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This Graduate Report Contain basic information regarding the Trip Distribution related to Urban Transportation Systems. I submitted this report during the 1st Semester of Masters in Town and Regional Planning.

Index

 

Sr. №             Description

 

1.0                   Introduction

2.0                   Trip distribution model, what and when?

3.0                   Methods of Trip distribution 

   3.1                Trip distribution using growth factor

      3.1.1         Uniform Factor Method 

      3.1.2         Average Factor Method 

      3.1.3         The Fratar Method 

      3.1.4         The Furness Method 

   3.2                Criticism of Growth Factor Methods  

   3.3                Trip Distribution using Synthetic Models 

      3.3.1         Gravity Model 

      3.3.2         Tanner’s Model 

      3.3.3         Opportunity Model

         3.3.3.1   Intervening Opportunity Model

         3.3.3.2   Competing Opportunity Model                        References                                                                                                

 

          1.0     Introduction

Over the years, modelers have used several different formulations of trip distribution. The first was the Fratar or Growth model (which did not differentiate trips by purpose). This structure extrapolated a base year trip table to the future based on growth, but took no account of changing spatial accessibility due to increased supply or changes in travel patterns and congestion. Over the years, modelers have used several different formulations of trip distribution.

The first was the Fratar or Growth model. This structure extrapolated a base year trip table to the future based on growth, but took no account of changing spatial accessibility due to increased supply or changes in travel patterns and congestion. The next models developed were the gravity model and the intervening opportunities model. Evaluation of several model forms in the 1960's concluded that "the gravity model and intervening opportunity model proved of about equal reliability and utility in simulating the 1948 and 1955 trip distribution for Washington, D.C.

The Fratar model was shown to have weakness in areas experiencing land use changes. As comparisons between the models showed that either could be calibrated equally well to match observed conditions, because of computational ease, gravity models became more widely spread than intervening opportunities models. Some theoretical problems with the intervening opportunities model were discussed by Whitaker and West concerning its inability to account for all trips generated in a zone which makes it more difficult to calibrate, although techniques for dealing with the limitations have been developed by Ruiter.

The next models developed were the gravity model and the intervening opportunities model. The most widely used formulation is still the gravity model. Evaluation of several model forms in the 1960s concluded that “the gravity model and intervening opportunity model proved of about equal reliability and utility in simulating the 1948 and 1955 trip distribution for Washington, D.C. (Heanue and Pyers 1966).” The Fratar model was shown to have weakness in areas experiencing land use changes. As comparisons between the models showed that either could be calibrated equally well to match observed conditions, because of computational ease, gravity models became more widely spread than intervening opportunities models. Some theoretical problems with the intervening opportunities model were discussed by Whitaker and West (1968) concerning its inability to account for all trips generated in a zone which makes it more difficult to calibrate, although techniques for dealing with the limitations have been developed by Ruiter (1969).

With the development of logit and other discrete choice techniques, new, demographically disaggregate approaches to travel demand were attempted. By including variables other than travel time in determining the probability of making a trip, it is expected to have a better prediction of travel behavior. The logit model and gravity model have been shown by Wilson (1967) to be of essentially the same form as used in statistical mechanics, the entropy maximization model. The application of these models differ in concept in that the gravity model uses impedance by travel time, perhaps stratified by socio-economic variables, in determining the probability of trip making, while a discrete choice approach brings those variables inside the utility or impedance function. Discrete choice models require more information to estimate and more computational time.

Ben-Akiva and Lerman (1985) have developed combination destination choice and mode choice models using a logit formulation for work and non-work trips. Because of computational intensity, these formulations tended to aggregate traffic zones into larger districts or rings in estimation. In current application, some models, including for instance the transportation planning model used in Portland, Oregon use a logit formulation for destination choice. Allen (1984) used utilities from a logit based mode choice model in determining composite impedance for trip distribution. However, that approach, using mode choice log-sums implies that destination choice depends on the same variables as mode choice. Levinson and Kumar (1995) employ mode choice probabilities as a weighting factor and develop a specific impedance function or “f-curve” for each mode for work and non-work trip purposes.

 

  • Trip

A trip is a one-way person movement by mechanized mode of transportation, having two trip ends, an origin and a destination.

 

  • Trip generation

It is the term used in Transportation Planning Process to cover the field of calculating the number of trip ends in a given area.

  • Trip purpose

Work

School

Business

Social or recreational, sports

Others

  • Factors Governing Trip Generations and Attraction Rates…

Income

Car ownership

Family size and composition

Land use characteristics

Distance of the zone from the town centre

Accessibility to public transport system and its efficiency

Employment opportunities

  • Trip distribution

It is a stage which determines the number of trips ti-j which would originate from zone I and terminate in zone j.

2.0            Trip distribution model, what and when?


           
In recent years, planners have developed methodologies for estimating the distribution of future traffic over an entire transportation network. These procedures, which have been used for both urban and statewide systems, involve the use of five typical types of models:

1.      Land use.

2.      Trip generation

3.      Trip distribution

4.      Traffic assignment

5.      Model Split

The models are mathematical equations and procedures that collectively relate travel patterns to land use, demographic characteristics, and parameters of the transportation system. The models are developed and “calibrated” for a given study area so as to reproduce existing travel patterns as determined from actual counts. Assuming the basic relationship between travel, land use and socio-economic characteristics remain constant over time, planners use the models to evaluate future alternative land use and transportation system.

After having obtained an estimate of the trips generated from and attracted to the various zones, it is necessary to determine the direction of travel. Trip distribution models begin with the number of trip ends generated by each zone and answer the question, “What zone are the trips going to and coming from?” The number of trips generated in every zone of the area under study has to be apportioned to the various zones to which these trips are attracted.

tripdistribution.jpg

Figure 1 shows the stages of general planning process involved in any traffic management proposal preparation.

 

 

3.       Methods of Trip Distribution

 

            In trip distribution, two known sets of trip ends are connected together, without specifying the actual route and sometimes without reference to travel mode, to form a trip matrix between known origins and destinations.

There are two types of trip distribution methods,

1.      Growth factor methods

2.      Synthetic methods

Growth factor methods have been used in earlier studies but have yielded place now to the more rational synthetic models. The following are the important growth factor methods:

1.      Uniform factor method

2.      Average factor method

3.      Fratar method

4.      Furness method

The synthetic methods are as give below:

1.      Gravity model

2.      Tanner model

3.      Intervening opportunities model

4.      Competing opportunities model

 

The distribution of trips between zones can be best understood by a matrix, shown in the figure below. The horizontal axis of the matrix represents the zones of attractions (destinations D), 1, 2, 3,……j…n and the vertical axis represents the zones of generations (origin, O), 1, 2, 3,…..i…n. The number of trips indicated at the intersection of any zone of origin and attraction e.g. t i-j represents the number of trips originating in zone i and terminating in zone j. The total of any individual row, i, represents the total number of trips generated in zone, i.e. pi. Similarly the total of any individual column, j, represents the number of terminating in zone j, i.e. aj.

odmatrix.jpg

 

3.1            Trip Distribution using Growth Factor

Growth factor methods assume that in the future the trip-making pattern will remain substantially the same as today but that the volume of trips will increase according to the growth of the generating and attracting zones. These methods are simpler than synthetic methods and for small towns where considerable changes in land-use and external factors are not expected, they have often been considered adequate.

 

3.1.1    Uniform Factor Method

This method also known as Constant Factor Method assumes that all zones will increase in a uniform manner and that the existing traffic pattern will be the same for the future when growth is taken into account. This was the earliest method to be used, the basic assumption being that the growth which is expected to take place in the survey area will have an equal effect on all the trips in the area. The relationship between present and future trips can be expressed by

t’ij = tij x E

Where,           t’ij is the future number of trips between zone (i) and zone (j)

tij is the present number of between zone (i) and zone (j)

E is the constant factor derived by dividing the future number of trip ends expected in the survey area by the existing number of trip ends.

This method suffers from the disadvantages that it will tend to overestimate the trips between densely developed zones, which probably have little development potential, and underestimate the future trips between underdeveloped zones, which are likely to be extremely developed in the future. It will also fail to make provi­sion for zones which are at present undeveloped and which may generate a considerable number of trips in the future.

 

3.1.2    Average Factor Method

This method attempts to take into account the varying rates of growth of trip-making which can be expected in the differing zones of a survey area. The average growth factor used is that which refers to the origin end and the destination end of the trip and is obtained for each zone as in the constant factor method. Expressed mathematically, this can be stated to be

t’ij = tij {(Ej + Ei)/2}

Where Ei = Pi/pi and Ej = Aj/aj

t’ij = future flow ab,

 

tij = present flow ab,

Pi = future production of zone i,

pi = present production of zone i,

Aj = future attraction of zone j,

aj = present attraction of zone j.

At the completion of the process attractions and productions will not agree with the future estimates and the procedure must be iterated using as new values for Ei and Ej the factors Pi/p’i and Aj/a’j where p’i and a’j are the total productions and attractions of zones i and j respectively, obtained from the first distribution of trips. The process is iterated using successive values of p’i and a’j until the growth factor approaches unity and the successive values of t’ij and tij are within 1 to 5 percent depending upon the accuracy required in the trip distribution.

The average factor method suffers from many of the disadvantages of the constant factor method, and in addition if a large number of iterations are required then the accuracy of the resulting trip matrix may be questioned.

 

3.1.3    The Fratar Method

This method was introduced by T. J. Fratar to overcome some of the disadvantages of the constant factor and average factor methods. According to this method, the total trips for each zone are distributed to the interzonal movements, as a first approximation, according to relative attractiveness of each movement. Thus, the future trips estimated for any zone would be distributed to the movements involving that zone in proportion to the existing trips between it and each other zone and in proportion to the expected growth of each other zone. This may be expressed as

fratormethod.jpg

Where,

            Ti-j        =          Future trips from zone i to zone j

            ti-j         =          Present trips from zone i to zone j

            Pi         =          Future trips produced at zone i

            pi         =          Present trips attracted at zone i

            Ai        =          Future trips attracted to zone j

            aj          =          Present trips attracted to zone j

            k          =          Total numbers of zones

 

When the future traffic into and out of all zones is similarly distributed, each interzonal trip has been assigned two tentative values – one the result of the distribution for one of the zones involved and the other, the result of the distribution for the other zone involved. As a first approximation those pairs of tentative values are averaged. A new ‘growth factor’ for each zone is then calculated and the distribution process is repeated.

The procedure is laborious except for simple problems, but can be conventionally tackled by a computer. It has the same drawbacks as other growth factor models and is unable to forecast trips for those areas which were predominantly under-developed during the base year. It does not take into account the effects of changes in accessibility for various portions of the study area.

 

3.1.1    The Furness Method

The method was devised by K. P. Furness is also iterative in nature. For this the estimates of future traffic originating and terminating at each zone are required, thus yielding origin growth factors and destination growth factors for each zone. The traffic movements are made to agree alternately with the future traffic originating in each zone and the estimated future terminating in each zone, until both these conditions are roughly satisfied.

The Furness method gives results similar to Fratar, but requires less computation.

 

3.2            Criticism of Growth Factor Methods

The following are some of the disadvantages of the growth factor methods:

1.                  Present trip distribution matrix has to be obtained first, for which large scale O–D studies with high sampling sizes are needed so as to estimate the smaller zone-to-zone movements accurately

2.                  The error in original data collected on specific zone-to-zone movements gets magnified.

3.                  None of the methods provide a measure of the resistance to travel and all imply that resistance to travel will remain constant. They neglect the effect of changes in travel pattern by the construction of new facilities and new network.

Despite the above shortcomings, the growth factor methods are relatively simpler to use and understand. They can be used for studies of small areas and for updating stable and uniform data.

 

3.3     Trip Distribution using Synthetic Models

In synthetic models of trip distribution, an attempt is made to discern the underlying causes of movements between places, and relationships are established between trips and measures of attraction, generation and travel resistance. Synthetic models have an important advantage that they can be used not only to predict future trip distribution but also to synthesis the base-year flows. The necessity of having to survey every individual cell in the trip matrix is thus obviated and the cost of data collection is reduced.

 

3.3.1  Gravity Model

            One of the well known synthetic models is the Gravity Model. Based in Newton’s concept of gravity, the model as proposed by Voorhees assumes that the interchange of trips between zones in an area is dependent upon the relative attraction between the zones and the spatial separation between them as measured by an appropriate function of distance. This function of spatial separation adjusts the relative attraction of each zone for the ability, desire or necessity of the trip maker to overcome the spatial separation. Whereas the trip interchange is directly proportional to the relative attraction between the zones, it is inversely proportional to the measure of spatial separation.

            A simple equation representing the above relationship is of the following form:

gravitymodel.jpg

Where,

Ti-j        =          Trips between zones i and j

Pi            =          Trips produced in zone i

Aj         =          Trips attracted to zone j

dij        =          Distance between zone i and j, or the time or cost of traveling between them

K         =          A constant, usually independent of i

n          =          An exponential constant, whose value is usually found to lie 1 and 3

k          =          Total number of zones

 

The following formulation was also used in earlier studies dispersing with the proportionality constant:

gravitymodel1.jpg

Where, Ti-j, Pi, Aj, di-k and n have the same meaning as given earlier.

            In order to simplify the computation requirements of the model, the following formulation has been frequently used;

gravitymodel2.jpg

Where,

Ti-j        =          Trips produced in Zone i and attracted to zone j

Pi         =          Trips produced in zone i

Aj         =          Trips attracted to zone j

Fi-j        =          Empirically derived travel time factor which expresses the average area-wide effect of spatial separation on trip interchange between zones i and j

Ki-j       =          A specific zone-to-zone adjustment factor to allow for the incorporation of the effect on travel patterns of defined social or economic linkages not otherwise accounted for in the gravity model formulation

k          =          Total number of zones

m         =          Iteration number

p          =          Trip purpose

           

The above relationship can be used for determining the trip interchange for each trip purpose and each mode of travel.

3.3.2  Tanner’s Model

            Tanner has suggested that the inverse of nth power, 1/(di-j)n in the gravity model formula cannot give valid estimates at both very small and very large distances. In this place he proposes the function eλd/dn, where λ and n are constants. The new formula suggested by him is of the form:

tannarmodel.jpg

Where,

t1-2                    =          Number of journey per day between the two places 1 and 2

m                     =          A constant

P and P2          =          Populations, or other measures of size of the two places

d1-2                        =          Distance between places 1 and 2 or the time or cost of traveling between them

C1 and C2        =          Constants, one for each place,

C1 being defined by

tannarmodel2.jpg

3.3.3  Opportunity Model

Opportunity models are based on the statistical theory of probability as the theoretical foundation. The concept has been pioneered by Schneider and developed by subsequent studies. The two well-known models are:

1.                  The intervening opportunities models;

2.                  The competing opportunities model.

The opportunity models can be represented by the general formula:

Tij = Oi P (Di)

Where,

Ti-j        =          Predicted number of trips from zone i to j.

Oi        =          Total number of trips originating in zone i.

P(Dj)    =          Calculated probability of a trip terminating in zone j.

Dj         =          Total trip destinations attracted to zone j.

 

 

3.3.3.1        Intervening Opportunities Model

            In the intervening opportunities model, it is assumed that the trip interchange between and origin and a destination zone is equal to the total trips emanating from the origin zone multiplied by the probability that each trip will find and acceptable terminal at the destination. It is further assumed that the probability that a destination will be acceptable is determined by two zonal characteristics: the size of the destination and the order in which it is encountered as trips proceed from the origin. The probability functions in above equation P(Dj), may then be expressed as the difference between the probability that the trip origins at i will find a suitable terminal in one of the destinations, ordered by closeness to i, up to and including j, and the probability that they will find a suitable terminal in the destination up to but excluding j. The following equation represents mathematically this concept:

interveningoppo.jpg

Where,

Ti-j        =          Predicted number of trips from zone i to j

Qi        =          Total number of trips originating in zone i

L          =          Probability density of destination acceptability at the point of consideration

A         =          Number of origins between i and j when arranged in order of closeness

B          =          Number of destinations between i and j when arranged in order of closeness

It may be noted that:

                                    A = B + Dj

 

3.3.3.2                  Competing Opportunities Model

            In the competing opportunities model, the adjusted probability of a trip ending in a zone  is the product of two independent probabilities, viz., the probability of a trip being attracted to a zone and the probability of a trip finding a destination in that zone. A form of this model is given below:

competingoppo.jpg

References

  • L. R. Kadiyali, Traffic Engineering and Transportation Planning; Sixth Edition (1997); Fourth Reprint (2003); Khanna Publishers, Delhi.
  • R. J. Salter and N. B. Hounsell; Highway Traffic Analysis and Design; Third Edition (1996); MACMILLAN press Limited, London.
  • R. C. Sharma; Principles and Practice of Highway Engineering; Second Edition (1999); Pushpak Press India Limited; Chandigarh.
  • Paul H. Wright and Randor J. Paquette; Highway Engineering; Fifth Edition (1985); John Wiley & Sons; Canada.