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प्रश्न
Solve the following problem :
A readymade garments manufacturer has to process 7 items through two stages of production, namely cutting and sewing. The time taken in hours for each of these items in different stages are given below:
| Items | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Time for Cutting | 5 | 7 | 3 | 4 | 6 | 7 | 12 |
| Time for Sewing | 2 | 6 | 7 | 5 | 9 | 5 | 8 |
Find the sequence in which these items are to be processed through these stages so as to minimize the total processing time. Also find the idle time of each machine.
उत्तर
| Items | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Time for Cutting | 5 | 7 | 3 | 4 | 6 | 7 | 12 |
| Time for Sewing | 2 | 6 | 7 | 5 | 9 | 5 | 8 |
Observe that Min (cutting, sewing) = 2, corresponds to item 1 on sewing.
∴ Item 1 is placed last in sequence.
| 1 |
Then the problem reduces to
| Items | 2 | 3 | 4 | 5 | 6 | 7 |
| Time for Cutting | 7 | 3 | 4 | 6 | 7 | 12 |
| Time for Sewing | 6 | 7 | 5 | 9 | 5 | 8 |
Now, Min (cutting, sewing) = 3, corresponds to item 3 on cutting.
∴ Item 3 is placed first in sequence.
| 3 | 1 |
Then the problem reduces to
| Items | 2 | 4 | 5 | 6 | 7 |
| Time for Cutting | 7 | 4 | 6 | 7 | 12 |
| Time for Sewing | 6 | 5 | 9 | 5 | 8 |
Now, Min (cutting, sewing) = 4, corresponds to item 4 on cutting.
∴ Item 4 is placed after 3 in sequence.
| 3 | 4 | 1 |
Item 4 is placed after 3 in sequence.
| Items | 2 | 5 | 6 | 7 |
| Time for Cutting | 7 | 6 | 7 | 12 |
| Time for Sewing | 6 | 9 | 5 | 8 |
Now, Min (cutting, sewing) = 5, corresponds to item 6 on sewing
∴ Item 6 is placed before 1 in sequence.
| 3 | 4 | 6 | 1 |
Then the problem reduces to
| Items | 2 | 5 | 7 |
| Time for Cutting | 7 | 6 | 12 |
| Time for Sewing | 6 | 9 | 8 |
Now, Min (cutting, sewing) = 6, corresponds to item 2 on sewing and item 5 on cutting.
∴ Item 2 is placed before 6 and item 5 is placed after 4 and item 7 on remaining in sequence.
∴ Optimal sequence is
| 3 | 4 | 5 | 7 | 2 | 6 | 1 |
Total elapsed time
| Item | Cutting | Sewing | ||
| In | Out | In | Out | |
| 3 (3, 7) | 0 | 3 | 3 | 10 |
| 4 (4, 5) | 3 | 7 | 10 | 15 |
| 5 (6, 9) | 7 | 13 | 15 | 24 |
| 7 (12, 8) | 13 | 25 | 25 | 33 |
| 2 (7, 6) | 25 | 32 | 33 | 39 |
| 6 (7, 5) | 32 | 39 | 39 | 44 |
| 1 (5, 2) | 39 | 44 | 44 | 46 |
∴ Total elapsed time = 46 hrs
Idle time for cutting = 46 44 = 2 hrs
Idle time for sewing = 3 + 1 = 4 hrs.
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