Transportation Planning

Fundamentals of transport modeling, forecasting & development

Domain

Explanation

Objectives
  • Appreciate complexities of transport systems (TS) as dynamic interconnected interacting system of entire socio-economic-political environment
  • Identify basic components, input options & output impacts
  • Methodologies of TS analysis & planning

Assignments
  • Various new TS within Singapore context
  • Forecasting & evaluation à general domain-indept. research

Contents
  • Overview of TS
  • Urban TP
  • Data collection & measurement
  • Modeling methodology
  • Trip generation
  • Trip distribution

Overview of TS
  • TS as inter-related entity within macro socio-economic systems
  • IC: activity A(0), TS T(0) à equilibrium flow: F(0) equating V(t)=D(A,S) & S(t)=J(T,V) à Shifts in response: A(t), T(t) à new flow, F(t)
  • TS:
  1. Multi-disciplinary
  2. Multi-problem
  3. Multi-sectoral
  4. Multi-modal
  • Short-term: equilibrium flows
  • Long-term shifts:
  1. Activity shifts: demand changes
  2. TS shifts: supply changes, operator decisions, resource changes
  • Variables:
  1. Options: inputs
  2. Impacts: user, operator, physical, functional, government
  3. Models:
    • Service models: level of service (supply)
    • Resource models: resources required
    • Demand models: volumes of travel
    • Equilibrium models: balanced volume & LoS
    • Activity shift models: long term changes to distribution & structure of activity system
  • Modeling: stability, convergence

Urban TP in Singapore
  • History:
  1. <1960: no systematic TP à colonial
  2. 1960-1980: problem-driven TP à no foresight, trial-and-error
  3. 1990-: vision-driven TP à foresight, solve future problems
  • Policies:
  1. Vehicle ownership policies
  2. Congestion management measures
  3. Parking controls
  • Operations:
  1. Buses: few à many private ones à merger into SBS (1973) à TIBS (1982)
  2. MRT: 1971 concept plan feasibility à 1978: confirmed à 1986: completed mains à 1996: Woodlands link à 2002: NE line
  • Organization:
  1. LTA: merger of statutory boards for vision-driven TP
  2. Integration of T services: more capacity, modes, range of services, coverage of public transport services
  • Future:
  1. Freight transport
  2. Leisure
  3. Multi-modal integration
  4. Safety
  5. Calming

Data
  • Sampling:
  • Variables à sample à inference
  • Types:
  1. Random sampling: samples equally likely & independent of each other
  2. Stratified random sampling: split entire population into homogeneous groups using a priori info (like census) à weights for groups à more accurate
  3. Choice-based sampling: 2-stage à 1st: randomly selecting groups à 2nd: randomly sampling within each group
  • Based on: 1) proportion; 2) values or means
  • Errors:
  • Measurement errors: inaccuracies related instrument, observation & data gathering à increases with realism & complexity
  • Sampling errors
  • Computational errors: numerical algorithms à stability, consistency, iteration, convergence
  • Specification errors: include irrelevant variables à omit relevant variables à exclude personal preferences à others: functional
  • Transfer errors: from different contexts à from US to here à spatial errors à temporal errors à combined
  • Aggregation errors: data aggregation à alternative aggregation à model aggregation
  • Dealing with errors: simpler model with appropriate complexity à aware of errors à reduce errors
  • Surveys as data collection:
  • Length: which stage, resources
  • Study horizon: tactical (short-term) or strategic (long-term)
  • Study area: zoning structure à influence results
  • Resources
  • Types of surveys:
  1. O-D survey: from where to where
  2. Travel time & delay
  3. Trip generation
  4. Parking surveys

TP modeling
  • History: Post-war à development à concepts à critical review à R&D issues
  • Databases:
  • Population
  • Land use
  • Economic activity
  • TS
  • Travel
  • Laws
  • Finances
  • Community values: prepare for the young à might model the young
  • Study aspects:
  • Time frame: period
  • Area: zoning structure
  • Network:
  1. Node: intersection
  2. Link: connection between adjacent nodes
  3. Zone centroid: area inside zone representative of all characteristics of this zone
  4. Centroid connectors: links from zone centroid to network
  • TP tools:
  • Strategic: sketch plan à many unknowns, many alternatives, low costs, more flexibility
  • Tactical: just sufficient details
  • Microanalysis: detailed à highest costs, lowest flexibility
  • Sketch plan example: simple need à background info à expand à narrow à infer road needs
  • Traditional 4-step model:
  1. Trip generation: forecasting required, least accuracy à whether to travel
  2. Trip distribution: where to travel to
  3. Modal choice: what to travel in
  4. Network assignment: most research, most accurate à where to travel on

Trip generation
  • Trips made: number, purpose, period
  • Definition:
  • 1-way journey: from Oà D, Dà O
  • Trip: 2-way return journeys Oà Dà O
  • Trip production: home end of home-based (HB) trip or origin of non home-based (NHB) trip
  • Trip attraction: destination of (HB) or (NHB)
  • Trip generation: how to determine the no. of trips made
  • Commuter peak
  • Peak hour: highest 1-hour flow of traffic
  • Peak hour of generator: peak hour @ origin ends of trips
  • PCU: passenger car unit: differs for different vehicles (1 PCU: 1 car/taxi/van)
  • Concept of trip making:
  • Concept: activity:
    • Derived demands
    • Characteristics of trips: 1) time, 2) Frequency
  • Classification of person & trips:
    • Trip purpose: (HB) – work, school, shop; (NHB) more variables & difficulty
    • Time of day: purpose; periods – morning, evening, weekend, holiday
  • Person type: socio-economic differences & attributes
  • Mode type: availability
  • History:
  • Components & concerns
  • Modeling issues:
    • Identify factors affecting generation: focus on person (trip production); focus on activity (trip attraction)
    • Modeling: functional form (how?: method/algorithm); degree of realism (validity & achieve purpose)
  • Factors affecting trip generation:
  • Production: Household (HH) level – family size, structure, employment category; Zonal level – land value, residential density, accessibility
  • Attraction: general land-use, employment factors, special factors (size of industry)
  • Trip type: work, school, shop, recreation, sports
  • Freight production & attraction: GFA (gross floor area), business related, no. of employees, no. of sales
  • Modeling methods:
  • Growth factor model: crude (many errors), but fast, simple & allows checking for refinement
  • Regression model: maths relationships based on statistical analysis
  • Cross-classification: category analysis
  • Regression modeling:
  • Linear regression:
    • Coefficient of determination: total deviation=(explained variation)+(unexplained variation)
  • Multiple linear regression:
    • Relative contribution of variables: correlation, covariance, R2
    • 1st step: univariate method
    • 2nd step: multivariate method à compare
  • Zonal-based multiple regression:
    • Objective: develop linear relationship based on information @ zone level
    • Limits: costs of data collection
  • Household-level multiple regression:
    • Advantage: more accurate than zonal-level & allows stratification
    • Done @ HH-level, but HH size not vary much
    • Appeal: discrete & available
  • Non-linearities:
    • Linearization: using Taylor’s series truncation
    • Discretization into linear segments
    • Use of dummy variables
    • Example: apply linear regression à check sign (correct direction) à R2 à check
  • Category analysis: UK
  • Introduction:
    • Regression: USA; requires linear specifications variable independence; assumptions: error distribution & independence
    • Suffers from lower quality of correlations & statistical validity
    • Category analysis: allows many variables, no need for linear &/or continuous
  • Advantages:
    • Aggregation of trips easier
    • No unified relationship between groups need or without need for classification
    • Functional relationship need no mathematical form: but can be derived from data (type)
    • Trip rates independent of zone partitioning: since either HH-based or person-based; no need/worry for zoning
  • Disadvantages:
    • Difficult assessment of validity & adequacy: no goodness of fit
    • Extrapolation beyond limit of observation impossible or interpolation inappropriate
    • Reliability depends on sample size
    • Poorer system of identification of variable available: can only guess this category is included, but never really know how important
  • Basis:
    • Table of trip rates: constant or time-varying, based on category attributes (HH or person)
    • Assumptions:
      • Trip rates static: applicable to all future forecasts (not realistic)
      • Reproducible
    • Representation:
      • tp(h): average no. of trips per day by (HH) type (h) for purpose (p)
  • Categorization:
    • Divided by trip purpose p & HH type p
    • h defned differently for different p
    • Trip rate: tp(h) trips per day
    • No. of HH of type (h)=a(h)
    • Total no. of trips=O= tp(h)*a(h)
    • at: total trips made at t-hour interval
    • hourly rates as % of day
    • vehicle rates from mode split & occupancy: different vehicles (taxi, bus,train)
  • Typical category: trips/day/DU (DU: dwelling unit)
    • Residential: density & trips/day/DU
    • Retail
    • Industrial/manufacturing
    • Commercial development
    • Others: trip production
  • Person category analysis:
    • Follows that of HH-category
    • Basis: conceptually similar
    • Advantages:
      • Increased basic modeling unit (disaggregate)
      • Variable more readily identified & defined
      • Easier to obtain samples since variations are less
      • Fewer variables needed
      • Age & trip-making more precise
    • Disadvantages: interaction effects cannot be modeled; examples like shared resources – cars, finance
    • Model development:
      • Identification of trip types (by personal mobility pattern)
      • Identification of personal attributes
      • Collection of data: fewer categories
      • Evaluation of trips: combine categories
    • Trip generation applications:
      • Land-use classification
      • Residential
      • Institutional
      • Religious
      • Educational
      • Industrial development
      • Commercial development
      • Community facilities & recreation
      • Transport terminals: inter-phasing of trips (not O-D)
  • Summary:
    • What is category analysis?
    • Advantages & disadvantages?
    • How trip tables are set up?

Trip distribution
  • Flow of traffic between O & D
  • Trip makers’ decisions on choice of destinations (spatial dimension):
  • 2 types:
    • Long term: stable, e.g. Home-Work
    • Short term: random process
  • tij: trips from zone i to zone j
  • Modeling approaches:
  • Growth factor models: crude, but simple & sketchy; based on growth rates (constant or varying)
  • O-D demand models: demand directly between O-D: use statistics to determine flow volume
  • Choice models: how people make choices: choice probabilities & factors:
    • Socio-economic factors
    • Personal attributes
    • Choice: modes or routes
  • Physical interaction models: based on spatial interaction analogous to physical system
    • Economic & statistics models: gravity & intervening opposition
    • Transportation network
    • Entropy maximisation
  • Principle of transportation distribution:
    • Zone of origin: production
    • Zone of destination: attraction
    • Linkage between O-D: impedance between zones; represents level of difficulty in getting through
    • How many trips from O end up in D?
  • Basic properties:
    • Basis: trip values internally consistent, logical & significant
    • Conservation:
      • : total from zone i
      • : total into zone j

: total from zone i to zone j

    • Non-negativity:
    • Divisibility & compressibility: seldom satisfied
  • General structure:
    • Maths:

magnitude, %, any factors

  • Growth factor models:
    • Growth factors can be assumed historical model implied
    • Existing distributed trips=Tij
    • New trips forecasted:
    • Problems:
      • Undeveloped zones difficult to forecast
      • ai=bj: use average growth rate (between O & D) --> requires Balancing
  • Balancing:
    • Applicable to all models
    • Flow constraints occur whenever growth rates are unequal
    • 3 ways:
      • Unconstrained model: can be unequal totals from O & to D
      • Singly constrained model: constrain either (i constraints) or (j constraints); apply growth rate ratio either @ origin or @ destination
      • Doubly constrained model: satisfy both &

(i+j constraints); requires iteration to converge to level of tolerance through the growth rate ratio

    • Balancing between zones (O&D) and total number D
  • Demand models:
    • General:
      • Utitlity: U=U(Xij) for
      • Cost: C=C(Xij) for
    • Modeling:
      • Stratified
      • Aggregate
      • Disaggregate
      • Other T.D. models: disaggregate into trip type (or person) & mode type à

à advantages: combine T.G. & T.D. à no need for balancing

    • Principle:
      • Utility maximization principle: Xi values to maximize U
      • Marginal U = Marginal C:
      • Distribution function: constant elasticity utility function [e.g. , constant] à needs calibration
      • Generation-distribution model:
        • : without utility
        • e.g.: California Alameda County model: Tij=o.45Pi0.719Ej1.128e-0.165cij [where P: population; E: employment; e: total time]
    • Example in notes:
      • Note trip generation: no costs & only with socio-economic characteristics
      • E.g.: table of data à T.D. Tij formula à Tij
  • Choice models:
    • General:
      • Dependent only on individual choice: individual utility maximization
      • Independent of all other variables (not true actually)
      • Choice probability:
    • Multinomial logit models:
      • Commonly-used choice model
      • Choice function:
        • Generalized travel cost Cgj
        • Measures of attractiveness of destination j
      • Trip computation:
      • Example: V=1.58e-0.3t à data (S,t,trips) à Vij à à
      • Influence of coefficient (1.58) above is little: since MNL choice model deals with relative form à hence, major drawback of choice model
  • Intervening opportunity model:
    • Old-dated: less often used
    • Choice of destination based competing opportunities (opportunity costs)
    • Rankings: ordering j:1 (nearest) … last (furthest)
    • Choice probability & derivation
  • Physical (spatial) models:
    • Aggregate model: using mechanics applied to trip distribution context
    • Similar to choice models
    • Types:
      • Gravity models:
        • Newton: à , : calibration constant
        • Generalization:
        • Constraints:
        • Example à impedance, F à total t=terminal t + travel t à table of data à Fij à DjFij à singly constrained
      • Entropy models:
        • Uses statistical or probabilistic mechanics of particle behaviour
        • Entities randomly arranged Iin no. of ways à combinations (maths)
        • Principle: maximum likelihood of occurrence
        • E.g.: all-or-nothing distribution or uniform distribution
      • Theoretical network model:
        • Electrical or flow network
  • Factors affecting trip distribution:
    • Socio-economic factors:
      • Affect trip-making potential
      • Demand Model

•Factors embedded in Trip Production and Trip Attraction variables

•Eg. Oi=kiPia , Dj=hjEj+ljNsjc
    • Choice Model

•Factors used only for Attraction variables and disaggregate grouping

•Eg. Πij = e-V(j)/∑ e-V(k) : V(j) = f(Ns)
    • Physical Model

•Factors may be embedded in Trip Production and Trip Attraction variables as in Demand Models
    • Transport supply factors:
      • Generalized cost function, cij=f(…):
        • Total travel time
        • Total travel cost
        • Schedule inconvenience
        • Discomfort & inconvenience in travel
        • Other factors: appeals
      • Forms of estimation for impedance function F:
        • Exponential:
        • Power:
        • Tanner’s:

Calibration
  • Stages:
    • Specification of model structure: formulation
    • Statistical estimation of parameter: hypothesis à not fine-tuning, but correcting model functions
    • Evaluation of parameter estimation: goodness of fit
  • Principles:
    • Empirical validation
    • Statistical methods
  • Procedure:
    • Model specifications:
      • Select independent variables: socio-economic & transport supply
      • Functional form
      • Statistical technique: depends on functional form
    • Model calibration:
      • Evaluation of goodness of fit
      • estimation
  • Models:
    • Demand models
    • Gravity models
    • Multinomial logit models
  • Calibrating modeling process:
    • Data à calibrate model parameters à needs iteration & balancing (not model-balancing as above) à calibrate modeling process
    • Principle: modeled average trip length = observed average trip length
    • Algorithm: Hyman algorithm

•Algorithm

–Set b 0 = 1/c* where c* is the mean cost (or impedance) from the observed trip length distribution (OTLD).

–For iteration m, perform trip distribution with constraints using the current value of b m-1.

–Obtain the cost of the modelled trip length distribution (MLTD), i.e., cm.

–Adjust b m successively until cm @ c*.

•Next Estimate

–Apply the adjustments:

•General procedure

–The procedure to equate cm of MLTD with c* of OLTD can be applied to any model with different functional forms.

–Most planning software allows cm to be output for such calibration purposes.

–The adjustment of more than one calibration constants may be done by trial and error.

–The calibration can be done separately for different trip purpose

Summary T.D.
  • •Trip distribution is an important procedure in transportation planning process
  • •The functional form of trip distribution models is generally similar regardless of behavioural differences
  • •The performance of trip distribution may require consideration of constraints on flow, which will affect the way trips are balanced through iterations
  • •Calibration of the functions can be done statistically
  • •Calibration of the modelled flows must be checked by equating the average trip lengths.

Prof Chin's assignments

Domain

Explanation

New paratransit

Personal rapid transit

Public transport for the elderly & disabled

Free & subsidized public transport

Freight demand forecasting

Leisure travel demand forecasting

Car ownership demand forecasting

Value of time

Dr Lee's assignments

Domain

Explanation

Assignments

Lee Der Horng
  • Topics:
  • Discrete choice models:
  • Modal split:
  • Mathematical programming:
  • Route choice:
  • Combined & feedback T.P.

Recap
  • T.P. studies:
    • Infrastructure
    • Operate & manage T.P. system
    • Analyze, predict & evaluate movement
  • Transportation:
  • Derived demand: w.r.t. space & time
  • Supply service with capacity
  • Aim of healthy T.P.:
    • To design, plan, predict & analyze T.P. to minimize externalities:
    • Congestion, accidents, pollutions, ecology, financial development & improper development
  • Background:
    • 1950’s: post-war boom, urbanization, rising SOL
    • 1950-1980: trial-&-error tools
    • 1980-: challenges: research, theory, behaviour & forecast, planning
  • Model:
    • Series of equations, inequalities & relationships
    • Concerns:
      • Land-use policies
      • Socio-economic conditions
      • Control policies
      • Trip, activity & equilibrium
    • Level of aggregation
    • Planning horizons
    • Modeling principles:
      • Stats
      • Optimization
      • Simulation
    • Modeling criteria:
      • Sensitivity: to forecast & distinguish policies & alternatives
      • Causal: behavioural links between T.S. & decisions
      • Flexible: simplify wide variety of planning using combined 4-step; data collection effort: more (disaggregate) ,less (aggregate: census, zones)
      • Transferable: applicable to re-use
      • Efficient: high forecasting accuracies with different objective functions

7 steps of T.P.
  • Organization & goal definition: funding
  • Base year inventory: database for planning; travel pattern (O-D surveys); land-use; socio-economic; assumptions: shortest path & route choice behaviour independent of people
  • Model analysis: debugging to establish relationships between quantities from base year inventory; calibrate these for base year; sequential:
  • Trip generation: Output – trips @ origin production & trips @ destination attraction
  • Trip distribution: Output – OD matrix
  • Mode choice: aim – determine portion of total number of trips made between O & D using different transport modes; Output – mode usage rate (trips/mode/purpose)
  • Route choice (assignment): aim – allocate OD trips to routes in network to estimate the resulting volume, travel time & pattern; Output – traffic flow & pattern
  • Travel forecast
  • Network evaluation
  • Implementation
  • Feedback: re-investigate previous steps to correct inconsistencies, model modification, parameter re-calibration & problem re-definition

Land-use (LU)
  • Optimize utilization of land for uses
  • Land Dd & Ss
  • Location & activity interaction problem
  • "black box": LU plan is assumed
  • Development drives T.P. development
  • Needs integrated LU-TP process
  • Tools for LU:
    • Micro-economics
    • Spatial interaction
    • RUM theory
    • Discrete choice
    • Spatial accounting
  • Strive for balance with equilibrium: state not easily changed (from micro-simulation)

Framework of LU & TP
  • Regional level: employment & population
  • Urban activity level: residential & employment
  • Urban transport level: 4-step combined process

Aggregate approach
  • Data aggregation
  • Less computing, required data, calibration
  • Models: statistics, optimization
  • 4-step process
  • Drawbacks:
  • Lack of behaviour emphasis
  • Lack of time dimension
  • Trip independence

Disaggregate approach
  • Individual microscopic data
  • More behavioural data, calibration, computing, data collection & difficulties
  • Models: econometrics (of which random utility theories are a part)

Discrete choice models
  • Readings: Ortuzar chapter 7 & McFadden chapter 3
  • Aim: describe individual’s choices between competing alternatives
  • Choices:
    • T.G. & T.D. determination by system: more stable over long term
    • Behaviour => choices: difficult to capture for analytical approaches (Quantitative approach)
    • Structured choices
  • Assumptions:
    • Individual chooses rationally
    • Subject to constraints
    • Preferences & satisfaction => attractiveness & utility
    • IIA: independence from irrelevant alternatives => no correlation between two or more alternatives in their unmeasured attributes

Utility
  • Measure satisfaction upon available alternatives
  • F(attributes,characteristics)
  • Rational: chooses highest utility
  • Components: measurable (analytically random) & un-measurable (preference, priorities, weightings)
  • Choice models: random probability with which alternatives are chosen
  • Probability of choosing this mode

Utility function
  • U=(Ui), i=1…k alternatives
  • a: vector of variables for observed attributes & observed characteristics of decision-maker
  • Perceived utility = measured utility + unobserved utility error
  • Uk(a)=Vk(a) + e(a)
  • Once error e(a) specified/assumed, determine distribution of utilities => stochastic distribution to be used

Logit models
  • Widely used for discrete models, e.g. MNL, NML
  • Both aggregate & disaggregate
  • Derived from RUM by assuming all random terms are IID Gumbel variables
  • Choice probability: ,
  • Readings:
    • Gumbel distribution: general shape approximates that of normal distribution; analytically convenient for closed-form analysis (derivation)
    • Derive logit models
  • Types:
    • Binary logit: only 2 alternatives; properties
    • Multinomial logit: many alternatives
  • Logit properties:
    • Parameter
    • : deterministic alternative information, no error
    • : lack of information, equal probabilities, independent of alternative characteristics
    • Sets relative lowest & highest expected demands:
  • Uses:
    • IIA: independence of addition or removal of other alternatives
    • Choice probabilities: only a function of respective utilities & independent of other alternatives
    • Properties:
    • Independent of composition of alternatives
  • IIA fails when alternative is not in the same choice set
  • Paradox of logit model

Nested logit models
  • Aim: to overcome limitation (IIA) of MNL
  • Choice is hierarchical: travel, where, t, mode, route
  • Apply utility maximization to joint alternative choice
  • Marginal or unconditional probability of 1st level alternative with maximum utility and is chosen:

Reflection of disaggregate models
  • Behaviour-based: more stable
  • Estimation using individual data: can be aggregate with less data
  • Flexible in variables
  • Only probability, not show which alternative chosen

Probit models
  • All random terms are IID normal variables
  • A lot of problems:
    • Intractable
    • Error term: normal distribution
    • jpf: multivariate normal (MVN)
    • choice probability: highest utility, but not closed-form

Theory of Individual Travel Demand
  • By McFadden
  • Purpose of formal consumer choice model: explicit considerations to guide the selection of variables & possible restrictions on demand function aiding in parametric estimation
  • Objective: base restrictions on fewest assumptions & make relationships explicit
  • Derived demand model determined by consumption activities of consumer:
    • Economic consumer within framework of Court-Griliches-Becker-Lancaster consumption-activity-household-production model
    • Assumes: individual with basic wants
    • Assumes: have a utility function defined to satisfy these wants
    • Demand utility function:
    • U is the utility, x is the finite vector attributes, s is the vector of individual social and demographic characteristics influencing tastes, B is the set of available alternatives
  • Derive restrictions by factoring the demand function into component parts:
    • is the separable additive utility component part i, x(i) is the attributes of part i, is the transformation matrix
    • Component parts i=1~7:
    • Trip mode choice
    • Time-of-day choice
    • Destination
    • Trip-no-trip choice
    • Choice of vehicle ownership
    • Choice of residential and work location
    • All other consumer choices

Mode choice
  • Chapter 6 (Ortuzar, 2002)
  • Introduction:
    • Modes available for O->D
    • Calculation based on utility of competing modes
    • Common tool: logit models
  • Factors:
    • Characteristics of trip makers: car, HH, income, residential density
    • Characteristics of journey: purpose, season, time of day, day of week
    • Characteristics of transport facility: quality of comfort/safety, quantity of t/costs/packaging
  • Decision factors:
    • Time: access t, waiting t, in-vehicle t, transfer t, out-of-vehicle t
    • Travel cost
  • Limitations:
    • Cost & t: disutilities
    • Factors not included: crime, safety, security
    • Importance of neglected factors
    • Access times simplified

Route choice

(Traffic Assignment)
  • Aim: to find the flow (& travel t) on density, speed, flow on each link, given:
    • Network graph
    • Link performance function: rep. Travel impedance or level of service (LoS)
    • O-D matrix
    • Route choice principles
  • Concerns:
    • Impacts of scenarios
    • Network traffic flow pattern
    • How to obtain & how to assign trips (vehicles) onto study network
    • Mimic flow pattern from O-D
    • Tool: route choice, micro-analysis
  • Study area: Zoning
    • Details of zoning affect complexity & computations
    • Zoning aim: determine where journeys begin & end
    • Factors influencing trip generation
    • Establishing main corridors of movement
    • Region with trade-off between data collection & resources required
    • Aspects:
      • Centroid: the start & end of everything within a zone
      • Centroid connectors: links from centroid to network
      • Node: sink or source
      • Link: flow connections
      • External zone centroid: rep. Everything outside of study area
  • Network representation:
    • Nodes + Links
    • Network connected: sequence of links/paths/routes
    • Dummy links: impose high travel t & low cost
  • Data collection:
    • Infrastructure survey: O-D
    • LU & socio-economic
    • How to collect: traffic estimation count, license plate, cordon line, home
  • Link performance functions:
    • Link impedance, cost or LoS
    • Plot: travel t vs V (flow, q) vs. capacity (C)
    • Nonlinear programming: minimum
  • Practical concerns:
    • Special numbering for nodes
    • Link costs for dummy links
    • Short-cut prevention: by dummy links (high travel t)
    • Geographical implications to link & node numbering

User equilibrium user optima
  • No traveler can improve t.t. by unilaterally changing routes
  • J.G. Wardrop 1952
  • 1st principle for route choice: travel time (costs) on all used paths <= travel time (costs) of any unused paths
  • All used routes between equal O-D pair have equal costs
  • Any unused route higher or equal costs/t.t
  • User equilibrium: optimize the link travel time
  • Competitive, user cannot unilaterally change routes
  • Static, symmetric (individual link only), deterministic (Wardrop perfect info), link-based, path-constrained, indept., fixed demand
  • UE assumptions:
    • Rational behaviour: shortest path
    • Perfect information of road condition: deterministic
    • Travellers identical in terms of route choice behaviour
  • Beckmann et al 1956: link optimization, path constrained
    • s.t.
    • : flow conservation
    • : non-negativity

: path-link incidence matrix (definitional constraint)

  • At optimality, travel time on all used paths connecting given OD pair are equal
  • Observation:
    • Objective function in terms of link flows
    • Constraints in terms of path flows
    • "bridge" for link & path flows
  • UE condition:
  • : travel time of shortest path
  • Optimality conditions:
    • Existence (Necessary): Lagrangian L(f,u)
    • Uniqueness (Sufficient): Hessian matrix (+def) or if fails, uses Frank-Wolfe algorithm

System optimal system optima
  • 2nd principle for route choice: average travel time is minimal
  • J.G. Wardrop 1952
  • System optimal: optimize the total network travel time
  • Equilibrium generally not reached using system optimal, due to the need of joint decisions (sarcrifice self-interest for system interest – violates assumption of rational maximization of random utility)
  • Non-competitive, allows user to unilaterally change routes
  • Static, symmetric (individual link only), deterministic (Wardrop perfect info), link-based, path-constrained, indept., fixed demand
  • Use: serve as benchmark comparison of flow pattern & network designs
  • System Optimal: network link travel time optimization, path constrained
    • s.t.
    • : flow conservation
    • : non-negativity

: path-link incidence matrix (definitional constraint)

  • At optimality, marginal total travel time on all used paths connecting given OD pair are equal
  • Observation:
    • Same constraints as UE
    • Differs in objective function: system network minimisation
  • UE condition:
  • : marginal total travel time of on OD pair mn
  • Optimality conditions:
    • Existence (Necessary): Lagrangian L(f,u)
    • Uniqueness (Sufficient): Hessian matrix (+def) or if fails, uses Frank-Wolfe algorithm

UE & SO
  • If congestion effects are neglected, then travel time is not function of link flow:
    • UE:
    • SO:

& same constraints: same answer

  • For the same network, link performance functions & required OD, UE mean travel time > SO mean travel time
  • Due to not objective of UE to minimize overall total travel time, SO considers overall global network travel time
  • UE: for same OD pair, the travel times of all used paths are equal & less than/equal to travel times of unused paths
  • SO: for same OD pair, marginal travel times of all used paths are equal & less than/equal to marginal travel times of unused paths

Frank-Wolfe Algorithm
  • Original:
    • Step 0: guess an initial solution
    • Step 1: choose a search direction for by finding , which is the solution of the following LP: such that the constraints (original), then the new direction is
    • Step 1.5: if is small enough (-->0), stop
    • Step 2: perform a line search in the direction, => max => t0: optimal step size
    • Step 3: back to step (1) with
  • Applied to traffic assignment:
    • Step 0: perform all-or-nothing (AON) assignment, based on ta=ta(0) (free-flow travel t), this yields {xa’} (old solution), then set counter: n=1
    • Step 1: Update ta’=ta(xa’): new solution
    • Step 2: Direction finding: perform AON based on {xan}, this will give us a set of auxiliary (new) {yan}
    • Step 3: determine step size from line search: find that solves such that (constraints: original)
    • Step 4: move
    • Step 5:

 

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