KomarudinMFKM2009

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

iii

To my God, Allah 'azza wa jalla

Then to my beloved parents, wife and daughter

iv

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.

v

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.

.

vi

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.

vii

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

viii

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

ix

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

xi

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

xii

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

xiii

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

xiv

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.

2

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

6

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)

8

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

9

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.

10

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.

11

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.

13

• 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,

14

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

REFERENCES

AbdelRazig, Y., El-Gafy, M., and Ghanem, A. (2006). Dynamic Construction Site

Layout Using Ant Colony Optimization. Transportation Research Board

(TRB) Annual Meeting. 22-26 January. Washington, D.C., USA: TRB.

Anjos, M.F., and Vannelli, A. (2006). A New Mathematical-Programming

Framework for Facility Layout Problem. INFORMS Journal on Computing

18(1), 111-118.

Armour, G..C., and Buffa, E.S. (1963). A Heuristic Algorithm and Simulation

Approach to Relative Allocation of Facilities. Management Science. 9(2),

294-300.

Balakrishnan, J., Jacobs, F. R., and Venkataramanan, M. A. (1992). Solutions for The

Constrained Dynamic Facility Layout Problem. European Journal of

Operational Research. 57(2), 280-286.

Baykasoglu, A., Dereli, T., and Sabuncu, I. (2006). An Ant Colony Algorithm for

Solving Budget Constrained and Unconstrained Dynamic Facility Layout

Problems. The International Journal of Management Science. 34(4), 385-396.

Bhríde, F. M. G., McGinnity, T. M., and McDaid, L. J. (2005). Landscape

Classification and Problem Specific Reasoning for Genetic Algorithms.

International Journal of Systems and Cybernetics. 34(9/10), 1469-1495.

99

Blum, C. (2005a). Ant Colony Optimization: Introduction and recent trends. Physics

of Live Reviews. 2(4), 353-373.

Blum, C. (2005b). Beam-ACO: Hybridizing ant colony optimization with beam

search: An application to open shop scheduling. Computers and Operations

Research 32(6), 1565–1591.

Blum, C., and Roli, A. (2003). Metaheuristics in Combinatorial Optimization:

Overview and Conceptual Comparison. ACM Computing Surveys. 35(3), 268-

308.

Bozer, Y.A., and Meller, R.D. (1997). A Reexamination of the Distance-Based

Facility Layout Problem. IIE Transactions. 29(7), 549-560.

Bozer, Y.A., Meller, R.D., and Erlebacher, S.J. (1994). An Improvement-Type layout

Algorithm for Single and Multiple Floor facilities. Management Science.

40(7), 918-932.

Bullnheimer, B., Hartl, R.F., and Strauss, C. (1999). A new rank-based version of the

Ant System: A computational study. Central European Journal for

Operations Research and Economics. 7(1), 25-38.

Chen, W. H., and Srivstava, B. (1994). Simulated annealing procedures for forming

machine cell in-group technology. European Journal of Operational

Research. 75(1), 100–111.

Corry, P., and Kozan, E. (2004). Ant Colony Optimisation for Machine Layout

Problems. Computational Optimization and Applications. 28(3), 287-310.

Dai, C. H., Zhu, Y.F. and Chen, W.R. (2006). Adaptive Probabilities of Crossover and

Mutation in Genetic Algorithms Based on Cloud Model. Information Theory

Workshop. 22-26 October. Chengdu, China: IEEE, 710-713.

100

De Jong, K.A. (1975). An analysis of the behavior of a class of genetic adaptive

systems. Doctor Philosophy. Department of Computer Science, University of

Michigan, Ann Arbor: USA.

Demirel N.C. and Toksari M.D. (2006) Optimization of the Quadratic Assignment

Problem using an Ant Colony Algorithm. Applied Mathematics and

Computation. 183(1), 427-435.

Dorigo, M., and Di Caro, G. (1999). Ant colony optimization: A new meta-heuristic.

Proceeding of the 1999 Congress on Evolutionary Computation. 6-9 July.

Washington, DC, USA: IEEE, 1470-1477.

Dorigo, M., and Gambardella, L.M. (1997). Ant Colony System: A cooperative

learning Approach to The Traveling Salesman Problem. IEEE Transactions

on Evolutionary Computation. 1(1), 53-66.

Dorigo, M., Maniezzo, V., and Colorni, A. (1996). Ant System: Optimization by a

colony of cooperating agents. IEEE Transactions on Systems, Man, and

Cybernetics – Part B. 26(1), 29-41.

Eiben, A.E., Hinterding, R. and Michalewicz, Z. (1999). Parameter Control in

Evolutionary Algorithms. IEEE Transactions on Evolutionary Computation.

3(2), 124-141.

Favuzza, S., Graditi, G. and Sanseverino, E.R. (2006). Adaptive and Dynamic Ant

Colony Search Algorithm for Optimal Distribution Systems Reinforcement

Strategy. Applied Intelligence. 24(1), 31-42.

Francis, R.L., and White, J.A. (1974). Facility Layout and Location: An Analytical

Approach. Englewood Cliffs, NJ: Prentice-Hall.

101

Frazelle, E.H. (1986). Material Handling: A Technology for Industrial

Competitiveness. Material Handling Research Center Technical Report,

Georgia Institute of Technology, USA.

Gambardella, L.M. and Dorigo, M. (2000). Ant Colony System hybridized with a

new local search for the sequential ordering problem. INFORMS Journal on

Computing. 12(3), 237–255.

Gambardella, L.M., Taillard, E.D. and Dorigo. M. (1997). Ant colonies for the QAP.

Technical Report 4-97, IDSIA, Lugano, Switzerland.

Gendreau, M., and Potvin, J.-Y. (2005). Metaheuristics: A bibliography. Annals of

Operations Research. 140(1), 189-213.

Glover, F., and Laguna, M. (1997). Tabu Search. Boston: Kluwer.

Grefenstette, J.J. (1986). Optimization of Control Parameters for Genetic Algorithms.

IEEE Transactions on Systems, Man, and Cybernetics. 16(1), 122-128.

Hani, Y., Amodeo, F., Yalaoui, F., and Chen, H. (2007). Ant Colony Optimization for

Solving an Industrial Layout Problem. European Journal of Operational

Research. 183(2), 633-642.

Hao, Z., Cai, R.C., and Huang, H. (2006). An Adaptive Parameter Control Strategy

for ACO. Proceeding of International Conference on Machine Learning and

Cybernetics. August 13-16. Dalian, China: 203-206.

Hao, Z., Huang, H., Qin, Y., and Cai, R. (2007). An ACO Algorithm with Adaptive

Volatility Rate of Pheromone Trail. Proceeding of International Conference

on Computational Science. 27-30May. Beijing, China: 1167-1170.

102

Hartono, P., Hashimoto, S. and Wahde, M. (2004). Labeled-GA with Adaptive

Mutation Rate. Proceeding of IEEE Congress on Evolutionary Computation.

20-23 June. Portland, Oregon: 1851-1858.

Hassan, M.M.D., Hogg, G.L., and Smith, D.R. (1986). Shape: A construction

algorithm for area placement evaluation. International Journal of Production

Research. 24(5), 1283-1295.

Hinterding, R. (1997). Self-adaptation using Multi-chromosomes. Proceeding of

IEEE International Conference on Evolutionary Computation.13-16 April.

Indianapolis, USA: 87-91.

Huang, H., Yang, X., Hao, Z., and Cai, R. (2006). A Novel ACO Algorithm with

Adaptive Parameter. Proceeding of International Conference on Intelligent

Computing. 16-19 August. Yunnan, China: 12-21.

Ibaraki, T. (1987). Combinatorial optimization problems and their complexity. In

Enumerative Approaches to Combinatorial Optimization - Part I. Annals of

Operations Research. 10, 1-49.

Johnson, R. V. (1982). Spacecraft for multi-floor layout planning. Management

Science. 28(40), 407-417.

Kim, J.-G., and Kim, Y.-D. (1999). A branch and bound algorithm for locating input

and output points of departments on the block layout. Journal of the

Operational Research Society. 50(5), 517-525

Ko, M-S., Tang, T-W. and Hwang, C-S. (1996). Adaptive Crossover Operator based

on Locality and Convergence. Proceedings of the 1996 IEEE International

Joint Symposia on Intelligence and System. 4-5 November. Rockville, USA:

IEEE, 18-22.

103

Konak, A., Kulturel-Konak, S., Norman, B.A., and Smith, A.E. (2006). A New

Mixed Integer Formulation for Optimal Facility Layout design. Operation

Research Letters. 34(6), 660-672.

Koopmans, T. C., and Beckman, M. (1957). Assignment problems and the location of

economic activities. Econometrica. 25(2), 53-76.

Koppen, M. (2004). No-Free-Lunch theorems and the diversity of algorithms.

Congress on Evolutionary Computation. 1, 235-241.

Liu, Q., and Meller, R.D. (2007). A sequence-pair representation and MIP-model-

based heuristic for the facility layout problem with rectangular departments.

IIE Transactions. 39(4), 377-394.

Maffoli, F. and Galbiati, G. (2000). Approximability of hard combinatorial

optimization problems: an introduction. Annals of Operations research. 96(1-

4), 221-236.

Maniezzo, V. (1999). Exact and approximate nondeterministic tree-search procedures

for the quadratic assignment problem. INFORMS Journal on Computing.

11(4), 358-369.

Maniezzo, V., and Colorni, A. (1999). The Ant System applied to the quadratic

assignment problem. IEEE Transaction of Knowledge and Data Engineering.

11(5),769-778.

Maniezzo, V., Colorni, A., and Dorigo, M. (1994). The ant system applied to the

quadratic assignment problem. Technical Report IRIDIA/94-28, Université

Libre de Bruxelles, Belgium.

104

McKendall Jr, A.R., and Shang, J. (2006). Hybrid Ant Systems for the Dynamic

Facility Layout Problem. Computers and Operations Research. 33 (3), 790-

803.

Meller, R.D. (1992). Layout Algorithms for Single and Multiple Floor Facilities.

Doctor of Philosophy Thesis, Department of Industrial and Operations

Engineering, University of Michigan, USA.

Meller, R.D., and Bozer, Y.A. (1996). A New Simulated Annealing Algorithm for The

Facility Layout Problem. International Journal of Production Research.

34(6), 1675-1692.

Meller, R.D., and Bozer, Y.A. (1997). Alternative approaches to solve the multi-floor

facility layout problem. Journal of Manufacturing Systems. 16(3), 192-203.

Meller, R.D., and Gau, K.Y. (1996). The facility layout problem: recent and emerging

trends and perspectives. Journal of Manufacturing Systems. 15(5), 351-366.

Meller, R.D., Chen, W., and Sherali, H.D. (2007). Applying the Sequence-pair

Representation to Optimal Facility Layout designs. Operations Research

Letters. 35(5), 651-659.

Meller, R.D., Narayanan, V., and Vance, P.H. (1999). Optimal Facility Layout

Design. Operations Research Letters. 23(3-5), 117–127.

Mendel, J. (1995). Fuzzy logic systems for engineering: a tutorial. Proceedings of the

IEEE. 83(3), 345-377.

Merkle, D., Middendorf, M., and Schmeck, H. (2002). Ant colony optimization for

resource-constrained project scheduling. IEEE Transaction on Evolutionary

Computation. 6(4), 333-346.

105

Montreuil, B. (1990). A Modelling Framework for Integrating Layout Design and

Flow Network Design. Proceedings of the Material Handling Research

Colloqium. Hebron: 43-58.

Montreuil, B., and Ratliff, H.D. (1988). Optimizing the location of input and output

stations within facilities layout. Engineering Costs and Production

Economics. 14(3), 177-187.

Mouhoub, M., and Wang, Z. (2006). Ant Colony with Stochastic local Search for the

Quadratic Assignment Problem. Proceedings of the 18th IEEE International

Conference on Tools with Artificial Intelligence. 13-15 November.

Washington D.C.: IEEE, 127 – 131.

Muther, R. (1955). Practical Plant Layout. New York: McGraw-Hill.

Osman, I.H., and Laporte, G. (1996). Metaheuristics: A bibliography. Annals of

Operations Research. 63(5), 513-628.

Pilat, M.L., and White, T. (2002). Using Genetic Algorithms to optimize ACS-TSP.

In M. Dorigo et al. (Eds): ANTS 2002. Lecture Notes in Computer Science.

2463, 282-287.

Qin, L., Luo, J., Chen, Z., Guo, J., Chen, L., and Pan, Y. (2006). Phelogenetic Tree

Construction using Self Adaptive Ant Colony Algorithm. Proceedings of the

First International Multi-Symposiums on Computer and Computational

Sciences - Volume 1. 20-24 June. Zhejiang, China: 179-187.

Ramkumar, A.S., and Ponnambalam, S.G. (2006). Hybrid Ant Colony System for

solving Quadratic Assignment Formulation of Machine Layout Problems.

Proceedings IEEE Conference on Cybernetics and Intelligent Systems. 7-9

June. IEEE:1-5.

106

Randall, M. (2004). Near Parameter Free Ant Colony Optimisation. Lecture Notes in

Computer Science. 3172, 374-381.

Ropke, S. and Pisinger, D. (2006). An Adaptive Large Neighborhood Search

Heuristic for the Pickup and Delivery Problem with Time Windows.

Transportation Science. 40, 455–472.

Rosenblatt, M.J. (1986). The dynamics of plant layout. Management Science. 32(1),

76-86.

Sawai, H. and Kizu, S. (1998). Parameter-free Genetic Algorithm Inspired by

Disparity Theory of Evolution. Proceedings of the 5th International

Conference on Parallel Problem Solving from Nature. 27-30 September.

Amsterdam, Holland: 702-711.

Scholz, D., Petrick, A., and Domschke, W. (2009). STaTS: A slicing tree and tabu

search based heuristic for the unequal area facility layout problem. European

Journal of Operational Research. 197(1), 166-178.

Shebanie II, C.R. (2004). An Integrated, Evolutionary Approach to Facility Layout

and Detailed Design. Master of Engineering, Department of Industrial

Engineering, University of Pittsburgh. Pittsburgh: USA.

Sherali, H.D., Fraticelli, B.M.P., and Meller, R.D. (2003). Enhanced Model

Formulations for Optimal Facility Layout. Operations Research. 51(4), 629-

644.

Shmygelska, A., and Hoos, H.H. (2005). An ant colony optimisation algorithm for

the 2D and 3D hydrophobic polar protein folding problem. Bioinformatics.

6(30), 1-22.

107

Singh, S.P., and Sharma R.R.K. (2005). A review of different approaches to the

facility layout problem. International Journal of Advanced Manufacturing

Technology. 30(5-6), 425-433.

Solimanpur, M., Vrat, P., and Shankar, R. (2004). Ant Colony Optimization

Algorithm to the inter-cell layout problem in cellular manufacturing.

European Journal of Operational Research. 157(3), 592-606.

Solimanpur, M., Vrat, P., and Shankar, R. (2005). An Ant algorithm for the single

row layout problem in flexible manufacturing systems. Computers and

Operations Research. 32(3), 583-598.

Stützle, T., and Dorigo, M. (1999). ACO algorithms for the quadratic assignment

problem. In D. Corne, M. Dorigo and F. Glover (eds) New Ideas in

Optimization. New York: McGraw-Hill.

Stützle, T., and Hoos, H.H. (2000). MAX –MIN Ant system. Future Generation

Computer Systems. 16(8), 889-914.

Syarif, A., Gen, M., and Yamashiro, M. (2004). Hybridized Parallel Genetic

Algorithm for Facility Location Problem. Research Reports, Ashikaga

Institute of Technology. 38, 65-71.

Talbi, E.G., Roux, O., Fonlupt, C. and Robillard, D. (2001). Parallel Ant Colonies for

Quadratic Assignment Problem. Future Generation Computer Systems. 17(4),

441-449.

Tam, K.Y. (1992). A Simulated Annealing Algorithm for Allocating Space to

Manufacturing Cells. International Journal of Production Research. 30(1),

63-87.

108

Tam, K.Y., and Li, S.G. (1991). A Hierarchical Approach to Facility Layout Problem.

International Journal Production Research. 29(1), 165-184.

Tate, D.M., and Smith, A.E. (1995). Unequal-Area Facility Layout by Genetic

Search. IIE Transactions. 27, 465-472.

Tompkins J.A., White J.A., Bozer J.A. and Tanchoco, J.M.A. (2003). Facilities

Planning. John Wiley & Sons: New York.

Tretheway, S.J. and Foote, B.L. (1994). Automatic Computation and Drawing of

Facility Layouts with Logical Aisle Structures. International Journal of

Production Research. 32(7), 1545-1556.

Vakharia, A.J., and Chang, Y.L. (1997). Cell formation in group technology: a

combinatorial search approach. International Journal of Production

Research. 35(7), 2025–2043

van Camp, D.J. (1989). A Nonlinear Optimization Approach for Solving Facility

Layout Problem. Master of Applied Science, Department of Industrial

Engineering, University of Toronto, Canada.

van Camp, D.J., Carter, M.W., and Vannelli, A. (1991). A Nonlinear Optimization

Approach for Solving Facility Layout Problem. European Journal of

Operation Research. 57(2), 174-189.

Wang, P. Y., Wang, G. S., Song, Y. H., and Johns, A. T. (1996). Fuzzy logic controlled

genetic algorithms. Proceedings of the Fifth IEEE International Conference

on Fuzzy Systems. 8-11 September. New Orleans, LA, USA: IEEE, 2, 314-17.

Watanabe, I., and Matsui, S. (2003). Improving the Performance of ACO Algorithms

by Adaptive Control of Candidate Set. Proceedings of Congress on

109

Evolutionary Computation (CEC '03). 8-12 December. Canberra, Australia:

IEEE, 2, 1355-1362.

Wolpert, D.H. and Macready, W.G. (1997). No Free Lunch Theorems for

Optimization. IEEE Transactions on Evolutionary Computation. 1(1), 67-82.

Wong, K.Y. and See, P.C. (2009). A New Minimum Pheromone Threshold Strategy

(MPTS) for Max-Min Ant System. Applied Soft Computing. 9 (3), 882-888.

Xue, F., Sanderson, A.C., Bonissone, P., and Graves, R.J. (2005). Fuzzy Logic

Controlled Multi-Objective Differential Evolution. Proceedings on the 14th

IEEE International Conference on Fuzzy Systems. 22-25 May. Reno NV,

USA: IEEE, 720-725.

Zadeh, L.A. (1988). Fuzzy Logic. IEEE Computer. 21(4), 83-93.

110

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.

2. Wong, K.Y. and Komarudin. (2008). Parameter tuning for ant colony

optimization: a review. Proceedings of the International Conference on

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

with Empty Spaces in Unequal Area Facility Layout Problems. Accepted for

publication in International Journal of Industrial and Systems Engineering,

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