https://file.aiquickdraw.com/mathbot/file/1728437059348erg6yl91.jpg

The problem depicted is an *Assignment Problem*, where the goal is to assign 5 salesmen to 5 districts in such a way that the total sales revenue is maximized. The table provided shows the expected sales revenue (in thousands of rupees) for each salesman in each district.

The most efficient way to solve this problem is by using the *Hungarian Algorithm*, which is specifically designed to solve assignment problems like this one by maximizing (or minimizing) the total outcome.

Here's the table extracted from the image:

| Salesman | District 1 (D₁) | District 2 (D₂) | District 3 (D₃) | District 4 (D₄) | District 5 (D₅) |
|----------|-----------------|-----------------|-----------------|-----------------|-----------------|
| S₁       | 40              | 46              | 36              | 48              | 48              |
| S₂       | 48              | 32              | 36              | 29              | 44              |
| S₃       | 49              | 35              | 41              | 38              | 45              |
| S₄       | 30              | 46              | 49              | 44              | 44              |
| S₅       | 37              | 41              | 48              | 43              | 47              |

### To solve the problem:

1. **Formulation:**
   - We need to assign each salesman (S₁ to S₅) to one of the districts (D₁ to D₅) such that no district gets more than one salesman and the total revenue is maximized.

2. **Optimization:**
   - The Hungarian Algorithm will reduce this problem into simpler steps and help us find the optimal assignment for the maximum sales revenue.

I'll solve it now using this approach.The optimal assignment of salesmen to districts, using the Hungarian algorithm, is as follows:

- **Salesman S₁** should be assigned to **District D₄**.
- **Salesman S₂** should be assigned to **District D₁**.
- **Salesman S₃** should be assigned to **District D₅**.
- **Salesman S₄** should be assigned to **District D₂**.
- **Salesman S₅** should be assigned to **District D₃**.

This assignment will maximize the total expected sales revenue, which is **235 thousand rupees**.

Would you like further details or clarifications?

Here are five related questions for further exploration:
1. How does the Hungarian Algorithm work step-by-step?
2. Can the problem be solved using any other optimization methods?
3. How would the assignment change if there were more salesmen or districts?
4. How can we modify the approach for minimizing instead of maximizing revenue?
5. What happens if the sales data contains negative values?

**Tip:** The Hungarian algorithm is efficient for solving assignment problems and guarantees the optimal solution in polynomial time.

A company plans to assign 5 salesmen to 5 districts in which it operates. Estimates of sales revenue in thousands of rupees for each salesman in different districts are given. What should be the placement of the salesmen to maximize the expected sales revenue?

This problem involves assigning 5 salesmen to 5 districts based on maximizing expected sales revenue. Using the Hungarian Algorithm for solving the assignment problem, the total expected revenue is maximized. Learn how the assignment yields optimal results.

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