Given:
The given table for number of productions and attractions in each district and travel time in minutes is
District | Productions | Attractions | Travel time (min) |
1 | 2 | 3 | 4 |
1 | 3400 | 2800 | 4 | 11 | 15 | 10 |
2 | 6150 | 6500 | 11 | 6 | 6 | 9 |
3 | 3900 | 2550 | 15 | 6 | 6 | 11 |
4 | 2800 | 4400 | 10 | 9 | 11 | 4 |
Table (1)
The given table for travel time versus friction factor is
Time(min) | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Fij | 1.51 | 1.39 | 1.30 | 1.22 | 1.13 | 1.04 | 0.97 | 0.92 | 0.85 | 0.80 | 0.75 | 0.71 |
Table (2)
Formula used:
The equation to calculate the number of attractions is given by
Ajk=AjCj( k−1)Aj(k−1) ...... (I)
Here, Ajk is the adjusted factor for attraction district, Cjk is the actual attraction total for district, Aj is the desired attraction total for district, j is attraction district number, n is number of district s, k is iteration number, and m is the number of iteration..
The equation to calculate the number of trip between each district is given by
Tij=Pi[AjFijKij∑ A j F ij K ij] ...... (II)
Here, Pi is the number of trips produced in district i, Aj is the number of trips attracted to district j, Fij is the inverse function of travel time, Kij is the socio-economic adjustment factor for interchange j and Tij is the number of trips that are produced in district i and attracted to district j
Calculation:
Trip distribution for district 1 is calculated as
Substitute 3400 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−1.
T1−1=3400[2800×1.51×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3400[422816286.5]=883
Substitute 3400 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−2.
T1−2=3400[6500×0.92×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3400[598016286.5]=1248
Substitute 3400 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−3.
T1−3=3400[2550×0.71×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3400[1810.516286.5]=378
Substitute 3400 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−4.
T1−4=3400[4400×0.97×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3400[426816286.5]=891
Trip distribution for district 2 is calculated as
Substitute 6150 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−1.
T2−1=6150[2800×1.51×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=6150[422816286.5]=1597
Substitute 6150 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−2.
T2−2=6150[6500×0.92×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=6150[598016286.5]=2258
Substitute 6150 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−3.
T2−3=6150[2550×0.71×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=6150[1810.516286.5]=684
Substitute 6150 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−4.
T2−4=6150[4400×0.97×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=6150[426816286.5]=1612
Trip distribution for district 3 is calculated as
Substitute 3900 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−1.
T3−1=3900[2800×1.51×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3900[422816286.5]=1013
Substitute 3900 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−2.
T3−2=3900[6500×0.92×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3900[598016286.5]=1432
Substitute 3900 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−3.
T3−3=3900[2550×0.71×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3900[1810.516286.5]=434
Substitute 3900 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−4.
T3−4=3900[4400×0.97×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=3900[426816286.5]=1022
Trip distribution for district 4 is calculated as
Substitute 2800 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−1.
T4−1=2800[2800×1.51×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=2800[422816286.5]=727
Substitute 2800 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−2.
T4−2=2800[6500×0.92×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=2800[598016286.5]=1028
Substitute 2800 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−3.
T4−3=2800[2550×0.71×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=2800[1810.516286.5]=311
Substitute 2800 for Pi, 2800, 6500, 2550, 4400 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−4.
T4−4=2800[4400×0.97×12800×1.51×1+6500×0.92×1+2550×0.71×1+4400×0.97×1]=2800[426816286.5]=734
The table for the above calculated values is shown below.
District | 1 | 2 | 3 | 4 | Computed P | Given P |
1 | 883 | 1248 | 378 | 891 | 3400 | 3400 |
2 | 1597 | 2258 | 684 | 1612 | 6151 | 3400 |
3 | 1013 | 1432 | 434 | 1022 | 3901 | 3900 |
4 | 727 | 1028 | 311 | 734 | 2800 | 2800 |
Computed A | 189 | 5966 | 1807 | 4263 | 16252 | 16250 |
Given A | 2800 | 6500 | 2550 | 4400 | 16250 | |
Table (3)
The adjusted value for district 1 is calculated as
Substitute 2800 for Aj, 2800 for Aj(k−1), and 4220 for Cj(k−1) in equation (I).
Ajk=28004220×2800=1858
The adjusted value for district 2 is calculated as
Substitute 6500 for Aj, 6500 for Aj(k−1), and 5966 for Cj(k−1) in equation (I).
Ajk=65005966×6500=7082
The adjusted value for district 3 is calculated as
Substitute 2550 for Aj, 2550 for Aj(k−1), and 1807 for Cj(k−1) in equation (I).
Ajk=25501807×2550=3599
The adjusted value for district 4 is calculated as
Substitute 4400 for Aj, 4400 for Aj(k−1), and 4263 for Cj(k−1) in equation (I).
Ajk=44004263×4400=4541
Carry out second iteration for each district with adjusted values.
Trip distribution for district 1 is calculated as
Substitute 3400 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−1.
T1−1=3400[1858×1.51×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3400[2805.5816290.28]=585
Substitute 3400 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−2.
T1−2=3400[7092×0.92×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3400[6524.6416290.28]=1362
Substitute 3400 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−3.
T1−3=3400[3599×0.71×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3400[2555.2916290.28]=533
Substitute 3400 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T1−4.
T1−4=3400[4541×0.9×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3400[4404.7716290.28]=919
Trip distribution for district 2 is calculated as
Substitute 6150 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−1.
T2−1=6150[1858×1.51×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=6150[2805.5816290.28]=1059
Substitute 6150 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−2.
T2−2=6150[7092×0.92×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=6150[6524.6416290.28]=2463
Substitute 6150 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−3.
T2−3=6150[3599×0.71×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=6150[2555.2916290.28]=965
Substitute 6150 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T2−4.
T2−4=6150[4541×0.9×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=6150[4404.7716290.28]=1663
Trip distribution for district 3 is calculated as
Substitute 3900 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−1.
T3−1=3900[1858×1.51×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3900[2805.5816290.28]=671
Substitute 3900 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−2.
T3−2=3900[7092×0.92×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3900[6524.6416290.28]=1562
Substitute 3900 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−3.
T3−3=3900[3599×0.71×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3900[2555.2916290.28]=612
Substitute 3900 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T3−4.
T3−4=3900[4541×0.9×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=3900[4404.7716290.28]=1055
Trip distribution for district 4 is calculated as
Substitute 2800 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−1.
T4−1=2800[1858×1.51×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=2800[2805.5816290.28]=482
Substitute 2800 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−2.
T4−2=2800[7092×0.92×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=2800[6524.6416290.28]=1122
Substitute 2800 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−3.
T4−3=2800[3599×0.71×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=2800[2555.2916290.28]=439
Substitute 2800 for Pi, 1858, 7092, 3599, 4541 for Aij, 1.51, 0.92, 0.71, 0.97 for Fij and 1 for Kij in equation (II) to calculate T4−4.
T4−4=2800[4541×0.9×11858×1.51×1+7092×0.92×1+3599×0.71×1+4541×0.97×1]=2800[4404.7716290.28]=757
The table for the above calculated values is shown below.
District | 1 | 2 | 3 | 4 | Computed P | Given P |
1 | 585 | 1362 | 533 | 919 | 3399 | 3400 |
2 | 1059 | 2463 | 965 | 1663 | 6150 | 3400 |
3 | 671 | 1562 | 612 | 1055 | 3900 | 3900 |
4 | 482 | 1122 | 439 | 757 | 2800 | 2800 |
Computed A | 2797 | 6509 | 2549 | 4394 | 16249 | 16250 |
Given A | 2800 | 6500 | 2550 | 4400 | 16250 | |
Table (4)
Conclusion:
Therefore, the trip distribution calculation using the gravity model for two iterations is shown above.