Production SchedulingP.C. Chang, IEM, YZU. 1 Parallel Machine Scheduling Baker p.114 Minimizing Makespan Problems N / 3 / Cmax meanT meanF Multiple Bin

Production Scheduling P.C. Chang, IEM, YZU.1

Parallel Machine SchedulingBaker p.114

Minimizing Makespan Problems

N / 3 / Cmax meanT meanF

Multiple Bin PackingKnapsack PackingVehicle Packing - assignment

M1

M2

M3


Lower Bound Approach

]t[max,tm

1MaxM j

j

n

1jj

*

For Example :

j 1 2 3 4 5 6 7 8

tj 1 2 3 4 5 6 7 8

123

821M*


McNaughton’s Algorithm

12M*

Baker p.115

1. Select some job to begin on machine 1 at time zero

2. Choose any unscheduled job and schedule it as early as possible on the same machine. Repeat this step in until the machine is occupied beyond time M* (or until jobs are scheduled.)

3. Reassign the processing scheduled beyond M* to the next machine instead, starting at time zero. Return to step 2.

1

87

6

432M1

M2

M3

For example:

5

5 7

12

Preemptive


Min M

If job preemption is prohibited, the problem of min makespan is somewhat more difficult. No direct algorithm has been developed for calculating the optimal makespan or for constructing optimal schedule.

LPT Heuristic

A simple effective heuristic procedure for constructing a schedule involves the use of LPT scheduling as a dispatching mechanism.

1. Construct an LPT ordering of the jobs.

2. Schedule the jobs in order, each time assigning a job to the machine with the least amount of processing already assigned.


LPT

LPT (Largest Processing Time) Rule

1

8

7

6 5

4

3 2M1

M2

M3

Sequence = 8-7-6-5-4-3-2-1

13

If SPT

1

8

7

6

5

4

3

2

M1

M2

M3 15


Integer Programming Formulation

integerand0X

11

mj10Xt-Y

S.T.

YMin

,0

,1

ij

1

n

1iiji

m

jij

ij

niX

otherwise

jmachinetoassignedijobX

Each job can only be processed by one machine


Hw.

8/3/Cmax

8 jobs to be scheduled on a machine cell will 3 machines.

Use LPT & IP to solve.

Software:LINGO

i 1 2 3 4 5 6 7 8

ti 2 5 3 8 7 6 4 6


…

Minimizing Mean Flow Timeti[j] : processing time of the j-th job in sequence on the i-th

machine.

Fi[j] : flow time of the j-th job in sequence on the machine.

ni : number of jobs processed by the i-th machine

Baker p.118

m

1i

n

1j]j[ii

m

1i

n

1j]j[i

ii

t)1jn(n

1F

n

1F

1 2 3 nMachine i

iji tn

iji t)1n( iji t)2n( ijt1

1111

222

33

4


Min Mean F with Parallel Identical Machine

1. Construct an SPT ordering of the jobs.

2. To the machine with the least amount of processing already allocated, assign the next job on the ordered list of jobs. (Break ties arbitrarily.) Repeat until all jobs assigned.

]n[B2]1[B)1m(m2

1)m(B

)nm1(machinesmforboundloweramachineoneonjobneomachinenforFofvalueimalminthe)n(B

machinesinglemachineoneforFofvalueimalminthe)1(B

F

w

w

w


Ex.

j 1 2 3 4 5 6 7 8

tj 1 2 3 4 5 6 7 8

5.6

5.421526

1

)8(B2)1(B)13(32

1)3(B

5.48/36)8(B

158120)1(B 8

1854637281

- Lower Bound

Find out LB

ii

m

i

PMax

nBBmm

B

m

PB

2112

1

1


Hm & H1

HmStep 1. Form a priority list of all unscheduled jobs according to some rule,

R.Step 2. Assign the first m jobs on the list to different machines. Repeat

Step 2 until all jobs are scheduled and then go to Step 3. Step 3. Apply WSPT sequencing to each machine.

H1Step 1. Form a priority list of all unscheduled jobs according to some rule,

R.Step 2. Assign the first job on the list to the machine with the least amoun

t of processing allocated. Repeat until all jobs have been assigned. Then go to Step 3.

Step 3. Apply WSPT sequencing to each machine.


Ex.j 1 2 3 4 5 6 7 8 9 10

tj 5 21 16 6 26 19 50 41 32 22

wj 4 5 3 1 4 2 5 4 3 2

tj/wj 1.2 4.2 5.3 6.0 6.5 9.5 10.0 10.2 10.7 11.0

Initial job list {10-9-8-…-2-1} WSPT from large to smallMethod 1.

Processing commitments Job(wi) machine

1.(0,0,0,0,0) 10(2) 1

9(3) 2

8(4) 3

7(5) 4

6(2) 5Processing commitments Job(wi) machine

2.(22,32,41,50,19) 5(4) 1

4(1) 4

3(3) 3

2(5) 5

1(4) 2Large weight matches small one

Arrange 5 jobs each time


Ex.

Machine Sequence

1 5-10

2 1-9

3 3-8

4 4-7

5 2-6

5M1

M2

M3

M4

M5

10

1 9

3 8

4 7

2 6 67.32Fw

26 48

566

5716

375

4021


Ex.

Method 2.

Processing commitments Job machine

( 0, 0, 0, 0, 0) 10 1

( 22, 0, 0, 0, 0) 9 2

( 22, 32, 0, 0, 0) 8 3

( 22, 32, 41, 0, 0) 7 4

( 22, 32, 41, 50, 0) 6 5

( 22, 32, 41, 50, 19) 5 5

( 22, 32, 41, 50, 45) 4 1

( 28, 32, 41, 50, 45) 3 1

( 44, 32, 41, 50, 45) 2 2

( 44, 53, 41, 50, 45) 1 3

( 44, 53, 46, 50, 45)

Arrange 1 job each time


Ex.

Machine Sequence

1 3-4-10

2 2-9

3 1-8

4 7

5 5-6

5

M1

M2

M3

M4

M5

10

1

9

3

8

4

7

2

6 42.32Fw

16 44

50

46

5321

4526

22

5


HW.

Use the following methods to solve the previous case.

1. Large weight matches large one2. Compare the difference between Arrange 1 job each

time and Arrange 5 job each time.


Parallel Identical Processors and Dependent Jobs Baker p.125

Out tree In tree Chain


Parallel Identical Processors and Dependent Jobs

Labeling Phase

Step1. Assign the label 0 to each terminal job.

Step2. For each job j, identify the unique k for which j<<k, and assign to job j the label

Scheduling Phase

Step3a. If the number of jobs without predecessors is less than or equal to m, schedule these job concurrently, leaving excess machines idle. Go to step4.

Step3b. If the number of jobs without predecessors exceeds m, schedule the m jobs with the largest labels (breaking ties arbitrarily). Go to step 4.

Step4. Remove the scheduled jobs from the problem and return to step 3 until all jobs are scheduled.

Baker p.125

1kj

Hu’s Algorithm


Ex.

17

1

4

2

3

7

6

5

11

10

9

8

16

15

14

13

124

3 2

1

0

Labeling

1

1

2

2

3

3

3

4

4

4

4

5


Ex.

17

1

4

2

3

7

6

5

11

10

9

8

16

15

14

13

12

1 23 4 5

6

7

17

15

16

12

13

14

8 5 2 4

9 6

10 11 7

3

1M1

M2

M3

Scheduling

Documents

Production SchedulingP.C. Chang, IEM, YZU. 1 Parallel Machine Scheduling Baker p.114 Minimizing Makespan Problems N / 3 / Cmax meanT meanF Multiple Bin