Given: Study area with four transportation analysis zones, and origin-destination survey results. Provide a trip distribution calculation using the gravity model for two iterations; assume Kij = 1.
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 = 3,400 × ((2,800 × 1.51) /
((2,800 × 1.51) + (6,500 × 0.92) + (2,550 × 0.71) + (4,400 × 0.97)))
T11 = 3,400 × (4,228 / 16287)
T11 = 883
T12 = 3,400 × ((6,500 × 0.92) /
((2,800 × 1.51) + (6,500 × 0.92) + (2,550 × 0.71) + (4,400 × 0.97)))
T12 = 3,400 × (5,980 / 16,287)
T12 = 1,248
T13 = 3,400 × ((2,550 × 0.71) /
((2,800 × 1.51) + (6,500 × 0.92) + (2,550 × 0.71) + (4,400 × 0.97)))
T13 = 3,400 × (1,811 / 16,287)
T13 = 378
T14 = 3,400 × ((4,400 × 0.97)/
((2,800 × 1.51) + (6,500 × 0.92) + (2,550 × 0.71) + (4,400 × 0.97)))
T14 = 3,400 × (4268 / 16,287)
T14 = 891
T21 = 6,150 × ((2,800 × 0.92) /
((2,800 × 0.92) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 1.04)))
T21 = 6,150 × (2,576 / 18,917)
T21 = 837
T22 = 6,150 × ((6,500 × 1.30) /
((2,800 × 0.92) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 1.04)))
T22 = 6,150 × (8,450 / 18,917)
T22 = 2747
T23 = 6,150 × ((2,550 × 1.30) /
((2,800 × 0.92) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 1.04)))
T23 = 6,150 × (3,315 / 18,917)
T23 = 1078
T24 = 6,150 × ((4,400 × 1.04) /
((2,800 × 0.92) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 1.04)))
T24 = 6,150 × (4576 / 18,917)
T24 = 1488
T31 = 3,900 × ((2,800 × 0.71) /
((2,800 × 0.71) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 0.92)))
T31 = 3,900 × (1,988 / 17,801)
T31 = 436
T32 = 3,900 × ((6,500 × 1.30) /
((2,800 × 0.71) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 0.92)))
T32 = 3,900 × (8,450 / 17,801)
T32 = 1851
T33 = 3,900 × ((2,550 × 1.30) /
((2,800 × 0.71) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 0.92)))
T33 = 3,900 × (3,315 / 17,801)
T33 = 726
T34 = 3,900 × ((4,400 × 0.92) /
((2,800 × 0.71) + (6,500 × 1.30) + (2,550 × 1.30) + (4,400 × 0.92)))
T34 = 3,900 × (4,048 / 17,801)
T34 = 887
T41 = 2,800 × ((2,800 × 0.97) /
((2,800 × 0.97) + (6,500 × 1.04) + (2,550 × 0.92) + (4,400 × 1.51)))
T41 = 2,800 × (2,716 / 18,466)
T41 = 412
T42 = 2,800 × ((6,500 × 1.04) /
((2,800 × 0.97) + (6,500 × 1.04) + (2,550 × 0.92) + (4,400 × 1.51)))
T42 = 2,800 × (6,760 / 18,466)
T42 = 1025
T43 = 2,800 × ((2,550 × 0.92) /
((2,800 × 0.97) + (6,500 × 1.04) + (2,550 × 0.92) + (4,400 × 1.51)))
T43 = 2,800 × (2,346 / 18,466)
T43 = 356
T44 = 2,800 × ((4,400 × 1.51) /
((2,800 × 0.97) + (6,500 × 1.04) + (2,550 × 0.92) + (4,400 × 1.51)))
T44 = 2,800 × (6,644 / 18,466)
T44 = 1007
Next, calculate the adjusted attraction factors using Equation 12.4.
Zone 1
Ajk = (2,800 / 2,568) × 2,800
Ajk = 3,053
Zone 2
Ajk = (6,500 / 6,871) × 6,500
Ajk = 6,149
Zone 3
Ajk = (2,550 / 2,538) × 2,550
Ajk = 2,562
Zone 4
Ajk = (4,400 / 4,273) × 4,400
Ajk = 4,531
Now apply the gravity model formula for Iteration 2 using the above adjusted
attraction factors.
Iteration 2
T11 = 3,400 × ((3,053 × 1.51) /
((3,053 × 1.51) + (6,149 × 0.92) + (2,562 × 0.71) + (4,531 × 0.97)))
T11 = 3,400 × (4,610 /16,481)
T11 = 951
T12 = 3,400 × ((6,149 × 0.92) /
((3,053 × 1.51) + (6,149 × 0.92) + (2,562 × 0.71) + (4,531 × 0.97)))
T12 = 3,400 × (5,657 /16,481)
T12 = 1167
T13 = 3,400 × ((2,562 × 0.71) /
((3,053 × 1.51) + (6,149 × 0.92) + (2,562 × 0.71) + (4,531 × 0.97)))
T13 = 3,400 × (1,819 /16,481)
T13 = 375
T14 = 3,400 × ((4,531 × 0.97) /
((3,053 × 1.51) + (6,149 × 0.92) + (2,562 × 0.71) + (4,531 × 0.97)))
T14 = 3,400 × (4,395 /16,481)
T14 = 907
T21 = 6,150 × ((3,053 × 0.92) /
((3,053 × 0.92) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 1.04)))
T21 = 6,150 × (2,809 / 18,845)
T21 = 917
T22 = 6,150 × ((6,149 × 1.30) /
((3,053 × 0.92) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 1.04)))
T22 = 6,150 × (7,994 / 18,845)
T22 = 2609
T23 = 6,150 × ((2,562 × 1.30) /
((3,053 × 0.92) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 1.04)))
T23 = 6,150 × (3,331 / 18,845)
T23 = 1087
T24 = 6,150 × ((4,531 × 1.04) /
((3,053 × 0.92) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 1.04)))
T24 = 6,150 × (4,712 / 18,845)
T24 = 1538
T31 = 3,900 × ((3,053 × 0.71) /
((3,053 × 0.71) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 0.92)))
T31 = 3,900 × (2,168 / 17,660)
T31 = 479
T32 = 3,900 × ((6,149 × 1.30) /
((3,053 × 0.71) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 0.92)))
T32 = 3,900 × (7,994 / 17,660)
T32 = 1765
T33 = 3,900 × ((2,562 × 1.30) /
((3,053 × 0.71) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 0.92)))
T33 = 3,900 × (3,331 / 17,660)
T33 = 736
T34 = 3,900 × ((4,531 × 0.92) /
((3,053 × 0.71) + (6,149 × 1.30) + (2,562 × 1.30) + (4,531 × 0.92)))
T34 = 3,900 × (4,169 / 17,660)
T34 = 921
T41 = 2,800 × ((3,053 × 0.97) /
((3,053 × 0.97) + (6,149 × 1.04) + (2,562 × 0.92) + (4,531 × 1.51)))
T41 = 2,800 × (2,961 / 18,555)
T41 = 447
T42 = 2,800 × ((6,149 × 1.04) /
((3,053 × 0.97) + (6,149 × 1.04) + (2,562 × 0.92) + (4,531 × 1.51)))
T42 = 2,800 × (6,395 / 18,555)
T42 = 965
T43 = 2,800 × ((2,562 × 0.92) /
((3,053 × 0.97) + (6,149 × 1.04) + (2,562 × 0.92) + (4,531 × 1.51)))
T43 = 2,800 × (2,357 / 18,555)
T43 = 356
T44 = 2,800 × ((4,531 × 1.51) /
((3,053 × 0.97) + (6,149 × 1.04) + (2,562 × 0.92) + (4,531 × 1.51)))
T44 = 2,800 × (6,842 / 18,555)
T44 = 1032
Observe that the computed attractions approximately equal the given attractions.
A total convergence would be expected in another iteration.
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