0. Parallel-Machine Models
Chapter 7
Elements of Sequencing and Scheduling by Kenneth R. Baker
Byung-Hyun Ha R1

1. Outline Introduction Minimizing the makespan Minimizing total flowtime
Nonpreemptable jobs
Dependent nonpreemptable jobs
Preemptable jobs
Minimizing total flowtime
Summary

2. Introduction Beyond single machine Parallel machines Some types
Parallel systems, serial systems (flow shop), hybrid systems (job shop)
Sequencing + resource allocation
Parallel machines
Basic model (Pm)
n jobs independent and simultaneously available at time zero
m identical parallel machines
A job can be processed by at most one machine at a time
e.g., Pm || Cj
More general models
Parallel machines with different speeds, or uniform machines (Qm)
Speed of machine i -- vi
Processing time of job j on machine i -- pij = pj /vi
Unrelated machines in parallel (Rm)
Speed of machine i on job j -- vij
pij = pj / vij

3. Minimizing the Makespan
Makespan problem for parallel machines
Resource allocation problem
 Sequence is not important
Preemptable jobs -- Pm | prmp | Cmax
Optimal makespan (McNaughton, 1959)
M* = max[j=1..n pj /m, max{pj}]
ALGORITHM 1 -- Minimizing M with m parallel, identical machines
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 until the machine is occupied beyond time M* (or until all jobs are scheduled).
3. Reassign the processing scheduled beyond M* to the next machine instead, starting at time zero. Return to Step 2.
Exercise
Three machines, jobs with processing times of 1, 2, 3, 4, 5, 6, 7, 8
M* ? Optimal schedule?

4. Minimizing the Makespan
Nonpreemptable jobs -- Pm || Cmax
NP-hard
PARTITION reduces to Pm || Cmax
 PARTITION -- NP-hard in ordinary sense
Given positive integers a1, ..., at , is there a subset X  {1, ..., t} such that 2iX ai = a at ?
Branch and bound? Not easy to obtain good lower bound
Dynamic programming? Extremely large number of states, commonly
List scheduling -- a plausible way
Procedure
1. Construct a list of jobs.
2. Remove the first job from the list and place it in the schedule as early as possible.
3. Repeat Step 2 without changing the existing partial schedule.
Dispatching mechanism for real-time decision, possibly
Optimal schedule can always be produced by some list scheduling! (why?)
Search space reduction to n!

5. Minimizing the Makespan
Nonpreemptable jobs (cont’d)
Performance guarantee
A (worst-case) bound on the performance of a particular solution procedure
General form with error bound r
z  rz*
where z -- objective from heuristic, z* -- that of optimal solution
Theorem 1
List scheduling for independent, nonpreemptable jobs yields a makespan satisfying M /M*  2 – 1/m.
Proof of Theorem 1
Consider a schedule by a list scheduling with makespan M.
Let k be a job that finishes at time M (start time of the job: M – pk).
At time (M – pk), all m machines must have been occupied since time zero.
Hence, m(M – pk)  j=1..n pj – pk  M  j=1..n pj /m + pk(m – 1)/m.
j=1..n pj /m  M* (from results on preemptable case)
pk  M*
Therefore, M  j=1..n pj /m + pk(m – 1)/m  M* /m + M*(m – 1)/m.

6. Minimizing the Makespan
Nonpreemptable jobs (cont’d)
LPT list scheduling
List scheduling by longest processing time (LPT)
Theorem 2 (Graham, 1969)
LPT list scheduling for independent, nonpreemptable jobs yields a makespan satisfying M /M*  4/3 – 1/3m.
Proof of Theorem 2 (adopted from Pinedo, 2008)
Suppose there exist counterexamples such that 4/3 – 1/3m  M /M*.
Then, there must exist an example with the smallest number of jobs.
Consider this smallest one and assume it has n jobs and job 1 is shortest.
Then, C1 = M. Otherwise, a counterexample with fewer jobs exists. (why?)
Since all other machines are busy at start of job 1, C1 – p1 = M – p1  j=2..n pj /m.
Hence, M  p1 + j=2..n pj /m = p1(1 – 1/m) + j=1..n pj /m.
Since M*  j=1..n pj /m, 4/3 – 1/3m  M /M*  p1(1 – 1/m)/M* + j=1..n pj /mM*  p1(1 – 1/m)/M* + 1.
It leads to M*  3p1 , which implies that the optimal schedule may results in at most two jobs on each machine. (why?)
However, LPT scheduling is optimal in that case (why?), which is a contradiction.

7. Minimizing the Makespan
Nonpreemptable jobs (cont’d)
First-fit decreasing (FFD)
To check whether makespan M is possible or not
Procedure
1. Order the jobs according to LPT
2. Attempt to assign the first job on the list to the first machine on which the job will fit (i.e., it completes on or before M). If no such machine exists, report FAIL.
3. Repeat Step 2 until all jobs have been scheduled.
Sometimes it fails, when a feasible schedule actually exists.
 Determining whether M is valid is no easier than solving the makespan problem itself
 It is from bin packing problem
 Best-fit decreasing  LPT list scheduling
Exercise
2 parallel machines, jobs with processing times 3, 3, 2, 2, 2, 2
M by LPT list scheduling? Result of FFD using that M?

8. Minimizing the Makespan
Nonpreemptable jobs (cont’d)
Multifit algorithm
Determining schedule by searching for the smallest feasible value of M, using FFD
Min. trial M -- LB from preemptable case
Max. trial M -- max[2j=1..n pj /m, max{pj}], or makespan of any feasible schedule
Theorem 3
The multifit algorithm for independent, nonpreemptable jobs yields a makespan satisfying M /M*  72/61, or about 1.18.

9. Minimizing the Makespan
Dependent nonpreemptable jobs -- Pm | prec | Cmax
Complexity
Strongly NP-hard, generally
Special cases
P1 | prec | Cmax
Easy (why?)
P | prec | Cmax
Project scheduling, polynomially solvable by CPM (critical path method)
Pm | pj=1, intree | Cmax , Pm | pj=1, outtree | Cmax
P2 | pj=1, prec | Cmax
intree (assembly tree)
outtree

10. Minimizing the Makespan
Dependent nonpreemptable jobs (cont’d)
Intree and unit-length jobs -- Pm | pj=1, intree | Cmax
Critical path (CP) rule gives optimal schedule
To give highest priority to the job at the head of the longest string of jobs in the precedence graph (ties may be broken arbitrarily)
A.k.a. highest level first rule
 Equivalent to ALGORITHM 2 + scheduling phase
Proof of optimality -- see p. 120 of Pinedo (2008)
 Analogous to LPT list scheduling
Exercise
Three parallel machines, following intree jobs with unit processing time 12 17 13 8 5 2 14 9 6 3 1 15 10 7 4 16 11

11. Minimizing the Makespan
Dependent nonpreemptable jobs (cont’d)
Arbitrary precedence with two machines -- P2 | pj=1, prec | Cmax
Critical path (CP) rule using labels of direct successors
To give priority to the job with the lexicographically smallest sequence of direct successors’ labels in nonincreasing order
 Equivalent to ALGORITHM 3 + scheduling phase
Exercise
Three parallel machines, unit-length jobs with following precedence 6 5 3 4 7 2 1

12. Minimizing the Makespan
Preemptable jobs
Arbitrary precedence with two machines -- P2 | prmp, prec | Cmax
A possible approach
Transforming to P2 | pj=1, prec | Cmax with chains
Example
Guarantee optimal?
No! The jobs must be divided into half-unit-length jobs. 5 6 7 1 1 1 2 2 2 5 6 7 5 6 7 1 1 1 4 4 1 1 1 2 3 1 2 3 1 1 1 1 1 1

13. Minimizing Total Flowtime
Minimizing F -- Pm || Cj
Notation
pi[j] -- processing time of the jth job in sequence on the ith machine
Fi[j] -- flowtime of the jth job in sequence on the ith machine
nj -- number of jobs processed by the ith machine
Objective function
F = i=1..m j=1..ni Fi[j] = i=1..m j=1..ni (ni – j + 1)pi[j]
Solution approach
F as scalar product of two vectors
Matching integer coefficients (ni – j + 1) with processing times pi[j]
F cannot be minimized unless ni differ by at most one. (why?)
SPT list scheduling
Dispatching procedure and adaptable dynamic job arrivals
Exercise
Two parallel machines and six jobs with processing times of 1, 2, 3, 4, 5, 6
Minimizing Fw -- Pm || wjCj
NP-hard

14. Summary More than one machine LPT list scheduling Flowtime measures
Allocation and sequencing
In case of makespan problems, allocation may be enough
Two-phase method, commonly, because of complexity
Allocation first, sequencing next considering as single machine
LPT list scheduling
Applicable to some situations
No precedence, intree, critical path
Flowtime measures
Total tardiness problems
Very limited optimization approaches