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Trip Distribution
Trip distribution is the second step in the four-step modeling process. Trip distribution is a process by which the trips generated in one zone are allocated to other zones in the study area. These trips may be within the study area (internal internal) or between the study area and areas outside the study area (internal external).
Trip distribution requires explanatory variables that are related to the impedance (generally a function of travel time and/or cost) of travel between zones, as well as the amount of trip-making activity in both the origin zone and the destination zone.
The inputs to trip distribution models include: The trip generation outputs (the productions and attractions by trip purpose for each zone) and, Measures of travel impedance between each pair of zones, obtained from the transportation networks. Socioeconomic and area characteristics are sometimes also used as inputs. The outputs are trip tables, production zone to attraction zone, for each trip purpose. Because trips of different purposes have different levels of sensitivity to travel time and cost, trip distribution is applied separately for each trip purpose, with different model parameters.
The distribution of trips between zones can be better illustrated in the form of a matrix.
There are two types of trip distribution matrices:
- Production- attraction PA matrix.
- Origin destination OD matrix.
In the PA matrix the rows and columns represent production and attraction zones respectively as shown in Table (1).
Table (1): Production-Attraction Matrix. Attraction zones
Production zones
1 ……… j ……… n sum 1... i... N
sum
- The cell element is the number of trips produced at zone i and attracted to zone j.
- Hence a PA matrix has no directional meaning and its cell element represent trip interchanges.
- The number of production zones generally equals the number of attraction zones, n.
- The row sum, is the total number of trips produced at zone i and the column sum is the total number of trips attracted to zone j.
- The sum of production equals the sum of attractions.
Converting a PA matrix into an OD Matrix
For directional traffic assignment, the PA matrix must be converted to an OD matrix. Consider the trip interchange between zones 1 and 2 shown in the following Figure below:
Let the proportion of trips originating from the production zone, denoted by , be 0.
Then, the general formulae for finding the cell elements of an OD matrix, from the cell elements of PA matrix, are:
λ) λ)
Conversion of PA matrix into OD matrix ( λ= 0.4)
O Zone 1
D Zone 2
D Zone 1
O Zone 2
A Zone 2
P Zone 1
𝒕𝟏𝟐 𝟐𝟎𝟎 𝒕𝒓𝒊𝒑𝒔
𝒕𝟏 𝟐 𝟖𝟎 𝒕𝒓𝒊𝒑𝒔 𝒕𝟐^ 𝟏𝟐𝟎𝟏 𝒕𝒓𝒊𝒑𝒔
1 2 sum 1 20 100 120 2 40 60 100 sum 60 160 220
1 2 sum 1 20 40+24=64 84 2 16+60=76 60 136 sum 96 124 220
Note that the sum and equals the sum and also, the
diagonal elements in both matrices are the same.
In practice the value of λ will be around 0.5 for a 24 hour matrix which will result in a symmetrical OD matrix.
Consider the previous example by taking λ= 0.
1 2 sum 1 20 100 120 2 40 60 100 sum 60 160 220
1 2 sum 1 20 50+20= 70
90 2 20+ 50 = 70
60 130 sum 90 130 220
Attraction zone
Production zone
Attraction zone
Production zone
Origin zone
Destination zone
2- Terminal times and costs. Terminal times represent impedances at the origin and destination of the trip such as the amount of time spent to walk to/from a transit mode, to park or access a parked car, to pay parking cost, and so on. Terminal times vary by the area type of the zone. The terminal times are directly proportional to the population and employment density of the zone.
3- Intrazonal impedance. Network models do not assign trips that are made within a zone (intrazonal trips). For that reason, when a network is skimmed, intrazonal times are not computed and must be added separately to this skim matrix. There are a number of techniques for estimating intrazonal times.
- Some of these methods use the average of the skim times to the nearest neighboring zones and define the intrazonal time as one- half of this average.Various mechanisms are used to determine which zones should be used in this calculation, including using a fixed number of closest zones or using all zones whose centroids are within a certain distance of the zone’s centroid.
- Other methods compute intrazonal distance based on a function of the zone’s area, for example, proportional to the square root of the area. Intrazonal time is computed by applying an average speed to this distance.
Friction Factors
Friction factor is the primary independent variable and the impact measure of the spatial separation and travel time between two zones in gravity model. Friction factors are inversely related to spatial separation between zones. As the travel time increases, the friction factors decreases. A number of different functional forms have been used to calculate friction factors including exponential, inverse power, and gamma functions. Exponential:
Inverse Power:
Gamma:
Where;
Fij the friction factors between zones i and j, a, b, and c model coefficients, tij the travel time between zone i and j, and e the base of natural logarithms.
The gamma function is the most and recommends one used in the transportation planning practices.
Note that since a is a scaling parameter that does not change the shape of the gamma function curve, it can be set at any value that proves convenient for the modeler to interpret the friction factors.
Figure 1. Home-based work trip distribution gamma functions.
Figure 2. Home-based nonwork trip distribution gamma functions.
Figure 3. Nonhome-based trip distribution gamma functions.
Example 1 Use of Calibrated F Values and Iteration To illustrate the application of the gravity model, consider a study area consisting of three zones. The data have been determined as follows: the number of productions and attractions has been computed for each zone by methods described in the section on trip generation, and the average travel times between each zone have been determined. Both are shown in Tables below. Assume Kij is the same unit value for all zones. Finally, the F values have been calibrated as previously described and are shown in Table below for each travel time increment. Note that the intrazonal travel time for zone 1 is larger than those of most other inter-zone times because of the geographical characteristics of the zone and lack of access within the area. This zone could represent conditions in a congested downtown area. Determine the number of zone-to-zone trips through two iterations.
Table 4: Zone-to-Zone Trips: First Iteration, Singly Constrained.
The results summarized in Table 4 represent a singly constrained gravity model. This constraint is that the sum of the productions in each zone is equal to the number of productions given in the problem statement. However, the number of attractions estimated in the trip distribution phase differs from the number of attractions given. For zone 1, the correct number is 300, whereas the computed value is
- Values for zone 2 are 270 versus 210, and for zone 3, they are 180 versus 161. To create a doubly constrained gravity model where
the computed attractions equal the given attractions, calculate the adjusted attraction factors according to the formula.
To produce a doubly constrained gravity model, repeat the trip distribution computations using modified attraction values so that the numbers attracted will be increased or reduced as required. For zone 1, for example, the estimated attractions were too great. Therefore, the new attraction factors are adjusted downward by multiplying the original attraction value by the ratio of the original to estimated attraction values.
Perform a second iteration using the adjusted attraction values.
Growth Factor Model Trip distribution can also be computed when the only data available are the origins and destinations between each zone for the current or base year and the trip generation values for each zone for the future year. This method was widely used when O-D data were available but the gravity model and calibrations for F factors had not yet become operational. Growth factor models are used primarily to distribute trips between zones in the study area and zones in cities external to the study area. The growth factor method cannot reflect changes in travel time between zones, as does the gravity model. The most popular growth factor model is the Fratar method , which is a mathematical formula that proportions future trip generation estimates to each zone as a function of the product of the current trips between the two zones Tij and the growth factor of the attracting zone Gj.
∑
Where Tij : number of trips estimated from zone i to zone j ti : present trip generation in zone i Gx : growth factor of zone x Ti : tiGi : future trip generation in zone i tix : number of trips between zone i and other zones x tij : present trips between zone i and zone j Gj : growth factor of zone j
Example 2 Forecasting Trips Using the Fratar Model A study area consists of four zones (A, B, C, and D). An O-D survey indicates that the number of trips between each zone is as shown in Table 1. Planning estimates for the area indicate that in five years the number of trips in each zone will increase by the growth factor shown in Table 2 and that trip generation will be increased to the amounts shown in the last column of the table. Determine the number of trips between each zone for future conditions. Table 1 : Present Trips between Zones.
Table 2 : Present Trip Generation and Growth Factors.
Solution: Using the Fratar formula, calculate the number of trips between zones A and B, A and C, A and D, and so forth. Note that two values are obtained for each zone pair (that is, TAB and TBA ). These values are averaged, yielding a value for TAB = ( TAB + TBA ) /2. The calculations are as follows.
expected to occur in five years and the trip generation estimated in the preceding calculation. The values are given in Table 4. Table 4 : Growth Factors for Second Iteration
The calculations for the second iteration are left to the student to complete and the process can be repeated as many times as needed until the estimate and actual trip generation values are close in agreement. Example 3 : A 3 by 3 trip table representing a total of 2500 trips is shown in the following table, which is for the base year. Zone 1 2 3 Total 1 1 4 2 7 2 6 2 3 11 3 4 1 2 7 Total 11 7 7 25
The next table indicates the origin and destination growth factors for horizon year:
Zone 1 2 3
Origin factor (production) 2 3 4
Destination factor (attraction) 3 4 2
Use the frater technique to distribute the trips in the horizon year.
Solution:-
∑ T1-1=72 (13) / (13)+(44)+(22)= T1-2=72 (44) / (13)+(44)+(22)= T1-3=72 (22) / (13)+(44)+(22)= T2-1=113 (63) / (63)+(24)+(32)= T2-2=113 (24) / (63)+(24)+(32)= T2-3=113 (32) / (63)+(24)+(32)= Similar calculation yield:
Zone 1 2 3
Estimated total trip generation
Actual trip generation ( Ti*Gi)
*New growth factor 1 2 10 2 14 14 1 2 19 8 6 33 33 1 3 17 6 6 29 28 0. Estimated total trip generation
Actual trip generation 33 28 14 *New growth factor
*Must be (0.95-1.05) ** Out of range.
Zone 1 2 3
Origin factor(production) 1 1 0.
Destination factor (attraction) 0.87 1.17 1
