Fall 2017 - MTAT .08.043

Transportation Theory and Applications

Lecture IV: Trip distribution

A. Hadachi

outline ❖ Our objective

❖ Introducing two main methods

❖ for trip generation

objective ❖ Trip generation is about destination choice.

❖ Introducing the problematic

❖ Knowing: Productions and attractions for each zone meaning the origin and destination

❖ Determine: the number of trips from each zone to all other zones.

❖ This means that we are interested in generating OD-matrix

objective ❖ This means that we are interested in estimating future

OD-matrix based on basic trip matrix t.

Tij are the number of trips from zone i to zone j

Qi are production potential of zone i

Conditions:

Trip distribution❖ 2 methods

❖ Growth-factor model

❖ Gravity model (based on analogy)

Growth-factor model

❖ Uniform growth-factor

❖ Singly constrained growth-factor

❖ Doubly constrained growth-factor

Growth-factor model

1. base matrix

2. Growth-factor

OD- matrix observed tripstraffic count

Estimated trips

In case the only information available is the general growth rate for the whole study area

Then, for each cell we can apply :

And

❖ Uniform Growth-factor

Growth-factor model❖ Example:

Let’s consider the matrix in the table below 4/4 yeas based. if we know that the growth in traffic is expected to grow by 20%. what will be the expected future matrix ?

Since the growth is expected to be 20% then

Growth-factor model❖ Singly Constrained Growth-factor

In case we have information about expected growth in specific trips originating from each zone, such as a shopping trips or working trips.

Thus, we have to apply the origin-specific growth factor to the appropriate row.

The same thing applies if we know extra information about the destination trips then we apply to the concerned column

for origin specific factor

for destination specific factor

Growth-factor model❖ Example:

Let’s suppose we have a table with predicted growth for origins

we have just to multiply with ratio:

Growth-factor model❖ Doubly constrained growth-factors

❖ Knowns as Fratar or Furness methodsAi and Bj are balancing factors

and growth factorselements of basic matrix t.

for simplification we will introduce factors: and

Hence,

and condition:

in some cases to respect the condition you may require correcting trip-end estimates produced by the trip generation models.

Growth-factor model❖ Process:

1.Set bj =1

2.with bj solve for ai to satisfy trip generation constraint

3.with ai solve bj to satisfy trip attraction constraint

4.update matrix and check for errors

5.repeat steps 2 and 3 till convergence.

Error is calculated as:

: actual productions from zone i: calculated productions from zone i: actual attraction from zone j: calculated attraction from zone j

Growth-factor model❖ Example:

Growth-factor model❖ Example:

Growth-factor model❖ Example:

Growth-factor model❖ Example:

Gravity model❖ Computing gravitational attraction between planets

Gij is gravitational force between i and j g is gravitational constant mi, mj is mass of planet i and jdij is the distance between i and j

Tij is number of trips from zone i to j is the measure of average trip intensityOi, Dj is production potential of zone i and attraction potential of zone jf(cij) is accessibility of j from i

Gravity model❖ Formula

❖ Assumptions

Tij is number of trips from zone i to j is the measure of average trip intensityOi, Dj is production potential of zone i and attraction potential of zone jf(cij) is accessibility of j from i

we will assume that the number of trips between an origin and destination is promotional to:

Production factor at the originorAttraction factor at the destination or factor depends on the cost

Gravity model❖ By introducing balancing factors the formula become

(Ai and Bj):

if we apply:

and

Hence,

Balancing factor:

Bj depends on Ai thus:

Gravity model❖ Process:

1.Set Bj=1, find Ai using

2.Find Bj using

3.Compute the error as

6.Again set Bj=1 and find Ai, also find Bj.

7.Repeat the steps until convergence.

: actual productions from zone i: calculated productions from zone i: actual attraction from zone j: calculated attraction from zone j

Gravity model❖ Example:

Gravity model❖ Example:

Gravity model❖ Example:

Gravity model❖ Example:

Gravity model❖ Example: