyperbolic Impedance Function
The simplest form:
θ = -2  Original gravity hypothesis
Overestimation of shorter trips
it increases quickly as c decreases and approaches infinity when c approaches zero
𝑓 𝑖𝑗 = 𝑐 𝑖𝑗 θ θ<0
Transportation Demand Analysis - Trip Distribution
𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗

35. Exponential Impedance Function
To correct the infinity problem
𝑓 𝑖𝑗 =𝜃1𝑒𝑥𝑝 𝜃2 𝑐 𝑖𝑗 𝜃1>0, 𝜃2<0
This function approaches θ1 when c approaches zero.
Transportation Demand Analysis - Trip Distribution

36. Comparison (hyperbolic vs. exponential)
Both are monotonically decreasing functions of c
θ1 and θ2 are related to the total (or average) trip cost in a system, although this relationship is usually not expressed explicitly
Transportation Demand Analysis - Trip Distribution

37. Gamma Function
fij is expected to approach zero as c does in following cases:
Models for vehicular trips (excluding walking trips)
Models for work trips or specialty shopping trips
The most commonly used function with these characteristics is a Gamma function fij = θ1cijexp-(θ2cij)
* θi is positive
fij
fij
Transportation Demand Analysis - Trip Distribution
fij

38. Real Cases
fij
Transportation Demand Analysis - Trip Distribution

39. Gravity Model Calibration
Aim:
Calibrate the impedance function (based on current trip matrix)
A try and error approach is suggested
fij
cij
Transportation Demand Analysis - Trip Distribution
Initial value f =1.0
1.0

40. Example
Calibrate the impedance function in 5min intervals for a 4-zone city such to:
Current Travel-time matrix (minutes)
Time 1 2 3 4 5 16 13 18 7 20 12 9
Transportation Demand Analysis - Trip Distribution
Current Trip interchange matrix (trips)
Trips 1 2 3 4 Pi
250
125
375 75
825
100
400 50
225
775
205 60
420
910
155
215
320
175
865 Aj
710
800
970
895
3375

41. Example Select the O-D pairs according to the requested intervals:
Time 1 2 3 4 5 16 13 18 7 20 12 9
Transportation Demand Analysis - Trip Distribution
Travel Time
OD Pairs
22,34,43
13,31,24,42
12,21,14,41,23,32
11,33,44

42. Example
Determine the observed trips for each of the requested intervals:
Trips 1 2 3 4 Pi
125
375 75
825
100
400 50
225
775
205 60
420
910
155
215
320
865 Aj
710
800
970
895
3375
250
225
175 =
T11+T33+T44
= 650
Transportation Demand Analysis - Trip Distribution
Travel Time
OD Pairs
Observed
F1 (assumed for 1st iteration)
650 1
22,34,43
1140
13,31,24,42
1020
12,21,14,41,23,32
565
11,33,44

43. Example 𝑇11= 825∗710∗𝐹1 710∗𝐹1+800∗𝐹1+970∗𝐹1+895∗𝐹1
Trips 1 2 3 4 Pi
173.56
195.56
237.11
218.78
825
163.04
183.70
222.74
205.52
775
191.44
215.70
261.54
241.32
910
181.97
205.04
248.61
229.39
865 Aj
710
800
970
895
3375
Transportation Demand Analysis - Trip Distribution
𝑇11= 825∗710∗𝐹1 710∗𝐹1+800∗𝐹1+970∗𝐹1+895∗𝐹1
= 825∗710∗1 710∗1+800∗1+970∗1+895∗1 =173.56

44. Example
Trips 1 2 3 4 Pi
173.56
195.56
237.11
218.78
825
163.04
183.70
222.74
205.52
775
191.44
215.70
261.54
241.32
910
181.97
205.04
248.61
229.39
865 Aj
710
800
970
895
3375
=664.48
Travel Time
OD Pairs
Calculated F2 ∆F
11,33,44
664.48
0.978
0.022
22,34,43
673.62
1.692
0.692
13,31,24,42
839.10
1.215
0.216
12,21,14,41,23,32
0.471
0.528
Transportation Demand Analysis - Trip Distribution
F2 = 𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 ×𝐹1= ×1=0.978
Convergence criterion: ∆F≤0.01

45. Example 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 =214.34 Trips 1 2 3 4 Pi 214.34
Travel Time
Zones
Calculated F2
11,33,44
664.48
22,34,43
673.62
1.692
13,31,24,42
839.10
12,21,14,41,23,32
Trips 1 2 3 4 Pi
173.56
195.56
237.11
218.78
825
163.04
183.70
222.74
205.52
775
191.44
215.70
261.54
241.32
910
181.97
205.04
248.61
229.39
865 Aj
710
800
970
895
3375
0.978
1.216
0.472
𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗
= ×
825×710×0.978
=214.34
0.472×800
1.216×970
0.472×895
Transportation Demand Analysis - Trip Distribution
Trips 1 2 3 4 Pi
214.34
116.46
363.9
130.29
825
80.25
324.41
109.63
260.69
775
212.04
92.71
233.12
372.12
910
75.749
219.95
371.28
198.01
865 Aj
753.53
961.11
3375

46. Example
Travel Time
OD Pairs
Calculated F3 ∆F
11,33,44
645
0.985
0.007
22,34,43
1068
1.807
0.114
13,31,24,42
1057
1.173
0.042
12,21,14,41,23,32
605
0.440
0.031
Trips 1 2 3 4 Pi
112.48
363.39
125.84
825
346.20
102.33
251.56
775
85.32
231.37
391.55
910
69.614
208.98
390.13
196.26
865 Aj
752.99
965.22
3375
Transportation Demand Analysis - Trip Distribution

47. Example
Travel Time
OD Pairs
Calculated F4 ∆F
11,33,44
651
0.984
0.001
22,34,43
1128
1.826
0.019
13,31,24,42
1026
1.167
0.007
12,21,14,41,23,32
570
0.436
0.004
Trips 1 2 3 4 Pi
224.22
112.03
363.41
125.33
825
74.12
349.64
101.26
249.97
775
200.20
84.31
230.55
394.91
910
68.77
207.32
393.37
195.51
865 Aj
567.32
753.32
966.74
3375
Transportation Demand Analysis - Trip Distribution

48. Example ∆F≤0.01 √ Transportation Demand Analysis - Trip Distribution
Travel Time
OD Pairs
Calculated F5 ∆F
11,33,44
650
0.9832
0.000
22,34,43
1138
0.003
13,31,24,42
1021
0.001
12,21,14,41,23,32
566
∆F≤ √
Transportation Demand Analysis - Trip Distribution

49. Notes f reflects the effect of transportation system in Gravity model
Modifications in transportation system could be captured by f
Distance-based calibration of f results in Fratar model (why?)
f is usually adjusted for generalized cost (e.g., time, cost,…)
f may represent by Gamma or Negative-binomial distribution
Transportation Demand Analysis - Trip Distribution

50. Demand Models of Trip Distribution
Transportation Demand Analysis
Trip Distribution
Demand Models of Trip Distribution

51. Definition 𝑈 𝑖𝑗 𝑘 :Utility of trip from i to j, for individual k
𝑝 𝑗 :Population of zone j (Attraction of j)
𝑇 𝑖𝑗 𝑘 :Number of trip by individual k between i and j
𝑑 𝑖𝑗 :Impedance factor
𝑀 𝑖 𝑘 :A property of zone i for individual k
Transportation Demand Analysis - Trip Distribution

52. Demand Models of Trip Distribution
fij=ln(Tij) 1
Tij
Transportation Demand Analysis - Trip Distribution

53. Demand Models of Trip Distribution
Transportation Demand Analysis - Trip Distribution

54. Destination choice models
Transportation Demand Analysis
Trip Distribution
Destination choice models

55. Determine the percentage distribution of trips
Concept
Determine the percentage distribution of trips
from a given origin to available destinations
not directly the flow of traffic
It follows the general structure of transportation choice model
based on the principle of individual utility maximization
Transportation Demand Analysis - Trip Distribution

56. Definitions
i: individual (or a homogenous group of individual trip makers)
J: destinations available for a particular trip purpose
Pi(j/J): Probability of choice of destination j among J destinations
𝜋 𝑖𝑗 = 𝑝 𝑖 𝑗 𝐽 =𝑓 𝐴 𝑖𝑗 ; 𝐴 𝑖𝑘 for all 𝑘∈𝐽
Aij is a vector of attributes of destination j for traveler i
attractiveness, travel cost, socioeconomic attributes
Transportation Demand Analysis - Trip Distribution

57. Multinomial logit structure 𝜋 𝑖𝑗 = 𝑝 𝑖 𝑗 𝐽 = 𝑒 𝑉 𝑖 (𝑗) 𝑘 𝑒 𝑉 𝑖 (𝑘)
Formulation
Multinomial logit structure
𝜋 𝑖𝑗 = 𝑝 𝑖 𝑗 𝐽 = 𝑒 𝑉 𝑖 (𝑗) 𝑘 𝑒 𝑉 𝑖 (𝑘)
Vi(j): Generalized travel cost of destination j for traveler i
a function of Attributes of destination j
tij: trip distribution
𝑡 𝑖𝑗 = 𝑎 𝑖 𝜋 𝑖𝑗
ai: number of trip makers
Transportation Demand Analysis - Trip Distribution

58. Intervening opportunity models
Transportation Demand Analysis
Trip Distribution
Intervening opportunity models

59. Concept According to Stouffer, 1940: 𝑇 𝑖𝑗 =𝑘 𝑎 𝑗 𝑉 𝑗
The probability of choice of a particular destination j is
proportional to
the opportunity for trip purpose satisfaction at that destination, (aj)
inversely proportional to
all such opportunities that are closer to the trip maker’s origin, (Vj)
(i.e., intervening opportunities)
Transportation Demand Analysis - Trip Distribution
𝑇 𝑖𝑗 =𝑘 𝑎 𝑗 𝑉 𝑗

60. Definitions dv v P(dv): Probability of satisfaction at dv opportunity
P(v): Probability of satisfaction after considering v opportunities
L : proportionality constant of accepting a destination opportunity
𝑃 𝑑𝑣 =𝐿 1−𝑃 𝑣 𝑑 𝑣
Assuming uniformity for the probability of satisfaction at destinations:
Therefore:
𝑃 𝑑𝑣 =𝑑𝑃 𝑣
Transportation Demand Analysis - Trip Distribution
𝑑𝑃(𝑣) 1−𝑃(𝑣) =𝐿𝑑 𝑣 ⇒ − ln 1−𝑃 𝑣 =𝑙𝑣+𝑐⇒ 1−𝑃 𝑣 = 𝑘𝑒 −𝑙𝑣
𝑉 𝑗 = total destination opportunities from origin zone i to the jth destination
𝑃 𝑣 𝑗 =1− 𝑘𝑒 −𝑙 𝑣 𝑗

61. Definitions 𝑃 𝑣 𝑗 =1−𝑘 𝑒 −𝑙 𝑣 𝑗
𝑃 𝑣 𝑗 =1−𝑘 𝑒 −𝑙 𝑣 𝑗
U(vj): Probability of dissatisfaction after considering vj opportunities and continue the trip to destination j+1
P(vj)+U(vj)=  U(vj)= ke-lvj
Opportunity of destination j: vj -vj-1
Probability of staying at destination j: U(vj-1)-U(vj)
Transportation Demand Analysis - Trip Distribution

62. Equation k= 1 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 )
Tij=Oi*(U(vj-1)-U(vj))
= Oi* 𝑘 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 )
 Oi* 𝑘 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) =𝑂i
k= 1 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 )
𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) = (𝑒 −𝑙 𝑣 −1 − 𝑒 −𝑙 𝑣 0 ) + (𝑒 −𝑙 𝑣 0 − 𝑒 −𝑙 𝑣 1 ) +…+ (𝑒 −𝑙 𝑣 𝐽−1 − 𝑒 −𝑙 𝑣 𝐽 ) = (𝑒 −𝑙 𝑣 −1 − 𝑒 −𝑙 𝑣 𝐽 )
Transportation Demand Analysis - Trip Distribution
v-1: opportunity before first destination (j=0): 0
𝑣 𝐽 : total destination opportunities for all J destinations
 Tij= Oi∗ (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) − 𝑒 −𝑙 𝑣 𝐽

63. Example Example travel-time Matrix (minutes)1 2 3 4 5 16 13 18 7 20 12 9
Example travel-interchange Matrix (trips)
Transportation Demand Analysis - Trip Distribution
Trips 1 2 3 4 Pi ?
825
775
910
865 Aj
710
800
970
895
3375

64. Example Order zones by travel time and subtended volumes1 2 3 4 5 16 13 18 7 20 12 9
→5<13<16<18 →
𝑇11<𝑇13<𝑇12<𝑇14
𝑂𝑟𝑑𝑒𝑟𝑒𝑑 𝑍𝑜𝑛𝑒:1,3,2,4
Transportation Demand Analysis - Trip Distribution
Origin Zone
Order 1 3 2 4 Vj
710
1680=
2480=
3375=
800
1695
2405
3375
970
1865
2575
895
2665

65. Example
Sample Calculations for “Calibrated” intervening opportunity Model
Origin Zone
 Destination Zone
  𝑂 𝑖
 1− 𝑒 −𝐿 𝑉 J
  𝑒 −𝐿 𝑉 𝑗−1
  𝑒 −𝐿 𝑉 𝑗
  𝑇 𝑖𝑗 1
825
248 3
264 2
775
259
141 4
910
252
865
262
Transportation Demand Analysis - Trip Distribution
L= 1 Number of current trips = =2.963× 10 −4

66. Example Estimated Trip-Interchange for intervening opportunity Model
Transportation Demand Analysis - Trip Distribution
Final estimated Trip-Interchange Matrix (using row-column factors)

67. Transportation Demand Analysis - Trip Distribution
Finish