### Transportation Network Analysis: User Equilibrium and System Optimal Flow#### Problem OverviewThree 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 ExplanationUnder 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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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.

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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