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Question
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.
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Solution
| 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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