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
