Answered

There are three possible paths to travel from Quackers Hills to Stanhope Gardens, two suburbs in the Western Sydney LGA (traffic zones). It is estimated that on an average 4000 car trips are made during the peak hour. Determine the user equilibrium flow and travel time on each path given the average travel time of the paths A, B and C are:
ta = 0.25Xa + 1
tb = 0.15Xb + 2
tc = 0.45Xc + 8
Where t is the travel time given in minutes and x is the flow given in thousands of vehicles/h on a path.
Flow on path A in vehicles/h:
Flow on path B in vehicles/h:
Flow on path C in vehicles/h:
User equilibrium travel time in minutes:



Answer :

To determine the user equilibrium flow and travel time on each path (A, B, and C), we need to find the flow on each path where the travel times are equalized for all paths given the total demand.

Given:

- Total demand (peak hour trips) = 4000 cars

- Travel time functions:

 - \( t_a = 0.25X_a + 1 \)

 - \( t_b = 0.15X_b + 2 \)

 - \( t_c = 0.45X_c + 8 \)

Let's find the flow on each path and the user equilibrium travel time step by step:

### Step 1: Set up the Equations for User Equilibrium

User equilibrium occurs when the travel time on each path is equalized:

\[ t_a = t_b = t_c \]

### Step 2: Express Equations in Terms of Total Demand

Convert the flow \( X \) from thousands of vehicles/h to vehicles/h:

For path A:

\[ t_a = 0.25X_a + 1 \]

For path B:

\[ t_b = 0.15X_b + 2 \]

For path C:

\[ t_c = 0.45X_c + 8 \]

### Step 3: Solve for Equilibrium Flow on Each Path

Set \( t_a = t_b = t_c \) and solve for \( X_a, X_b, X_c \).

From \( t_a = t_b \):

\[ 0.25X_a + 1 = 0.15X_b + 2 \]

\[ 0.25X_a - 0.15X_b = 1 \]

\[ 5X_a - 3X_b = 20 \quad \text{(Multiply by 20)} \quad \longrightarrow