Gravity Model

Gravity Model

• The gravity model is used in trip distribution and is the most widely used model.
• The number of trips between two zones is directly proportional to the number of trip attractions generated by the destination zone.
• The number of trips between two zones is inversely proportional to a function of travel time between the two zones.
• Tij = Pi * (Aj * Fij * Kij) / (Σj Aj * Fij * Kij), where:
  • Tij equals the number of trips produced in zone i and attracted to zone j.
  • Pi equals the total number of trips produced in zone i.
  • Aj equals the number of trips attracted to zone j.
  • Fij equals a value that is an inverse function of travel time.
  • Kij equals a socioeconomic adjustment factor for interchange ij.

Singly Constrained

• Singly constrained models are used when information is available about the expected growth trips originating in each zone only or trips attracted to each zone only.

Doubly Constrained

• Doubly constrained models are used when information is available on the future number of trips originating and terminating in each zone.
• The adjusted attraction factors are computed with the formula:
  • Ajk = Aj(k-1) / Cj(k-1) * Aj, where:
    • Ajk = adjusted attraction factor for attraction zone (column) j, iteration k (Ajk = Aj when k = 1)
    • Cjk = actual attraction (column) total for zone j, iteration k
    • A = desired attraction total for attraction zone (column) j
    • j = attraction zone number, j = 1, 2, ..., n
    • k = iteration number, m = number of iterations

Calibrating a Gravity Model

• Calibration involves developing friction factors and socioeconomic adjustment factors.
• Friction factors reflect the effect travel time of impedance has on trip making.
• A trial-and-error adjustment process is generally adopted for calibration.
• Another way to calibrate is to use the factors from a past study in a similar urban area.

Input Items

• Three items are used as input to the gravity model for calibration:
  • Production-attraction trip table for each purpose.
  • Travel times for all zone pairs, including intrazonal times.
  • Initial friction factors for each increment of travel time.

Calibration Process

• The calibration process involves adjusting the friction factor parameter until the planner is satisfied that the model adequately reproduces the trip distribution as represented by the input trip table.
• Data such as the trip-time frequency distribution and the average trip time are used.
• Total trip attractions at all zones j, as calculated by the model.
• If this comparison shows significant differences, the attraction Aj is adjusted for each zone, where a difference is observed.
• The model is rerun until the calculated and observed attractions are reasonably balanced.
• The model's trip table and the input travel time table are used for two comparisons which are trip-time frequency distribution and the average trip time.
• If there are significant differences, the process begins again.

Standard Gravity Model

• Tij = aOiDjf(cij), where f(cij) is a generalized function of the travel costs with one or more parameters for calibration.
• A useful function is a 'deterrence function' that represents the disincentive to travel as distance (time) or cost increases:
  • Exponential function: f(cij) = exp(-βcij)
  • Power function: f(cij) = cij^-n
  • Combined function: f(cij) = cij^-n exp(-βcij)

Singly and Doubly Constrained Model

• Restrictions need factor by replacing the single proportionality factor α by two sets of balancing factors A; and Bᵢ.
  • Tij = AᵢOᵢBᵢDjf(cij)
• Single constrained versions (either origin or destination constrained), can be produced by making one set of balancing factors A; and B; equal to one.

Balancing Factors

• The need to ensure restrictions are met replaces the single proportionality factor α by two sets of the balancing factors.
  • Tij = AiOiBjDjf(cij)
• The O and D terms could go into factors which rewrite the model to:
  • Tij = aibjf(cij)
• Can produce singly constrained versions, either origin or destination constrained, by making one set of balancing factors A and B equal to one.

Singly and Doubly Constrained Model

• Singly Constrained Model
  • For an origin-constrained model: B = 1.0
    • Ai = 1/(Σj Djf(cij)
  • For a destination-constrained model: A = 1.0
    • Bj = 1/(Σi Oif(cij))
• Doubly Constrained Model
  • Ai = 1/(Σj BjDjf (cij))
  • Bj = 1/(Σi AiOif(cij))
• Balancing factors are interdependent and the calculation of one set requires the values of the other.
• An iterative process is suggested to given set of values for the deterrance function f(c₁₁), start with all B₁ = 1, solve for A; and then use these values to reestimate the B;'s; repeat until convergence is achieved.

Modal Split Analysis

• Modal split analyzes people's decisions regarding the mode of travel.
• Mode usage analysis can be done at various points in the forecasting process.
• A typical approach is to do it after trip distribution because it allows relationships to compare the alternative transportation services.

Mode Choice Factors

• Trip makers characteristics include family income, autos available, family size and residential density.
• Trip characteristics include travel distance, time of day, purpose.
• Transport system characteristics are riding time, waiting time, cost.

Logit Model

• A share model divides persons between modes based on each mode's relative desirability for any given trip.
• Modes are more desirable if they are faster, cheaper, or otherwise favorable than competitive modes.
• The better the mode is, the more utility it has for potential travelers.
• Um = . Where:
  • Um = utility of mode m
  • n = number of attributes
  • X = attribute value (time, cost, etc.)
  • β = coefficient value for attribute i
• The equations
  • P₁ = e^U(i) / Σn r=1 e^U(r)
    • WHERE: U(i) = utility of mode i, U(r) = utility of mode r, n = number of modes.

Estimating the Utility Functions of Choice

• A logit model of choice has been developed between the modes automobile and bus, one variable of the model is total travel time, T.
• The deterministic component of the model's utility function is:
  • V = aT where a is a constant coefficient.

Hierarchical/Nested Logit

• As a modeling tool the HL is useful as being presented fashion.
• Structure is characterized by grouping all subsets of correlated options within groups.
• Each nest is represented by a composite alternative which competes with the others available to the individual.
• The introduction of lower nests uses the utilities of composite alternatives and has two components:
  • Consists of the expected maximum utility (EMU) of the lower nest options.
  • Considers the vector z of attributes common to all members of the nest.

EMU Portion

• EMU = log(Σj exp(Wj))
  • Where Wj is the utility of alternative Aj in the nest, therefore the composite utility of the nest is:
    • V1 = φEMU + az
    • Where & and a are parameters to be estimated.

Trip Assignment

• Trip assignment predicts the paths the trips will take.

Minimum-Path Techniques

• Based on the assumption that travelers want to use the minimum impedance route between two points.
• Trips are loaded onto the links making up the minimum path.
• Also called all-or-nothing because all trips between a given origin and destination are loaded on the corresponding links which comprise the minimum path, and nothing on other links.
• Can cause some links to be assigned more travel volume than the link has capacity at the original assumed speed.

Minimum-Path with Capacity Restraint

• Based on the findings that with increasing traffic flow, speed decreases.
• A positive relationship between impedance and flow for all types of highways.
• Attempts to balance the assigned volume, the capacity of a facility, and the related speed.

Bureau of Public Roads (BPR) Method

• The equation is:
  • TQ = T0[1 + α(Q/Qmax)^β]
    • Where:
      • TQ = travel time at traffic flow Q
      • T0 = “zero-flow” travel time
      • Q = traffic flow (veh/hr)
      • Qmax = practical capacity, equal to ¾ saturation flow
      • α, β = parameters

Davidson's Method

• The equation is: TQ = T0[(1 - (1-τ)(Q/Qmax)) / (1 - (Q/Qmax))]
  • Where:
    • TQ = travel time at traffic flow Q
    • T0 = “zero-flow” travel time.
    • Q = traffic flow (veh/hr)
    • Qmax = saturation flow
    • τ = level of service (LOS) parameter, suggested values of 0 to 0.2 for freeways, 0.4 to 0.6 for urban arterials, and 1 to 1.5 for collector roads.

Drew's Technique

• Employs an iterative procedure between travel time ratios and volume to capacity ratios.
• Volume of trips is:
  • Vmrij volume of trips from i to j using mode m over road r
• Travel time ratios
  • t^mrij / t^mr0ij = f(V^mrij / Q^mrij)
• From the capacity restraint of highway facility
  • t^mrij / t^mr0ij = 1 - ((V^mrij/Q^mrij )*(1-kr))/[1 - ((V^mrij / Q^mr ij))]
  • k = a level of service factor which varies for different types of highway facility, with
    • [0, ½] for freeway or expressway
    • [½, 1] for arterial routes
    • 0 for a "perfect" highway facility

Computational (iterative) procedure:

• Find Vc1ij, Vc2ij, ... Vcr ij

1. V^c1ij + V^c2ij + ..... V^cr i = Vij
2. t^c1 j = t^c2 ij = ...... t^cr = t^cr ij

Links in Series

• To find the cost function for travel using links in series shown, is for Cij is the cost of travel on link ij and fij is the flow along link ij:
  • fAB = fBC = fAC
    • CAC = CAB + CBC
    • CAC = (3 + fAB) + (2 + 2fBC) = CAC = 5 + 3f AC

Links in Parallel

• To find the cost of travel from A to B using links are connected in parallel following:
• fAB = f1+f2
• C₁ = C2 = C

Incremental Assignment

• Divide the total trip matrix T into a number of fractional matrices by applying a set of proportional factors p such that ∑n ρn = 1.
• The fractional matrices are loaded, incrementally, onto successive trees, each calculated using link costs from the last accumulated flows.
• Typical values for pn are: 0.4, 0.3, 0.2, and 0.1.

Increment Assignment

• It is easy to program and its results are interpreted to build up congestion for the peak period.
• Limitation - once a flow has been assigned to a link it is not removed and loaded onto another one.
  • So initial interation may assign too much flow on a link for Wardrop's equilibrium conditions and the algorithm won't converge to the correct solutions otherwise.
  • The algorithm doesn't necessarily converge to Wardrop's equilibrium solution.

Method of Successive Averages

• Iterative algorithms developed to overcome the problem of allocating too much traffic to low-capacity links.
• In an iterative assignment algorithm, calculating a current flow on a link combines the current flow on the previous iteration and an auxiliary flow coming from an all-or-nothing assignment.
• These Algorithms are close to the correct equilibrium solutions, and the greater the iterations in a more realistically network, makes for more convergence.
• Fixing the maximum number of iterations is not a good approach due to the evaluation that Link and total costs can vary considerably in successive iterations.