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.• 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!)m possible schedules.

– For example, for n=5 and m=5, there are 24,833x1010 , possible schedules.

2 jobs-2 machines ExampleMachine 1 Machine 2

Job I 4 1Job J 1 4

Machine 1Machine 1Machine 2Machine 2

I JI J

4 5 9

Machine 1Machine 1Machine 2Machine 2

IJIJ

1 5 6

Machine 1Machine 1Machine 2Machine 2

65 10

IJI J

1

Machine 1Machine 1Machine 2Machine 2

I JIJ

4 5 9 10

Tota

l Flo

w T

ime

(or M

akes

pan)

Mea

n Fl

ow T

ime

Mea

n Id

le T

ime

9

6

10

10

(5+9)/2= 7

(5+6)/2=5.5

(6+10)/2=8

(10+9)/2=9.5

(4+4)/2= 4

(1+1)/2=1

(5+5)/2=5

(5+5)/2=5

Example 8.5Job Machine A Machine B

1 5 2

2 1 6

3 9 7

4 3 8

5 10 4

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)– min Ai ≥ max Bi or min Ci ≥ max Bi

– Define Ai‘ = Ai + Bi and define Bi‘ = Bi + Ci

Example 8.5Job Machine A Machine B Machine C

1 4 5 8

2 9 6 10

3 8 2 6

4 6 3 7

5 5 4 11

What is the optimal job sequence ?

1 4 5 2 3

min Ai = 4max Bi = 6min Ci = 6

Check the conditions min Ai ≥ max Bi or min Ci ≥ max Bi Required condition is satisfied.

Job Machine A Machine B

1 9 13

2 15 16

3 10 8

4 9 10

5 9 15

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.

1 2 3 4 5 6 7 8 9 10 11 12

AA

BB

CC13 14 15

Job 11 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

5

6

7

8

9

10

Job 2

Example 8.7Operation Time

A 3

B 4

C 5

Operation Time

A 3

B 4

C 5

Job 1 Job 2

AA

BB

CC

A1 A2B1 B2

C1 C2