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Question Bank [2022] — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Question Bank [2022]3 Marks
Question
Find the optimal sequence that minimizes total time required to complete the following jobs in the order ABC. The processing times are given in hours.
Jobs I II III IV V VI VII
Machine A 6 7 5 11 6 7 12
Machine B 4 3 2 5 1 5 3
Machine C 3 8 7 4 9 8 7
✓

Answer

Job I II III IV V VI VII
Machine A 6 7 5 11 6 7 12
Machine B 4 3 2 5 1 5 3
Machine C 3 8 7 4 9 8 7

Here min A = 5, max B = 5, min C = 3

Since min A ≥ max B is satisfied, the problem can be converted into a two machine problem.

Let G and H be two fictitious machines such that G = A + B and H = B + C

Then the problem can be written as

Job I II III IV V VI VII
Machine G 10 10 7 16 7 12 15
Machine H 7 11 9 9 10 13 10

Observe that Min (G, H) = 7, corresponds to job I on machine H, job III on machine G and job V on machine G.

∴ Job I is placed last in sequence, Job III and V are placed either first or second in sequence.

V III I
OR
III V I

Then the problem reduces to

Job II IV VI VII
Machine G 10 16 12 15
Machine H 11 9 13 10

Now, Min (G, H) = 9, corresponds to Job IV on machine H.

∴ Job IV is placed second last in the sequence.

V III IV I
OR
III V IV I

Then the problem reduces to

Job II VI VII
Machine G 10 12 15
Machine H 11 13 10

Now, Min (G, H) = 10, corresponds to Job II on machine G and Job VII on machine H.

∴ Job II is placed third in the sequence, Job VII is placed third last in the sequence and remaining Job VI is placed fourth in the sequence.

V III II VI VII IV I
OR
III V II VI VII IV I

∴ We consider the optimal sequence as V – III – II – VI – VII – IV – I

Total elapsed time

Job Machine
A		 Machine
B		 Machine
C
In Out In Out In Out
V (6, 1, 9) 0 6 6 7 7 16
III (5, 2, 7) 6 11 11 13 16 23
II (7, 3, 8) 11 18 18 21 23 31
VI (7, 5, 8) 18 25 25 30 31 39
VII (12, 3, 7) 25 37 37 40 40 47
IV (11, 5, 4) 37 48 48 53 53 57
I (6, 4, 3) 48 54 54 58 58 61

Total elapsed time = 61 hrs

Idle time for Machine A = 61 – 54 = 7 hrs

Idle time for Machine B = (61 – 58) + 6 + 4 + 5 + 4 + 7 + 8 + 1 = 38 hrs

Idle time for Machine C = 7 + 1 + 6 + 1 = 15 hrs.

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