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Question
For the following problem, find the sequence that minimizes total elapsed time (in hours) required to complete jobs on two machines M1 and M2 in the order M1-M2 Also find the minimum elapsed time T.
|
Jobs |
A |
B |
C |
D |
E |
|
Machine M1 |
5 |
1 |
9 |
3 |
10 |
|
Machine M2 |
2 |
6 |
7 |
8 |
4 |
Solution
Given :
|
Jobs |
A |
B |
C |
D |
E |
|
Machine M1 |
5 |
1 |
9 |
3 |
10 |
|
Machine M2 |
2 |
6 |
7 |
8 |
4 |
finding the optimal sequence :
The minimum processing time on two machines is 1 which correspond to task B on machine M1. This shows that task B will be preceding first. After assigning task B, we are left with 4 tasks on two machines
|
Jobs |
A |
C |
D |
E |
|
Machine M1 |
5 |
9 |
3 |
10 |
|
Machine M2 |
2 |
7 |
8 |
4 |
| B |
Minimum processing time in this reduced problem is 2 which correspond to job A on machine M2 therefore task A will be processed in the last.
| B | A |
The problem now reduces to following 3 tasks on two machines with processing time as follows:
|
Jobs |
C |
D |
E |
|
Machine M1 |
9 |
3 |
10 |
|
Machine M2 |
7 |
8 |
4 |
Minimum processing time in this reduced problem is 3 which correspond to job D on machine M1 therefore task D will be processed in the next to B.
| B | D | A |
The problem now reduces to following 2 tasks on two machines with processing time as follows:
|
Jobs |
C |
E |
|
Machine M1 |
9 |
10 |
|
Machine M2 |
7 |
4 |
Minimum processing time in this reduced problem is 4 which correspond to job E on machine M2 therefore task E will be processed in the next to last.
| B | D | C | E | A |
Further, it is also possible to calculate the minimum elapsed time corresponding to the optimal sequencing B→D→C→E→A
|
Job Sequence |
Machine M1 |
Machine M2 |
||
|
Time In |
Time Out |
Time In |
Time Out |
|
| B |
0 |
1 |
1 |
7 |
| D |
1 |
4 |
7 |
15 |
| C | 4 | 13 | 15 | 22 |
| E | 13 | 23 | 23 | 27 |
| A | 23 | 28 | 28 | 30 |
Hence the total elapsed time for this proposed sequence staring from job B to completion of job A is 30 hours .During this time machine M1 remains idle for 2 hours and the machine M2 remains idle for 3 hours
Minimum time elapse : 30 Hours
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