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:

Question

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?

Answer 1

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.