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# Sequencing problems in Operations Research

Sequencing of jobs on two and three machines...

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### Sequencing problems in Operations Research

1. 1. Sequencing Problems Abu Bashar
2. 2. Sequencing??? • It is the selection of an appropriate order in which a number of jobs (Operations) can be assigned to to a finite number of service facilities (Machines or equipments) so as to optimize the outputs in terms of time, cost or profit.
3. 3. Common Sequencing Rules • FCFS. First Come First Served. Jobs processed in the order they come to the shop. • SPT. Shortest Processing Time. Jobs with the shortest processing time are scheduled first. • EDD. Earliest Due Date. Jobs are sequenced according to their due dates. • CR. Critical Ratio. Compute the ratio of processing time of the job and remaining time until the due date. Schedule the job with the largest CR value next.
4. 4. Two Work Center Sequencing • Johnson’s Rule: technique for minimizing completion time for a group of jobs to be processed on two machines or at two work centers. • Minimizes total idle time • Several conditions must be satisfied
5. 5. Johnson’s Rule Conditions • Job time must be known and constant • Job times must be independent of sequence • Jobs must follow same two-step sequence • Job priorities cannot be used • All units must be completed at the first work center before moving to the second
6. 6. Sequencing Problems • One machine, many jobs • Two machines, many jobs • Three machines, many jobs • Many machines, many jobs
7. 7. One machine, many jobs • Total time is independent of sequence • Sequencing minimizes idle time
8. 8. Two machines, many jobs • All Jobs follow same sequence – Johnson’s Rule • Jobs have different sequence – Jackson’s Rule • Johnson’s 3 machine rule
9. 9. Johnson’s Rule • Johnson’s rule: A procedure that minimizes idle time when scheduling a group of jobs on two workstations. • Step 1. Find the shortest processing time among the jobs not yet scheduled. If two or more jobs are tied, choose one job arbitrarily. • Step 2. If the shortest processing time is on workstation 1, schedule the corresponding job as early as possible. If the shortest processing time is on workstation 2, schedule the corresponding job as late as possible. • Step 3. Eliminate the last job scheduled from further consideration. Repeat steps 1 and 2 until all jobs have been scheduled.
10. 10. Eliminate M3 from consideration. The next shortest time is M2 at Workstation 1, so schedule M2 first. Eliminate M5 from consideration. The next shortest time is M1 at workstation #1, so schedule M1 next. Eliminate M1 and the only job remaining to be scheduled is M4. Example 16.5 Johnson’s RuleJohnson’s Rule at theat the Morris Machine Co.Morris Machine Co. Time (hr)Time (hr) MotorMotor Workstation 1Workstation 1 Workstation 2Workstation 2 M1M1 1212 2222 M2M2 44 55 M3M3 55 33 M4M4 1515 1616 M5M5 1010 88 Sequence =Sequence = M1M1M2M2 M3M3M4M4 M5M5 Shortest time is 3 hours at workstation 2, so schedule job M3 last. Eliminate M2 from consideration. The next shortest time is M5 at workstation #2, so schedule M5 next to last.
11. 11. Workstation M2 (4) M1 (12) M4 (15) M5 (10) M3 (5) Idle—available for further work 0 5 10 15 20 25 30 Day 35 40 45 50 55 60 65 Idle2 M2 (5) M1 (22) M4 (16) M5 (8) M3 (3)Idle 1 Gantt Chart for the Morris Machine Company Repair Schedule The schedule minimizes the idle time of workstation 2 and gives the fastest repair time for all five motors. No other sequence will produce a lower makespan. Example 16.5 Johnson’s RuleJohnson’s Rule at theat the Morris Machine Co.Morris Machine Co.
12. 12. Problem Job 1 2 3 4 5 6 7 8 A 4 5 8 10 6 5 10 6 B 8 10 2 4 5 7 6 6
13. 13. Example
14. 14. Johnson’s 3 Machine Rule • Domination – Largest on B is not greater than smallest on A or C. Domination guarantees optimality • Conversion – Convert into a 2 M/C problem as M1 = A + B, M2 = B + C • Find optimal sequence for this 2 machine problem • Schedule on 3 machines based on the sequence above
15. 15. Job 1 2 3 4 5 6 A 8 6 10 7 9 7 B 7 4 6 5 5 4 C 13 12 11 12 13 10 M1 15 10 16 12 14 11 M2 20 16 17 17 18 14 Classwork
16. 16. j 1 2 3 4 5 tj1 8 11 7 6 9 tj2 3 2 5 7 11 tj3 6 5 7 13 10 Classwork
17. 17. Thank You Very Much