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Interactive Audio Lesson
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Understanding User Equilibrium Assignment
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Today, we're going to explore user equilibrium assignment, which is crucial in understanding how traffic flows are managed. Can someone tell me what user equilibrium means?
Is it when no driver can reduce their travel cost by changing routes?
Exactly! That's a key principle. In user equilibrium, each driver is assumed to make the best decision based on available information. Let's remember this with the acronym 'NEE' - *No Extra Expense*.
What happens if one route gets congested?
Good question! If a route becomes congested, drivers will shift to alternate routes until a new equilibrium is reached. This leads us to how we calculate flows and travel times.
Calculating Travel Times
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Imagine we have two paths with different travel times. How do we set those in a calculation?
We need to use the travel time functions provided for each link!
Exactly! Each link has a function determining travel time based on flow. We’ll use equations like `t = 10 + 3x_1` for our calculations. How would we set this up for equilibrium?
We need to combine the flows and ensure the total flow equals the demand.
Spot on! And since we're working with a constant total flow of 12, it brings us to maintaining that equilibrium through our equations.
Verifying User Equilibrium
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Once we've calculated our flows, how do we verify these are at user equilibrium?
We check if the travel times along all used paths are equal!
Exactly! It also means that any unused routes will have a higher travel time. Let's practice this with the flows we calculated earlier.
So if all travel times are equal, we’ve reached user equilibrium?
That's correct! This ensures our theory matches reality in our traffic models.
Introduction & Overview
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Quick Overview
Standard
The section provides problems focused on user equilibrium assignment in traffic flows, including instructions for calculating system travel time and verifying flows at user equilibrium. It engages students with applied examples to deepen their understanding of traffic assignment methods.
Detailed
Problems in Traffic Assignment
In this section, we tackle the essential problems associated with user equilibrium assignment in traffic systems. The main objective is to calculate the system travel time and link flows by implementing user equilibrium assignment for a given network. Students are tasked with using the equations of travel time in order to derive optimal flow values, ensuring that these values conform to user equilibrium conditions. This exercise not only adopts analytical skills but also emphasizes the practical application of theoretical concepts in the realm of traffic assignment.
Audio Book
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User Equilibrium Assignment Problem
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- Calculate the system travel time and link flows by doing user equilibrium assignment for the network in the given figure. Verify that the flows are at user equilibrium.
Detailed Explanation
This problem is asking you to perform a user equilibrium assignment for a specific transport network. User equilibrium is where no driver can reduce their travel costs by switching routes. To solve this, you need to calculate both the total travel time across the network and the flow of traffic on each road link based on the provided travel time equations and any constraints set by the problem. This involves using the equations from the previous sections to derive a function that minimizes the travel time, considering the trip distribution and the capacities of each route. Finally, you verify the results to ensure the flows align with the user equilibrium condition.
Examples & Analogies
Think of it like a busy marketplace where each stall represents a route. If customers want to buy items, they will naturally choose the stalls that have the best prices (low travel time). However, if one stall gets too crowded, customers may switch to a less popular stall that offers similar products at comparable prices. The user equilibrium ensures that no customer can find a quicker route to their desired stall without causing congestion elsewhere.
Solution Derivation
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Solution Solution Substituting the travel time in equation 10.4 to 10.8 yield to
x1 x2
min Z(x)= 12+3x dx + 10+5x dx
1 1 2 2
∫0 ∫0
3x2 5x2
=12x + 1 +10x + 2
1 2
2 2
subject to
x +x =12
1 2
Substituting x =12 x , in the above formulation will yield the unconstrained formulation as below:
2 1
−
3x2 5(12 x )2
min Z(x)=12x + 1 +10(12 x)+ − 1 min Z(x)=4x2 58x +480.
Detailed Explanation
In this chunk, the derivation of the travel time function and flow assignment is presented. You substitute the given travel time equations into the optimization problem Z(x), which represents the total travel time as a function of traffic flow. You then integrate these equations to establish a formula that calculates the overall travel time. After substituting specific constraints, you arrive at a minimization problem that helps to determine the flow on each route. The goal here is to find the optimal distribution of traffic that satisfies both the total demand and the user equilibrium conditions.
Examples & Analogies
Imagine a delivery service needing to optimize routes for multiple packages. They can calculate the fastest delivery times if they adjust traffic based on current order volumes—like how you might reconfigure the routes drivers take based on current street congestions. By updating their routes according to new deliveries, they minimize overall time—which is exactly the process we’re following in the problem statement.
Solving the Optimization Problem
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Differentiate the above equation with respect to x and equating to zero, leads to the solution for x1 and x2: x1 = 7.25, x2 = 4.75
Detailed Explanation
To solve for the optimal values of x1 and x2, you differentiate the travel time function Z(x) with respect to x and set the derivative equal to zero. This mathematical technique finds the points at which the total travel time is minimized. After performing this differentiation and solving the resulting equation, you find the specific flow values for x1 (traffic flow on the first route) and x2 (traffic flow on the second route). These represent the optimal traffic distribution that minimizes total travel time while maintaining user equilibrium.
Examples & Analogies
Consider a person trying to decide how long to spend on two different tasks to minimize their total time. By evaluating the time benefit of working on Task A versus Task B and adjusting their focus, they find a sweet spot, where changing their focus further would only lead to increased time—a balance that mirrors our search for optimal route usage in traffic distribution.
Verification of User Equilibrium
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Travel times are t1 = 12 + 3(7.25) = 33.75, t2 = 10 + 5(4.75) = 33.75, i.e., t1 = t2
Detailed Explanation
To confirm that the calculated flows are at user equilibrium, you need to check that the travel times on both routes are equal (t1 = t2). This is a key characteristic of user equilibrium—when flows are balanced, drivers are indifferent between routes, as neither presents a faster option at the current volume level. By substituting the values of x1 and x2 back into the travel time equations, you find that both travel times equate, validating your solution.
Examples & Analogies
Think about two friends heading to the same destination from different starting points. If one friend assures the other that they will arrive simultaneously after considering traffic, it highlights user equilibrium—neither route offers a faster alternative given the current circumstances. By confirming both routes have the same travel time, you verify that drivers would be equally satisfied choosing either path.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Traffic Assignment: The process of distributing trip interchanges to transportation systems.
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User Equilibrium: A situation where no driver can benefit from changing their chosen route based on current conditions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Calculating flow on two parallel links and showing that travel time is minimized.
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Using travel time functions to verify that the chosen path in a network is optimal.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When flows align and paths are true, travel times stay equal for me and you.
📖 Fascinating Stories
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Imagine a city where drivers exchange secrets about the best routes. They always find the path that costs the least time, ensuring they're all happy travelers.
🧠 Other Memory Gems
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Remember 'CUT' - Congestion, Unused, Travel Times to verify user equilibrium in paths.
🎯 Super Acronyms
Use 'UE' for User Equity to recall that no single route should be superior for drivers.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: User Equilibrium Assignment
Definition:
A method in traffic assignment where no driver can reduce their travel cost by unilaterally changing routes.
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Term: Travel Time Function
Definition:
An equation that relates the travel time on a link to the flow of traffic on that link.
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Term: Congestion
Definition:
A situation where demand for road space exceeds the capacity, leading to increased travel times.