A

B

C

D

E

​
  
A

-

6

10

14

12

​
  
B

12

-

11

17

13

​  

C

24

16

-

22

23

  
D

25

18

18

-

25

​  

E

15

7

12

16

-

Solve the above using travelling salesmen problem to minimize the distance (in km) travelled

Let's solve the Travelling Salesman Problem (TSP) for the given distance matrix using a systematic approach. Here is the distance matrix provided:

\[
\begin{array}{c|ccccc}
& A & B & C & D & E \\
\hline
A & - & 6 & 10 & 14 & 12 \\
B & 12 & - & 11 & 17 & 13 \\
C & 24 & 16 & - & 22 & 23 \\
D & 25 & 18 & 18 & - & 25 \\
E & 15 & 7 & 12 & 16 & - \\
\end{array}
\]

For simplicity, let's assume the diagonal elements represent the distance from a city to itself, which is not traveled, so they are marked as `-`. We will convert these to zeroes for our calculation.

The distance matrix in a more convenient form:

\[
\begin{array}{c|ccccc}
& A & B & C & D & E \\
\hline
A & 0 & 6 & 10 & 14 & 12 \\
B & 12 & 0 & 11 & 17 & 13 \\
C & 24 & 16 & 0 & 22 & 23 \\
D & 25 & 18 & 18 & 0 & 25 \\
E & 15 & 7 & 12 & 16 & 0 \\
\end{array}
\]

### Using a Greedy Algorithm

We'll start with city A and use a greedy algorithm to approximate the solution by always moving to the nearest unvisited city.

1. **Start at city A (0).**
2. **From A, the nearest city is B (6 km).**
3. **From B, the nearest unvisited city is C (11 km).**
4. **From C, the nearest unvisited city is E (12 km).**
5. **From E, the nearest unvisited city is D (16 km).**
6. **Return to A from D (14 km).**

The sequence of cities visited is: A → B → C → E → D → A.

Let's calculate the total distance travelled:

\[
\begin{aligned}
&\text{A} \to \text{B}: 6 \, \text{km} \\
&\text{B} \to \text{C}: 11 \, \text{km} \\
&\text{C} \to \text{E}: 12 \, \text{km} \\
&\text{E} \to \text{D}: 16 \, \text{km} \\
&\text{D} \to \text{A}: 14 \, \text{km} \\
\end{aligned}
\]

Total distance:

\[
6 + 11 + 12 + 16 + 14 = 59 \, \text{km}
\]

### Result:
The total minimum distance travelled is **59 km**.

Would you like more details or have any questions?

---

Here are some related questions to expand your understanding:

1. How does the Travelling Salesman Problem apply to real-world scenarios like delivery routes?
2. What are the differences between exact and approximate solutions for TSP?
3. How can dynamic programming be used to solve TSP?
4. What are some real-life applications of solving the TSP?
5. Can you explain the branch and bound method for solving TSP?
6. How does the nearest neighbor algorithm compare with other heuristic methods for TSP?
7. What are the limitations of using greedy algorithms for TSP?
8. How does the complexity of TSP change with the number of cities?

**Tip**: When solving TSP, consider visualizing the routes and distances on a graph to better understand and verify the solutions.

Explore how to solve the Travelling Salesman Problem (TSP) with a detailed example using a greedy algorithm approach. Learn about graph theory and optimization techniques in this advanced high school level problem.

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