1. Parallel Machine Scheduling
Baker p.114
Minimizing Makespan Problems M1 M2 M3
N / 3 / Cmax
meanT
meanF
Multiple Bin Packing
Knapsack Packing
Vehicle Packing - assignment

2. Lower Bound Approach
For Example : j 1 2 3 4 5 6 7 8 tj

3. McNaughton’s Algorithm
Baker p.115
Select some job to begin on machine 1 at time zero
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.)
Reassign the processing scheduled beyond M* to the next machine instead, starting at time zero. Return to step 2.
For example:
Preemptive M1 M2 M3 1 2 3 4 5 5 6 7 7 8 12

4. 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.
Construct an LPT ordering of the jobs.
Schedule the jobs in order, each time assigning a job to the machine with the least amount of processing already assigned.

5. LPT Sequence = 8-7-6-5-4-3-2-1 M1 M2 M3 M1 M2 M3
LPT (Largest Processing Time) Rule
Sequence = M1 M2 M3 8 3 2 7 4 1 6 5 13
If SPT M1 M2 M3 1 4 7 2 5 8 3 6 15

6. Integer Programming Formulation
Each job can only be processed by one machine

7. 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

8. Minimizing Mean Flow Time
Baker p.118
ti[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 1 2 3 4 1 2 3 1 2 1
Machine i 1 2 3 … n

9. Min Mean F with Parallel Identical Machine
Construct an SPT ordering of the jobs.
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.

10. Ex. j 1 2 3 4 5 6 7 8 tj
Find out LB
- Lower Bound

11. Hm & H1 Hm
Step 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. H1
Step 2. Assign the first job on the list to the machine with the least amount of processing allocated. Repeat until all jobs have been assigned. Then go to Step 3.

12. Ex. j tj wj Method 1. Arrange 5 jobs each time 1 2 3 4 5 6 7 8 9 10 2116 26 19 50 41 32 22 wj
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 { …-2-1} WSPT from large to small
Method 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) 5
Processing commitments
Job(wi)
machine
2.(22,32,41,50,19)
5(4) 1
4(1) 4
3(3) 3
2(5) 5
1(4) 2
Arrange 5 jobs each time
Large weight matches small one

13. Ex. M1 M2 M3 M4 M5 5 10 1 9 3 8 4 7 2 6 Machine Sequence 1 5-10 2 1-9
3-8 4
4-7 5
2-6 M1 M2 M3 M4 M5 5 10 26 48 1 9 5 37 3 8 16 57 4 7 6 56 2 6 21 40

14. Processing commitments
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)
( 22, 32, 41, 50, 45)
( 28, 32, 41, 50, 45)
( 44, 32, 41, 50, 45)
( 44, 53, 41, 50, 45)
( 44, 53, 46, 50, 45)
Arrange 1 job each time

15. Ex. M1 M2 M3 M4 M5 3 4 10 2 9 1 8 7 5 6 Machine Sequence 1 3-4-10 2
2-9 3
1-8 4 7 5
5-6 M1 M2 M3 M4 M5 3 4 10 16 22 44 2 9 21 53 1 8 5 46 7 50 5 6 26 45

16. HW. Use the following methods to solve the previous case.
Large weight matches large one
Compare the difference between Arrange 1 job each time and Arrange 5 job each time.

17. Parallel Identical Processors and Dependent Jobs
Baker p.125
In tree
Chain
Out tree

18. Parallel Identical Processors and Dependent Jobs
Baker p.125
Hu’s Algorithm
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.

19. Ex. 17 12
Labeling 5 4 8 5 2 3 2 1 13 4 14 9 6 3 1 4 3 2 1 15 10 4 3 7 4 2 16 11 1 4 3

20. Ex. 1 2
Scheduling 3 4 5 17 12 8 5 2 13 14 9 6 3 1 7 15 10 7 4 16 11 6 M1 M2 M3 17 12 8 5 2 4 1 15 13 9 6 3 16 14 10 11 7