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The problem in the image is a traveling salesperson problem (TSP) where the sales director, starting from city A, must visit cities B, C, D, and E, and then return to city A. We need to use the brute force method to find the Hamiltonian circuit with the minimum total airfare cost.

### Steps to solve using the brute force method:
1. **List all possible Hamiltonian circuits** starting from city A. These circuits will involve visiting each city exactly once and returning to A at the end.
   
   The cities involved are A, B, C, D, E. The Hamiltonian circuits starting and ending at A are as follows:
   - A → B → C → D → E → A
   - A → B → C → E → D → A
   - A → B → D → C → E → A
   - A → B → D → E → C → A
   - A → B → E → C → D → A
   - A → B → E → D → C → A
   - A → C → B → D → E → A
   - A → C → B → E → D → A
   - A → C → D → B → E → A
   - A → C → D → E → B → A
   - A → C → E → B → D → A
   - A → C → E → D → B → A
   - A → D → B → C → E → A
   - A → D → B → E → C → A
   - A → D → C → B → E → A
   - A → D → C → E → B → A
   - A → D → E → B → C → A
   - A → D → E → C → B → A
   - A → E → B → C → D → A
   - A → E → B → D → C → A
   - A → E → C → B → D → A
   - A → E → C → D → B → A
   - A → E → D → B → C → A
   - A → E → D → C → B → A

2. **Calculate the total airfare cost** for each of these circuits by adding the weights (costs) associated with the edges in the graph.

Let me calculate the total costs for these routes to determine the optimal Hamiltonian circuit.The optimal Hamiltonian circuit using the brute force method is:

**A → C → D → B → E → A**, with a total airfare cost of **1177**.

Would you like a further breakdown of the calculations or any additional details?

Here are some related questions:
1. How does the brute force method work for other cities and distances?
2. What is the difference between the brute force method and other TSP-solving methods?
3. Can this method be applied to larger graphs, and how does complexity change?
4. How would the result change if one of the distances was increased or decreased?
5. How can optimization techniques like dynamic programming improve the solution process?

**Tip:** The brute force method guarantees finding the optimal solution but becomes computationally expensive as the number of cities increases.

A sales director who lives in city A is required to fly to regional offices in cities B, C, D, and E. The weighted graph shows the one-way airfares between any two cities. Use the Brute Force Method to find the optimal solution. Which Hamiltonian circuit is the optimal solution?

Learn how to solve the Traveling Salesperson Problem (TSP) using the brute force method. This problem involves finding the Hamiltonian circuit that minimizes travel costs between cities A, B, C, D, and E. Suitable for students studying advanced high school math or college-level courses.

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