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  # Trip Distribution
  
  Trip Distribution
  Two known sets of trip ends are connected together to form a tip matrix
  between origins and destinations.
  In the literature there are two basic methods by which this connection can be
  achieved.
  
  ## 1. Growth factor method
  
  Constant factor method
  Average factor method
  Fratar method
  Furness method
  2. Synthetic Methods
  Gravity model
  Opportunity model
  
  ## Growth Factor Methods
  
  Assumptions:
  
  The trip making pattern will remain the same in the future as it is in the
  base year
  
  The volume will increase according to the growth in the generating and
  attracting zones
  
  Assumes:
  
  ## That growth in all zones will increase in an uniformed manner
  
  That the existing traffic pattern will be the same for the future but that the
  volume will change
  
  That the growth which is expected to take place in the survey area will
  have an equal effect on all trips made in the area
  
  ## The equation used in this method is as follows:
  
  t 'ij = tij E
  Where: tij is the future number of trips from zone i to zone j
  tij is the present number of trips from zone i to zone j
  E is the constant factor derived by dividing the number of trip ends in
  the future year by the base year
  1. Tends to overestimate the trips between densely populated zones which
  probably have little further development potential
  2. Tends to underestimate the future trips between underdeveloped zones
  which could be extensively populated in the future
  
  Given:
  
  ## 4x4 matrix = demand in the base year
  
  Uniform growth rate = 20%
  
  Method:
  Each cell in the matrix will be multiplied by 1.2 (to get a 20% increase).
  
  ## Singly constrained growth factor methods
  
  Used where information is available on the expected growth in trips originating
  from zones.
  Consider the following trip matrix
  
  Method
  Multiply each cell in Row 1 by 400/355
  Multiply each cell in Row 2 by 460/455
  Multiply each cell in Row 3 by 400/255
  Multiply each cell in Row 4 by 702/570
  
  ## Average factor method
  
  This method tries to take into account the varying rates of growth in trip
  making in different zones.
  The equation is as follows:
  Where
  
  t 'ij = tij ( Ei + E j ) / 2
  
  Ei = Pi / pi and E j = Aj / a j
  Pi = future trip production of zone i
  pi = present trip production of zone i
  Aj = future trip attractions of zone j
  aj = present trip attractions of zone j
  
  Note:
  
  After the first iteration, trip productions will not match trip attractions
  
  ## Futher iterations will be necessary to achieve accuracy of 1-5%
  
  1. This method suffers the same disadvantages as the constant growth factor
  method
  2. If many iterations need to be preformed then the accuracy of the resulting
  trips matrix may be questioned.
  
  k
  
  Pi Aj
  t 'ij = tij . .
  pi a j
  
  ik
  
  (A
  
  / ak )tik
  
  trips
  
  ## The Furness Method (Doubly constrained growth factor method)
  
  The model is represented as follows
  
  t 'ij = tij Pi / pi
  t ''ij = t 'ij ( Aj /sum of trips attracted to j in the first iteration)
  Example of the Fratar Method
  
  Pa
  Ab
  Ac
  =2
  =3
  =4
  =2
  pa
  ab
  ac
  
  ## 200 + 400 + 600
  
  t 'ab 200*2*3[
  ]
  200*3 + 400*4 + 600*2
  
  = 423
  
  ## 200 + 400 + 600
  
  t 'ac = 400*2*4[
  ]
  200*3 + 400*4 + 600*2
  = 1129
  
  ## 200 + 400 + 600
  
  ]
  200*3 + 400*4 + 600*2
  
  = 847
  
  ## There are four steps to the calculation
  
  1. Total the outgoing trips for each zone and multiply by the zonal growth
  factor to obtain the predicted origin out going totals
  2. Multiply the lines in the matrix by the appropriate growth factor
  3. Total the incoming trips into each zone and divide by the predicted
  incoming totals to obtain the destination factors
  4. Repeat the iteration process until the origin or destination factor being
  calculated is sufficiently close to unity (within 5%)
  
  ## Examples of the Furness Method
  
  Example 1:
  Consider the following trip matrix with the target origins and destinations for
  the future year included.
  
  ## Base year trip matrix for doubly constrained problem.
  
  Step 1 is already done, sum totals for each row.
  
  Step 2
  Multiply each cell in row 1 by 400/355
  Multiply each cell in row 2 by 460/455
  Multiply each cell in row 3 by 400/255
  Multiply each cell in row 4 by 702/570
  
  The table above shows the trip matrix after application of ratio of target to
  existing totals to the rows
  Step 3
  Multiply column 1 by 260/257.7
  Multiply column 2 by 400/464.6
  Multiply column 3 by 500/529.5
  Multiply column 4 by 802/701.2
  
  Step 4
  Continue iterations as in step 2 and step 3 until the origin and destination
  factors are within 5% of unity
  
  ## The accuracy is dependent upon the accuracy achieved in defining the
  
  growth factor used.
  
  Growth factor process is not related to the factors which influence trip
  makers.
  
  Synthetic Models
  
  ## Try to include the causes influencing present day travel patterns
  
  Assume that these underlying causes will remain the same in the future
  
  ## Trip distribution in the NTA model
  
  The trip distribution model (TDM) is used to determine the pattern of trips
  between sets of trip generators and trip attractions.
  This is a sub-model of the full NTA model.
  The function of the model is to determine to what zones the trips are
  generated at any particular origin will travel. As the model has 666 zones,
  the matrix of trip patterns is 666 x 666.
  
  The TAGM model also uses trip distributions from the base year. These
  are based upon observed travel patterns from the GDA household and
  education surveys and from the POWCAR dataset.
  The travel patterns represented by the base year matrices are also
  calibrated in the GDA transport model to ensure that when assigned to the
  transport networks, the model outputs closely match the observed network
  characteristics.
  This process - known as base year model calibration.
  The calibration of the base year model also generates base year travel
  costs that can (optionally) be fed back into the TDM to impact on travel
  patterns.
  Therefore the TDM mode:
  - creates an all day forecast year trip generations and attractions from the
  TAGM
  - Base year trip distribution matrices for the am-peak and off-peak periods.
  
  The TDM uses a combination of two methods to derive forecast year travel
  patterns in the form of 666 x 666 trip matrices for each modelled time
  period:
  Factoring: Where well established trip patterns exist in the base year, this
  pattern is retained and simply factored up to the new forecast year trip
  generations and attractions.
  Sectoring: The GDA Transport Model has the facility to aggregate trip
  patterns using a 75 strategic zone system (called sectors). In the case of a
  green field site development in a zone or where there is insufficient data in
  the base year to determine the pattern of trips, the base year trip pattern of
  the 75 zone sector containing the green field zone is used to give the
  equivalent forecast year trip pattern.
  In addition to using Factoring and Sectoring, the TDM can also (optionally)
  use the travel costs that are output during the route choice / trip
  assignment stage of the model to influence travel patterns in the forecast
  year. If this option is chosen, the TDM uses a form of gravity model to
  determine travel patterns for green field zones that have development in
  the forecast year, or for zones where there is a major change in population
  or travel costs between the base and forecast years.
  
  ## The Gravity Distribution is based on the Newtons gravitational formula,
  
  and in modelling trip distribution takes the form:
  Tij = Oi . Dj . f(Cij)
  Where Tij is the number of trips between origin O and destination D. Oi is
  the total trips generated at origin O and Dj is the number of trips attracted
  to destination D. f(Cij) is called the deterrence function based on the cost
  of travel between O and D.
  The TDM uses a log-normal form of the deterrence function as follows:
  f(Cij) = EXP( . Cij)
  Where is a measure of peoples sensitivity to travel costs.
  The outputs from the TDM are in the form of trip matrices (666 x 666) for
  each of the six journey purposes divided by car available and car not
  available persons (i.e. twelve trip matrices). These trip matrices are the
  essential inputs into the next three stages of the model described in the
  next section.
  
  ## The Gravity Distribution Model
  
  Distribution models of a different kind have been developed to assist in the
  forecasting future trip patterns when important changes in the network take
  place.
  These models make assumptions about group trip making behaviour and
  the way this is influenced by external factors such as total trip ends and
  distance travelled. The most widely used of these models is the gravity
  model.
  This model estimates trips for each cell in the matrix without directly using
  the observed trip pattern.
  The gravity model takes the following functional form:
  
  ! Pi Pj
  Tij = 2
  dij
  
  Where :
  Pi and Pj = populations of the towns O-D pair
  dij = distance between i and j
  
  ! = proportionality factor
  
  This approach was seen to be too simplistic, and improvements have been
  introduced. One such improvement was the assumption that the effect of
  distance (or separation) could be modelled better using a decreasing function.
  
  ## Tij = !Oi D j f (cij )
  
  where: f (cij ) is a generalised function of the travel cost with one or more parameters for calibration.
  
  ## This function is often called a deterrence function.
  
  Properties of the Gravity Model
  1. It provides a more rigorous way of specifying the mathematical properties of
  the resulting model
  2. The use of a mathematical programming framework also facilities the use of a
  standard tool-kit of solution methods and the analysis of the efficiency of
  alternative algorithms
  3. The theoretical framework used to generate the model also assists in
  providing an improved interpretation of the solutions generated by it
  4. Just because the gravity model can be generated in a number of ways does
  not make it correct. The appropriateness of the model depends upon the
  acceptability of the assumptions required for its generation and interpretation.
  
  This approach was seen to be too simplistic, and improvements have been
  introduced. One such improvement was the assumption that the effect of
  distance (or separation) could be modelled better using a decreasing function.
  
  ## Tij = !Oi D j f (cij )
  
  where: f (cij ) is a generalised function of the travel cost with one or more parameters for calibration.
  
  ## This function is often called a deterrence function.
  
  Properties of the Gravity Model
  1. It provides a more rigorous way of specifying the mathematical properties of
  the resulting model
  2. The use of a mathematical programming framework also facilities the use of a
  standard tool-kit of solution methods and the analysis of the efficiency of
  alternative algorithms
  3. The theoretical framework used to generate the model also assists in
  providing an improved interpretation of the solutions generated by it
  4. Just because the gravity model can be generated in a number of ways does
  not make it correct. The appropriateness of the model depends upon the
  acceptability of the assumptions required for its generation and interpretation.
explanation: The document is a detailed explanation of trip distribution methods used in transportation planning, including growth factor methods and synthetic methods such as the gravity model. It describes various techniques and their mathematical formulations, along with examples and the assumptions behind each method. The document does not contain a specific question or a question and answer.
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  # Trip Distribution
  
  Trip Distribution
  Two known sets of trip ends are connected together to form a tip matrix
  between origins and destinations.
  In the literature there are two basic methods by which this connection can be
  achieved.
  
  ## 1. Growth factor method
  
  Constant factor method
  Average factor method
  Fratar method
  Furness method
  2. Synthetic Methods
  Gravity model
  Opportunity model
  
  ## Growth Factor Methods
  
  Assumptions:
  
  The trip making pattern will remain the same in the future as it is in the
  base year
  
  The volume will increase according to the growth in the generating and
  attracting zones
  
  Assumes:
  
  ## That growth in all zones will increase in an uniformed manner
  
  That the existing traffic pattern will be the same for the future but that the
  volume will change
  
  That the growth which is expected to take place in the survey area will
  have an equal effect on all trips made in the area
  
  ## The equation used in this method is as follows:
  
  t 'ij = tij E
  Where: tij is the future number of trips from zone i to zone j
  tij is the present number of trips from zone i to zone j
  E is the constant factor derived by dividing the number of trip ends in
  the future year by the base year
  1. Tends to overestimate the trips between densely populated zones which
  probably have little further development potential
  2. Tends to underestimate the future trips between underdeveloped zones
  which could be extensively populated in the future
  
  Given:
  
  ## 4x4 matrix = demand in the base year
  
  Uniform growth rate = 20%
  
  Method:
  Each cell in the matrix will be multiplied by 1.2 (to get a 20% increase).
  
  ## Singly constrained growth factor methods
  
  Used where information is available on the expected growth in trips originating
  from zones.
  Consider the following trip matrix
  
  Method
  Multiply each cell in Row 1 by 400/355
  Multiply each cell in Row 2 by 460/455
  Multiply each cell in Row 3 by 400/255
  Multiply each cell in Row 4 by 702/570
  
  ## Average factor method
  
  This method tries to take into account the varying rates of growth in trip
  making in different zones.
  The equation is as follows:
  Where
  
  t 'ij = tij ( Ei + E j ) / 2
  
  Ei = Pi / pi and E j = Aj / a j
  Pi = future trip production of zone i
  pi = present trip production of zone i
  Aj = future trip attractions of zone j
  aj = present trip attractions of zone j
  
  Note:
  
  After the first iteration, trip productions will not match trip attractions
  
  ## Futher iterations will be necessary to achieve accuracy of 1-5%
  
  1. This method suffers the same disadvantages as the constant growth factor
  method
  2. If many iterations need to be preformed then the accuracy of the resulting
  trips matrix may be questioned.
  
  k
  
  Pi Aj
  t 'ij = tij . .
  pi a j
  
  ik
  
  (A
  
  / ak )tik
  
  trips
  
  ## The Furness Method (Doubly constrained growth factor method)
  
  The model is represented as follows
  
  t 'ij = tij Pi / pi
  t ''ij = t 'ij ( Aj /sum of trips attracted to j in the first iteration)
  Example of the Fratar Method
  
  Pa
  Ab
  Ac
  =2
  =3
  =4
  =2
  pa
  ab
  ac
  
  ## 200 + 400 + 600
  
  t 'ab 200*2*3[
  ]
  200*3 + 400*4 + 600*2
  
  = 423
  
  ## 200 + 400 + 600
  
  t 'ac = 400*2*4[
  ]
  200*3 + 400*4 + 600*2
  = 1129
  
  ## 200 + 400 + 600
  
  ]
  200*3 + 400*4 + 600*2
  
  = 847
  
  ## There are four steps to the calculation
  
  1. Total the outgoing trips for each zone and multiply by the zonal growth
  factor to obtain the predicted origin out going totals
  2. Multiply the lines in the matrix by the appropriate growth factor
  3. Total the incoming trips into each zone and divide by the predicted
  incoming totals to obtain the destination factors
  4. Repeat the iteration process until the origin or destination factor being
  calculated is sufficiently close to unity (within 5%)
  
  ## Examples of the Furness Method
  
  Example 1:
  Consider the following trip matrix with the target origins and destinations for
  the future year included.
  
  ## Base year trip matrix for doubly constrained problem.
  
  Step 1 is already done, sum totals for each row.
  
  Step 2
  Multiply each cell in row 1 by 400/355
  Multiply each cell in row 2 by 460/455
  Multiply each cell in row 3 by 400/255
  Multiply each cell in row 4 by 702/570
  
  The table above shows the trip matrix after application of ratio of target to
  existing totals to the rows
  Step 3
  Multiply column 1 by 260/257.7
  Multiply column 2 by 400/464.6
  Multiply column 3 by 500/529.5
  Multiply column 4 by 802/701.2
  
  Step 4
  Continue iterations as in step 2 and step 3 until the origin and destination
  factors are within 5% of unity
  
  ## The accuracy is dependent upon the accuracy achieved in defining the
  
  growth factor used.
  
  Growth factor process is not related to the factors which influence trip
  makers.
  
  Synthetic Models
  
  ## Try to include the causes influencing present day travel patterns
  
  Assume that these underlying causes will remain the same in the future
  
  ## Trip distribution in the NTA model
  
  The trip distribution model (TDM) is used to determine the pattern of trips
  between sets of trip generators and trip attractions.
  This is a sub-model of the full NTA model.
  The function of the model is to determine to what zones the trips are
  generated at any particular origin will travel. As the model has 666 zones,
  the matrix of trip patterns is 666 x 666.
  
  The TAGM model also uses trip distributions from the base year. These
  are based upon observed travel patterns from the GDA household and
  education surveys and from the POWCAR dataset.
  The travel patterns represented by the base year matrices are also
  calibrated in the GDA transport model to ensure that when assigned to the
  transport networks, the model outputs closely match the observed network
  characteristics.
  This process - known as base year model calibration.
  The calibration of the base year model also generates base year travel
  costs that can (optionally) be fed back into the TDM to impact on travel
  patterns.
  Therefore the TDM mode:
  - creates an all day forecast year trip generations and attractions from the
  TAGM
  - Base year trip distribution matrices for the am-peak and off-peak periods.
  
  The TDM uses a combination of two methods to derive forecast year travel
  patterns in the form of 666 x 666 trip matrices for each modelled time
  period:
  Factoring: Where well established trip patterns exist in the base year, this
  pattern is retained and simply factored up to the new forecast year trip
  generations and attractions.
  Sectoring: The GDA Transport Model has the facility to aggregate trip
  patterns using a 75 strategic zone system (called sectors). In the case of a
  green field site development in a zone or where there is insufficient data in
  the base year to determine the pattern of trips, the base year trip pattern of
  the 75 zone sector containing the green field zone is used to give the
  equivalent forecast year trip pattern.
  In addition to using Factoring and Sectoring, the TDM can also (optionally)
  use the travel costs that are output during the route choice / trip
  assignment stage of the model to influence travel patterns in the forecast
  year. If this option is chosen, the TDM uses a form of gravity model to
  determine travel patterns for green field zones that have development in
  the forecast year, or for zones where there is a major change in population
  or travel costs between the base and forecast years.
  
  ## The Gravity Distribution is based on the Newtons gravitational formula,
  
  and in modelling trip distribution takes the form:
  Tij = Oi . Dj . f(Cij)
  Where Tij is the number of trips between origin O and destination D. Oi is
  the total trips generated at origin O and Dj is the number of trips attracted
  to destination D. f(Cij) is called the deterrence function based on the cost
  of travel between O and D.
  The TDM uses a log-normal form of the deterrence function as follows:
  f(Cij) = EXP( . Cij)
  Where is a measure of peoples sensitivity to travel costs.
  The outputs from the TDM are in the form of trip matrices (666 x 666) for
  each of the six journey purposes divided by car available and car not
  available persons (i.e. twelve trip matrices). These trip matrices are the
  essential inputs into the next three stages of the model described in the
  next section.
  
  ## The Gravity Distribution Model
  
  Distribution models of a different kind have been developed to assist in the
  forecasting future trip patterns when important changes in the network take
  place.
  These models make assumptions about group trip making behaviour and
  the way this is influenced by external factors such as total trip ends and
  distance travelled. The most widely used of these models is the gravity
  model.
  This model estimates trips for each cell in the matrix without directly using
  the observed trip pattern.
  The gravity model takes the following functional form:
  
  ! Pi Pj
  Tij = 2
  dij
  
  Where :
  Pi and Pj = populations of the towns O-D pair
  dij = distance between i and j
  
  ! = proportionality factor
  
  This approach was seen to be too simplistic, and improvements have been
  introduced. One such improvement was the assumption that the effect of
  distance (or separation) could be modelled better using a decreasing function.
  
  ## Tij = !Oi D j f (cij )
  
  where: f (cij ) is a generalised function of the travel cost with one or more parameters for calibration.
  
  ## This function is often called a deterrence function.
  
  Properties of the Gravity Model
  1. It provides a more rigorous way of specifying the mathematical properties of
  the resulting model
  2. The use of a mathematical programming framework also facilities the use of a
  standard tool-kit of solution methods and the analysis of the efficiency of
  alternative algorithms
  3. The theoretical framework used to generate the model also assists in
  providing an improved interpretation of the solutions generated by it
  4. Just because the gravity model can be generated in a number of ways does
  not make it correct. The appropriateness of the model depends upon the
  acceptability of the assumptions required for its generation and interpretation.
  
  This approach was seen to be too simplistic, and improvements have been
  introduced. One such improvement was the assumption that the effect of
  distance (or separation) could be modelled better using a decreasing function.
  
  ## Tij = !Oi D j f (cij )
  
  where: f (cij ) is a generalised function of the travel cost with one or more parameters for calibration.
  
  ## This function is often called a deterrence function.
  
  Properties of the Gravity Model
  1. It provides a more rigorous way of specifying the mathematical properties of
  the resulting model
  2. The use of a mathematical programming framework also facilities the use of a
  standard tool-kit of solution methods and the analysis of the efficiency of
  alternative algorithms
  3. The theoretical framework used to generate the model also assists in
  providing an improved interpretation of the solutions generated by it
  4. Just because the gravity model can be generated in a number of ways does
  not make it correct. The appropriateness of the model depends upon the
  acceptability of the assumptions required for its generation and interpretation.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed explanation of trip distribution methods used in transportation planning, including growth factor methods and synthetic methods such as the gravity model. It describes various techniques and their mathematical formulations, along with examples and the assumptions behind each method. The document does not contain a specific question or a question and answer.
  
  Label: no-question