Given: Small town with three transportation analysis zones, and origin-destination survey results. Provide a trip distribution calculation using the gravity model for two iterations; assume Kij = 1. The following table shows the number of productions and attractions in each zone:
The survey’s results for the zones’ travel time in minutes were as follows:
The following table shows travel time versus friction factor.
What will be an ideal response?
The mathematical formulation for the gravity model as provided as Equation 12.3:
Since Kij = 1, this factor does not affect calculations. The iterative application of
Equation 12.3 is as follows:
Iteration 1
T11 = 250 × ((395 × 26) / ((395 × 26) + (180 × 41) + (425 × 52)))
T11 = 250 × (10,270 / 39,750)
T11 = 65
T12 = 250 × ((180 × 41) / ((395 × 26) + (180 × 41) + (425 × 52)))
T12 = 250 × (7,380 / 39,750)
T12 = 46
T13 = 250 × ((425 × 52) / ((395 × 26) + (180 × 41) + (425 × 52)))
T13 = 250 × (22,100 / 39,750)
T13 = 139
T21 = 450 × ((395 × 52) / ((395 × 52) + (180 × 13) + (425 × 50)))
T21 = 450 × (20,540 / 44,130)
T21 = 209
T22 = 450 × ((180 × 13) / ((395 × 52) + (180 × 13) + (425 × 50)))
T22 = 450 × (2,340 / 44,130)
T22 = 24
T23 = 450 × ((425 × 50) / ((395 × 52) + (180 × 13) + (425 × 50)))
T23 = 450 × (21,250 / 44,130)
T23 = 217
T31 = 300 × ((395 × 82) / ((395 × 82) + (180 × 50) + (425 × 39)))
T31 = 300 × (32,390 / 57,965)
T31 = 168
T32 = 300 × ((180 × 50) / ((395 × 82) + (180 × 50) + (425 × 39)))
T32 = 300 × (9,000 / 57,965)
T32 = 46
T33 = 300 × ((425 × 39) / ((395 × 82) + (180 × 50) + (425 × 39)))
T33 = 300 × (16,575 / 57,965)
T33 = 86
Next, calculate the adjusted attraction factors using Equation 12.4.
Zone 1
Ajk = (395 / 442) × 395
Ajk = 353
Zone 2
Ajk = (180 / 116) × 180
Ajk = 279
Zone 3
Ajk = (425 / 442) × 425
Ajk = 409
Now apply the gravity model formula for Iteration 2 using the above adjusted
attraction factors.
Iteration 2
T11 = 250 × ((353 × 26) / ((353 × 26) + (279 × 41) + (409 × 52)))
T11 = 250 × (9,178 / 41,885)
T11 = 55
T12 = 250 × ((279 × 41) / ((353 × 26) + (279 × 41) + (409 × 52)))
T12 = 250 × (11,439 / 41,885)
T12 = 68
T13 = 250 × ((409 × 52) / ((353 × 26) + (279 × 41) + (409 × 52)))
T13 = 250 × (21,268 / 41,885)
T13 = 127
T21 = 450 × ((353 × 52) / ((353 × 52) + (279 × 13) + (409 × 50)))
T21 = 450 × (18,356 / 42,433)
T21 = 195
T22 = 450 × ((279 × 13) / ((353 × 52) + (279 × 13) + (409 × 50)))
T22 = 450 × (3,627 / 42,433)
T22 = 38
T23 = 450 × ((409 × 50) / ((353 × 52) + (279 × 13) + (409 × 50)))
T23 = 450 × (20,450 / 42,433)
T23 = 217
T31 = 300 × ((353 × 82) / ((353 × 82) + (279 × 50) + (409 × 39)))
T31 = 300 × (28,946 / 58,847)
T31 = 148
T32 = 300 × ((279 × 50) / ((353 × 82) + (279 × 50) + (409 × 39)))
T32 = 300 × (13,950 / 58,847)
T32 = 71
T33 = 300 × ((409 × 39) / ((353 × 82) + (279 × 50) + (409 × 39)))
T33 = 300 × (15,951 / 58,847)
T33 = 81
Observe that the computed attractions approximately equal the given attractions.
A total convergence would be expected in another iteration.
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