Formula Sheet: Transportation Planning

Trip Generation

Trip Production and Attraction Models

Linear Regression Model: \[T = a + b_1X_1 + b_2X_2 + \ldots + b_nX_n\]

• T = number of trips produced or attracted
• a = regression constant
• b_i = regression coefficient for variable i
• X_i = independent variable i (e.g., household size, income, auto ownership)

Cross-Classification (Category Analysis) Method: \[T_i = \sum_{j} (HH_{ij} \times TR_j)\]

• T_i = total trips for zone i
• HH_ij = number of households in zone i and category j
• TR_j = average trip rate for category j

Trip Rate Analysis

Average Trip Rate: \[R = \frac{T}{U}\]

• R = trip rate (trips per unit)
• T = total number of trips
• U = number of units (households, employees, dwelling units, etc.)

Trip Distribution

Gravity Model

Basic Gravity Model: \[T_{ij} = P_i A_j \frac{F_{ij}}{\sum_{j} A_j F_{ij}}\]

• T_ij = trips from zone i to zone j
• P_i = total trip productions in zone i
• A_j = total trip attractions in zone j
• F_ij = friction factor or impedance function between zones i and j

Doubly Constrained Gravity Model: \[T_{ij} = A_i B_j P_i A_j F_{ij}\] Where: \[A_i = \frac{1}{\sum_{j} B_j A_j F_{ij}}\] \[B_j = \frac{1}{\sum_{i} A_i P_i F_{ij}}\]

• A_i = balancing factor for productions in zone i
• B_j = balancing factor for attractions in zone j

Friction Factors

Inverse Power Function: \[F_{ij} = t_{ij}^{-n}\]

• t_ij = travel time or distance between zones i and j
• n = calibration parameter (typically 1 to 3)

Exponential Function: \[F_{ij} = e^{-\beta t_{ij}}\]

• β = calibration parameter
• t_ij = travel time or distance between zones i and j

Gamma Function: \[F_{ij} = t_{ij}^a e^{-\beta t_{ij}}\]

• a = calibration parameter
• β = calibration parameter

Growth Factor Methods

Uniform Growth Factor: \[T_{ij}^f = T_{ij}^p \times F\]

• T_ij^f = future trips between zones i and j
• T_ij^p = present (base year) trips between zones i and j
• F = uniform growth factor

Average Growth Factor (Fratar Method): \[T_{ij}^f = T_{ij}^p \times \frac{G_i + G_j}{2}\]

• G_i = growth factor for zone i = (Future Productions_i / Present Productions_i)
• G_j = growth factor for zone j = (Future Attractions_j / Present Attractions_j)

Detroit Method (Iterative): \[T_{ij}^{(k+1)} = T_{ij}^{(k)} \times \frac{E_i^f}{E_i^{(k)}} \times \frac{E_j^f}{E_j^{(k)}}\]

• k = iteration number
• E_i^f = future trip ends for zone i
• E_i^(k) = calculated trip ends for zone i at iteration k

Mode Choice

Logit Models

Binomial Logit Model (Two Modes): \[P_i = \frac{e^{U_i}}{e^{U_i} + e^{U_j}}\]

• P_i = probability of choosing mode i
• U_i = utility of mode i
• U_j = utility of mode j (alternative mode)

Multinomial Logit Model: \[P_i = \frac{e^{U_i}}{\sum_{k=1}^{n} e^{U_k}}\]

• P_i = probability of choosing mode i
• U_i = utility of mode i
• n = total number of available modes

Utility Functions

General Utility Function: \[U_i = \alpha_0 + \alpha_1 t_i + \alpha_2 c_i + \alpha_3 X_i\]

• U_i = utility of mode i
• α₀ = mode-specific constant
• α₁ = coefficient for travel time
• t_i = travel time for mode i (minutes)
• α₂ = coefficient for cost
• c_i = travel cost for mode i (dollars)
• α₃ = coefficient for other variables
• X_i = other mode-specific variables

Modal Split

Diversion Curve Method:

• Uses empirical curves plotting percentage of travelers using a mode versus a service ratio
• Service ratio typically defined as ratio of travel times or costs

\[SR = \frac{t_{auto}}{t_{transit}}\]

• SR = service ratio
• t_auto = automobile travel time
• t_transit = transit travel time

Traffic Assignment

All-or-Nothing Assignment

• All trips between an origin-destination pair are assigned to the minimum cost (shortest) path
• No consideration of congestion or capacity constraints
• Volume on link a from zone i to zone j:

\[V_a = \sum_{i,j} T_{ij} \delta_{ij,a}\]

• V_a = volume on link a
• T_ij = trips from zone i to zone j
• δ_ij,a = 1 if link a is on the shortest path from i to j, 0 otherwise

User Equilibrium Assignment

Wardrop's First Principle:

• At equilibrium, travel times on all used routes between an O-D pair are equal and less than or equal to travel time on any unused route

Volume-Delay Function (BPR Function): \[t_a = t_0 \left[1 + \alpha \left(\frac{V_a}{C_a}\right)^\beta\right]\]

• t_a = travel time on link a (minutes)
• t₀ = free-flow travel time on link a (minutes)
• V_a = traffic volume on link a (vehicles per hour)
• C_a = capacity of link a (vehicles per hour)
• α = calibration parameter (typically 0.15)
• β = calibration parameter (typically 4.0)

Capacity Restraint (Incremental Assignment)

• Trips are assigned in increments (e.g., 25%, 25%, 25%, 25%)
• Link travel times are updated after each increment based on accumulated volume
• Routes are re-evaluated for each increment

System Equilibrium Assignment

Wardrop's Second Principle:

• At equilibrium, average journey time is minimized (system optimal)
• Used for optimal traffic control rather than prediction

Travel Demand Forecasting

Four-Step Model Process

1. Trip Generation: Estimate number of trips produced and attracted by each zone
2. Trip Distribution: Distribute trips between origin and destination zones
3. Mode Choice: Determine travel mode for each trip
4. Traffic Assignment: Assign trips to specific routes in the network

Peak Hour Factor

Peak Hour Factor (PHF): \[PHF = \frac{V_h}{4 \times V_{15}}\]

• PHF = peak hour factor (dimensionless, ≤ 1.0)
• V_h = total volume during peak hour (vehicles)
• V₁₅ = volume during peak 15-minute period within the peak hour (vehicles)

Peak Hour Volume: \[V_{peak} = AADT \times K \times D\]

• V_peak = directional design hour volume (vehicles per hour)
• AADT = annual average daily traffic (vehicles per day)
• K = proportion of AADT occurring in peak hour (typically 0.08 to 0.12)
• D = directional distribution factor (proportion in peak direction, typically 0.5 to 0.7)

Design Hour Volume

Directional Design Hour Volume (DDHV): \[DDHV = AADT \times K \times D\]

• DDHV = directional design hour volume (vehicles per hour)
• K = K-factor (decimal representing proportion of AADT in design hour)
• D = directional factor (decimal, typically 0.5 to 0.7)

Level of Service and Performance Measures

Vehicle Hours of Travel

Vehicle Hours of Travel (VHT): \[VHT = \sum_{a} V_a \times t_a\]

• VHT = total vehicle hours of travel (vehicle-hours)
• V_a = volume on link a (vehicles)
• t_a = travel time on link a (hours)

Vehicle Miles of Travel

Vehicle Miles of Travel (VMT): \[VMT = \sum_{a} V_a \times L_a\]

• VMT = total vehicle miles of travel (vehicle-miles)
• V_a = volume on link a (vehicles)
• L_a = length of link a (miles)

Average Travel Speed

Network Average Speed: \[S_{avg} = \frac{VMT}{VHT}\]

• S_avg = average travel speed (miles per hour)
• VMT = vehicle miles of travel
• VHT = vehicle hours of travel

Population and Employment Forecasting

Linear Growth Method

\[P_f = P_0 + r(t_f - t_0)\]

• P_f = future population
• P₀ = current (base year) population
• r = constant rate of growth (persons per year)
• t_f = future year
• t₀ = base year

Exponential Growth Method

\[P_f = P_0 e^{k(t_f - t_0)}\]

• P_f = future population
• P₀ = current population
• k = exponential growth rate constant
• t_f - t₀ = number of years from base to future year

Growth Rate Calculation: \[k = \frac{1}{t_2 - t_1} \ln\left(\frac{P_2}{P_1}\right)\]

• P₁ = population at time t₁
• P₂ = population at time t₂

Geometric Growth Method

\[P_f = P_0(1 + r)^{t_f - t_0}\]

• P_f = future population
• P₀ = current population
• r = constant growth rate (decimal)
• t_f - t₀ = number of years

Growth Rate Calculation: \[r = \left(\frac{P_2}{P_1}\right)^{\frac{1}{t_2 - t_1}} - 1\]

Declining Growth (Logistic) Method

\[P_f = \frac{P_s}{1 + e^{-a - bt_f}}\]

• P_f = future population
• P_s = saturation population (maximum sustainable population)
• a, b = constants determined from historical data
• t_f = future year

Accessibility and Connectivity Measures

Network Connectivity

Beta Index: \[\beta = \frac{L}{N}\]

• β = beta index (dimensionless)
• L = number of links (edges) in network
• N = number of nodes (vertices) in network
• For connected network: β ≥ 1

Gamma Index: \[\gamma = \frac{L}{L_{max}} = \frac{L}{3(N-2)}\]

• γ = gamma index (0 ≤ γ ≤ 1)
• L = number of links in network
• L_max = maximum possible links = 3(N - 2) for planar network
• N = number of nodes
• γ = 1 indicates maximum connectivity

Alpha Index: \[\alpha = \frac{C}{C_{max}} = \frac{L - N + 1}{2N - 5}\]

• α = alpha index (0 ≤ α ≤ 1)
• C = actual number of circuits (loops) = L - N + 1
• C_max = maximum possible circuits = 2N - 5 for planar network
• α = 0 indicates tree network (no circuits)
• α = 1 indicates maximum circuitry

Transit Planning Measures

Service Area and Coverage

Service Coverage Ratio: \[SCR = \frac{A_{served}}{A_{total}}\]

• SCR = service coverage ratio (decimal or percent)
• A_served = area within service buffer (e.g., 0.25 mile of transit stop)
• A_total = total study area

Transit Service Frequency

Headway: \[H = \frac{60}{f}\]

• H = headway (minutes between consecutive vehicles)
• f = frequency (vehicles per hour)

Fleet Size: \[N = \frac{C}{H}\]

• N = number of vehicles required
• C = round-trip cycle time (minutes)
• H = headway (minutes)

Transit Capacity

Maximum Persons per Hour per Direction: \[P = \frac{3600 \times c \times L}{H}\]

• P = persons per hour per direction
• c = capacity per vehicle (persons)
• L = load factor (typically 0.85 for standing capacity)
• H = headway (seconds)

Parking Analysis

Parking Accumulation

Parking Accumulation: \[A(t) = A_0 + \sum_{t_0}^{t} (E_i - X_i)\]

• A(t) = accumulation at time t (vehicles)
• A₀ = initial accumulation (vehicles)
• E_i = entries during interval i (vehicles)
• X_i = exits during interval i (vehicles)

Parking Turnover

Parking Turnover Rate: \[T = \frac{V}{S}\]

• T = parking turnover rate (vehicles per space per day)
• V = total number of vehicles parked during study period
• S = number of parking spaces

Parking Duration

Average Parking Duration: \[D = \frac{\sum_{i} d_i}{n}\]

• D = average parking duration (hours)
• d_i = duration of individual parking event i (hours)
• n = number of parking events

Parking Occupancy

Parking Occupancy Rate: \[O = \frac{A_{peak}}{S} \times 100\%\]

• O = occupancy rate (percent)
• A_peak = peak accumulation (vehicles)
• S = total number of spaces

Traffic Impact Analysis

Trip Generation Rates

ITE Trip Generation Equation: \[T = aX + b\] or \[T = aX^b\]

• T = number of trips
• X = independent variable (e.g., floor area, dwelling units, employees)
• a, b = regression coefficients from ITE Trip Generation Manual

Pass-By and Diverted Trips

Net New External Trips: \[T_{net} = T_{total} - T_{internal} - T_{passby} - T_{diverted}\]

• T_net = net new external trips
• T_total = total site-generated trips
• T_internal = internal capture trips
• T_passby = pass-by trips
• T_diverted = diverted trips

Air Quality and Emissions

Mobile Source Emissions

Total Emissions: \[E = VMT \times EF\]

• E = total emissions (grams or pounds)
• VMT = vehicle miles of travel
• EF = emission factor (grams per mile or pounds per mile)

Regional Emissions: \[E_{total} = \sum_{i} (VMT_i \times EF_i)\]

• E_total = total regional emissions
• VMT_i = vehicle miles of travel for vehicle class or road type i
• EF_i = emission factor for class/type i

Travel Time and Delay

Average Travel Time

Link Travel Time: \[t = \frac{L}{S}\]

• t = travel time (hours)
• L = link length (miles)
• S = average speed (miles per hour)

Path Travel Time: \[T_{path} = \sum_{a \in path} t_a\]

• T_path = total path travel time
• t_a = travel time on link a in the path

Delay Analysis

Total Delay: \[D = (t_{actual} - t_{freeflow}) \times V\]

• D = total delay (vehicle-hours)
• t_actual = actual travel time (hours)
• t_freeflow = free-flow travel time (hours)
• V = traffic volume (vehicles)

Benefit-Cost Analysis for Transportation Projects

Travel Time Savings

Value of Travel Time Savings: \[B_{time} = \Delta t \times V \times VOT\]

• B_time = benefit from travel time savings (dollars)
• Δt = change in travel time (hours per trip)
• V = number of trips (trips per year)
• VOT = value of time (dollars per hour)

Vehicle Operating Cost Savings

Operating Cost Savings: \[B_{VOC} = \Delta VMT \times C_{operating}\]

• B_VOC = benefit from vehicle operating cost savings (dollars)
• ΔVMT = change in vehicle miles of travel (vehicle-miles per year)
• C_operating = vehicle operating cost (dollars per vehicle-mile)

Accident Cost Savings

Accident Cost Reduction: \[B_{accident} = \Delta A \times C_A\]

• B_accident = benefit from accident reduction (dollars per year)
• ΔA = reduction in number of accidents (accidents per year)
• C_A = average cost per accident (dollars)

Benefit-Cost Ratio

Benefit-Cost Ratio: \[BCR = \frac{PV_{benefits}}{PV_{costs}}\]

• BCR = benefit-cost ratio (dimensionless)
• PV_benefits = present value of all benefits (dollars)
• PV_costs = present value of all costs (dollars)
• Project is economically justified if BCR > 1.0

Net Present Value: \[NPV = PV_{benefits} - PV_{costs}\]

• NPV = net present value (dollars)
• Project is economically justified if NPV > 0

Present Value Calculations

Present Value of Future Amount: \[PV = \frac{FV}{(1 + i)^n}\]

• PV = present value (dollars)
• FV = future value (dollars)
• i = discount rate (decimal)
• n = number of years

Present Value of Uniform Annual Series: \[PV = A \times \frac{(1 + i)^n - 1}{i(1 + i)^n}\]

• PV = present value (dollars)
• A = uniform annual amount (dollars per year)
• i = discount rate (decimal)
• n = number of years