AN IMPROVED ANT SYSTEM ALGORITHM FOR UNEQUAL AREA
FACILITY LAYOUT PROBLEMS
OCTOBER 2009
A thesis submitted in fulfillment of the
requirements for the award of the degree of
Master of Engineering (Mechanical Engineering)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
KOMARUDIN
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To my God, Allah 'azza wa jalla
Then to my beloved parents, wife and daughter
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ACKNOWLEDGEMENTS
In preparing this thesis, I was in contact with many people, researchers, and
academicians. They have contributed towards my understanding and thoughts. In
particular, I wish to express my sincere appreciation to my main thesis supervisor,
Dr. Wong Kuan Yew, for his encouragement, guidance, critics and friendship. I am
also indebted to Universiti Teknologi Malaysia (UTM) for funding my Master study.
My fellow friends from the Indonesian Student Association (PPI) should also
be recognized for their support. My sincere appreciation also extends to all my
colleagues and others who have provided assistance at various occasions. Their
views and tips are useful indeed. Unfortunately, it is not possible to list all of them in
this limited space. I am grateful to all my family members.
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ABSTRACT
To date, a formal Ant Colony Optimization (ACO) based metaheuristic has
not been applied for solving Unequal Area Facility Layout Problems (UA-FLPs).
This study proposes an Ant System (AS) algorithm for solving UA-FLPs using the
Flexible Bay Structure (FBS) representation. In addition, this study proposes an
improvement to the FBS representation when solving problems which have empty
spaces. The proposed algorithm uses several types of local search to improve its
search performance. It was extensively tested using 20 well-known problem
instances taken from the literature. The proposed algorithm is effective and can
produce all of the best FBS solutions (or even better) except for 2 problem instances.
In addition, it can improve the best-known solution for 7 problem instances. The
improvement gained by the proposed algorithm is up to 21.36% compared to
previous research. Evidently, the proposed algorithm is also proven to be effective
when solving large problem sets with 20, 25, and 30 departments. Furthermore, this
study has implemented a Fuzzy Logic Controller (FLC) to automate the tuning of the
AS algorithm. The experiments involved tuning four parameters individually, i.e.
number of ants, pheromone information parameter, heuristic information parameter,
and evaporation rate, as well as tuning all of them at once. The results showed that
FLC could be used to replace manual parameter tuning which is time consuming.
The results also showed that instead of using static parameter values, FLC has the
potential to help the AS algorithm to achieve better objective function values.
.
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ABSTRAK
Sehingga kini, metaheuristik Ant Colony Optimization (ACO) belum lagi digunakan
untuk menyelesaikan Unequal Area Facility Layout Problems (UA-FLPs). Matlamat
projek ini adalah untuk membangunkan satu algoritma Ant System (AS) untuk
menyelesaikan UA-FLPs dengan menggunakan model Flexible Bay Structure (FBS).
Selain itu, kajian ini juga membuat pembaikan kepada model FBS apabila
menyelesaikan UA-FLPs yang mengandungi ruang kosong. Algoritma yang dibina
tersebut menggunakan beberapa jenis pencarian tempatan (local search) untuk
meningkatkan prestasi pencariannya. Ianya diuji secara luas menggunakan 20
masalah terkenal yang diambil daripada literatur. Algoritma tersebut adalah berkesan
dan boleh menghasilkan semua solusi FBS terbaik (bahkan lebih baik) kecuali untuk
2 set masalah. Selain itu, algoritma tersebut juga boleh memperbaiki solusi terbaik
untuk 7 set masalah. Pembaikan yang diperolehi mencapai 21.36% bila dibandingkan
dengan kajian-kajian yang lepas. Jelasnya, algoritma ini juga berkesan untuk
menyelesaikan masalah bersaiz besar yang mempunyai 20, 25, dan 30 jabatan.
Tambahan lagi, projek ini telah menerapkan Fuzzy Logic Controller (FLC) untuk
menala secara otomatik nilai parameter-parameter di dalam algoritma AS.
Eksperimen dengan FLC membabitkan penalaan empat parameter secara berasingan,
iaitu number of ants, pheromone information parameter, heuristic information
parameter, dan evaporation rate, dan penalaan semua empat parameter secara
sekaligus. Hasilnya menunjukkan bahawa FLC boleh digunakan untuk menggantikan
penalaan parameter secara manual. Hasilnya juga menunjukkan bahawa FLC
berpotensi untuk membantu algoritma AS untuk mencapai fungsi objektif yang lebih
baik daripada menggunakan nilai parameter yang statik.
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TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES x
LIST OF FIGURES xi
LIST OF ABBREVIATIONS xii
LIST OF APPENDICES xv
1 INTRODUCTION 1
1.1 Overview 1
1.2 Background of Research 2
1.3 Problem Statement 3
1.4 Objective of Study 4
1.5 Scope of Study 4
1.6 Organization of Thesis 5
1.7 Conclusions 6
2 LITERATURE REVIEW 7
2.1 Introduction 7
2.2 Combinatorial Optimization Problem 7
2.2.1 Complexity of Combinatorial Optimization 8
2.2.2 Intensification and Diversification in
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Metaheuristic 10
2.2.3 Parameter Tuning for Metaheuristics 12
2.3 Facility Layout Problems 15
2.4 Unequal Area Facility Layout Problems 20
2.4.1 Exact procedures 26
2.4.2 Heuristic Procedures 30
2.4.3 Metaheuristic Algorithm 32
2.5 Ant Colony Optimization 35
2.5.1 Ant Colony Optimization Framework 36
2.5.2 ACO Applications in Facility Layout Problems 38
2.5.3 Parameter Tuning for Ant Colony Optimization 41
2.6 Conclusions 45
3 METHODOLOGY 47
3.1 Introduction 47
3.2 Methodology 47
3.2.1 Ant System Development 50
3.2.2 Fuzzy Logic Controller (FLC) Development 52
3.3 Research Hypothesis 54
3.4 Data Collection 54
3.5 Conclusions 56
4 ANT SYSTEM FORMULATION 57
4.1 Introduction 57
4.2 Structure of the Proposed Ant System (AS) Algorithm 57
4.2.1 Solution representation of Unequal Area Facility
Layout Problems 59
4.2.2 Objective function 62
4.2.3 Ant Solutions Construction 63
4.2.4 Local Search Procedures 66
4.2.5 Pheromone update scheme 67
4.2.6 Stopping criteria 68
4.3 Implementation details of the algorithm 68
4.4 Conclusions 72
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5 PARAMETER TUNING USING FUZZY LOGIC 73
5.1 Introduction 73
5.2 Manual Parameter Tuning 73
5.3 Fuzzy Logic Controller 75
5.4 FLC-AS Formulation 77
5.5 Conclusions 81
6 EVALUATION OF THE PROPOSED ALGORITHM 82
6.1 Introduction 82
6.2 Problem Sets and Parameter settings used to evaluate the
Proposed Algorithm 82
6.3 Evaluation of the Proposed Algorithm with Manual
Parameter Tuning 85
6.4 Evaluation of the Proposed Algorithm with Fuzzy Logic
Controller 88
6.5 Conclusions 93
7 CONCLUSIONS 94
7.1 Introduction 94
7.2 Conclusions 95
7.3 Future Work 97
REFERENCES 98
PUBLICATIONS 110
Appendices A-B 111-136
x
LIST OF TABLES
TABLE NO. TITLE PAGE
Table 2.1 Comparisons of FLP classes 17
Table 2.2 Ant Colony Optimization general form 36
Table 2.3 Applications of ACO algorithms in FLPs 39
Table 2.4 ACO parameter values in various optimization problems 42
Table 2.5 Parameter Tuning for ACO algorithms 45
Table 3.1 UA-FLP problem sets used in this research 55
Table 3.2 Characteristics of problem sets used in previous research 56
Table 5.1 Manual parameter tuning results 75
Table 5.2 Rule bases for (a) number of ants and evaporation rate, and (b)
pheromone information parameter and heuristic information
parameter 79
Table 6.1 Problem set data 83
Table 6.2 Statistical data on results and computation time of the proposed
algorithm 85
Table 6.3 Results and comparison of the proposed algorithm with previous
research 86
Table 6.4 Results and comparison of the proposed algorithm with the best-FBS
and best-known solutions 87
Table 6.5 Results and comparison of the proposed algorithm with Fuzzy Logic
Controller 89
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LIST OF FIGURES
FIGURE NO. TITLE
PAGE
Figure 2.1 Exponential complexity and its blow-up point (modified from Ibaraki,
1987) 9
Figure 2.2 An illustration of intensification and diversification in metaheuristics
(adapted from Blum and Roli, 2003) 11
Figure 2.3 (a) Block Layout and (b) Detailed Layout 22
Figure 2.4 The solution of UA-FLP with continuous representation by Anjos and
Vannelli (2006) 23
Figure 2.5 The solution of UA-FLP with QAP model by Hardin and Usher 24
Figure 2.6 The solution of UA-FLP with FBS model in Shebanie II (2004) 25
Figure 2.7 The solution of UA-FLP with STS model in Shebanie II (2004) 25
Figure 2.8 Demonstration of the shortest path finding capability of an ant colony
(Blum, 2005a) 36
Figure 3.1 Research methodology main steps 48
Figure 3.2 FLC-AS development methodology 51
Figure 4.1 The proposed AS algorithm 58
Figure 4.2 An Example of Solution Representation in the Proposed Algorithm 59
Figure 4.3 Layout generated from the ant solution representation in Figure 4.2 60
Figure 4.4 An example of layout solution with (a) original FBS and (b) modified-
FBS 61
Figure 4.5 Details of the proposed AS algorithm 69
Figure 5.1 Scheme for parameter tuning in metaheuristics 76
Figure 5.2 The proposed FLC-AS algorithm 78
Figure 5.3 Graphs showing membership functions for (a) input parameters, and
(b) output parameters 79
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LIST OF SYMBOLS
A - the total area of the facility
cij - the cost per unit distance per unit material from department i to
department j
Cik - department i located in location candidate k (department i located in
the kth element of the department sequence)
dij - the distance between departments i and j
f(s) - ant objective function
F(s) - quality function
fij - the number of material flow from departments i to department j
H - the height of the facility
hi - the height of department i
l i - length of department i
lmin - minimum length of department
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mi - sum of material flow from and to department i
n - the number of departments
N(s) - available departments which have not been used in the corresponding
ant
p(Cik) - probability to locate department i to location candidate k
pinf - the number of infeasible departments
Supd - the set of ants that is used for updating the pheromone
Vall - the best overall objective function value found
Vfeas - the best feasible objective function value found
w - weight for the corresponding ant solution s.
W - the width of the facility
wi - the width of department i
x - the centroid of an area below fuzzy membership function µc.
xk - x axis value for the center of location candidate k
xo - the crisp value
yk - y axis value for the center of location candidate k
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α - pheromone information parameter
αmax - maximum aspect ratio
β - heuristic information parameter
λk - the parameter for location candidate k
µc - fuzzy membership function
ρ - evaporation rate
τik - pheromone value associated with the solution to locate department i to
location candidate k
xv
LIST OF APPENDICES
APPENDIX TITLE PAGE
A Data Sets for UA-FLPs 112
B Best layout obtained by the proposed AS
algorithm
127
CHAPTER 1
INTRODUCTION
1.1 Overview
Determining the physical organization of a production system is defined as a
Facility Layout Problem (FLP). The term production system does not only point to
manufacturing, but it also concerns service and communication systems (Meller and
Gau, 1996). The objective of an FLP is mainly to minimize the total material
handling cost of the production system. It can also include or combine various
objectives such as to maximize space utilization, maximize flexibility, and maximize
employee satisfaction and safety (Muther, 1955). In general, FLPs can include
various layouts such as manufacturing, hospital, office, and construction layouts.
One of the important FLPs is the Unequal Area Facility Layout Problem (UA-
FLP) which was originally formulated by Armour and Buffa (1963). In UA-FLPs,
there is a fixed rectangular facility with dimensions H and W, where H is the height
and W is the width. A number of departments which do not need to have the same
area requirement must be arranged according to the following criteria: (1) all
departments must be located inside the facility, (2) all departments must not overlap
with each other, and (3) the final dimensions of the departments must fulfill some
maximum ratio constraints and/or minimum value restrictions (Meller and Gau,
1996). The goal of the problem is to partition the facility into departments so as to
minimize the total material movement cost. This objective is based on the material
handling principle that material handling cost increases proportionally with the
distance which must be traveled by the materials.
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A UA-FLP is a complete Nondeterministic Polynomial (NP) problem and
thus, recent research using exact algorithms can only optimally solve up to 11
departments (Meller et al., 2007). In recent years, researchers have proposed several
metaheuristic approaches to obtain high quality solutions for large UA-FLPs (Meller
and Gau, 1996; Singh and Sharma, 2005). Researchers have also proposed several
UA-FLP representations such as Flexible Bay Structure (FBS), Slicing Tree Structure
(STS), and Quadratic Assignment Problem (QAP) to reduce the complexity of UA-
FLPs (Meller and Gau, 1996).
1.2 Background of Research
As a metaheuristic, the Ant Colony Optimization (ACO) family was
introduced by M. Dorigo and his colleagues in the early 1990s (Dorigo et al., 1996).
The first ACO algorithm developed was Ant System (AS) and subsequently, many
variants have been developed in order to improve the algorithm performance. Among
other ACO variants are Elitist AS (EAS), Rank-based AS (RAS), MAX-MIN Ant
System (MMAS), and Ant Colony System (ACS). ACO was initially tested for
solving the Traveling Salesman Problem (TSP), and then it has been implemented to
solve various kinds of optimization problems such as Facility Layout Problem (FLP),
scheduling, vehicle routing, timetabling, data mining, bioinformatics problems, etc.
(Blum, 2005a).
One area which has been successfully solved by ACO is FLP. It is a very hard
optimization problem and can be applied in many application areas (Meller and Gau,
1996). These reasons have encouraged many researchers to study it. The use of ACO
to solve FLPs is mainly based on two reasons (Stützle and Dorigo, 1999). Firstly,
ACO is suitable to solve discrete optimization problems because its solution structure
uses a discrete solution representation. Secondly, ACO can use heuristic information
to exploit the problems. This heuristic information can help ACO to search and
examine promising solution regions.
3
Unfortunately, the application of ACO in solving FLPs is mainly limited to
Quadratic Assignment Problems (QAPs) as will be discussed later in the literature
review section. Only little research has been done to solve other classes of FLPs. For
this reason, this research aims to implement ACO for solving another class of FLP,
i.e. UA-FLPs.
When using metaheuristic algorithms, one of the important factors is the
balance between intensification and diversification in the search space. To provide
this balance, determining the right values for metaheuristic parameters is one of the
critical issues (Eiben et al., 1999). To handle this matter, this research implements a
Fuzzy Logic Controller (FLC) to automatically tune the ACO’s parameters.
1.3 Problem Statement
UA-FLP as one of the well-known problems in facility planning has been
studied for years. It is a very difficult problem because of the constraints contained.
Conventional methods can only optimally solve UA-FLPs with up to 11 departments
(Meller et al., 2007). Meanwhile, the size of the problem still grows and the largest
problem currently has 35 departments (Liu and Meller, 2007). Therefore, researchers
have proposed several metaheuristic approaches to solve UA-FLPs especially for the
large problems (Tate and Smith, 1995; Liu and Meller, 2007; Scholz et al., 2009).
Basically, metaheuristic approaches can only report the best-obtained solutions since
they may not guarantee to obtain the optimum solutions. Therefore, there is an
opportunity to use other metaheuristic approaches to improve the best-known
solutions for UA-FLPs.
On the other hand, ACO has not been used for solving UA-FLPs although it is
considered as one of the state of the art algorithms for solving FLPs, particularly
QAPs. Therefore, it is advantageous to implement ACO for solving UA-FLPs in
order to show that it can be used as an alternative to build a facility layout. Such an
implementation may achieve better results compared to approaches proposed by
4
previous studies. This will then encourage the use of ACO for solving UA-FLPs,
with more confidence.
The selection of parameter values for ACO is essential to its performance.
Past methods which have been used for parameter tuning in ACO have been
reviewed but no one has used FLC to tune ACO’s parameter values. On the other
hand, FLC performed well when it was implemented to tune the parameter values of
other metaheuristic algorithms. Therefore, the use of FLC to automatically tune the
ACO’s parameters is considered necessary because it may benefit ACO’s
performance. If the results are as good as those obtained from manual tuning, then
FLC can be used to replace manual tuning which is time consuming.
1.4 Objective of Study
The objectives of this research are to:
i. Formulate an AS algorithm with FLC for solving UA-FLPs. Particularly, FLC is
used to automatically tune the parameter values in AS.
ii. Evaluate the performance of the algorithm in solving UA-FLPs. The evaluation is
based on comparing its performance, technically the best objective function
values achieved with those obtained from previous studies.
1.5 Scope of Study
This study covers the design, development, and evaluation of the proposed
algorithm for solving UA-FLPs. In addition, this research is restricted by the
limitations listed below:
5
i. This study chooses the AS algorithm for the ACO implementation and it does not
consider other ACO variants. AS is chosen because this research is an initial
effort to use ACO for solving UA-FLPs and furthermore, AS is the most basic
algorithm of ACO.
ii. In order to simplify UA-FLPs, this research uses the FBS model representation.
The FBS model is selected because it has been thoroughly studied and thus, it
provides a good basis for comparison.
iii. All the case problems are taken from previous studies which have been published
in the literature. The smallest problem contains 7 departments whereas the largest
one has 35 departments.
iv. This study is only interested in solving UA-FLPs which have a rectangular
facility with fixed dimensions.
v. The total department areas must not exceed the facility area. In addition, this
research will also solve UA-FLPs with empty spaces.
vi. All the departments are considered as rectangular and they have certain
constraints that bound their dimensions with minimum and maximum values. In
addition, this research does not take into account problems which have
departments with fixed dimensions or locations.
vii. Rectilinear distance is used in material movement cost calculation.
1.6 Organization of Thesis
This thesis is structured as follows. Chapter 2 describes the literature review
which lays down the fundamentals of UA-FLP and ACO. It provides a
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comprehensive review on the previous approaches used to solve UA-FLPs and
clearly highlights that ACO has not been used for solving them. Chapter 3 discusses
the methodology used in conducting this research. Chapter 4 introduces the
formulation of the proposed AS algorithm for solving UA-FLPs. Meanwhile, chapter
5 presents the FLC formulation applied to the AS algorithm. In chapter 6, evaluation
results obtained from the computational experiment are presented and discussed.
Finally, chapter 7 gives several conclusions for the research and directions for future
work.
1.7 Conclusions
In this chapter, the background of the research, its objectives and scopes have
been described. The idea of using AS for solving UA-FLPs is a relatively novel
concept. It is indicated that the use of FLC may improve the performance of AS by
balancing its search intensification and diversification. This research is expected to
produce an AS algorithm for solving UA-FLPs. This research is also expected to
become a new paradigm for tuning the parameters in AS by using FLC. In the next
chapter, a thorough review of the related literature will be provided.
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
This chapter provides the basic knowledge of metaheuristics, Ant Colony
Optimization (ACO), Unequal Area Facility Layout Problems (UA-FLPs) and
parameter tuning for ACO. This chapter becomes the foundation for formulating an
Ant System (AS) for solving UA-FLPs. This chapter also gives the starting point to
develop a Fuzzy Logic Controller (FLC) to automatically tune the AS's parameters.
As will be described later, AS has not been used for solving UA-FLPs.
Furthermore, it is also shown that FLC has not been used for automatically tuning the
AS's parameters. These two considerations have motivated the author to conduct this
research on using AS and FLC for solving UA-FLPs.
2.2 Combinatorial Optimization Problem
The term Combinatorial Optimization (CO) is composed of combinatorial
and optimization. An optimization problem P is generally defined as follows (Ibaraki,
1987):
P: maximize (minimize) )(xf subject to Sx∈ (2.1)
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where XS⊂ denotes a feasible region in the underlying space X. Namely, S
is the set of feasible solutions satisfying the imposed constraints. The
function RSf →: , where R is the set of real number, is called the objective
function. A feasible solution Sx∈ is optimal if no other feasible solution y satisfies
)()( xfyf > in the case of maximization (or )()( xfyf < in the case of
minimization). Maximization can be expressed as minimization since maximizing
)(xf is equal to minimizing )(xf− , and vice versa. The sets X and S take a
variety of forms in application.
P is a CO problem if X and S are discrete or combinatorial (implying that
X and Sare discrete sets of finite elements or countable infinite elements) (Ibaraki,
1987). From the definition, it can be concluded that many problems such as, Linear
Programming (LP), Traveling Salesman Problem (TSP), Quadratic Assignment
Problem (QAP), Vehicle Routing Problem (VRP), and more importantly, Unequal
Area Facility Layout Problem (UA-FLP) are in the CO domain.
2.2.1 Complexity of Combinatorial Optimization
The complexity of an optimization problem can be determined with two
important measures, time complexity and space complexity (Ibaraki, 1987; Maffoli
and Galbiati, 2000). Time complexity is related to the number of computation steps
such as additions, multiplications, comparisons, and so forth which is required to
solve the optimization problem, whereas space complexity is associated with the size
of the required space when performing the computation, i.e., the input data space and
the working space.
An optimization problem is regarded as easy if it can be solved by an
algorithm with time complexity )( kNO for a constant k, where N is the size of the
problem. Such a complexity is said to have a polynomial order, and the algorithm is
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called a polynomial time algorithm. On the other hand, if any algorithm for solving
the problem requires a complexity not bounded by a polynomial N, it is considered to
be difficult. Typical algorithms which do not have a polynomial order complexity are
having a logarithmic order )( log NNO or exponential order )( NkO .
Unfortunately, many CO problems turn out to be difficult to solve. Many CO
problems show typical behaviors of exponential functions; they blow-up at certain
points as shown in Figure 2.1. In other words, algorithms with exponential
complexity can be practical only when they are used for solving a problem with size
N smaller than its blow-up point. So it is more convenient to use approximation
algorithms for solving a problem with size higher than its blow-up point.
Problem Size
Com
puta
tion
Tim
e
Blow up point
Figure 2.1 Exponential complexity and its blow-up point (modified from Ibaraki,
1987)
Approximation algorithms do not guarantee optimal solutions but can find
near-optimal solutions. One of the approximation algorithms which is extensively
explored nowadays is metaheuristic. Osman and Laporte (1996) defined
metaheuristic as an iterative generation process which guides a subordinate heuristic
by combining intelligently different concepts for exploring and exploiting the search
space, and learning strategies are used to structure information in order to find near-
optimal solutions efficiently.
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For a broaden view, Gendreau and Potvin (2005) have proposed a unifying
view of metaheuristics. They argued that every metaheuristic can be divided, but not
always into a small number of algorithmic components. They are:
• Construction. This is used by all metaheuristics for creating initial solution(s).
• Recombination. This component generates new solutions from the current ones
through the recombination process.
• Random Modification. This is used to modify current solution(s) through random
perturbations.
• Improvement. This is used to explicitly improve the solutions(s); for example, by
selecting the best solution in the neighborhood, by applying a local search or by
projecting the solution in the feasible region.
• Memory Update. This component updates either the standard memories (Tabu
Search), pheromone trails (ACO), populations (Genetic Algorithm) or reference
sets (Scatter Search).
• Parameter/Neighborhood Update. This component adjusts the parameter values
or modifies the neighborhood structures.
2.2.2 Intensification and Diversification in Metaheuristic
The effectiveness of metaheuristics is measured by two considerations, i.e.
quality of solution, and computation time needed. A good metaheuristic can produce
a solution as close as possible to the optimal solution and still minimizing the
computation time needed.
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To deal with these two contradictive objectives, the terms intensification and
diversification have been introduced (Glover and Laguna, 1997). Occasionally these
terms are called in different names, such as exploitation and exploration (Eiben et
al., 1999), or local search and global search (Blum and Roli, 2003). These terms
help us to understand the nature of metaheuristic algorithms and at the same time,
give guidelines to improve them.
A very good explanation about these terms has been delivered by Glover and
Laguna (1997). According to them, intensification focuses on examining neighbors
of elite solutions while diversification encourages the search process to examine
unvisited regions and generates solutions that differ in significant ways from those
which have been found before. These descriptions are illustrated in Figure 2.2.
Figure 2.2 An illustration of intensification and diversification in metaheuristics
(adapted from Blum and Roli, 2003)
As can be seen in Figure 2.2, during the intensification, the metaheuristic
algorithm excessively tries to find a neighborhood solution which is better than its
current elite solution. If the search only relies on the intensification component, it
12
will trap the algorithm in the local optimum since no other neighborhood solution is
better than the local optimum. The search should avoid wasting excessive computing
time in the local optimum region and thus diversification should be activated. So, the
algorithm must also incorporate the diversification component to allow it to leave the
local optimum by jumping the search process to an unvisited region. The notions
“neighbor”, “locality”, and “area” (or “region”) are subjected to the characteristics of
the solution space.
It can be concluded that the intensification and diversification components
play an important role in the success of metaheuristic algorithms. In order to have the
ability to control intensification or diversification, researchers have proposed several
methods (Blum and Roli, 2003). One method which implicitly takes control of this
role is parameter tuning (Eiben et al., 1999). Usually, this activity is done during the
end of an iteration as described in the previous section (Gendreau and Potvin, 2005).
This technique will guide the search process whether it should activate intensification
or diversification.
2.2.3 Parameter Tuning for Metaheuristics
Parameter tuning can be identified as a process to find the correct
combination of an algorithm's parameters for each individual problem (Bhríde et al.,
2005). It should be noted that parameter tuning is different from parameter setting
(Eiben et al., 1999). The reasons for parameter tuning are (Eiben et al., 1999;
Wolpert and Macready, 1997):
• Generally, the performances of all metaheuristic algorithms depend on their
parameter values. Small size optimization problems are usually not too
influenced by their parameter values.
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• There are no universal parameter values which can solve all optimization
problems effectively and efficiently, and thus, parameter tuning is needed.
Research has shown that parameter tuning has several purposes. It can be
used for one or more of these purposes (Bhríde et al., 2005; Dai et al., 2006; Hartono
et al., 2004; Røpke and Pisinger, 2006; Syarif et al., 2004):
• To balance between intensification and diversification during the search process
• To speed-up the algorithm by giving good operators more probabilities to occur
• To maintain the solution feasibility by controlling the penalty function parameter
Parameter tuning methods can be differentiated by examining their conditions
when they are used in a metaheuristic algorithm. If the parameter values being tuned
are always same or not changing during the execution, the method can be considered
as a static parameter tuning system. For this system, parameter values are tuned
before the running of the algorithm and they remain static during the execution. On
the other hand, a parameter tuning method which has its parameter values being
changed during the running of the algorithm is called a dynamic parameter tuning
system. Apart from this two systems, there can be a mix system (Bhríde et al. 2005),
which tunes the parameters before running the algorithm, and the parameter values
are dynamic during the execution of the algorithm.
By examining the properties of the methods, the author classifies the static
parameter tuning system into two classes, experimentation method, and landscape
analysis method. Meanwhile a dynamic parameter tuning method can be categorized
into four classes; random value method, deterministic method, adaptive method, and
self-adaptive method.
Experimentation method or manual parameter tuning is the earliest method
used in parameter tuning. Since the initial development of metaheuristic algorithms,
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researchers have realized about the importance of parameter values. Different
parameter values might lead to different optimum results and/or computation time.
For this reason, researchers performed several experimentations to execute
metaheuristic algorithms with various parameter values. Usually, the values which
yielded the best result were claimed as the optimum parameter values. It could be
observed that the works done by De Jong (1975) and Grefenstette (1986) were the
earliest attempts to use manual parameter tuning. Although it has several drawbacks,
this method is still used nowadays because of its simplicity.
Another method to tune parameter values in metaheuristic algorithms is
landscape analysis method. This method is influenced by the No Free Lunch
Theorem (NFLT) (Wolpert and Macready, 1997). The theorem argues that one single
algorithm will not perform better than other algorithms for all kinds of optimization
problems and their instances. This is because every optimization problem has its own
characteristics and behaviors (Koppen, 2004). Within an optimization problem, it has
many problem instances that may have different difficulties and properties.
Therefore, a metaheuristic algorithm must be able to deal with different CO problems
and their instances in a different way. Thus, Bhríde et al. (2005) proposed to classify
the problem landscape before effective parameter adaptation may occur in a
metaheuristic.
Another approach dealing with parameter tuning is using random parameter
values. Mostly, researchers who called their method as parameter free algorithm did
this (Sawai and Kizu, 1998). They argued that due to the robustness of their method,
there is no need to tune the parameter values. While other researchers proposed the
parameter values to be dynamic, this method fixes several parameters and changes
the rest dynamically by using random functions. Usually they use a narrow scale of
parameters to be randomized. For example, if the mutation rate in Genetic Algorithm
(GA) has a value between 0 and 1, this method randomizes it between 0.6 and 0.8.
This method has helped people to deal with parameter tuning, but its efficiency and
effectiveness can be hard to prove.
15
Deterministic method changes the parameter values with deterministic rules.
The parameters constantly change, either ascending or descending or change
according to other deterministic functions. The change is not related to the algorithm
performance; it just depends on the number of iterations. It can be argued that
Simulated Annealing (SA) has a similar behavior to this method.
Adaptive method is extensively used in tuning the parameters of
metaheuristic algorithms. In this method, the results from earlier iterations will be
extracted to become useful information for guiding the parameter tuning process.
This method tries to adapt the state of a running metaheuristic algorithm and takes an
action which can balance intensification and diversification in that particular
metaheuristic. Many studies have been done with this method (Ko et al., 1996;
Hartono et al., 2004; Bhríde et al., 2005; Dai et al., 2006; Røpke and Pisinger 2006).
Another interesting method for parameter tuning is self-adaptive parameter
tuning. This method tries to treat parameters as decision variables which must be
solved. This method usually comes from the evolutionary computation family. In this
method, parameters are encoded into chromosomes and processed with mutation and
recombination. The better values of the encoded parameters will lead to better
individuals, which in turn are more likely to survive and produce offsprings, and
hence propagate these better parameter values. This method was shown to perform
effectively in Hinterding (1997).
2.3 Facility Layout Problems
Determining the physical organization of a production system is defined as a
Facility Layout Problem (FLP). The term production system does not only point to
manufacturing but it also concerns service and communication systems (Meller and
Gau, 1996). The objective of an FLP is mainly to minimize the total material
handling cost of the production system. It can also include or combine various
16
objectives such as to maximize space utilization, maximize flexibility, and maximize
employee satisfaction and safety (Muther, 1955).
Facility layout plays an important role in the manufacturing environment
since it determines the physical relationships of manufacturing activities. It
represents the locations of sub-facilities where materials will be moved from and to.
Therefore, it determines the traveling distance of materials in manufacturing
processes. Material movement can be considered as a non-value added process in
manufacturing. Thus, facility layout must be considered as an approach to control
material movement cost.
In manufacturing, material movement cost is considered as a part of material
handling cost. In a typical manufacturing industry, material handling accounts for
25% of all employees, 55% of all factory space, and 87% of production time
(Frazelle, 1986). In addition, material handling is estimated to represent between
15% and 70% of the total cost of a manufactured product (Tompkins et al., 2003).
The decrease of material movement cost will reduce the material handling cost and
will eventually cut down the manufacturing cost.
FLP is one of the old engineering problems and still an active area nowadays.
Muther (1955) has developed a systematic procedure for designing plant layout in
the 1950s. Research on facility layout still continues until nowadays, including
innovative and developmental works to integrate all components in facilities
planning. Two thorough literature reviews on FLPs were done by Meller and Gau
(1996) and Singh and Sharma (2005). Meller and Gau (1996) reviewed nearly 100
papers published in 10 years before 1996, and Singh and Sharma (2005) reviewed
nearly 140 papers published in 20 years before 2005. Both of them discussed recent
trends in facility layout research in their respective time. They highlighted a tendency
to integrate the design of a facility layout with the design of a material handling
system. In addition, they also concluded the need for a stochastic facility layout
rather than a static one.
17
Muther (1955) has listed the objectives of plant layout. Although the
objectives are dedicated for plant layout, they are also relevant for facility layout.
The objectives can be summarized as follows:
• to reduce the traveling distances of materials.
• to have a regular flow of the parts and products and to recucenot permitting
bottleneck in the production.
• to effectively utilize the space occupied by the facilities.
• to enhance satisfaction and safety of the workers.
• to obtain flexibility that can be easily readjusted for changing conditions.
FLP can be divided into three main categories, namely, (1) QAP, (2) UA-FLP,
and (3) Machine Layout Problem (MLP). The comparisons of these three categories
are summarized in Table 2.1
Table 2.1 Comparisons of FLP classes
NoProblem
typeDepartment size Location candidate Facility area
1 QAPSame size, fixed dimension or ignored
Fixed location Fixed or ignored
2 UA-FLPVaried size, decision variables
Decision variables≥ Total department areas
3 MLPVaried size, fixed dimension
Decision variables≥ Total department areas or free dimension
QAP was introduced by Koopmans and Beckman (1957) to model the
problem of locating interacting facilities of equal areas. The objective of QAP is to
find an assignment of all interacting facilities to all locations such that the total cost
of assignment is minimized. QAP is a special case of FLP in which all departments
have equal areas and all locations are fixed and known.
18
UA-FLP was originally formulated by Armour and Buffa (1963). The goal is
to partition a region into departmental sub-regions so as to minimize the total
material movement costs. The department areas do not need to be the same but there
are dimension ratio constraints or minimum dimension restrictions. Researchers have
been proposing several representations to reduce the complexity of UA-FLP. Two
successful representations are the Flexible Bay Structure (FBS) model and Slicing
Tree Structure (STS) model (Meller and Gau, 1996). These allow researchers to use
discrete optimization algorithms to solve UA-FLPs.
MLP has several fixed size machines and a number of products which need to
be produced in specific processes. The objectives are to determine the location of
machines in a given space while at the same time minimizing material handling
(Tompkins et al., 2003).
In recent research, some publications have discussed special cases of FLP.
Below are several special cases of FLP which have been published in the literature:
1. Multi-Floor Facility Layout Problem
The problem is to allocate sub-facilities in a certain facility which has many
floors. In addition, there are interactions between floors causing material handling in
vertical movement. The objectives of this problem are to determine the dimensions
and locations of departments (including which floor they will be placed), as well as
the vertical material handling systems. The algorithms for solving Multi-Floor FLP
have been discussed in Johnson (1982), Bozer et al. (1994), Meller and Bozer (1996)
and Meller and Bozer (1997).
2. Facility Layout Problem with Input/Output Points
Rather than using the centroid of each sub-facility to calculate distance, the
algorithm must try to find the exact location of the flow in and out points. Usually
the points are located on the boundaries of each sub-facility. In addition, research has
been done to use the aisle structure to calculate actual distance rather than rectilinear
19
distance. The research on this subject can be referred in Montreuil and Ratliff (1989)
and Kim and Kim (1999).
3. Dynamic Facility Layout Problem
This problem uses dynamic production demand as a replacement for static
production demand. Facility layout is not optimized for a certain period only, but it is
optimized for a number of given periods (or in stochastic demand conditions). The
optimum facility layout will be based on the material handling cost in each period
and also the re-arrangement cost between periods if there are changes in the facility
layout. The research on this issue has appeared in Rosenblatt (1986) and
Balakrishnan et al. (1992).
4. Facility Layout Problem in Flexible Manufacturing System (FMS)
Flexible Manufacturing System (FMS) is a production system in which a set
of machines and flexible material-handling equipment like robots, Automated
Guided Vehicles (AGVs), etc. are linked and controlled by a central computer. FMS
is different from the classical manufacturing system due to the higher degree of
automation, less number of machines, frequent setups, higher volume and flow of
information, etc. The objective of this type of problem is to produce a layout of
machines so that it can provide flexibility in FMS. The work of Solimanpur et al.
(2005) has tried to solve a single row layout problem in the FMS environment.
5. Facility Layout Problem in Cellular Manufacturing System (CMS)
Cellular Manufacturing System (CMS) is a mix production system strategy
between mass production and job shop manufacturing. In CMS, similar parts are
classified into part families which will be assigned into an appropriate machine cell.
This has a cost-effectiveness advantage because it reduces the part traveling
distances since the manufacturing processes of each part are predominantly
performed in one machine cell. It can still balance the machine loads because the
loads are distributed among machine cells and products are not bounded in a certain
20
cell. On the other hand, this system has the flexibility to adjust production schedules
because product families have similarity in processes so that the sequence will not
change much. The objective of this type of layout problem is to configure/group
machine cells and their locations in a facility. The works of Chen and Srivstava
(1994) and Vakharia and Chang (1997) have attempted to solve this problem using
metaheuristic approaches.
2.4 Unequal Area Facility Layout Problems
Unequal Area Facility Layout Problem (UA-FLP) is primarily used to model
a manufacturing layout. The problem arises when there is a need to make or modify a
facility layout. The need can be caused by (i) development of a new manufacturing
facility, (ii) expansion of an old manufacturing facility, (iii) alteration of the
production amounts and sequences, etc.
UA-FLP was originally formulated by Armour and Buffa (1963). They
assumed that there is a given fixed rectangular region, or facility, with dimensions H
and W, where H is the height and W is the width of the facility. There are also several
departments which need to be assigned to the facility. The number of departments,
the area of each department, and the material flow values associated with each pair of
departments are assumed to be known. The objective function for minimizing the
total material movement cost is given by:
∑ ∑= ≠=
=
n
i
n
jijijijij cfdCostTotal
1 ,1
(2.2)
n = the number of departments
fij = the total material flow between department i and department j, where
i, j = 1, 2, …, n
21
dij = the distance between department i and department j, where i, j = 1, 2,
…, n
cij = the cost of moving one unit material per unit distance from
department i to department j, where i, j = 1, 2, …, n
The goal is to partition the facility into departmental sub-regions so as to
minimize the total material movement cost. This objective is based on the material
handling principle that material handling cost increases with traveling distance. The
distance is measured in a variety of ways; here are two ways which are widely used:
Input-output (I/O) points distance: The distance is measured between
specified I/O points of two departments and in some cases, it is measured along the
aisles of the facility (Tretheway and Foote, 1994). The major drawback of this
accurate measure is that the locations of the I/O points and/or the aisles are still
unknown until the detailed layout has been obtained.
Centroid-to-centroid (CTC) distance: The distance is calculated between the
centers (centroids) of two departments. The shortcoming of the CTC distance is
department shapes can become very long and narrow if the shape restrictions are
loose. This is because an algorithm based on the CTC distance attempts to align the
department centroids as close as possible (Tate and Smith, 1995). Another weakness
of the CTC distance is an L-shaped department may have a centroid that falls outside
of the department (Francis and White, 1974).
Two calculation methods can be used to compute the two distance measures
above. Rectilinear distance is the most common calculation method used. It
computes distance using paths which are parallel to the perpendicular (orthogonal)
axes. Armour and Buffa (1963), Tate and Smith (1995), and Liu and Meller (2007)
are among the researchers who used rectilinear distance to compute the distance
between two departments. The second calculation method is Euclidean distance. This
method computes distance by using a straight-line path connecting two points.
Euclidean distances were used in Tam and Li (1991) and van Camp et al. (1991).
22
UA-FLP is very applicable to model real-world manufacturing layout
problems, especially those which use a process layout approach. Several case
problems found in the literature were originally taken from real-world manufacturing
problems (van Camp, 1989; Meller, 1992; Liu and Meller, 2007). In addition, Meller
(1992) stated that he used his proposed algorithm to solve the Ohio lamp plant layout
belonging to the General Electric Company.
Solving an UA-FLP will produce a block layout, which specifies the relative
location of each department (see figure 2.3a). The result can be expanded to obtain a
detailed layout, which specifies the exact department locations, aisle structures,
input/output (I/O) point locations, and the layout within each department (see figure
2.3b).
1
4
2
3
1
4
2
3
(a) (b)
Figure 2.3 (a) Block Layout and (b) Detailed Layout
UA-FLP can be modeled using various representations. Each representation
has its particular characteristics and complexities. There are four representation
models which are categorized based on the characteristic of the final layout. They are
the Continuous model, QAP model, Flexible Bay Structure (FBS) model, and Slicing
Tree Structure (STS) model.
1. Continuous model
The Continuous model has the characteristic that the departments' placements
and dimensions in the facility are not restricted to any rule other than the problem
itself. This ensures the model has the possibility to generate all layout solutions
23
including the exact optimal solution. Nevertheless, this model is very hard to be
solved when the solution space becomes large. This leads to the development of
other models which are easier to be solved. Generally, the continous model has been
used to solve UA-FLP using Mixed Integer Linear Programming (MILP) (Meller and
Gau, 1996) and mathematical programming (Anjos and Vannelli, 2006). Figure 2.4 is
a final layout example modeled as a continuous representation which cannot be
produced by other representations.
Figure 2.4 The solution of UA-FLP with continuous representation by Anjos and
Vannelli (2006)
2. Quadratic Assignment Model (QAP) model
Several researchers have used the QAP model for solving UA-FLP (Armour
and Buffa, 1963; Hardin and Usher, 2005). All departments in the problem are
broken into small grids which have the same area. Then the algorithm tries to gather
the grids which come from the same department into an interconnected region. This
can be achieved by adding large artificial flows among grids which come from
similar departments. As shown in Figure 2.5, this model often produces non-
24
rectangular blocks in the final layout. In addition, Bozer and Meller (1997) showed
that this model is ineffective because it implicitly adds department constraints.
Figure 2.5 The solution of UA-FLP with QAP model by Hardin and Usher (2005)
3. Flexible Bay Structure (FBS) model
In the FBS formulation, the placement of departments in an UA-FLP
generates columns or bays. Each bay does not need to have the same width and same
number of departments. The width of a bay is automatically adjusted by the number
of departments it contained. Therefore, the problem becomes simpler and easier to be
solved. The problem complexity is reduced into determining the departments'
placement order and the total number of departments that each bay contains. FBS has
an advantage in that the bays will become candidates for aisle structures and this
facilitates users to transform the model into an actual facility design (Konak et al.,
2006).
25
Figure 2.6 The solution of UA-FLP with FBS model in Shebanie II (2004)
4. Slicing Tree Structure (STS) model
In the STS model, the final UA-FLP layout can be described as a tree
representation. The tree representation contains departments as nodes and also “u”,
“b”, “r”, and “l” as connecting nodes. The connecting nodes act as the slicing
operators in the formation of sub-facility regions. The four slicing operators are “u”
for up cut, “b” for bottom cut, “r” for right cut, and “l” for left cut. The department
nodes and the slicing operators define relative department locations and adjacencies,
as well as the partition sequences. The STS model has been used by Scholz et al.
(2009).
Figure 2.7 The solution of UA-FLP with STS model in Shebanie II (2004)
26
Besides developing models to represent an UA-FLP, many researchers have
proposed their methodologies to solve it. These methodologies can be classified into
exact procedures, heuristic and improvement procedures, and metaheuristic
algorithms.
2.4.1 Exact procedures
Montreuil (1990) introduced a Mixed-Integer Programming (MIP)
formulation for solving UA-FLP. The formulation uses a continuous model to
represent the UA-FLP layout. The notations and formulation are shown below.
Notations:
(The first six items of the notations are parameters, while the last four items
are decisions variables.)
1. The building is L units in the x-direction and W units in the y-direction.
2. The indices i, j will be used for departments, where i, j = 1, 2, … N.
3. Each department i has an area requirements ai.
4. The upper and lower limits on the length or the width of department i are denoted
by ubi and lbi respectively.
5. The minimum and maximum perimeters of department i are denoted by pi and Pi
respectively.
27
6. The set of positive flows is denoted by { }ijfF = . That is Fjif ij ∈∀> ,0 and
furthermore MF = . The mth positive flow, fm, originates from department i(m)
and terminates at department j(m).
7. The rectilinear distance between department i and department j is expressed as
the sum of the distance in the x-direction, dijx, and the distance in the y-direction,
dijy. Note that for flow m, the distances in the x and y directions are denoted by
dmx and dm
y respectively.
8. The location of department i is indicated by its centroid, which is denoted by (xi,
yi).
9. Each department is rectangular shaped; department i has dimension 2l i in the x-
direction and dimension 2wi in the y-direction.
10. The relative location decision variables are denoted by zxij and zy
ij. In general, the
zij decision variables determine whether one department is to the north, south,
east, or west of another department in the layout.
Montreuil's formulation:
enforcednotisimpositionabovetheif
jofleftwestthetobemustiifzx
ij 0
)(1
enforcednotisimpositionabovetheif
jofabovenorththetobemustiifzy
ij 0
)(1
min ( )∑ +m
ym
xmm ddf (2.3)
s.t. )()( mjmixm xxd −≥ m∀ (2.4)
28
)()( mimjxm xxd −≥ m∀ (2.5)
)()( mjmiym yyd −≥ m∀ (2.6)
)()( mimjym yyd −≥ m∀ (2.7)
iii lLxl −≤≤ i∀ (2.8)
iii wWyw −≤≤ i∀ (2.9)
iii ubllb ≤≤ 2 i∀ (2.10)
iii ubwlb ≤≤ 2 i∀ (2.11)
32 ≤+++≤ yji
yij
xji
xij zzzz jiji <∀ ;, (2.12)
xijjjii Lzlxlx +−≤+ ji,∀ (2.13)
xjiiijj Lzlxlx +−≤+ ji,∀ (2.14)
yijjjii Wzwywy +−≤+ ji,∀ (2.15)
yjiiijj Wzwywy +−≤+ ji,∀ (2.16)
( ) iiii Pwlp ≤+≤ 4 i∀ (2.17)
The objective (Equation 2.3) is based on the multiplication of the material
flow cost and the rectilinear distance between department centroids. A standard linear
29
programming trick is used to linearize the absolute values in the distance function
(see Eqs. 2.4-2.7). In Eqs. (2.8) and (2.9) each department is constrained to be within
the facility, while in Eqs. (2.10) and (2.11) the maximum and minimum lengths of
the department rectangles are constrained. The constraint set (Equation 2.12) ensures
that the relative department location constraints are relaxed in two or three directions.
In Eqs. (2.13) - (2.16), the relative location decision variables are utilized to ensure
that departments do not overlap. Finally, in Equation (2.17), a bounded perimeter
constraint is used as a surrogate area constraint because the actual area constraint,
4liwi = ai, is nonlinear. Although this MIP approach is powerful and promising, only
problems with six or less departments can be solved optimally (Meller and Gau,
1996). This is due to the extensive computing time requirements for large-scale
problems.
Meller et al. (1999) pointed out that Montreuil’s perimeter constraint is
modeled such that an increase in the UA-FLP aspect ratio directly corresponds to the
increases in errors between the actual and solved areas. The difference between the
original and surrogate areas can be as much as 11%, 25%, 36%, and 44% for aspect
ratios of 2, 3, 4, and 5, respectively. Thus Meller et al. (1999) proposed an improved
surrogate perimeter constraint that provides a more realistic and effective
implementation by forcing a department area to adhere rigorously to changes in its
perimeter. They reported that their approach could reach optimal solutions for UA-
FLPs with up to eight departments.
Sherali et al. (2003) provided a similar approach that significantly reduces
errors in department areas by using a polyhedral outer approximation of the area
constraints and branching priorities. Using the polyhedral approximation and other
innovative techniques, they reported optimal solutions for UA-FLPs with up to nine
departments. Meller et al. (2007) have presented a new formulation for UA-FLPs
based on the sequence-pair representation. They tightened the structure of the
problems, and extended the solvable solution space from problem with nine
departments to problem with eleven departments.
30
In addition, Konak et al. (2006) introduced an MIP approach for UA-FLPs
with Flexible Bay Structure (FBS). In their method, the nonlinear department area
constraints are modeled in a continuous plane without using any surrogate constraint.
The department dimension is managed by controlling the width of the bay. The width
of each bay is calculated as the total area of the departments assigned to that bay
divided by the length of the facility in the y –axis direction. Their method is
extensively tested and it could find optimal solutions for problems with up to 15
departments.
2.4.2 Heuristic Procedures
The majority of UA-FLP algorithms can be identified as either construction
heuristic algorithms or improvement heuristic algorithms. In the former category, the
solutions are constructed from scratch. It is considered the simplest heuristic
approach to solve UA-FLPs, but the quality of the solution produced is generally
unsatisfactory. On the other hand, improvement heuristic algorithms use iterations to
improve the initial solution. Improvement methods can be easily combined with
construction methods. In general, several heuristic approaches have been published
in the literature including CRAFT, SHAPE, NLT, and MULTIPLE (Meller and Gau,
1996).
CRAFT (Computerized Relative Allocation of Facilities Technique) is the
oldest improvement-type approach developed by Armour and Buffa (1963). CRAFT
works by improving the initial layout through the exchange of departments. It
performs two-way or three-way exchanges of the centroids of non-fixed departments
that are equal in area or adjacent. For each exchange, CRAFT calculates an estimated
cost reduction and it chooses the exchange with the largest estimated cost reduction
(steepest descent). It then performs the department exchanges and continues until no
estimated cost reduction can be obtained. Constraining the department exchanges to
those departments that are adjacent or equal in area is likely to affect the solution
31
quality. The procedure has also been criticized because it can produce departments
with irregular shapes (Tompkins et al., 2003).
SHAPE developed by Hassan et al. (1986), is a construction algorithm that
utilizes a discrete representation and an objective function based on rectilinear
distances between department centroids. The department selection sequence is
dependent on a ranking, which is based on each department's flow and a user-defined
critical flow value. Department placement begins at the center of the layout.
Subsequent department placement is based on the objective function value, with
departments placed on each of the layout's four sides. The algorithm is easy to
implement; however, because the department shape is controlled by the objective
function, the shape of the facility may deteriorate towards the end.
NLT (Nonlinear Optimization Layout Technique) is a construction algorithm
developed by van Camp et al. (1991). It is based on nonlinear programming
techniques and it utilizes Euclidean distance to calculate distances between
department centroids. In NLT, there are three constraints: departments cannot
overlap, cannot be located outside the facility, and cannot be assigned to a less than
required area. The constrained model is transformed into an unconstrained form by
an exterior point quadratic penalty method. With a three-stage approach, a difficult
problem is solved using the solution from the previous stage (as an initial solution
point). The department shapes are all rectangular.
MULTIPLE (MULTI-floor Plant Layout Evaluation) is a single or multi-floor
improvement-type algorithm developed by Bozer et al. (1994). MULTIPLE uses a
discrete representation and enhances CRAFT by increasing the number of exchanges
considered in each iteration. In addition, MULTIPLE can restrict the irregularity of
department shape by using an irregularity measure based on the perimeter and area of
each department. However, since it uses a discrete representation, the department
shapes may not be rectangular.
Anjos and Vannelli (2006) introduced a two-phase mathematical
programming method to solve UA-FLPs. The first phase used in their method is a
32
relaxation of UA-FLPs, which intends to find a good starting point for algorithm
implementation. In the second phase, an iterative algorithm is used to solve the exact
formulation of the problems as a non-convex Mathematical Program with
Equilibrium Constraints (MPEC). Although this two-phase mathematical
programming method achieves better objective function values (as compared to those
accomplished by other previous research) when solving UA-FLPs, it has been tested
only on one UA-FLP instance introduced by Armour and Buffa (1963).
2.4.3 Metaheuristic Algorithm
Several metaheuristics such as SA, GA, Tabu Search (TS), and Particle
Swarm Optimization (PSO) have been used to approximate the solutions for very
large UA-FLPs. The SA algorithm originated from the theory of statistical mechanics
and is based upon the analogy between the annealing of solid metals and solving
optimization problems. GA iteratively searches for the global optimum by generating
new solutions through processes similar to genetic reproduction approaches. PSO
uses several particles to represent solutions and allows them to fly in a solution space
so that they cooperate to converge to the optimum solution.
In terms of the application of metaheuristics, Tam (1992) used LOGIC
(Layout Optimization using Guillotine-Induced Cut) which is an SA algorithm to
solve UA-FLPs with STS representation. With a given slicing tree and department
area values, the layout can be determined by recursively partitioning a rectangular
area and placing the departments into the areas according to the four specific
branching operators. Since this approach is likely to produce long and narrow
department shapes, two shape constraints are added as penalties to the objective
function. The algorithm uses two-way exchanges of branching operators as an
attempt to find a better layout.
Tate and Smith (1995) proposed a GA for solving UA-FLPs using FBS
representation. A dynamic or adaptive penalty function (as shown in Equation 2.18)
33
is used to guide the search into feasible solution regions. This method searches a
solution space by varying department-to-bay assignments or by adding or removing
bay breakpoints. The algorithm generates good layouts and it is shown to outperform
CRAFT and NLT.
)()()( allfeask VVmmp −= (2.18)
where:
p(m) = the penalty function
m = the number of infeasible departments
k = parameter that determines the severity of the penalty function
Vfeas = the best feasible objective function value found
Vall = the best overall objective function value found
Hardin and Usher (2005) used PSO to solve UA-FLPs with QAP
representation. Each department is divided into grids/tiles with equal size. The
solution construction is performed by managing tiles based on two different rules.
The first rule attempts to move tiles to their own department centroid, and the second
rule attempts to move tiles to the centroid of other departments which have high
volume relationship. Then new solutions are constructed based on the two rules
above. This method shows an improvement when compared to the CRAFT method.
Recently, Liu and Meller (2007) proposed an approach to solve UA-FLPs
represented as sequence pairs, by using GA and MIP. Conceptually, Liu and Meller
(2007) used GA to modify the solutions represented as sequence-pairs. Thereafter,
the sequence-pair representation is transformed to become a feasible solution with
MIP. The main purpose of using the sequence-pair representation is to eliminate all
34
infeasible binary variables which make large UA-FLPs difficult to solve. Liu and
Meller (2007) claimed that their method could achieve optimal solutions for
problems with up to eleven departments. Besides this, they have shown some
improvements when solving problem instances with larger data sets.
More recently, Scholz et al. (2009) proposed a TS algorithm with STS
representation for solving UA-FLPs. They incorporated a bounding curve for dealing
with fixed departments and free dimension facility in UA-FLPs. Their algorithm
incorporated four types of neighborhood moves to find better solutions. They
compared their algorithm with previous research and showed large improvements.
In order to enhance the UA-FLP methods to produce a more realistic layout
solution, a Maximum Aspect Ratio (MAR) has been introduced by van Camp et al.
(1989), which is given by,
{ }{ }
=
ii
ii
wl
wl
,min
,maxmaxα (2.19)
Where:
αmax = maximum aspect ratio
l i = length of department i, i = 1, 2, 3 ..n
wi = width of department i, i = 1, 2, 3 ..n
The MAR ensures that the UA-FLP methods produce an acceptable physical
layout solution. A normal MAR is usually between two and seven. As the MAR
becomes smaller, a UA-FLP becomes highly constrained since the allowable
department dimensions become more restricted. On the other hand, a MAR higher
than seven requires significant computing time because there are more candidate
solutions to be considered.
35
From the previous literature review, it can be said that UA-FLP is still an
active area. It also shows that Ant Colony Optimization (ACO) or particularly Ant
System (AS) has not been utilized to solve UA-FLPs. This encourages the author to
apply AS for solving them.
2.5 Ant Colony Optimization
Ant Colony Optimization (ACO) was introduced by Marco Dorigo and his
colleagues in the early 1990s (Blum, 2005a). ACO works by imitating the foraging
behavior of ants when searching the shortest path to reach their food. Figure 2.8
shows an experiment that demonstrates the shortest path finding capability of ant
colonies. In the initial condition, all ants are in their nest and there exists two paths of
different lengths between the ants’ nest and the food source. When searching for
food, ants initially explore both paths in equal probability. 50% of the ants take the
short path and 50% of them take the long path to the food source. The ants that have
taken the short path have arrived earlier at the food source. Therefore, the probability
to take again this path is higher. The pheromone trail on the short path receives, in
probability, a stronger reinforcement, and the probability to take this path grows.
Finally, due to the pheromone evaporation on the long path, the whole colony will, in
probability, use the short path.
ACO was initially developed for solving the TSPs. Thereafter, its application
was extended to solve various kinds of discrete optimization problems including the
QAPs, VRPs, Job-Shop Scheduling Problems (JSPs), etc.
ACO is currently among the state-of-the-art algorithms for solving, for
example, the sequential ordering problem (Dorigo and Gambardella, 2000), resource
constrained project scheduling problem (Merkle et al., 2002), open shop scheduling
problem (Blum, 2005b), and 2D and 3D hydrophobic polar protein folding problem
(Shmygelska and Hoos, 2005).
36
Figure 2.8 Demonstration of the shortest path finding capability of an ant colony
(Blum, 2005a)
2.5.1 Ant Colony Optimization Procedure
After parameter initialization and problem data input, every ACO iteration
consists of ant solutions construction, local search procedures (optional), and
pheromone information update (see Table 2.2).
Table 2.2 Ant Colony Optimization procedure
Set parameters, initialize pheromone trails
while termination conditions not met do
ConstructAntSolutions
ApplyLocalSearch {optional}
UpdatePheromones
end while
37
The explanations of these three components in ACO are as follows (Blum,
2005a):
ConstructAntSolutions(). A set of m artificial ants construct solutions from
elements of a finite set of available solution components. Solution construction starts
with an empty partial solution. Then, at each construction step, the current partial
solution is extended by adding a feasible solution component without violating any
problem constraint. The choice of a solution component is selected probabilistically
at each construction step. The rules for the probabilistic choice of solution
components vary across different ACO variants but the common rule used in AS
(Dorigo et al., 1996) is:
[ ] [ ][ ] [ ] )(,
)(.
)(.)|(
)(
scc
cscp i
scii
iii
i
Ν∈∀= ∑Ν∈
βα
βα
ητητ
(2.20)
where τi is the pheromone value associated with the solution component ci.
η(ci) is an optional weight value for each feasible solution component ci ∈ N(s).
Furthermore, α and β are positive parameters that determine the pheromone
information parameter and the heuristic information parameter.
ApplyLocalSearch(). Once solutions have been constructed, and before
updating the pheromones, often some optional actions may be required. These are
often called daemon actions, and can be used to implement problem specific and/or
centralized actions, which are intended to improve the solutions obtained by the ants.
UpdatePheromones(). The aim of the pheromone update is to avoid a too
rapid convergence in sub-optimal regions and to increase the pheromone values
associated with good or promising solutions. Usually, these are achieved by (i)
decreasing all the pheromone values through pheromone evaporation, and (ii)
increasing the pheromone values associated with a chosen set of solution(s) Supd:
38
{ }∑ ∈∈
+−←scSssii
iupd
sFw|
)(.).1( ρτρτ (2.2)
where Supd is the set of solutions that is used for the update, ρ ∈ (0, 1] is a
parameter called evaporation rate, F: S → R+ is called the quality function such that
f (s) < f (s') → F (s) ≥ F (s'), ∀s ≠ s' ∈ S, and w is the weight for corresponding
solution s.
To date, a few types of ACO variants exist (e.g. Ant System (AS), Elitist Ant
System (EAS), Rank-based Ant System (RAS), Max-Min Ant System (MMAS) and
Ant Colony System (ACS)) and there are little differences among them. In AS,
pheromone values are updated by all the m ants that have built a solution. EAS
updates the pheromone values by using all ants in the respective iteration, as well as
the best-so-far ant (Dorigo et al., 1996). Instead of using all the ants, RAS only uses
several high quality ants plus the best-so-far ant for updating the pheromone values
(Bullnheimer et al. 1999). In MMAS, pheromone update only comes from the
iteration best ant or the best-so-far ant, and the value of the pheromone is bounded
(Stützle and Hoos, 2000). ACS differs by introducing a parameter qo as the
probability for choosing a solution component which maximizes [τi]α.[η(ci)]
β or
using 1-qo as the probability to perform a construction step according to Equation
2.20 (Dorigo and Gambardella, 1997). In addition, ACS uses the best-so-far solution
update rule and introduces local pheromone update.
2.5.2 ACO Applications in Facility Layout Problems
As mentioned earlier, ACO has successfully solved many CO problems
(Blum, 2005a). ACO is suitable to be used for solving CO problems since it uses a
discrete representation. ACO also has another advantage in using heuristic
information to exploit the problems so that it can solve them in a shorter computation
time. Moreover, ACO often uses local search to gain an intensification advantage just
like other metaheuristics.
39
One area that has been solved successfully by ACO is FLP. FLP is a very hard
optimization problem and can be applied in many areas. These reasons have
encouraged many researchers to solve it. Unfortunately, the applications of ACO in
solving FLPs are mainly limited to QAP. Only little research has been conducted to
solve other FLP classes. Table 2.3 shows the applications of ACO in solving FLPs.
Table 2.3 Applications of ACO algorithms in FLPs
No Reference FLP class ACO variant1 Maniezzo et al. (1994) QAP AS-QAP2 Gambardella et al. (1997) QAP HAS-QAP3 Maniezzo and Colorni (1999) QAP AS-QAP4 Maniezzo (1999) QAP ANTS-QAP5 Stützle and Hoos (2000) QAP MMAS6 Talbi et al . (2001) QAP Parallel Ant Colonies7 Corry and Kozan (2004) MLP Ant Colony System8 Solimanpur et al . (2004) QAP Modified ACO9 AbdelRazig et al . (2005) QAP Modified ACO
10 Baykasoglu et al . (2005) QAP Modified ACO11 Solimanpur et al . (2005) QAP Modified ACO12 Demirel and Toksari (2006) QAP Hybrid ACO-SA13 McKendall Jr. and Shang (2006) QAP Hybrid Ant System
14 Mouhoub and Wang (2006) QAP Ant Colony with stochastic local search
15 Ramkumar and Ponnambalam (2006)
QAP Hybrid Ant Colony System
16 Hani et al . (2007) QAP ACO with guided local search
17 Wong and See (2009) QAP MMAS
ACO has been implemented to solve QAP since its early development. The
first developed algorithm is Ant System (AS), and then several kinds of ACO
variants have been reported to solve QAP such as ANTS-QAP, MAX-MIN Ant
System (MMAS-QAP), and Hybrid Ant System (HAS-QAP). These algorithm share
two similar characteristics: (1) the pheromone trail of ACO is concerned with the
desirability to assign facility i to location j, and (2) local search procedures are used
to improve the ant solutions.
The differences among the ACO variants above can be summarized as
follows (Stützle and Dorigo, 1999): In AS, the heuristic information of facility i to be
located in location j is inversely proportional to the sum of distances from location j
40
to all other locations and the sum of flows from facility i to all other facilities.
ANTS-QAP computes heuristic values dynamically by assigning lower bounds on
the solution costs of the completed partial solutions. ANTS-QAP does not use
pheromone evaporation, but it uses a mechanism to increase or decrease the
pheromone values depending on the quality of the solutions. MMAS does not use
heuristic information but it relies on local search to gain high quality solutions. In
addition, in MMAS, the pheromone values are bounded to specific upper and lower
bounds. The pheromone values are assigned to the upper and lower bound values if
an operation causes them to go beyond these limits. HAS-QAP does not use
pheromone trails to construct solutions, but it tries to modify the current solution by
swapping two facilities which are chosen randomly or selected based on probabilities
taken from pheromone trail values. In HAS-QAP, only the global best ant is allowed
to update the pheromone values.
The computational results among the ACO variants show that ANTS-QAP is
the best ACO variant on all the problem instances tested. In addition, computational
results show that ACO based algorithm is currently among the best algorithms for
solving real-life and structured QAP instances (Stützle and Dorigo, 1999).
In addition, ACO has been used by Corry and Kozan (2004) for solving the
dynamic machine layout problem. The solution representation used consists of three
parts: locations, orientations, and the order in which machines are positioned. The
locations are represented in grid blocks to provide a framework for depositing
pheromone trail information. An artificial ant constructs a solution by choosing a
machine and its location. The pheromone trail is recorded as the ant moves while
constructing the solution. Then this information is used for updating the pheromone
values. The model has shown better result than the heuristically reduced integer
programming method.
41
2.5.3 Parameter Tuning for Ant Colony Optimization
In contrast to the development of ACO's structure, parameter tuning in ACO
has only received little attentions. Only several papers have been reported to develop
and improve this issue (e.g. Dorigo et al., 1996; Dorigo and Di Caro, 1999;
Watanabe and Mitsui, 2003). Several researchers have tuned the various parameters
in ACO, while others have experimented with tuning the ACO's parameters when it
is combined with another metaheuristic such as GA and PSO.
For example, Dorigo et al. (1996) have experimented with parameter tuning
in ACO to solve TSP. They gave guidelines on the boundary of the basic parameters
in ACO:
α: relative importance of the pheromone information, α ≥ 0
β: heuristic information parameter, β ≥ 0
ρ: evaporation rate, 0 ≤ ρ < 1
In another study, Dorigo and Di Caro (1999) experimentally concluded that
good parameter values for ACO are; m = problem size (for TSP), α = 1, β = 5, and ρ
= 0.5.
As mentioned earlier, ACO has been successfully used to solve the sequential
ordering problem, resource constrained project scheduling problem, open shop
scheduling problem, and 2D and 3D hydrophobic polar protein folding problem. It is
found that for the cases above, the optimum ACO parameters used are different and
they are not always the same with Dorigo and Di Caro (1999)'s suggestion. The
parameter values used for the cases above can be seen in Table 2.4. Note that
parameter β = 0 shows that the algorithm does not use heuristic information, and ant
number m = 1 because of hybridization with beam search.
42
Table 2.4 ACO parameter values in various optimization problems
No Optimization problem m α β ρ Reference1 Sequential ordering
problem10 1 0 0.1 Gambardella and Dorigo
(2000)2 Project scheduling
problem5 1 1 0.975 Merkle et al. (2002)
3 Open shop scheduling problem
1 10 0 0.1 Blum (2005b)
4 Bioinformatic problem 10-15 1 2 0.6 Shmygelska and Hoos (2005)
From the fact that researchers tried to tune their parameter values in ACO and
they concluded different parameter values as optimal values, two points can be
concluded:
• The performance of ACO is influenced by its parameter values. This is because
they implicitly influence search intensification and diversification in the
algorithm.
• There are no universal parameter values for ACO that can be used to solve all
optimization problems efficiently and effectively. The differences in 'the optimal
parameter values' come from the differences in problem size, optimization
problem type, and problem instance characteristic.
The mistakes in setting parameter values may lead the algorithm to bias and
ruin its performance. This is because the algorithm cannot diversify enough to reach
the optimum region and/or cannot intensify in the correct direction. Instead of using
static parameter tuning, several researchers dynamically tune the parameter values in
ACO (Watanabe and Mitsui, 2003; Qin et al., 2003; Favuzza et al., 2006). This is
done to gain control of the intensification and diversification mechanisms in ACO.
Basically, the parameter tuning methods in ACO can be classified into static
parameter tuning, adaptive parameter tuning, and self-adaptive parameter tuning.
43
In terms of the first category, Dorigo and Gambardella (1997) have proposed
the optimum number of ants in ACS. They showed that the optimum number of ants
is influenced by the problem size dimension. Nevertheless, they concluded that ACS
works well when m = 10 based on their experimental observation.
Gambardella and Dorigo (2000) proposed that the qo value (see section 2.5.1)
is a function of the problem size, i.e. qo = 1 – s/n. This makes qo dependent on the
problem size n. s is the expected number of nodes selected with the probabilistic rule.
Pilat and White (2002) proposed two hybrid methods incorporating GA into
ACS. The first algorithm uses a population of ants that encodes decision variables in
ACS. In every iteration, four ants will be selected to execute the ACS and GA
operations. Each selected ant will first run the ACS algorithm to generate a solution.
Then through crossover and mutation, the two worst ants will be replaced by the new
ones. They compared this hybrid algorithm with ACS and did not find significant
improvement in finding optimum solutions. The second hybrid algorithm is used to
find ACS’s optimum parameter values (as alternative to Dorigo and Gambardella
(1997)'s proposal). This algorithm is quite similar with the first one except that the
population consists of one complete ACS algorithm rather than an ant. They
concluded that the parameter values would influence the algorithms’ performance
and no magic parameter values can be used for all optimization problems.
Watanabe and Matsui (2003) proposed a method to dynamically tune the size
of the candidate set (ants) in ACS. The candidate set is used to limit the search space
only to promising regions. They restricted the candidate set by only considering 90%
accumulation of the sorted pheromone concentration. With this method, it is not
necessary to set the size of the candidate set in advance. The computational results
with several graph coloring problem instances indicated that the proposed control
mechanism can potentially improve the efficiency of ACO algorithms, especially for
large optimization problems.
Qin et al. (2006) used an adaptive ant colony algorithm to solve phelogenetic
tree construction problem. The term adaptive in this algorithm refers to dynamically
44
tuning α and β which are based on pheromone distributions. At the initial stage of the
algorithm, ants should select solution components based mainly on heuristic
information β. This can be achieved by assigning a relatively large value to β. After
some iterations, the pheromone values for the solution components are increased,
thus their influence becomes more and more important. Therefore the value of α
should be increased so that the pheromone influence can be gained. Experimental
results show that the proposed method has better performance than GA. It can
converge faster and obtain better quality results than GA.
Hao et al. (2006) introduced the use of adaptive ACS by incorporating PSO.
The role of PSO is to find the correct values for ACO parameters by formulating
them as particles. PSO is executed only when the new best solution is found. The
proposed solution was tested with TSP problem instances, and compared with pure
ACS. The results showed that PSO-ACS performs better than ACS. In addition, Hao
with different groups of researchers have also conducted dynamic parameter tuning
on the heuristic information parameter β (Huang et al., 2006) and pheromone trail
evaporation rate ρ (Hao et al., 2007). The results indicated improvement as
compared to the traditional ACO.
Favuzza et al. (2006) used a varying qo based on the number of unimproved
iterations. If the algorithm reaches a local convergence indicated by a high number of
unimproved iterations, then the value of qo will be decreased, thus allowing the
algorithm to heighten the diversification process. As soon as the algorithm leaves the
local convergence, the qo value will be increased again allowing the intensification
process to happen. The proposed algorithm has proven to be robust in finding the
optimal reinforcement strategy for distribution systems.
Randall (2004) proposed a near parameter free ACO. Particularly, the
parameter search process is integrated within the running of the algorithm. In other
classes of metaheuristics, this method is called self-adaptive parameter tuning (Eiben
et al., 1999). The proposed method has comparable results to the standard
implementation of ACO. In addition, this method removes the need to tune the
parameters by hand.
45
In Table 2.5, the various parameter tuning methods for ACO together with
their drawbacks are summarized. The significance of these methods is dependent on
their implementation. More research is needed to examine and compare previous
studies using the same optimization problem, problem size, and problem instances.
As described in Table 2.5, major drawback of these methods is the incapability to
control search intensification and diversification. In addition, there is no previous
research which has used Fuzzy Logic Controller (FLC) to tune the parameters of
ACO.
Table 2.5 Parameter Tuning for ACO algorithms
No Method Problem Drawbacks Reference1 Proposes a value for ant
numberTSP Only for ant number, does not
have the ability to balance exploitation and exploration
Dorigo and Gambardella (1997)
2 Determines q o based on problem size
Sequential ordering
Only changes the tuning need from q o to s
Gambardella and Dorigo (2000)
3 Genetic Algorithm TSP Overheating the algorithm, does not show satisfied results
Pilat and White (2002)
4 Candidate Set Graph Coloring Only useful for limiting diversification and increasing intensification, cannot control diversification
Watanabe and Matsui (2003)
5 Integrates parameter search within metaheuristic
TSP and QAP Not useful for balancing diversification and intensification
Randall (2004)
6 changes q o based on the number of unimproved iterations
Distribution System
Diversification may become bias depending on the distribution of pheromone trail and heuristic information
Favuzza et al. (2006)
7 PSO TSP Does not have the ability to balance exploitation and exploration
Hao et al. (2006)
8 Dynamic tuning based on pheromone diversity
Phelogenetic Tree
Does not give concern to solutions which are trapped in local optima
Qin et al. (2006)
2.6 Conclusions
This chapter has discussed the importance of FLPs along with a review on the
applications of ACO in FLPs. It is found that ACO has not been implemented to
solve UA-FLPs which have been used for years to model manufacturing layout
46
problems. To alleviate the complexities of UA-FLPs, this chapter has described
several layout representations. One of the important layout representations is FBS
which will divide a facility into bays, and then departments will be assigned to them.
In addition, FBS has an advantage because the bays may become aisle candidates in
the final manufacturing layout.
In addition, balancing intensification and diversification in the search space is
an important key element in metaheuristics. This can be achieved by dynamically
tuning the parameter values. This chapter has reviewed the parameter tuning methods
that have been used for ACO. It is found that the previous parameter tuning methods
have not been able to effectively balance intensification and diversification.
CHAPTER 3
METHODOLOGY
3.1 Introduction
The methodology used in this research begins with problem definition. After
an extensive literature review, several research hypotheses are generated. Then, the
necessary data are collected before starting the developmental work. The
development of the proposed algorithm covers the algorithm design, codification,
and verification. Following this, the algorithm is tested using numerous case
problems taken from the literature. Based on the results obtained from the evaluation,
analysis will be carried out.
3.2 Methodology
The methodology used to complete this research consists of seven main steps,
which are shown in Figure 3.1.
1. Problem definition
In this step, the problem definition is determined along with the research
boundaries and research objectives. Firstly, the problem to be solved is identified in
this step. In order to ensure the research is completed on schedule, research
48
boundaries are clearly determined. The research objectives are also determined
because they act as the guidance for the rest of the steps.
Figure 3.1 Research methodology main steps
2. Literature review
In this step, an extensive literature review is conducted. Previous studies are
reviewed especially in the areas of Unequal Area Facility Layout Problems (UA-
FLPs), Ant Colony Optimization (ACO), and parameter tuning in ACO. From the
literature review, the current trend related to this research can be found. It also shows
the research opportunity which should be tackled in this research.
49
3. Research hypothesis
Hypotheses are then determined based on the research objectives and
literature review. These hypotheses represent preliminary conclusions that must be
supported or rejected when completing the research.
4. Data collection
The next step is to determine and collect what are needed to conduct the
experiment and test the hypotheses. The important data needed are test problems
which have been published in the literature. Test problems act as inputs for algorithm
design and testing. On the other hand, previous results found in the literature are used
to evaluate the performance of the proposed algorithm.
5. Development of the ACO algorithm for UA-FLPs
The development of the algorithm is divided into two phases. Phase I is
intended to design the basic ACO formulation for solving UA-FLPs and phase II is
intended to construct the Fuzzy Logic Controller (FLC) which will dynamically tune
the parameter values. The algorithm development uses the C++ programming
language since it is fast and yet easy to be implemented.
6. Testing and evaluation of the algorithm
The final algorithm is tested using test problems taken from the literature. The
test problems are chosen from those which are well-known and widely used in the
literature. It is intended that the selected test problems cover the whole range of
problem size (from small until large). In addition, they are tested using conditions
which are similar to those from previous research. Then the results are evaluated by
comparing them with previous research in terms of the best known solution and best
Flexible Bay Structure (FBS) solution.
50
7. Result analysis and conclusion
Analysis is conducted to evaluate the performance of the algorithm. This
includes analyzing its advantages and disadvantages. This step also highlights some
important notes for developing the algorithm and gives some future directions for
continuing the research. Finally, the hypotheses will be answered and conclusions
will be drawn.
3.2.1 Ant System Development
This research uses the Ant System (AS) framework for implementing ACO.
The detailed algorithm development methodology is shown in Figure 3.2 and the
steps involved are discussed below:
1. Algorithm formulation
The algorithm is formulated using the Ant System (AS) framework because it
is the most fundamental concept in ACO. The structure and implementation of AS is
designed based on paradigms obtained from the literature. Although AS has been
implemented to solve other classes of FLPs such as Quadratic Assignment Problems
(QAPs), it still needs modification in solving UA-FLPs. Besides this, there are
several optional components that can be considered in the implementation of the
algorithm such as local search, global update rule, etc. In addition, to alleviate the
complexity of UA-FLPs, Flexible Bay Structure (FBS) is used to model them.
2. Algorithm codification
Then, the AS algorithm is codified using the C++ language. C++ is chosen
because it is proven superior in computation time and yet easy to be programmed.
Due to maintenance reasons, the codification uses the structural programming
51
paradigm. In structural programming, one big program is divided into small
programs in order to decrease redundancy and increase maintenance easiness.
Formulation of AS Algorithm
Manual parameter
tuning
Verification of Algorithm
Start
Testing of AlgorithmTest problems from literature
Codification of Algorithm
End
Integration of FLC with AS
Formulation of FLC
Codification of FLC-AS
Test problems from literature
Verification of Final Algorithm
Testing of Final Algorithm
Figure 3.2 FLC-AS development methodology
3. Algorithm verification
In conjunction with the algorithm codification, the program will also be
inspected for bugs and mistakes. Each part of the program will be tested whether
there is any problem such as division by zero, variable not found, etc. After all parts
52
of the program can run properly, small test problems will be input to the program.
The first iteration will be examined to check if there is any problem or bug. Several
random testings will also be conducted to ensure that the program runs without any
problem.
4. Algorithm testing and manual parameter tuning
After the program can be properly used, it will be tested using test problems
taken from the literature. In the initial stage, the best parameter setting will be
determined using manual parameter tuning on one test problem. Manual parameter
tuning will be conducted on five different levels for several parameters. The
parameter values which can achieve the best objective function will be used as the
optimal parameter settings. These settings will be used for conducting the experiment
for the rest of the test problems.
3.2.2 Fuzzy Logic Controller (FLC) Development
This section is intended to develop the FLC and integrate it with AS. To
complete this part, this research uses the following steps:
1. FLC Formulation
The objective of this step is to design the FLC which consists of three main
components; input parameters along with their fuzzy membership functions, output
parameters along with their fuzzy membership functions, and fuzzy inference engine.
Input parameters are associated with the convergence and/or divergence level of the
algorithm. Meanwhile, output parameters are algorithm parameters that will control
the searching process of the algorithm. Fuzzy inference engine will become a bridge,
which converts the convergence/divergence level into a specific action that must be
taken to adjust the convergence/divergence level.
53
2. Integration of FLC with AS
The function of this step is to integrate the FLC with the pure AS. This step
includes determining where and when to put the FLC into the algorithm. It also
decides the frequency of executing the FLC.
3. FLC-AS codification
In this step, the complete FLC-AS concept is transformed into the C++
language. In the same way as AS programming, the codification uses the structural
programming paradigm. FLC becomes a new module and it will be inserted into the
main AS algorithm.
4. FLC-AS verification
This step ensures that the FLC is functioning. It also examines the connection
between FLC and AS. The FLC module will be tested whether there is any problem
such as division by zero, variable not found, etc. After all parts of the program can
run properly, small test problems will be input to the program. The first iteration will
be examined to check if there is any problem or bug. Several random testings will
also be conducted to ensure that the program runs without problems.
5. FLC-AS testing
After the final program can be properly used, it will be tested using test
problems found in the literature. The initial parameter values for running the
algorithm will be taken from the manual parameter tuning. For each test problem,
five replications will be conducted.
54
3.3 Research Hypothesis
Solving UA-FLPs using AS is a complex research. The need for obtaining
high quality solutions and balancing search intensification and diversification is a
challenge. To address these problems, the following hypotheses are proposed:
• High quality UA-FLP solutions can be achieved by implementing an initial AS
algorithm with FBS representation.
• FLC can be used to replace manual parameter tuning and to automatically tune
the parameter values of AS.
3.4 Data Collection
The initial AS and FLC-AS algorithms are tested using numerous problem
sets taken from the literature. The list of the problem sets can be found in Table 3.1.
Note that αmax is the maximum aspect ratio constraint and lmin is the minimum side
length constraint. The complete data for the problem sets can be found in Appendix
A.
In order to allow the use of FBS representation on problem sets M11s, M11a,
M15s, M15a and M25, the author have modified them so that they do not have a
fixed size department. In addition, the fixed location constraint is changed into a
sequence constraint. The modified fixed size department then becomes the last
department that will be assigned in the FBS representation. As for BA12TS and
BA14TS, they are the modified problem sets initially provided by Tate and Smith
(1995) to change the empty space to dummy department(s).
55
Table 3.1 UA-FLP problem sets used in this research
Width Height1 O7 7 8.54 13.00 αmax = 4 Meller et al. (1998)
2 FO7 7 8.54 13.00 αmax = 5 Meller et al. (1998)
3 FO8 8 11.31 13.00 αmax = 5 Meller et al. (1998)
4 O9 9 12.00 13.00 αmax = 5 Meller et al. (1998)
5 V10s 10 25.00 51.00 lmin = 5 van Camp (1989)
6 V10a 10 25.00 51.00 αmax = 5 van Camp (1989)
7 M11s 11 6.00 6.00 lmin = 1 Bozer et al. (1994)
8 M11a 11 3.00 2.00 αmax = 5 Bozer et al. (1994)
9 M15s 15 15.00 15.00 lmin = 1 Bozer et al. (1994)
10 M15a 15 15.00 15.00 αmax = 5 Bozer et al. (1994)
11 M25 25 15.00 15.00 αmax = 5 Bozer et al. (1994)
12 NUG12 12 3.00 4.00 αmax = 5 van Camp (1989)
13 NUG15 15 3.00 5.00 αmax = 5 van Camp (1989)
14 BA12 12 7.00 9.00 lmin = 1 van Camp (1989)
15 BA12TS 16 7.00 9.00 lmin = 1 Tate and Smith (1995)
16 BA14 14 6.00 10.00 lmin = 1 van Camp (1989)
17 BA14TS 14 6.00 10.00 lmin = 1 Tate and Smith (1995)
18 AB20 20 2.00 3.00 αmax = 1.75 Armour and Buffa (1963)
19 SC30 30 12.00 15.00 αmax = 5 Liu and Meller (2007)
20 SC35 35 15.00 16.00 αmax = 4 Liu and Meller (2007)
ReferenceProblem
setNo
Common shape constraint
Facility sizeNumber of departments
The test problems are chosen from those which are well-known and widely
used. Table 3.2 can be used to compare the characteristics of this research's problem
sets with those used in previous research. As can be seen, different researchers have
applied different type and number of problem sets in their evaluation. Moreover, not
all problem instances can be obtained because their data are not published.
Nevertheless, by comparing Table 3.1 and Table 3.2, it can be said that the problem
sets used in this study are adequate and comparable to previous research (20 problem
instances are used as compared to a minimum of 1 and a maximum of 27). Most of
the previous studies have only tested less than 20 problem sets. The smallest problem
instance used in this study is just one department larger than the smallest problem
instance found in the literature. In addition, the proposed algorithm will be tested
using several large problem instances which have appeared in the literature.
56
Table 3.2 Characteristics of problem sets used in previous research
No LiteratureNumber of problem
sets testedMinimum
problem sizeMaximum
problem size1 Armour & Buffa (1963) 1 20 202 van Camp (1989) 10 5 303 Bozer et al. (1994) 3 11 254 Tate and Smith (1995) 4 10 205 Sherali et al. (2003) 15 6 96 Konak et al. (2006) 21 7 207 Liu and Meller (2007) 27 6 358 Scholz et al . (2009) 7 7 20
3.5 Conclusions
This chapter has discussed about the methodology used in this research. The
methodology consists of seven main steps, (i) problem definition, (ii) literature
review, (iii) research hypothesis, (iv) data collection, (v) algorithm development, (vi)
algorithm testing and evaluation, and (vii) analysis and conclusion. The development
of the algorithm is achieved by two steps, (i) AS development, and (ii) FLC-AS
development and integration. This chapter has also provided the research hypotheses
which will be tested in this research. Finally, it has also provided the problem sets
which will be used to evaluate the proposed algorithm.
CHAPTER 4
ANT SYSTEM FORMULATION
4.1 Introduction
This chapter presents the Ant Colony Optimization (ACO) formulation for
solving Unequal Area Facility Layout Problems (UA-FLPs). As mentioned earlier,
Ant System (AS) is used as the basic ACO formulation in this study. The structure of
the AS algorithm comprises three main parts, i.e. construct ant solutions, apply local
search, and update pheromones. Specifically, ant solutions are constructed based on
both pheromone information and heuristic information. The proposed algorithm uses
five types of local search to improve the search. In addition, the algorithm uses the
iteration-best ant and global-best ant to update the pheromone values.
4.2 Structure of the Proposed Ant System (AS) Algorithm
The proposed AS algorithm follows the ACO general form, as presented in
Table 2.2. In this implementation, the main looping of the proposed AS algorithm
consists of (i) construct ant solutions, (ii) apply local search procedures and (iii)
update pheromone values. The details are summarized in Figure 4.1.
The proposed AS algorithm uses the Flexible Bay Structure (FBS)
representation to transform the continuous characteristic of UA-FLPs into a discrete-
based optimization problem. The advantages of the FBS representation are: (i) it
58
decreases the problem complexity and (ii) it generates aisle candidates in the final
layout. The layout infeasibility issue is handled by adopting the adaptive penalty
objective function used by Tate and Smith (1995).
Figure 4.1 The proposed AS algorithm
The proposed algorithm constructs ant solutions based on pheromone
information and heuristic information. The heuristic information is obtained from
processing the material flow data and relative department locations. Meanwhile the
pheromone information is collected during each iteration from the global-best ant
and iteration-best ant.
59
4.2.1 Solution representation of Unequal Area Facility Layout Problems
The proposed AS algorithm uses FBS to represent an UA-FLP. Conceptually,
FBS divides the facility into vertical or horizontal bays. The bay width is flexible,
depending on the departments that it contained. In addition, the number of bays and
the number of departments in each bay are also flexible. The proposed algorithm
tries to find the optimum value for the number of bays, the number of departments
contained in each bay, and the department-placement order that could minimize the
objective function value.
Each ant solution consists of two parts. The first part represents the
department sequence. This sequence is the order of n departments represented by
integer numbers which will be placed into the facility. The second part represents
break points which manage the number of bays and the number of departments
contained in each bay. This second part is represented by n-1 binary numbers. 1
represents a bay break and 0 otherwise. The bay width is adjusted by dividing the
sum of department areas contained by the facility length.
Figure 4.2 shows an example of an ant solution which is used in the proposed
algorithm. This ant solution is for an UA-FLP with n = 7. The first part contains the
order of seven departments (5-3-2-1-6-4-7) which will be placed into the facility. The
second part contains six binary numbers (0-0-1-0-1-0) which represent break bay
locations.
Figure 4.2 An Example of Solution Representation in the Proposed Algorithm
This representation will be transformed into a layout solution in Figure 4.3.
Specifically, the departments are placed from left to right, and bottom to top based on
the sequence 5-3-2-1-6-4-7. For the bay breaks, number 1 appears two times, in the
60
3rd and 5th places. These show that the breaks are generated after the third and fifth
departments. So there are 3 bays, the first bay contains departments {5, 3, 2}, the
second bay contains departments {1, 6} and the third bay contains departments {4,
7}.
With FBS representation, the proposed algorithm only searches solutions that
comprise a combination of department sequence and bay break sequence. Hence, the
complexity of UA-FLPs using FBS representation is n!2n-1. This representation also
has the advantage in that the bays will become aisle candidates and this facilitates
users to transform the model into an actual layout.
Figure 4.3 Layout generated from the ant solution representation in Figure 4.2
In order to deal with problem sets with empty spaces, the author proposed
two improvements for FBS representation: (1) extending the feasible solution space
by using empty space to fulfill department constraints and (2) improving the
objective function by recursively filling the bay with empty space. The author refers
to this representation as modified-FBS (mFBS). An example of mFBS representation
is shown in Figure 4.4.
The transformation from FBS representation into mFBS solution is based on
two procedures:
61
1. Determine the bay height based on the total area of departments which are
located in a bay. Whenever departments that violate department constraints are
found, the bay height will be increased by adding an available empty space into
the bay to satisfy the constraints. Whenever an empty space is added, it will be
proportionally placed in the left and right side of the bay.
(a) (b)
Figure 4.4 An example of layout solution with (a) original FBS and (b) modified-
FBS
2. After all department constraints in each bay are satisfied and if there is still an
unoccupied empty space, select the middle bay and add the available empty space
until the maximum bay height is fulfilled. Examine if this improves the objective
function value. If there is an improvement, the modification will be accepted.
Otherwise, the original layout will be restored. Whenever an empty space is
added, it will be proportionally placed in the left and right side of the bay. This
procedure will be repeated for all bays by selecting the nearest bay from the
center point of the facility. In addition, this procedure will be executed only if the
empty space is still available.
62
4.2.2 Objective function
The goal of the proposed algorithm is to minimize the total material handling
cost. The cost (objective function) for an ant is calculated based on Equation 4.1.
)(inf1 ,1
allfeas
n
i
n
jijijijij VVpdfcFunctionObjectiveAnt −+=∑ ∑
= ≠=
(4.1)
where:
n = the number of departments
cij = the cost of moving one unit material per unit distance from
department i to department j, where i, j = 1, 2, …, n
fij = the number of material flows from department i to department j,
where i, j = 1, 2, …, n
dij = the distance between departments i and j, where i, j = 1, 2, …, n
pinf = the number of infeasible departments
Vfeas = the best feasible objective function value found
Vall = the best overall objective function value found
The distance is measured using rectilinear distance from centroid-to-centroid
of each department. In addition, the objective function incorporates an adaptive
penalty function (Tate and Smith, 1995) to guide the search process towards feasible
solution regions. Specifically, an ant solution is given a penalty value which is
proportional to the number of infeasible departments that it contains. A department is
63
considered as infeasible if it violates the department constraints, i.e. either the
maximum aspect ratio or minimum side length.
4.2.3 Ant Solutions Construction
The proposed algorithm employs the pheromone information and heuristic
information for constructing ant solutions. In this respect, the two parts of an ant
solution are not generated concurrently. The bay breaks are generated first and their
constructions are solely based on pheromone information. The generated bay breaks
are transformed into a dummy layout which will be used to calculate the heuristic
information for that particular ant solution. Following this, the department sequences
are constructed based on pheromone information and heuristic information.
During execution, the proposed algorithm keeps the pheromone information
for the two parts of the ant solution (department sequences and bay breaks). For each
element of the department sequences, pheromone information is the relative
probability to choose department 1 until department n. While for each element of the
bay breaks, pheromone information corresponds to the desirability to select a bay
break or not.
When adding a solution component, the proposed algorithm only chooses the
solution component which has not appeared in that particular ant solution to avoid
duplication. The proposed algorithm cannot guarantee that the generated ant solution
is feasible because the feasibility can only be known after a complete layout solution
is formed. Instead, it uses an adaptive penalty function to guide the search process to
the feasible regions as mentioned in section 4.2.2.
After the bay breaks have been generated, the proposed algorithm creates a
dummy layout solution assuming that all departments have the same area (obtained
by dividing the facility area with the number of departments). The dummy layout
specifies the location candidates which represent the relative locations of
64
departments in the final layout solution. The location candidates will be used as a
basis for determining the heuristic information for department sequences
construction.
The proposed algorithm uses the heuristic information to help achieves a
higher quality solution in a shorter time. The heuristic information helps to guide the
proposed algorithm to place a department with high material flow in a location
candidate nearer to the center of the facility. In order to provide such heuristic
information, the proposed algorithm introduces λ as a parameter for location
candidate. Assume that point (0, 0) is located at the bottom left of the facility. The
parameter for each location candidate is calculated based on Equation 4.2.
|2
|2
|2
|2
Wk
yWL
kx
Lk
−−+−−=λ (4.2)
where:
λk = the parameter for location candidate k, where k = 1,2,…, n
L = the facility length
W = the facility width
xk = x axis value for the center of location candidate k
yk = y axis value for the center of location candidate k
Equation 4.2 ensures that a location candidate which is nearer to the center of
the facility will have a higher λ value. Apparently, the equation is a linear function
since the proposed algorithm uses the rectilinear distance as the measuring
parameter.
65
By using the pheromone information and heuristic information, the proposed
algorithm generates the department sequences by randomly choosing them based on
probabilities calculated using Equation 4.3. The τik acts as the pheromone
information collected in previous search to guide the proposed algorithm to ‘good
solution regions that have been explored’. The product of m and λ represents the
heuristic information which heuristically guides the proposed algorithm to place a
department with high material flow in a location candidate nearer to the center of the
facility.
[ ] [ ][ ] [ ] )(,
..
..)|(
)(
scm
msCp ik
sckiik
kiikik
ik
Ν∈∀= ∑Ν∈
βα
βα
λτλτ
(4.3)
where
p(Cik) = probability to locate department i to location candidate k
Cik = department i located in location candidate k (department i located in
the kth element of the department sequences)
τik = pheromone value associated with the solution to locate department i to
location candidate k
mi = sum of material flow from and to department i
λk = parameter for location candidate k
α = pheromone information parameter
β = heuristic information parameter
66
N(s) = available departments which have not been used in the corresponding
ant
4.2.4 Local Search Procedures
In order to enhance the search performance of AS, the author incorporates
five types of local search as improvement procedures into the algorithm. They are:
1. Swap between department sequence, which randomly chooses two different
departments and exchanges their locations.
2. 1-insert procedure on department sequence, which randomly chooses one
department and moves it to a new location in the department sequence.
3. 2-opt procedure on department sequence. This local search randomly chooses a
subset chain of the department sequence and rearranges it in the opposite
direction.
4. Bay break change procedure, which randomly changes the bay break (from 0 to 1
or vice versa).
5. Bay break swap procedure, which randomly swaps the bay break with its
neighbor.
The local search procedures can be classified into two categories. Both are
used to provide a effective search for the proposed algorithm. The first category
(procedures 1-3) is a neighborhood search to find a good department sequence and
the second category (procedures 4-5) is a neighborhood search to find a good bay
layout. All the procedures are used because of the large solution space of certain
problem instances. All of them have the same probability to occur since the solution
67
space of one problem instance may differ from those of other problem instances and
thus, it is hard to predict the most effective local search procedure.
During implementation, the algorithm continuously probes the neighborhood
of the solution search space (using local search procedures) to find a solution with a
better quality. Whenever a better solution (refers to a solution that either (1) improves
the solution feasibility (indicated by a decrease in the number of infeasible
departments), or (2) improves the objective function value without degrading the
solution feasibility) is found, the neighborhood that contains such a solution will be
explored in the next local search step.
Although the proposed algorithm incorporates five types of local search, it
randomly selects one of them in an implementation, and recursively repeats it until a
stopping criterion is met. As in this case, the stopping criteria for the local search are
specified as: (1) the maximum number of steps pre-specified by users, and (2) the
number of steps where the local search does not improve the solution quality.
Whenever one of these criteria is met, the local search will be terminated.
4.2.5 Pheromone update scheme
The proposed algorithm uses the standard pheromone updating strategy to
avoid premature search convergence and to bias the search process into a solution
search space that contains good or promising solutions. It uses the global-best and
iteration-best ant solutions to update the pheromone value. The pheromone updating
process is shown in Equation 4.4.
{ }∑ ∈∈
+−←scSssikik
ikupd
sFw|
)(.).1( ρτρτ (4.4)
where:
68
Supd = the set of ants that is used for updating the pheromone
ρ = evaporation rate, ρ ∈ (0, 1)
F(s) = quality function such that f (s) < f (s') ⇒ F (s) ≥ F (s'), ∀ s ≠ s' ∈ S,
with f(s) is the objective function. The proposed algorithm uses F(s) =
1/f(s)
w = weight for the corresponding ant solution s. The proposed algorithm
uses w = 1 for all Supd.
4.2.6 Stopping criteria
The proposed algorithm uses two stopping criteria to terminate its search: (1)
the maximum number of iterations, which sets an upper limit for the number of steps
involved in a single implementation and, (2) the maximum number of iterations
where AS could not further improve the best-so-far solution. The latter helps the
algorithm to terminate the search if a better solution could not be found. Whichever
stopping criterion is met first, the proposed algorithm will be stopped.
4.3 Implementation details of the algorithm
The detailed implementation of the proposed algorithm is shown in Figure
4.5. In short, the algorithm is modified from the general AS framework to
accommodate the FBS representation, adaptive penalty function, ant solution
construction, local search procedures and pheromone update scheme for UA-FLPs.
69
Start
Set parameter values
m,
Set algorithm and local search
stopping criteriaimax, unmax, LSmax, unLSmax
Set algorithm parameter
i =0, un=0, cp=0
i = iteration numberun = unimproved iteration number
cp = computation time
m = ant number = pheromone trail
parameter = heuristic information
parameter = evaporation rate
Input problem dataData : L, W, n, ai, ubi, lbi, fij
L = facility length in x-directionW = facility width in y-direction
n = department numberai = area for department iubi = upper bound for the
length and width of department i
lbi = lower bound for the length and width of department i fij = material flow from
department i to department jCalculate partial
heuristic information table sfi
Initialize pheromone value
table ik
sfi = sum of flow from department i
to all other departments.
ik =pheromone value for placing department i to
location k
A
Set global best objective function and solution
f(s)* = infinityS* = {}
Update iteration counter
Initialize ant number counter d
i = i + 1
d = 0
D
Set iteration bestobjective function and solution
f(i)* = infinitySi* = {}
a
Figure 4.5 Details of the proposed AS algorithm
70
Figure 4.5 Details of the proposed AS algorithm (continued)
71
B
Select ant Supd, for pheromone update
Update pheromone value
i = imax? ORun = unmax?
End
yes
no
D
Check the global best ant,
f(i)* < f(s)* ?
un = 0f(s)* = f(i)*S* = Si*
yes un = un + 1
no
C
Ant number counter d = ant number m ?
Check the best ant in iteration i,
f(d-th) < f(i)* ?
f(i)* = f(d-th)Si* = S(d-th)
yes
no
yes
no
a
Figure 4.5 Details of the proposed AS algorithm (continued)
72
4.4 Conclusions
This chapter has presented an AS algorithm with FBS representation for
solving UA-FLPs. The main algorithm
consists of three parts, i.e. construct ant solutions, apply local search
procedures, and update pheromones. In addition, this chapter has also proposed an
improvement to the FBS representation to solve UA-FLPs with empty spaces.
CHAPTER 5
PARAMETER TUNING USING FUZZY LOGIC
5.1 Introduction
This chapter discusses the parameter setting of the proposed algorithm. This
chapter mentions the manual parameter tuning conducted on the proposed algorithm.
In addition, it also explains the formulation of the Fuzzy Logic Controller (FLC) to
automatically tune the parameters of the proposed algorithm. FLC is chosen because
it can handle imprecise propositions which represent the state of the running
proposed algorithm. This facilitates the proposed algorithm to balance its
intensification and diversification efforts. By balancing these efforts, it ensures that
the proposed algorithm does not waste too much computation effort in an
unattractive region and has the ability to escape from local optima. In addition, FLC
can be used as an alternative for manual parameter tuning which is time consuming.
5.2 Manual Parameter Tuning
In the case of manual parameter tuning, the parameter values are determined
based on a randomly selected test problem AB20. This tuning is conducted on five
different levels for the four main parameters of the proposed algorithm. Initially,
these parameters are assigned with the following values recommended by Dorigo et
al. (1999):
74
• Number of ants m = problem size,
• Pheromone information parameter α = 1,
• Heuristic information parameter β = 5,
• Evaporation rate ρ = 0.5.
Then each parameter value is varied on five different levels. This is done by
changing one parameter at a time while the others are held constant. Since the tuning
involves four parameters, 20 experiments are needed. The parameter values which
achieve the best objective function (minimum objective function) will be used as the
optimal parameter settings.
The results of the manual parameter tuning are shown in Table 5.1. From the
table, it can be seen that, the best parameter values are: number of ants = n (n =
number of departments), pheromone information parameter = 1, heuristic
information parameter = 5, and evaporation rate = 0.1. These values will be used for
testing the rest of the problem sets.
Having conducted the manual parameter tuning, the next section will discuss
about the use of FLC to dynamically tune the parameter values.
75
Table 5.1 Manual parameter tuning results
No Number of ants
Pheromone information parameter
Heuristic information parameter
Evaporation rate
AS best AS worst AS average
1 5 1 5 0.5 5699.20 6065.68 5866.812 10 1 5 0.5 5677.83 6065.68 5784.963 20 1 5 0.5 5677.83 5699.20 5686.384 30 1 5 0.5 5677.83 6065.68 5832.975 35 1 5 0.5 5677.83 5804.28 5703.126 20 1 5 0.5 5677.83 5699.20 5686.387 20 3 5 0.5 5677.83 6065.68 5755.408 20 5 5 0.5 5677.83 6065.68 5910.549 20 10 5 0.5 5677.83 6065.68 5755.40
10 20 15 5 0.5 5677.83 5699.20 5686.3811 20 1 1 0.5 5677.83 5699.20 5686.3812 20 1 3 0.5 5677.83 6065.68 5759.6713 20 1 5 0.5 5677.83 6065.68 5837.2414 20 1 10 0.5 5677.83 6065.68 5759.6715 20 1 15 0.5 5677.83 5699.20 5686.3816 20 1 5 0.1 5677.83 5699.20 5682.1017 20 1 5 0.3 5677.83 5699.20 5690.6518 20 1 5 0.5 5677.83 6065.68 5837.2419 20 1 5 0.7 5677.83 5699.20 5686.3820 20 1 5 0.9 5677.83 6065.68 5755.40
5677.83 5699.20 5682.10Minimum
5.3 Fuzzy Logic Controller
Zadeh (1988) introduced the fuzzy membership function µ to handle imprecise
propositions. Unlike the traditional logic which only has true (1) or false (0) value,
the fuzzy membership function may have values between the interval [0, 1]. For
example, temperature would be a linguistic variable in the control of a heater. It can
take on linguistic values such as high, low, quite low, etc. A temperature value of
40oC may have a membership value of 0.8 for “hot” because it is above the normal
room temperature, and it could get higher. These imprecise propositions can be used
to represent the state of a running algorithm (whether intensifying or diversifying).
The fuzzy logic for tuning metaheuristic parameters can be regarded as a
control system. Thus, the implementation of this Fuzzy Logic Controller (FLC) can
be depicted in Figure 5.1. To implement it, several things must be decided; input
parameters along with their membership function, output parameters along with their
membership function, and the fuzzy rule base which specifies the relationships
between the input and output parameters.
76
Figure 5.1 Scheme for parameter tuning in metaheuristics
There are several papers which have used FLC to tune parameter values in
metaheuristics. Most of them are implemented to tune evolutionary algorithms. For
example, Wang et al. (1996) proposed a fuzzy logic controlled genetic algorithm
(FCGA) to adaptively adjust the crossover rate and mutation rate during the
optimization process. This algorithm was tested using a power economic dispatch
problem. The result showed that FCGA has much better performance than the
conventional genetic algorithm. Xue et al. (2005) proposed the use of FLC to
dynamically tune parameters in multi-objective differential evolution (MODE). They
used two inputs for FLC, i.e. population diversity (PD) and generation percentage
(GP) already performed. For the outputs, they used greediness and perturbation
factor of the reproduction operator. These two parameters control explicitly the
exploitation and exploration of the evolutionary algorithm. They showed that FLC-
MODE obtained better results in 80% of the testing examples compared to the
conventional MODE.
The significance of parameter values in metaheuristics has been laid down by
Eiben et al. (1999). Although their discussion was aimed at the evolutionary
algorithm, their work was also applicable to AS since both metaheuristics are
stochastic based algorithms. They stated that parameter dependencies, time
consuming efforts, and unnecessarily optimal parameter values are the technical
drawbacks of manual parameter tuning. In brief, they suggested the use of dynamic
parameter setting.
77
Based on these reasons, this research studies the effect of FLC on
dynamically tuning the parameters of AS. This study involves four basic parameters
of AS; number of ants, pheromone information parameter, heuristic information
parameter, and evaporation rate.
5.4 FLC-AS Formulation
Basically, FLC is added to automatically tune the proposed AS algorithm’s
parameter values. FLC updates the parameter values in every iteration of the
algorithm. It takes the convergence and divergence states of the running algorithm as
inputs, and then feedbacks the parameter values changes as outputs into the main AS
loop. The proposed FLC-AS algorithm can be seen in Figure 5.2.
The role of FLC is to balance the search intensification and diversification of
the algorithm. The input parameters for the FLC are the standard deviation of
solutions and the number of unimproved iterations. The former can represent how
solutions in one iteration differ from those in another iteration. Meanwhile the latter
represents the degree of which the current search is trapped in local optima. In other
words, these two input parameters help to indicate the state of intensification and
diversification of the algorithm. Before they are transformed into fuzzy sets, they are
normalized by dividing them with their respective maximum value. The membership
function for the input parameters is shown in Figure 5.3(a).
78
Figure 5.2 The proposed FLC-AS algorithm
The proposed algorithm utilizes four output parameters which are aimed to
influence its search intensification and diversification. These parameters are the
number of ants, pheromone information parameter, heuristic information parameter,
and evaporation rate. In this research, there are four experiments which tune the
parameters individually and one experiment which tunes all of them at once. The
membership function for the output parameters is shown in Figure 5.3(b).
The input parameters will be processed to produce the appropriate action by
applying the fuzzy rule base. These rule bases usually takes the form of IF-THEN
79
statements. The rule bases for the number of ants and evaporation rate is shown in
Table 5.2(a), whereas the one for the pheromone information parameter and heuristic
information parameter is shown in Table 5.2(b). The abbreviations for the input
parameter values are VL = very low, L = low, M = medium, H = high, and VH = very
high, and those for the output parameter values are NL = negative large, N =
negative, MV = maintain value, P = positive, and PL = positive large.
0
1
0 0.25 0.5 0.75 1
Normalized input param eter value
Mem
ber
ship
val
ue Very Low
Low
Medium
High
Very High
(a)
0
1
-1 -0.5 0 0.5 1
Normalized output param eter value
Mem
ber
ship
val
ue Negative Large
Negative
Maintain Value
Positive
Positive large
(b)
Figure 5.3 Graphs showing membership functions for (a) input parameters, and (b)
output parameters
Table 5.2 Rule bases for (a) number of ants and evaporation rate, and (b) pheromone
information parameter and heuristic information parameter
VL L M H VH
VL PL PL P P MV
L PL P P MV N
M P P MV N N
H P MV N N NL
VH MV N N NL NL
Standard deviation of solutions
Num
ber
of
unim
prov
ed it
erat
ions
Output
(a)
VL L M H VH
VL NL NL N N MV
L NL N N MV MV
M N N MV P P
H N MV P P P
VH MV P P PL PL
OutputStandard deviation of solutions
Num
ber
of
unim
prov
ed it
erat
ions
(b)
80
The rule bases are designed to control the search intensification and
diversification of the proposed algorithm. It is made to provide the algorithm with an
ability to jump out from local optima. In addition, the rule bases are intended to limit
the search when it is performed in a too wide region. The rule bases above can be
interpreted as follows:
• When the algorithm is trapped in local optima (standard deviation of solutions is
low), the FLC will try to widen the algorithm's search by increasing the number
of ants.
• When the algorithm searches in a very wide region (the number of unimproved
iterations is low), then the FLC will help to focus the search by increasing the
evaporation rate.
• If the current search does not help the algorithm to escape from local optima,
then the FLC will try to change the direction of the search. The direction is
altered by modifying the pheromone information parameter and/or heuristic
information parameter.
After applying the fuzzy rule base, the fuzzy outputs will be defuzzified to be
transformed into crisp values. This research uses the centroid method (Mendel, 1995)
as the defuzzification technique. The centroid method can be seen in Equation 5.1 in
which xo is the crisp value, and x is the centroid of the area below the fuzzy
membership function µc. The changes in the output parameters are limited to 10% of
the gap between their respective maximum and minimum value. The FLC is only
executed after 10 iterations to allow the algorithm to sample the distribution of the
input parameter values.
∫∫=
dxx
dxxxx
C
C
o).(
.).(
µ
µ (5.1)
81
5.5 Conclusions
This chapter has described the parameter setting for the proposed AS
algorithm. Initially, this chapter discusses the implementation of manual parameter
tuning. Then it describes the implementation of the Fuzzy Logic Controller (FLC) to
automatically tune the proposed algorithm. The input parameters for the FLC are the
standard deviation of solutions and the number of unimproved iterations. In addition,
the proposed algorithm utilizes four output parameters which are aimed to influence
its search intensification and diversification. These parameters are the number of
ants, pheromone information parameter, heuristic information parameter, and
evaporation rate.
CHAPTER 6
EVALUATION OF THE PROPOSED ALGORITHM
6.1 Introduction
This chapter presents the evaluation of the proposed algorithm. At first, the
proposed algorithm is tested using the parameter values obtained from manual
parameter tuning (refer to section 5.2). After that, the proposed algorithm is tested
together with the Fuzzy Logic Controller (FLC) as described in section 5.4. Four
basic parameters of the AS algorithm are tuned with FLC, i.e. number of ants,
pheromone information parameter, heuristic information parameter, and evaporation
rate. Four experiments are conducted to tune each of the parameters individually and
one experiment is conducted to tune all of them at once. Finally, this chapter
provides a discussion about the results obtained.
6.2 Problem Sets and Parameter settings used to evaluate the Proposed
Algorithm
The proposed algorithm is tested using numerous problem sets taken from the
literature. The list of problem sets can be found in Table 6.1. Note that αmax is the
maximum aspect ratio constraint, lmin is the minimum side length constraint and both
best-FBS solution and best-known solutions are in cost unit.
83
In order to allow the use of FBS representation on problem sets M11s, M11a,
M15s, M15a and M25, the author has modified them so that they do not have a fixed
size department. In addition, the fixed location constraint is changed into a sequence
constraint. The modified fixed size department then becomes the last department that
will be assigned in the FBS representation. As for BA12TS and BA14TS, they are
the modified problem sets initially provided by Tate and Smith (1995) to change the
empty space to dummy department(s).
Table 6.1 Problem set data
Problem data Best FBS solution Best known solution1 O7 7 αmax = 4 - 131.63 Meller et al. (1998) - Sherali et al. (2003)
2 FO7 7 αmax = 5 23.12 20.95 Meller et al. (1998) Konak et al. (2006) Sherali et al. (2003)
3 FO8 8 αmax = 5 22.39 22.39 Meller et al. (1998) Konak et al. (2006) Sherali et al. (2003)
4 O9 9 αmax = 5 241.06 235.95 Meller et al. (1998) Konak et al. (2006) Sherali et al. (2003)
5 V10s 10 lmin = 5 22,899.65 19,994.10 van Camp (1989) Konak et al. (2006) Scholz et al . (2009)
6 V10a 10 αmax = 5 21,463.07 21,463.07 van Camp (1989) Konak et al. (2006) Konak et al. (2006)
7 M11s 11 lmin = 1 1,317.79 1,317.79 Bozer et al. (1994) Konak et al. (2006) Konak et al. (2006)
8 M11a 11 αmax = 5 1,225.00 1,185.20 Bozer et al. (1994) Konak et al. (2006) Gau and Meller (1999)
9 M15s 15 lmin = 1 27,781.95 27,781.95 Bozer et al. (1994) Konak et al. (2006) Konak et al. (2006)
10 M15a 15 αmax = 5 31,779.09 29,157.60 Bozer et al. (1994) Konak et al. (2006) Gau and Meller (1999)
11 M25 25 αmax = 5 - 1,588.37 Bozer et al. (1994) - Gau and Meller (1999)
12 NUG12 12 αmax = 5 265.60 265.60 van Camp (1989) Konak et al. (2006) Konak et al. (2006)
13 NUG15 15 αmax = 5 526.75 526.75 van Camp (1989) Konak et al. (2006) Konak et al. (2006)
14 BA12 12 lmin = 1 8,801.33 8,180.00 van Camp (1989) Konak et al. (2006) Scholz et al . (2009)
15 BA12TS 16 lmin = 1 8,600.33 8,600.33 Tate and Smith (1995) Konak et al. (2006) Konak et al. (2006)
16 BA14 14 lmin = 1 5,004.55 4,712.33 van Camp (1989) Konak et al. (2006) Scholz et al. (2009)
17 BA14TS 14 lmin = 1 4,927.69 4,927.69 Tate and Smith (1995) Konak et al. (2006) Konak et al. (2006)
18 AB20 20 αmax = 1.75 6,890.82 6,890.82 Armour and Buffa (1963) Konak et al. (2006) Konak et al. (2006)
19 SC30 30 αmax = 5 - 3,707.00 Liu and Meller (2007) - Liu and Meller (2007)
20 SC35 35 αmax = 4 - 3,604.00 Liu and Meller (2007) - Liu and Meller (2007)
Best known solution*
ReferenceProblem set
NoBest FBS solution*
Depart-ments
Common shape constraint
* Values in the 5th and 6th columns represent cost
It can be said that the problem sets used can represent the UA-FLP domain.
The problem size varies from 7 to 35 departments. To date, the largest problem set
known is SC35 (with 35 departments) which was originated from a recent research
done by Liu and Meller (2007). Twenty problem sets are used in this research
whereas the largest number of problem sets which have been used in previous
research is 27 (as shown in Table 3.2). This study can only test less than 27 problem
sets because some of them cannot be obtained (their data are not published).
Most of the problem sets do not have a too restrictive department constraint.
Most of them have a maximum aspect ratio of 4 or 5, or a minimum side length of 1.
84
These constraints bound the department solution space so that the departments will
not be too long and narrow. One problem set which has a restrictive constraint is
AB20. This problem set has a maximum aspect ratio of 1.75. This constraint
generates a little solution space because it restricts the department dimensions and
allows little combinations of departments to be placed in one bay.
Table 6.1 also provides the best FBS solutions and best known solutions.
These can be used to compare the performances of the proposed algorithm with those
from previous research. This comparison is also important to examine whether the
proposed algorithm can achieve the optimal solutions or not.
All of the best FBS solutions are obtained from Konak et al. (2006). Their
research used Mixed Integer Programming (MIP) to solve UA-FLPs with FBS
representation. They claimed that their method could generate optimal solutions for
problems with up to 14 departments. However, it could not be used to solve larger
problems.
On the other hand, the best known solutions are obtained from various
research. They are originated from the MIP approach (Meller et al., 1999; Sherali et
al., 2003), continuous representation approach (Liu and Meller, 2007), Slicing Tree
Structure (STS) representation approach (Scholz et al., 2009) and FBS representation
approach (Konak et al., 2006).
The author uses a maximum number of local search steps = 1000n, and a
maximum number of local search steps spent when the solution quality could not be
improved = 100n. The proposed algorithm is replicated five times with a maximum
number of iterations = 1000 and a maximum number of unimproved iterations = 500.
The algorithm is coded with C++ and compiled using GCC 4.3.0. It is tested using an
Intel Centrino Duo processor (1.7 GHz) and a Linux operating system.
85
6.3 Evaluation of the Proposed Algorithm with Manual Parameter Tuning
In this experiment, the proposed algorithm uses the parameter values obtained
from manual parameter tuning; number of artificial ants = n (n = number of
departments), pheromone information parameter = 1, heuristic information parameter
= 5, and evaporation rate = 0.1.
The statistical results obtained for the proposed algorithm are summarized in
Table 6.2. From the table, it can be shown that the proposed algorithm is effective
since the difference between the best and worst solutions is relatively low. It can also
be seen that the computational time of the proposed algorithm is acceptable for
facility layout planning purposes.
Table 6.2 Statistical data on results and computation time of the proposed algorithm
Best time Total time(1 replication) (5 replications)
1 O7 136.58 136.58 136.58 38 194 2 FO7 23.12 23.12 23.12 17 97 3 FO8 22.39 22.39 22.39 26 134 4 O9 241.06 241.06 241.06 44 228 5 V10s 22,899.64 22,899.64 22,899.64 57 316 6 V10a 21,463.10 21,463.10 21,463.10 61 309 7 M11s 1,321.35 1,321.35 1,321.35 83 457 8 M11a 1,204.15 1,204.15 1,204.15 77 396 9 M15s 23,197.80 23,232.12 23,369.40 210 1,330
10 M15a 27,545.30 27,682.00 27,809.20 259 1,604 11 M25 1,496.42 1,508.11 1,525.19 2,215 9,567 12 NUG12 262.00 262.00 262.00 126 637 13 NUG15 536.75 536.75 536.75 328 1,792 14 BA12 8,786.00 8,786.00 8,786.00 164 890 15 BA12* 8,299.50 8,299.50 8,299.50 301 1,960 16 BA12TS 8,587.05 8,587.05 8,587.05 445 3,121 17 BA14 5,004.55 5,004.55 5,004.55 313 2,009 18 BA14* 4,913.22 4,913.22 4,913.22 498 2,913 19 BA14TS 4,927.69 4,932.78 4,936.18 401 1,836 20 AB20 5,677.83 5,690.65 5,699.20 1,055 7,640 21 SC30* 3,679.85 3,716.56 3,749.46 17,935 74,533 22 SC35* 3,962.72 4,061.89 4,148.29 28,671 148,616
No Problem setObjective function value Computation time (second)
AS Best AS mean AS worst
*the problem is solved using mFBS
86
The comparisons of the proposed algorithm with previous research can be
examined in Table 6.3 and Table 6.4. The algorithm performs better than the
approach proposed by Tate and Smith (1995). Although both methods use FBS
representation, the proposed algorithm utilizes a high number of local search
procedures which help it to focus on finding good results. In general, the proposed
algorithm produces the same or better results compared to the method proposed by
Konak et al. (2006) except for problem sets M11s and NUG15. This is probably
because they used Mixed Integer Programming which is weak in solving large
problem instances. The proposed algorithm can produce better results than the
method used by Liu and Meller (2007) on BA12*, BA14*, and SC30* because of the
use of mFBS. On the other hand, the proposed algorithm cannot outperform the
method proposed by Scholz et al. (2009) since they use an STS representation which
can produce more solution candidates.
Table 6.3 Results and comparison of the proposed algorithm with previous research
No Problem Set
Tate and Smith (1995)
Konak et al. (2006)
Liu and Meller (2007)
Scholz et al. (2009)
AS Best objective function value
1 O7 - - 131.63 132.00 136.58 2 FO7 - 23.12 20.73 - 23.12 3 FO8 - 22.39 22.31 - 22.39 4 O9 - 241.06 235.95 239.07 241.06 5 V10s - 22,899.65 19,997.00 19,994.10 22,899.64 6 V10a - 21,463.07 - - 21,463.10 7 M11s - 1,317.79 - - 1,321.35 8 M11a - 1,225.00 - - 1,204.15 9 M15s - 27,781.95 - - 23,197.80
10 M15a - 31,779.09 - - 27,545.30 11 M25 - - - - 1,496.42 12 NUG12 - 265.60 - - 262.00 13 NUG15 - 526.75 - - 536.75 14 BA12 - 8,801.33 8,702.00 8,264.00 8,786.00 15 BA12* - 8,801.33 8,702.00 8,264.00 8,299.50 16 BA12TS 8,861.00 8,600.33 - - 8,587.05 17 BA14 - 5,004.55 5,004.00 4,712.33 5,004.55 18 BA14* - 5,004.55 5,004.00 4,712.33 4,913.22 19 BA14TS 5,080.10 4,927.69 - - 4,927.69 20 AB20 7,205.40 6,890.82 - - 5,677.83 21 SC30* - - 3,706.83 - 3,679.85 22 SC35* - - 3,604.12 - 3,962.72
*the proposed algorithm uses mFBS for solving the problems
In addition, as shown in Table 6.4, the proposed algorithm is able to produce
the best FBS solution (or even better) for all the problem sets except only for M11s
87
and NUG15. These outstanding results can be achieved because of the use of the
right representation in AS and the contribution of local search procedures.
Meanwhile, when a comparison with the best known solution is made, the proposed
algorithm can improve 7 problem sets, i.e. M15s, M15a, M25, NUG12, BA12TS,
AB20, and SC30. Despite this, it is unable to improve the previous best known
solution for 10 problem sets. This is due to the nature of the FBS representation that
restricts the solution space (Konak et al., 2006).
Table 6.4 Results and comparison of the proposed algorithm with the best-FBS and
best-known solutions
Best FBS** Best known**1 O7 136.58 - -3.62%2 FO7 23.12 0.00% -9.39%3 FO8 22.39 0.00% 0.00%4 O9 241.06 0.00% -2.12%5 V10s 22,899.64 0.00% -12.69%6 V10a 21,463.10 0.00% 0.00%7 M11s 1,321.35 -0.27% -0.27%8 M11a 1,204.15 1.73% -1.57%9 M15s 23,197.80 19.76% 19.76%
10 M15a 27,545.30 15.37% 5.85%11 M25 1,496.42 - 6.14%12 NUG12 262.00 1.37% 1.37%13 NUG15 536.75 -1.86% -1.86%14 BA12 8,786.00 0.17% -6.90%15 BA12* 8,299.50 6.05% -1.44%16 BA12TS 8,587.05 0.15% 0.15%17 BA14 5,004.55 0.00% -5.84%18 BA14* 4,913.22 1.86% -4.09%19 BA14TS 4,927.69 0.00% 0.00%20 AB20 5,677.83 21.36% 21.36%21 SC30* 3,679.85 - 0.74%22 SC35* 3,962.72 - -9.05%
Problem SetObjective function value
AS BestPercent Difference with
No
*the proposed algorithm uses mFBS for solving the problems
**positive values in the last two columns represent improvements
Furthermore, the results between BA12 and BA12*, and BA14 and BA14*
can be compared to evaluate the effectiveness of mFBS. The results show that the
mFBS (which is used in BA12* and BA14*) performs better than the original FBS
(which is used in BA12 and BA14). As stated in the previous section, mFBS can
88
extend the solution space by using free space to help the layout to fulfill department
constraints. It also tries to improve the solution by adding empty space to the bays.
The proposed algorithm is effective when solving large problem sets. As
shown, it performs better on problem sets M25, AB20, and SC30 which have 25, 20,
and 30 departments respectively. The results reflect the possibility to improve other
UA-FLP representation techniques when solving these problems. When the proposed
algorithm is used for solving SC35, it is unable to improve the best known solution.
Although the problem is similar to SC30, SC35 has a more restricted problem
constraint (the maximum aspect ratio is = 4). This problem constraint restricts the
algorithm to find a collection of departments which can be placed in one bay, thus
simultaneously decreases the objective function.
6.4 Evaluation of the Proposed Algorithm with Fuzzy Logic Controller
In this experiment, the proposed algorithm uses FLC to tune the parameter
values. For the initial parameter values, this research uses the following settings:
number of artificial ants = n (n = number of departments), pheromone information
parameter = 1, heuristic information parameter = 5, and evaporation rate = 0.1. When
applying FLC, the following limits for the parameter values are used: number of ants
- between 5 and 2n-5, pheromone information parameter and heuristic information
parameter - between 1 and 15, and evaporation rate - between 0.1 and 0.9.
The results obtained from testing the AS algorithm with fuzzy logic are
shown in Table 6.5. In the third column, there are six types of implementations.
FLC0 means AS without FLC (manual parameter tuning), FLC1 means AS with FLC
on number of ants, FLC2 means AS with FLC on pheromone information parameter,
FLC3 means AS with FLC on heuristic information parameter, FLC4 means AS with
FLC on evaporation rate and FLC5 means AS with FLC on all the four parameters at
once. The computation times needed for these implementations are not shown
because they do not differ much from those given in Table 6.2
89
Table 6.5 Results and comparison of the proposed algorithm with Fuzzy Logic
Controller
AS best AS mean AS worst AS best* AS mean* AS worst*FLC0 136.58 136.58 136.58 - - -FLC1 136.58 136.58 136.58 0.0% 0.0% 0.0%FLC2 136.58 136.58 136.58 0.0% 0.0% 0.0%FLC3 136.58 136.58 136.58 0.0% 0.0% 0.0%FLC4 136.58 136.58 136.58 0.0% 0.0% 0.0%FLC5 136.58 136.58 136.58 0.0% 0.0% 0.0%FLC0 23.12 23.12 23.12 - - -FLC1 23.12 23.12 23.12 0.0% 0.0% 0.0%FLC2 23.12 23.12 23.12 0.0% 0.0% 0.0%FLC3 23.12 23.12 23.12 0.0% 0.0% 0.0%FLC4 23.12 23.12 23.12 0.0% 0.0% 0.0%FLC5 23.12 23.12 23.12 0.0% 0.0% 0.0%FLC0 22.39 22.39 22.39 - - -FLC1 22.39 22.39 22.39 0.0% 0.0% 0.0%FLC2 22.39 22.39 22.39 0.0% 0.0% 0.0%FLC3 22.39 22.39 22.39 0.0% 0.0% 0.0%FLC4 22.39 22.39 22.39 0.0% 0.0% 0.0%FLC5 22.39 22.39 22.39 0.0% 0.0% 0.0%FLC0 241.06 241.06 241.06 - - -FLC1 241.06 241.06 241.06 0.0% 0.0% 0.0%FLC2 241.06 241.06 241.06 0.0% 0.0% 0.0%FLC3 241.06 241.06 241.06 0.0% 0.0% 0.0%FLC4 241.06 241.06 241.06 0.0% 0.0% 0.0%FLC5 241.06 241.06 241.06 0.0% 0.0% 0.0%FLC0 22,899.64 22,899.64 22,899.64 - - -FLC1 22,899.64 22,899.64 22,899.64 0.0% 0.0% 0.0%FLC2 22,899.64 22,899.64 22,899.64 0.0% 0.0% 0.0%FLC3 22,899.64 22,899.64 22,899.64 0.0% 0.0% 0.0%FLC4 22,899.64 22,899.64 22,899.64 0.0% 0.0% 0.0%FLC5 22,899.64 22,899.64 22,899.64 0.0% 0.0% 0.0%FLC0 21,463.10 21,463.10 21,463.10 - - -FLC1 21,463.10 21,463.10 21,463.10 0.0% 0.0% 0.0%FLC2 21,463.10 21,463.10 21,463.10 0.0% 0.0% 0.0%FLC3 21,463.10 21,463.10 21,463.10 0.0% 0.0% 0.0%FLC4 21,463.10 21,463.10 21,463.10 0.0% 0.0% 0.0%FLC5 21,463.10 21,463.10 21,463.10 0.0% 0.0% 0.0%FLC0 1,321.35 1,321.35 1,321.35 - - -FLC1 1,321.35 1,321.35 1,321.35 0.0% 0.0% 0.0%FLC2 1,321.35 1,321.35 1,321.35 0.0% 0.0% 0.0%FLC3 1,321.35 1,321.35 1,321.35 0.0% 0.0% 0.0%FLC4 1,321.35 1,321.35 1,321.35 0.0% 0.0% 0.0%FLC5 1,321.35 1,321.35 1,321.35 0.0% 0.0% 0.0%FLC0 1,204.15 1,204.15 1,204.15 - - -FLC1 1,204.15 1,204.15 1,204.15 0.0% 0.0% 0.0%FLC2 1,204.15 1,204.15 1,204.15 0.0% 0.0% 0.0%FLC3 1,204.15 1,204.15 1,204.15 0.0% 0.0% 0.0%FLC4 1,204.15 1,204.15 1,204.15 0.0% 0.0% 0.0%FLC5 1,204.15 1,204.15 1,204.15 0.0% 0.0% 0.0%
7 M11s
8 M11a
5 V10s
6 V10a
3 FO8
4 O9
O7
No
1
2 FO7
Percent difference with FLC0Problem set
Objective function valueMethod
* positive values in the last three columns represent improvements
90
Table 6.5 Results and comparison of the proposed algorithm with Fuzzy Logic
Controller (continued)
AS best AS mean AS worst AS best* AS mean* AS worst*FLC0 23,197.80 23,232.12 23,369.40 - - -FLC1 23,197.80 23,197.80 23,197.80 0.0% 0.1% 0.7%FLC2 23,197.80 23,197.80 23,197.80 0.0% 0.1% 0.7%FLC3 23,197.80 23,264.72 23,369.40 0.0% -0.1% 0.0%FLC4 23,197.80 23,232.12 23,369.40 0.0% 0.0% 0.0%FLC5 23,197.80 23,232.12 23,369.40 0.0% 0.0% 0.0%FLC0 27,545.30 27,682.00 27,809.20 - - -FLC1 27,545.30 27,545.30 27,545.30 0.0% 0.5% 1.0%FLC2 27,545.30 27,638.72 27,701.00 0.0% 0.2% 0.4%FLC3 27,545.30 27,669.86 27,701.00 0.0% 0.0% 0.4%FLC4 27,545.30 27,607.58 27,701.00 0.0% 0.3% 0.4%FLC5 27,545.30 27,669.86 27,701.00 0.0% 0.0% 0.4%FLC0 1,496.42 1,508.11 1,525.19 - - -FLC1 1,522.69 1,533.35 1,550.23 -1.7% -1.6% -1.6%FLC2 1,483.48 1,499.01 1,525.91 0.9% 0.6% 0.0%FLC3 1,505.53 1,522.58 1,543.17 -0.6% -1.0% -1.2%FLC4 1,484.82 1,503.93 1,524.06 0.8% 0.3% 0.1%FLC5 1,502.23 1,530.72 1,568.93 -0.4% -1.5% -2.8%FLC0 262.00 262.00 262.00 - - -FLC1 262.00 262.00 262.00 0.0% 0.0% 0.0%FLC2 262.00 262.00 262.00 0.0% 0.0% 0.0%FLC3 262.00 262.00 262.00 0.0% 0.0% 0.0%FLC4 262.00 262.00 262.00 0.0% 0.0% 0.0%FLC5 262.00 262.00 262.00 0.0% 0.0% 0.0%FLC0 536.75 536.75 536.75 - - -FLC1 536.75 536.75 536.75 0.0% 0.0% 0.0%FLC2 536.75 536.75 536.75 0.0% 0.0% 0.0%FLC3 536.75 536.75 536.75 0.0% 0.0% 0.0%FLC4 536.75 536.75 536.75 0.0% 0.0% 0.0%FLC5 536.75 536.75 536.75 0.0% 0.0% 0.0%FLC0 8,299.50 8,299.50 8,299.50 - - -FLC1 8,299.50 8,299.50 8,299.50 0.0% 0.0% 0.0%FLC2 8,299.50 8,299.50 8,299.50 0.0% 0.0% 0.0%FLC3 8,299.50 8,299.50 8,299.50 0.0% 0.0% 0.0%FLC4 8,299.50 8,299.50 8,299.50 0.0% 0.0% 0.0%FLC5 8,299.50 8,299.50 8,299.50 0.0% 0.0% 0.0%FLC0 8,587.05 8,587.05 8,587.05 - - -FLC1 8,587.05 8,587.05 8,587.05 0.0% 0.0% 0.0%FLC2 8,587.05 8,596.08 8,632.19 0.0% -0.1% -0.5%FLC3 8,587.05 8,587.05 8,587.05 0.0% 0.0% 0.0%FLC4 8,587.05 8,587.05 8,587.05 0.0% 0.0% 0.0%FLC5 8,587.05 8,587.05 8,587.05 0.0% 0.0% 0.0%FLC0 4,913.22 4,913.22 4,913.22 - - -FLC1 4,913.22 4,913.22 4,913.22 0.0% 0.0% 0.0%FLC2 4,913.22 4,913.22 4,913.22 0.0% 0.0% 0.0%FLC3 4,913.22 4,913.22 4,913.22 0.0% 0.0% 0.0%FLC4 4,913.22 4,913.22 4,913.22 0.0% 0.0% 0.0%FLC5 4,913.22 4,913.22 4,913.22 0.0% 0.0% 0.0%
MethodObjective function value Percent difference with FLC0
15 BA12TS
16 BA14
13 NUG15
14 BA12
11 M25
12 NUG12
9 M15s
10 M15a
NoProblem
set
* positive values in the last three columns represent improvements
91
Table 6.5 Results and comparison of the proposed algorithm with Fuzzy Logic
Controller (continued)
AS best AS mean AS worst AS best* AS mean* AS worst*FLC0 4,927.69 4,932.78 4,936.18 - - -FLC1 4,927.69 4,929.39 4,936.18 0.0% 0.1% 0.0%FLC2 4,927.69 4,927.69 4,927.69 0.0% 0.1% 0.2%FLC3 4,927.69 4,929.39 4,936.18 0.0% 0.1% 0.0%FLC4 4,927.69 4,932.78 4,936.18 0.0% 0.0% 0.0%FLC5 4,927.69 4,927.69 4,927.69 0.0% 0.1% 0.2%FLC0 5,677.83 5,690.65 5,699.20 - - -FLC1 5,677.83 5,857.68 6,167.87 0.0% -2.9% -7.6%FLC2 5,677.83 5,832.97 6,065.68 0.0% -2.4% -6.0%FLC3 5,677.83 5,677.83 5,677.83 0.0% 0.2% 0.4%FLC4 5,677.83 5,682.10 5,699.20 0.0% 0.2% 0.0%FLC5 5,677.83 5,910.54 6,065.68 0.0% -3.7% -6.0%FLC0 3,679.85 3,716.56 3,749.46 - - -FLC1 3,654.75 3,737.90 3,795.27 0.7% -0.6% -1.2%FLC2 3,700.19 3,746.88 3,820.45 -0.5% -0.8% -1.9%FLC3 3,649.41 3,707.04 3,762.74 0.8% 0.3% -0.4%FLC4 3,727.14 3,746.45 3,762.34 -1.3% -0.8% -0.3%FLC5 3,662.67 3,726.04 3,804.61 0.5% -0.3% -1.4%FLC0 3,962.72 4,061.89 4,148.29 - - -FLC1 3,941.12 4,052.00 4,170.52 0.5% 0.2% -0.5%FLC2 4,006.41 4,069.06 4,135.07 -1.1% -0.2% 0.3%FLC3 3,965.57 4,047.76 4,136.80 -0.1% 0.3% 0.3%FLC4 3,962.43 4,067.36 4,132.52 0.0% -0.1% 0.4%FLC5 4,009.60 4,104.60 4,174.42 -1.2% -1.0% -0.6%
Percent difference with FLC0
20 SC35
MethodObjective function value
18 AB20
19 SC30
NoProblem
set
17 BA14TS
* positive values in the last three columns represent improvements
The table provides the best, the mean (average), and the worst objective
function values gathered from 5 replications for each problem instance. The last three
columns provide the differences between AS with FLC and without FLC (FLC0). In
addition, modified FBS (mFBS) is used for solving problems with empty space
(BA12, BA14, SC30 and SC35).
When comparing the results in the AS-best column, AS with FLC performs as
good as AS with manual parameter tuning on 17 problem sets since all the
implementations of FLC on these particular problems produce the same best
objective function values as those generated by AS with manual parameter tuning. In
general, all of these 17 problem sets do not have a too large number of departments.
This concludes that the parameter values have a relatively low influence on small
problem instances.
92
The difference starts when AS with FLC is used for solving M25, SC30, and
SC35 which have 25, 30, and 35 number of departments respectively. Nevertheless,
the results of using FLC are still encouraging because the differences are very small
which are below 2%. The high number of departments means a wider solution space
must be explored. Hence, it can be concluded that parameter values influence the
performance of the AS algorithm when it is used for solving large problem instances.
The performances of the five types of FLC implementations are not too
varied. When FLC is used to tune only one parameter (FLC1, FLC2, FLC3, and
FLC4), the results are comparable with those obtained when it is not used (FLC0).
These can be achieved because the values of the other fixed parameters are taken
from manual parameter tuning. On the other hand, when FLC is used to tune the four
parameters at once, good quality results are also attained. This proves that FLC can
be utilized to produce good solutions for AS by automating the tuning of its
parameters. This brings an advantage in using FLC because it omits the need to
perform manual parameter tuning which is time consuming.
Additionally, the use of FLC for tuning AS's parameters resulted in both
improvement and deterioration on M25, SC30, and SC35 problem instances. FLC
can find better results on 6 cases whereas it performs poorer on 8 cases. The
variability in performance of the AS algorithm is a basic characteristic of a
metaheuristic. Although metaheuristic algorithms have their own strengths, they still
rely on random numbers to conduct their search.
Furthermore, it cannot be decided which type of FLC implementation (FLC1,
FLC2, FLC3, FLC4, FLC5) is the best since the results are varied. For example,
FLC1 can improve problem sets SC30 and SC35 but it performs a little poorer when
solving M25. However, the most useful FLC implementation is FLC5 since it does
not need any parameter value from manual parameter tuning. FLC5 can certainly be
used as a parameter tuning method for the AS algorithm.
93
6.5 Conclusions
This chapter has evaluated the proposed AS algorithm with FBS
representation for solving UA-FLPs. The proposed algorithm is effective and can
produce all of the best FBS solutions (or even better) except for M11s and NUG15.
In addition, it can improve the best known solution for 7 problem instances, i.e.
M15s, M15a, M25, NUG12, BA12TS, AB20, and SC30. This research has also
proposed an improvement to the FBS representation by using free or empty space.
The improvement obtained is justified on problem sets BA12 and BA14. Evidently,
the proposed algorithm is also proven to be effective when solving large problem
sets: M25, AB20, and SC30.
Furthermore, this chapter has described the testing of the FLC to automate the
tuning of the proposed algorithm. The experiments involved tuning four parameters
individually, i.e. number of ants, pheromone information parameter, heuristic
information parameter, and evaporation rate, as well as tuning all of them at once.
The results show that FLC can be used to replace manual parameter tuning which is
time consuming. The results also show that instead of using static parameter values,
FLC has the potential to help the AS algorithm to achieve better objective function
values.
CHAPTER 7
CONCLUSIONS
7.1 Introduction
This study attempts to investigate the performance of Ant System (AS) when
solving Unequal Area Facility Layout Problems (UA-FLPs). To date, a formal Ant
Colony Optimization (ACO) based metaheuristic has not been applied to solve them.
Recent research has mainly used it to deal with Quadratic Assignment Problems
(QAPs). Generally, ACO has been proven to perform outstandingly when solving
QAPs, which are the backbone problems for UA-FLPs (Stützle and Dorigo, 1999).
This motivates the author to create a new AS for solving UA-FLPs, and to investigate
its performance.
In addition, parameter values play important roles in a metaheuristic
algorithm. The significance of parameter values in metaheuristics has been laid down
by Eiben et al. (1999). They stated that parameter dependencies, time consuming
efforts, and unnecessarily optimal parameter values are the technical drawbacks of
manual parameter tuning. In brief, they suggested the use of dynamic parameter
setting.
Based on these reasons, this study investigates the effect of Fuzzy Logic
Controller (FLC) on dynamically tuning the parameters of AS. This study involves
four basic parameters of AS; number of ants, pheromone information parameter,
heuristic information parameter, and evaporation rate.
95
The proposed algorithm with and without FLC are tested using numerous
UA-FLPs which are represented as Flexible Bay Structures (FBSs). The problem
instances comprehensively represent the UA-FLP domain since their sizes range
from 7 to 35 departments. In addition, the problem instances chosen have been
heavily used in previous research.
7.2 Conclusions
This study has involved an in-depth literature review on several domains, i.e.
ACO, facility layout, and parameter tuning for metaheuristics. In addition, this study
has formulated an AS algorithm for solving UA-FLPs. It has also designed a Fuzzy
Logic implementation for tuning the algorithm's parameter values. Then the
algorithm is tested using well-known problem instances from the literature. In short,
the following key points can be made:
1. This research has formulated an AS algorithm with FBS representation for
solving UA-FLPs. The formulation of the proposed algorithm follows the past
characteristics of AS in solving QAPs. The main looping of the proposed
algorithm consists of three main parts: (1) construct ant solutions, (2) apply local
search procedures, and (3) update pheromone values. The FBS representation is
represented by department sequence and bay break sequence. The proposed
algorithm uses several local search procedures to enhance the search on
department sequence and bay structure layout. In this study, the global-best
solution and iteration-best solution are used to update the pheromone values in
every iteration. In addition, the study proposes an improvement to the FBS
structure (modified-FBS or mFBS) by using empty space to fulfill department
constraints.
2. This study is the first that applies the AS concept for solving UA-FLPs. The ACO
family has been used to solve Facility Layout Problems (FLPs) but its
implementation was intended to solve QAPs and Machine Layout Problems
(MLPs). This condition is shown in the literature review section (Table 2.3). In
96
addition, this study is the initial attempt to use FLC to automatically tune the
parameter values of AS. Previous research has used dynamic parameter tuning
but none has tried to utilize FLC as shown in Table 2.4.
3. This research has described the implementation of FLC to automate the tuning of
the proposed algorithm. The input parameters for the FLC are the standard
deviation of solutions and the number of unimproved iterations. The former can
represent how solutions in one iteration differ from those in another iteration.
Meanwhile the latter represents the degree of which the current search is trapped
in local optima. The proposed algorithm utilizes four output parameters which
are aimed to influence its search intensification and diversification. These
parameters are the number of ants, pheromone information parameter, heuristic
information parameter, and evaporation rate. Then several rule bases are
designed to control the search intensification and diversification of the algorithm.
They are made to provide the algorithm with an ability to jump out from local
optima. In addition, the rule bases are intended to limit the search when it is
performed in a too wide region.
4. This study has evaluated the proposed AS algorithm with FBS representation for
solving UA-FLPs. The proposed algorithm is effective and can produce all of the
best FBS solutions (or even better) except for M11s and NUG15. In addition, the
proposed algorithm can improve the best known solution for 7 problem instances,
i.e. M15s, M15a, M25, NUG12, BA12TS, AB20, and SC30. The improvement
obtained by using the mFBS representation is justified on problem sets BA12 and
BA14. Evidently, the proposed algorithm is also proven to be effective when
solving large problem sets: M25, AB20, and SC30.
5. This study has described the testing of FLC to automate the tuning of the
proposed algorithm. The experiments involved tuning four parameters
individually, i.e. number of ants, pheromone information parameter, heuristic
information parameter, and evaporation rate, as well as tuning all of them at once.
The results show that FLC can be used to replace manual parameter tuning which
is time consuming. The results also show that instead of using static parameter
values, FLC has the potential to help the AS algorithm to achieve better objective
function values.
97
7.3 Future Work
In terms of future work, this study can be expanded in the following ways:
1. It can be extended to include additional constraints of UA-FLPs such as fixed
department location, fixed department dimension, and free facility dimension.
2. It can include special cases of UA-FLPs, such as multiple-floor UA-FLPs, multi-
period UA-FLPs, stochastic UA-FLPs, or UA-FLPs with input-output points.
3. The proposed AS algorithm can be substituted with other ACO variants such as
Max-Min Ant System (MMAS), Ant Colony System (ACS), etc. The substitution
with other ACO variants is expected because they have been proposed by
researchers to complement and improve the performance of the basic AS
algorithm.
Finally, based on the results obtained, it is believed that this research will
bring a significant advancement to the FLP domain.
98
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PUBLICATIONS
1. Wong, K.Y., Ahmad, R. and Komarudin. (2007). An evaluation of parameters
tuning methods in metaheuristic algorithms. Proceedings of the Regional
Conference on Advanced Processes and Systems in Manufacturing. Putrajaya,
Malaysia, pp.41-48.
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Computer and Communication Engineering. Kuala Lumpur, Malaysia,
pp.542-545.
3. Wong, K.Y. and Komarudin. (2008). Ant colony optimization in solving
facility layout problems. Proceedings of the International Conference on
Mechanical and Manufacturing Engineering. Johor Bahru, Malaysia.
4. Komarudin and Wong, K.Y. (2009). Applying Ant System for Solving
Unequal Area Facility Layout Problems. Accepted for publication in
European Journal of Operational Research.
5. Komarudin, Wong, K.Y., and See, P. C. (2009). Solving Facility Layout
Problems using Flexible Bay Structure Representation and Ant System
Algorithm. Submitted to Expert Systems with Applications.
6. Wong, K.Y. and Komarudin. (2010). Comparison of Techniques for Dealing
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Vol.6, No.3.
111
APPENDICES
112
APPENDIX A
Data Sets for UA-FLPs
A.1 Summary of UA-FLP Data Sets
Table A.1 Summary of data for UA-FLPs
Width Height1 O7 7 8.54 13.00 αmax = 4 Meller et al. (1998)
2 FO7 7 8.54 13.00 αmax = 5 Meller et al. (1998)
3 FO8 8 11.31 13.00 αmax = 5 Meller et al. (1998)
4 O9 9 12.00 13.00 αmax = 5 Meller et al. (1998)
5 V10s 10 25.00 51.00 lmin = 5 van Camp (1989)
6 V10a 10 25.00 51.00 αmax = 5 van Camp (1989)
7 M11s 11 6.00 6.00 lmin = 1 Bozer et al. (1994)
8 M11a 11 3.00 2.00 αmax = 5 Bozer et al. (1994)
9 M15s 15 15.00 15.00 lmin = 1 Bozer et al. (1994)
10 M15a 15 15.00 15.00 αmax = 5 Bozer et al. (1994)
11 M25 25 15.00 15.00 αmax = 5 Bozer et al. (1994)
12 NUG12 12 3.00 4.00 αmax = 5 van Camp (1989)
13 NUG15 15 3.00 5.00 αmax = 5 van Camp (1989)
14 BA12 12 7.00 9.00 lmin = 1 van Camp (1989)
15 BA12TS 16 7.00 9.00 lmin = 1 Tate and Smith (1995)
16 BA14 14 6.00 10.00 lmin = 1 van Camp (1989)
17 BA14TS 14 6.00 10.00 lmin = 1 Tate and Smith (1995)
18 AB20 20 2.00 3.00 αmax = 1.75 Armour and Buffa (1963)
19 SC30 30 12.00 15.00 αmax = 5 Liu and Meller (2007)
20 SC35 35 15.00 16.00 αmax = 4 Liu and Meller (2007)
ReferenceProblem
setNo
Common shape constraint
Facility sizeNumber of departments
A.2 Data Set O7
Table A.2 Area requirement for problem set O7
Dept Area Maximum Aspect Ratio
1 16 42 16 43 16 44 36 45 9 46 9 47 9 4
113
Table A.3 Material flow matrix for problem set O7
From To Material Flow
1 4 51 7 12 4 32 7 13 4 23 7 14 5 44 6 45 7 26 7 1
A.3 Data Set FO7
Table A.4 Area requirement for problem set FO7
Dept Area Maximum Aspect Ratio
1 16 52 16 53 16 54 36 55 9 56 9 57 9 5
Table A.5 Material flow matrix for problem set FO7
From To Material Flow
1 2 12 3 13 4 14 5 15 6 16 7 1
A.4 Data Set FO8
Table A.6 Area requirement for problem set FO8
Dept Area Maximum Aspect Ratio
1 16 52 16 53 16 54 36 55 36 56 9 57 9 58 9 5
114
Table A.7 Material flow matrix for problem set FO8
From To Material Flow
1 2 12 3 13 4 14 5 15 6 16 7 17 8 1
A.5 Data Set O9
Table A.8 Area requirement for problem set O9
Dept Area Maximum Aspect Ratio
1 16 52 16 53 16 54 36 55 36 56 9 57 9 58 9 59 9 5
Table A.9 Material flow matrix for problem set O9
From To Material Flow
1 4 51 5 51 9 12 4 32 5 32 9 13 4 23 5 23 9 14 6 44 7 45 6 35 9 46 9 27 9 1
115
A.6 Data Set V10s
Table A.10 Area requirement and material flow matrix for problem set V10s
Dept 1 2 3 4 5 6 7 8 9 10 Area Min Side1 - 0 0 0 0 218 0 0 0 0 238 12 - 0 0 0 148 0 0 296 0 112 13 - 28 70 0 0 0 0 0 160 14 - 0 28 70 140 0 0 80 15 - 0 0 210 0 0 120 16 - 0 0 0 0 80 17 - 0 0 28 60 18 - 0 888 85 19 - 59.2 221 110 - 119 1
A.7 Data Set V10a
Table A.11 Area requirement and material flow matrix for problem set V10a
Dept 1 2 3 4 5 6 7 8 9 10 Area Max Aspect Ratio1 - 0 0 0 0 218 0 0 0 0 238 52 - 0 0 0 148 0 0 296 0 112 53 - 28 70 0 0 0 0 0 160 54 - 0 28 70 140 0 0 80 55 - 0 0 210 0 0 120 56 - 0 0 0 0 80 57 - 0 0 28 60 58 - 0 888 85 59 - 59.2 221 510 - 119 5
A.8 Data Set M11s
Table A.12 Area requirement and material flow matrix for problem set M11s
Dept 1 2 3 4 5 6 7 8 9 10 11 Area Min Side 1 0 10 0 0 140 90 20 0 40 0 0 3 1
2 0 0 10 0 0 0 0 0 0 0 0 2 1
3 0 0 0 10 0 0 0 0 0 0 0 4 1
4 0 0 0 0 0 0 0 0 0 0 4 5 1
5 0 10 0 0 0 0 40 0 0 20 0 2 1
6 0 0 10 0 0 0 0 0 20 0 0 3 1
7 0 0 0 0 0 0 0 10 0 0 0 4 1
8 0 0 0 0 0 0 0 0 0 0 11 5 1
9 0 0 0 0 0 0 0 0 0 20 0 1 1
10 0 0 0 0 0 0 0 0 0 0 20 1 1
11 146 0 0 0 0 0 0 0 0 0 0 6 1
116
A.9 Data Set M11a
Table A.13 Area requirement and material flow matrix for problem set M11a
Dept 1 2 3 4 5 6 7 8 9 10 11 Area Max Aspect Ratio1 0 10 0 0 140 90 20 0 40 0 0 3 5
2 0 0 10 0 0 0 0 0 0 0 0 2 5
3 0 0 0 10 0 0 0 0 0 0 0 4 5
4 0 0 0 0 0 0 0 0 0 0 4 5 5
5 0 10 0 0 0 0 40 0 0 20 0 2 5
6 0 0 10 0 0 0 0 0 20 0 0 3 5
7 0 0 0 0 0 0 0 10 0 0 0 4 5
8 0 0 0 0 0 0 0 0 0 0 11 5 5
9 0 0 0 0 0 0 0 0 0 20 0 1 5
10 0 0 0 0 0 0 0 0 0 0 20 1 5
11 146 0 0 0 0 0 0 0 0 0 0 6 5
A.10 Data Set M15s
Table A.14 Area requirement and material flow matrix for problem set M15s
Dept 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Area Min side
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 240 15 1
2 240 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 1
3 0 0 0 1200 0 0 0 0 0 0 0 0 0 0 0 9 1
4 0 0 0 0 0 0 0 0 0 1200 0 0 0 0 0 7 1
5 0 0 0 0 0 0 0 0 0 0 0 0 0 600 0 9 1
6 0 0 0 0 0 0 0 480 0 0 0 0 0 0 0 25 1
7 0 0 0 0 0 0 0 480 0 0 0 0 0 0 0 25 1
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 120 15 1
9 0 0 0 0 0 0 0 0 0 600 0 0 0 0 0 10 1
10 0 0 0 0 0 0 0 0 0 0 0 600 0 0 0 25 1
11 0 0 0 0 0 0 480 0 0 0 0 0 0 0 0 10 1
12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 600 15 1
13 0 0 0 0 0 0 480 0 0 0 0 0 0 0 0 6 1
14 0 0 0 0 0 0 0 0 0 0 0 600 0 0 0 19 1
15 0 10 50 0 25 40 0 0 25 0 40 0 20 0 0 25 1
117
A.11 Data Set M15a
Table A.15 Area requirement and material flow matrix for problem set M15a
Dept 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Area Max aspect ratio
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 240 15 5
2 240 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 5
3 0 0 0 1200 0 0 0 0 0 0 0 0 0 0 0 9 5
4 0 0 0 0 0 0 0 0 0 1200 0 0 0 0 0 7 5
5 0 0 0 0 0 0 0 0 0 0 0 0 0 600 0 9 5
6 0 0 0 0 0 0 0 480 0 0 0 0 0 0 0 25 5
7 0 0 0 0 0 0 0 480 0 0 0 0 0 0 0 25 5
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 120 15 5
9 0 0 0 0 0 0 0 0 0 600 0 0 0 0 0 10 5
10 0 0 0 0 0 0 0 0 0 0 0 600 0 0 0 25 5
11 0 0 0 0 0 0 480 0 0 0 0 0 0 0 0 10 5
12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 600 15 5
13 0 0 0 0 0 0 480 0 0 0 0 0 0 0 0 6 5
14 0 0 0 0 0 0 0 0 0 0 0 600 0 0 0 19 5
15 0 10 50 0 25 40 0 0 25 0 40 0 20 0 0 25 5
A.12 Data Set M25
Table A.16 Area requirement and material flow matrix for problem set M25
Dept 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Area Max aspect ratio
1 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 5
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 5
3 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5
4 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 1 5
6 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5
7 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 5 5
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 0 3 5
9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 2 5
10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 5
11 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5
12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 2 5
13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 4 5
14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 5 5
15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 80 0 0 0 0 0 2 5
16 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 5
17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 5 0 0 1 5
18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 0 0 0 5 5
19 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5
20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 4 5
21 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5
22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 4 5
23 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 5 5
24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 2 5
25 4 5 4 0 0 16 0 4 0 0 0 16 0 8 8 8 0 0 0 0 0 0 0 0 0 4 5
118
A.13 Data Set NUG12
Table A.17 Area requirement for problem set NUG12
Dept Area Max aspect ratio
1 1 52 1 53 1 54 1 55 1 56 1 57 1 58 1 59 1 5
10 1 511 1 512 1 5
Table A.18 Material flow matrix for problem set NUG12
From To Material flow
From To Material flow
1 2 5 4 8 101 3 2 4 11 51 4 4 4 12 51 5 1 5 6 101 8 6 5 10 51 9 2 5 11 11 10 1 5 12 11 11 1 6 7 51 12 1 6 8 12 3 3 6 9 12 5 2 6 10 52 6 2 6 11 42 7 2 7 8 102 9 4 7 9 52 10 5 7 10 23 8 5 7 11 33 9 5 7 12 33 10 2 8 11 53 11 2 9 11 103 12 2 9 12 104 5 5 10 11 54 6 2 11 12 24 7 2
119
A.14 Data Set NUG15
Table A.19 Area requirement for problem set NUG15
Dept Area Max aspect Ratio1 1 52 1 53 1 54 1 55 1 56 1 57 1 58 1 59 1 5
10 1 511 1 512 1 513 1 514 1 515 1 5
Table A.20 Material flow matrix for problem set NUG15
From To Material Flow
From To Material Flow
From To Material Flow
1 2 10 3 11 2 6 14 51 4 5 3 12 2 6 15 101 5 1 3 13 5 7 8 61 7 1 3 14 5 7 10 11 8 2 3 15 5 7 11 51 9 2 4 5 1 7 12 51 10 2 4 6 1 7 13 51 11 2 4 7 5 7 14 11 13 4 4 10 2 8 9 52 3 1 4 11 1 8 10 22 4 3 4 13 2 8 11 102 5 2 4 14 5 8 13 52 6 2 5 6 3 9 11 102 7 2 5 7 5 9 12 52 8 3 5 8 5 9 13 102 9 2 5 9 5 9 15 22 11 2 5 10 1 10 12 42 13 10 5 12 3 10 15 52 14 5 5 14 5 11 12 53 4 10 5 15 5 11 14 53 5 2 6 7 2 12 13 33 7 2 6 8 2 12 14 33 8 5 6 9 1 13 14 103 9 4 6 10 5 13 15 23 10 5 6 13 2 14 15 4
120
A.15 Data Set BA12
Table A.21 Area requirement for problem set BA12
Dept Area Min Side
1 9 1
2 8 1
3 10 1
4 6 1
5 4 1
6 3 1
7 3 1
8 4 1
9 2 1
10 2 1
11 1 1
12 1 1
Table A.22 Material flow matrix for problem set BA12
Dept 1 2 3 4 5 6 7 8 9 10 11 12
1 - 288 180 54 72 180 27 72 36 0 0 9
2 - 240 54 72 24 48 160 16 64 8 16
3 - 120 80 0 60 120 60 0 0 30
4 - 72 18 18 48 24 48 12 0
5 - 12 12 64 16 16 4 8
6 - 18 24 6 12 3 3
7 - 0 6 6 3 6
8 - 16 16 16 4
9 - 4 4 2
10 - 2 2
11 - 2
12 -
A.16 Data Set BA12TS
Table A.23 Area requirement for problem set BA12TS
Dept Area Min Side
1 9 1
2 8 1
3 10 1
4 6 1
5 4 1
6 3 1
7 3 1
8 4 1
9 2 1
10 2 1
11 2 1
12 2 1
13 2 0
14 1 0
15 1 0
16 1 0
121
Table A.24 Material flow matrix for problem set BA12TS
Dept 1 2 3 4 5 6 7 8 9 10 11 12
1 - 288 180 54 72 180 27 72 36 0 0 9
2 - 240 54 72 24 48 160 16 64 8 16
3 - 120 80 0 60 120 60 0 0 30
4 - 72 18 18 48 24 48 12 0
5 - 12 12 64 16 16 4 8
6 - 18 24 6 12 3 3
7 - 0 6 6 3 6
8 - 16 16 16 4
9 - 4 4 2
10 - 2 2
11 - 2
12 -
A.17 Data Set BA14
Table A.25 Area requirement for problem set BA14
Dept Area Min Side
1 9 1
2 8 1
3 9 1
4 10 1
5 6 1
6 3 1
7 3 1
8 3 1
9 2 1
10 3 1
11 2 1
12 1 1
13 1 0
14 1 0
Table A.26 Material flow matrix for problem set BA14
Dept 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 - 72 162 90 108 27 0 0 18 27 18 0 0 0
2 - 72 80 0 48 0 48 32 0 16 8 0 0
3 - 45 54 27 27 27 0 27 0 9 18 0
4 - 30 0 30 30 20 0 20 10 10 0
5 - 18 0 18 12 18 24 0 0 0
6 - 9 9 0 0 6 6 6 0
7 - 9 12 9 6 3 0 0
8 - 6 9 0 3 0 0
9 - 6 4 6 2 0
10 - 6 3 6 0
11 - 2 0 0
12 - 4 0
13 - 0
14 -
122
A.18 Data Set BA14TS
Table A.27 Area requirement for problem set BA14TS
Dept Area Min Side
1 9 1
2 8 1
3 9 1
4 10 1
5 6 1
6 3 1
7 3 1
8 3 1
9 2 1
10 3 1
11 2 1
12 1 1
13 1 0
14 3 0
Table A.28 Material flow matrix for problem set BA14TS
Dept 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 - 72 162 90 108 27 0 0 18 27 18 0 0 0
2 - 72 80 0 48 0 48 32 0 16 8 0 0
3 - 45 54 27 27 27 0 27 0 9 18 0
4 - 30 0 30 30 20 0 20 10 10 0
5 - 18 0 18 12 18 24 0 0 0
6 - 9 9 0 0 6 6 6 0
7 - 9 12 9 6 3 0 0
8 - 6 9 0 3 0 0
9 - 6 4 6 2 0
10 - 6 3 6 0
11 - 2 0 0
12 - 4 0
13 - 0
14 -
123
A.19 Data Set AB20
Table A.29 Area requirement and material flow matrix for problem set AB20
Dept 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 AreaMax
aspect ratio
1 - 18 12 0 0 0 0 0 0 10.4 11.2 0 0 12 0 0 0 0 0 0 0.27 1.752 18 - 9.6 244.5 7.8 0 139.5 0 12 13.5 0 0 0 0 0 0 0 0 69 0 0.18 1.753 12 9.6 - 0 0 22.1 0 0 31.5 39 0 0 0 130.5 0 0 0 0 136.5 0 0.27 1.754 0 244.5 0 - 10.8 57 75 0 23.4 0 0 14 0 0 0 0 0 15 157.5 0 0.18 1.755 0 7.8 0 10.8 - 0 22.5 13.5 0 15.6 0 0 0 0 13.5 0 0 0 0 0 0.18 1.756 0 0 22.1 57 0 - 61.5 0 0 0 0 4.5 0 0 0 0 0 10.5 0 0 0.18 1.757 0 139.5 0 75 22.5 61.5 - 240 0 18.7 0 0 0 9.6 0 0 0 16.5 0 37.5 0.09 1.758 0 0 0 0 13.5 0 240 - 0 0 0 0 6 0 0 0 0 0 75 334.5 0.09 1.759 0 12 31.5 23.4 0 0 0 0 - 0 0 0 0 75 0 0 75 0 0 0 0.09 1.75
10 10.4 13.5 39 0 15.6 0 18.7 0 0 - 3.6 120 0 186 19.2 0 0 0 52.5 0 0.24 1.7511 11.2 0 0 0 0 0 0 0 0 3.6 - 22.5 0 30 9.6 225 0 0 0 0 0.6 1.7512 0 0 0 14 0 4.5 0 0 0 120 22.5 - 0 0 16.5 0 150 0 84 0 0.42 1.7513 0 0 0 0 0 0 0 6 0 0 0 0 - 80 10.4 60 0 0 0 0 0.18 1.7514 12 0 130.5 0 0 0 9.6 0 75 186 30 0 80 - 97.5 0 0 9 0 0 0.24 1.7515 0 0 0 0 13.5 0 0 0 0 19.2 9.6 16.5 10.4 97.5 - 0 52.5 0 0 0 0.27 1.7516 0 0 0 0 0 0 0 0 0 0 0 0 60 0 0 - 120 0 0 0 0.75 1.7517 0 0 0 0 0 0 0 0 75 0 0 150 0 0 52.5 120 - 0 75 0 0.64 1.7518 0 0 0 15 0 10.5 16.5 0 0 0 0 0 0 9 0 0 0 - 46.5 0 0.41 1.7519 0 69 136.5 157.5 0 0 0 75 0 52.5 0 84 0 0 0 0 75 46.5 - 0 0.27 1.7520 0 0 0 0 0 0 37.5 334.5 0 0 0 0 0 0 0 0 0 0 0 - 0.45 1.75
A.20 Data Set SC30
Table A.30 Area requirement for problem set SC30
Dept AreaMax Aspect
RatioDept Area
Max Aspect Ratio
1 3 5 25 1 52 4 5 26 4 53 4 5 27 6 54 16 5 28 1 55 4 5 29 14 56 5 5 30 4 57 2 5 31 1 08 3 5 32 1 09 5 5 33 1 0
10 6 5 34 1 011 2 5 35 1 012 24 5 36 1 013 5 5 37 1 014 3 5 38 1 015 11 5 39 1 016 6 5 40 1 017 2 5 41 1 018 8 5 42 1 019 4 5 43 1 020 5 5 44 1 021 4 5 45 1 022 3 5 46 1 023 1 5 47 1 024 3 5
124
Table A.31 Material flow matrix for problem set SC30
From ToMaterial
FlowFrom To
Material Flow
1 24 2.95 10 12 38.031 25 6.32 11 12 8.841 26 1.26 12 13 12.971 27 2.11 12 14 4.671 28 1.26 12 15 53.421 30 13.91 12 16 1.832 6 20.38 13 15 12.972 19 4.5 14 15 4.672 20 3.63 15 4 63.952 21 2.93 15 18 190.742 22 1.29 16 15 4.582 23 1.43 17 15 0.764 3 394.11 18 4 190.745 11 4.09 19 29 9.185 12 33.09 20 29 7.46 5 8.92 21 29 5.976 7 2.07 22 29 2.636 8 4.84 23 29 2.926 10 4.56 24 29 5.96 12 1.83 25 29 12.656 17 0.61 26 29 2.537 8 12.97 27 29 4.228 9 43.23 28 29 2.539 11 4.76 29 4 190.139 12 38.47 30 29 59.64
125
A.21 Data Set SC35
Table A.32 Area requirement for problem set SC35
Dept AreaMax Aspect
RatioDept Area
Max Aspect Ratio
1 3 4 31 9 42 5 4 32 14 43 4 4 33 10 44 14 4 34 4 45 4 4 35 3 46 5 4 36 2 07 2 4 37 2 08 3 4 38 2 09 5 4 39 2 0
10 6 4 40 2 011 2 4 41 2 012 6 4 42 2 013 5 4 43 2 014 3 4 44 2 015 13 4 45 2 016 6 4 46 2 017 2 4 47 2 018 10 4 48 2 019 4 4 49 2 020 5 4 50 2 021 4 4 51 2 022 3 4 52 2 023 1 4 53 2 024 3 4 54 2 025 1 4 55 2 026 4 4 56 2 027 6 4 57 2 028 1 4 58 2 029 18 4 59 2 030 4 4
126
Table A.33 Material flow matrix for problem set SC35
From ToMaterial
FlowFrom To
Material Flow
1 24 2.95 11 12 8.841 25 6.32 12 32 72.891 26 1.26 13 15 12.971 27 2.11 14 15 4.671 28 1.26 15 4 63.951 30 13.91 15 18 190.742 6 20.38 16 15 4.582 19 4.5 17 15 0.762 20 3.63 18 4 190.742 21 2.93 19 29 9.182 22 1.29 20 29 7.42 23 1.43 21 29 5.974 3 225.65 22 29 2.635 11 4.09 23 29 2.925 12 33.09 24 29 5.96 5 8.92 25 29 12.656 7 2.07 26 29 2.536 8 4.84 27 29 4.226 10 4.56 28 29 2.536 12 1.83 29 33 190.136 17 0.61 30 29 59.646 31 3.93 31 32 22.927 8 12.97 32 13 12.978 9 43.23 32 14 4.679 11 4.76 32 15 53.429 12 38.47 32 16 1.83
10 12 38.03 33 34 168.45
127
APPENDIX B
Best layout obtained by the proposed AS algorithm
B.1 Best layout obtained for problem set O7
B.2 Best layout obtained for problem set FO7
128
B.3 Best layout obtained for problem set FO8
B.4 Best layout obtained for problem set O9
129
B.5 Best layout obtained for problem set V10s
B.6 Best layout obtained for problem set V10a
130
B.7 Best layout obtained for problem set M11s
B.8 Best layout obtained for problem set M11a
B.9 Best layout obtained for problem set M15s
131
B.10 Best layout obtained for problem set M15a
B.11 Best layout obtained for problem set M25
B.12 Best layout obtained for problem set NUG12
132
B.13 Best layout obtained for problem set NUG15
B.14 Best layout obtained for problem set BA12
133
B.15 Best layout obtained for problem set BA12 (mFBS)
B.16 Best layout obtained for problem set BA12TS
134
B.17 Best layout obtained for problem set BA14
B.18 Best layout obtained for problem set BA14 (mFBS)
B.19 Best layout obtained for problem set BA14TS
135
B.20 Best layout obtained for problem set AB20
B.21 Best layout obtained for problem set SC30
136
B.22 Best layout obtained for problem set SC35
Recommended