10381067 . solution methodology based genetic algorithm for multi-objective facility layout problem

7/28/2019 10381067 . Solution Methodology Based Genetic Algorithm for Multi-Objective Facility Layout Problem

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Zagazig University

Faculty of Engineering

Solution Methodology Based Genetic

Algorithm For Multi-Objective FacilityLayout Problem

A Thesis submitted in fulfillment of the requirements for the degree of Master of

Philosophy to Industrial Engineering and Systems Department

By

Raafat Hussien El-ShaerIndustrial and Systems Engineering Department, Faculty of Engineering,

Zagazig University.

Supervisors:

Prof. Dr. Gamal Mohamed NawaraIndustrial and Systems Engineering Department, Faculty of Engineering,

Zagazig University.

Prof. Dr. Mohamed Abbas ShoumanOperations Research and Decision Support Department, Faculty of Computers

and Informatics, Zagazig University.

Dr. Hisham Mohamed El awadyIndustrial and systems Engineering Department, Faculty of Engineering,

Zagazig University.


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Approval Sheet:

Solution Methodology Based Genetic Algorithm ForMulti-Objective Facility Layout Problem

A Thesis

By

Raafat Hussien El-Shaer

Submitted in fulfillment of the requirements for the degree of Master of

Philosophy to Industrial Engineering and Systems Department

Approved as to style and content by:

Prof. Dr. Faten Faheem Mahmoud

Industrial and Systems Engineering Department, Faculty ofEngineering, Zagazig University.

Prof. Dr. Mohamed Nashaat ForseIndustrial Engineering Department, Faculty of Engineering,

Alexandria University.

Prof. Dr. Gamal Mohamed NawaraIndustrial and Systems Engineering Department, Faculty of

Engineering, Zagazig University.

Prof. Dr. Mohamed Abbas ShoumanOperations Research and Decision Support Department,

Faculty of Computers and Informatics, Zagazig University.


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I

Acknowledgements

No words can express my limitless gratitude and most sincere

thanks to Allah against what He has given us. Special thanks

and great appreciation are to Prof. Gamal Mohamed Nawara

for his great tolerance and strong support. Special thanks and

great appreciation are Also to Prof. Mohamed Abbas Shouman

for his keen supervision and to the precious and valuable time he

gave me. Thanks to his insightful comments, suggestions and

valuable feedback he kindly offered, which helped me so much to

accomplish my thesis.

Also special thanks are due to Dr. Hisham Mohamed El awady

for his great tolerance and sincere help, which guided me to

fulfill this work.Moreover, my thanks are also due to many colleagues for various

help .

Finally, I would like to say thank-you to my family and my

wife who encourage me and give me a quiet times to achieve

this thesis.


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II

Abstract

The facility layout problem (FLP) concerns finding the most effectivearrangement of facilities, personals, and any resources required so as to

minimize the costs associated with projected interaction among them. These

costs may reflect material handling costs or preferences regarding

adjacencies among departments. Because its one of the truly difficult ill-

structured, multi-criteria, multi-objectives, and NP-completeness problem; the

Artificial Intelligence techniques are one of the most powerful tools to deal

with this problem category. Therefore, the prime objective of this study is to

apply the Genetic Algorithm, (GA) as an efficient AI tool, capable of dealing

with this problem and generating reasonable good solutions within a suitable

CPU time.

In order to achieve this objective, the following points have been fulfilled:

1. A comprehensive review of literature on facilities layout is presented to

show the progressive changes that have occurred over the past

decades in techniques, the concepts and models. Especially, in Multi-

Objectives facility layout problem (MOFLP).

2. A brief overview of GAs is presented, which shows the motivated idea

of the algorithm, the relevant operators, the encoding ways, the

selection methods, crossover methods and mutation methods.

3. GA is proposed for the problem under consideration, with

comprehensive illustration for its building operators and parameters are

discussed in details.

4. GAMOFLP Software, written with C++ language, is developed for the

proposed GA in point (3).

This work addresses the second design stage of the cellular layout (i.e. inter-

cell layout problem), which is a phase of the cellular manufacturing design,

and is an important step before the design of finalized layout. The inter-cell

layout problem seeks the best arrangement of cells based on their inter-

relationships within the available area. In The proposed GA, a special

permutation encoding is used to represent the chromosome and the


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III

generation-based reproduction system with Elitism is applied using the

tournament selection method, two-points crossover method and order change

mutation method.

In this work, the problem planning time is short and one floor is considered,

and the available floor space can have restricted areas. The problem is

affected by multiple criteria and has multiple objectives. The scaling problem

among objectives is solved by a reasonable quadratic model.

A solution methodology based GA for multi-objectives FLP (GAMOFLP) is

developed to handle the conflicting criteria and normalize the objectives

variables that affect the problem in order to get fair and reasonable good

solution.

The proposed GAMOFLP program is designed to tackle the following types of

FLPs:

1. Establish a layout for new facilities.

2. Establish a layout for new facilities subject to restricted areas.

3. Select the best position to add new facility (facilities) to an existing

layout (plant).

The validation of the proposed GAMOFLP system is verified using solved 20

problems that have published in previous researches. The results have

proved the efficiency and effectiveness of the proposed system and its

capability of getting acceptable suboptimal solutions within convenient time. In

details, related to the final solutions, the experiments show that the proposed

GAMOFLP capable of finding better solutions than that those published or at

least equal. Also comparatively with the required time to find the final

solutions, in spite of its not fair tool for comparing, the results show the

superiority of the system in providing better solutions within less CPU time

than have published or at most in its limits.


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IV

Contents

Page

Acknowledgement I

Abstract II

List of Tables VII

List of Figures VIII

List of Abbreviations X

Chapter 1

Introduction to Facility Layout Problem and Genetic Algorithms 1

1.1 Facility Layout Problem 1

1.2 Genetic Algorithms (GAs) 3

1.3 Thesis Outline 4

Chapter 2

A Review and survey of FLPs 6

2.1 Introduction 6

2.2 FLP Formulation 6

2.3 Computational Complexity 7

2.4 Optimal Algorithms and Suboptimal Algorithms 9

2.4.1 Optimal Algorithms 9

2.4.2 Suboptimal Algorithms 9

2.4.2.1 Constructive approaches 10

2.4.2.2 Improving approaches 11

2.4.2.3 Hybrid approaches 11

2.4.2.4 Graphical approaches 11

2.5 Intelligent Techniques for FLP 12

2.5.1 Expert Systems and MCFLP 12

2.5.2 Fuzzy Aystems and MCFLP 13

2.5.3 Genetic Algorithms for MCFLP 14

2.5.4 Intelligent Hybrid System for MCFLP 14

2.6 General Comments 15


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V

Chapter 3 Page

Facility Layout Problem Formulation 16

3.1 Introduction 16

3.2 Quantitative Approach 17

3.3 Qualitative Approach 18

3.4 Multi-Objectives Approaches 19

Chapter 4

A Review of Genetic Algorithms 24

4.1 What is a Genetic Algorithm? 24

4.2 GA Parameters and Performance 25

4.2.1 Crossover and Mutation 25

4.2.2 Population Size 26

4.2.3 Selection Methods 26

4.2.4 Reproduction Methods 27

4.3 On-line, Off-line and Best Individual Performance 28

4.4 GA Application 29

4.4.1 Layout Problems with GAs 29

4.4.2 FLPs with GAs 30

4.5 The Need for Good Solver Techniques 31

Chapter 5

Proposed Genetic Algorithm 32

5.1 Why Genetic Algorithm? 33

5.2 The Proposed Genetic Algorithm 33

5.2.1 Chromosome Representation 34

5.2.1.1 Representation of special layout configuration 34

5.2.2 Initial Population 35

5.2.3 Selection Method 35

5.2.4 Crossover Operator 36

5.2.5 Mutation Operator 37

5.2.6 Reproduction System 37


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VI

Page

5.2.7 Evaluation Function 37

5.2.8 Termination Criteria 37

5.2.9 Algorithm Steps 38

5.2.10 The default Values of The Proposed GA Parameters 39

5.3 User Interface 41

5.3.1 GAMOFLP Features 41

5.3.2 GAMOFLP Screens 42

Chapter 6

System Evaluation and Validation 51

6.1 Performance Evaluation 51

6.2 Harraz's Procedure Comparison 51

6.3 Comparison with Chen and Shas Procedure 60

6.4 Common Test Problems 62

Chapter 7Conclusions and Future Work 65

7.1 Snapshot About Thesis 65

7.2 Conclusion 65

7.3 Scope and Recommendation of Future Work 66

References 68

Appendix ( A ): Hill-Climbing Investigation Method 76Appendix ( B ): Test Problems Data 79

Appendix ( C ): Proposed GAMOFLP Code 84


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VII

List of Tables

Page

Table 2.1 : Classification of FLP solvers. 8

Table 2.2 : Constructive approaches. 10

Table 5.1 : GA parameters to be investigated. 40

Table 5.2 : Hill-climbing investigation procedure. 40

Table 6.1 : Best solution qualities for test problem in Harraz [64]. 52




Table 6.5 : Best solution qualities for test problem in Harraz [64]. 56Table 6.6 : Best solution qualities for test problem in Harraz [64]. 57



Table 6.9 : Best solution qualities for test problems in Chen and Sha [21]. 60

Table 6.10: Best solution qualities for test problem in Chen and Sha [21]. 61

Table 6.11: Best solution qualities for the eight test problems in Nugent et al.[41]. 63

Table 6.12: Average solution qualities for test problems in Nugent et al. [41]. 64


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VIII

List of Figures

Page

Figure 3.1: Two algorithms performance. 28

Figure 3.2 : Chromosome representation for an identical FLP. 30

Figure 5.1 : Phenotype representation layout. 34

Figure 5.2 : The available layout space with restricted area. 35

Figure 5.3 : Available layout space of restricted area and pre-specified facilities. 35

Figure 5.4 : PMX crossover method. 37

Figure 5.5 : Genetic algorithm flow chart. 39

Figure 5.6 : Some Basic restricted shapes for the FLPs available space. 42

Figure 5.7 : Available space and pre specified facilities. 42

Figure 5.8 : Screen no. ( 1 ) Genetic Algorithm parameters. 43

Figure 5.9 : Screen no. ( 2 ) Criteria type. 43

Figure 5.10: Screen no. ( 3 ) Layout space. 44

Figure 5.11: Screen no. ( 4 ) Available layout space. 45

Figure 5.12: Screen no. ( 5 ) Pre-specified facilities. 45

Figure 5.13: Screen no. ( 6 ) Entering relationship matrix. 46

Figure 5.14: Screen no. ( 7 ) Entering flow matrix. 46Figure 5.15: Screen no. ( 8 ) Entering cost matrix. 46

Figure 5.16: Screen no. ( 9 ) Entering the objectives weight. 47

Figure 5.17: Screen no. (10) Confirming relationship matrix data. 47

Figure 5.18: Screen no. (11) Adjusting coding system values. 48

Figure 5.19: Screen no. (12) confirming flow matrix data. 48

Figure 5.20: Screen no. (13) Solution progress. 49

Figure 5.21: Screen no. (14) Final solution and improvement path. 49

Figure 5.22: Best illustrative example solution after 500 generation. 50

Figure 6.1: Five-department problem % improvement in the OFV and

CPU computation time. 52

Figure 6.2: Six-department problem% improvement in the OFV and


Figure 6.3: Seven department problem % improvement in the OFV and



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IX

Page

Figure 6.4: Eight department problem% improvement in the OFV and


Figure 6.5: Nine department problem % improvement in the OFV and


Figure 6.6: Ten department problem % improvement in the OFV and


Figure 6.7: Eleven department problem % improvement in the OFV and


Figure 6.8: Twelve department problem % improvement in the OFV and


Figure 6.9: 20-department problem % improvement in the OFV. 62

Figure A.1 : Flow, adjacency and distance data for a randomly generated

25-cell problem 76

Figure B.1 : Flow, adjacency and distance data for a 5-cell problem. 79








Figure B.9 : Flow, adjacency and distance data for an 8-cell problem. 81

Figure B.10: Flow, adjacency and distance data for a 12-cell problem. 81




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X

List of Abbreviations

AI Artificial Intelligence

AHP Analytic Hierarchy Process

ALDEP Automated Layout Design ProgramANNs Artificial Neural Networks

AS Ant System

C++ C++ Language

CM Cellular Manufacturing

CORELAP Computerized Relationship Layout Planning

CRAFT Computerized Relative Allocation of Facilities Technique

DEA Data Envelopment Analysis

DM Decision-Maker

ES Expert Systems

FATE A new Construction Algorithm for Facilities Layout

FDMS Fuzzy Decision-Making SystemsFL Fuzzy Logic

FLEXPERT Fuzzy-integrated Expert System for Facility Layout

FLP Facility Layout Problem

GA Genetic Algorithm

HC Hill-Climbing Method

HSs Hybrid Systems

KBDSS Knowledge-Based Decision

LB Lower Bound

LPA Linear Placement Algorithm

MAT Modular Allocation Technique

Max.Iter Maximum Iteration NumberMCDM Multi-Criteria Decision-Making

MCFLP Multi-Criteria Facility Layout Problem

MOFLP Multi-Objectives Facility Layout Problem

N Total Number of Facilities

NP Non-Deterministic Polynomial

OFV Objective Function Value

OV Objective Value

Pc Crossover Probability

PLANET Plant Layout Analysis and Evaluation Technique

Pm Mutation Probability

PMX Partially Matched Crossover

PopSize Population Size

QAP Quadratic Assignment Problem

QSP Quadratic Set-covering Problem

SA Simulated Annealing

SBL Shape-based Block Layout

TFC Total Flow Cost

TNR Total Numerical Rating

Ts Tournament Size

VLSI Very Large Scale Integrated

WIP Work-in-process


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Chapter 1

Introduction to: Facility Layout Problem

and Genetic Algorithms

1.1 Facility Layout Problemhe facility layout problem, FLP, deals with finding the most effective

physical arrangement of facilities, personal, and any resources required

to facilitate the production of goods and services. It has attracted the attention

of many researchers due to its practical utility and interdisciplinary importance.

The production function of a manufacturing company is significantly affected

by the layout of its manufacturing shop. While a well designed layout can

considerably improve the efficiency of the shop, a poor one leads to increased

work-in-process (WIP) overloading the material handling system and

contribute to inefficient set-ups, longer queues, etc. Ham et al. [1]. Material

handling and layout related costs have been estimated to be about 20%-50%

of the operating expenses in manufacturing systems, Tompkins and White [2].Historically, two basic approaches have most commonly been used to

generate desirable layouts: a qualitative one and a quantitative one. These

approaches are usually used one at a time when solving a facility layout

problem.

With qualitative approaches, layout designers provide subjective evaluations

of desired closeness between departments. Then, overall subjective

closeness ratings between various departments are maximized. Thesesubjective closeness ratings are: A (absolutely necessary), E (essentially

T

1


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Chapter 1 : Introduction to FLP & GA 2

important), I (important), O (ordinary), U (unimportant) and X (undesirable), to

indicate the respective degrees of necessity that two given departments be

located close together. Layout designers may then assign numerical values to

the ratings such that they have the ranking A > E > I > O > U> X. Seehof and

Evans [3], Lee and More [4], Muther and McPherson [5] and Muther [6] have

developed algorithms based on qualitative criteria to obtain final layouts.

These different qualitative approaches are distinguished primarily by the

scoring methods used for the closeness ratings. In this aspect, the numerical

values used by Sule [7] and Harmonosky and Tothero [8] for ratings are A= 4,

E= 3, I= 2, O= 1, U= 0 and X= -1. Also, the ALDEP procedure presented by

Seehof and Evans [3] used the numerical values: A= 64, E= 16, I= 4, O= 1,

U= 0 and X= -1024.

Quantitative approaches involve primarily the minimization of material

handling costs between various departments. The quadratic assignment

problem (QAP) formulation for assigning N facilities to N mutually exclusive

locations is the most typical model used. Gilmore [9], Lawler [10] and, Gavett

and Plyter [11] have offered exact solution procedures using branch-and-

bound techniques. However, The QAP formulation belongs to the class of NP-

complete problems Garey and J ohnson [12], and no definite method can

achieve optimal solution in a reasonable time when 15 or more facilities are

considered. Consequently, many heuristic algorithms have been developed

for achieving a trade-off between computation time and the efficiency of the

final solution Kusiak and Heragu [13]. The proposed approach in this work is

also a heuristic one.

Many researchers are seeking about the appropriateness of a single criterion

objective selection in solving the facility layout problem due to the drawbacks

of using qualitative and quantitative approaches separately. The major

limitations on quantitative approaches are that they consider only relationships

that can be quantified and no qualitative factors are considered. The

shortcoming of qualitative approaches is their strong assumption that all

qualitative factors can be aggregated into one criterion. In real life, the facility

layout problem must consider quantitative and qualitative criteria and this falls

into the category of the multi-objective facility layout (MOFL) problem.


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The primary purpose in solving the MOFL problem is to generate efficient

alternatives that can be presented to the decision maker for his or her

selection. Malakooti and Tsurushima [14] classified three methods for solving

the MOFL problem:

a) Generate a set of efficient layout alternatives and then present it to the

decision maker;

b) Assess the decision makers preferences first, and then generate the

best layout alternative; and,

c) Use an interactive method to find the best layout alternative.

The proposed approach in this work falls into the category of the type (a)

methods in terms of generating good-quality solutions using an effective

heuristic algorithm. As a matter of fact, Rosenblatt [15], Dutta and Sahu[16],

Fortenberry and Cox [17], Waghodekar and Sahu [18], Urban [19] and

Houshyar [20], Harmonosky and Tothero [8] all developed QAP formulations

by specifying different objective weights to generate the best layout [see

section (3.4)]. However, there are two inadequacies in these approaches:

1. All factors are not represented on the same scale.

2. Measurement units used for objectives are incomparable.

Chen and Sha [21] developed a heuristic approach to overcome the above-

mentioned inadequacies by reasonably normalizing all objectives of MOFLP,

and handling qualitative and quantitative information in a similar fashion. In

the same direction, in this work, an approach based genetic algorithm using

Chen and Shas model to deal with this problem and comparative analysis

have been presented.

1.2 Genetic Algorithms (GAs)Genetic algorithms attempt to mimic the biological evolution process for

discovering good solutions. They are based on a direct analogy to Darwinian

natural selection and mutations in biological reproduction and belong to a

category of heuristics known as randomized heuristics. They employ

randomized choice operators in their search strategy and do not depend on


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complete a priori knowledge of the features of the domain. These operators

have been conceived through abstractions of natural genetic mechanisms

such as crossover and mutation and have been cast into algorithmic forms

Arunkumar and Chockalingam [22]. Repetitive executions of these heuristics

need not yield the same solution.

A genetic algorithm maintains a collection or population of solutions

throughout the search. It initializes the population with a pool of potential

solutions to the problem and seeks to produce better solutions by combining

the better of the existing ones through the use of one or more genetic

operators. solutions are chosen at each iteration with a bias towards those

with the best objective values. With various mapping techniques and an

appropriate measure of fitness of solutions (i.e., objective function value), a

genetic algorithm can be tailored to evolve a solution for many types of

problems, including optimization of a function or determination of the proper

order of a sequence.

Theoretical analyses suggest that genetic algorithms can quickly locate

high performance regions in extremely large and complex search spaces. In

addition, the distributed and repeated sampling can lead to some natural

insensitivity to noisy feedback. The genetic algorithm based heuristics are

highly suited for application to large instances of problems that are hard to

model and for which no satisfactory tailored algorithms are available. There

are many variations and refinements, but basically, any genetic algorithm

contains reproduction, crossovers and mutation.

GAs have been used to find good solution to various NP-complete

problems such as time-tabling problems Ross et al [23] and cable routing

problems Kloske and Smith [24]; and have shown good performance in many

applications. Actually some papers such as Cohoon et al [25] and Tam [26]

suggest GAs superiority in FLPs. Therefore, GAs are a promising approach to

FLPs.

1.3 Thesis OutlineThe scope of the work is not to solve a planning problem, it is to handle the

multi-criteria aspect that affects the idealized relative location of cells within a


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floor plan. Due to the combinatorial and ill-structured nature of the problem, a

solution methodology based GA heuristic is developed to suggest an

acceptable solution. In the proposed GA, a special chromosome

representation is applied to represent some constraints such as restricted

locations and pre-specified facilities. Moreover, an investigation using Hill-

climbing method is applied to tune up the parameters of the proposed GA.

The approach is validated by many recommended test problems taken from

previous papers and compared with other multi-objective approaches.

This thesis consists of seven chapters including this chapter. In Chapter 2,

FLPs are reviewed and surveyed especially, multi criteria. The Genetic

Algorithms and their features are reviewed in Chapter 3. In Chapter 4, the FLP

formulation is reviewed and proposed formulation is mentioned. The

implementation details of the proposed GA are described in Chapter 5.

Chapter 6 presents comparative study between the proposed approach and

other MOFLP approaches. Finally, in Chapter 7, the conclusions of this

research work and some recommendations for future work are highlighted.


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Chapter 2

A Review and survey of FLPs

2.1 Introduction

any sorts of layout can be mentioned. For example, bin-packing

problems such as Smith [27] and Falkenauer [28] try to maximize the

storage packets number; VLSI chip layout problems like Larmore et al [29]

and Saab and Rao [30] aim to minimize the area occupied by chips. In

particular, facility layout problems (FLPs) may be one of the largest fields.

This is because there is usually a problem to minimize total traffic cost

between facilities in a building. For example, FLP can be applied to

manufacturing machines in a factory Souilah [31], to people traffic in an office

Steadman [32], and to backboard wiring on an electrical board Steinberg [33].

In FLPs, under given conditions including facility size and flow between each

pair of them, the problem is to minimize the total flow cost.

2.2 FLP Formulation

Though some other models have been suggested as shown in Kusiak and

Heragu [13], the quadratic assignment problem (QAP) described in Koopmans

and beckmann [34] may be regarded as the basic representation of FLPs. The

M

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Chapter 2 : A Review and survey of FLPs 7

name of (QAP) was so given because the objective function is a second-

degree function of variables and the constraints are linear functions of

variables. In spite of the QAP has been frequently used to model the FLP, this

does not mean that all FLPs can be formulated as a QAP. In the case of

machine layout problem (MLP) in which the locations of machines are not

known initially, QAP cannot be formulated.

The second formulation for FLP is a quadratic set-covering problem (QSP)

Bazaraa [35]. In this formulation, the total area occupied by all facilities is

divided into a number of blocks where each facility is assigned to exactly one

location and each block is occupied by at most one facility. A disadvantage of

QSP formulation is that the problem size increases as the total area occupied

by all the facilities is divided into smaller blocks Bazaraa [35].

The FLP is formulated using several integer-programing formulations. Lawler

[10] was the first to formulate the facility layout problem as a linear integer-

programing problem; Love and Wong [36] proposed a simple integer

programming formulation to the QAP. Kaufman and Broeckx [37] developed a

linear mixed integer program to formulate the FLP and Ritzman et al [38]

formulated a large mixed integer goal programming model for assigning

offices in buildings.

Flouds and Robinson [39] developed a Graph-theoretic approach model. in

this approach, it is assumed that the desirability of locating each pair of

facilities adjacent to each other is known. In this model, a closeness rating

indicating desirability of locating facility i adjacent to facility j is assumed. The

model seeks to maximize the closeness rating of the facilities.

In addition of the above formulations, Rosenblatt [15] developed a model,

which minimize the flow cost of material and maximize a closeness rating

measure (multi-objective FLP). These two objectives are conflicting

objectives. Rosenblatt [15], Dutta and Sahu [16] developed heuristic

algorithms to solve the model.

2.3 Computational Complexity

Because of there are N! ways of putting N facilities into N locations the QAPs

objective function can take N! different values at most. As a result, in order to


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get the best layout, all the N! patterns should be estimated. Nevertheless,

since N! grows extremely large ifNgoes large, it is impossible to search for all

patterns in polynomial time. So, Sahni and Gonzalez [40] showed that the FLP

consisting of many facilities is a NP-complete problem. To find an optimal

solution to the fifteen-facility problem in Nugent et al [41], more than an hour

of CPU time was required on a CDC CYBER 76. Among the eight test

problems in Nugent et al [41], the largest problem for which an optimal

solution was found was the fifteen-facility layout problem. This is considered

as a major constraint.

Between the late 1950s and the mid 1990's, a number of algorithms have

been developed to solve the FLP. These algorithms may be classified as:

Optimal algorithms,

Suboptimal algorithms.

Table 2.1 lists FLP solvers as optimal and suboptimal procedures.

Table 2.1: Classification of FLP solvers Kazuhiro [42].

Procedure Description

Optimal algorithms

Looking for the best layout (How to prune the

redundant alternatives is important.) [43],[11].

Suboptimal algorithms Looking for reasonably good layout

Constructive approach Putting one facility to another

(Usually facilities having heavier load are

considered prior to lighter ones) [44],[45].

Improving approach From a random layout, exchanging some

selected facilities (selection strategy is

important.) [46],[47],[41]

Hybrid approach A combination of constructive and improving

approach. [48],[49],[50]

Graphical approach Analytical approach (a sort of constructive

approach) [51],[52]


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2.4 Optimal Algorithms and Suboptimal Algorithms

The next subsections present a review about the two procedures.

2.4.1 Optimal Algorithms

Optimal algorithms search for the best solution by some heuristics like branch

and bound Land [43], Bazaraa and Elshafei [53] or Gavett and Plyter [11] and

cutting plane Bazaraa and Sherali [54], whereas suboptimal procedures look

for reasonably good solutions by various strategies. However, owing to the

NP-completeness, the approaches for optimal solution may be impractical

especially when number of facilities increases Hanan and Kurtzberg[55]. The

computational results reported in Lavalle and Roucairol [56] indicate that the

parallel branch and bound algorithm requires high computation time for layout

problems with twelve or more facilities. Moreover, the largest facility layout

problems are solved optimally by a cutting plane algorithm is a layout problem

with eight facilities.

A common experience of optimal algorithms is that optimal solution is found

early in the branching process but is not verified until a substantially high

number of solutions have been enumerated Burkard and Stratmann [48],

Bazaraa and Kirca [49]. This prompted researchers to terminate the branch

and bound process prematurely without verifying optimality. This resulted in

heuristic branch and bound algorithms.

Burkard [57] listed two criteria for premature termination of branch and bound

process; termination based on time limits and termination based on upper

bounds quality.

2.4.2 Suboptimal algorithms

The optimal algorithms discussed above have the following disadvantages [13]:

1. Computational time requirement is high.

2. The largest problem size solved optimally is a problem of fifteen facilities.

This directs the researchers to concentrate on suboptimal algorithms for

solving the FLP. According to Kusiak and Heragu [13], the suboptimal

algorithms can be categorized into four approaches as exhibited in Table 2.1.


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2.4.2.1 Constructive approaches

Some algorithms like MAT in Edwards et al [44] and LPA in Neghabat

[45] are called constructive approaches, since they make physical layouts by

adding one facility to another. In these approaches, deciding the order of

putting facilities is generally important. In this aspect, LPA begins by placing

the two facilities of the highest flow among each other; then another facility is

placed, which has the highest flow with the located facilities, as close as

possible to them; and so on. MAT ranks pairs of facilities according to their

flow values and location pairs according to their distance values and uses this

information to determine the layout. Also there are many constructive

approaches which use Rel-chart to determine the ordering of facilities to

construct a layout like ALDEP in Seehof and Evans [3], CORLAP in Lee and

Moore [4], and RMA Comp I in Muther and McPherson [5]. Table 2.2 lists

constructive approaches summary.

Table 2.2 : Construct ive approaches.

Use the following data to adjust

facilities building orderConstruction

algorithmFrom-to-chart Rel-chart

Year,Ref.

HC66 * 1966,[47]

ALDEP * 1967,[3]

CORLAP * 1967,[4]

RMA Comp I * 1970,[5]

MAT * 1970,[44]

PLANET * 1972,[58]

LSP * 1972,[59]

LPA * 1974,[45]

FATE * * 1978 [60]

INLAYT * 1980,[61]

FLAT * 1986,[62]


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2.4.2.2 Improving approaches

In improving approaches, there is always an initial solution, which is

often randomly generated. Based on this initial solution, systematic

exchanges between facilities are made in order to get better solutions. The

exchanges continue until either a reasonably good solution is obtained or a

certain allowed time passes. Hence, the solution quality of these approaches

depends upon the initial layout evaluation, and the selection strategy of

exchanging facilities is critical. In this aspect, while CRAFT in Buffa et al [46]

has no limitation of facilities exchange, HC63-66 in Hiller and Connors [47]

restricte the facilities exchange in the aim of computation time reduction. Also,

some algorithms like Biased Sampling Method in Nugent et al [41] select

facilities stochastically to save time.

2.4.2.3 Hybrid approaches

hybrid approaches are used to obtain the merits of both approaches

above. This is, starting from a layout created by a constructive algorithm; the

layout is modified by some improving algorithms. For instance, Burkard and

Stratmann [48] create an initial layout by branch and bound technique under

time limit; then, improve it by exchanging two or three facilities at one time.

2.4.2.4 Graphical approaches

Graphical approachesput importance on facilities adjacencies. In these

approaches, each facility and each adjacency of two facilities are represented

by a node and an arc, respectively. Therefore, a feasible layout must be a

planar graph, which can be drawn on a plane without any intersections of

arcs. Thus, as Kusiak and Heragu [13] suggested, these approaches

classified as sorts of construction approaches because they construct a larger

graph by adding a new node to a planar graph so that the new graph can be

still planar and that its flow cost can be as small as possible. However,

according to Foulds [51], these approaches may be able to solve FLPs with at

most fifteen facilities in reasonable computation time.

The major drawbacks of the aforementioned approaches lie in the fact that thesearch for the best layout is not very efficient, is not easily obtainable, and the


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multi-objective nature are not considered in the problem Hiller and Connors

[47]. As a matter of fact, facility layout problem can be considered one of the

truly difficult ill-structured, multi-criteria decision making and combinatorial

optimization problem. According to Malakooti and Tsurushima [14] the layout

problem is an ill-structured problem because:

There are multiple criteria that must be considered to test alternate

layouts.

It is hard to determine a problem space that can represent all

characteristic of the problem.

There are many domain-specific and problem-specific constraints.

Due to the combinatorial nature of facility layout problem, finding an optimal

solution is very complex and tedious task, especially if the number of facilities

is large. Still finding out for new and recent developments rather than

conventional approaches to overcome the aforementioned drawbacks.

Intelligent techniques such as Expert systems (ESs), Fuzzy logic (FLs),

Genetic algorithms (GAs), Ant systems (ASs) , Artificial neural networks

(ANNs) and Aybrid systems (HSs) have been used as new advancements for

the tackled problem.

2.5 Intelligent Techniques for FLP

Intelligent techniques were introduced to the field of facilities layout in

the early 1980s. Most of these systems are briefly reviewed taking into

account the multi-criteria concept (MCFLP).

2.5.1 Expert Systems and MCFLP

Kumara et al [63] have developed a heuristic-based ES. They have

defined the facilities layout problem as a multi-objective problem and have

outline a methodology to handle the qualitative constraints in conjunction with

heuristic procedures for quantitative parameters.

Malakooti and Tsurushima [14] have developed an ES for multi-criteria facility

layout. Their approach is based on expert systems and multi-criteria decision-

making (MCDM). The expert system interacts with the decision-maker (DM),


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and reflects the DMs preferences in the selection of rules and priorities. The

approach consists of two parts: (1) construction of a layout based on a set of

rules and restrictions, and (2) improvement of the layout based on interaction

with DM. The MCDM expert system approach considers and incorporates the

multiple-criteria in these two parts as follows: First, it uses priorities on the

selection of rules, adjacency of departments, and departments for

construction purposes. Second, it uses different objectives such as material

handling cost, flexibility, and material-handling time for paired comparison of

generated layouts for improvement purposes.

Harraz [64] has developed a knowledge-based decision support system for

multi-criteria facility layout. The system works in a tandem mode. It combines

a rule-based module with an optimization module. The rule-based module

enables the user to assign different priorities for criteria and generates a

layout based on a set of rules. The resultant layout is seeded optionally to the

improvement algorithm to find a better configuration for the solution. The

improvement module is based on the simulated annealing (SA) global

optimization algorithm. The proposed algorithm in the current work will use the

test problems used in Harraz [64], as a comparison.

2.5.2 Fuzzy Systems and MCFLP

Raoot and Rakshit [65] have developed a linguistic pattern approach

for multiple criteria facility layout problems. A multiple criteria model is

formulated using the basic concept of linguistic pattern and a heuristic

procedure is proposed to generate a set of efficient alternative layouts. The

problem of selection of the facility layout from a set of alternatives, which

satisfy different objectives and restrictions to known degrees, is considered as

a MCDM problem and the ELECTRE method based on out ranking relations

approach, is used to select the best layout. Dweiri and Meier [66] have

established a vigorous methodology, based on fuzzy set theory, to improve

the facility layout process. The AHP is used to find the weights of both

qualitative and quantitative factors, which affects the closeness rating

between departments in a plant. FUZZY, a computer program developed

based on the fuzzy decision-making systems (FDMS), is used to generate the


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activity relationship charts. These charts are used by FZYCRLP, a modified

version of CORELAP, to develop the layouts. FELAP, another program based

on FDMS, is used to evaluate the layouts. This evaluation method uses the

distances and the relationships between departments to score the layout.

2.5.3 Genetic Algorithms for MCFLP

Hamamoto et al [67] developed a genetic algorithm based- facility

layout method with an embedded simulation model for the pharmaceutical

industry. This method allows the user to select the objectives that are

important in each particular layout design in the pharmaceutical industry.

2.5.4 Intelligent Hybrid Systems for MCFLP

Badiru and Arif [68] have developed, FLEXPERT, a fuzzy-integrated

expert system for facility layout. FLEXPERT considers the multi-criteria nature

of the layout problem and the fuzziness of the input data through the

integration of an expert system and a fuzzy algorithm with a commercial

facility layout program (BLOCPLAN). The system generates the best layout

that satisfies the qualitative as well as the quantitative constraints on the

layout problem. Yang and Kuo [69] proposed a hierarchical analytic hierarchy

process (AHP) and data envelopment analysis (DEA) approach to solve a

plant layout design problem. A computer-aided layout-planning tool was used

to generate a considerable numbers of layout alternatives as well as to

generate quantitative decision-making unit (DMU) outputs. The qualitative

performance measures were weighted by AHP. DEA was then used to solve

the multiple-objective layout problem. Lee and Lee [70] presented a shape-

based block layout (SBL) approach for solving facility layout problem. The

SBL approach employs hybrid genetic algorithm to find good solution. The

objective function of SBL approach minimizes total material handling cost and

maximizes space utilization. Azadivar and Wang [71] presented a facility

layout optimization technique, using simulation and genetic algorithms, that

takes into consideration the dynamic characteristics and operational

constraints of the system, and is able to solve the facility layout design

problem based on a system's performance measures, such as the cycle time


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and productivity. Genetic algorithms are used to optimize the layout for

manufacturing effectiveness while simulation serves as a system performance

evaluation tool.

2.6 General Comments

As a result from the survey, it is concluded that the facility layout

problem still requires effective approach that overcomes the mentioned

inadequacies. In this work, an effective approach using the Chen and Shas

formulation is presented. The approach reasonably normalizes all objectives

of the MOFL problem, and handling qualitative and quantitative and any other

objectives information in similar fashion. Because of existing optimizationmethods are computationally inefficient when large numbers of facilities are

involved, stochastic heuristic methods are more appropriate for generating

effective layouts.


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Chapter 3

Facility Layout Problem Formulation

3.1 Introduction:

he facility layout problem formulation considers the arrangement ofN

facilities toMlocations (N


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Chapter 3: Facility Layout Problem Formulation 17

])()[( 22),( klklkl yyxxD

Two approaches are exist for solving facility layout problems, quantitative

approach and qualitative approach [72]. These approaches are:

3.2 Quantitative Approach

This approach tries to assign N facilities to M locations in order to

minimize the material handling cost between departments. In this approach,

FLP is formulated as Quadratic Assignment Problem (QAP) [34] because the

objective function is a second-degree polynomial function of the variables, and

the constraints are identical of the constraints of the assignments problems.

The objective function is defined as:

The value W(i,j) is the number of pallets trips required for transferring all the

parts, which use the two facilities consecutively. Note that the quantity W(i,j)

may also be weighted by any combination of parameters to consider special

material handling requirements, part weights, actual material handling costs,

etc. to quantify the distanceD(l,k) between the locations landk. Two traditional

measures can be used:

Cartesian distance; i.e.:

otherwise.0and,locationtoassignedisfacilityif1ofnumberbinaryis

and;andlocationbetweendistancetheis

.facilityandfacilitybetweenflowmaterialofnumbertheis

Where

________________

,10

(3),...,2,11

(2),...,2,11

:_

(1).

),(

),(

),(

),(

1 ),(

1 ),(

1

1 1 ),(),(),(),(11

vuX

klD

jiW

MN

vuallfororX

MvX

NuX

tosubject

XXDWQMin

vu

kl

ji

vu

N

u vu

M

v vu

N

i

N

ij kjliklji

M

l

M

lk


30/143


Or the rectilinear distance; i.e.:

Where (xl,yl) and (xk,yk) are the coordinates of the geometric centers of

locations whose facility i and facilityj respectively. Since the coordinate space

considered is discrete and finite, each point in the space is assigned a unique

position number in order to simplify the analysis. The problem, thus, consists

of determining the position number corresponding to each ofN facilities, in

order to minimize the objective function ofequation (1). Constraint (2) ensures

that the facilities do not overlap when they are assigned to adjacent position.

Furthermore, constraint (3) ensures each position cannot be occupied by

more than one facility.

3.3 Qualitative Approach

It includes assigning N facilities to M locations in order to locate

departments that utilize common materials or utilities adjacent to one another,while separating departments for reasons of safety, noise, heat, etc. The

approach uses QAP formulation [34] and employs relationship chart (REL

chart) as input to this process. The using of lettering system provides the

desirability associated of each pair of departments located adjacently. The

objective function is defined as:

)()(),( klklkl yyxxD

MN

vuallfororX

MvX

NuX

tosubject

XXLRQMax

vu

N

u vu

M

v vu

N

i

N

ij

M

l

M

lk kjliklji

,10

)6(,...,2,11

)5(,...,2,11

:_

)4(.

),(

1 ),(

1 ),(

1

1 1 1 ),(),(),(),(2


31/143


3.4 Multi-Objectives Approaches

Some techniques have been presented for solving the problem with

both qualitative and quantitative objectives. These involved combining the two

formulations into one single formulation and consider both quantitative and

qualitative information.

J acobs [73] and Shang [74] formulated the problem as multi-criteria and

consider only one quantitative or qualitative factor in the objective function.

The problem is solved using computer heuristics similar to those used for

single goal problems.

Other formulation allow problem to be solved using two quantitative and

qualitative factors simultaneously such as those proposed by Harraz [64],

Sarin et al [75], and Chen and Sha [21]. The following section will present an

overview of MOFLP approaches:

The QAP formulation of the MOFL problem is shown in equations (7) to (10):

.locationatfacilityandlocationatfacilitylocatingofcostThe

otherwise,

,locationtoasssignedisfacilityif

0

1

Where

(10),10

(9),...,2,11

(8),...,2,11

:_

)7(.

),(

),(

1 ),(

1 ),(

1

1 1 1 ),(),(

kjliA

vu

X

MN

vuallfororX

MvX

NuX

tosubject

XXAQMin

ijlk

vu

vu

N

u vu

Mv vu

N

i

N

ij

M

l

M

lk kjliijlk

otherwise.(0)and,locationtoassignedisfacilityif)1(esnumber takBinary

otherwise

adjacentarelocationstwowhen the

0

1

siteandsitebetweeneightLocation w

andfacilitybetweenvalueratingCloseness

Where

),(

),(

),(

vuX

(k)(l)L

jiR

vu

kl

ji


32/143


Aijlk in equation (7) is a cost variable representing the combination of

quantitative and qualitative measures in MOFL models. Equation (8) ensures

that each location contains only one facility. Equation (9) ensures that each

facility is assigned to only one location. As mentioned Chen and Sha [21],

these models are classified into four categories:

(1) Rosenblatt [15] and Dutta and Sahu [16] defined the cost term as:

Where Cijlk is the total material handling cost, Rijlk is the total

closeness rating score, andFCand WRare weights assigned to the

total material handling cost and to the total rating score.(2) Foretenberry and Cox [17] defined the cost term as:

WhereFij is the workflow between two facilities, Dlk is the distance

between two locations andRij is the closeness rating desirability of

the two facilities.

(3) Urban [19] defined the cost term as:

Where Cis a constant weight that determines the importance of the

closeness rating of workflow.

(4) Khare et al [76] defined the cost term as:

Where W1 and W2 are weights assigned to the work flow and to

the closeness rating.

The listed models are similar in nature, and vary only in stating the

relationship between the cost term Aijlk and the quantitative and qualitative

measures. Although, these models have been applied to the MOFL problem,

they all have two inadequacies:

1. All factors may not be represented on the same scale: for example,values for work flow may range from zero to a tremendous amount,

,ijlkRijlkCijlk RWCFA

,ijlkijijlk RDFA

,)(lkijijijlk DCRFA

,)( 21 lkijijijlk DRWFWA


33/143


while closeness rating values may range from -1 to 4. Hence, the

closeness ratings would be dominated by workflow and have little

impact on the final layout Harmonosky and Tothero [8].

2. Measurement units used for objectives are incomparable: the

closeness rating represents an order preference indicating the

necessity that given facilities be located close together. The total

closeness rating score is only an ordinal value; on the other hand, the

material flow handling is measured according to cost. Combining

these two values with different measurement units in an algebraic

operation is unsuitable Chen and Sha [21].

For the reasons cited above, Harmonosky and Tothero [8] suggested an

approach that normalizes all factors, before combining them. To normalize a

factor, each relationship value is divided by the sum of all relationship values

for that factor, as shown in equation (11).

Where Sijmis the relationship value between departments i and j for factor m,

and Tijm is the normalized relationship value between departments i and j for

factor (objective) m.

Next, all values are multiplied by weights representing the relative importance

of each factor m. Then, the sum of all values for each departments pair is

calculated.

The resulting objective function is shown in equation (12).

(11)

N

u

N

v uvm

ijm

ijm

S

ST

s).(objectivefactorsofNo.Tandm,factorforwieghttheis

(12). ),(),(

m

kjlilkijm

N

i

N

j

M

l

M

k

T

m m

where

XXDTQMin


34/143


Harmonosky and Tothero [8] proposed a methodology for normalizing all

factors into comparable units on the same scale. However, the scaling

problem remains unresolved. Note first that values for work flow may range

from zero to a very large positive value, while closeness rating values may

range from a negative value to a positive value. After using equation (11),

most normalized relationship values of larger scaling factor are lower than

those of smaller scaling factor. As a result, the larger scaling factor would

have very little effect on the final layout. Second, different scoring values for

closeness ratings may cause some inadequacies as mentioned in Chen and

Sha [21] and presented in the literature. However, Chen and Sha [21]

proposed an approach that normalizes all factor before combining them. In

order to achieve normalization, they subtract the mean of the layout cost

distribution from each objective value and divide the result by the standard

deviation of all feasible layout costs, as exhibited in equation (13).

The variance expression Vm, has been proposed by Khare et al [77]. The

meanMm of all feasible layout costs can also be computed Nugent et al. [41].

Chen and Sha [21] considered only those objectives with distance-weightedattributes such as flow and closeness rating. For this reason, they proposed

this approach for solving the MOFL problem, which is based on the

minimization of distance-weighted objectives. This is achieved by minimization

of total flow cost (TFC) and minimization of total numerical rating (TNR) that is

presented by Khare et al. [76]. Hence, all distance-weighted objective

functions can be characterized as a normal distribution. Using proposed

equation (13), they reasonably normalize all objectives, and resolve both the

different scale and measurement unit problems. The values obtained are then

.objectiveforvaluenormalizetheis

.objectiveforondistributicostlayouttheofvariancetheis

.objectiveforondistributicostlayouttheofmean valuetheis

)21.(objectiveforlocationatfacilityandlocationatfacilitylocatingofvalueobjectivetheis

(13),2/1m

mH

mV

mM

,...,t,mmkjliS

where

V

MSH

m

m

m

ijlkm

m

m

N

i

N

j

M

k

M

l ijlkm


35/143


multiplied by weights (Wm) representing the relative importance of each

objective. In their proposed model, the resulting objective function is shown in

equation (14):

Owing to the proposed model of Chen and Sha [21], as mentioned, that has

the superiority of the previous models; it is used as the evaluation criteria for

the solution quality in the current work.

(14). t

m mmHWQMin


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Chapter 4

A Review of Genetic Algorithms

4.1 What is a Genetic Algorithm?

enetic algorithms (GAs ) are a problem solving technique hinted at by

living creatures evolution Whitley [78]. In GAs, chromosomes, linear

encodings of a problems possible solution, are selected from a population,

operations such as crossovers and mutations are applied; and they survive in

higher probability if they are regarded as a better ones. That is, a GAs

mechanism is similar to natures one in which superior individuals can

produce more descendants in the future.

In order to use GAs for solving a problem, important points are considered:

Representation of the chromosomes;

Design of crossover and mutation operators; and

Fitness functions.

G4


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C h a p t e r 4 : A R e v i e w o f G e n e t i c A l g o r i t h m s 25

4.2 GA Parameters and Performance

In GAs, there are many kinds of parameters, which influence the GAs

behavior. The next section, some important GA parameters and their

influences will be briefly reviewed.

4.2.1 Crossover and Mutation: crossover usually takes two parents and

can reproduce one, two or more children. In actual GAs, the allele of either

parent is simply copied into the corresponding place of the childs

chromosome. There are some variations e.g. one-point crossover, two-point

crossover, and uniform crossover.

One-point crossover first specifies a split point on a chromosome at

random; then copies the alleles between the head and the splitting

point of one parent and those between the splitting point and the tail of

the other parent.

Two-point crossover initially choose two splitting points at random; then

duplicates the alleles between the splitting points of one parent, and

the other alleles from the other parent.

Uniform crossover randomly picks each allele from either of the two

parents.

If two parents 1-2-3-4-5 and 6-7-8-9-0 are selected, a child 1-2-8-9-0 may be

created by one-point crossover, 1-7-8-4-5 may be produced by two-point

crossover, and uniform crossover may generate 1-7-8-4-0.

In contrast, mutation may happen to one allele or probabilistically to every

allele. A chromosome 1-2-3-4-5 may be changed to 1-6-3-4-5.

In GAs, crossover rate and mutation rate are applied. For instance, if

crossover rate is .4 and mutation rate is .01, crossover will happen with 40%

probability and mutation will occur on each allele with 1% probability.

Therefore, if the chromosome consists ofL alleles, a chromosomes changing

probability by the mutation is 1-(1-M)LLMwhereMis the mutation rate andM

can be assumed to be less than 1/L Goldberg [79].


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4.2.2 Population Size: the number of chromosomes is often called

population size, and it may also influence the GA. As mentioned in Goldberg

[79], a GA of large population size may have better solutions ultimately

because of large number of chromosomes may include good schemata insome chromosome. On the other hand, GAs with smaller population can

change rapidly; therefore, it may show better performance in the early stages

rather than those with a larger population.

4.2.3 Selection Methods: There are many kinds of selection methods; three

methods are introduced here:

Rank method Baker [80] first sorts all the chromosomes in order of fitnessvalues; then, the probability of selecting a particular chromosome is

proportional to the inverse for the order rather than the fitness itself. If there

are five chromosomes whose fitness values are 1, 6, 3, 8, 5; then the rank

ordering is 5th, 2nd, 4th, 1st, 3rd and the probabilities of selection are 1/15,

4/15, 2/15, 5/15, 3/15.

In contrast, tournament selection Brindle [81] is as follows. First, a particular

numberS is decided as the size of tournament. Second, S chromosomes are

uniformly chosen from all the chromosomes (population size). Finally, the best

one among the S chromosomes is selected as a parent. Of course, two

parents are required in normal GAs; therefore, the above process is usually

done 2N times, where N is the population size. In the tournament selection,

the same chromosome may be chosen more than once Whitley [78].

However, the tournament selection with large Scauses a strong pressure to

choose very fit chromosomes. And, this leads premature convergence of

chromosomes, which usually produce only poor solutions. To explain it, let us

consider the probabilities of being chosen as a parent for the five

chromosomes: the best one in the generation; the 75th percentile; the median;

the 25th percentile; and the worse one. In tournament selection, each

candidate for parents has to win against (S-1) competitors in a group to

become a parent. Because the probabilities of meeting a weaker chromosome

for the five chromosomes are 100%, 75%, 50%, 25% and 0%; the

probabilities of becoming a parent by winning against (S-1) competitors for the


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five are 1, (0.75)s-1, (0.5)s-1, (0.25)s-1 and 0. Thus, ifS= 2, they are 1, 0.75, 0.5,

0.25 and 0; and ifS=5, they are 1, 0.32, 0.06, 0.004 and 0. Hence, we can see

that very fit chromosomes will be frequently chosen as parents in largeS.

Modified tournament selection method may be useful in some cases to

cope with this defect as mentioned in Ross and Hallam [82]. In this method, a

chromosome is first chosen as the first candidate at random; secondly, the

chromosome is compared with at most (S-1) chromosomes randomly chosen.

If a better chromosome than the first candidate is found from the (S-1)

chromosomes, then the better one is selected as a parent immediately;

however, if all the (S-1) chromosomes are worse than the first candidate, then

the first one is selected as a parent. Therefore, other candidates than the first

one can become the parent only by beating the first candidate. Hence, the

strong pressure to choose very fit chromosome observed in tournament

selection should become weaker in this modified tournament selection.

4.2.4 ReproductionMethods there are some variations of reproduction

methods. In the next section, two methods are explained:

Genitor method Whitley [83], is considered as one of the ( + ) evolution

strategy Baeck et al [84], in which offspring are produced from parents,

and the best chromosomes of ( + ) are retained. Because the best

chromosomes are always retained in this strategy, the population may

converge gradually without drastic drifts. On the other hand, the generation-

based GA is regarded as one of the (, ) evolution strategy, in which

offspring are produced from parents and the best chromosomes of are

retained. Because the best chromosomes may be lost in this strategy, the

population may dramatically drift in search space for solutions. Therefore, all

the chromosomes in Genitor may converge quicker than those in generation-

based GA, Davis [85]; but Genitor may produce only poor solutions due to

premature convergence Whitley [78]. For more information about genetic

algorithms Schaffer et al. [86], Goldberg [79], and Holland [87].


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4.3 On-line, Off-line and Best Individual Performance

According to Baker [80], there are three types of criteria to evaluate the GA

performance. They are:

1. On-line performance, the average of all results that have appeared;

2. The off-line performance, the average of best results of each

generation; and

3. The best individual performance, the best result that has appeared.

While the off-line and the best individual performances only take account of

the best chromosome in each generation, the on-line performance reflects the

performance of chromosomes other than the best one as well. Therefore, the

GAs showing good on-line performance may not produce remarkable

chromosomes.

Between the off-line and the best individual performances, the off-line

one can take the convergence speed into account unlike the best individual

performance. If two algorithms X and Y show the best individual performance

as shown in Figure 3.1, the off-line performance of algorithm X is better than Y

at time T1, though the best individual performance of both algorithms are

same. But, at time T2, the off-line performance of algorithm X is still betterthan Y, although its best individual performance is worse. That is, off-line

performance is generally influenced by the past records. However, because

the quality of the best solution may be important in practical applications, the

best individual performance may be more useful than the other performance

measures. Hence, the best individual is used in the current thesis.

Figure 3.1 Two algorithms performance.

Time

Fitness

of

the

bestindividual

T1 T2

Algorithm Y

Algorithm X


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4.4 GA Application

As Glover and Greenberg [88] mentioned, GAs may be effective

approaches for NP-complete problems as ones of stochastic approaches. For

example, Fang et al [89] and Ross et al [23] introduced GAs efficiency on job-

shop scheduling problems and time tabling problems, respectively. Similarly,

various layout problems and FLPs are solved by GAs. Here, some of them will

be reviewed.

4.4.1 Layout Problems with GAs: Regarding layout-related research,

Kloske and Smith [24] tackled cable routing problems with a GA. In the GA, a

chromosome consists of the index of each cables routing alternatives. For

example, if there are three cables to be routed, a chromosome will have three

alleles. And, if a chromosome is 2-3-2; the first cable will be routed by the

second possible way for the cable, the second cable will be routed by the third

possible way for the cable, and the third cable will be routed by the second

possible way for the cable. Although Kloske and Smith [24] did not report any

details how each cables alternative ways are produced, they state that the

GA worked well.

Whereas Kloske and Smith [24] were able to use traditional crossovers

and mutations,while Smith [27] had to use a modified crossover to cope with

his chromosome encoding in a bin-packing problem. In his research, a

chromosome represents a list of packing order of objects. For instance, if a

chromosome is 4-1-3-2; then it represents that object No. 4 will be first

packed, object No. 1 will be second packed, and so on. Therefore, ordinary

crossover may produce nonsense children. If 4-1-3-2 and 1-2-3-4 are one-point crossed over, and if splitting point is set between the second and third

genes; then, it may produce twin children 4-1-3-4 and 1-2-3-2, which do not

represent solutions of this problem. In order to tackle this problem, he used

modified crossover, which keeps the genes before splitting point of the first

parent and applies the order in the second parent for the rest of objects.

Therefore, in the above example, 4-1-2-3 and 1-2-4-3 may be created instead.

Furthermore, Falkenauer [28] took account of the redundancy of the

representation of chromosomes. For example, in the encoding of Smith [27],


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1-2-3-4 and 4-3-2-1 may be different solutions. Nevertheless, if objects 1 and

2 fill a bin and if object 3 and 4 fill another bin, these two chromosomes

virtually represent the same solution. Because higher redundancy makes the

GAs search space larger and GAs power weaker, he suggested another

encoding method to reduce the redundancy. Although it might be highly

dependent on the problems domain, we may be able to see the fact that

chromosomes representation and design of crossovers and mutations will

much influence GA performance.

4.4.2 FLPs with GAs: As regards FLPs, various kinds of chromosome

representations and crossovers and mutations have been also suggested.

Cohoon and Paris [90] introduced an original crossover method for the cell

assignment representation. In a 3x3 FLP, a chromosome 1-2-5-6-7-9-3-8 may

represent a layout shown in Figure 3.2. Since the conventional crossover

tends to favor shorter schemata more Whitley [78], the relation of facilities No.

5 and No. 6 may be kept in higher probability than that of No. 2 and No. 6, in

this example. Cohoon and Paris [90] introduced another special crossover

which can take into account such two dimensional adjacencies.

1 2 3

5 6 7

4 8 9

Figure 3.2 Chromosome representation for an identical FLP.

Smith and Tate [91] used flexible bay structure representation to tackle non-

identical FLPs. In the representation, a physical layout is represented by two

chromosomes. The first one specifies the order of putting facilities into cells,

and the second chromosome specifies how many cells are included in each

bay (row). Smith and Tate [91] used the crossover as follows: the childs first

chromosome is produced from the first chromosomes of parents by the same

method of Smith [27]; and the childs second chromosome is copied from

either parents second one. As for the mutation, one of the following three

types is applied. If the first type (MU1) is applied, a bay chosen at random is


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divided into two bays. If the second type (MU2) is applied, two sequential bays

chosen at random are merged into one bay. If the third type (MU3) is applied,

a part of genes will be reversed. That is, MU1 and MU2 affect the second

chromosome, whereas MU3 affects the first chromosome. Smith and Tate [91]

set the probability ratio of MU1, MU2, and MU3 occurring to be 1:1:2.

While the flexible bay structure requires each cell to lie in rows (bays),

the slicing tree structure (STS) can generate more various physical layouts.

However, if Polish expression, which corresponds to a tree structure ant to a

layout, is directly used as a chromosomes representation, ordinary

crossovers and mutations can not be applied, because a combination

operators and operands in random order may not be a valid Polish expression

(solution).

In order to use Polish expression as chromosomes representation,

Cohoon et al [25] suggested several types of special crossover and mutation

methods. For example, one of crossover methods creates a child so that it

can inherit the trees structure from one parent and that it can inherit the

operators in the Polish expression from the other parent. As for mutations,

Cohoon et al [25] used Wong and Lin [92]s methods, which are used, for

solutions move in simulated annealing. They are, swapping adjacent

operands; switching a sequence of adjacent operators; and swapping an

operator and a neighborhood operand.

On the other hand, in order to use conventional crossovers and mutations,

Tam [26] suggested a method where the tree structure is fixed and a

chromosome includes only operators of the Polish expression. In other words,

Tam [26] limited the search space, while Cohoon et al [25] did not.

In conclusion, there have been many methods for representation,

crossovers, and mutations; and this might suggest that better methods may

appear in the future. The proposed GA and its operators will be explained in

details in the next chapter.

4.5 The Need for Good Solver Techniques

From the previous chapter, it is seen that the QAP is capable of

representing the inter-cell layout problem, but the complexity of the situation


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appear from the fact that the QAP formulation belongs to the class of NP-

complete problems as it is cited in Garey and Johnson [12], and no definite

method can arrive at an optimal solution in a reasonable time when 15 or

more facilities are considered. Consequently, many heuristic algorithms have

been developed for achieving a trade-off between computation time and the

efficiency of the final solution Kusiak and Heragu [13].

In search for effective solution procedures, the Genetic algorithm, an offshoot

of the artificial intelligence (AI) submerged as an effectual tool capable of

treating the NP-hard FLP problem.


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Chapter 5

Proposed Genetic Algorithm

5.1 Why Genetic Algorithm?

GAs differ from traditional optimization and search procedure in fourways Davis [85]:

They work with a coding of the parameter, not the parameters

themselves;

Start search from a set of population points, not single point;

Use payoff information, not derivatives or other auxiliary knowledge; and

Use probabilistic transition rules, not deterministic rules.

5.2 The Proposed Genetic Algorithm

In the following subsections, the proposed genetic algorithm variables

will be discussed in details. Such chromosome representation, initial

population, selection method, crossover and mutation operator, and

reproduction systemetc.

5


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C h a p t e r 5 : P r o p o s e d G e n e t i c A l g o r i t h m 34

5.2.1 Chromosome Representation:

Chromosome representation maps feasible solutions of the problem.

The effectiveness of the crossover operator depends greatly on the

representation scheme used. The representation should be such that the

crossover operator preserves high performing partial arrangements

(schemata) of strings, and minor changes in the chromosome translate into

minor changes in the corresponding solution.

From the three FLP representation methods, mentioned in the previous

sections, the cell assignment representation (permutation method) is used to

represent the facilities layout, such that the allele ( j ) at position ( i ) in the

genome indicates that facility ( j ) is assigned to location ( i ). For example,Individual 3,4,7,1,8,2,9,6,5 represents a solution for the shown layout in

Figure 5.1.

Figure 5.1 Phenotype representation layout.

5.2.1.1 Representation of special layout configuration

First, if the layout area has some restricted location, at this situation,

each restricted location in the layout is encoded to 1 in the genotype. For

instance, Figure 5.2 is represented by this schema:(#,#,#,#,#,#,#,-1,#,#,-1,-1,-

1,#,#).

Figure 5.2 Available layout space with restricted area.

Where #: means any alleles can be assigned to this position except what has

assigned before.

Note that all chromosomes have this schema.

3 4 7

1 8 2

9 6 5


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Second, how can represent the chromosome if there are any pre-specified or

occupied facilities. Figure 5.3 is represented by this schema:

(#,#,7,#,#,#,#,#,3,#,-1,-1,-1,#,#).

7

3

Figure 5.3 Available layout space of restricted area

and pre-specified facilities.

Note that all chromosomes have this schema.

5.2.2 Ini t ia l Population:

The initial population of chromosome is generated randomly. For not

generating illegal chromosome, each process of generating a random gene

(facility) checks the previous genes (facilities), which are once picked in the

previous generations and choose a gene that has not been chosen before in

the chromosome. This procedure ensures that each gene is only once chosen

in a chromosome. This initialization keeps proceeding until the number ofchromosome reaches the pre-set population size (PopSize). In the current

study, PopSize considered is 200.

5.2.3 Select ion Method

The selection criterion is used to select the two parents to apply the

crossover operator. The appropriateness of selection method for a GA

depends upon the other GA operators chosen. In the literature, a typical

selection method gives a higher priority to fitter individuals since this leads to

a faster convergence of the GA. Nevertheless, the overall results are obtained

when the selection is not biased towards fitter individuals. So, the tournament

selection method is applied in order to control convergence speed by the

tournament size, Ts. In the current study, Ts considered is 4.


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5.2.4 Crossover Operator

The crossover operation corresponds to the concept of mating. It is

hoped that the crossover (mating) of good parents may produce good

offspring. Thus, the crossover operation is a simple yet powerful way of

exchanging information and creating new solutions. Partially matched

crossover (PMX) [93] operator is employed with probability Pc here to

generate two offspring. The PMX operator starts as follows, at first two cut

points are chosen at random for the parents. Then the genes of the father

string bounded by the cut points will be copied to the same positions of the

first offspring, and the remaining genes of the offspring will be filled up by the

mother string in the same order. In such case, the offspring may not be legalbecause of possibilities of repeated genes. In this case, the repeated genes

will be replaced by genes corresponding to the mapping of the father and

mother spring bounded by the cut points. Suppose the father is (2, 4, 5, 3, 8,

9, 6, 1, 7), the mother is (3, 9, 8, 6, 5, 4, 2, 7, 1) and the cut points lie after the

third gene and before the seventh gene. Thus, the father genes inside the cut

points are 3, 8, 9, the mother genes outside the cut points are 3, 9, 8, 2, 7, 1.

The offspring is (3, 9, 8, 3, 8, 9, 2, 7, 1). This would be illegal because 3, 8, 9,

are repeated genes. Note that 3, 8, 9 of the father genes map to 6, 5, 4, thus

we replace 3 by 6, 8 by 5, and 9 by 4, to get the new offspring as (6, 4, 5, 3, 8,

9, 2, 7, 1). The second offspring is generated by the same manner but it uses

the mother genes inside the cut points, and the remaining genes are filled up

by father string in the same order. Figure 5.4 shows the PMX method to

generate feasible solutions. In the current study,Pc considered is 0.7.

Cut points

Father 2 4 5 3 8 9 6 1 7

Mother 3 9 8 6 5 4 2 7 1

First offspring 2 4 5 6 5 4 6 1 7

Second offspring 3 9 8 3 8 9 2 7 1

Legal 1st

offspring 2 9 8 6 5 4 3 1 7

Legal 2nd

offspring 6 4 5 3 8 9 2 7 1

Figure 5.4 PMX crossover method.


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5 .2 .5 Mutation Operator

If the entire population has only one type of string (or strings are verysimilar), then the crossover of two strings does not produce any new strings.

To escape from this scenario, the mutation operator may be used. In our GA,

The mutation operation may happen with probability (Pm, mutation rate), to

swap every allele (gene) and selected allele at random (order change

mutation). For instance, chromosome 1, 5, 6,3, 4, 2 may be change to 1, 3, 6,

5, 4, 2. In the current study,Pm considered is 0.1.

Note that pre-specified or occupied facilities and/or restricted locations are not

affected by the crossover and/or mutation operators.

5.2.6 Reproduction System

The generation-based system is used. That is, offsprings from

parents are produced and the best chromosome ofare retained. Owing to

making new population only by new offspring can cause lost of the best

chromosome from the last population. This is true, so called Elitism is often

used. This means, that at least one best solution is copied without changes t

10381067 . solution methodology based genetic algorithm for multi-objective facility layout problem

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