Transportation Engineering Dr. Hana Naghawi 1 Transportation Engineering 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 is the study of the movement of people and goods 2 Transportation System (TS) TS is the service provided for the movement of people and goods between different locations We all have a personal experience as a user of transportation system - car driver - bus passenger
- elevator user - sidewalk user 3 Components of (TS) 1- physical facility 2- Fleet 3- operating facility 4- organization 5- Operating strategy 4 Why Do People Travel?
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 All humans are interacting over distance and time as a result the spatial distribution of activities generates travel demand 5 Goods Transportation All agricultural & industrial raw materials, products, equipment are needed to be transported from one place to another
6 The Importance of Transportation The Importance of Transportation is in the development of a country. A countrys economic status depends upon how well served the country by different modes of transportation 7 Challenges for Transportation Engineer 1- Traffic congestion 2- Traffic safety
3- Environmental protection 4- Incorporating new technology 5- Funding 6- Equal accessibility 7- Developing institutional arrangements 8 Transportation in an Urban Setting 9 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 10 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 11 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
12 Trip Purpose Trip characteristics Trips 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 13 Temporal Distribution of Trips Time 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 14
Typical Time-of-Day Distribution 15 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 16 Spatial Distribution of Travel Each 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 17 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 18 Spatial Distribution of Trips 19 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
20 Modes of Transportation 1- Land Transportation - highway - rail - urban transit 2- Air Transportation - domestic - international 3- Water Transportation 4- Pipelines 5- Others
When 2 or more modes are combined to provide utility & service to public, the combination is known as a multimodal system 21 Modal Distribution The various modes have different shares by 1- Purpose and trip length 2- Effectiveness in providing the service accessibility, mobility and productivity 3- Cost 4- Specialization There are other variables that affect modal distribution- age, gender, vehicle ownership. 22
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 23 Theory of Consumer Behavior Goods (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 24 Supply and Demand 25
Transportation Supply The 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
26 space is commonly used to quantify transport supply 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
27 Transportation Demand Note: demand is the dependent variable and the cost is the independent variable 28 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) 29 Movement & Transportation Connection between Land Use and Transportation 30 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 31 Elasticity Price elasticity of demand is the change in demand following a unit change in price = d/d d/d = p/d___ d/d p/p d/d p/d/d d
Note: elasticity is not the slop of the demand curve, it is p/d divided by the slop of the demand curve 32 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 33
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 34 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 35 Transportation Planning Process
Approach Define Goals and Objectives Identify Problems Generate Alternatives Evaluate Alternatives Select Optimal Alternative 36 Travel Forecasting: Inventory
37 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.) 38
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 39 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 40 Inventory Content Overall 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 41 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) 42 Random Samples The basic statistical definition of random is that every unit of the population has an equal probability of being selected Randomness is a means to ensure representativeness
Random sampling can be achieved only through considerable care in the sampling process Random 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 43 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 44 Data Expansion This 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 45 Methods of Data Collection There are a number of alternative methods for collecting personal data: Diary methods vs. retrospective methods Trip or travel diary Activity diary
Time use diary Personal interview and CAPI Mail out-mail back Mail out with telephone or CATI retrieval 46 Methods of Data Collection Roadside interviews and counts (not popular anymore) Interviews can establish: Number of occupants Trip purpose Frequency Origin and destination
Counts can determine only directions and number of vehicles Roadside counts and interviews are used for cordons screening 47 Methods of Data Collection On-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 48 Methods of Data Collection Screenlines, cutline, and cordon surveys These are surveys undertaken at boundaries or along imaginary lines These surveys involve count of all crossing The surveys may involve interviews of a sample of vehicle drivers and/or passengers Some surveys may be done using license plat
techniques 49 Methods of Data Collection Speed/flow surveys are also required These are usually done with traffic counters There are problems with multi-axle vehicle Speeds may also be measured using one of several techniques such as floating car techniques 50 Methods of Data Collection Collection of data on goods movements involves
many problems Confidentiality of freight data Burden of task Most goods movement data collection has been done with small samples collecting truck movements 51 Time and Cost Issues Preparation 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 52 Travel Forecasting: Data Management 53 Data-Collection Plans Data need to be consistent with respect to time period Hence, data from household interview survey need to be for the same time periods as data for:
Highway counts On-board transit survey Cordon and screenline surveys Etc., 54 Data-Collection Plans Supplementary data may also be used to provide additional information (e.g. employment data) Data expansion requires supplementary data Ongoing data collection Highway and transit counting programs Land-use updating
Etc., 55 How current Should Data be? Data should be as up-to-date as possible Older data have many problems: Changes in household structure Changes in travel pattern Changes in technology Difference in congestion levels Etc., 56
Travel Forecasting: Zones and Networks 57 Traffic Analysis Zones- TAZs Why do we need zones? To analyze travel pattern in the aggregate level As means to provide characteristics of a neighborhood that affect travel To represent sources and sinks of trips (O-D) What are zones? Zones are contiguous geographic areas within the
metropolitan region Zones are geographic units with variety of characteristics 58 How to do zones Zones should be: Homogenous with respect to LU Interzonal trips must be minimized Roughly equal size Physical, political, historical barriers should be recognized Logical shapes to specify logical centroid which will represent the zone in aggregate
59 Networks What is a network A network is a computer represent-action of a transportation system It consists of nodes representing intersections and links representing traveled way between intersections Nodes Have a location Have no other attributes in standard networks 60
Networks Links Have no position information Represent the traveled way or route between two nodes Have attributes that relate to performance Some basic parameters of network are: Capacity Speed Time Area type Facility type One-way, two way 61
Networks There are differences and similarities between highway and transit networks Highway networks: Consists of the physical street facilities and attributes Have capacities that affect loading (assignment) Transit networks Represent bus and rail routes over the physical streets or rail lines Do not have capacities that relate to loading Nodes represent bus stops or rail stations not intersections Networks include access and egress links for walking and auto
62 Local walk networks may be required Travel Demand Estimation In general, travel demand forecasting attempts to quantify the amount of travel on the transportation system 63 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 64 Travel Demand Estimation The 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 65 Travel Demand Estimation The 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: 66 Travel Demand Estimation The 4-step Process Whether to make the trip Trip Generation Where to go Trip Distribution How to get there Mode Choice What rout to choose Network Assignment 67
1- Trip Generation Travel 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 68 Trip Generation
Trips are defined as either home-based or nonhome-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 69 Trip Generation The concept of home-basing and productions and attractions are linked as follows: Purpose Productions
Home-based Home Non-home-based origin Attractions Non-home Destination Productions and attractions have no directional content 70 Trip Generation
There is: one home-work o-d trip, one work-home o-d trip, but two home-work p-a trip 71 Trip Production Production 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 72 Trip Production Two 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) 73 Regression Models Linear regression models are of the form: Pi = a + aX + aX +aX + + 74 Regression Models The model assumes linearity of effects on trip productions
Note that: Pi = aX lnPi = X = alnX 75 Regression Models Linear regression models assume normality of the error term (the same variance over X) 76 Linear Regression Review
77 Statistical Model We assume the response variable y is related to predictor variable x by: yi = a + b xi + i , 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) 1 , , n are the unknown error. These are unobservable
random variables, which we assume are independently and normally distributed with mean 0 and unknown standard deviation . So y1 , , yn are also normal random variables. 78 Least Squares find the straight line that minimizes D=S((yi-a-bxi)2 y=a + b x Predicted value at xi yi resi = yi a-bxi
xi 79 Some Notations For Calculating Estimators n n S x xi S y yi 1
x Sx n 1 y Sy n i 1 SS n x
x n i 1 n 2 i SS y y i2 i 1
i 1 SPxy xi yi i 1 S x2 S xx ( xi x ) SS x n i 1 n n
2 S yy ( yi y ) 2 SS y i 1 S y2 n n S xy ( xi x )( yi y ) SPxy i 1 SxSy
n 80 Formulas For Estimators n ( x x )( y b i 1 i i
y) n (x x) i 2 S xy S xx
i 1 a y bx resi yi a bxi 2 ( res ) i n 2
81 Standard Errors SE (b) 1 S xx 2 1 x SE (a ) n S xx
82 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 83 T Statistics For Null Hypothesis H0: b=0 ;
t b SE (b) b 1 S xx d.f.= n - 2 84
T Statistics For Null Hypothesis H0: a=0 ; a a t 2 SE (a ) 1 x n S xx
d.f.= n - 2 85 Goodness of Fit The strength of linear association between two variables is measured by 2 r S 2 xy
S xx S yy between 0 and +1 if r2 small, the straight line does not give a good fit. 86 What to Look at in your Regression Model Sign of each parameter and value of the intercept Significance of each parameter, t-test R,coefficent of determination, how much the
variation in the data is explained by the model F-value: measure if all parameters differ from zero 87 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 88 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 89 Cross Classification 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 Disadvantages Does not permit extrapolation
No goodness of fit measures Requires large sample size 90 One step Cross classification model HBW 2007 eq.* 0-$8000 $8K-$16K $16K-$32K $32K-$56K $56K plus
* Note: US avg. median HH income = $30K in 1990 is $50,000 (2007) From: Amarillo 1990 model One step Cross classification model NHB 2007 eq. 0-$8000 $8K-$16K $16K-$32K $32K-$56K $56K plus From: Amarillo 1990 model
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
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 95 Trip Attraction Attractions 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 96 Trip Attraction Intensity 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 97 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 98
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 99 Balancing Procedure 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 100 Balancing Residential zone (1) commercial zone (2)
1000hhs 100 hh 1 person 400 hh 2 persons 500 hh 3 persons 20 business 10 employ 5 persons each 10 employ 10 persons each 10hhs 4 hh 2 persons 6 hh 3 persons 100 business 60 employ 5 persons each
40 employ 10 persons each Productions/hh/day =1 + 2(hh size) Attractions in a zone/day = 100 + 5(# of employees in the zone) 101 Balancing Productions/hh/day =1 + 2(hh size) P ( productions zone (1) = 100 ( 1+ 2(1)) + 400 ( 1+ 2(2)) + 500 ( 1+ 2(3)) = 5800 P ( productions zone (2) = 4 ( 1+ 2(2)) + 6 ( 1+ 2(3)) = 62 Total productions = 5800 + 62 = 5862
102 Balancing - Example Attractions in a zone/day = 100 + 5(# of employees in the zone) a ( attractions zone (1) = 100 + 5(10 * 5 + 10 *10) = 850 a (attractions s zone (2) = 100 + 5( 60 * 5 +40 * 10) = 3600 Total attractions = 850 + 3600 = 4450 Balancing Productions and Attractions: a = 850 * 5862/4450 = 1119 = P a = 3600 * 5862/4450 = 4742 = P 103
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 104 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 105 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
jT = P 2- The sum over all zones i of the trips attracted to j and produced at i must equal the attractions at j T = Aj 106 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 jT = P = jAj = T
107 Growth Factor Model The 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 108 Growth Factor Model
All growth factor models have as their basic structure: T = E*t Where: T: future trip interchange between zones i and j E: expansion factor between zones i and j t: 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, E, is formulated in terms of growth factor 109 Growth Factor Model
Note that in trip distribution, trip generation is already complete so P and Aj are known. For growth factor models you use t, the existing trip matrix, as the starting point to grow the new matrix. Thus, for all growth factor models, Pi, Aj, and t are assumed known for all i,j 110 Growth Factor Model Growth 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 111 Growth Factor Model Using the convention that uppercase letters depicts future values and lowercase letters present values Growth factors for zone i are: Fi = Pi/pi Fi = 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 = Ti /tti
112 Growth Factor Model Fratar Model The model proposes that the expansion factor is the product of the production-specific growth factor (Fi ) and the attraction-specific growth factor (Fj) divided by the weighted average of all attraction-specific factors 113 Fratar Model
Formulation of the Fratar Model E = F __Fj_ = F Fj j t jFj t jFj t j t Thus, T = t F Fj j t j Fj t 114 Fratar Model Formulation of the Fratar Model Note, the expansion factor is Fi times the Fj divided by the
weighted average of Fj As always, F = E * t 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 115 Fratar Model Example 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, t, a necessary component for any growth factor model. From the table we can get all pis and ajs by summing the rows and columns FROM (i) A B C D
A 0 12 14 20 TO (j) B 12 0 10 8
C 10 14 0 10 D 18 6 14 0 116 Fratar Model
Example FROM (i) A B C D aj Aj Fj A 0
12 14 20 46 70 1.53 TO (j) B C 12 10 0 14
10 0 8 10 30 34 46 50 1.53 1.50 D 18 6
14 0 38 114 3 pi 40 32 38 38 148 Pi
80 50 110 40 Fi 2 1.6 2.9 1.1 280 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 117 Fratar Model - Example T = t F Fj j t j Fj t 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) 118 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) 119 Fratar Model - Example FROM (i) A
B C D aj Aj Fj A 0 16.32 29.84 22.05 68.21 70
1.03 2nd rule NOT satisfied TO (j) B 16.80 0 21.46 8.88 47.14 46
0.98 C 13.74 18.78 0 10.88 43.40 50 1.15 D pi 49.45 79.99
16.10 51.2 58.90 110.20 0 41.81 124.45 283.20 114 0.92 Pi 80 50 110 40
Fi 1.00 0.98 0.998 0.96 1st rule satisfied 280 3rd rule satisfied
Note that it satisfied conservation rules 1 & 3 but rule 2 was not satisfied, a 2nd iteration was done to satisfy rule 2 120 Fratar Model - Example FROM (i) A B C D aj
Aj Fj A 0 15.85 31.90 20.75 68.50 70 1.02 2nd rule satisfied
TO (j) B 16.94 0 21.83 7.95 46.72 46 0.985 C 16.25 20.36
0 11.43 48.04 50 1.04 D pi 46.80 79.99 13.97 50.18 56.25 109.98 0 40.13 117.02 280.28
114 0.97 Pi 80 50 110 40 Fi 1.00 0.996 1.00 0.997
1st rule satisfied 280 3rd rule satisfied 121 Detroit Growth Factor Model Formulation E = F __F j_ =
j F j n n: number of zones T = t F F j F j/tn This is a simplification of the Fratar Model in that it uses the arithmetic average rather than the weighted average used in Fratar Model The Detroit model does not satisfy any conservation rules but can be made to approach satisfying them by using the 122 iterative approach Detroit Growth Factor Model
Homework Apply Detroit Growth Factor Methods to the previous example Iterate the solutions till the adjusted growth factors approaches unity 123 Gravity Model The most common trip distribution model The gravity factor model is hypothesized as an analogy to Newtons 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 124 The Basic Gravity Model The form of Newtons Low of Gravitation is: f12 = GM1M2/ D12
Where: f12: the gravitational force between objects 1 & 2 G: the gravitational constant Pallin adapted Newtons Low to transportation using population and distance for one-way trips: T = (K*Hi *Hj)/R Where: T: the trips between zones i & j Hi, Hj: population of zones i & j R: distance between zones i & j K, n: parameters to be estimated 125 Theoretical Development of the Gravity Model
Pallins model does not meet the trip conservation rules Applying the trip-conservation rules produces a highly restricted model A single value for the exponent n was found to be difficult to obtain To 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(R)...... Function of impedance 126
Revised Gravity Model In general gravity model became: T = K Pi Aj f(R) 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 T = Pi Aj Bi Cj f(R) Bi = { j[Cj Aj f(R)]} Cj = { i[Bj Pi f(R)]} 127 Gravity Model
A 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) 128 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 129 3- Mode-Choice Represents 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 shares e.g. it is incorrect to state that .the modal split is 5% when the meaning is that 130 transit has 5% of the market 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 distribution
It 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 131 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 132 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 133 Utility Function It 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 socioeconomic characteristics of the chooser and the attribute of the alternative. Ignored influences are assumed to cause random but unbiased variation in behavior 134 Parameters of Choice Attributes 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 135 Income Sample Utility Function Vauto = + (auto travel time) + (auto travel cost) Vtransit = (transit travel time) + {(transit travel cost)/household income} Where,
, , , , and are parameters that are estimated from travel survey data and reflect the relative value assigned to each attribute (affect mode choice) 136 Logit Model These are models that relate the probability of an outcome to a (usually) linear function of parameters in an exponential function Pauto = e____ e+ e Insert shape of logit model
137 Logit Model - Example A 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 0.40 Auto 2.00 For a particular 0-D trip, the cost and time are as follows: Mode Cost(JD) Time (min) Auto
25 15 Rail 1.5 20 138 Bus 1.0 30 Logit Model - Example UA = 2 0.3(2.5)-0.02(15) = 0.95 UR = -0.45 UB = -0.90
PA = e/(e+ e+ e) = 0.712 PR = e/(e+ e+ e) = 0.176 PB = e/(e+ e+ e) = 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 139 4- Network Assignment Use 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 140 Network Assignment All assignment based on the principle that travelers will choose the shortest path First network assignment method to be used was All-ornothing 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 141 Network Assignment First 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% 142 Network Assignment Early 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 143 Network Assignment Stochastic 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
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Chemical Reactions Aqueous Solubility of Ionic Compounds Not
Aqueous Solubility of Ionic Compounds. Not all compounds dissolve in water. Solubility varies from compound to compound. Soluble ionic compounds dissociate.