CE 2710

Introduction to Transportation Engineering

Homework 5 Solution

  1. Given the following data use the gravity model to calculate all the interchange volumes (in other words, get the trip distribution matrix).

Zone / Production, Pi / Attractiveness, Aj
1 / 2000 / 2
2 / 1000 / 5
3 / 2000 / 1

Impedance Factor, Wij

J
I / 1 / 2 / 3
1 / 5 / 20 / 10
2 / 20 / 5 / 10
3 / 10 / 10 / 5

Calibration Factor, c = 1.5Socio-economic factor, kij = 1.0

Solution

Step 1- Calculate Friction Factors

Based on the equation we can calculate friction factor for example:

Then the following matrix is prepared:

Fij (friction factor)

J
TAZ / 1 / 2 / 3
I / 1 / 0.0894 / 0.0112 / 0.0316
2 / 0.0112 / 0.0894 / 0.0316
3 / 0.0316 / 0.0316 / 0.0894

Step 2- Find Denominator of Gravity Model Equation {AjFijKij}

Find denominator of Gravity Model equations for each pair of origin and destination by this relationAjFijKij

For example for i=1 and j=2 we have :

Based on the calculations the following matrix is completed:

AjFijKij

J
TAZ / 1 / 2 / 3 / Σ
I / 1 / 0.1789 / 0.0559 / 0.0316 / 0.2664
2 / 0.0224 / 0.4472 / 0.0316 / 0.5012
3 / 0.0632 / 0.1581 / 0.0894 / 0.3108

Step 3- Find Probability that Trip i will be attracted to Zone j, {pij}

In this step, by using the following formula and based on the calculated values in step 2 we can find the probability of travelled that will be attracted between each pair of TAZ.

For example the probability of travel between TAZ 1 and 2 is calculates as below:

Consequently we can complete the following matrix:

pij

J
TAZ / 1 / 2 / 3
I / 1 / 0.6715 / 0.2098 / 0.1187
2 / 0.0446 / 0.8923 / 0.0631
3 / 0.2035 / 0.5087 / 0.2878

Step 4- Find Trip Interchanges, {Qij}

Based on the total amount of trip production in each TAZ (given number) and the calculated probabilities in previous step we can find “Trip Interchanges”, Qij

The matrix of trip interchanges is as below:

Qij

J
TAZ / 1 / 2 / 3 / Σ
I / 1 / 1343 / 420 / 237 / 2000
2 / 45 / 892 / 63 / 1000
3 / 407 / 1017 / 576 / 2000
Σ / 1795 / 2329 / 876 / 5000
  1. Use the gravity model to estimate the trip distribution matrix for this planning year.

Given:

Zone / Production, Pi / Attractiveness, Aj
1 / 2000 / 10
2 / 5000 / 8

Impedance Factor, Wij

J
I / 1 / 2
1 / 2 / 10
2 / 9 / 1

Calibration Factor, c = 1.5

Kij

J
1 / 2
I / 1 / 1 / 1.2
2 / 1.2 / 1

Solution:

Step 1- Calculate Friction Factors

Fij

J
TAZ / 1 / 2
I / 1 / 0.3536 / 0.0316
2 / 0.0370 / 1.0000

Step 2- Find Denominator of Gravity Model Equation {AjFijKij}

AjFijKij

J
TAZ / 1 / 2 / Σ
I / 1 / 3.5355 / 0.3036 / 3.8391
2 / 0.4444 / 8.0000 / 8.4444

Step 3- Find Probability of travel between TAZ i and TAZ j, {pij}

The probability matrix

Pij

J
TAZ / 1 / 2
I / 1 / 0.9209 / 0.0791
2 / 0.0526 / 0.9474

Step 4- Find Trip Interchanges, {Qij}

Qij

J
TAZ / 1 / 2 / Σ
I / 1 / 1842 / 158 / 2000
2 / 263 / 4737 / 5000
Σ / 2105 / 4895 / 7000
  1. Calculate the market shares for the following modes using the utility function given.

uk= ak– 0.003 X1– 0.04 X2

ak, modal constant / X1, travel cost in cents / X2, travel time in minutes
Auto / -0.20 / 120 / 30
BRT / -0.40 / 60 / 45
Regular Bus / -0.60 / 30 / 55

Solution:

Step 1- Calculation utility values for each mode:

Step 2- Finding market share of each mode:

  1. Calculate the market shares for auto and light rail using the utility function given.

uk= ak– 0.05 Ta– 0.04 Tw– 0.02 Tr– 0.01 C

ak, modal constant / Ta – access time / Tw – waiting time / Tr – riding time / C – out of pocket cost
Light Rail / -0.05 / 10 / 10 / 45 / 50
Auto / -0.05 / 5 / 5 / 30 / 100

Solution:

Step 1- Calculation utility values for each mode:

Step 2- Finding market share of each mode: