Solve the following problem : A foreman wants to process 4 different jobs on three machines: a shaping machine, a drilling machine and a tapping machine, the sequence of operations being shaping-dri - Mathematics and Statistics

Sum

Solve the following problem :

A foreman wants to process 4 different jobs on three machines: a shaping machine, a drilling machine and a tapping machine, the sequence of operations being shaping-drilling-tapping. Decide the optimal sequence for the four jobs to minimize the total elapsed time. Also find the total elapsed time and the idle time for every machine.

 Job Shaping (Minutes) Drilling (Minutes) Trapping (Minutes) 1 13 3 18 2 18 8 4 3 8 6 13 4 23 6 8

Solution

 job 1 2 3 4 Shaping (Minutes) 13 18 8 23 Drilling (Minutes) 3 8 6 6 Trapping (Minute) 18 4 13 8

Here, Min Shaping = 8, Max Drilling = 8, Min Trapping = 4.
Since Min Shaping ≥ Max Drilling is satisfied, the problem can be converted into a two machine problem.
Let G and H be two fictitious machines such that G = Shaping + Drilling and H = Drilling + Trapping
Then the problem can be written as

 Job 1 2 3 4 Machine G 16 26 14 29 Machine H 21 12 19 14

Observe that Min (G, H) = 12, corresponds to Job 2 on machine H.
∴ Job 2 is placed last in sequence.

 2

Then the problem reduces to

 Job 1 3 4 Machine G 16 14 29 Machine H 21 19 14

Now, Min (G, H) = 14, corresponds to Job 3 on machine G and Job 4 on machine H.
∴ Job 3 is placed first in the sequence, Job 4 is placed second last in the sequence and remaining Job 1 is placed second in the sequence.
∴ Optimal sequence is

 3 1 4 2

Total elapsed time

 Job Shaping Drilling Trapping In Out In Out In Out 3 (8, 6, 13) 0 8 8 14 14 27 1(13, 3, 18) 8 21 21 24 27 45 4 (23, 6, 8) 21 44 44 50 50 58 2 (18, 8, 4) 44 62 62 70 70 74

∴ Total elapsed time = 74 mins
Idle time for Shaping = 74 – 62 = 12 mins
Idle time for Drilling = (74 – 70) + 8 + 7 + 20 + 12 = 51 mins
Idle time for Trapping = 14 + 5 +12 = 31 mins.

Concept: Types of Sequencing Problem
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Chapter 7: Assignment Problem and Sequencing - Part II [Page 130]

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Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 7 Assignment Problem and Sequencing
Part II | Q 5 | Page 130
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