A 2 L Heuristic Algorithm for Scheduling Tasks on Non-Identical Machines

A2L Heuristic Algorithm for Scheduling Tasks on Non-Identical

Machines

Alexander C. Tubbs Alison L. Steppig Laura A. Hemker

Southern Illinois University Edwardsville

Introduction

The A2L heuristic algorithm provides an efficient solution for scheduling tasks on non-identical machines without preemption. Specifically, the objective of this algorithm’s performance measure is to minimize the average flow time of a given set of processing times. Shortest processing time (SPT) and a form of weighted SPT contribute to the optimal solution. The step-by-step procedure provides a straightforward method for application.

Overview

Step 1: List all the minimum processing times for each job j. For n jobs having a single SPT on a single machine, assign to the specified machine i.

Step 2: Create a Gantt chart with these assigned jobs. (If all jobs are assigned, continue to Step 12)

Step 3: List all machine-job, ij, combinations with their given processing times, Pj.

Step 4: Rank all the combinations by Pj in ascending order.

Overview, continued

Step 5: Create a system of rounds 1. Select the jobs that appear in the first occurrence of each

machine, label as Round 12. If not all Jobs appear, select the jobs that appear in the

second occurrence of each machine and label as Round 2.

3. Continue with Rounds until all jobs appear.4. Select a final round of jobs after all the jobs appear.

Step 6: List all possible combinations found in the rounds for the jobs that were not assigned in Step 1.

Overview, continued

Step 7: Identify the shortest processing time, Pj, for each unassigned job.

Step 8: Starting with the job having the largest SPT, evaluate the Fj of all possible combinations for that job. Assign job to the machine having the shortest Fj. If a tie occurs when evaluating Fj, do not assign job and continue with next largest SPT job as described in the next step. Update Gantt chart.

Step 9: Repeat Step 8, selecting the job having the next highest SPT value, and continue until all un-assigned jobs are evaluated.

Overview, continued

Step 10: Re-Evaluate those jobs that had a tie within Fj since new jobs have been assigned. Assign these jobs if ties are broken, for those jobs that ties did not break, create a multi-solution set.

Step 11: Create a complete Gantt chart containing all final selections in the order in which they were assigned.

Step 12: In the Gantt chart, if a machine is not in use, go back and list all Pj for each job on the open machine. On the Gantt chart, select those jobs running sequentially on a machine and note their Cj for that machine. If Pj < Cj, select the job that has the largest difference between the values and re-assign to the open machine.

Overview, continued

Step 13: Update complete Gantt chart.

Step 14: Arrange the machines in SPT order on the Gantt chart if multiple jobs are to run on the same machine.

Step 15: Calculate the average flow-time of the final Gantt chart.

Project Problem, 4 jobs & 5 machines

  Job 1 Job 2 Job 3 Job 4

M 1 3 1 4 2

M 2 4 3 2 9

M 3 6 5 1 8

M 4 8 7 10 7

M 5 2 6 5 3

Step 1:

• List all the minimum processing times for each job j. For n jobs having a single SPT on a single machine, assign to the specified machine i.

SPT Assign to

Job1 2 M5

Job2 1 M1

Job3 1 M3

Job4 2 M1

Step 2:

Create a Gantt chart with these assigned jobs. Since all the jobs are assigned, skip to Step 12.

M5

J1, Pj=2

M4

M3

J3, Pj=1

M2

M1

J2, Pj=1 J4, Pj=2

Step 12:

In the Gantt chart, if a machine is not in use, go back and list all Pj for each job on the open machine. On the Gantt chart, select those jobs running sequentially on a machine and note their Cj for that machine. If Pj < Cj, select the job that has the largest difference between the values and re-assign to the open machine.

  J3 J2 J1 J4

M2 2 3 4 9

Pj Given Cj on Current

J4 9 > 3

J2 3 > 1

  J2 J4 J1 J3

M4 7 7 8 10

Pj Given Cj on Current

J4 7 > 3

J2 7 > 1

Step 13:

Update complete Gantt chart.

M5

J1

M4

M3

J3

M2

M1

J2 J4

Step 14:

Arrange the machines in SPT order on the Gantt chart if multiple jobs are to run on the same machine.

M5

J1

M4

M3

J3

M2

M1

J2 J4

Step 15:

Calculate the average flow-time of the final Gantt chart.

75.14

7

4

21)21(1

F

8 jobs & 5 machines

  Job 1 Job 2 Job 3 Job 4 Job 5 Job 6 Job 7 Job 8

M 1 2 2 6 1 1 3 3 10

M 2 5 9 9 8 10 3 7 3

M 3 5 9 4 10 9 10 1 5

M 4 7 5 9 2 6 8 7 8

M 5 6 1 4 4 6 4 7 7

Step 1:

List all the minimum processing times for each job j. For n jobs having a single SPT on a single machine, assign to the specified machine i.

SPT Assign to

Job1 2 M1

Job2 1 M5

Job4 1 M1

Job5 1 M1

Job7 1 M3

Job8 3 M2

Step 2:

Create a Gantt chart with these assigned jobs. (If all jobs are assigned, continue to Step 12)

M5J2

M4

M3J7

M2J8

M1J1 J4 J5

Step 3:

List all machine-job, ij, combinations with their given processing times, pj.

PjM1(J1) 2M1(J2) 2M1(J3) 6M1(J4) 1M1(J5) 1M1(J6) 3M1(J7) 3M1(J8) 10M2(J1) 5M2(J2) 9M2(J3) 9M2(J4) 8M2(J5) 10M2(J6) 3M2(J7) 7M2(J8) 3

M3(J1) 5M3(J2) 9M3(J3) 4M3(J4) 10M3(J5) 9M3(J6) 10M3(J7) 1M3(J8) 5M4(J1) 7M4(J2) 5M4(J3) 9M4(J4) 2M4(J5) 6M4(J6) 8M4(J7) 7M4(J8) 8

M5(J1) 6M5(J2) 1M5(J3) 4M5(J4) 4M5(J5) 6M5(J6) 4M5(J7) 7M5(J8) 7

Step 4:

Rank all the combinations by Pj in ascending order. PjM1(J4) 1M1(J5) 1M3(J7) 1M5(J2) 1M1(J1) 2M1(J2) 2M4(J4) 2M1(J6) 3M1(J7) 3M2(J6) 3M2(J8) 3M3(J3) 4M5(J3) 4M5(J4) 4M5(J6) 4M2(J1) 5M3(J1) 5M3(J8) 5M4(J2) 5

M1(J3) 6M4(J5) 6M5(J1) 6M5(J5) 6M2(J7) 7M4(J1) 7M4(J7) 7M5(J7) 7M5(J8) 7M2(J4) 8M4(J6) 8M4(J8) 8M2(J2) 9M2(J3) 9M3(J2) 9M3(J5) 9M4(J3) 9M1(J8) 10M2(J5) 10M3(J4) 10M3(J6) 10

Step 5:

Create a system of rounds.1. Select the jobs that appear in the first occurrence of each

machine, label as Round 1

2. If not all Jobs appear, select the jobs that appear in the second occurrence of each machine and label as Round 2.

3. Continue with Rounds until all jobs appear.

4. Select a final round of jobs after all the jobs appear.

Step 5 continued:

Pj

M1(J4) 1

M1(J5) 1

M3(J7) 1

M5(J2) 1

M1(J1) 2M1(J2) 2M4(J4) 2M1(J6) 3M1(J7) 3M2(J6) 3M2(J8) 3M3(J3) 4M5(J3) 4M5(J4) 4M5(J6) 4M2(J1) 5M3(J1) 5M3(J8) 5M4(J2) 5

M1(J3) 6M4(J5) 6M5(J1) 6M5(J5) 6M2(J7) 7M4(J1) 7M4(J7) 7M5(J7) 7M5(J8) 7M2(J4) 8M4(J6) 8M4(J8) 8M2(J2) 9M2(J3) 9M3(J2) 9M3(J5) 9M4(J3) 9M1(J8) 10M2(J5) 10M3(J4) 10M3(J6) 10

Jobs Idenitified

R1: 4, 6, 7, 2

R2: 5, 8, 3

R3: 1

FINAL

Step 6:

List all possible combinations found in the rounds for the jobs that were not assigned in Step 1.

Pj

M1(J4) 1

M1(J5) 1

M3(J7) 1

M5(J2) 1

M1(J1) 2

M1(J2) 2

M4(J4) 2

M2(J6) 3

M2(J8) 3

M3(J3) 4

M5(J3) 4

M5(J4) 4

M5(J6) 4

M2(J1) 5

M3(J1) 5

M3(J8) 5

M4(J2) 5

M4(J5) 6

M2(J7) 7

M4(J1) 7

Job1 Job2 Job3 Job4 Job5 Job6 Job7 Job8

ASSIGNED ASSIGNED M3(J3) ASSIGNED ASSIGNED M2(J6) ASSIGNED ASSIGNED

M5(J3) M5(J6)

Step 7:

Identify the shortest processing time, Pj, for each unassigned job.

Job3 Pj Job6 Pj

M3(J3) 4 M2(J6) 3

M5(J3) 4 M5(J6) 4

Step 8:

Starting with the job having the largest SPT, evaluate the Fj of all possible combinations for that job. Assign job to the machine having the shortest Fj. If a tie occurs when evaluating Fj, do not assign job and continue with next largest SPT job as described in the next step. Update Gantt chart.

Pj = 4

Tie: M3 and M5

M3:1 + 4 6

M5:1 + 4 6

Step 9:

Repeat Step 8, selecting the job having the next highest SPT value, and continue until all un-assigned jobs are evaluated.

Pj = 3

M2:3 + 3 9

M5:1 + 4 6

Assign J6 on M5

M5 J2 J6

M4

M3 J7

M2 J8

M1 J1 J4 J5

Step 10:

Re-Evaluate those jobs that had a tie within Fj since new jobs have been assigned. Assign these jobs if ties are broken, for those jobs that ties did not break, create a multi-solution set.

M3:1 + 4 6

M5:1 + 4 + 4 15

Likewise, J3 will then be on M3

Step 11:

Create a complete Gantt chart containing all final selections in the order in which they were assigned.

M5J2 J6

M4

M3J7 J3

M2J8

M1J1 J4 J5

Step 12:

In the Gantt chart, if a machine is not in use, go back and list all Pj for each job on the open machine. In the Gantt chart, if a machine is not in use, go back and list all Pj for each job on the open machine. If Pj < Cj, select the job that has the largest difference between the values and re-assign to the open machine.

  Job 1 Job 2 Job 3 Job 4 Job 5 Job 6 Job 7 Job 8

M 4 7 5 9 2 6 8 7 8

Pj Given Cj on CurrentJ1 7 > 2J2 5 > 1J3 9 > 4J4 2 < 3J5 6 > 4J6 8 > 5J7 7 > 1

Step 13:

Update complete Gantt chart.

M5J2 J6

M4J4

M3J7 J3

M2J8

M1J1 J5

Step 14:

Arrange the machines in SPT order on the Gantt chart if multiple jobs are to run on the same machine.

M5J2 J6

M4J4

M3J7 J3

M2J8

M1J5 J1

Step 15:

Calculate the average flow-time of the final Gantt chart.

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Conclusions

The A2L heuristic algorithm proved to be capable of minimizing average flow-time for every scheduling problem of non-identical machines without preemption that we applied it to. We believe that it could be efficiently applied to larger-scale problems if made into a computer program.

References

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New Jersey Baker, K. R. (1974). Introduction to Sequencing and

Scheduling, John Wiley & Sons, 0471045551,New York

Randhawa, S. U. & Kuo, C. H. (1997). EvaluatingScheduling Heuristics for Non-Identical

Parallel Processors. International Journal of Production Research, Vol. 35, No. 4, (April 1997), pp. 969 981, 0020-7543