Let's analyze the flow network to answer part a.

From the diagram:

• We observe flows coming into and out of various points (A, B, C, D, E) with specified flow values.

• Let's define the flow variables at each branch:

  • $x_1$: Flow from A to B
  • $x_2$: Flow from B to C
  • $x_3$: Flow from E to B
  • $x_4$: Flow from B to D
  • $x_5$: Flow from C to D
  • $x_6$: Flow from E to C

Flow at Nodes (Conservation of Flow):

• At Node A: Incoming flow = 80, outgoing flow through $x_1$. $x_1 = 80$

• At Node E: Incoming flow = 50, outgoing flows through $x_3$ and $x_6$. $50 = x_3 + x_6$

• At Node B: Incoming flows = $x_1 + x_3$, outgoing flows = $x_2 + x_4$. $x_1 + x_3 = x_2 + x_4$

• At Node C: Incoming flows = $x_2 + x_6$, outgoing flow = $x_5$. $x_2 + x_6 = x_5$

• At Node D: Incoming flows = $x_4 + x_5$, outgoing flow = 90. $x_4 + x_5 = 90$

Using these conservation equations:

1. $x_1 = 80$
2. $50 = x_3 + x_6$
3. $x_1 + x_3 = x_2 + x_4$ or $80 + x_3 = x_2 + x_4$
4. $x_2 + x_6 = x_5$
5. $x_4 + x_5 = 90$

Now, solving these for minimum flows:

• From equation (2), we know that $x_3 = 50 - x_6$.
• Substitute $x_3$ into equation (3): $80 + (50 - x_6) = x_2 + x_4$ gives $130 - x_6 = x_2 + x_4$ (equation 6).
• From equation (4), $x_2 = x_5 - x_6$.
• Substitute into equation (6): $130 - x_6 = (x_5 - x_6) + x_4$.
• Simplify to get $130 = x_5 + x_4$.

From equation (5), $x_4 + x_5 = 90$. Therefore, both $x_4$ and $x_5$ must solve $x_4 + x_5 = 90$ and $x_4 + x_5 = 130$, meaning these must be free variables.

Based on this, the correct configuration is option B, where $x_1 = 80$ and the rest are free.

Would you like a more detailed explanation, or do you have any specific questions?

Related questions:

1. How does the conservation of flow apply to networks with more nodes?
2. What happens if the flow directions change?
3. How do we compute maximum flow using the same principles?
4. Can this problem be generalized for larger networks?
5. What is the significance of identifying free variables in network flows?

Tip: When solving flow network problems, always start by analyzing nodes with known inflows and outflows first to simplify the system of equations.