Heuristic Algorithm for Uncertain Permutation Flow-shop Problem
Jerzy Józefczyk and Michał Ćwik
Faculty of Computer Science and Management, Wroclaw University of Technology,
Wybrzeze Wyspianskiego 27, Wroclaw, Poland
Keywords: Decision Making, Optimization, Uncertainty, Interval Data, Minmax Regret, Heuristic Algorithms.
Abstract: A complex population-based solution algorithm for an uncertain decision making problem is presented. The
uncertain version of a permutation flow-shop problem with interval execution times is considered. The
worst-case regret based on the makespan is used for the evaluation of permutations of tasks. The resulting
complex minmax combinatorial optimization problem is solved. The heuristic algorithm is proposed which
is based on the decomposition of the problem into three sequential sub-problems and employs a paradigm of
evolutionary computing. The proposed algorithm solves the sub-problems sequentially. It is compared with
the fast middle point heuristic algorithm via computer simulation experiments. The results show the
usefulness of this heuristic algorithm for instances up to five machines.
1 INTRODUCTION
Investigation of uncertain versions of decision
making problems has a long history. Such problems
being closer to real-world applications are more
complex and difficult to solve. Three issues (Is) are
crucial when considering uncertain problems: the
representation of uncertainty (I1), the evaluation of
arisen uncertain decision making problems (I2), and
the determination of corresponding solution
algorithms (I3). A variety of approaches are
presented and discussed in the literature on all
mentioned issues. Their particular combinations lead
to plenty specific and mostly difficult complex
decision making problems. It is worth noting when
referring to issue I1 that representations of
uncertainty in the form of probability distributions
and the ones based on fuzzy sets and logic seem to
be predominant (Dutt and Kurian, 2013; Aayyub and
Klir, 2006; Klir, 2006). It is assumed that a
probability distribution exists over the space of all
values of corresponding random variables. This
representation is treated as the objective one as the
probability distribution and derivative descriptions
can be empirically verified. Substantial difficulties
with the determination and (or) the estimation of
mentioned probabilistic descriptions, which require
considerable and credible empirical data, is the main
disadvantage of this popular representation. Other
important drawbacks are discussed in (Kouvelis and
Yu, 1997). For the fuzzy approach, the availability
of experimental data on the uncertainty can be
replaced by experts’ opinions in the form of
corresponding membership functions. In a
consequence, the quality of this subjective
representation as well as of following activities
based on it strongly depends on an expert’s quality.
Mentioned shortcomings of both popular
representations have motivated many researchers to
develop other approaches which cope with the
uncertainty more adequately. One of such an
approach is used in the paper. The main idea of the
evaluation of uncertainty as the second mentioned
issue (I2) consists first of all in the substantiation
(determinization) of the uncertainty. The main idea
of such a substantiation consists in transformation of
an uncertain problem into its deterministic
counterpart. Taking the expected value for the
probabilistic representation, when selected
probabilistic distribution reflects the uncertainty, is a
good example of the substantiation. Other frequently
used operators of the substantiation can be found
e.g. in (Yager, 1988). Having the deterministic
problem as the result the issue I3 arises. In fact, all
feasible solution methods and resultant solution
algorithms can be considered, i.e. exact,
approximate as well as heuristic ones.
It is obvious that the choice and justification of
the formalization approach and the solution
algorithm for an unceratin decision making problem
Józefczyk, J. and
´
Cwik, M.
Heuristic Algorithm for Uncertain Permutation Flow-shop Problem.
In Proceedings of the 1st International Conference on Complex Information Systems (COMPLEXIS 2016), pages 119-127
ISBN: 978-989-758-181-6
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
119
depend strongly on the problem per se as well as its
complexity and computational difficulty.
Considerations in the paper are limited to the area of
uncertain combinatorial optimization problems with
the parametric uncertainty. It means that for a
combinatorial optimization problem not all
parameters are known, precise, evident or given. The
permutation flow-shop with unlimited buffers to
minimize the makespan is considered, e.g. (Pinedo,
2008). It is one of the most important task
scheduling problems with many applications mainly
in manufacturing, production, logistic and service
systems, but also in information and computing
systems. Generally speaking, the problem deals in
the execution of a finite number of complex tasks by
a finite number of executors (machines, processors).
Each task requires the carrying-out of the same
number of operations, being parts of tasks, which
equals the number of executors. The exact mapping
of executors to operations within tasks is given. A
permutation of tasks is sought minimizing defined
criterion for a given execution times of operations
by corresponding executors. The completion time of
the last task last operation in the permutation plays
often a role of such criterion. An assembly process
of a product performed along a production line is a
good example of the investigated flow-shop
problem. Then, the order of carried-out products
should be determined to minimize the total
production time. The non-deterministic versions of
flow-shop without full information on execution
times are also a subject of many research works, e.g.
(Pinedo and Schage, 1982; Kouvelis et al, 2000;
Averbakh, 2006; Kasperski and Zielinski, 2008).
A specific junction of the mentioned three issues
I1, I2 and I3 is proposed in the paper to solve the
uncertain optimal decision making problem
(optimization problem). Namely, it is assumed that
execution times of tasks by machines are uncertain
(not fully known). However, the information on their
ranges in the form of intervals is only given. The
uncertainty in execution times cause
straightforwardly the uncertainty of the criterion
being the deterministic evaluation of the flow-shop
problem considered. The regret based approach is
proposed to make possible the evaluation of
resultant optimization problem. The notion ‘regret’
assesses the difference between the value of criterion
for fixed realization (scenario) of uncertain
parameters and the optimal value of criterion – for a
given decision (optimization variable). The
application of the regret based approach is
recommended for the interval uncertainty (Kouvelis
and Yu, 1997; Aissi et al, 2009), however, resulting
deterministic combinatorial optimization problem is
extremely complex and difficult. As it has been
pointed out, the regret requires substantiation of the
criterion evaluating a decision with respect to all
feasible scenarios of uncertainty to have the
deterministic evaluation of a decision. In the paper,
the substantiation via maximization is proposed
which expresses the utmost pessimism od a decision
maker (in fact, a decision algorithm) with respect to
scenarios of uncertainty (execution times for the
considered flow-shop) which can occur but are not
known while making a decision. It leads to worst-
case i.e. robust decisions on the one hand but safe
decisions on the other hand. The solution algorithm
determined on such a basis will perform well
irrespective of the actual scenario of uncertainty.
However, it can work fairly when medium scenarios
of uncertainty will take place. The substantiation via
averaging seems to be more adequate for such cases
which, however, can give poor results for extreme
scenarios of uncertainty. The substantiation via
maximization is used hereinafter.
In the paper, a bespoke hybrid heuristic solution
algorithm is proposed to solve the uncertain problem
(issue I3). As it is presented in following sections,
the consequent unceratin flow-shop is extremelly
difficult combinatorial optimization problem, at least
NP-hard one. The rechearches have been focused on
developing of time-effective solution algorithms
appropriated for real-world applications. A hybrid
heuristic algorithm is the result of presented
investigations. The evolutionary computing as an
important paradigm of the computational
intelligence has been employed as the basis for the
developed algorithm.
The uncertain flow-shop problem has been firstly
stated in (Kouvelis and Yu, 1997). Then it has been
investigated in some works. Its NP-hardness was
proved in (Kouvelis et al, 2000) where a branch-
and-bound algorithm and a heuristic procedure
based on a local improvement were also developed.
Particular attention has been paid in this paper to the
elaboration of approximate and heuristic time-
efficient solution algorithms. Some computational
complexity properties were also investigated in
(Kasperski et al, 2012) for the case with discrete
bounded and unbounded scenario sets. The case of
the problem with only two tasks and m machines
was presented in (Averbakh, 2006) where a linear-
time algorithm is given. The evolutionary heuristic
algorithm for the case of three machines was
considered in (Ćwik and Józefczyk, 2015).
The main contribution of the paper deals with
proposing and experimentally evaluating of a time
COMPLEXIS 2016 - 1st International Conference on Complex Information Systems
120
effective hybrid heuristic solution algorithm for
more than three machines. Former works on minmax
regret problems, in general, and on minmax regret
flow-shop problems, in particular, considered mainly
theoretical issues for special cases, e.g. (Averbakh,
2000; Lebedev and Averbakh, 2006; Conde, 2010;
Volgenant and Duin, 2010; Lu et al, 2012). The
results, which can be found there, do not allow us to
have constructive tools for solving real-world
problems, in general, and permutation flow-shop
problems with interval execution times to minimize
the makespan, in particular. The algorithm presented
in the paper fills this gap for the uncertain flow-shop
and refers to analogous works where time-effective
algorithms for other minmax regret problems with
interval uncertainty are presented, e.g. (Józefczyk,
2008; Józefczyk and Siepak, 2013a,b; Siepak and
Józefczyk, 2014; Averbakh and Pereira,. 2011;
Pereira, 2016)
To sum up, the decision making under
uncertainty is a subject of the paper. In particular,
the complex combinatorial NP-hard optimization
problem is solved via hybrid heuristic population-
based solution algorithm. This algorithm can be
treated as a complex tool of computational
intelligence.
The reminder of the paper is organized as
follows. Deterministic and uncertain versions of the
flow-shop investigated are stated in Section 2 as the
combinatorial optimization problems. The idea for
the solution algorithm together with the detailed
presentation of its three component sub-algorithms
are given in Section 3. The next section presents
results of computer simulation experiments
evaluating the algorithm. Section 5 with conclusions
completes the paper.
2 PROBLEM FORMULATION
The flow-shop task scheduling problem is
investigated with a set
}...,,,,,{=
21 mi
MMMM M
of m machines which are assigned for performing n
tasks constituting a set of tasks
}...,,,,,{=
21 nj
JJJJ J
. Each task from the set J
needs to be sequentially carried out by all executors
from the set M. For the permutation version of flow-
shop, which is only investigated in the paper, each
machine executes tasks in the same order. Moreover,
the version with unlimited buffers is considered
which means that the task after being completed by
the current machine can leave it and wait if
necessary for the service by the next machine at the
buffer located there. Let us denote by
ij
p
execution
times of task j by executor i for
mi ...,,2,1=
and
nj ..,,2,1=
which form the matrix
nj
miij
pp
,...,2,1
,...,2,1
][
=
=
=
.
A part of the task performed by a single machine is
called often an operation. Then the problem consists
in the determination of a permutation of tasks
)...,,...,,,(
21 nj
π
π
π
π
π
=
to minimize the
makespan, i.e. the completion time of the last task
from permutation
π
by the last machine. The
current element
j
π
of
π
is the number of task
executed as the jth in turn, and
Π
π
where
Π
is
the set of all
!n
feasible permutations, i.e.
}},...,,2,1{,,:{ kjnkj
kj
=
π
π
π
Π
. The
makespan
),(
max
π
pC
as the function of p and
π
can be written in different ways (e.g. Pinedo, 2008).
However, the following recursive form seems to be
the most popular. Let
),(
,
π
π
pC
j
i
be the time
moment when ith machine finishes the execution of
task
j
π
:
],),(,),(max[=),(
1
,,1,,
π
π
π
ππππ
pCpCppC
jjjj
iiii
+
mi ...,,3,2=
,
nj ...,,3,2=
.
(1)
Moreover,
,...,,2,1,=),(
1
,,
11
mippC
i
k
ki
=
=
ππ
π
(2)
and
....,,2,1,=),(
1
,1,1
njppC
j
k
kj
=
=
ππ
π
(3)
In a consequence,
),(),(
,max
π
π
π
pCpC
n
m
=
. The
deterministic flow-shop task scheduling problem
deals with the determination of such
Π
π
to
minimize
),(
max
π
pC
. The optimal permutation
as well as the optimal makespan
),()(
maxmax
π
=
Δ
pCpC
is obtained. This problem is
NP-hard for
2>m
. The polynomial optimal
Johnson’s algorithm exists for
2=m
(Garey et al,
1976).
Let assume now that crisp values of execution
times
ij
p
are not known and given. Instead,
intervals for
ij
p
with possible values of
corresponding execution times are available as only
information of these times. Namely,
ij
ij
ij
ij
ij
ppppp ],,[
. Now, matrix p called a
Heuristic Algorithm for Uncertain Permutation Flow-shop Problem
121
scenario is an element of the Cartesian product of all
scenarios
],[...],[
11
mn
mn
ij
pppp ××=P
, i.e.
Pp
.
The vagueness of p makes it impossible to directly
use the makespan
),(
max
π
pC
as the evaluation
function of
π
. The approach based on regret is
applied. The regret
),(min),(
)(),(
maxmax
maxmax
σπ
π
σ
pCpC
pCpC
Π
=
(4)
is the difference between the value of makespan and
the optimal value of the makespan for fixed p. It is
calculated for current p and
π
. The regret requires
substantiation according to scenarios. The worst-
case with respect to p has been chosen in the form of
the maximization which gives the criterion
)],(min),([max)(
maxmax
σ
π
π
σ
pCpCz
p
Π
=
P
.
(5)
Consequently, the uncertain version of the flow-shop
task scheduling consists in finding such optimal
permutation
π
that
=
)],(min),([maxmin)(
maxmax
*
σππ
σπ
pCpCz
p
ΠΠ
P
(6)
for given M, J,
,..,,2,1,,
,
,
mipp
ji
ji
=
nj ..,,2,1=
.
3 SOLUTION ALGORITHM
The problem (6) called hereinafter P is NP-hard
even for
2=m
(Lebedev and Averbakh, 2006). It is
easy to see that (6) is composed of three nested
optimization sub-problems:
SP1: inner minimization of
),(
max
σ
pC
with
respect to permutation
σ
, being the deterministic
flow-shop,
SP2: maximization with respect to feasible
scenarios p, i.e. searching for so called worst-case
scenario, and at the same time calculating the value
of criterion for SP3,
– SP3: outer minimization with respect to
permutation
π
.
The majority of results presented in the
corresponding literature refer to cases where SP1 is
easy, i.e. solvable by polynomial algorithms as well
as worst-case scenarios as the solution of SP2 can be
determined either in an intuitional way or by the
reduction to easy optimization problems.
Nevertheless, it is not a case of this paper’s problem.
Now, the problem (6) is a composition of
individually difficult optimization sub-problems
making P extremely difficult issue. In a
consequence, it is easy to justify the lack of any
approximate algorithm for P (Ćwik and Józefczyk,
2015). So, heuristic algorithms are only possible to
solve problem P in a reasonable time. Such an
heuristic algorithm called ALG is proposed in the
paper. To develop it, some simplifications have been
assumed while solving SP1-SP3. First of all, SP1-
SP3 are solved independently and successively by
corresponding individual sub-algorithms. As the
result, we have admittedly a heuristic algorithm but
working in a reasonable and acceptable time. The
main idea for the algorithm ALG solving P is based
on the assumption that SP3 is solved by known
metaheuristics with the dedicated way of calculating
the value of
)(
π
z
being the criterion for the
metaheuristics. Let us present the sub-algorithms
successively.
3.1 Sub-algorithm for SP1
The solution of SP1 is replaced by its lower bound
in the form
)minmin(max
)(
1
,
,
,1
1
,
1
1
,
,1
,1
LBmax,
+=
=
=
=
=
=
++=
m
ki
li
l
nl
n
j
jk
k
i
ji
nj
mk
ppp
pC
(7)
where first and third elements of the maximum
reflect respectively the following properties:
at least one task needs to be processed on all
machines indexed from 1 to k 1 before the kth
machine starts processing operation, unless
1k =
,
– after ith machine completes the processing, there is
at least one task that needs to be processed on
machines indexed from k + 1 to m, unless
km=
.
We can take the highest value with respect to all
machines as the fact that (7) holds for all machines.
The details on this lower evaluation of
)(
max
pC
can
be found in (Ćwik and Józefczyk, 2015). Such a
form of the lower bound is used by the procedure
calculating approximate value of (5) in SP2.
3.2 Sub-algorithm for SP2
The purpose for SP2 is to calculate the value of
function z for the worst-case scenario. For many
minmax regret problems, it is enough only to
consider extreme point scenarios while searching the
worst-case scenarios, i.e.:
}),{(,
ij
p
ij
p
ij
pji
.
(8)
COMPLEXIS 2016 - 1st International Conference on Complex Information Systems
122
Calculation of
max
(,)Cp
π
can be easily
substituted with the determination of length of the
longest path between
1
1,
v
π
and
,
n
m
v
π
in a directed
graph G (V,A) defined by the solution π where V
and A are set of vertexes and arcs, respectively. Each
vertex
,
j
i
v
π
corresponds exactly to an individual
execution time of task
j
π
executed by machine i.
Therefore, its weight is equal to
,
j
i
p
π
. Set A consists
of arcs connecting vertices corresponding to
subsequent operations of the same task
),(
,1,
jj
ii
vv
ππ
+
, for 0 < i< m) and of operations
performed subsequently on the same machine
),(
1
,,
+jj
ii
vv
ππ
, for 1< j< n). Assuming that processing
times of operations are unknown, all possible paths
between
1
1,
v
π
and
,
n
m
v
π
represent all possible ways of
calculating
max
(,)Cp
π
. Therefore, we can associate
each of those paths with a separate scenario. Let us
denote path r as a sequence of vertices that belong to
it. Then, we can construct scenario
r
p
as follows:
}.for,
for,:{
rvpp
rvpppp
ij
ij
ij
ijijijij
r
=
==
(9)
As it has been shown in (Ćwik and Józefczyk,
2015), this way we can limit the search from
checking all
2
mn
extreme points scenarios to
()
()()
2!
1! 1!
mn
mn
⎛⎞
+−
⎜⎟
⎜⎟
−−
⎝⎠
paths that exist in G. Then, a simple
enumeration has been applied for such decreased
number of paths, which consists in checking all
feasible paths. This approach turned out effective
only for three machines (
3=m
). Therefore, the
heuristic time-effective procedure is now proposed
to enable its applicability for more than three
machines. The idea is as follows. For each vertex
,
j
i
v
π
in graph G, we construct a path from vertex
1
1,
v
π
.
Obviously, there is only one possible path for i = 1 or
j = 1. Therefore, the construction is trivial. For i > 1
and j > 1, we can see that the path to
,
j
i
v
π
needs to
contain one of the two vertices:
1,
j
i
v
π
,
1
,
j
i
v
π
.
Assuming that the path to each of them has already
been constructed, we add the last vertex
,
j
i
v
π
and
choose the path which yields a greater value of z as a
constructed path for vertex
,
j
i
v
π
. After processing all
the vertexes in such a manner, the procedure returns
constructed path for
,
n
m
v
π
together with the
corresponding value of z denoted as
z
~
. In a
consequence, sub-algorithm for SP2 comprising this
procedure is summarized by the following pseudo-
code.
Input: Graph G(V,A) generated by
feasible solution
.
Output: Heuristic value of
)(
π
z
denoted
as
)(
~
π
z
.
1: Set empty matrix cp
2: for
n
i
3: for
mj
4: if i==1 or j==1
5: cp[i][j]:= p_r(v[1][1],v[i][j])
6: else:
7: prev1 := cp[i-1][j]
8: prev2 := cp[i][j-1]
9: c1 := prev1 + v[i][j]
10: c2 := prev2 + v[i][j]
11: if z(c1)>z(c2):
12: cp[i][j]:=c1
13: else:
14: cp[i][j]:=c2
15: endif
16 endif
17: endfor
18: endfor
19: return
])][[(
~
mncpz
3.3 Sub-algorithm for SP3
A standard simple evolutionary algorithm has been
applied for solving SP3 as it has been presented in
(Ćwik and Józefczyk, 2015).
The values of permutation
π
play directly a role
of a chromosome. Due to the complexity of
()z
π
,
the outcome of the sub-algorithm for SP2 in the
form of
)(
~
π
z
is used as the fitness function.
The initial population is constructed via the
random generation of N permutations
(chromosomes). The solution of the middle interval
metaheuristics (MIH) is the basis for generation of a
half of the population. Namely, a MIH-based
permutation undergoes (N/2) – 1 independent
random mutations to have in total (N/2)–element
part of the initial population. The remainder of the
initial population is filled with random permutations
generated uniformly from the search space. MIH
consists in the brute conversion of the uncertain
interval problem into its deterministic counterpart by
assuming the middles of intervals as deterministic
processing times, i.e.
2/)(
MIH
ij
ij
ij
ppp +=
.
The order crossover operator (Goldberg, 1989)
has been employed to avoid the generation of
Heuristic Algorithm for Uncertain Permutation Flow-shop Problem
123
unfeasible permutations, so no repair algorithm is
required. This operator is characterized by parameter
cross
P
being the probability of crossing over two
selected chromosomes. Accordingly, 1
cross
P
is the
probability of passing the two chromosomes to the
next generation without any changes. The value of
parameter requires tuning to ensure the best
algorithm performance.
A simple mutation operator has been also used.
Firstly, it is randomly determined if the chromosome
undergoes the mutation. The probability of the
mutation
mut
P
is considered another parameter of the
algorithm which is tuned. If the chromosome is
decided to undergo the mutation, two random genes
are swapped.
All chromosomes from the population are
selected to generate a new population. The
population is sorted according to decreasing values
of the fitness function. First two chromosomes are
removed from the list and the result of their
crossover is added to the new population. This
process is repeated until the list is empty.
The algorithm terminates when after 5
subsequent iterations no correction is being observed
between the best chromosomes from each
population. The best chromosome of the last
generation is returned as the solution
π
. The sub-
algorithm for SP3 is presented in the form of
corresponding pseudo-code.
Input: Matrices
p
and
p
of size
n
m
×
containing respectively lower and upper
bounds of processing times.
Output: Permutation
π
~
and value of
criterion
)
~
(
~
π
z
.
1: pop = Initialpopulation()
2: SortByFitFunc(pop)
3: repeat
4: nextgen = list of chromosomes
5: add(nextgen,pop[0])
6: add(nextgen,pop[1])
7: remove(pop,pop[0])
8: remove(pop,pop[1])
9: repeat:
10: ind=RemvoveFromPop(pop[0],pop[1])
11: Sibs = Crossover(ind[1],ind[2])
12: Mutate(Sibs[1])
13: Mutate(Sibs[2])
14: add(nextgen,Sibs[1])
15: add(nextgen,Sibs[2])
16: until(pop is empty)
17: pop = nextgen
18: SortByFitFunc(pop)
19: until (Stop condition is fulfilled)
20: return(pop[0],
([])
zpop0
)
The algorithm for P referred to as ALG obtained
after merging the described sub-algorithms solving
SP1-SP3 is in fact hybrid heuristic population-based
one.
4 NUMERICAL EVALUATION
OF THE ALGORITHM
The algorithm proposed is evaluated via computer
simulation experiments performed using a PC with
Intel Core i5 CPU processor of 2,53 GHz with 4GB
of RAM. According to our best knowledge, there are
no benchmarks in the literature for this problem
which could be used for the comparison. Therefore,
the own instances have been generated in the
following way. For each of mn uncertain parameters
(execution times), the lower bound
,ij
p
is randomly
generated from the finite interval [0,K] with the
uniform distribution. The upper bound
,ij
p
is then
generated from the finite interval
,,
,
ij ij
p
pC
⎡⎤
+
⎣⎦
also
with the uniform distribution where K and C are
parameters of the experiments. The default values of
algorithm parameters and problem parameters have
been assumed as:
85.0
cross
=P
,
15.0
mut
=P
, N = 20,
K = 100, C = 200.
Values of criterion (6) as well as run times of
algorithms are the basis for evaluation. The
algorithm ALG is compared with the heuristic
deterministic algorithm MIH. As it has been
mentioned, MIH generates a solution to any interval
data uncertain problem by solving the deterministic
counterpart which is obtained by substituting all
interval execution times of the problem with its
interval middle points. The arisen deterministic
problem is also NP–hard. Therefore, the NEH
algorithm is applied as the solution tool (Nawaz et
al, 1983). The solution of deterministic problem
MIH
π
is returned as the solution of the uncertain
problem together with the heuristic value of the
criterion
)(
~
MIH
π
z
calculated by the sub-algorithm
solving SP2. The results are presented in Tables 1–
3. Every numerical result in tables is the mean value
of five independent instances randomly generated
for fixed n and m from given matrices
p
and
p
of
the bounds of interval processing times. Due to
randomness of the evolutionary algorithm each
instance is additionally repeated for this algorithm
five times and the mean result is taken for every of
five instances. In a consequence, every result of
COMPLEXIS 2016 - 1st International Conference on Complex Information Systems
124
Table 1: Computational results for m = 3.
n
MIH
()z
π
)
~
(
~
π
z
MIH
T
T
5 447.4 351.36 <0.001 0.309
6 372.2 318.96 0.001 0.402
7 419.8 372.16 0.001 0.600
8 555.0 361.28 0.002 0.785
9 639.0 405.0 0.003 1.093
10 575.6 424.0 0.003 1.361
11 667.6 470.8 0.004 1.294
12 762.2 474.7 0.006 1.724
13 730.8 473.3 0.007 1.892
14 626.6 474.7 0.009 2.085
15 834.8 509.5 0.011 2.459
16 360.4 476.6 0.013 2.913
17 839.4 510.8 0.016 3.643
18 869.2 516.2 0.017 3.812
19 999.6 510.0 0.021 3.876
20 964.2 511.6 0.026 4.089
21 1057.8 522.7 0.029 5.371
22 918.0 538.6 0.032 5.623
23 892.2 548.7 0.037 5.895
24 922.8 522.3 0.040 5.273
25 965.4 537.1 0.044 6.977
26 1207.8 550.9 0.052 6.419
27 886.4 540.6 0.056 8.583
28 1252.8 553.6 0.063 8.548
29 1433.4 559.8 0.070 8.941
30 993.2 564.9 0.079 10.043
Table 2: Computational results for m = 4.
n
)(
~
MIH
π
z
)
~
(
~
π
z
MIH
T
T
5 488.6 392.4 <0.001 0.594
6 599.6 476.1 <0.001 0.772
7 765.8 643.4 0.002 1.147
8 696.6 605.3 0.002 1.293
9 667.4 548.3 0.003 1.686
10 789.8 650.2 0.004 2.116
11 791.6 621.2 0.005 2.941
12 815.4 702.4 0.007 2.929
13 1076.8 753.5 0.008 3.22
14 906.2 723.2 0.010 3.350
15 903.6 737.0 0.012 4.031
16 1042.6 772.8 0.014 5.109
17 1129.2 808.6 0.018 6.182
18 1173.8 779.4 0.020 6.729
19 1218 812.9 0.023 6.383
20 1242 831.4 0.029 7.312
21 1140.6 820.3 0.033 8.593
22 1336.6 811.2 0.036 10.425
23 1394.8 825.8 0.040 10.433
24 1468.2 847.5 0.045 12.819
25 1252.4 858.4 0.051 12.970
26 1645.8 891.4 0.059 14.852
27 1448.8 881.5 0.064 16.267
28 1255.4 898.4 0.072 18.853
29 1500.8 857.5 0.079 14.276
30 1463.8 856.4 0.087 19.002
Table 3: Computational results for m = 5.
n
)(
~
MIH
π
z
)
~
(
~
π
z
MIH
T
T
5
622.4 544.7 <0.001 0.790
6
742.2 629.8 0.002 0.987
7
786.6 677.0 0.002 1.619
8
917.8 790.0 0.003 1.758
9
848.4 753.2 0.003 2.428
10
909.4 806.1 0.005 2.607
11
1032.6 845.7 0.006 3.541
12
1051.4 879.6 0.007 4.259
13
1121.8 983.3 0.009 5.238
14
1073.2 940.6 0.012 5.007
15
1241.4 1018.6 0.014 5.668
16
1315.8 1088.4 0.017 7.327
17
1357.4 1099.3 0.020 9.004
18
1453.0 1068.6 0.024 10.628
19
1381.8 1113.3 0.028 12.190
20
1587.6 1153.0 0.030 12.271
21
1616.4 1148.0 0.036 16.190
22
1630.0 1158.2 0.044 18.645
23
1681.8 1155.0 0.046 18.397
24
1594.6 1175.2 0.054 17.910
25
1586.4 1180.0 0.063 20.780
26
1875.2 1214.9 0.067 23.18
27
1721.6 1199.9 0.074 24.570
28
1832.2 1201.7 0.083 26.622
29
1917.0 1237.2 0.092 32.713
30
2163.8 1207.3 0.104 40.000
ALG is, in fact, the mean value of 25 independent
random instances. The values of criterion as well as
the run times in seconds for MIH and ALG are
respectively denoted in corresponding columns of
tables as
)(
~
MIH
π
z
and
)
~
(
~
π
z
as well as
MIH
T
and T.
The results confirmed the usefulness of ALG in
comparison with MIH from both the criterion
z
~
and
the computational time points of view. It is
necessary to point out that the values of criterion (5)
for the resulting permutations
MIH
π
and
π
~
were
calculated by the sub-algorithm SP2, i.e. the values
of
z
~
instead of z are the basis for this comparison. It
turned out that ALG is substantially better than
MIH. Namely, the relative improvement calculated
as the ratio
)
~
(
~
)(
~
MIH
π
π
z
z
(10)
fluctuates from 1,11 to 1.79 (except the instance for
m = 3 and n = 16 when MIH is better). The mean
improvement for all instances is equal to 1.36. The
computational time of ALG is fully acceptable
however much longer than for MIH.
Heuristic Algorithm for Uncertain Permutation Flow-shop Problem
125
5 CONCLUSIONS
The complex and extremely difficult combinatorial
optimization problem, i.e. the uncertain permutation
flow-shop with the makespan as criterion, has been
investigated. The case of minmax regret with
interval execution times has been considered. The
nested three sub-problems have been solved
independently by heuristic sub-algorithms. The
algorithm of the whole optimization problem as the
consecutive composition of all three sub-algorithms
has been compared with the MIH metaheuristics
which is often applied for minmax regret problems
with interval data. MIH has good properties for
many simple optimization problems. For example, it
is 2-approximate algorithm for the special case of
flow-shop problem considered in the paper with only
two machines (m = 2). Therefore, this metaheuristics
has been chosen as the basis for comparison. It is
worth noting that the calculation of criterion for
SP3, i.e. for the whole problem is also NP-hard. The
comparison uses its approximate value
z
~
.
Unfortunately, the relation between z and
z
~
is not
known. This issue requires more profound studies.
The preliminary considerations show the usefulness
of
z
~
as the approximate value of the criterion for
solution
π
~
. Namely, four very simple instances
have been considered for m = 3 and 4 as well as
n = 4 and 5. These instances have been solved by
ALG as well as their optimal solutions have been
derived by the brute force algorithm. The relative
differences between both results in the sense of ratio
)(/)
~
(
~
**
ππ
zz
analogous to (10) are presented in
Table 4. The values of criterion
z
~
are worse than
*
z
no more than 1.30.
The research of this problem is now continuing.
For example, the instances for greater m and n are
solved as well as more effective and faster sub-
algorithm for SP3 is verified.
Table 4: Values of
)(/)
~
(
~
**
ππ
zz
for different m and n.
m\n 4 5
3
1.21 1.13
4
1.30 1.14
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