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Question
Three different aeroplanes are to be assigned to carry three cargo consignments with a view to maximize profit. The profit matrix (in lakhs of ₹) is as follows :
| Aeroplanes | Cargo consignments | ||
| C1 | C2 | C3 | |
| A1 | 1 | 4 | 5 |
| A2 | 2 | 3 | 3 |
| A3 | 3 | 1 | 2 |
How should the cargo consignments be assigned to the aeroplanes to maximize the profit?
Solution
Given problem is of maximization Given matrix is
| C1 | C2 | C3 | |
| A1 | 1 | 4 | 5 |
| A2 | 2 | 3 | 3 |
| A3 | 3 | 1 | 2 |
Above problem can be converted into minimization problem by subtracting each of the elements in the table from the largest element (5 here)
Maximum from the above matrix is 5.
∴ Matrix for minimization problem is
| C1 | C2 | C3 | |
| A1 | 4 | 1 | 0 |
| A2 | 3 | 2 | 2 |
| A3 | 2 | 4 | 3 |
Subtracting row minimum from each row
| C1 | C2 | C3 | |
| A1 | 4 | 1 | 0 |
| A2 | 1 | 0 | 0 |
| A3 | 0 | 2 | 1 |
Minimum in each column is zero
∴ There is no change in the above matrix.
∴ Allocation is
∴ Assignment is A1 → C3 , A2 → C2, A3 → C1
∴ Maximum profit is = 5+3+3
= 11 lakhs
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