https://file.aiquickdraw.com/mathbot/file/1730256839476dbvl07h7.jpg

To solve this problem using the **Nearest Neighbor Method**, let's follow the instructions, starting from vertex \( A \) and moving to the nearest unvisited vertex until we've formed a circuit back to \( A \).

### Given Edge Weights:
- \( BD = 140 \)
- \( BE = 110 \)
- \( AC = 120 \)
- \( CD = 180 \)
- \( BD = 140 \) (Note: this might be a typo and only needs to be stated once.)

### Nearest Neighbor Method Steps:
1. **Start at \( A \):**
   - Nearest neighbors from \( A \) are:
     - \( B = 100 \)
     - \( C = 120 \)
     - \( D = 150 \)
   - **Choose \( B \) (100)**.

2. **Move to \( B \):**
   - Nearest neighbors from \( B \) are:
     - \( E = 110 \)
     - \( C = 50 \)
     - \( D = 140 \)
   - **Choose \( C \) (50)**.

3. **Move to \( C \):**
   - Nearest neighbors from \( C \) are:
     - \( E = 80 \)
     - \( D = 180 \)
   - **Choose \( E \) (80)**.

4. **Move to \( E \):**
   - Nearest neighbors from \( E \) are:
     - \( D = 110 \)
   - **Choose \( D \) (110)**.

5. **Move to \( D \):**
   - The only unvisited vertex is \( A \) with a distance of \( 150 \).
   - **Return to \( A \) (150)**.

### Estimated Hamiltonian Circuit:
The circuit, starting and ending at \( A \), is:
\[
A \to B \to C \to E \to D \to A
\]

### Total Distance:
Adding up the weights:
\[
100 + 50 + 80 + 110 + 150 = 490
\]

### Answer:
The estimated cheapest Hamiltonian circuit, using the nearest neighbor method, has a total weight of **490**.

Would you like further details on the solution or explanation?

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#### Additional Questions for Deeper Understanding:
1. What are other methods to find Hamiltonian circuits apart from the nearest neighbor?
2. How does the nearest neighbor method compare to the brute-force approach?
3. How would the result change if we started from a different vertex?
4. What are the limitations of the nearest neighbor method?
5. Can we verify this solution by trying alternative paths?

#### Tip:
In optimization problems, always consider multiple heuristic methods for comparison to ensure the best solution.

Use the nearest neighbor method to estimate the cheapest Hamilton circuit, starting at A. (Given edge weights: BD = 140, BE = 110, AC = 120, CD = 180, BD = 140).

Learn how to estimate the cheapest Hamiltonian circuit starting from point A using the nearest neighbor method. This problem involves graph theory and optimization, providing a detailed step-by-step solution to calculate the circuit with minimum weight.

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