CE 254 Transportation Engineering The Four-Step Model: II. Trip Distribution Wes Marshall, P.E. University of Connecticut February 2008

The Basic Transportation Model… Study Area Zones Attributes of Zones Socioeconomic Data Land Use Data “Cost” of Travel btw. Zones The Road Network Inputs Traffic Volume by Road Link Mode Splits Emissions Outputs

What’s in the Black Box? The Four-Step Model

The Four-Step Modeling Process… Trip Generation Trip Distribution Mode Choice Trip Assignment WHY?

The Four-Step Model The main reason we use the four-step model is: To predict roadway traffic volumes & traffic problems such as congestion and pollution emissions In turn, we typically use the models to compare several transportation alternatives

Nevertheless, most agencies still use the good ol’ four-step model The Four-Step Model Originally developed in the 1950s with the interstate highway movement Since the 1950s, researchers have developed a multitude of advanced modeling techniques Nevertheless, most agencies still use the good ol’ four-step model

Overview of the Four-Step Model

} } Gravity Model TRIP GENERATION Input: - No. of Housing Units - Office, Industrial SF - HH Size - Income - No. of Cars Household Socioeconomic Data Land Use Data Examples of HH socioeconomic data Survey Data } Examples of land use data Iterative Process Process: Model residential trip productions and non-residential trip attractions w/ - Regression Models - Trip-Rate Analysis - Cross-Classification Models Output: - i.e. columns of trip productions and trip attractions Trip Ends by purpose TRIP DISTRIBUTION Input: Trip Ends by purpose Process: Growth Factor Models Not as accurate as Gravity Model Used for external trips or short-term planning Gravity Model Used for regional or long-term planning Output: - A matrix of trips between each TAZ… also called a “trip table” Trip Interchanges MODE CHOICE Input: Trip Interchanges Output: Trip Table by Mode TRIP ASSIGNMENT Input: Trip Table by Mode Output: Daily Link Traffic Volumes - i.e. traffic flows on network, ridership on transit lines

Trip Interchance Model MODE CHOICE Iterative Process Input: Trip Interchanges Process: Finds trip interchanges between i & j for each mode - Function of Trip Maker, Journey, and Transport Facility Trip End Model Mode plays role in trip ends Typically used for small cities with little traffic and little transit No accounting for the role that policy decisions play in mode choice Trip Interchance Model Use when LOS is important, transit is a true choice, highways are congested, and parking is limited Output: Trip Table by Mode TRIP ASSIGNMENT Input: Trip Table by Mode Process: Allocate trips to links between nodes i & j - Function of Path to Destination and Minimum Cost (time & money) Identify Attractive Routes via Tree Building Shortest Path Algorithm or Dijkstra’s Algorithm Assign Portions of Matrix to Routes / Tree User Equilibrium, Heuristic Methods, Stochastic Effects w/ Logit Search for Convergence Output: - i.e. traffic flows on network, ridership on transit lines Daily Link Traffic Volumes

Some General Problems with the Conventional Methodology Huge focus on vehicular traffic A transit component is typical in better models Typically forecasts huge increases in traffic Leads to engineers building bigger roads to accommodate “forecast” traffic Which leads to induced traffic and congestion… right back where we started when we needed the bigger roads in the first place

Some General Problems with the Conventional Methodology Pedestrians and bicyclists are rarely included Level of geography is difficult for non-motorized modes Network scale is insignificant Input variables are too limited

Preparing for a Four-Step Model Before jumping into trip generation, we first have to set up our project… Define study area and boundaries Establish the transportation network Create the zones

Defining the Study Area 3 Basic Types Regional Statewide or a large metro area Used to predict larger patterns of traffic distribution, growth, and emissions Corridor Major facility such as a freeway, arterial, or transit line Used to evaluate traffic Site or Project Proposed development or small scale change (i.e. intersection improvement) Used to evaluate traffic impact

Establish the Network Roads are represented by a series of links & nodes Links are defined by speed and capacity Turns are allowed at nodes Node Link

Establish the Network Typically only main roads and intersections are included Even collector roads are often excluded This practice is becoming less common as the processing power of computers has increased

Creating Zones Create Traffic Analysis Zones (TAZ) Uniform land use Bounded by major roads Typically small in size (about the size of a few neighborhood blocks) The State of Connecticut model has ~2,000 zones that cover 5,500 square miles and over 3.4 million people

Creating Zones All modeled trips begin in a zone and are destined for a zone Zones are usually large enough that most pedestrian and bicycle trips start and end in the same zone (and thus not modeled) Also, the typical data we collect about zones in terms of population and employment information is not enough to predict levels of walking and biking

Trip Generation

Trip Generation Using socioeconomic data, we try to estimate how many trips are “produced” by each TAZ For example, we might use linear regression to estimate that a 2-person, 2-car household with a total income of $90,000 makes 2 home-based work trips per day Using land use data, we estimate how many trips are “attracted” to each TAZ For example, an 3,000 SF office might bring in 12 work trips per day

Trip Generation The process considers the total number of trips Thus, walking and biking trips have not been officially excluded (although most models ignore them completely) The trips are generated by trip purpose such as work or shopping Recreational or discretionary trips are difficult to include

Trip Generation Input: Socioeconomic Data Land Use Data Output: 1 2 3 4 5 6 8 7 Output: Trip Ends by purpose (i.e. work) in columns of productions & attractions

Trip Generation Trip Distribution The question is… how do we allocate all the productions among all the attractions? Trip Matrix or Trip Table Zone 1

Trip Distribution

Trip Distribution We link production or origin zones to attraction destination zones A trip matrix is produced The cells within the trip matrix are the “trip interchanges” between zones

Trip Interchanges Decrease with distance between zones In addition to the distance between zones, total trip “cost” can include things such as tolls and parking costs Increase with zone “attractiveness” Typically includes square footage of retail or office space but can get much more complicated

Trip Distribution Similar to Trip Generation, all the modes are still lumped together by purpose (i.e. work, shopping) This creates a problem for non-vehicular trips because distance affects these trips very differently Additionally, many walking and biking trips are intra-zonal & difficult to model

Basic Criteria for TD Criteria for allocating all the productions among all the attractions Cost of trip Travel Time Actual Costs Attractiveness Quantity of Opportunity Desirability of Opportunity

How to Distribute the Trips? Growth Factor Models Gravity Model

Growth Factor Models Growth Factor Models assume that we already have a basic trip matrix Usually obtained from a previous study or recent survey data

Growth Factor Models The goal is then to estimate the matrix at some point in the future For example, what would the trip matrix look like in 10 years time? Trip Matrix, t (2008) Trip Matrix, T (2018)

Some of the More Popular Growth Factor Models Uniform Growth Factor Singly-Constrained Growth Factor Average Factor Detroit Factor Fratar Method

Uniform Growth Factor Model

Uniform Growth Factor i = I = Production Zone j = J = Attraction Zone Tij = τ tij for each pair i and j Tij = Future Trip Matrix tij = Base-year Trip Matrix τ = General Growth Rate

Uniform Growth Factor If we assume τ = 1.2, then… Tij = τ tij Tij = τ tij for each pair i and j Tij = Future Trip Matrix tij = Base-year Trip Matrix τ = General Growth Rate Uniform Growth Factor If we assume τ = 1.2, then… Trip Matrix, t (2008) Tij = τ tij = (1.2)(5) = 6 Trip Matrix, T (2018)

Uniform Growth Factor The Uniform Growth Factor is typically used for 1 or 2 year horizons However, assuming that trips grow at a standard uniform rate is a fundamentally flawed concept

Singly-Constrained Growth Factor Model

Singly-Constrained Growth Factor Method Similar to the Uniform Growth Factor Method but constrained in one direction For example, let’s start with our base matrix, t… attractions, j zones productions, i

Singly-Constrained Growth Factor Method Instead of one uniform growth factor, assume that we have estimated how many more or less trips will start from our origins… Now all we have to do is multiply each row by the ratio of (Target Pi) / (Σj)

Singly-Constrained Growth Factor Method Tij = tij (Target Pi) / (Σj) = 5 (400 / 355) = 5.6

Singly-Constrained Growth Factor Method Can also perform the singly-constrained growth factor method for a destination specific future trip table By multiplying each column by the ratio of (Target Aj) / (Σi)

Overview of the Singly-Constrained Growth Factor Methodology One of the simplest trip distribution techniques Used with existing trip table & future trip ends Typically, we balance flows after processing This means that the total number of productions equals the total number of attractions (or in terms of origins & destinations) Tij = Tji But there are more advanced growth factor models…

Average Growth Factor Model

Average Growth Factor Function Fik + Fjk g ( )= 2 Fik Fjk g ( )= F.k F = Growth Factor = Ratio of Target Trips to Previous Iteration Trips k = Iteration Number F.k= ΣTi Σtik-1 = Σ(Future Trips) Σ(Current Iteration Trips)

Previous Iteration Trip End The Basic Steps… Collect Inputs Matrix of Existing Trips, {tij} Vector of Future Trips Ends, {Ti} Compute Growth Factor for each zone Compute Inter-zonal Flows Compute Trips Ends If tik = Ti for each zone i, then stop… otherwise, go back to “Step 1” > ΣTi Target Trip End Fik= = Σtik-1 Previous Iteration Trip End > tijk = tijk-1 [g(Fik, Fjk, …)] for each ij pair tik = Σtijk for each zone i

Growth Factor Models: Average Factor Example 1.330= 1.41+1.25 2

Fratar Method Growth Factor Model

Fratar Method tjik-1 Fjk tijk=Ti Replace Step 2 with… Σtizk-1Fzk Tijk=Tjik= tijk + tjik 2 Balance Matrix with…

Growth Factor Models: Fratar Method Example Ti Future Trips 115 Fi(1) = = = = 2.09 ti(0) Current Trips 55 Σtizk-1Fzk = (t110)(F11)+(t120)(F21)+(t130)(F31)+(t140)(F41) =(0)(2.09) + (25)(1.30) + (10)(1.59) +(20)(1.46) = 78 Σtizk-1Fzk = (t210)(F11)+(t220)(F21)+(t230)(F31)+(t240)(F41) =(25)(2.09) + (0)(1.30) + (60)(1.59) +(30)(1.46) = 191

Growth Factor Models: Fratar Method Example t431 = (t340)(T4)(F31)/Σ[(t4z0)(Fz1)] = (15)(95)(1.59) / (105) = 22 t121 = (t210)(T1)(F21)/Σ[(t1z0)(Fz1)] = (25)(115)(1.30) / (78) = 48

Now onto Iteration 2… Balance Flows = (tij1 + tji1) / 2 = (22 + 24) / 2 = 23 Now onto Iteration 2…

Iteration 2… Balance Flows… & Compute Values

Iteration 3… Balance Flows… & Compute Values

Limitations of the Fratar Model Breaks down mathematically with a new zone Convergence to the target year not always possible The model does not reflect travel times or cost of travel between zones Thus, this model as well as the other growth factor models are only used for External trips through the zones or Short-term horizon years

The Gravity Model

The Inspiration for the Gravity Model The big idea behind the gravity model is Newton’s law of gravitation… The force of attraction between 2 bodies is directly proportional to the product of masses between the two bodies and inversely proportional to the square of the distance M1 M2 F = k r2

The Inspiration for the Gravity Model In terms of transportation planning and trip distribution: The zones correspond to the objects The attributes of the zones in terms of the relative proportion of productions and attractions represent the mass of the objects The distance between the zones is captured by the distance between the objects

Some of the Variables Tij = Trips between i & j i = production zone Tij = Qij = Trips Volume between i & j Fij =1/Wcij = Friction Factor Wij = Generalized Cost (including travel time, cost) c = Calibration Constant pij = Probability that trip i will be attracted to zone j kij = Socioeconomic Adjustment Factor Tij = Trips between i & j i = production zone j = attraction zone Tij = f (Pi, Aj, Cij) Cij = Generalized cost of trip from i to j

The Gravity Model Pi Aj FijKij Tij = Qij = = Pipij ΣAzFizKij = (Productions)(Attractions)(Friction Factor) = Sum of the (Attractions x Friction Factors) of the Zones Fij = 1 / Wcij & ln F = - c ln W The bigger the friction factor, the more trips that are encouraged

2 Types of Gravity Models Parametric Fits equation to curve Non-parametric Uses look-up table for bars % Trips Time (min)

To Apply the Gravity Model What we need… Productions, {Pi} Attractions, {Aj} Skim Tables {Wij) Target-Year Interzonal Impedances

Gravity Model Example 8.2 Given: Find: Target-year Productions, {Pi} Relative Attractiveness of Zones, {Aj} Skim Table, {Wij} Calibration Factor, c = 2.0 Socioeconomic Adjustment Factor, K = 1.0 Find: Trip Interchanges, {Qij}

Attractions vs. Attractiveness The number of attractions to a particular zone depends upon the zone’s attractiveness As compared to the attractiveness of all the other competing zones and The distance between them Two zones with identical attractiveness may have a different number of attractions due to one’s remote location Thus, substituting attractions for attractiveness can lead to incorrect results

1 Fij = Wcij Find Trip Interchanges, {Qij} Find Denominator of Gravity Model Equation {AjFijKij} Calculate Friction Factors, {Fij} Find Probability that Trip i will be attracted to Zone j, {pij} Given… Given… Target-Year Inter-zonal Impedances, {Wij} Calibration Factor c = 2.0 Socioeconomic Adj. Factor K = 1.0 Calculate Friction Factors, {Fij} 1 1 F11= 1 52 = 0.04 Fij = = Fij = Wcij Wcij Find Denominator of Gravity Model Equation {AjFijKij} AjFijKij=A4F34K34 = (5)(0.01)(1.0) = 0.05 Find Probability that Trip i will be attracted to Zone j, {pij} AjFijKij 0.05 pij = = = 0.3125 Σ(AzFizKij) 0.16 Find Trip Interchanges, {Qij} Qij = Pipij = (2600)(0.3125) = 813

Keep in mind that the socioeconomic factor, K, can be a matrix of value rather than just one value

Calibration of the Gravity Model When we talk about calibrating the gravity model, we are referring to determining the numerical value c The reason we do this is to fix the relationship between the travel-time factor and the inter-zonal impedance

Calibration of the Gravity Model Calibration is an iterative process We first assume a value of c and then use: [ ] [ ] Aj Fij Aj / Wcij Qij = Pi Qij = Pi Σ(Ax Fix) Σ(Ax/ Wcix) Qij = Tij = Trips Volume between i & j Fij =1 / Wcij = Friction Factor Wij = Generalized Cost (including travel time, cost) c = Calibration Constant

Calibration of the Gravity Model The results are then compared with the observed values during the base year If the values are sufficiently close, keep c The results are expressed in terms of the appropriate equation relating F and W with c If not, then adjust c and redo the procedure

Trip-Length Frequency Distribution Compares the observed and computed Qij values

Gravity Model Calibration Example Given: Zone 2 Zone 3 Zone 4 Zone 5 Zone 1 10 5 10 5 Target-Year Inter-zonal Impedances, {Wij} 15 15 Base-Year Trip Interchange Volumes, {tij} 5 TAZ City Find: c and Kij to fit the base data

Relating F & W 1 Fij = Wcij ln F = - c ln W Starting with: Take the natural log of both sides Now c is the slope of a straight line relating ln F and ln W 1 Fij = Wcij ln F = - c ln W ln F ln W

Trip-Length Frequency Distribution Target-Year Inter-zonal Impedances, {Wij} Base-Year Trip Interchange Volumes, {tij} Group zone pairs by max Wij Sum Interchange Volumes for each set of zone pairs Find f = (Σtix) / (Σtij)

First Iteration First Iteration 1 = F13= = 0.04 52 Let’s assume c = 2.0 Target-Year Inter-zonal Impedances, {Wij} Base-Year Trip Interchange Volumes, {tij} Friction Factor, {Fij} with c = 2.0 1 = F13= = 0.04 52

Friction Factor, {Fij} with c = 2.0 First Iteration First Iteration c = 2.0 Friction Factor, {Fij} with c = 2.0 Aj Fij

First Iteration First Iteration c = 2.0

First Iteration Second Iteration 1 = F13= = 0.089 51.5 Let’s assume c = 1.5 Target-Year Inter-zonal Impedances, {Wij} Base-Year Trip Interchange Volumes, {tij} Friction Factor, {Fij} with c = 1.5 1 = F13= = 0.089 51.5

Friction Factor, {Fij} with c = 1.5 First Iteration Second Iteration c = 1.5 Friction Factor, {Fij} with c = 1.5 Aj Fij

First Iteration Second Iteration c = 1.5

K Factors 1-Xi Kij = Rij 1 - XiRij Even after calibration, there will typically still be discrepancies between the observed & calculated data To “fine-tune” the model, some employ socioeconomic adjustment factors, also known as K-Factors The intent is to capture special local conditions between some zonal pairs such as the need to cross a river 1-Xi Kij = Rij 1 - XiRij Rij = ratio of observed to calculated Qij (or Tij) Xi = ratio of the base-year Qij to Pi (total productions of zone i)

K Factor Example Rij = = R13 = = 1.20 Xi = = X1 = = 0.60 1-Xi 1-0.6 Observed Qij 300 Rij = = R13 = = 1.20 Calculated Qij 249 Base-Year Qij 300 Xi = = X1 = = 0.60 Pi 500 1-Xi 1-0.6 Kij = Rij = K13 = 1.20 = 1.71 1 - XiRij 1–(0.6)(1.2)

The Problem with K-Factors Although K-Factors may improve the model in the base year, they assume that these special conditions will carry over to future years and scenarios This limits model sensitivity and undermines the model’s ability to predict future travel behavior The need for K-factors often is a symptom of other model problems. Additionally, the use of K-factors makes it more difficult to figure out the real problems

Limitations of the Gravity Model Too much of a reliance on K-Factors in calibration External trips and intrazonal trips cause difficulties The skim table impedance factors are often too simplistic to be realistic Typically based solely upon vehicle travel times At most, this might include tolls and parking costs Almost always fails to take into account how things such as good transit and walkable neighborhoods affect trip distribution No obvious connection to behavioral decision-making

Limitations of the Gravity Model The model fails to reflect the characteristics of the individuals or households who decide which destinations to choose in order to satisfy their activity needs White Collar Jobs Zone 3 Income=20000 Zone 2 Zone 1 Zone 4 Income=75000 Blue Collar Jobs The gravity model does not take this type of situation into account without using K-Factors… which leads back into another whole set of problems