3. Three routes connect an origin-destination pair with performance functions: tị =20 +0.571 t₂ = 4 + 2x2 t3 = 3 +0.2x² with t in minutes and in thousand vehicles per hour. (a) Determine the User Equilibrium flow on each route if q = 4000veh/h. (b) What is the minimum q (origin-destination demand) to ensure that all the three routes are used under user equilibrium? (c) Suppose that Route 1 is closed for repair. Find the system optimal flow on routes 2 and 3 and compute the total travel times.
3. Three routes connect an origin-destination pair with performance functions: tị =20 +0.571 t₂ = 4 + 2x2 t3 = 3 +0.2x² with t in minutes and in thousand vehicles per hour. (a) Determine the User Equilibrium flow on each route if q = 4000veh/h. (b) What is the minimum q (origin-destination demand) to ensure that all the three routes are used under user equilibrium? (c) Suppose that Route 1 is closed for repair. Find the system optimal flow on routes 2 and 3 and compute the total travel times.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![### Transportation Network Analysis: User Equilibrium and System Optimal Flow
#### Problem Overview
Three routes connect an origin-destination pair with performance functions:
\[ t_1 = 20 + 0.5x_1 \]
\[ t_2 = 4 + 2x_2 \]
\[ t_3 = 3 + 0.2x_3^3 \]
with \( t \) in minutes and \( x \) in thousand vehicles per hour.
#### Tasks
(a) **Determine the User Equilibrium flow on each route if \( q = 4000 \text{ vehicles/hour} \).**
(b) **Find the minimum \( q \) (origin-destination demand) to ensure that all three routes are used under user equilibrium.**
(c) **Suppose that Route 1 is closed for repair. Find the system optimal flow on routes 2 and 3, and compute the total travel times.**
#### Detailed Explanation
Under user equilibrium, no driver can reduce their travel time by switching routes. This implies that the travel times on all routes that are used are equal.
##### Part (a)
**Given: \( q = 4000 \text{ vehicles/hour} \)**
The equations for travel time become:
\[ t_1 = 20 + 0.5x_1 \]
\[ t_2 = 4 + 2x_2 \]
\[ t_3 = 3 + 0.2x_3^3 \]
Since the total traffic flow is 4000 vehicles/hour:
\[ x_1 + x_2 + x_3 = 4000 \]
##### Part (b)
Determine the minimum total demand \( q \) such that all routes are used in equilibrium. For all routes to be used, the travel times \( t_1, t_2, \) and \( t_3 \) must be equal. Set up the system of equations:
\[ 20 + 0.5x_1 = 4 + 2x_2 = 3 + 0.2x_3^3 \]
Solve these equations simultaneously to find the minimum \( q \).
##### Part (c)
**Given: Route 1 is closed.**
Now, routes 2 and 3 share the total demand \( q \):
\[ x_2 + x_3 = q \]
The objective is to minimize the total travel time](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F216015d9-daff-4e98-b3d1-7f3f5202e820%2F82e53a40-9d84-4a68-8dd1-ae737b52ab91%2F122989_processed.png&w=3840&q=75)
Transcribed Image Text:### Transportation Network Analysis: User Equilibrium and System Optimal Flow
#### Problem Overview
Three routes connect an origin-destination pair with performance functions:
\[ t_1 = 20 + 0.5x_1 \]
\[ t_2 = 4 + 2x_2 \]
\[ t_3 = 3 + 0.2x_3^3 \]
with \( t \) in minutes and \( x \) in thousand vehicles per hour.
#### Tasks
(a) **Determine the User Equilibrium flow on each route if \( q = 4000 \text{ vehicles/hour} \).**
(b) **Find the minimum \( q \) (origin-destination demand) to ensure that all three routes are used under user equilibrium.**
(c) **Suppose that Route 1 is closed for repair. Find the system optimal flow on routes 2 and 3, and compute the total travel times.**
#### Detailed Explanation
Under user equilibrium, no driver can reduce their travel time by switching routes. This implies that the travel times on all routes that are used are equal.
##### Part (a)
**Given: \( q = 4000 \text{ vehicles/hour} \)**
The equations for travel time become:
\[ t_1 = 20 + 0.5x_1 \]
\[ t_2 = 4 + 2x_2 \]
\[ t_3 = 3 + 0.2x_3^3 \]
Since the total traffic flow is 4000 vehicles/hour:
\[ x_1 + x_2 + x_3 = 4000 \]
##### Part (b)
Determine the minimum total demand \( q \) such that all routes are used in equilibrium. For all routes to be used, the travel times \( t_1, t_2, \) and \( t_3 \) must be equal. Set up the system of equations:
\[ 20 + 0.5x_1 = 4 + 2x_2 = 3 + 0.2x_3^3 \]
Solve these equations simultaneously to find the minimum \( q \).
##### Part (c)
**Given: Route 1 is closed.**
Now, routes 2 and 3 share the total demand \( q \):
\[ x_2 + x_3 = q \]
The objective is to minimize the total travel time
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