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To solve this problem using the **Hungarian Method**, which is designed for solving the assignment problem (minimizing cost in a bipartite graph), we need to adapt it to this Traveling Salesman Problem (TSP). Here's the step-by-step procedure:

### Step 1: Problem Representation
The matrix provided is:

| From\To | A   | B   | C   | D   | E   |
|---------|-----|-----|-----|-----|-----|
| **A**   | ∞   | 4   | 7   | 3   | 4   |
| **B**   | 4   | ∞   | 6   | 3   | 4   |
| **C**   | 7   | 6   | ∞   | 7   | 5   |
| **D**   | 3   | 3   | 7   | ∞   | 7   |
| **E**   | 4   | 4   | 5   | 7   | ∞   |

Replace the infinite values (representing that a city cannot be revisited) with a very large number (say ∞ = 999) to handle them mathematically:

\[
\begin{bmatrix}
999 & 4 & 7 & 3 & 4 \\
4 & 999 & 6 & 3 & 4 \\
7 & 6 & 999 & 7 & 5 \\
3 & 3 & 7 & 999 & 7 \\
4 & 4 & 5 & 7 & 999
\end{bmatrix}
\]

### Step 2: Row and Column Reduction

#### Row Reduction
- For each row, subtract the smallest value in that row from all other elements of that row.

\[
\begin{bmatrix}
996 & 1 & 4 & 0 & 1 \\
1 & 996 & 3 & 0 & 1 \\
2 & 1 & 994 & 2 & 0 \\
0 & 0 & 4 & 996 & 4 \\
0 & 0 & 1 & 3 & 996
\end{bmatrix}
\]

#### Column Reduction
- Now, subtract the smallest value in each column from all other elements in that column.

\[
\begin{bmatrix}
996 & 1 & 3 & 0 & 1 \\
1 & 996 & 2 & 0 & 1 \\
2 & 1 & 993 & 2 & 0 \\
0 & 0 & 3 & 996 & 4 \\
0 & 0 & 0 & 3 & 995
\end{bmatrix}
\]

### Step 3: Cover all Zeros
- Use the minimum number of horizontal and vertical lines to cover all the zeros in the matrix.
- After covering the zeros, check if the number of lines equals the number of rows (or columns). If not, proceed to adjust the matrix by the Hungarian Method's steps for uncovering zeros and repeating the reduction steps.

### Step 4: Optimal Assignment
Once all zeros are covered with exactly 5 lines, the assignment corresponding to the zeros will give the optimal travel plan.

Let's summarize the optimal assignments and compute the total cost.

I'll calculate the assignments and total travel time using Python next.The optimal travel plan using the Hungarian method is as follows:

- From City A to City B
- From City B to City D
- From City C to City E
- From City D to City A
- From City E to City C

The total minimum travel time for this route is **20 hours**.

Would you like further details or have any questions?

### 5 Follow-up Questions:
1. How does the Hungarian Method work in general for optimization problems?
2. Can the Hungarian method be used for maximizing profit rather than minimizing cost?
3. How would we solve the Traveling Salesman Problem for more than five cities?
4. What are other methods to solve the Traveling Salesman Problem, and how do they compare?
5. How would this problem change if a city could be revisited?

### Tip:
The Traveling Salesman Problem becomes computationally intensive as the number of cities increases. Approximations or heuristics like genetic algorithms or simulated annealing are often used for larger problems.

A Salesman has to travel to five cities in a month without revisiting any city. Determine an optimal travel plan using the given time matrix (in hours). Solve by Hungarian method.

This problem addresses the Traveling Salesman Problem (TSP) using a time matrix for five cities. Learn how to use the Hungarian Method to determine an optimal travel plan without revisiting any city. The solution includes the core concepts of combinatorics, matrix reduction, and optimization.

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