An Impact of Model Parameter Uncertainty on Scheduling Algorithms
Radosław Rudek
1
, Agnieszka Rudek
2
, Andrzej Kozik
3
and Piotr Skworcow
4
1
Institute of Business Informatics, Wrocław University of Economics, Wrocław, Poland
2
Department of Systems and Computer Networks, Wrocław University of Technology, Wrocław, Poland
3
Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, Wrocław, Poland
4
De Montfort University, Water Software Systems, Leicester, U.K.
Keywords:
Scheduling, Flowshop, Learning, Heuristic, Robustness.
Abstract:
This short paper presents a preliminary analysis of the impact of model parameter uncertainty on the accu-
racy of solution algorithms for the scheduling problems with the learning effect. We consider the maximum
completion time minimization flowshop problem with job processing times described by the power functions
dependent on the number of processed jobs. To solve the considered scheduling problem we propose heuristic
(NEH based) and metaheuristic (simulated annealing) algorithms. The numerical experiments show that NEH
and simulated annealing are robust for this problem with respect to model parameter uncertainty.
1 INTRODUCTION
Classical flowshopscheduling problems are perceived
to be more interesting in a theoretical context than as
a practical research (Gupta and Stafford, 2006). It fol-
lows from observations that algorithms constructed
on the basis of the classical models usually provide
unsatisfactory (unstable) solutions for real-life flow-
shop problems, since these models do not take into
consideration additional factors such as the learning
effect that is significant in practice (Biskup, 2008),
(Lee and Wu, 2004), (Rudek, 2011).
A schedule for a real-life problem (e.g., in man-
ufacturing or computer systems) is calculated on the
basis of a model and values of its parameters. Due
to the possible differences between estimated and real
valuesof problem parameters (e.g., shape of the learn-
ing curve, job processing times), the algorithms that
are efficient for the modelled problem do not have
to be accurate for the real problem. Therefore, it is
crucial to evaluate how values of parameters (uncer-
tainty) affect the quality of solutions provided by such
algorithms, thereby determine their usefulness.
Thus, in this paper, we will analyse the impact of
values of parameters on the accuracy of solution algo-
rithms for the scheduling problems with the learning
effect. In particular, we will consider the maximum
completion time minimization flowshop problem with
job processing times described by the power functions
dependent on the number of processed jobs.
This paper is organized as follows. Next section
contains the problem formulation. Approximation al-
gorithms with the analysis of their efficiency are given
subsequently. The last section concludes the paper.
2 PROBLEM FORMULATION
There are given a set J = {1,. . ., n} of n jobs and m
machines, namely M={M
1
,. . ., M
m
}. Each job j con-
sists of a set O = {O
1, j
, . .. , O
m, j
} of m operations.
Each operation O
z, j
has to be processed on machine
M
z
(z=1,. . ., m). Moreover operation O
z+1, j
may start
only if O
z, j
is completed. It is assumed that machines
have to process jobs in the same order, i.e., a permuta-
tion flowshop, and each machine can process one op-
eration at a time. There are no precedence constraints
between jobs, operations are non-preemptive and are
available for processing at time 0 on M
1
. Further, in-
stead of operation O
z, j
, we say job j on machine M
z
.
Due to the learning effect the processing time
˜p
(z)
j
(v) of job j processed as the vth in a sequence
on machine M
z
is described by a non-increasing pos-
itive function dependent on the number of previously
processed operations (v−1), i.e., on its position v in a
sequence. The function ˜p
(z)
j
(v) of the job processing
time that models the learning effect is called the learn-
ing curve. Moreover,each job j is characterized by its
normal processing ˜p
(z)
j
time on machine M
z
that is de-
353
Rudek R., Rudek A., Kozik A. and Skworcow P..
An Impact of Model Parameter Uncertainty on Scheduling Algorithms.
DOI: 10.5220/0004117603530356
In Proceedings of the 14th International Conference on Enterprise Information Systems (ICEIS-2012), pages 353-356
ISBN: 978-989-8565-10-5
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

fined as the time required to perform a job if the ma-
chine is not affected by learning, i.e., p
(z)
j
, ˜p
(z)
j
(1).
Following (Mosheiov and Sidney, 2003), in this
paper, we focus on a problem, where the processing
time of job j processed as the vth on machine M
i
is
described by:
˜p
(z)
j
(v) = p
(z)
j
v
α
(z)
j
, (1)
where p
(z)
j
and α
(z)
j
are the normal processing time
and the learning index, respectively, of job j on ma-
chine M
z
. Moreover, we will analyse the problem
with the special cases of (1), where α
(z)
j
= α for
j = 1, . . . , n and z = 1, . . . , m.
For the m-machine permutation flowshop prob-
lems the schedule of jobs on the machines can be un-
ambiguouslydefined by their sequence (permutation).
Let π =

π(1), ..., π(i), ..., π(n)

denote the sequence
(permutation) of the n jobs where π(i) is the job in
position i of π. Also, let Π be the set of all job per-
mutations. Thus, for each job π(i), i.e., scheduled in
the ith position in π, we can determine its completion
time C
(z)
π(i)
on machine M
z
as follows:
C
(z)
π(i)
= max
n
C
(z−1)
π(i)
,C
(z)
π(i−1)
o
+ ˜p
(z)
π(i)
(i), (2)
where C
(0)
π(1)
= C
(z)
π(0)
= 0 for z = 1, . . . , m and C
(1)
π(i)
=
∑
i
l=1
˜p
(1)
π(l)
(l) is the completion time of a job placed
in position i in the permutation π on M
1
. On this ba-
sis, the maximum completion time (makespan) for the
given π can be defined as C
max
(π)=C
(m)
π(n)
.
The objective is to find such a schedule
π
∗
of jobs on the machines that minimizes
the maximum completion time (makespan):
π
∗
, argmin
π∈Π
n
C
max
(π)
o
. For convenience,
the problem according to the three field no-
tation scheme X | Y | Z will be denoted as
Fm| ˜p
j
(v) = p
j
v
α
j
|C
max
and its special case (α
(z)
j
= α)
as Fm| ˜p
j
(v) = p
j
v
α
|C
max
.
3 ALGORITHMS
In this section, we will briefly describe the algo-
rithms that are analysed in the further part of this
paper. Namely, we present the extensive search
algorithm (ESA), the random schedule algorithm
(RND), the shortest processing time (SPT) rule, NEH
(Nawaz et al., 1983) and simulated annealing (SA)
(Kirkpatrick et al., 1983). Note that the problem
Fm| ˜p
j
(v) = p
j
v
α
j
|C
max
is strongly NP-hard even
without the learning effect for m ≥ 3, and it seems
to be strongly NP-hard for m = 2 with the learning
effect.
The extensive search algorithm (ESA) is an exact
method that searches the total solution space, which
size is O(n!).
The random schedule algorithm (RND) provides a
random schedule (permutation) as a solution; its com-
plexity is O(n).
The shortest processing time (SPT)rule constructs
the solution according to the non-decreasing order of
the normal processing times of jobs on machine M
1
,
i.e., p
(1)
j
; its computational complexity is O(nlogn).
The NEH algorithm (Algorithm 1) is based on
the method introduced by (Nawaz et al., 1983). It
starts with an initial solution π
initial
that determines
the order of jobs that are subsequently inserted into
the resulting solution π
∗
such that the criterion value
C
max
(π
∗
) is minimized. The computational complex-
ity of this algorithm is O(mn
3
).
Algorithm 1: NEH.
1: Determine the initial sequence of jobs
in π
initial
and set π
∗
:=
/
0
2: Get the first job j from π
initial
3: Insert j in such a position in π
∗
for which C
max
(π
∗
) is minimal
4: Remove j from π
initial
5: If π
initial
6=
/
0 Then go to Step 2
6: The permutation π
∗
is the given solution
Algorithm 2: SA.
1: Determine initial solution π
init
and π=π
∗
=π
init
, T =T
0
2: For i = 1 to Iterations
3: Choose π
′
by a random interchange of
two jobs in π
4: Assign π = π
′
with probability
P(T, π
′
, π) = min

1, exp

−
C
max
(π
′
)−C
max
(π)
T

5: If C
max
(π) < C
max
(π
∗
) Then π
∗
= π
6: T =
T
1+λT
7: The permutation π
∗
is the given solution
The presented simulated annealing (SA) algo-
rithm (Algorithm 2), that is based on (Kirkpatrick
et al., 1983), starts with an initial solution π
initial
and
generates in each iteration a new permutation π
′
based
on the current solution π by interchanging of two ran-
domly chosen jobs in π. This new solution π
′
re-
places π (i.e., π = π
′
) with the following probability
P(T, π
′
, π) = min

1, exp

−
C
max
(π
′
)−C
max
(π)
T

, where
T is the temperature that decreases in a logarithmical
manner T =
T
1+λT
, and the values of the initial tem-
perature T
0
and of the parameter λ are chosen empir-
ically. The solution π
∗
with the minimal found cri-
ICEIS2012-14thInternationalConferenceonEnterpriseInformationSystems
354

terion value C
max
(π
∗
) is also stored. The algorithm
stops after Iterations steps, thus, its overall computa-
tional complexity is O(Iterations· mn).
4 NUMERICAL ANALYSIS
In practice, a schedule for a real-life problem (e.g.,
in manufacturing systems) is calculated on the basis
of a model and values of its parameters. Due to the
possible differences between estimated and real val-
ues of problem parameters (e.g., shape of the learning
curve, job processing times), the algorithms that are
efficient for the modelled problem do not have to be
accurate for the real problem. Therefore, it is crucial
to evaluate how uncertain values of parameters affect
the quality of solutions provided by such algorithms.
Some of the analysed algorithms were described in
(Rudek, 2011).
Let REAL denote the flowshop problem
Fm| ˜p
j
(v) = p
j
v
α
j
|C
max
, where job processing
times are described by (1) and the values of the
job parameters (p
(z)
j
, α
(z)
j
) are precise. However, in
practice it is usually difficult to obtain such accurate
values and solution methods are based on uncertain
(estimated) values. Following this, let ESTIM denote
the flowshop scheduling problem, where the exact
values of job parameters are unknown. In this case,
job parameters are estimated, and we assume that job
processing times are described by
b
˜p
(z)
j
(v) = bp
(z)
j
v
b
α
,
where bp
(z)
j
and
b
α are the estimated values of p
(z)
j
and
α
(z)
j
, respectively.
In the further part of this section, we provide the
numerical analysis of the presented algorithms con-
cerning the impact of the imprecise model on their
efficiency. It is done according to the following steps.
First, we draw values of job parameters for the prob-
lem REAL. Next, we solve the problem REAL us-
ing an algorithm A, which find a schedule π with
criterion value C
max
(π). Based on the parameters
of the problem REAL, we draw or determine values
of parameters for the problem ESTIM (Fm| ˜p
j
(v) =
p
j
v
α
|C
max
), which simulates their estimation. Next,
we use the algorithm A to calculate a schedule
b
π for
the problem ESTIM. For this schedule, we calcu-
late the corresponding criterion value C
max
(
b
π) for the
problem REAL. The differenceC
max
(
b
π)−C
max
(π) in-
forms about the usefulness of the algorithm A in case
of imprecise values of job parameters. Algorithms
with smaller differences are more stable (robust), than
those with greater.
The values of parameters for the problem REAL
are generated as follows. For each pair of n ∈
{10, 25, 50} and m ∈ {2, 3}, 100 random instances are
generated from the uniform distribution in the fol-
lowing ranges of parameters: p
(z)
j
∈ [1, 10], α
(z)
j
∈
[−0.51, −0.15] for j = 1, . . . , n and z = 1, . . . , m. In
all experiments in this paper, p
j
are integers and
α
j
are rational values with accuracy of two deci-
mal place, e.g., for α
(z)
j
∈ [−0.51, −0.15] it is α
(z)
j
∈
{−0.51, −0.50, −0.49, . . ., −0.15}. The values of
α
(z)
j
∈ [−0.51, −0.15] corresponds to the learning
curves in range between 70% and 90%, which are
most common in practice (Biskup, 2008).
The values of the normal processing times for ES-
TIM are bp
(z)
= p
(z)
j
(1+∆
p
), where ∆
p
is the estima-
tion error, which allows us to control precision of pa-
rameters for the analysis; it simulates the estimation
process. The values of ∆
p
and
b
α are provided for par-
ticular experiments in Table 1.
Let A
R
= {ESA, ESA
max
, RND, SPT, NEH, SA}
denote the algorithms that calculate the schedule for
the problem REAL, where ESA
max
is the algorithm
that calculates the schedule with the maximum pos-
sible criterion value (opposite to ESA). ESA and
ESA
max
clearly show the place of the errors provided
by the analysed algorithms in reference to the opti-
mum and the worst criterion values. Note that the
algorithms RND provide the same solution (sched-
ule) for REAL and ESTIM. On the other hand, let
A
E
= {
d
ESA,
d
SPT,
[
NEH,
c
SA} denote the correspond-
ing algorithms from A
R
that calculate the schedule for
the problem ESTIM.
The initial solution for NEH and SA is a random
permutation (in this case natural) and values of the
parameters of SA were chosen empirically as follows:
Iterations= 1000000, T
0
=1000000 and λ=0.01.
1
The algorithms are evaluated, for each
instance I, according to the relative error
δ
A
(I) =

C
max
(π
A
I
)
C
max
(π
∗
I
)
− 1

· 100%, where C
max
(π
A
I
)
denotes the criterion value provided by algorithm
A ∈ {A
R
, A
E
} for instance I and C
max
(π
∗
I
) is the
optimal solution of instance I (if n = 10) or the best
found solution of instance I (if n ≥ 25) provided by
the considered algorithms. The optimal solution is
provided by ESA for the problem REAL. The results
concerning the percentage values of mean, minimum
and maximum relative errors and mean criterion
values
¯
C
max
(rounded to integer) provided by the
analysed algorithms are presented in Table 1.
First, we discuss the results provided by the
heuristic and metaheuristic algorithms for the prob-
1
All algorithms were coded in C++ and simulations
were run on PC, Intel
r
Core
TM
i7–2600K Processor and
8GB RAM.
AnImpactofModelParameterUncertaintyonSchedulingAlgorithms
355

Table 1: The impact of model parameter uncertainty on
the errors of the algorithms for p
(z)
j
∈ [1, 10], α
(z)
j
∈
[−0.51, −0.15], ∆
p
∈ [−0.25, 0.25],
b
α = −0.322.
n m Algorithms
¯
C
max
Errors
Mean Min Max
10 2 ESA 36 0.00 0.00 0.00
ESA
max
54 44.35 21.37 74.72
RND 44 19.58 4.05 46.09
SPT 39 5.20 0.00 18.63
NEH 37 1.60 0.00 8.09
SA 36 0.00 0.00 0.00
d
ESA 38 3.09 0.00 16.41
d
SPT 39 5.60 0.15 21.73
[
NEH 38 3.45 0.00 17.11
c
SA 38 3.05 0.00 16.41
3 ESA 41 0.00 0.00 0.00
ESA
max
62 49.64 28.56 80.89
RND 52 21.85 6.16 48.14
SPT 45 10.31 0.54 34.50
NEH 43 2.20 0.00 10.15
SA 41 0.01 0.00 0.44
d
ESA 43 4.31 0.00 16.21
d
SPT 46 10.93 0.25 30.58
[
NEH 43 5.21 0.57 19.90
c
SA 43 4.18 0.00 16.21
25 2 RND 82 19.57 7.95 32.41
SPT 75 6.65 0.10 19.38
NEH 72 2.34 0.12 5.82
SA 70 0.00 0.00 0.00
d
SPT 75 6.74 0.32 18.47
[
NEH 73 4.93 0.55 12.76
c
SA 73 3.90 0.26 13.73
3 RND 84 22.71 3.30 41.19
SPT 77 12.77 4.96 29.46
NEH 71 3.51 0.51 7.61
SA 70 0.00 0.00 0.00
d
SPT 78 12.90 5.05 28.37
[
NEH 75 7.53 2.11 19.88
c
SA 75 6.10 1.12 18.19
50 2 RND 129 19.32 9.61 31.39
SPT 119 9.35 0.20 20.08
NEH 113 3.09 0.12 7.18
SA 109 0.00 0.00 0.00
d
SPT 118 9.44 0.43 20.15
[
NEH 116 6.47 0.51 13.04
c
SA 113 4.19 0.42 11.86
3 RND 141 20.41 10.33 31.11
SPT 133 14.21 5.87 28.94
NEH 122 3.78 1.02 7.41
SA 117 0.00 0.00 0.00
d
SPT 134 13.93 6.46 30.50
[
NEH 125 7.66 2.95 15.01
c
SA 125 5.96 1.40 14.29
lem REAL, for which the exact values of model pa-
rameters are known. It can be seen in Table 1 that
SA finds solutions with criterion values close to the
optimum. On the other hand, the differences between
the mean relative errors provided by SA and NEH is
about 3.5% and for the maximum errors 10%; for SPT
it is about 14% for mean and 35% for maximum er-
rors. The random solution is usually equally between
the optimal and the worst case (n = 10) and provides
mean and maximum errors (in reference to SA) about
20% and 45%, respectively.
However, if the applied algorithms are based on
uncertain values of model parameters (solve the prob-
lem ESTIM), then their accuracy decreases in refer-
ence to the criterion value found by the algorithms,
which are based on exact values (solve the problem
REAL). It can be seen in Table 1 (for n = 10), that
SA is more robust with respect to model parameter
uncertainty than ESA. Namely, solutions obtained for
ESTIM by
c
SA have lower criterion values (in refer-
ence to REAL) than provided by
d
ESA. Note that the
mean relative errors of NEH and SA increase about 3-
5% if model parameters are uncertain. The exception
is SPT, which is robust to the analysed model param-
eter uncertainty, however, it provides solutions with
relative errors greater than
[
NEH and
c
SA. Note that
the considered algorithms calculate schedules that are
significantly lower than a random solution (RND).
From the numerical analysis followsthat NEH and
SA can be efficiently applied to solve the considered
real-life problem even if the model parameters are un-
certain.
5 CONCLUSIONS
In this paper, we analysed the impact of model pa-
rameter uncertainty on the accuracy of solution al-
gorithms for the makespan minimization flowshop
scheduling problem with job processing times de-
scribed by the power functions dependent on the num-
ber of processed jobs. We showed that the considered
algorithms are efficient even if the values of problem
parameters are not precisely identified.
REFERENCES
Biskup, D. (2008). A state-of-the-art review on schedul-
ing with learning effects. European Journal of Oper-
ational Research, 188:315–329.
Gupta, J. N. D. and Stafford, J. E. F. (2006). Flowshop
scheduling research after five decades. European
Journal of Operational Research, 169:699–711.
Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P. (1983). Op-
timization by simulated annealing. Science, 220:671–
680.
Lee, W.-C. and Wu, C.-C. (2004). Minimizing total com-
pletion time in a two-machine flowshop with a learn-
ing effect. International Journal of Production Eco-
nomics, 88:85–93.
Mosheiov, G. and Sidney, J. B. (2003). Scheduling with
general job–dependent learning curves. European
Journal of Operational Research, 147:665–670.
Nawaz, M., Enscore, J. E. E., and Ham, I. A. (1983). A
heuristic algorithm for m-machine, n-jobs Flow-shop
sequencing problem. Omega, 11:91–95.
Rudek, R. (2011). Computational complexity and solution
algorithms for flowshop scheduling problems with the
learning effect. Computers & Industrial Engineering,
61:20–31.
ICEIS2012-14thInternationalConferenceonEnterpriseInformationSystems
356