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Proceedings ArticleDOI

Permutation flowshop scheduling by genetic local search

02 Sep 1997-pp 232-238
TL;DR: An approximation method that would make use of a 'big valley' structure, where local optima occur in clusters over the landscape, is proposed by using a critical block-based neighbourhood structure, and a genetic local search method called MSXFGA, previously developed for the job shop scheduling problem.
Abstract: In this paper, the landscape for the permutation flowshop scheduling problem (PFSP) with stochastic local search and a critical block-based neighbourhood structure has been investigated. Numerical experiments using small benchmark problems show that there are good correlations between the makespans of local optima! the average distances to other local optima and the distances to the known global optima. These correlations suggest the existence of a 'big valley' structure, where local optima occur in clusters over the landscape. An approximation method for PFSP that would make use of this big valley structure is proposed by using a critical block-based neighbourhood structure, and a genetic local search method called MSXFGA, previously developed for the job shop scheduling problem. Computational experiments using more challenging benchmark problems demonstrate the effectiveness of the proposed method.

Summary (3 min read)

1 Introduction

  • It is well known that Genetic Algorithms (GAs) can be enhanced by incorporating constructive heuristics or pointbased local search methods.
  • Section 3 explains the GLS approach based on the stochastic local search and MSXF.
  • Section 4 investigates the existence of a ‘big valley structure’ for the PFSP with stochastic local search and the representative neighbourhood.
  • Experimental results demonstrate the effectiveness of the proposed method.

2 The permutation flowshop scheduling problem

  • The permutation flowshop scheduling problem (PFSP) is often designated by the symbolsn=m=P=Cmax, wheren jobs have to be processed onmmachines in the same order.
  • P indicates that only permutation schedules are considered, where the order in which each machine processes the jobs is identical for all machines.
  • More sophisticated operators are obtained by limiting the size of the neighbourhood by using the notion of critical blocks.
  • For each jobin a critical block, letSaj be a set of moves that shift the jobj to some position in the next block; similarlySbj shiftsj to the previous block.
  • In this paper two well-known distances are considered as follows: precedence-based distance:.

3.2 Multi-step crossover fusion

  • The genetic crossover operator has two functions, which the authors denote by F1 and F2.
  • MSXF carries out a short term ‘navigated’ local search starting from one of the parent solutions to find new good solutions (F2), where the other parent is used as a reference point so that the search direction is biased toward it and therefore the search is limited between the parents (F1).
  • Let the parent solutions bep0 andp1, and let the distance between any two individualsx andy in any representation be d(x; y).
  • Hered(yi; p1) can be estimated easily if d(x; p1) and the direction of the transition fromx to yi are Algorithm 3.2 Multi-Step Crossover Fusion (MSXF) Let p0; p1 be parent solutions.
  • As previously suggested, the termination condition can be given by, for example, a fixed number of iterationsL in the outer loop.

4 Landscape analysis

  • The link between landscape and search algorithm is given by the NS operators used in the algo- rithm.
  • For the same PFSP but with simpler NS operators, similar experiments reported in Reeves [6] found such a landscape did occur.
  • As discussed in [4, 2, 6], the existence of a big valley structure can be examined by first generating a set of random local optima and then observing the correlation be- tween their objective function values and their distances to the nearest global optimum, and/or their average distances to other local optima.
  • Extensive preliminary experiments found only two distinct global optima for the ta011 problem, very close to each other in terms of the precedence-based distance (the distance is two) and only one global optimum for ta021 problem; however one cannot rule out the possibility of finding other different global optima by continuing the search.
  • Therefore, the use of the easily-computed precedence-based distance appears to be justified, and the ‘big valley’ structure can be assumed to hold for this neighbourhood.

5 MSXF-GA for PFSP

  • MSXF-GA provides a framework for traversing local op- tima without being trapped, by concentrating its attention on the area between the parent solutions and thus eventually finding a very good solution under the assumption of a ‘big valley’.
  • MSXF-GA was applied to PFSP using the representative neighbourhood described in Section 2 and the precedence-based distance.
  • Do 1. Select two schedulesp0; p1 probabilistically from the population with a probability inversely proportional to their ranks.
  • If q’s makespan is less than the worst in the population, and no member of the current population has the same makespan asq, replace the worst individual withq.

6 Experimental results

  • In Section 4, the existence of a big valley structure became clear for the relatively small-size PFSP instances.
  • An adaptive multi-start method (AMS) in which new local search Parent0 Parent1 Offspring MSXF Figure 4: Navigated local search by MSXF-GA: A new search is started from one of the parents and while no other good solutions are found, the search ‘navigates’ towards the other parent.
  • Table 1 summarizes the performance statistics of MSXF-GA for a subset of Taillard’s benchmark problems together with the results found by Nowicki and Smutnicki using their tabu search implementation[5] and the lower and upper bounds, taken from the OR-library [1].
  • In all, 30 runs were completed for each problem under the same conditions but with different random number seeds.
  • The results for larger problems are not as impressive as those of50 20 problems, but still good enough to support their hypothesis.

7 Conclusions

  • The landscape for the Permutation Flowshop Scheduling Problem with stochastic local search and the representative neighbourhood structure has been investigated.
  • The experimental analysis using20 10 and20 20 Taillard benchmark problems shows the existence of a ‘big valley’ structure for PFSP.
  • This suggests a well-designed AMS method, such as MSXF-GA in which new local search is concentrated in a region between previously found local optima should be effective in finding near-optimal solutions.
  • MSXF-GA for the PFSP is implemented using the neighbourhood operator and applied to more challenging benchmark problems.
  • Experimental results demonstrates the effectiveness of the proposed method.

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Permutation Flowshop Scheduling by Genetic Local Search
Takeshi Yamada
1
and Colin Reeves
2
1
NTT Communication Science Labs., Kyoto, Japan
Email: yamada@cslab.kecl.ntt.co.jp
2
Coventry University, UK
Email: CRReeves@coventry.ac.uk
Abstract
In this paper, the landscape for the permutation owshop
scheduling problem (PFSP) with stochastic local search
and a critical block-based neighbourhood structure has
been investigated. Numerical experiments using small
benchmark problems show that there are good correlations
between the makespans of local optima, the average dis-
tances to other local optima and the distances to the known
global optima. These correlations suggest the existence
of a ‘big valley’ structure, where local optima occur in
clusters over the landscape. An approximation method
for PFSP that would make use of this big valley struc-
ture is proposed by using a critical block-based neighbour-
hood structure, and a genetic local search method called
MSXF-GA, previously developedfor the job shop schedul-
ing problem. Computational experiments using more chal-
lenging benchmark problems demonstrate the effective-
ness of the proposed method.
1 Introduction
It is well known that Genetic Algorithms (GAs) can be en-
hanced by incorporating constructive heuristics or point-
based local search methods. The incorporation is often re-
ferred as Genetic Local Search (GLS) or population-based
local search. Basically, GLS can be viewed as a variant of
Adaptive Multi-Start (AMS) methods in which new start-
ing points are generated adaptively based on previously
found local optima. Yamada and Nakano have extended
the idea of GLS and proposedMulti-Step Crossover Fusion
(MSXF) [12]. Instead of generating a new starting point
from parents by a recombination operator, MSXF uses one
of the parents itself as a new starting point and carries out
a ‘navigated’ local search where the search direction is bi-
ased toward the other parent.
It has been shown that the success of AMS as well as
GAs strongly depends on some global structure in the cost
This work was undertaken when the first author was staying at
Coventry University.
surface or, in GA terminology, fitness landscape. For ex-
ample, Boese et al. [2] have suggested that, in the cases
of the travelling salesman problem (TSP) and graph bi-
section for which AMS worked well, local optima tend to
be relatively close to each other (in terms of a plausible
metric) and to the known global optimum. This structure,
where local optima occur in clusters, has been called a ‘big
valley’ structure. Jones and Forrest [4] have shown that
many GA-easy problems have strong correlation between
fitness and distance to the global optimum. More recently,
Reeves [7] has re-formulated the landscape concept in the
context of an associated neighbourhood structure and con-
firmed that the landscape of the (makespan minimizing)
permutation flowshop sequencing problem(PFSP) induced
by rather simple neighbourhood operators also exhibits a
big valley structure under a position-based metric.
In this paper, the PFSP landscape induced by a more so-
phisticated neighbourhood operator is investigated. This
neighbourhood focuses on critical blocks and uses a
stochastic local search based on the Metropolis criterion.
Section 2 gives the problem definitions and explains a crit-
ical block-based neighbourhood structure and the distance
measures. Section 3 explains the GLS approach based on
the stochastic local search and MSXF. Section 4 investi-
gates the existence of a ‘big valley structure’ for the PFSP
with stochastic local search and the representative neigh-
bourhood.
Assuming that a ‘big valley’ structure holds for a wide
range of PFSP landscapes induced by this neighbourhood
operator, it is expected that the use of an adaptive multi-
start method in which new local search is concentrated on
a region between previously found local optima should be
effective in finding near-optimal solutions even for more
difficult problems. Section 5 shows an implementation of
MSXF-GA for the PFSP. In Section 6 MSXF-GA is ap-
plied to more challenging benchmark problems. Exper-
imental results demonstrate the effectiveness of the pro-
posed method.

2 The permutation flowshop
scheduling problem
The permutation flowshop scheduling problem (PFSP) is
often designated by the symbols
n=m=P =C
max
,where
n
jobs have to be processedon
m
machinesin the same order.
The processing of each job on each machine is an opera-
tion which requires the exclusive use of the machine for
an uninterrupted duration called the processing time.
P
indicates that only permutation schedules are considered,
where the order in which each machine processes the jobs
is identical for all machines. Hence a schedule is uniquely
represented by a permutation of jobs. The objective is to
find a schedule that minimizes the makespan
C
max
,the
time at which the last job is completed on the last machine.
In general, a neighbourhood
N
(
x
)
of a point
x
in a
search space can be defined as a set of new points that
can be reached from
x
by exactly one transition or move
(a single perturbation of
x
). Several transition operators
have been proposed for PFSP. The simplest one is the adja-
cent pairwise exchange operator which exchanges the po-
sitions of two adjacent jobs. The shift operator which takes
a job from its current position and re-inserts it to another
position is shown to be the most efficient among simple
operators[6]. More sophisticated operators are obtained by
limiting the size of the neighbourhood by using the notion
of critical blocks. Nowicki and Smutnicki have proposed
the representative neighbourhood method[5], where a re-
duced neighbourhood is generated from the original criti-
cal block-based neighbourhood by clustering its members
and picking up the best move (representative) from each
cluster. A new neighbourhood is the set of all represen-
tative moves. They proposed a sophisticated tabu search
algorithm using this neighbourhoodstructure and an inten-
sification mechanism called ‘back jump tracking’.
In this paper, a simplified form of their representative
neighbourhood is adopted. For each job
j
in a critical
block, let
S
a
j
be a set of moves that shift the job
j
to some
position in the next block; similarly
S
b
j
shifts
j
to the previ-
ous block. Evaluate schedules obtained from each move in
S
a
j
and denote the best one by
s
a
j
. Similarly
s
b
j
is obtained
from
S
b
j
. Then the representative neighbourhoodis defined
as a set of all schedules obtained by representative moves
f
s
a
j
;s
b
j
g
for all jobs
j
in all critical blocks (see Figure 1).
This correspondsto the case of
=1
in the notation of [5].
The difference between two permutation schedules
S
and
T
can be measured by an appropriately defined dis-
tance. For example, an adjacency-based distance is most
commonly used for TSP, where relative ordering is more
important than absolute position in the sequence. On the
other hand, a distance which respects absolute position
more than relative ordering is more suitable for PFSP. In
this paper two well-known distances are considered as fol-
lows:
precedence-based distance: This distance counts the
the best move is selected
as a representative move
the job moved to
the next block
the job moved to
the previous block
j
S
a
j
S
b
j
s
b
j
s
a
j
Figure 1: The best move to the next/previous block is se-
lected as a representative.
number of job pairs
f
i; j
g
in which
j
is preceded by
i
in
S
but not in
T
.
position-based distance: This distance sums up the posi-
tional differences for each job in
S
and
T
.
As shown in the later section, these two distances are
strongly correlated with each other, and also with an ap-
proximation to the minimal number of steps of the neigh-
bourhood operator to move from
S
to
T
. In this paper the
precedence-based distance is used.
3 Genetic local search
It is well known that GAs are not well suited for fine-
tuning structures that are very close to optimal solutions,
and that it is essential to incorporate local search meth-
ods, such as neighbourhood search, into GAs. The re-
sult of such incorporation is often called Genetic Local
Search (GLS) [10]. This approach can be viewed as a vari-
ant of Adaptive Multi-Start (AMS) methods in which local
search is applied repeatedly, each time a new starting point
being generated adaptivelybasedon previouslyfound local
optima [2]. The Multi-Step Crossover Fusion (MSXF) GA
proposed by Yamada and Nakano [12] is one such GLS
method, and it has been applied successfully to job-shop
scheduling problems. This section briefly reviews neigh-
bourhood search and the MSXF.
3.1 Neighbourhood search
An outline of a neighbourhood search (NS) for minimiz-
ing
V
(
x
)
is described in Algorithm 3.1, where
x
denotes a
point in the search space,
V
(
x
)
denotes its objective func-
tion value and
N
(
x
)
its neighbourhood. The termination
condition can be given, for example, as a fixed number of
iterations
L
.
Step 1 in Algorithm 3.1 defines the NS operator: the
main part of NS. This operator is categorized by the way
a point is selected from
N
(
x
)
, which is called the choice
criterion. For example, a descent method selects a point
y
2
N
(
x
)
such that
V
(
y
)
<V
(
x
)
. A stochastic method
probabilisticallyselects a point accordingto the Metropolis

Algorithm 3.1 Neighbourhood search
Select a starting point:
x
best
=
x
=
x
0
.
do
1. Select a point
y
2
N
(
x
)
according to the given
criterion based on the value
V
(
y
)
.Set
x
=
y
.
2. If
V
(
x
)
<V
(
x
best
)
then set
x
best
=
x
.
until some termination condition is satisfied.
criterion, i.e.
y
2
N
(
x
)
is selected with probability 1 if
V
(
y
)
<V
(
x
)
; otherwise, with probability:
P
T
(
y
)=
exp
(
,
V=T
)
;
where
V
=
V
(
y
)
,
V
(
x
)
:
(1)
Here
P
T
is called the acceptance probability. Simulated
Annealing (SA) is a method in which the parameter
T
(called the temperature) decreases to zero following an an-
nealing schedule as the number of iterations increases.
Although SA is a well-known stochastic method and
has been successfully applied to many problems includ-
ing scheduling problems, it would be unrealistic to apply
a full SA search within a GA because it would consume
too much time. Therefore a restricted search with a fixed
temperature parameter
T
=
c
is used in MSXF.
3.2 Multi-step crossover fusion
The genetic crossover operator has two functions, which
we denote by F1 and F2. Firstly (F1) it focuses attention
on a region between the parents in the search space; sec-
ondly (F2), it picks up possibly good solutions from that
region. Unlike traditional crossover operators, MSXF is
more search oriented: it is designed as an extension of lo-
cal search algorithm described in Algorithm 3.1, but has
the functions F1 and F2, and it is still called ’crossover’.
MSXF carries out a short term ‘navigated’ local search
starting from one of the parent solutions to find new good
solutions (F2), where the other parent is used as a reference
point so that the search direction is biased toward it and
therefore the search is limited between the parents (F1). A
stochastic local search algorithm is used for its base algo-
rithm. A similar idea is described under the title ‘path re-
linking’ [3] in the context of tabu search. MSXF is defined
in a problem-independent manner using a neighbourhood
structure and a distance measure, both of which are very
common for most combinatorial optimization problems.
Let the parent solutions be
p
0
and
p
1
, and let the distance
between any two individuals
x
and
y
in any representation
be
d
(
x; y
)
. A short term local search is carried out start-
ing from
p
0
and using
p
1
as a reference point as follows.
First
x
is set to
p
0
. All members in
N
(
x
)
are sorted so
that
y
i
2
N
(
x
)
with a smaller index
i
has a smaller dis-
tance
d
(
y
i
;p
1
)
. Here
d
(
y
i
;p
1
)
can be estimated easily if
d
(
x; p
1
)
and the direction of the transition from
x
to
y
i
are
Algorithm 3.2 Multi-Step Crossover Fusion (MSXF)
Let
p
0
;p
1
be parent solutions.
Set
x
=
q
=
p
0
.
do
For each member
y
i
2
N
(
x
)
, calculate
d
(
y
i
;p
1
)
.
Sort
y
i
2
N
(
x
)
in ascending order of
d
(
y
i
;p
1
)
.
do
1. Select
y
i
from
N
(
x
)
probabilistically ac-
cording to a probability inversely propor-
tional to the index
i
.
2. Calculate
V
(
y
i
)
if
y
i
has not yet been vis-
ited.
3. Accept
y
i
with probability one if
V
(
y
i
)
V
(
x
)
, and with
P
c
(
y
i
)
otherwise.
4. Change the index of
y
i
from
i
to
n
,andthe
indices of
y
k
(
k
2f
i
+1
;i
+2
;:::;n
g
) from
k
to
k
,
1
.
until
y
i
is accepted.
Set
x
=
y
i
.
If
V
(
x
)
<V
(
q
)
then set
q
=
x
.
until some termination condition is satisfied.
q
is used for the next generation.
known; it is not necessary to generate and evaluate
y
i
.One
of the members
y
i
2
N
(
x
)
is selected with a probability
inversely proportional to the index
i
.Then
y
i
is accepted
according to the Metropolis criterion using
T
=
c
in Equa-
tion (1). MSXF is described in outline in Algorithm 3.2.
As previously suggested, the termination condition can be
given by, for example, a fixed number of iterations
L
in the
outer loop. The best solution
q
is used for the next genera-
tion.
MSXF is not applicable if the distance between
p
0
and
p
1
is too small compared to the number of iterations. If
this happens very often, it means that the population has
already converged to a specific region of the search space.
In such a case, a mutation operator called Multi-Step Muta-
tion Fusion (MSMF) is applied to help diversify the search
again. MSMF can be defined in the same manner as MSXF
except that
N
(
x
)
members are sorted in descending order
of
d
(
y
i
;p
1
)
in Algorithm 3.2, and the most distant solution
is stored and used in the next generation instead of
q
,if
q
does not improve the parent solutions.
4 Landscape analysis
According to H¨ohn and Reeves [7], a landscape is defined
by a triple of a search space, an objective function and a
distance measure. The link between landscape and search
algorithm is given by the NS operators used in the algo-

ta011(a)
OBJFN
MEAND
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
40 50 60 70 80 90
ta011(b)
OBJFN
BESTD
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
0 20406080100
ta021(a)
OBJFN
MEAND
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
70 80 90 100 110
ta021(b)
OBJFN
BESTD
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
20 40 60 80 100 120 140
Figure 2: 1841 distinct local optima obtained from 2500 short term local search for the ta011 (
20
10
) problem and 2313
distinct local optima for the ta021 (
20
20
) problem are plotted in terms of (a) average distance from other local optima
and (b) distance from global optima (
x
-axis), against their relative objective function values (
y
-axis).
rithm. Because these operators generate new points in the
search space relative to a given point, they define a distance
d
N
(
s; t
)
on the search space given by the minimum num-
ber of applications of operator
N
that will convert element
t
into element
s
.
One can understand the degree of difficulty of the given
combinatorial optimization problem by looking at its land-
scape: if the landscape is simple and has only one peak,
it is very easy to find the global optimum by using simple
best ascent search. Unfortunately most
NP
-hard combi-
natorial optimization problems, including PFSP, have very
‘rugged’ landscapes with many false peaks under any NS
operator.
Recently, Boese et al. [2] have shown that an appropri-
ate choice of NS operator introduces some neat structure
into the landscape. In this ‘big valley’ structure, local op-
tima occur in clusters good candidate solutions are usu-
ally to be found ‘fairly close’ to other good solutions. If
a landscape has this structure, it would support the idea
of generating new starting points for search from a previ-
ous local optimum rather than from a random point in the
search space.
Before we apply our GLS method to PFSP, we investi-
gate whether there is a big valley structure for the PFSP
and the NS operator using the representative neighbour-
hood and a stochastic search based on Equation (1) at
T
=
c
(constant temperature). For the same PFSP but
with simpler NS operators, similar experiments reported
in Reeves [6] found such a landscape did occur.
As discussed in [4, 2, 6], the existence of a big valley
structure can be examined by first generating a set of ran-
dom local optima and then observing the correlation be-

tween their objective function values and their distances to
the nearest global optimum, and/or their average distances
to other local optima. The distance used here should be
d
N
for an operator
N
. However this distance is difficult to
compute, and precedence-based distance is used here as an
approximation.
Figure 2 shows a scatter plot of random local optima
for problems ta011 and ta021, being respectively the first
of Taillard’s
20
10
and
20
20
groups of problems [9].
Each local optimumis generated by runningthe neighbour-
hood search described in Algorithm 3.1 with
L
= 5000
based on the stochastic method with acceptance probabil-
ity
P
c
,
c
= 5
. Extensive preliminary experiments found
only two distinct global optima for the ta011 problem, very
close to each other in terms of the precedence-based dis-
tance (the distance is two) and only one global optimum
for ta021 problem; however one cannot rule out the possi-
bility of finding other differentglobal optima by continuing
the search. However, more than 2500 global optima were
found for the smaller ta001 (
20
5
) problem by spending
the same amount of CPU time.
The
x
-axis in Figure 2 represents (a) the average
precedence-based distance from other local optima (ME-
AND), and (b) the precedence-based distance from one of
the nearer global optima (BESTD). The
y
-axis represents
their objective function values relative to the global opti-
mum. These plots clearly show that there are good corre-
lations between the distances and objective function val-
ues. The calculated correlation coefficients for each plot
are: ta011(a): 0.74, ta011(b): 0.50, ta021(a): 0.62 and
ta021(b): 0.44. These values are statistically significant
at the 0.1% level, on the basis of 1000 replications in a
randomization test [6]. These high correlations suggest
that the local optima are radially distributed in the problem
space with the global optima as the centre, and the more
distant are the local optima from the centre, the worse are
their objective function values. Hence, by tracing local op-
tima step by step, moving from one optimum to nearby
slightly better one, without being trapped, one can eventu-
ally reach a near global optimal solution.
In the analysis above, the precedence-based distance is
used as a surrogate for
d
N
, because the minimum number
of steps for the neighbourhoodoperator to reach the global
optimum is difficult to compute. Although the precedence-
based distance seems to be a good alternative, the approx-
imation still need to be justified. For this purpose, the ap-
proximate number of steps to reach the global optimum
from each local optimum was calculated by choosing the
closest move to the global optimum each time from the
neighbourhood. While this does not necessarily give the
best distance between two points, it seems likely to give a
fairly close upper bound.
Figure 3 (a) shows the correlation between the
precedence-based distance and the approximate number
of steps for the local optima shown in Figure 2 ta011(a)
(correlation coefficient is 0.66). Figure 3 (b) shows that
(a)
STEPS
PREC
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
0 20406080100
(b)
POSN
PREC
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
0 20406080100
Figure 3: analysis of 1000 random local optima for
20
10
flowshop problem. The x-axis shows precedence-based
distance from the global minimum and the y-axis shows
the makespan.
there is a strong correlation between the precedence-based
distance and the position-based distance for the same lo-
cal optima (correlation coefficient is 0.91). Thus it does
not matter which distance is used. The same kind of ex-
periments were carried out for all Taillard’s
20
10
and
20
20
benchmarks, and similar results were obtained
in every case. Therefore, the use of the easily-computed
precedence-based distance appears to be justified, and the
‘big valley’structure can be assumed to hold for this neigh-
bourhood.
5 MSXF-GA for PFSP
As described in Section 3.2, the MSXF operator is de-
signed to find a new local optimum basedon previous ones.
MSXF-GA provides a framework for traversing local op-

Citations
More filters
Book
31 Jul 1997
TL;DR: This book explores the meta-heuristics approach called tabu search, which is dramatically changing the authors' ability to solve a host of problems that stretch over the realms of resource planning, telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics, pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservation and scores of other problems.
Abstract: From the Publisher: This book explores the meta-heuristics approach called tabu search, which is dramatically changing our ability to solve a hostof problems that stretch over the realms of resource planning,telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics,pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservationand scores of other problems. The major ideas of tabu search arepresented with examples that show their relevance to multipleapplications. Numerous illustrations and diagrams are used to clarifyprinciples that deserve emphasis, and that have not always been wellunderstood or applied. The book's goal is to provide ''hands-on' knowledge and insight alike, rather than to focus exclusively eitheron computational recipes or on abstract themes. This book is designedto be useful and accessible to researchers and practitioners inmanagement science, industrial engineering, economics, and computerscience. It can appropriately be used as a textbook in a masterscourse or in a doctoral seminar. Because of its emphasis on presentingideas through illustrations and diagrams, and on identifyingassociated practical applications, it can also be used as asupplementary text in upper division undergraduate courses. Finally, there are many more applications of tabu search than canpossibly be covered in a single book, and new ones are emerging everyday. The book's goal is to provide a grounding in the essential ideasof tabu search that will allow readers to create successfulapplications of their own. Along with the essentialideas,understanding of advanced issues is provided, enabling researchers togo beyond today's developments and create the methods of tomorrow.

6,373 citations


Cites background from "Permutation flowshop scheduling by ..."

  • ...…Design – Consiglio and Zenios (1999) • Neural Network Training – Kelly, Rangaswamy and Xu (1996) • Job Shop Scheduling – Yamada and Nakano (1996) • Flow Shop Scheduling – Yamada and Reeves (1997) • Graph Drawing – Laguna and Marti (1999) • Linear Ordering – Laguna, Marti and Campos (1997) •…...

    [...]

Journal Article
TL;DR: The features of Scatter Search and Path Relinking are described, which set them apart from other evolutionary approaches, and that offer opportunities for creating increasingly more versatile and effective methods in the future.
Abstract: —The evolutionary approach called Scatter Search, and its generalized form called Path Relinking, have proved unusually effective for solving a diverse array of optimization problems from both classical and real world settings. Scatter Search and Path Relinking differ from other evolutionary procedures, such as genetic algorithms, by providing unifying principles for joining solutions based on generalized path constructions (in both Euclidean and neighborhood spaces) and by utilizing strategic designs where other approaches resort to randomization. Scatter Search and Path Relinking are also intimately related to the Tabu Search metaheuristic, and derive additional advantages by making use of adaptive memory and associated memory-exploiting mechanisms that are capable of being adapted to particular contexts. We describe the features of Scatter Search and Path Relinking that set them apart from other evolutionary approaches, and that offer opportunities for creating increasingly more versatile and effective methods in the future.

801 citations

Journal Article
TL;DR: This work identifies a template for scatter search and path relinking methods that provides a convenient and user friendly basis for their implementation and describes Illustrative forms of these subroutines that make it possible to create methods for a wide range of optimization problems.
Abstract: Scatter search and its generalized form called path relinking are evolutionary methods that have recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints, these methods use strategies for combining solution vectors that have proved effective for scheduling, routing, financial product design, neural network training, optimizing simulation and a variety of other problem areas. These approaches can be implemented in multiple ways, and offer numerous alternatives for exploiting their basic ideas. We identify a template for scatter search and path relinking methods that provides a convenient and user friendly basis for their implementation. The overall design can be summarized by a small number of key steps, leading to versions of scatter search and path relinking that are fully specified upon providing a handful of subroutines. Illustrative forms of these subroutines are described that make it possible to create methods for a wide range of optimization problems.

711 citations


Cites background from "Permutation flowshop scheduling by ..."

  • ...…– Consiglio and Zenios (1996) Neural Network Training – Kelly, Rangaswamy and Xu (1996) Job Shop Scheduling – Yamada and Nakano (1996) Flow Shop Scheduling – Yamada and Reeves (1997) Graph Drawing – Laguna and Marti (1997) Linear Ordering – Laguna, Marti and Campos (1997) Unconstrained…...

    [...]

Book ChapterDOI
01 Oct 1997
TL;DR: In this article, the authors present a template for scatter search and path relinking methods that provides a convenient and "user friendly" basis for their implementation, which can be summarized by a small number of key steps, leading to versions of scatter search that are fully specified upon providing a handful of subroutines.
Abstract: Scatter search and its generalized form called path relinking are evolutionary methods that have recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints, these methods use strategies for combining solution vectors that have proved effective for scheduling, routing, financial product design, neural network training, optimizing simulation and a variety of other problem areas. These approaches can be implemented in multiple ways, and offer numerous alternatives for exploiting their basic ideas. We identify a template for scatter search and path relinking methods that provides a convenient and “user friendly” basis for their implementation. The overall design can be summarized by a small number of key steps, leading to versions of scatter search and path relinking that are fully specified upon providing a handful of subroutines. Illustrative forms of these subroutines are described that make it possible to create methods for a wide range of optimization problems.

273 citations

Journal ArticleDOI
TL;DR: This paper examines recent developments in the field of evolutionary computation for manufacturing optimization with a wide range of problems, from job shop and flow shop scheduling, to process planning and assembly line balancing.
Abstract: The use of intelligent techniques in the manufacturing field has been growing the last decades due to the fact that most manufacturing optimization problems are combinatorial and NP hard. This paper examines recent developments in the field of evolutionary computation for manufacturing optimization. Significant papers in various areas are highlighted, and comparisons of results are given wherever data are available. A wide range of problems is covered, from job shop and flow shop scheduling, to process planning and assembly line balancing.

264 citations


Cites methods from "Permutation flowshop scheduling by ..."

  • ...The only exceptions are Reeves [75], Yamada and Reeves [107], and Ross and Tuson [103], who presented results on standard benchmark problems taken from Tailard [112]....

    [...]

  • ...NRX is, in fact, a neighborhood search algorithm, as the MSFX operator proposed by Yamada and Reeves [107]....

    [...]

  • ...Among them, Reevest al....

    [...]

  • ...only exceptions are Reeves [75], Yamada and Reeves [107], and Ross and Tuson [103], who presented results on standard benchmark problems taken from Tailard [112]....

    [...]

References
More filters
Book
31 Jul 1997
TL;DR: This book explores the meta-heuristics approach called tabu search, which is dramatically changing the authors' ability to solve a host of problems that stretch over the realms of resource planning, telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics, pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservation and scores of other problems.
Abstract: From the Publisher: This book explores the meta-heuristics approach called tabu search, which is dramatically changing our ability to solve a hostof problems that stretch over the realms of resource planning,telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics,pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservationand scores of other problems. The major ideas of tabu search arepresented with examples that show their relevance to multipleapplications. Numerous illustrations and diagrams are used to clarifyprinciples that deserve emphasis, and that have not always been wellunderstood or applied. The book's goal is to provide ''hands-on' knowledge and insight alike, rather than to focus exclusively eitheron computational recipes or on abstract themes. This book is designedto be useful and accessible to researchers and practitioners inmanagement science, industrial engineering, economics, and computerscience. It can appropriately be used as a textbook in a masterscourse or in a doctoral seminar. Because of its emphasis on presentingideas through illustrations and diagrams, and on identifyingassociated practical applications, it can also be used as asupplementary text in upper division undergraduate courses. Finally, there are many more applications of tabu search than canpossibly be covered in a single book, and new ones are emerging everyday. The book's goal is to provide a grounding in the essential ideasof tabu search that will allow readers to create successfulapplications of their own. Along with the essentialideas,understanding of advanced issues is provided, enabling researchers togo beyond today's developments and create the methods of tomorrow.

6,373 citations

Journal ArticleDOI
TL;DR: This paper proposes 260 randomly generated scheduling problems whose size is greater than that of the rare examples published, and the objective is the minimization of the makespan.

2,173 citations

Proceedings Article
01 Jun 1989

2,164 citations


"Permutation flowshop scheduling by ..." refers methods in this paper

  • ...1 describes the outline of the MSXF-GA routine for the PFSP using the steady state model proposed in [8, 11]....

    [...]

Journal ArticleDOI
TL;DR: A system (OR-Library) that distributes test problems by electronic mail (e-mail) that has available test problems drawn from a number of different areas of operational research.
Abstract: In this note we present a system (OR-Library) that distributes test problems by electronic mail (e-mail). This system currently has available test problems drawn from a number of different areas of...

1,939 citations


"Permutation flowshop scheduling by ..." refers methods in this paper

  • ...Table 1 summarizes the performance statistics of MSXF-GA for a subset of Taillard’s benchmark problems together with the results found by Nowicki and Smutnicki using their tabu search implementation[5] and the lower and upper bounds, taken from the OR-library [1]....

    [...]

  • ...It can be seen that the results for50 20 problems are remarkable: the solution qualities of our best results are improved over those found in [5] for most of the problems, and some results (marked in bold letters) are even better than the existing best results reported in the OR-library....

    [...]

Proceedings Article
01 Jun 1989
TL;DR: This paper reports work done over the past three years using rank-based allocation of reproductive trials to suggest that allocating reproductive trials according to rank is superior to tness proportionate reproduction.
Abstract: This paper reports work done over the past three years using rank-based allocation of reproductive trials. New evidence and arguments are presented which suggest that allocating reproductive trials according to rank is superior to tness proportionate reproduction. Ranking can not only be used to slow search speed, but also to increase search speed when appropriate. Furthermore, the use of ranking provides a degree of control over selective pressure that is not possible with tness proportionate reproduction. The use of rank-based allocation of reproductive trials is discussed in the context of 1) Holland's schema theorem, 2) DeJong's standard test suite, and 3) a set of neural net optimization problems that are larger than the problems in the standard test suite. The GENITOR algorithm is also discussed; this algorithm is speciically designed to allocate reproductive trials according to rank.

1,314 citations


"Permutation flowshop scheduling by ..." refers methods in this paper

  • ...1 describes the outline of the MSXF-GA routine for the PFSP using the steady state model proposed in [8, 11]....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "Permutation flowshop scheduling by genetic local search" ?

In this paper, the landscape for the permutation flowshop scheduling problem ( PFSP ) with stochastic local search and a critical block-based neighbourhood structure has been investigated. These correlations suggest the existence of a ‘ big valley ’ structure, where local optima occur in clusters over the landscape.