AP.2. Mode Choice and Route Assignment.pdf

AP.2. Mode Choice and Route Assignment.

1. AP.2. MODE CHOICE AND
ROUTE ASSIGNMENT
Prepared by: Saurav Barua
Ph.D. Student, Dept. of Civil Engineering,
Chulalongkorn University, Bangkok, Thailand

2. Contents
❑ModeChoice:Explainthelogitmodel(e.g.,
multinomialornestedlogit)anditsapplicationin
predictingtheprobabilityofchoosingaspecific
travelmode.
❑TrafficAssignment:Discusstheconceptofuser
equilibriumandhowitisappliedintraffic
assignmentmodels.
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3. Mode Choice
✓Mode choice analysis, the third step in the four-step
transportation model, is where we predict which mode
of transportation travelers will use.
✓The logit model is the most prominent tool for this task,
grounded in economic theories of choice and utility.
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4. The Core Concept: Random Utility Theory
✓Before diving into the models, it's essential to
understand the principle they are built on: Random
Utility Theory (RUT) . This theory states that when an
individual chooses an option from a set of alternatives
(in this case, travel modes), they will select the one that
provides them with the highest utility .
✓Utility is a measure of the attractiveness or satisfaction a
person derives from an alternative. In mode choice, it is
assumed to have two components:
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5. Random Utility Theory (Contd.)
1.Systematic Utility (V
im​): This is the observable, measurable
part of utility. It is a function of the attributes of the traveler
(i) and the travel mode (m), such as travel time, cost, and
convenience. We can write a formula for it.
2.Random Utility (ϵ
im​): This is the unobservable part. It
captures all the factors that affect a choice but are unknown
or unmeasurable by the planner, such as personal
preferences, perceptions of comfort and safety, or a sudden
change in mood.
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6. Random Utility Theory (Contd.)
The total utility is the sum of these two parts: U
im​=V
im​+ϵ
im
A person will choose mode m if its utility is greater than
the utility of all other available modes (j): U
im​>U
ijfor all
j≠m. Since we cannot observe the random component ϵ,
we cannot predict the choice with certainty. Instead, we
calculate the probability that a given mode will have the
highest utility.
This is where the logit model comes in.
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7. The Multinomial Logit (MNL) Model
The Multinomial Logit (MNL) model is the simplest and most common form of a
discrete choice model. It calculates the probability of an individual choosing a
specific travel mode from a set of available alternatives.
Mathematical Formulation
The probability that individual i chooses mode m (P
im) from a choice set C is
given by:
P
im​=∑j∈C
ie
Vij
​/e
Vim
​​, Where: P
im​ is the probability of choosing mode m, e is the
base of the natural logarithm (Euler's number), V
im​ is the systematic utility of
mode m for individual i.
The denominator is the sum of the exponentiated systematic utilities of all
available modes j in the choice set C
i(e.g., Car, Bus, Rail).
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8. The Multinomial Logit (MNL) Model
The systematic utility (V
im​) is typically a linear function of the attributes:
V
im​=β
0​+β
1X
im1​+β
2X
im2​+⋯+β
nX
imn
X
im1​,X
im2​,… are the values of the attributes for mode m (e.g., in-vehicle
travel time, out-of-vehicle travel time, travel cost).
β
1​,β
2​,… are the coefficients (or weights) estimated from observed travel
data. These coefficients represent the importance of each attribute. For
example, the coefficient for travel cost is expected to be negative, as
higher costs reduce utility.
β
0 is the alternative-specific constant, which captures the average
influence of all unmeasured factors for that mode.
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9. A Critical Limitation: The IIA Property
The MNL model relies on a significant assumption known as the
Independence of Irrelevant Alternatives (IIA) . This property states that
the ratio of probabilities of choosing between two alternatives depends
only on the utility of those two alternatives, not on any other "irrelevant"
alternatives.
This leads to the classic "red bus/blue bus" problem :
1.Imagine a traveler has two choices: Car and Bus (a red one), with a 50%
probability for each.
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10. The IIA Property(Contd.)
2. Now, a new mode, a "blue bus," is introduced, which is identical to the
red bus in every way (time, cost, etc.).
3. Intuitively, the probability of choosing Car should remain 50%, and the
50% probability for Bus should be split between the red and blue buses
(25% each).
4. However, the MNL model would predict that the new alternative draws
share proportionally from the originals, resulting in probabilities of 33.3%
for Car, 33.3% for the red bus, and 33.3% for the blue bus. This is
unrealistic.
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11. The Nested Logit (NL) Model
The Nested Logit (NL) model was developed to overcome the IIA limitation
of the MNL model. It does this by grouping similar alternatives into
"nests."
Structure and Application
The NL model organizes the choice as a tree structure. For a typical
commute, the structure might look like this:
•Top Level (Nest Choice): Private Vehicle vs. Public Transit
•Lower Level (Elemental Choice):
•If "Private Vehicle" is chosen: Car Driver (only option)
•If "Public Transit" is chosen: Bus vs. Rail
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12. NL model (Contd.)
The choice is now modeled in two stages:
1.First, a model predicts the choice between the nests (Private vs. Public).
2.Second, another model predicts the choice of a specific mode within the
chosen nest (e.g., if Public Transit is chosen, what's the probability of
choosing Bus vs. Rail?).
This structure allows for correlation between the unobserved factors (ϵ) of
alternatives within the same nest. In our example, it correctly assumes
that the unobserved factors affecting the choice between Bus and Rail
(e.g., preference for shared transport, dislike of driving in traffic) are more
similar to each other than to the unobserved factors for choosing a Car.
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13. NL model (Contd.)
This resolves the red bus/blue bus problem. The two buses would be in
the same "Bus" nest, and the introduction of the blue bus would only split
the probability share within that nest, without affecting the probability of
the "Car" nest.
In summary, logit models provide a powerful and flexible framework for
predicting travel behavior. While the MNL is simpler, the NL model offers
greater realism by accounting for the fact that some alternatives are closer
substitutes than others.
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14. Traffic Assignment
✓Traffic assignment is the fourth and final step of the conventional
transportation planning model.
✓It simulates how travelers choose their routes and predicts the resulting
traffic flow on the transportation network.
✓The central concept governing most traffic assignment models is User
Equilibrium.
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15. The Concept of User Equilibrium (UE)
User Equilibrium is a state of network traffic defined by Wardrop's First
Principle:
No driver can unilaterally reduce their travel time by switching to another
route.
In a network at user equilibrium, every driver is acting in their own self-interest
to find the quickest path from their origin to their destination, given the choices
of all other drivers. This leads to a stable condition where:
1.Equal Travel Times: For any given origin-destination (O-D) pair, all routes that
are actually used have the same travel time.
2.Unused Routes are Longer: Any route between the O-D pair that is not used
has a travel time greater than or equal to the travel time on the used routes.
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16. User Equilibrium (Contd.)
Think of it as a "selfish equilibrium." Drivers will keep switching from
slower routes to faster ones until there is no personal advantage to be
gained by switching. At that point, the system stabilizes.
It's important to contrast this with Wardrop's Second Principle , or System
Optimum (SO) , where the total system-wide travel time is minimized. In
an SO scenario, some drivers might be forced (e.g., by tolls) to take slightly
longer routes to prevent bottlenecks and improve the overall average
travel time for everyone. However, in the absence of such system-wide
control, traffic tends to settle into a User Equilibrium.
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17. The Key Component: The Link Performance Function
To model user equilibrium, we must first understand that travel time on a road segment
(a "link") is not constant. It increases as the volume of traffic on the link increases. This
relationship is defined by a Link Performance Function (LPF) .
The most famous LPF is the Bureau of Public Roads (BPR) function :
t=t
0​[1+α(CV​)
β
]
Where: t = The congested travel time on the link, t
0​ = The free-flow travel time (at zero
volume).
V = The volume of traffic on the link, C = The capacity of the link.
α and β = Model parameters calibrated from real-world data (typical values are α=0.15
and β=4).
This function is critical because it creates the feedback loop: traffic volume affects travel
time, and travel time, in turn, affects route choice, which determines traffic volume.
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18. How User Equilibrium is Applied in Traffic Assignment Models
✓Because of the chicken-and-egg relationship between volume and travel
time, UE cannot be solved directly.
✓It must be found using an iterative algorithm.
✓The goal of the algorithm is to load the origin-destination trip matrix
(from the trip distribution step) onto the network model until the
equilibrium condition is met.
✓Here is a simplified overview of the most common method, an iterative
assignment algorithm (based on the Frank-Wolfe algorithm):
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19. User Equilibrium in Traffic Assignment (contd.)
Step 0: Initialization
•Assume the network is empty, so all links are at their free-flow travel time (t
0​).
•Perform an "All-or-Nothing" (AON) assignment . For each O-D pair, find the
single shortest path based on free-flow times and assign all trips for that O-D
pair to that one path. This gives an initial (and unrealistic) set of link volumes.
Step 1: Update Link Travel Times
•Using the link volumes from the previous step, apply the Link Performance
Function (e.g., the BPR function) to every link in the network. This calculates
the new, congested travel time for each link.
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20. User Equilibrium in Traffic Assignment (contd.)
Step 2: Find New Shortest Paths
•Based on these new, congested link times, find the new shortest path for every O-D
pair.
Step 3: Determine the Optimal Shift
•This is the core of the algorithm. You don't move all traffic to the new shortest path.
Instead, you determine an optimal fraction (λ) of the traffic volume from each O-D
pair to shift from the previous set of routes to the new shortest path found in Step 2.
The algorithm finds the value of λ that moves the system closer to equilibrium.
Step 4: Update Link Volumes
•A new set of link volumes is calculated by combining the old volumes with the fraction
of traffic shifted to the new shortest paths. The new flow is a weighted average of the
previous flow and the new AON flow.
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21. User Equilibrium in Traffic Assignment (contd.)
Step 5: Check for Convergence
•The algorithm checks if the network has reached a stable equilibrium. This is
usually done by measuring the change in link flows between iterations. If the
change is very small (below a predefined tolerance), the process stops.
•If the solution has not converged, the algorithm loops back to Step 1 and
repeats the process with the newly updated link volumes.
This loop continues until the flows on the network stabilize, meaning no driver
can find a better route, and Wardrop's First Principle is satisfied.
The final output is a set of traffic volumes and congested travel times for every
link in the network, representing the predicted traffic flow at User Equilibrium.
This information is invaluable for identifying future bottlenecks, testing the
impact of new roads, or analyzing changes in travel patterns.
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