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Solve the following problem : Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order AB. - Mathematics and Statistics

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Solve the following problem :

Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order AB.

Machines Jobs (Processing times in minutes)
I II III IV V VI VII
Machine A 12 6 5 11 5 7 6
Machine B 7 8 9 4 7 8 3

Determine the sequence for the jobs so as to minimize the processing time. Find the total elapsed time and the idle times for both the machines.

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Solution

Machines Jobs (Processing times in minutes)
I II III IV V VI VII
Machine A 12 6 5 11 5 7 6
Machine B 7 8 9 4 7 8 3

Observe that Min (A, B) = 3, corresponds to job VII on machine B.

∴ Job VII is placed last in sequence.

            VII

Then the problem reduces to

Machines Jobs (Processing times in minutes)
I II III IV V VI
Machine A 12 6 5 11 5 7
Machine B 7 8 9 4 7 8

Now, Min (A, B) = 4, corresponds to job IV on machine B

∴ Job IV is placed before VII in sequence.

          IV VII

Then the problem reduces to

Machines Jobs (Processing times in minutes)
I II III V VI
Machine A 12 6 5 5 7
Machine B 7 8 9 7 8

Now, Min (A, B) = 5, corresponds to job III and V on machine A.

∴ Job III and V is placed either first or second in sequence.

III V       IV VII
OR
V III       IV VII

Then the problem reduces to

Machines Jobs (Processing times in minutes)
I II VI
Machine A 12 6 7
Machine B 7 8 8

Now, Min (A, B) = 6, corresponds to job II on machine A

∴ Job II is placed on third place in sequence.

III V II     IV VII
OR
V III II     IV VII

Then the problem reduces to

Machines Jobs (Processing times in minutes)
I VI
Machine A 12 7
Machine B 7 8

Now, Min (A, B) = 7, corresponds to job I on machine B and VI on machine A.

∴ Job I is placed before IV and job VI on remaining in sequence.

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

We take the optimal sequence as,

III V II VI I IV VII

Total elapsed time

Job Machine A Machine B
In Out In Out
III (5, 9) 0 5 5 14
V (5, 7) 5 10 14 21
II (6, 8) 10 16 21 29
VI (7, 8) 16 23 29 37
I (12, 7) 23 35 37 44
IV (11, 4) 35 46 46 50
VII (6, 3) 46 52 52 55

∴ Total elapsed time = 55 mins

Idle time for machine A = 55 – 52 = 3 mins

Idle time for machine B = 5 + 2 + 2 = 9 mins.

Concept: Types of Sequencing Problem
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