Gravity Model

A study area consists of three zones. The data have been determined as shown in the following tables. Assume a Kij =1.

Zone Productions and Attractions

 Zone 1 2 3 Total Trip Productions 140 330 280 750 Trip Attractions 300 270 180 750

Travel Time between zones (min)

 Zone 1 2 3 1 5 2 3 2 2 6 6 3 3 6 5

Travel Time versus Friction Factor

 Time (min) F 1 82 2 52 3 50 4 41 5 39 6 26 7 20 8 12

Determine the number of trips between each zone using the gravity model formula and the data given above. Note that while the Friction Factors are given in this problem, they will normally need to be derived by the calibration process described in the Theory and Concepts section.

[Solution Shown Below]

Solution

First, determine the friction factor for each origin-destination pair by using the travel times and friction factors given in the problem statement.

 Fij as Determined from Travel Time Zone 1 2 3 1 39 52 50 2 52 26 26 3 50 26 39

Once you have the friction factors for each potential trip, you can begin solving the gravity model equation as shown below. Solving for the A*F*K term in a tabular form makes this process easier. Study the equation below and the following table.

Where:
Tij = number of trips that are produced in zone I and attracted to zone j
Pi = total number of trips produced in zone I
Aj = number of trips attracted to zone j
Fij = a value which is an inverse function of travel time
Kij = socio economic adjustment factor for interchange ij

 AjFijKij 1 2 3 sum 1 11700 14040 9000 34740 2 15600 7020 4680 27300 3 15000 7020 7020 29040

Once the A*F*K terms for each origin-destination are tabulated, you can insert these values into the gravity model equation and determine the number of trips for each origin-destination. The following table illustrates this.

Zone to Zone First Iteration:

 zone 1 2 3 P 1 47 57 36 140 2 189 85 57 330 3 145 68 68 280 A 380 209 161 750 given A 300 270 180 750

Since the total trip attractions for each zone don’t match the attractions that were given in the problem statement, we need to adjust the attraction factors. Calculate the adjusted attraction factors according to the following formula:

Ajk =

Where:
Ajk = adjusted attraction factor for attraction zone (column) j iteration k.
Ajk = Aj when k=1
Cjk = actual attraction (column) total for zone j, iteration k
Aj= desired attraction total to attraction zone (column) j
j= attraction zone number
n= number of zones
k = iteration number

To produce a mathematically correct result, repeat the trip distribution computation using the modified attraction values.

For example, for zone 1:

 Zone 1 2 3 Aj1 380 209 161 Given A 300 270 180 Aj2 237 349 201 AjFijKij 1 2 3 sum 1 9237 18138 10062 37437 2 12316 9069 5232 26617 3 11842 9069 7848 28759

Zone to Zone Second Iteration:

 zone 1 2 3 P 1 35 68 38 140 2 153 112 65 330 3 115 88 76 280 A 303 269 179 750 given A 300 270 180 750

Upon finishing the second iteration, the calculated attractions are within 5% of the given attractions. This is an acceptable result and the final summary of the trip distribution is shown below.

The resulting trip table is:

 zone 1 2 3 1 35 68 38 2 153 112 65 3 115 88 76