Advertisements
Advertisements
Question
A company has a team of four salesmen and there are four districts where the company wants to start its business. After taking into account the capabilities of salesmen and the nature of districts, the company estimates that the profit per day in rupees for each salesman in each district is as below:
| Salesmen | District | |||
| 1 | 2 | 3 | 4 | |
| A | 16 | 10 | 12 | 11 |
| B | 12 | 13 | 15 | 15 |
| C | 15 | 15 | 11 | 14 |
| D | 13 | 14 | 14 | 15 |
Find the assignment of salesmen to various districts which will yield maximum profit.
Solution: It is a maximization problem. Subtract all the elements from `square`.
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(4, 3, 1, 1),(1, 1, 5, 2),(3, 2, 2, 1)]`
∴ Subtract the smallest element of each row from the elements of that row:
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`
∴ Subtract the smallest element of each column from the elements of that column:
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`
∴ Since the number of lines covering zeros is equal to the order of the matrix, the optimal solution has reached:
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`
The optimal solution is obtained.
| Salesmen | Districts | Profits (₹) |
| A | 1 | 16 |
| B | `square` | `square` |
| C | `square` | `square` |
| D | 4 | 15 |
∴ Total profit = ₹ `square`
Advertisements
Solution
It is a maximization problem. Subtract all the elements from \[\boxed{16}\].
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(4, 3, 1, 1),(1, 1, 5, 2),(3, 2, 2, 1)]`
∴ Subtract the smallest element of each row from the elements of that row:
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`
∴ Subtract the smallest element of each column from the elements of that column:
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`
∴ Since the number of lines covering zeros is equal to the order of the matrix, the optimal solution has reached:
`{:(1, 2, 3, 4):}`
`{:("A"), ("B"), ("C"), ("D"):}[(0, 6, 4, 5),(3, 2, 0, 0),(0, 0, 4, 1),(2, 1, 1, 0)]`
The optimal solution is obtained.
| Salesmen | Districts | Profits (₹) |
| A | 1 | 16 |
| B | \[\boxed{3}\] | \[\boxed{15}\] |
| C | \[\boxed{2}\] | \[\boxed{15}\] |
| D | 4 | 15 |
∴ Total profit = ₹ \[\boxed{61}\].
