Sequencing algorithms for multiple machines

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Sequencing algorithms for multiple machines. Operations scheduling , Nahmias. Sequencing Algorithms for multiple machines. Assume that n jobs are to be processed through m machines . The number of possible schedule is staggering, even for moderate values of both n and m .

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Sequencingalgorithmsformultiplemachines

Operationsscheduling, Nahmias

Sequencing Algorithms for multiple machines
• Assume that njobs are to be processed through mmachines.
• The number of possible schedule is staggering, even for moderate values of both n and m.
• For each machine there are n! different ordering of the jobs.
• If the jobs may be processed on the machines in any order, it follows that there are (n!)mpossible schedules.
• For example, for n=5 and m=5, there are 24,833x1010 , possible schedules.
2 jobs-2 machines Example

Mean Idle Time

Mean Flow Time

Total Flow Time

(or Makespan)

Machine 1

I

J

Machine 2

I

J

9

(5+9)/2= 7

(4+4)/2= 4

4

5

9

Machine 1

J

I

J

I

Machine 2

6

(5+6)/2=5.5

(1+1)/2=1

1

5

6

Machine 1

I

J

Machine 2

I

J

1

5

6

10

10

(6+10)/2=8

(5+5)/2=5

Machine 1

I

J

Machine 2

J

I

10

(10+9)/2=9.5

(5+5)/2=5

4

5

9

10

Example 8.5

What is the optimal job sequence ?

2

4

3

5

1

Extension to three machines
• The problem of scheduling jobs on three machines is considerably more complex.
• The three machine problem can be reduced to a two machine problem if the following condition is satisfied:
• Label the machines A, B and C
• Ai = Processing time of job i on machine A. (Bi ,Ci are defined as similar)
• minAi≥ max Bior min Ci≥ max Bi
• Define Ai‘ = Ai+ Biand define Bi‘ = Bi+ Ci
Example 8.5

minAi= 4

max Bi = 6

min Ci= 6

Check the conditions

minAi≥ max Bior min Ci≥ max Bi

Required condition is satisfied.

What is the optimal job sequence ?

1

4

5

2

3

The two-shop Flow shop problem
• Assume that two jobs are to be processed through m machines.
• Each job must be processed by the machines in a particular order, but sequences for the two jobs need not to be the same.
• A graphical procedure for solving this problem is developed by Aker (1954)
Aker’s Algorithm
• Draw a Cartesian coordinate system.
• Processing times for first job on the horizontal axis
• Processing times for second job on the vertical axis
• On each axis, mark off the operation times in the order in which the operations must be performed for that job.
• Block out areas corresponding to each machine at the intersection of the intervals marked for that machine on the two axis.
• Determine a path from origin to the end of final block that does not intersect any of the blocks and that minimizes the vertical movement.
Example 8.7

Job 1

Job 2

10

C

9

8

7

B

6

5

Job 2

4

3

2

A

1

1

2

3

4

5

6

7

8

9

10

11

12

Job 1

A1

A2

A

B1

B2

B

C

C1

C2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15