HOW USEFUL ARE PREEMPTIVE SCHEDULES

1
HOW USEFUL ARE PREEMPTIVE SCHEDULES?

• Peter Brucker
• Silvia Heitmann
• University of Osnabrück
• Johann Hurink
• University of Twente

2
HOW USEFUL ARE PREEMPTIVE SCHEDULE?

• Consider scheduling problems with one or m
• identical processors.
• For which problems does preemption provide
• a better schedule than the best nonpreemptive
• one?
• In this case preemption is nonredundant.
• Otherwise it is redundant

3
HOW USEFUL ARE PREEMPTIVE SCHEDULES?

• Single machine problems
• Problems with identical parallel machines
• Concluding remarks
• All data are assumed to be integers

4
SINGLE MACHINE PROBLEMS

• Redundant
• 1/prec/f
• 1/p(j)1r(j)prec/f
• 1/r(j)prec/C_max
• f regular

• Nonredundant
• 1/r(j)/L_max
• 1/r(j)/ S C(j)

5
SINGLE MACHINE PROBLEMS
1/prec/f redundant
x
j
y
j
z
j
. . .
. . .
x
y
z
j
. . .
6
SINGLE MACHINE PROBLEMS
1/p(j)1r(j)prec/f redundant
j
k
l
j
l
j
l
k
C(j)
C(l)
C(k)
r(j)
r(j)1
l
k
j
k
l
l
l
k
7
SINGLE MACHINE PROBLEMS
1/r(j)prec/C_max redundant
. . .
C_max
r(j)
If in each block jobs are scheduled
nonpreemptively in an order of
nondecreasing r(j)-values then the block
structure will not change.
8
SINGLE MACHINE PROBLEMS
1/r(j)/L_max and 1/r(j)/SC(j)
nonredundant
p(1)4 r(1)0 d(1)5 p(2)1 r(2)1
d(2)2
1
2
1
preemptive schedule
0 1 2
5
L_max 0 S C(j) 7
2
1
optimal nonpreemptive schedule
0 1 2
6
L_max 1 S C(j) 8
9
PARALLEL IDENTICAL MACHINES

• Redundant
• P//Sw(j)C(j)
• McNaughton (1959)
• P/chains/Sw(j)C(j)
• Du et al. (1991)
• P/p(j)1r(j)outtree/SC(j)
• Brucker et al. (2002)

• Nonredundant
• P/r(j)/SC(j)
• P2/p(j)1outtree/Sw(j)C(j)
• P2/p(j)1intree/SC(j)
• P2/p(j)2/C_max

10
PARALLEL IDENTICAL MACHINES
P2/p(j)2/C_max nonredundant
2
1
1
3
2
2
3
0 1 2
3
0 1 2
3 4
C_max3
C_max4
preemptive schedule optimal
nonpreemptive schedule
11
PARALLEL IDENTICAL MACHINES NEW RESULTS

• Redundant
• P/p(j)1intree/Sw(j)U(j)
• P/p(j)1r(j)/Sw(j)U(j)
• P/p(j)1r(j)/Sw(j)T(j)

• Open
• P/p(j)1chainsr(j)/Sw(j)U(j)
• P/p(j)1outtree/Sw(j)U(j)

12
PARALLEL IDENTICAL MACHINES NEW RESULTS
P/p(j)1r(j)/Sw(j)U(j) redundant
Lemma 1 The time window problem
P/p(j)1r(j)d(j)/- has a feasible solution if
and only if the corresponding preemptive problem
has a feasible solution.
Theorem 1 For problem P/p(j)1r(j)/Sw(j)U(j)
preemptions provide no better solution.
13
PARALLEL IDENTICAL MACHINES NEW RESULTS
P/p(j)1r(j)/Sw(j)U(j) redundant
Proof of Lemma 1
Jobs
Periods k,k1)
1
s
smin r(j)
-1
m
r(j)
tmax d(j)
m
u_jk1
j
k
T
b(T)n
b(j)-1
d(j)-1
x_jk processing time of job j in k,k1)
m
t-1
n
t-1
-1
14
PARALLEL IDENTICAL MACHINES NEW RESULTS
P/p(j)1r(j)/Sf_j redundant
f_j monotone nondecreasing and linear between
consecutive integer time points
Theorem 2 For P/r(j)pmtn/Sf_j an optimal
solution exists in which preemptions only occour
at integer time points.
Redundancy is an immediate consequence of this
theorem.
15
PARALLEL IDENTICAL MACHINES NEW RESULTS
P/p(j)1r(j)/S f_j redundant
Proof of Theorem 2 Consider optimal preemptive
schedule S with finishing times C(j)
Jobs
Periods k,k1)
1
s
Smin r(j)
- p(1)
t upper bound for C(j)-values
m
r(j)
m
0 1
c_jk0 u_jk1
m
j
k
T
S p(j)
S

-p(j)
m
C(j)
slope C(j),C(j)1) 1
x_jk processing time of job j in k,k1)
m
t-1
n
t-1
-p(n)
16
PARALLEL IDENTICAL MACHINES NEW RESULTS
P/p(j)1r(j)/S f_j redundant
Proof of Theorem 2
S optimal preemptive schedule F flow
corresponding with S F optimal integer flow S
schedule corresponding with F
We have cost of S S f_j(C(j)) cost of F gt
S f_j(C(j)) cost of F gt cost of S. Thus,
cost of S cost of S.
17
CONCLUDING REMARKS

• We have three classes of problems
• Problems for which the preemptive and nonpre-
  emptive versions have the same complexity.
• Problems for which the preemptive versions are
  easier (e.g. P2//C_max).
• Problems for which the preemptive versions are
  harder (e.g. P/p(j)p/Sw(j)U(j), BruckerKrav-
• chenko1999).
• What are the reasons for such a behavior?