3. Example-3
Find solution of Processing 2 Jobs Through m Machines Problem
| Job-1 | A | B | C | D | E | | Machine-1 | 3 | 4 | 2 | 6 | 2 | | Job-2 | B | C | A | D | E | | Machine-2 | 5 | 4 | 3 | 2 | 6 |
Solution:
1. We are given the job sequences and processing time of 2 jobs at 5 machines. We follow the graphical method to find the minimum total elapsed time from starting the first job at the first machine to completion of the second job at the last machine.
2. We first represent the processing time of job 1 on different machines, i.e, `3,4,2,6,2` along the x-axis and the processing time of job 2, i.e., `5,4,3,2,6` along the y-axis.
We draw the first vertical line at `3` hrs, the second at `3 + 4=7` hrs, the third at `7 + 2=9` hrs, and so on.
Similarly, we draw the horizontal lines at `5` hrs, the second at `5 + 4=9` hrs, the third at `9 + 3=12` hrs, and so on.
3. We draw the rectangular blocks by pairing the same machines as shown in the Figure
(1) For machine A, 0 to 3 hours on x-axis and 9 to 12 hours on y-axis
(2) For machine B, 3 to 7 hours on x-axis and 0 to 5 hours on y-axis
(3) For machine C, 7 to 9 hours on x-axis and 5 to 9 hours on y-axis
(4) For machine D, 9 to 15 hours on x-axis and 12 to 14 hours on y-axis
(5) For machine E, 15 to 17 hours on x-axis and 14 to 20 hours on y-axis
4. Avoiding the rectangular blocks, draw the line starting from origin O to the end point, moving horizontally, vertically and diagonally along a line which makes an angle `45deg` with the horizontal line.
Moving horizontally along this line indicates that the job 1 is under process while the job 2 is idle.
Similarly moving vertically along this line indicates that the job 2 is under process while the job 1 is idle.
The diagonal movement along this line shows that both jobs are under process. Since simultaneous processing of both jobs on a machine is not possible, therefore a diagonal movement through rectangle areas is not allowed.
(1) we move diagonally upto (3,3), which means both the jobs 1 and 2 are being processed simultaneously
(2) we move vertically upto (3,5), which means Job 2 is under process amd Job 1 is idle for 2 hrs
(3) we move diagonally upto (9,11), which means both the jobs 1 and 2 are being processed simultaneously
(4) we move vertically upto (9,14), which means Job 2 is under process amd Job 1 is idle for 3 hrs
(5) we move diagonally upto (15,20), which means both the jobs 1 and 2 are being processed simultaneously
(6) we move horizontally upto (17,20), which means Job 1 is under process and Job 2 is idle for 2 hrs
5. An optimum path minimizes the idle time for both the jobs.
Idle time of job 1 `=2+3=5` hours.
Idle time of job 2 `=2` hours.
6. The elapsed time is obtained by adding the idle time for either job to the processing time for that job.
Elapsed time, job 1 = processing time of job 1 + idle time of job 1 `=(3+4+2+6+2) + (5) = 17 + 5=22` hours.
Elapsed time, job 2 = processing time of job 2 + idle time of job 2 `=(5+4+3+2+6) + (2) = 20 + 2=22` hours.
This material is intended as a summary. Use your textbook for detail explanation.
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