To solve the traffic flow in the network, we will use the method of flow balance (or conservation of flow). The idea is to set up a system of linear equations based on the following principle:

At each node, the sum of incoming flow equals the sum of outgoing flow.

Here is the step-by-step approach:

Define Variables

The traffic flows are labeled as $x_1, x_2, x_3, x_4, x_5$. We'll use these in the equations.

Node-Based Equations

At Node 1 (top-left): Incoming: 300 Outgoing: $x_1, x_2$

300 = x_1 + x_2 \tag{1}

At Node 2 (top-right): Incoming: $x_1$ Outgoing: 150 and $x_4$

x_1 = 150 + x_4 \tag{2}

At Node 3 (bottom-left): Incoming: $x_2, x_5$ Outgoing: 200

x_2 + x_5 = 200 \tag{3}

At Node 4 (bottom-right): Incoming: $x_4$ Outgoing: $x_3, x_5$

x_4 = x_3 + x_5 \tag{4}

Flow Constraint: The total inflow equals the total outflow:

300 + 1000 = 150 + 200 + 350 \tag{5}

This is satisfied, so we focus on solving the equations.

Solve the System of Equations

We now solve the equations (1), (2), (3), and (4) for $x_1, x_2, x_3, x_4, x_5$.

$1. \, x_1 + x_2 = 300$ $2. \, x_1 = 150 + x_4$ $3. \, x_2 + x_5 = 200$ $4. \, x_4 = x_3 + x_5$

Let me solve this system. sympy import symbols, Eq, solve

Define variables

x1, x2, x3, x4, x5 = symbols('x1 x2 x3 x4 x5')

Define equations

eq1 = Eq(x1 + x2, 300) # Node 1 eq2 = Eq(x1, 150 + x4) # Node 2 eq3 = Eq(x2 + x5, 200) # Node 3 eq4 = Eq(x4, x3 + x5) # Node 4

Solve the system of equations

solution = solve([eq1, eq2, eq3, eq4], (x1, x2, x3, x4, x5)) solutionThe solution to the system of equations is expressed in terms of $x_5$ (a free variable):

1. $x_1 = x_5 + 100$
2. $x_2 = 200 - x_5$
3. $x_3 = -50$ (a negative value, implying reverse flow or a modeling adjustment may be needed)
4. $x_4 = x_5 - 50$
5. $x_5$ remains free to assign specific values.

Key Insights:

• The negative $x_3$ suggests an issue with how the system is modeled or that $x_5$ needs proper bounds.
• You can assign specific values to $x_5$ (e.g., $x_5 = 50$) to fully determine the flows.

Let me know if you'd like to explore specific scenarios or further refine this solution. Here are some related questions to expand this topic:

1. How does assigning a specific value to $x_5$ impact the traffic flow?
2. What could a negative flow ($x_3 = -50$) indicate in a real-world context?
3. Can we adjust the system to ensure non-negative flow values for all paths?
4. How can sensitivity analysis be performed on this traffic network?
5. What methods can be used to validate traffic flow models in practice?

Tip: Always double-check flow balance equations at each node to confirm the correctness of your solution.