Transportation Theory and Applications
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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 ODmatrix
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 ❖
Uniform Growth-factor
OD- matrix observed trips traffic count
Estimated trips
In case the only information available is the general growth rate Then, for each cell we can apply :
And
1. base matrix
2. Growth-factor
for the whole study area
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 methods Ai and Bj are balancing factors and growth factors elements 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 j dij is the distance between i and j
Tij is number of trips from zone i to j is the measure of average trip intensity Oi, Dj is production potential of zone i and attraction potential of zone j f(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 intensity Oi, Dj is production potential of zone i and attraction potential of zone j f(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 origin or Attraction 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 Balancing factor: Hence,
Bj depends on Ai thus:
Gravity model ❖
Process: 1.Set Bj=1, find Ai using 2.Find Bj using 3.Compute the error as
: actual productions from zone i : calculated productions from zone i : actual attraction from zone j : calculated attraction from zone j
6.Again set Bj=1 and find Ai, also find Bj. 7.Repeat the steps until convergence.
Gravity model ❖
Example:
Gravity model ❖
Example:
Gravity model ❖
Example:
Gravity model ❖
Example:
Gravity model ❖
Example:
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 ODmatrix
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 ❖
Uniform Growth-factor
OD- matrix observed trips traffic count
Estimated trips
In case the only information available is the general growth rate Then, for each cell we can apply :
And
1. base matrix
2. Growth-factor
for the whole study area
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 methods Ai and Bj are balancing factors and growth factors elements 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 j dij is the distance between i and j
Tij is number of trips from zone i to j is the measure of average trip intensity Oi, Dj is production potential of zone i and attraction potential of zone j f(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 intensity Oi, Dj is production potential of zone i and attraction potential of zone j f(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 origin or Attraction 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 Balancing factor: Hence,
Bj depends on Ai thus:
Gravity model ❖
Process: 1.Set Bj=1, find Ai using 2.Find Bj using 3.Compute the error as
: actual productions from zone i : calculated productions from zone i : actual attraction from zone j : calculated attraction from zone j
6.Again set Bj=1 and find Ai, also find Bj. 7.Repeat the steps until convergence.
Gravity model ❖
Example:
Gravity model ❖
Example:
Gravity model ❖
Example:
Gravity model ❖
Example:
Gravity model ❖
Example:
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