Answer
The least-squares solution for the flow variables is:
𝑥1=92.5
𝑥2=107.5
𝑥3=15
𝑥4=37.5
𝑥5=22.5
Work Step by Step
Let's go step by step to solve the flow network problem, following the same process as the example.
Step 1: Writing the equations for each node
From the flow diagram and the conservation rules, we have the following equations:
1. Node A: \( x_1 = x_3 + x_4 + 40 \)
2. Node B: \( 200 = x_1 + x_2 \)
3. Node C: \( x_2 + x_3 = x_5 + 100 \)
4. Node D: \( x_4 + x_5 = 60 \)
Now, let's rearrange these equations so that all variables are on the left side and constants on the right.
Step 2: Rearranging the equations
Rewriting the equations:
1. \( x_1 - x_3 - x_4 = 40 \)
2. \( x_1 + x_2 = 200 \)
3. \( x_2 + x_3 - x_5 = 100 \)
4. \( x_4 + x_5 = 60 \)
Step 3: Creating the matrix form
We can now express this system of equations in matrix form. Each row corresponds to one equation, with the coefficients of \( x_1 \), \( x_2 \), \( x_3 \), \( x_4 \), and \( x_5 \) in the matrix. The constants on the right-hand side are stored in a separate vector.
The matrix form is:
\[
\begin{bmatrix}
1 & 0 & -1 & -1 & 0 \\
1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5
\end{bmatrix}
=
\begin{bmatrix}
40 \\
200 \\
100 \\
60
\end{bmatrix}
\]
### Step 4: Solving the system of equations
We solve this matrix system to find the values of \( x_1 \), \( x_2 \), \( x_3 \), \( x_4 \), and \( x_5 \). Since this is an underdetermined system (4 equations, 5 variables), we will use the least-squares method to find a solution that best satisfies the equations.
After solving, we get:
- \( x_1 = 92.5 \)
- \( x_2 = 107.5 \)
- \( x_3 = 15 \)
- \( x_4 = 37.5 \)
- \( x_5 = 22.5 \)
This means that for the flow to balance at each node, these are the minimum flows required for the given directions and conditions.
### Step 5: Verifying the solution
We can substitute these values back into the original equations to check:
1. For Node A: \( 92.5 = 15 + 37.5 + 40 \) → True
2. For Node B: \( 92.5 + 107.5 = 200 \) → True
3. For Node C: \( 107.5 + 15 = 22.5 + 100 \) → True
4. For Node D: \( 37.5 + 22.5 = 60 \) → True
Thus, the solution is correct and satisfies all the flow constraints.
This is the step-by-step solution to your problem.
