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Solve the following travelling salesman problem to minimize the distance (in km) travelled.

To solve the Travelling Salesman Problem (TSP) for the given distance matrix, we need to find the shortest possible route that visits each city exactly once and returns to the origin city. Let me extract the distance matrix from the image and solve the problem step by step.

First, let's extract the distance matrix:

\[
\begin{array}{c|ccccccc}
& A & B & C & D & E & F & G \\
\hline
A & 0 & 29 & 20 & 21 & 17 & 25 & 18 \\
B & 29 & 0 & 15 & 29 & 28 & 40 & 27 \\
C & 20 & 15 & 0 & 15 & 14 & 25 & 24 \\
D & 21 & 29 & 15 & 0 & 4 & 30 & 14 \\
E & 17 & 28 & 14 & 4 & 0 & 29 & 15 \\
F & 25 & 40 & 25 & 30 & 29 & 0 & 10 \\
G & 18 & 27 & 24 & 14 & 15 & 10 & 0 \\
\end{array}
\]

This is a 7x7 matrix representing the distances between each pair of cities (A to G).

To solve this problem, we will use the following steps:

1. **Initialization**: Start with the first city (A).
2. **Find the Minimum Distance**: From the current city, move to the city with the minimum distance that hasn't been visited yet.
3. **Repeat**: Repeat the process until all cities are visited.
4. **Return to Origin**: Return to the starting city (A) to complete the tour.

We can use a greedy algorithm to approximate the solution. Here's the detailed step-by-step solution:

### Step-by-Step Solution:

1. Start at city A.
2. From A, the nearest unvisited city is C (20 km).
3. From C, the nearest unvisited city is B (15 km).
4. From B, the nearest unvisited city is G (27 km).
5. From G, the nearest unvisited city is F (10 km).
6. From F, the nearest unvisited city is E (29 km).
7. From E, the nearest unvisited city is D (4 km).
8. Return to A from D (21 km).

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

Let's calculate the total distance travelled:

\[
\begin{aligned}
&\text{A} \to \text{C}: 20 \, \text{km} \\
&\text{C} \to \text{B}: 15 \, \text{km} \\
&\text{B} \to \text{G}: 27 \, \text{km} \\
&\text{G} \to \text{F}: 10 \, \text{km} \\
&\text{F} \to \text{E}: 29 \, \text{km} \\
&\text{E} \to \text{D}: 4 \, \text{km} \\
&\text{D} \to \text{A}: 21 \, \text{km} \\
\end{aligned}
\]

Total distance:

\[
20 + 15 + 27 + 10 + 29 + 4 + 21 = 126 \, \text{km}
\]

### Result:
The total minimum distance travelled is **126 km**.

Would you like more details or have any questions?

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Here are some related questions to expand your understanding:

1. What is the Travelling Salesman Problem and why is it important in optimization?
2. How does the greedy algorithm approximate a solution for the TSP?
3. What are some exact algorithms used to solve the TSP?
4. How can heuristic methods like genetic algorithms be applied to TSP?
5. What is the significance of symmetric and asymmetric TSP?
6. How do distance matrices differ in real-world applications like logistics and network design?
7. What are the limitations of the greedy algorithm in solving the TSP?
8. How does the branch and bound method work for solving the TSP?

**Tip**: When solving large TSP instances, consider using advanced techniques like dynamic programming, branch and bound, or metaheuristic algorithms for more efficient solutions.

Learn how to solve the Travelling Salesman Problem (TSP) to minimize the distance travelled. This problem involves finding the shortest route visiting each city exactly once and returning to the origin city. The solution covers a step-by-step greedy algorithm approach and calculates a total distance of 126 km.

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