Gravity Model

Example 1

Trip distribution between two zones with the gravity model — Gravity Model

Zone i produces 3,099 trips. Zone 1 has 2,421 attractions with a friction factor of 0.80; zone 2 has 1,327 attractions with a friction factor of 0.45. Use the gravity model to distribute the trips.

Given

P_i = 3,099 trips
$A_{1} = 2,421, F_{1} = 0.80$
$A_{2} = 1,327, F_{2} = 0.45$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2421(0.80) = 1,937$
Numerator (zone 2)
$1327(0.45) = 597.2$
Denominator
$1,937 + 597.2 = 2,534$
Substituting — T_i1 = 3099(1,937/2,534) = 2,369 trips
Balance — T_i2 = 3099 − 2,369 = 730.3 trips

Answer:

Zone 1 receives 2,369 trips; zone 2 receives 730.3 trips

Why the other options are there

1,550 trips each (attractions ignored)
2,002 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 2

Trip distribution between two zones with the gravity model — Gravity Model (2)

Zone i produces 3,353 trips. Zone 1 has 2,976 attractions with a friction factor of 0.70; zone 2 has 2,476 attractions with a friction factor of 0.70. Use the gravity model to distribute the trips.

Given

P_i = 3,353 trips
$A_{1} = 2,976, F_{1} = 0.70$
$A_{2} = 2,476, F_{2} = 0.70$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2976(0.70) = 2,083$
Numerator (zone 2)
$2476(0.70) = 1,733$
Denominator
$2,083 + 1,733 = 3,816$
Substituting — T_i1 = 3353(2,083/3,816) = 1,830 trips
Balance — T_i2 = 3353 − 1,830 = 1,523 trips

Answer:

Zone 1 receives 1,830 trips; zone 2 receives 1,523 trips

Why the other options are there

1,677 trips each (attractions ignored)
1,830 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 3

Trip distribution between two zones with the gravity model — Gravity Model (3)

Zone i produces 2,510 trips. Zone 1 has 2,203 attractions with a friction factor of 0.40; zone 2 has 2,910 attractions with a friction factor of 0.45. Use the gravity model to distribute the trips.

Given

P_i = 2,510 trips
$A_{1} = 2,203, F_{1} = 0.40$
$A_{2} = 2,910, F_{2} = 0.45$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2203(0.40) = 881.2$
Numerator (zone 2)
$2910(0.45) = 1,310$
Denominator
$881.2 + 1,310 = 2,191$
Substituting — T_i1 = 2510(881.2/2,191) = 1,010 trips
Balance — T_i2 = 2510 − 1,010 = 1,500 trips

Answer:

Zone 1 receives 1,010 trips; zone 2 receives 1,500 trips

Why the other options are there

1,255 trips each (attractions ignored)
1,081 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 4

Trip distribution between two zones with the gravity model — Gravity Model (4)

Zone i produces 3,520 trips. Zone 1 has 1,974 attractions with a friction factor of 0.35; zone 2 has 886 attractions with a friction factor of 0.95. Use the gravity model to distribute the trips.

Given

P_i = 3,520 trips
$A_{1} = 1,974, F_{1} = 0.35$
$A_{2} = 886, F_{2} = 0.95$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1974(0.35) = 690.9$
Numerator (zone 2)
$886(0.95) = 841.7$
Denominator
$690.9 + 841.7 = 1,533$
Substituting — T_i1 = 3520(690.9/1,533) = 1,587 trips
Balance — T_i2 = 3520 − 1,587 = 1,933 trips

Answer:

Zone 1 receives 1,587 trips; zone 2 receives 1,933 trips

Why the other options are there

1,760 trips each (attractions ignored)
2,430 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 5

Trip distribution between two zones with the gravity model — Gravity Model (5)

Zone i produces 3,940 trips. Zone 1 has 2,276 attractions with a friction factor of 0.65; zone 2 has 817 attractions with a friction factor of 0.45. Use the gravity model to distribute the trips.

Given

P_i = 3,940 trips
$A_{1} = 2,276, F_{1} = 0.65$
$A_{2} = 817, F_{2} = 0.45$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2276(0.65) = 1,479$
Numerator (zone 2)
$817(0.45) = 367.7$
Denominator
$1,479 + 367.7 = 1,847$
Substituting — T_i1 = 3940(1,479/1,847) = 3,156 trips
Balance — T_i2 = 3940 − 3,156 = 784.2 trips

Answer:

Zone 1 receives 3,156 trips; zone 2 receives 784.2 trips

Why the other options are there

1,970 trips each (attractions ignored)
2,899 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 6

Trip distribution between two zones with the gravity model — Gravity Model (6)

Zone i produces 2,033 trips. Zone 1 has 1,914 attractions with a friction factor of 0.50; zone 2 has 1,264 attractions with a friction factor of 0.85. Use the gravity model to distribute the trips.

Given

P_i = 2,033 trips
$A_{1} = 1,914, F_{1} = 0.50$
$A_{2} = 1,264, F_{2} = 0.85$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1914(0.50) = 957.0$
Numerator (zone 2)
$1264(0.85) = 1,074$
Denominator
$957.0 + 1,074 = 2,031$
Substituting — T_i1 = 2033(957.0/2,031) = 957.8 trips
Balance — T_i2 = 2033 − 957.8 = 1,075 trips

Answer:

Zone 1 receives 957.8 trips; zone 2 receives 1,075 trips

Why the other options are there

1,017 trips each (attractions ignored)
1,224 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 7

Trip distribution between two zones with the gravity model — Gravity Model (7)

Zone i produces 3,459 trips. Zone 1 has 1,703 attractions with a friction factor of 0.85; zone 2 has 2,165 attractions with a friction factor of 0.90. Use the gravity model to distribute the trips.

Given

P_i = 3,459 trips
$A_{1} = 1,703, F_{1} = 0.85$
$A_{2} = 2,165, F_{2} = 0.90$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1703(0.85) = 1,448$
Numerator (zone 2)
$2165(0.90) = 1,949$
Denominator
$1,448 + 1,949 = 3,396$
Substituting — T_i1 = 3459(1,448/3,396) = 1,474 trips
Balance — T_i2 = 3459 − 1,474 = 1,985 trips

Answer:

Zone 1 receives 1,474 trips; zone 2 receives 1,985 trips

Why the other options are there

1,730 trips each (attractions ignored)
1,523 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 8

Trip distribution between two zones with the gravity model — Gravity Model (8)

Zone i produces 3,781 trips. Zone 1 has 713 attractions with a friction factor of 0.75; zone 2 has 1,915 attractions with a friction factor of 0.95. Use the gravity model to distribute the trips.

Given

P_i = 3,781 trips
$A_{1} = 713, F_{1} = 0.75$
$A_{2} = 1,915, F_{2} = 0.95$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$713(0.75) = 534.8$
Numerator (zone 2)
$1915(0.95) = 1,819$
Denominator
$534.8 + 1,819 = 2,354$
Substituting — T_i1 = 3781(534.8/2,354) = 858.9 trips
Balance — T_i2 = 3781 − 858.9 = 2,922 trips

Answer:

Zone 1 receives 858.9 trips; zone 2 receives 2,922 trips

Why the other options are there

1,891 trips each (attractions ignored)
1,026 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 9

Trip distribution between two zones with the gravity model — Gravity Model (9)

Zone i produces 1,557 trips. Zone 1 has 2,868 attractions with a friction factor of 0.70; zone 2 has 2,225 attractions with a friction factor of 0.90. Use the gravity model to distribute the trips.

Given

P_i = 1,557 trips
$A_{1} = 2,868, F_{1} = 0.70$
$A_{2} = 2,225, F_{2} = 0.90$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2868(0.70) = 2,008$
Numerator (zone 2)
$2225(0.90) = 2,003$
Denominator
$2,008 + 2,003 = 4,010$
Substituting — T_i1 = 1557(2,008/4,010) = 779.5 trips
Balance — T_i2 = 1557 − 779.5 = 777.5 trips

Answer:

Zone 1 receives 779.5 trips; zone 2 receives 777.5 trips

Why the other options are there

778.5 trips each (attractions ignored)
876.8 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 10

Trip distribution between two zones with the gravity model — Gravity Model (10)

Zone i produces 2,230 trips. Zone 1 has 2,226 attractions with a friction factor of 0.70; zone 2 has 2,621 attractions with a friction factor of 0.80. Use the gravity model to distribute the trips.

Given

P_i = 2,230 trips
$A_{1} = 2,226, F_{1} = 0.70$
$A_{2} = 2,621, F_{2} = 0.80$

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2226(0.70) = 1,558$
Numerator (zone 2)
$2621(0.80) = 2,097$
Denominator
$1,558 + 2,097 = 3,655$
Substituting — T_i1 = 2230(1,558/3,655) = 950.7 trips
Balance — T_i2 = 2230 − 950.7 = 1,279 trips

Answer:

Zone 1 receives 950.7 trips; zone 2 receives 1,279 trips

Why the other options are there

1,115 trips each (attractions ignored)
1,024 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Core formulas for this FE topic

Worked exam-style examples

Gravity Model

Core formulas for this FE topic

Tij=Pi>/Tij = Pi > /Tij=Pi>/

Tij=numberoftripsthatareproducedinZoneiandattractedtoZonejTij = number of trips that are produced in Zone i and attracted to Zone jTij=numberoftripsthatareproducedinZoneiandattractedtoZonej

Pi=totalnumberoftripsproducedinZoneiPi = total number of trips produced in Zone iPi=totalnumberoftripsproducedinZonei

Aj=numberoftripsattractedtoZonejAj = number of trips attracted to Zone jAj=numberoftripsattractedtoZonej

Fij=frictionfactorthatisaninversefunctionoftraveltimebetweenZonesiandjFij = friction factor that is an inverse function of travel time between Zones i and jFij=frictionfactorthatisaninversefunctionoftraveltimebetweenZonesiandj

Kij=socioeconomicadjustmentfactorfortravelbetweenZonesiandjKij = socioeconomic adjustment factor for travel between Zones i and jKij=socioeconomicadjustmentfactorfortravelbetweenZonesiandj

Worked exam-style examples