Subordinates differ-in efficiency and tasks differ in their intrinsic difficulty. His estimate of time each man would take to perform each task is given in the matrix below :
TASKS SUBORDINATE 1 2 3 4
A
8
26
17
11
B
13
28
14
26
C
38
19
18
15
D
19
26
24
10
How should the tasks be allotted, one to a man, so as to minimize the total manhours
To determine how the tasks should be allotted to minimize the total manhours, we can use the Hungarian Algorithm. This is a combinatorial optimization algorithm specifically used to solve assignment problems like this, where the goal is to assign tasks to workers in such a way that the total cost (in this case, the total manhours) is minimized.
We are given a matrix representing the time (in manhours) each subordinate would take to perform each task:
| TASKS/SUBORDINATES | 1 | 2 | 3 | 4 |
|-------------------|----|----|----|----|
| A | 8 | 26 | 17 | 11 |
| B | 13 | 28 | 14 | 26 |
| C | 38 | 19 | 18 | 15 |
| D | 19 | 26 | 24 | 10 |
Each row represents a task (A, B, C, D) and each column represents a subordinate (1, 2, 3, 4). The value at each intersection represents the time required for that subordinate to complete the corresponding task.
### Step 1: Subtract Row Minimums
To begin, subtract the smallest value in each row from all the entries in that row.
- For Task A: The minimum is 8. Subtract 8 from all elements in row A:
(8-8, 26-8, 17-8, 11-8) = (0, 18, 9, 3)
- For Task B: The minimum is 13. Subtract 13 from all elements in row B:
(13-13, 28-13, 14-13, 26-13) = (0, 15, 1, 13)
- For Task C: The minimum is 15. Subtract 15 from all elements in row C:
(38-15, 19-15, 18-15, 15-15) = (23, 4, 3, 0)
- For Task D: The minimum is 10. Subtract 10 from all elements in row D:
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