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Transportation EngineeringDr. Hana Naghawi

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Transportation Engineering as defined by the Institute of Transportation Engineers (ITE) is the application of technology and scientific principles to planning, functional design, operation and management of facilities for any mode of transportation in order to provide for the safe, rapid, comfortable, convenient, economical and environmentally compatible movement for people and goods

Transportation Engineering

Transportation is the study of the movement of people and goods

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TS is the service provided for the movement of people and goods between different locations

Transportation System (TS)

We all have a personal experience as a user of transportation system- car driver- bus passenger- elevator user- sidewalk user

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Components of (TS)

1- physical facility2- Fleet 3- operating facility4- organization5- Operating strategy

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Very little travel is done for its own sake, we travel to satisfy needs that we cannot meet at home ( food, shelter, work, business, recreation), as well as the need to leave the home

Why Do People Travel?

All humans are interacting over distance and time as a result the spatial distribution of activities generates travel demand

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All agricultural & industrial raw materials, products, equipment are needed to be transported from one place to another

Goods Transportation

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The Importance of Transportation is in the development of a country. A country’s economic status depends upon how well served the country by different modes of transportation

The Importance of Transportation

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Challenges for Transportation Engineer

1- Traffic congestion2- Traffic safety 3- Environmental protection4- Incorporating new technology5- Funding6- Equal accessibility7- Developing institutional arrangements

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Transportation in an Urban Setting

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Urban Travel Characteristics

Urban transportation is the movement of people and goods between origin and destination within an urban area

Urban transportation is a trip from an origin to a destination to accomplish some activity at the destination

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Urban Travel Characteristics

Every day millions of trips are made in urban areas, satisfying a wide range of individual needs and using a variety of transportation means/modes

The 5 urban travel characteristics of this trip making behavior that worth special attention

- trip purpose- temporal distribution - spatial distribution- mode choice- cost

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Definition of a Trip

Trip One way movement from origin to destination Each trip has two ends

Typical trips Work, shop, school, business, social, recreational, serve passenger

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Trip Purpose Trip characteristicsTrips conducted for different purposes have different characteristics (e.g. common times of departure, days, frequency, length, etc.).

Trip purpose determined by objective of trip (e.g. go to work, shop, school, etc.) quantified by whether one end of the trip was home or not

When one end of trip is home, it is “home-base” trip and when neither end is home, it in non-home base

Common trip purpose classification:HBW, HBO, NHB

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Temporal Distribution of TripsTime of the Day

Peak occurs in peaks and troughs

Main peaks are in the morning and evening

Peaks vary in length with urban area size and system supply

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Typical Time-of-Day Distribution

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Influences on Time-of-Day Distribution

Dual peaks for commuters

Freight travel tends to peak later in the morning, stay high, and decline over night

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Spatial Distribution of TravelEach trip has an origin and destination

We need to understand spatial distribution to be able to determine where the mobility needs are

The CBD area remains the main attractor in most cities. There is a decreasing proportion of CBD oriented trips, although it remains the single most concentrated trip destination

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Spatial Distribution of Travel

The network impacts spatial distribution of trips. There are substantial differences between radial and grid systems

Spatial distribution of travel can be described graphically as shown in the next slid

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Spatial Distribution of Trips

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Trip Length

Distribution of trip length Varies with trip purpose Skewed to short distances

There are significant differences between work and non-work trip length distribution

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1- Land Transportation- highway- rail - urban transit

2- Air Transportation- domestic- international

3- Water Transportation4- Pipelines5- Others

When 2 or more modes are combined to provide utility & service to public, the combination is known as a multimodal system

Modes of Transportation

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Modal Distribution

The various modes have different shares by 1- Purpose and trip length2- Effectiveness in providing the service accessibility, mobility and productivity3- Cost4- Specialization

There are other variables that affect modal distribution- age, gender, vehicle ownership….

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Economic Theory in Transportation

Five economic concepts that are important in travel demand estimation: Theory of consumer behavior Demand and supply Derived demand Equilibrium Elasticity

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Theory of Consumer BehaviorGoods (or services) have utility

Consumers can distinguish the utility of goods

More of a good (commodity) is better than less of a good -Utility maximization

quantity consumed of one good = function Price of that good The quantity of alternative good The price of alternative good Available budget

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Supply and Demand

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Transportation SupplyThe capacity of transportation infrastructures or modes (operation system) Infrastructure and/or equipment (e.g. roads, terminals,

traffic control systems, etc.) Performance of transportation system (e.g. travel time,

headway, etc.) Operation of a transportation system (e.g. frequency of

transit service, hours of service, parking restrictions, etc.)

Supply is expressed in terms of infrastructures (capacity), services (frequency) and networks (coverage). The number of passengers, volume (for liquids or containerized traffic), or mass (for goods) that can be transported per unit of time and space is commonly used to quantify transport supply

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Transportation Demand

One of the most important areas of analysis in urban transportation planning is the estimation of travel demand (needs) for transportation facility and services

Transportation needs, even if those needs are satisfied, fully, partially or not at all. Similar to transportation supply, it is expressed in terms of number of people, volume, or tons per unit of time and space

It is a function of (cost of that service, cost of competing service, quantity of other service consumed and available budget

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Transportation Demand

Note: demand is the dependent variable and the cost is the independent variable

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Derived Demand

Travel is not consumed for its own sake but for the utility (at the destination) of what can be achieved by making the trip

No transportation demand analysis (travel estimation process) can be performed without considering the socioeconomic activity system at the trip end (land use)

The utility of the activity at the trip end is a function of the cost of making the trip ( the benefit of making the trip and the cost of making the trip are integrally linked)

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Movement & Transportation Connection between Land Use and Transportation

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Equilibrium Supply curve shows the change in supply/quantity of goods a producer is willing to offer at a given price. Demand curve shows the change in demand (by the consumer) with changing in price (e.g. bus seats at a given price)

Equilibrium: Demand = Supply

By increasing supply more demand is satisfied

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Elasticity

Price elasticity of demand is the change in demand following a unit change in price

ε = ∆d/d = p/d___ ∆p/p ∆p/∆d

Note: elasticity is not the slop of the demand curve, it is p/d divided by the slop of the demand curve

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Elasticity

Can get demand elasticity with changes in other things than price (e.g. frequency of transit service) Elasticity value less than 1 are termed “inelastic” and those equal or greater than 1 are “elastic” Elasticity is often expressed as percentages

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Cross Elasticity

The price cross elasticity of demand is the change in demand of one commodity following a unit change in the price of another (e.g. the change in the ridership of bus following a unit change in the price of rail)

Cross elasticity is always between pairs of commodities which are usually modes in transportation

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Transportation Planning and Engineering Process (TPEP)

TPEP is concerned with supplying infrastructure and/or equipments that satisfies the future demand

Development of facilities Planning Preliminary design (alternatives) Detailed design (best alternative) Construction Operation On going management Planning

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Transportation Planning Process Approach

Define Goals and Objectives

Identify Problems

Generate Alternatives

Evaluate Alternatives

Select Optimal Alternative

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Travel Forecasting:Inventory

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What is an Inventory and Why is it Needed?

An inventory is defined as an information system (or data base) of what is there

An inventory is needed to provide information on both current demand and supply

The inventory provides a description of current system and provides input for estimation of forecast models (i.e. current travel = f( current demand, price, performance of existing system, etc.)

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Components of an Inventory

Data are needed for: Network description

Present supply characteristics Present level of use (demand)

Zone characteristics and description Model update Model calibration Model validation

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Different Data Required in an Inventory

Physical inventory of highway and transit network

Inventory of land use

Inventory of population characteristics Inventory of travel pattern Internal trip making over 24-hour weekday period Internal-external trip-making External-external trip making

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Inventory ContentOverall data needs for using or developing forecasting procedure are: Sample data on trip making Network based data on travel characteristics (vol., time, speed…) Socioeconomic data of trip makers (INC, HHS, ..) Geographic referencing for all data (spatially determined)

Data needs for updating and validation are different from those for calibration Updating and validation do not require detailed data on trip making Updating and validation do require data on overall traffic volumes

by location and direction Updating and validation can be done on much smaller samples than

calibration

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Sampling

Samples versus censuses Sampling are designed to allow Samples are less than 100% of the population Census are generally not necessary

Sampling are designed to allow collection of a small sample of data rather than performing a census, which is very expensive

The basis of sampling is RANDOM samples (randomness)

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Random Samples The basic statistical definition of random is that every unit of the population has an equal probability of being selectedRandomness is a means to ensure representativenessRandom sampling can be achieved only through considerable care in the sampling processRandom DOES NOT equal HAPHAZARD Among the required properties of samples are: They should be representative of the total population They should not exclude any relevant subset of the

population (coverage area, telephone-based) They should have known statistical properties that allow

for proper expansion They should be unbiased

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Data Reduction

Data reduction is the process of taking data and entering it into a machine-readable form This involves coding data to numeric values (male,

female (0,1)) Good questionnaire design should provide simple,

direct encoding for most responses Address information requires geocoding (biggest

and most difficult) Open-ended questions also require coding

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Data ExpansionThis involves multiplying each observation in the sample by a number that represents the frequency of occurrence of this observation in the total population

Expansion is actually the inverse of sampling rate

Additional factors may be required to correct for under-or over- representation (Expansion and weighting factors)

Only expanded data should be used in modeling

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Methods of Data CollectionThere are a number of alternative methods for collecting personal data:Diary methods vs. retrospective methodsTrip or travel diaryActivity diaryTime use diaryPersonal interview and CAPIMail out-mail backMail out with telephone or CATI retrieval

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Methods of Data CollectionRoadside interviews and counts (not popular anymore)Interviews can establish:Number of occupants Trip purposeFrequencyOrigin and destination

Counts can determine only directions and number of vehiclesRoadside counts and interviews are used for cordons screening

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Methods of Data CollectionOn-board surveys, ride checks, and farebox counts on transit On-board surveys involve passengers being

interviewed or completing self-administrated surveys (This can establish boarding and alighting, origin and destination, mode of access, socioeconomic data, etc.)

Ride checks involve counting ons, offs along the route or at every stop

Farebox counts involve counting fare types and total fare revenue

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Methods of Data CollectionScreenlines, cutline, and cordon surveys

These are surveys undertaken at boundaries or along imaginary linesThese surveys involve count of all crossingThe surveys may involve interviews of a sample of vehicle drivers and/or passengersSome surveys may be done using license plat techniques

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Methods of Data CollectionSpeed/flow surveys are also required

These are usually done with traffic countersThere are problems with multi-axle vehicleSpeeds may also be measured using one of several techniques such as floating car techniques

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Methods of Data CollectionCollection of data on goods movements involves many problemsConfidentiality of freight dataBurden of task

Most goods movement data collection has been done with small samples collecting truck movements

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Time and Cost IssuesPreparation time including pretesting is about 3 to 4 months

Execution time is about 6 month

Data coding, cleaning, and preparation is about 18 to 24 months

Hence, total time to completion can be about 3 years

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Travel Forecasting: Data Management

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Data-Collection PlansData need to be consistent with respect to time periodHence, data from household interview survey need to be for the same time periods as data for:Highway countsOn-board transit surveyCordon and screenline surveysEtc.,

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Data-Collection PlansSupplementary data may also be used to provide additional information (e.g. employment data)Data expansion requires supplementary dataOngoing data collectionHighway and transit counting programsLand-use updatingEtc.,

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How current Should Data be?Data should be as up-to-date as possibleOlder data have many problems:Changes in household structureChanges in travel patternChanges in technologyDifference in congestion levelsEtc.,

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Travel Forecasting: Zones and Networks

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Traffic Analysis Zones- TAZsWhy do we need zones?To analyze travel pattern in the aggregate levelAs means to provide characteristics of a

neighborhood that affect travelTo represent sources and sinks of trips (O-D)

What are zones?Zones are contiguous geographic areas within the

metropolitan regionZones are geographic units with variety of

characteristics

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How to do zonesZones should be:Homogenous with respect to LUInterzonal trips must be minimizedRoughly equal sizePhysical, political, historical barriers should be recognizedLogical shapes to specify logical centroid which will represent the zone in aggregate

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Networks What is a networkA network is a computer represent-action of a transportation systemIt consists of nodes representing intersections and links representing traveled way between intersections

Nodes Have a locationHave no other attributes in standard networks

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Networks LinksHave no position informationRepresent the traveled way or route between two

nodesHave attributes that relate to performance

Some basic parameters of network are:CapacitySpeedTimeArea typeFacility typeOne-way, two way

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Networks There are differences and similarities between highway and transit networksHighway networks:Consists of the physical street facilities and attributesHave capacities that affect loading (assignment)

Transit networksRepresent bus and rail routes over the physical streets or rail linesDo not have capacities that relate to loadingNodes represent bus stops or rail stations not intersectionsNetworks include access and egress links for walking and autoLocal walk networks may be required

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Travel Demand Estimation

In general, travel demand forecasting attempts to quantify the amount of travel on the transportation system

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The Conventional 4-step Transportation Forecasting Process

The overall approach is to define the travel decision process as a set of models that include the various decisions that people make

The decisions that are made include: Whether or not to make a trip for a given reason(s) Where to go How to get there What route to take

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Travel Demand EstimationThe 4-step Process

Additional decisions may also be involved such as: When to make the trip With whom to make the trip How often to make the trip

These decisions are most probably made more or less simultaneously, or at least interactively and not in a specific sequence …… shortcoming of the process

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Travel Demand EstimationThe 4-step Process

The modeling of these decisions as a simultaneous model is very complex

In order to simplify the problem, transportation planners proposed a set of sequential models to represent the decision

This sequence of decision is shown in the next slid:

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Travel Demand EstimationThe 4-step Process

Whether to make the tripTrip Generation

Where to goTrip Distribution

How to get thereMode Choice

What rout to chooseNetwork Assignment

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1- Trip GenerationTravel demand modeling uses the concept productions and attractions rather than origins and destinations

The production end of a trip is home end if either end is home

The attraction end is the non-home end of a trip with either end is at home

For a trip with neither end at home, the production is the origin

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Trip GenerationTrips are defined as either home-based or non-home-based

A trip is home-based if one or other end of the trip is at home

Home is important b/c it defines the characteristics of the household and persons within it This also helps define the need to travel and the

available resources and constrains on travel

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Trip Generation

The concept of home-basing and productions and attractions are linked as follows:

Purpose Productions Attractions Home-based Home Non-home

Non-home-based origin Destination

Productions and attractions have no directional content

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Trip Generation

There is: one home-work o-d trip, one work-home o-d trip, but two home-work p-a trip

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Trip ProductionProduction is the home end for HB trips or origin for NHB trips. It estimates the tendency of HH to travel Function of HH characteristics and accessibility

Trip production models estimate the number of trips produced by household by purpose of trip

There are two primary approaches to modeling trip productions: Regression approach Cross-classification approach

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Trip ProductionTwo alternative dependent variables Trip rates (more common, disaggregated model,

person, HH) Trip totals (aggregated model, zonal)

Potential independent variables are: Household characteristics Vehicle ownership Income Neighborhood characteristics (avg. price of homes)

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Regression ModelsLinear regression models are of the form:

Pi = a₀ + a₁X₁ + a₂X₂ +a₃X₃ + …… + ε

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Regression ModelsThe model assumes linearity of effects on trip productions

Note that:Pi = aXlnPi = Xᵃ = alnX

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Regression ModelsLinear regression models assume normality of the error term (the same variance over X)

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Linear Regression Review

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Statistical Model We assume the response variable y is related to predictor variable x

by:

yi = a + b xi + εi , i=1 ,…, n

where:(1) yi denotes the response corresponding to the ith experiment run in which the predictor variable x is set at the value xi.(2) The parameters a and b , which locate the straight line, are unknown.(3) ε1 , … , εn are the unknown error. These are unobservable random variables, which we assume are independently and normally distributed with mean 0 and unknown standard deviation σ. So y1 , … , yn are also normal random variables.

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Least Squares

yi

Predicted value at xi

resi = yi – a-bxi

find the straight line that minimizes D= (S yi-a-bxi)2

xi

y=a + b x

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Some Notations For Calculating Estimators

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Formulas For Estimators

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Standard Errors

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Inferences

• with these estimators and standard errors, you can test hypotheses about the true parameters a and b.

– based on the t-distribution with n-2 degrees of freedom

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T Statistics

• For Null Hypothesis H0: b=0 ;

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T Statistics

• For Null Hypothesis H0: a=0 ;

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Goodness of Fit

• The strength of linear association between two variables is measured by

• between 0 and +1• if r2 small, the straight line does not give a good

fit.

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xy

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Sr

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What to Look at in your Regression Model

Sign of each parameter and value of the intercept

Significance of each parameter, t-test

R²,coefficent of determination, how much the variation in the data is explained by the model

F-value: measure if all parameters differ from zero

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Problems with Regression

Non-linearity of relationships between trips and most independent variables

Problems of ease of adding irrelevant variables

Be careful of … co-linearity

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Cross Classification Cross classification procedures measure the change in one variable (trips) when other variables (land use) are accounted for

It resembles regression techniques

Sometimes called “Category Analysis”

Applied for trip rates only

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Cross Classification

Disadvantages Does not permit extrapolation No goodness of fit measures Requires large sample size

Involves setting up matrices of trip rates for categories of HH, defined by one or more variables

Non-linearity of the relationship between trips and most independent variables

One step Cross classification model

HBW

From: Amarillo 1990 model

* Note: US avg. median HH income = $30K in 1990 … is $50,000 (2007)

0-$8000

$8K-$16K

$16K-$32K

$32K-$56K

$56K plus

2007 eq.*

NHB

From: Amarillo 1990 model

One step Cross classification model

0-$8000

$8K-$16K

$16K-$32K

$32K-$56K

$56K plus

2007 eq.

Triple Rate Cross classification model

Cross classification model

Some typical trip rate information:Person trips HBW Trips: 1.63 per day per worker HBW Trips: 0.75 per day per person Total Trips : 4.5 per day per person

Household trips HBW Trips: 2.5 per day HBNW Trips: 0. 5 per day NHB Trips: 2.5 per day Total Trips : 10 per day

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Calibration Model is calibrated for each trip purpose by determining the average trip rate for households in the specific category

Model may be designed and calibrated using Analysis of Variance (ANOVA)

ANOVA provides a statistic of goodness of fit

ANOVA will help determine the best classification

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Trip AttractionAttractions is the non-home end in a HB trips or the destination for NHB

Trip attraction models define the number of trips attracted by non-home land uses and by households other than those of the trip maker

Assumes that trip attractions are related to type and intensity of land use

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Trip AttractionIntensity measures could be square feet of industrial or commercial area or the number of persons employed

Trip attractions are generally estimated in terms of trips per square foot or per employee or resident

Trip attraction models may also use two different approaches mathematically: Regression approach Cross-classification approach

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Trip Attraction

Both methods use either trips per employee or trips per square foot

Attraction-production balancing is necessary, b/c there are two independent models producing what must total the same for the study region

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Balancing

Balancing of productions and attractions is based on what is more accurate

Alternatives are balanced to: Productions Attractions The mean of productions and attractions

Productions are generally estimated more accurately than attractions

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BalancingProcedure for production balancing is as follows: Sum total productions Sum total attractions For each zone, multiply attractions by the ratio of

total productions to total attractions New sum of attractions will equal sum of

productions

Procedure for attraction balancing is identical, except using the attractions as the control total

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Balancing

Residential zone (1) commercial zone (2)

Productions/hh/day =1 + 2(hh size)Attractions in a zone/day = 100 + 5(# of employees in the zone)

1000hhs100 hh 1 person400 hh 2 persons500 hh 3 persons

20 business 10 employ 5 persons each10 employ 10 persons each

10hhs4 hh 2 persons6 hh 3 persons

100 business 60 employ 5 persons each40 employ 10 persons each

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Balancing

Productions/hh/day =1 + 2(hh size)

P₁ ( productions zone (1) = 100 ( 1+ 2(1)) + 400 ( 1+ 2(2)) + 500 ( 1+ 2(3)) = 5800

P₂ ( productions zone (2) = 4 ( 1+ 2(2)) + 6 ( 1+ 2(3)) = 62

Total productions = 5800 + 62 = 5862

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Balancing - Example

a₁ ( attractions zone (1) = 100 + 5(10 * 5 + 10 *10) = 850

a₂ (attractions s zone (2) = 100 + 5( 60 * 5 +40 * 10) = 3600

Total attractions = 850 + 3600 = 4450

Balancing Productions and Attractions:

a₁ = 850 * 5862/4450 = 1119 = P₁

a₂ = 3600 * 5862/4450 = 4742 = P₂

Attractions in a zone/day = 100 + 5(# of employees in the zone)

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2- Trip Distribution Trip generation estimates the amount of travel, while the trip distribution describe the travel pattern in an area

Trip distribution is the process of allocating the trip production ends to the trip attraction ends to define the two ends of a trip

The allocation procedure must leave the total number of trips unchanged

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Trip Distribution Because travel patterns differ for different trip purposes, it is common to produce different trip distribution models for different trip purposes

There are two common methods and much less common method for performing trip distribution

The common methods are: Growth factor methods -Fratar, Detroit Average Gravity models

The less common method is: Intervening Opportunities Models

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Trip Conservation Rules

1- The sum over all zones j of the trips produced at i and attracted to j must equal the productions at i

ΣjTij = Pᵢ

2- The sum over all zones i of the trips attracted to j and produced at i must equal the attractions at j

ΣᵢTij = Aj

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Trip Conservation Rules

3- The sum over all production zones i and attraction zones j of the trips produced at i and attracted to j must equal the sum over i of productions at i, which must equal the sum over j of attractions at j, which equals the total trips

ΣᵢΣjTij = ΣᵢPᵢ = ΣjAj = T

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Growth Factor ModelThe growth factor models are based simply on determining a growth factor for each of the production and attraction ends and estimating future trips on the basis of present trips multiplied by a function of the growth factor

Growth factors are usually determined as a ratio of future productions and attractions to present productions and attractions

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Growth Factor ModelAll growth factor models have as their basic structure:

Tij = Eij*tij

Where:Tij: future trip interchange between zones i and jEij: expansion factor between zones i and jtij: existing trip interchange between zones i and j

Note: future values are shown in upper case and current values in lower case letters

Growth factor models differ only in the manner in which the expansion factor, Eij, is formulated in terms of growth factor

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Growth Factor ModelNote that in trip distribution, trip generation is already complete so Pᵢ and Aj are known. For growth factor models you use tij, the existing trip matrix, as the starting point to grow the new matrix. Thus, for all growth factor models, Pi, Aj, and tij are assumed known for all i,j

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Growth Factor ModelGrowth factors describe the growth that is expected to occur in zones

Ideally, growth factors are specified separately for productions and attractions, making two growth factors per zone

Growth factors are sometime allowed to alter from iteration to iteration as the model converges to a solution

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Growth Factor ModelUsing the convention that uppercase letters depicts future values and lowercase letters present values

Growth factors for zone i are: Fᴾi = Pi/pi

Fᵃi = Ai/ai

or, if a common growth factor is used for each zone,

Fi = Ti/ti = (Pi+Ai)/(pi+ai)

Single growth factor for a region: F = ΣTi /Σti

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Growth Factor ModelFratar Model

The model proposes that the expansion factor is the product of the production-specific growth factor (Fᴾi ) and the attraction-specific growth factor (Fᵃj) divided by the weighted average of all attraction-specific factors

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Fratar ModelFormulation of the Fratar Model

Eij = Fᵢ __Fj_ = Fᵢ Fj Σj tij

ΣjFj tij ΣjFj tij

Σj tij

Thus,Tij = tij Fᵢ Fj Σj tij

Σj Fj tij

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Fratar ModelFormulation of the Fratar Model

Note, the expansion factor is Fi times the Fj divided by the weighted average of Fj

As always, Fij = Eij * tij

The fratar model is probably the most used growth factor model

Fratar model satisfies the 1st and 3rd conservation rules, but not the 2nd. You can iterate the solution to get it to approach satisfying the 2nd conservation rule by adjusting the growth factors one each iteration

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Fratar ModelExample

FROM (i)

TO (j)

A B C D

A 0 12 10 18

B 12 0 14 6

C 14 10 0 14

D 20 8 10 0

Say that we have the following hypothetical P-A matrix showing the existing trips between the four zones A, B, C, and D. Note, the trip matrix is not symmetrical- for example more trips are produced in C that are attracted to A (14) than are produced in A and attracted to C (10)This is the existing trip matrix, tij, a necessary component for any growth factor model. From the table we can get all pi’s and aj’s by summing the rows and columns

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Fratar ModelExample

FROM (i)

TO (j)

A B C D pi Pi Fi

A 0 12 10 18 40 80 2

B 12 0 14 6 32 50 1.6

C 14 10 0 14 38 110 2.9

D 20 8 10 0 38 40 1.1

aj 46 30 34 38 148

Aj 70 46 50 114 280

Fj 1.53 1.53 1.50 3

If we sum the rows we get pi, the existing productions in each zone. Summing the columns gives us aj, the existing attractions in each zone Pi and Aj can be obtained from trip generation model

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Fratar Model - Example

TAB = __________12*2*1.53*40_________ = 16.80 (0*1.52)+(12*1.53)+(10*1.5)+(18*3)

TAC = __________10*2*1.5*40___________= 13.74 (0*1.52)+(12*1.53)+(10*1.5)+(18*3)

TAD = __________18*2*3*40______________= 49.45 (0*1.52)+(12*1.53)+(10*1.5)+(18*3)

TBA = __________12*1.6*1.52*32_________ = 16.32 (12*1.52)+(0*1.53)+(14*1.5)+(6*3)

TBC = __________14*1.6*1.5*32_________ = 18.78 (12*1.52)+(0*1.53)+(14*1.5)+(6*3)

Tij = tij Fᵢ Fj Σj tij Σj Fj tij

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Fratar Model - Example TBD = __________6*1.6*3*32____________ = 16.10

(12*1.52)+(0*1.53)+(14*1.5)+(6*3)

TCA = __________14*2.9*1.52*38_________ = 29.84 (14*1.52)+(10*1.53)+(0*1.5)+(14*3)

TCB = __________ 10*2.9*1.53*38 _________ = 21.46 (14*1.52)+(10*1.53)+(0*1.5)+(14*3)

TCD = _________ 14*2.9*3*38 ______________= 58.90 (14*1.52)+(10*1.53)+(0*1.5)+(14*3)

TDA = __________20*1.1*1.52*38_________ = 22.05 (20*1.52)+(8*1.53)+(10*1.5)+(0*3)

TDB = __________ 8*1.1*1.53*38 _________ = 8.88 (20*1.52)+(8*1.53)+(10*1.5)+(0*3)

TDC = _________ 10*1.5*1.1*38 ______________= 10.88 (20*1.52)+(8*1.53)+(10*1.5)+(0*3)

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Fratar Model - Example

FROM (i)

TO (j)

A B C D pi Pi Fi

A 0 16.80 13.74 49.45 79.99 80 1.00

B 16.32 0 18.78 16.10 51.2 50 0.98

C 29.84 21.46 0 58.90 110.20 110 0.998

D 22.05 8.88 10.88 0 41.81 40 0.96

aj 68.21 47.14 43.40 124.45 283.20

Aj 70 46 50 114 280

Fj 1.03 0.98 1.15 0.92

Note that it satisfied conservation rules 1 & 3 but rule 2 was not satisfied, a 2nd iteration was done to satisfy rule 2

1st rule satisfied

3rd rule satisfied

2nd rule NOT

satisfied

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Fratar Model - Example

FROM (i)

TO (j)

A B C D pi Pi Fi

A 0 16.94 16.25 46.80 79.99 80 1.00

B 15.85 0 20.36 13.97 50.18 50 0.996

C 31.90 21.83 0 56.25 109.98 110 1.00

D 20.75 7.95 11.43 0 40.13 40 0.997

aj 68.50 46.72 48.04 117.02 280.28

Aj 70 46 50 114 280

Fj 1.02 0.985 1.04 0.97

1st rule satisfied

3rd rule satisfied

2nd rule satisfied

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Detroit Growth Factor ModelFormulation

Eij = Fᵢ __F j_ = Σj F j

nn: number of zonesTij = tij Fᵢ F j

F j/nThis is a simplification of the Fratar Model in that it uses the arithmetic average rather than the weighted average used in Fratar ModelThe Detroit model does not satisfy any conservation rules but can be made to approach satisfying them by using the iterative approach

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Detroit Growth Factor ModelHomework

Apply Detroit Growth Factor Methods to the previous example Iterate the solutions till the adjusted growth factors approaches unity

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Gravity ModelThe most common trip distribution model

The gravity factor model is hypothesized as an analogy to Newton’s Law of Gravitation The masses of two bodies are replaced by the

productions and attractions of two zones The distance squared is replaced by a function

factor or a function of travel time or function of travel impedance

Impedance is usually defined as travel time or composite of time and cost

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The Basic Gravity ModelThe form of Newton’s Low of Gravitation is:

f12 = GM1M2/ D12²Where: f12: the gravitational force between objects 1 & 2

G: the gravitational constant

Pallin adapted Newton’s Low to transportation using population and distance for one-way trips:

Tij = (K*Hi *Hj)/Rijⁿ

Where: Tij: the trips between zones i & j Hi, Hj: population of zones i & j Rij: distance between zones i & j K, n: parameters to be estimated

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Theoretical Development of the Gravity ModelPallin’s model does not meet the trip conservation rulesApplying the trip-conservation rules produces a highly restricted modelA single value for the exponent n was found to be difficult to obtainTo correct these problems, the population of zones i & j was replaced with: Productions in i Attractions in j

The simple exponent was replaced by a general function of the form f(Rij)...... Function of impedance

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Revised Gravity ModelIn general gravity model became:

Tij = K Pi Aj f(Rij)

A doubly contrained gravity model is obtained by simultaneously satisfying both the 1st and 2nd trip conservation rules and the single constant K is replaced by two sets of constants Bi & Cj

Tij = Pi Aj Bi Cj f(Rij)

Bi = { Σj[Cj Aj f(Rij)]}¯¹

Cj = { Σi[Bj Pi f(Rij)]}¯¹

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Gravity ModelA number of different methods are available to estimate the impedance function Specific functional forms such as:

Gamma function Exponential function Power function Discrete functions (most common, function

factors)Travel times (and costs) are derived from the highway network (or highway and transit)

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Gravity Model

Network input data are therefore very important, b/c the network is the source of the impedance function

Actual travel times should be reflected on the network, including congested speeds for peak periods

The problem here is that congested speeds are unknown until the entire model steam has been run

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3- Mode-ChoiceRepresents the process of choices between alternative modes of travel

Purpose is to distribute the estimated generated trips among the various modes of transport

Simplest is a binary model reflecting the choice between auto and transit

Models were developed originally as “modal-split” models

It is incorrect to use the term “modal-split” to refer to mode shares or market sharese.g. it is incorrect to state that “….the modal split is 5%…” when the meaning is that transit has 5% of the market

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Mode-Choice Models Originally, this step was not present in earliest models which were highway planning models

After recognizing that some estimate was needed for transit trips, initial modes were proposed in the form of trip-end-modal-split models

A trip-end-modal-split model comes before trip distributionIt is based on the notion that demographics determine transit use, not LOS (ch of traveler not ch of car, transit,…)Many of these models were regression models or cross classification models

Subsequently, trip interchange models were developed These models follow trip distribution and apply to each trip interchange They allow mode choice to define not just as a function of demographics, but also LOS

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Mode-Choice Models

Different methods are still somewhat appropriate for different size cities

Small cities with limited transit could still use a trip end model

Any city with significant transit service or interested in planning for transit investments needs a trip interchange model

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Discrete Choice (disagg. Level)Most common discrete choice model is the Multinominal Logit Model

Discrete choice models relate the probability of choosing a particular mode to the utility the traveler will gain from choosing that mode relative to the utility of choosing any other modes available to the traveler

The models are usually constructed for three trip purposes: HBW, HBNW, NHB

Note: discrete choice models can be used in transportation for modeling any of the 4-step-process

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Utility FunctionIt is assumed that: Travelers are utility maximizes when it comes to the

choices they make in transportation Utility is a function of the attributes of an alternative as

well as the characteristics of the person making the choice Persons with the same socio-economic characteristics

have the same tastes and values in estimating the utility of alternatives

The utility is assumed to be assessed entirely by the socio-economic characteristics of the chooser and the attribute of the alternative. Ignored influences are assumed to cause random but unbiased variation in behavior

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Parameters of ChoiceAttributes that affect choice probably include: Time Cost Comfort Convenience Reliability Safety Etc….

The characteristics of the individual and household also affect choice, e.g., Vehicle ownership Trip purpose Age Income

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Sample Utility Function

Vauto = α + β(auto travel time) + ϒ(auto travel cost)

Vtransit = δ(transit travel time) + ε{(transit travel cost)/household income}

Where, α, β, ϒ, δ, and ε are parameters that are estimated from travel survey data and reflect the relative value assigned to each attribute (affect mode choice)

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Logit ModelThese are models that relate the probability of an outcome to a (usually) linear function of parameters in an exponential function

Pauto = eᵁᵅ____ Σeᵁᵀ+ eᵁᵅ

Insert shape of logit model

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Logit Model - ExampleA market segment consist of 500 individuals, a Multinominal Logit Model is calibrated for the market segment resulting in the following function:

Um =Bm – o.3C – 0.02T

Where C: out of pocket cost (JD) and T: travel time (min) Bm: Bus 0.00

Rail 0.40Auto 2.00

For a particular 0-D trip, the cost and time are as follows:Mode Cost(JD) Time (min)Auto 25 15Rail 1.5 20Bus 1.0 30

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Logit Model - ExampleUA = 2 – 0.3(2.5)-0.02(15) = 0.95UR = -0.45UB = -0.90

PA = e·⁹⁵/(e·⁹⁵+ e¯·⁴⁵+ e¯·⁹⁰) = 0.712 PR = e¯·⁴⁵/(e·⁹⁵+ e¯·⁴⁵+ e¯·⁹⁰) = 0.176 PB = e¯·⁹⁰/(e·⁹⁵+ e¯·⁴⁵+ e¯·⁹⁰) = 0.112

Note: PA + PR + PB = 1

# of trips expected to use Auto = 500* 0.712 = 356 trips# of trips expected to use Rail = 500* 0.176 = 88 trips# of trips expected to use Bus = 500*0.112 = 56 trips

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4- Network AssignmentUse output from trip distribution and mode choice to allocate highway vehicle trips to the network

Task of network assignment is to assign interzonal flows to specific routes (route choice decision)

The needed inputs are O-D totals for combined purposes

Assignment can be by hour, period, or Average Daily Traffic (ADT)

Highway and transit network assignment are done separately

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Network AssignmentAll assignment based on the principle that travelers will choose the “shortest path”

First network assignment method to be used was “ All-or-nothing” assignment All travel between each zone pair is loaded onto the

shortest path or path of minimum impedance Permits quick solution but tends to be inaccurate on

lightly traveled links

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Network AssignmentFirst improvement in assignment methodology was introduction of “capacity restraint”

Capacity restraint is the incorporation into the assignment process the fact that link impedance alters with the volume of traffic on the link

Early capacity restraint procedures: Incremental assignment- apply, for example, 40% of

interzonal flows using all-or-nothing, recalculate link travel times, assign 30% of remaining interzonal flows, recalculate link travel and apply 20%, recalculate, and then apply last 10%

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Network AssignmentEarly capacity restraint procedures: Sequential assignment- assign some interzonal flows using

all-or-nothing, recalculate link travel times, assign some more zones, and repeat the process until all interzonal flows have been assigned

Current procedure - Equilibrium assignment: User equilibrium - no traveler can alter his/her route

without increasing their travel times System equilibrium - total travel time in the system

minimized

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Network AssignmentStochastic assignment – when it is assumed that travelers do not have perfect information about travel times and therefore will behave probabilistically: Early stochastic assignment model developed by Robert

Dail using logit model to probabilistically assign trip interchange to routes based on relative travel times of valid routes

Stochastic user equilibrium assignment models are available in modern packages