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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.
AnImpactofModelParameterUncertaintyonSchedulingAlgorithms
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