Optimization of Midship Section for Bulk Carriers based on Common Structural Rules using Insect Antennae Algorithm

九州大学学術情報リポジトリ

Kyushu University Institutional Repository

昆虫アンテナ法を用いた共通構造規則に基づくばら 積み貨物船の船体中央部の最適化

趙, 珉徹

https://doi.org/10.15017/1932009

出版情報：Kyushu University, 2017, 博士（工学）, 論文博士 バージョン：

権利関係：

i

Optimization of Midship Section for Bulk Carriers based on Common Structural Rules using Insect Antennae Algorithm

Doctoral Dissertation by

Mincheol CHO

Department of Civil and Structural Engineering Graduate School of Engineering

Kyushu University

2018

i

ACKNOWLEDGEMENTS

My sincere gratitude goes to Professor Koji GOTOH, who has provided guidance and counseling throughout the course of this research. This work would not have been possible without him.

I am grateful to Professor Hiroyuki Kajiwara and Professor Takao Yoshikawa of the Department of Marine Systems Engineering, Faculty of Engineering, Kyushu University, for helping me in writing this dissertation through their valuable suggestions and comments.

I am grateful to Professor Jewoong Park of the Department of Naval Architecture and Ocean Engineering of Chosun University, for helping me in studying the optimization algorithm through his valuable comments.

I am greatly indebted to Mr. Yasuhiro IMAZU and Mr. Kunitaka FUKUOKA of Design Administration Department, Design Division, Tsuneishi Shipbuilding Co. Ltd., for their ungrudging support about system development project.

I also wish to thank all the members of the Laboratory of Fracture, Welding Mechanics and Production Systems, Department of Civil and Structural Engineering, Kyushu University.

A very special thanks goes to my family. Thanks to love and care from my parents (Sungcheon Cho and Sunbok Kim, and Jintaek Lim and Heesuk Hwang), my brothers (Minkwan Cho, Hyeongyong Choi, and Suyoung Park), sisters (Younghee Cho, Jinbeom Cho, Myoungha Lim, and Myoungjin Lim).

Finally, I would like to express my gratitude to my lovely wife (Myoungeui Lim) who has been on my side for good times and bad times.

This dissertation is dedicated to my family.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ... i

TABLE OF CONTENTS ... ii

LIST OF TABLES ... vi

LIST OF FIGURES ... viii

CHAPTER 1. INTRODUCTION ... 1

1.1 B

M

... 1

1.2 O

... 4

1.3 O

... 4

CHAPTER 2. INSECT ANTENNAE ALGORITHM ... 6

2.1 I

... 6

2.2 I

A

A

... 7

2.2.1 Inspiration and Implementation ... 7

2.2.2 Search Strategy ... 8

2.2.3 Organization of Antennae and Body ... 14

2.2.4 Parameters ... 17

2.2.5 Attraction Rate and Convergence Value ... 18

2.3 N

R

... 30

2.3.1 Benchmark Functions ... 30

2.3.1.1 Ackley Function ... 30

2.3.1.2 Drop-Wave Function ... 30

2.3.1.3 Goldstein and Price Function ... 31

2.3.1.4 Griewank Function ... 31

iii

2.3.1.5 Rastrigin Function ... 31

2.3.1.6 Rosenbrock Function ... 32

2.3.1.7 Schaffer Function ... 32

2.3.1.8 Schwefel Function ... 32

2.3.1.9 Shubert Function ... 32

2.3.2 Constrained Functions ... 37

2.3.2.1 Test Function I ... 39

2.3.2.2 Test Function II ... 41

2.3.2.3 Test Function III ... 42

2.3.2.4 Test Function IV ... 44

2.4 C

... 46

CHAPTER 3. ENGINEERING DESIGN WITH INSECT ANTENNAE ALGORITHM ... 47

3.1 I

... 47

3.2 E

O

P

... 50

3.2.1 Tension/Compression Spring Design ... 50

3.2.2 Welded Beam Design ... 51

3.2.3 Pressure Vessel Design ... 53

3.2.4 Speed Reducer Design ... 55

3.2.5 Stepped Cantilever Beam Design ... 57

3.2.6 Fifteen-Bar Planar Truss Design ... 59

3.3 C

... 63

CHAPTER 4. DEVELOPMENT OF OPTIMAL DESIGN SUPPORT SYSTEM FOR MIDSHIP SECTION OPTIMIZATION OF BULK CARRIER ... 64

4.1 I

... 64

4.2 F

O

D

S

S

... 65

4.2.1 Optimization Algorithm ... 65

4.2.2 Parallel Computing ... 73

iv

4.2.3 Stiffener Grouping ... 75

4.2.4 Adjustment of Seam Location ... 76

4.3 G

U

I

S

I

... 77

4.3.1 Main Dimensions Data ... 78

4.3.2 Hold and Tank Data ... 79

4.3.3 Bulk Head Position Data ... 80

4.3.4 Class Notation and Loading Conditions Data ... 81

4.3.5 Bending Moment Data ... 81

4.3.6 Frame Space Data ... 82

4.3.7 MidShip Section Data ... 83

4.3.8 Seam Location Data ... 84

4.3.9 Optimal Design Support System with Kamsarmax data ... 85

4.4 O

F

... 86

4.4.1 Design Variables ... 86

4.4.2 Constraint Conditions ... 92

4.4.3 Objective Function and Penalty Functions ... 96

4.5 N

R

... 98

4.5.1 Overall of Midship Section ... 99

4.5.2 Hull Girder - Section Modulus ... 101

4.5.3 Hull Girder - Bending Moment ... 101

4.5.4 Hull Girder - Shear Force ... 102

4.5.5 Hull Girder - Ultimate Strength ... 103

4.5.6 Local Strength of Plates ... 105

4.5.7 Local Strength of Stiffeners (Longitudinal Features) ... 106

4.5.8 Local Strength of Stiffeners (Minimum Requirements) ... 107

4.6 C

... 110

CHAPTER 5. CONCLUSION ... 111

v

BIBLIOGRAPHY ... 113

APPENDICES ... 120

A

A. S

C

I

A

A

(C, C++) ... 120

Appendix A.1. Classes and Functions ... 120

Appendix A.2. Main function ... 127

Appendix A.3. Object Function for Welded Beam Design ... 128

A

B. R

F

I

A

A

... 131

Appendix B.1. Setup File (Setuo_IAA.txt) for Welded Beam Design ... 131

Appendix B.2. Result File (Fitness.txt) of Welded Beam Design ... 132

Appendix B.3. Convergence History of Welded Beam Design ... 140

VITA ... 141

PUBLICATIONS ... 142

vi

LIST OF TABLES

Table Page

Table 2.1 Optimal results of the benchmark functions. ... 33

Table 2.2 Parameter values for constrained functions. ... 37

Table 2.3 Optimal results of test function I. ... 40

Table 2.4 Constraint values of test function I calculated by the IAA. (present study) ... 40

Table 2.5 Optimal results of test function II. ... 42

Table 2.6 Constraint values of test function II calculated by the IAA. (present study) .. 42

Table 2.7 Optimal results of the function III... 43

Table 2.8 Constraint values of test function III calculated by the IAA. (present study) . 44 Table 2.9 Optimal results of test function IV. ... 45

Table 2.10 Constraint values of test function IV calculated by the IAA. (present study) ... 46

Table 3.1 Parameter values for the engineering design problems. ... 47

Table 3.2 Optimal results of the tension and compression spring design problem. ... 51

Table 3.3 Optimal results of the welded beam design problem. ... 53

Table 3.4 Optimal results of the pressure vessel design problem. ... 55

Table 3.5 Optimal results of the speed reducer design problem... 57

Table 3.6 Optimal results of the stepped cantilever beam design problem. ... 59

Table 3.7 Optimal results of the 15-bar planar truss design problem for the three cases. ... 62

Table 3.8 Optimal results of the 15-bar planar truss design problem. ... 63

Table 4.1 Types of plate thickness. (type: 41, unit: mm) ... 87

Table 4.2 Types of web plate for T-section stiffener. (type: 462, unit: mm) ... 88

Table 4.3 Types of face plate for T-section stiffener. (type: 15, unit: mm) ... 90

Table 4.4 Types of angle stiffener. (type: 8, unit: mm) ... 91

vii

Table 4.5 Types of flat bar stiffener. (type: 20, unit: mm) ... 92

Table 4.6 Section area and hull girder section modulus. ... 101

Table 4.7 Hull girder calculation. ... 101

Table 4.8 Hull girder requirement calculation. ... 102

Table 4.9 Hull girder shear force calculation. ... 102

Table 4.10 Hull girder ultimate strength calculation... 103

Table 4.11 Local strength of plates. ... 105

Table 4.12 Local strength of stiffeners. (longitudinal features) ... 106

Table 4.13 Local strength of stiffeners for minimum requirements. (stdKSM) ... 107

Table 4.14 Local strength of stiffeners for minimum requirements. (woCUS) ... 108

Table 4.15 Local strength of stiffeners for minimum requirements. (wCUS) ... 109

viii

LIST OF FIGURES

Figure Page

Figure 2.1 Discretization of a body and two antennae of an insect: (a) natural antennae of the insects (e.g., longicorn beetles); (b) robot(R1) with their own two antennae; and (c) robot(R1) with randomly selected two antennae. ... 8 Figure 2.2 Pseudo code of the Insect Antennae Algorithm. ... 9 Figure 2.3 Movement procedure of the ith robot: (a) initial status; (b) select two

robots randomly; (c) rank three robots by using fitness of each robot; (d) calculate the three vectors V*; (e) calculate a new position PX; (f) calculate a vector Q; (g) calculate a velocity V using attraction rate; and (h) update the position of the ith robot. ... 13 Figure 2.4 Number of antennae for a body: (a) a body without antenna; (b) a body

with an antenna; (c) a body with two antennae; (d) a body with three antennae; and (e) a body with four antennae. ... 14 Figure 2.5 Convergence trend according to the number of antennas: (a) a body

without antenna; (b) a body with one antenna; (c) a body with two antennae; (d) a body with three antennae; (e) a body with four antennae;

and (f) comparison of average and standard deviation for each case. ... 17 Figure 2.6 Example of parameter settings. ... 18 Figure 2.7 Convergence trend according to the fixed attraction rates: (a), (b), (c), (d),

and (e) show the results of evaluation with attraction rates of 0.1, 0.2, 0.3,

0.35, and 0.4, respectively; (f), (g), and (h) indicate that the global solution

is searched successfully with 0.45, 0.5, and 0.55; (i), (j), (k), (l), and (m)

show the results of evaluation with attraction rates of 0.6, 0.65, 0.7, 0.8,

and 0.9, respectively; (n) and (o) show average and standard deviation

graphs scaled for case A to case H and case I to case M, respectively. ... 24

ix

Figure 2.8 Convergence trend according to the range of attraction rates: (a) and (b) are needed enough convergence process; (c), (d) and (e) are reached to the exact solution; and (f) and (g) are shown early convergence to the local optima; (h) and (i) show average and standard deviation graphs scaled for

case A to case E and case F to case G, respectively. ... 28

Figure 2.9 Convergence trend according to the reduction types of ranged attraction rate: (a) fixed ranged attraction rate, (b) linearly reduced ranged attraction rate; and (c) stepped ranged attraction rate... 29

Figure 2.10 Benchmark functions: (a) Ackley function; (b) Drop-Wave function; (c) Goldstein and Price function; (d) Griewank function; (e) Rastrigin function; (f) Rosenbrock function; (g) Schaffer function; (h) Schwefel function; and (i) Shubert function. ... 37

Figure 2.11 Convergence history of four constrained functions: (a) 5-d problems with three constraints; (b) 7-d problems with four constraints; (c) 8-d problem with six constraints; and (d) 10-d problem with eight constraints. ... 39

Figure 3.1 Convergence history of six engineering design problems: (a) tension/compression spring design; (b) welded beam design; (c) pressure vessel design; (d) speed reducer design; (e) stepped cantilever beam design; and (f) 15-bar planar truss design. ... 50

Figure 3.2 Schematic of spring design problem. ... 51

Figure 3.3 Schematic of welded beam design problem. ... 53

Figure 3.4 Schematic of pressure vessel design problem. ... 54

Figure 3.5 Schematic of speed reducer design problem. ... 57

Figure 3.6 Schematic of stepped cantilever beam design problem. ... 58

Figure 3.7 Schematic of the 15-bar planar truss design problem. ... 60

Figure 3.8 Analysis results of the optimal solution of three cases for a 15-bar planar truss: (a) case1; (b) case2; and (c) case3. ... 62

Figure 4.1 Hybrid IAA with Simulated Annealing. ... 66

x

Figure 4.2 Convergence tendency as per the increasing of iteration. ... 67

Figure 4.3 Variation of the number of robots corresponding to the each iteration on the ordinary IAA. ... 67

Figure 4.4 Variation of the number of robots corresponding to the each iteration on the adaptive population based IAA. ... 67

Figure 4.5 Two types of Levy functions: (a) 10 dimensions and (b) 20 dimensions. ... 69

Figure 4.6 Variation of temperature as per the T

and I

. ... 73

Figure 4.7 The basic concept of parallel computing using multithreading based on MPI. ... 74

Figure 4.8 Performance of parallel computing on ODSS. ... 75

Figure 4.9 Grouping according to the stiffeners location. ... 76

Figure 4.10 Adjustable seam location of Kamsarmax. ... 77

Figure 4.11 Input dialog for main dimensions data. ... 79

Figure 4.12 Input dialog for hold and tank data. ... 80

Figure 4.13 Input dialog for bulkhead position data. ... 80

Figure 4.14 Input dialog for class notation and loading conditions data. ... 81

Figure 4.15 Input dialog for bending moment data ... 82

Figure 4.16 Input dialog for frame space data. ... 82

Figure 4.17 Input dialog for midship section data. ... 84

Figure 4.18 Input dialog for seam location data. ... 84

Figure 4.19 Optimal Design Support System with Kamsarmax data... 85

Figure 4.20 Range of design variable for seams ... 86

Figure 4.21 Dimension of T-section stiffener. ... 87

Figure 4.22 Dimension of angle stiffener. ... 91

Figure 4.23 Dimension of flat bar stiffener. ... 91

Figure 4.24 Design standard for seam location considering stiffener and slot. ... 93

Figure 4.25 Flow chart of objective function. ... 97

Figure 4.26 Overall of midship section of standard Kamsarmax. ... 99

xi

Figure 4.27 Overall of midship section without considering ultimate strength ... 100

Figure 4.28 Overall of midship section with considering ultimate strength ... 100

Figure 4.29 Curve bending moment capacity vs curvature (overall) ... 103

Figure 4.30 Curve bending moment capacity vs curvature (hogging) ... 104

Figure 4.31 Curve bending moment capacity vs curvature (sagging) ... 104

1

CHAPTER 1. INTRODUCTION

1.1 Background and Motivation

Since the genetic algorithm proposed by Holland in 1975[1] was introduced, a large number of metaheuristic optimization algorithms have been presented to solve various real world optimization problems, as well as engineering optimum design problems. Most of the engineering optimization problems are highly non-linear and include very complex constraints and many design variables. For instance, the optimization of the midship section for a bulk carrier required 69 design variables and 228 constraints. Furthermore, several design variables concerning the stiffener require certain discrete values [19]. Such design problems that have many design variables and complicated constraints are sensitive to initial conditions. Therefore, a simple and efficient algorithm for obtaining the best global solution is required to overcome these difficulties. Many heuristic algorithms have been proposed in order to solve highly complicated optimization problems. These algorithms are inspired by nature and can be classified as swarm-intelligence-based, bio-inspired, physics-based, or chemistry- based, depending on the source of inspiration [50]. The state-of-the-art algorithms introduced in the past few decades are as follows:

Genetic Algorithms (GA), proposed by Holland [1], is based on the evolutionary theory

of Charles Darwin, and GA was improved by Goldberg [2]. There are three operations

such as selection, crossover, and mutation to find an optimal solution in GA. Simulated

Annealing (SA), presented by Kirkpatrick et al. [3], is a probabilistic technique for

approximating the global optimum, and it is often used to make up for the weak

points in other algorithms. SA was inspired from the process of annealing in metal

work. Tabu Search, proposed by Glover in 1986 [4], provides an efficient search

method in a local domain to find the global optimum. TS use memory space to

memorize the local optimal solution in order to avoid the local minima. Ant Colony

2

Optimization, introduced by Dorigo [5], is a probabilistic technique based on the

behavior of ants seeking a path between their colony and a source of food. Usually,

Ant Colony Optimization (ACO) is used to find optimal paths and sequences. Particle

Swarm Optimization (PSO), proposed by Kennedy and Eberhart [6], is inspired by the

movement behavior of a bird flock or a fish school. This algorithm has been

successfully applied to many optimization problems. PSO consists of a swarm of

particles, and the location of particles is updated by personnel best solution and social

best solution with each acceleration coefficient. Differential Evolution (DE), presented

by Storn and Prince [7], is a population-based algorithm, like GA and SA, and it is

improved via parameter tunes. DE consists of mutation, recombination, and selection

like GA, but DE use mutation as the primary search mechanism. Harmony Search (HS),

introduced by Geem et al. [8], is a phenomenon-mimicking algorithm inspired by the

improvisation process of musicians. Artificial Bee Colony (ABC), proposed by Karaboga

[9], is based on the intelligent foraging behavior of honeybee swarms. ABC consists of

three essential components such as employed and unemployed foraging bees, and

food sources. Firefly Algorithm (FA), introduced by Yang [10], is based on the idealized

behavior of the flashing characteristics of fireflies. Cuckoo Search (CS), proposed by

Yang and Deb [11], is inspired by the obligate brood parasitism displayed by some

cuckoo species by laying their eggs in the nests of other host birds. Bat Algorithm (BA),

presented by Yang [12], is inspired by the echolocation behavior of microbats with

varying pulse rates of emission and loudness. BA consists of position, velocity, average

loudness at each stage, and cooling factor. Water Cycle Algorithm (WCA), by Eskandar

et al. [13], is inspired by the observation of the water cycle process and movements of

rivers and streams toward the sea. Symbiotic Organisms Search (SOS), introduced by

Cheng and Prayogo [14], is based on the symbiotic interaction strategies adopted by

organisms to survive and propagate in the ecosystem. SOS simulates the symbiotic

interactions within a paired organism relationship that are used to search for the

fittest organism. There are three phases that resemble the real-world biological

3

interaction model are introduced: mutualism phase, commensalism phase, and parasitism phase. Ant Lion Optimizer (ALO), proposed by Mirjalili [15], is inspired by the hunting mechanism of antlions. ALO simulates five steps of hunts in larvae as follows: random walk of ants, building traps, entrapment of ants in traps, catching preys, and re-building traps. Sine Cosine Algorithm (SCA), introduced by Mirjalili [16], is a population-based algorithm, and the convergence mechanism is that the initial random candidate solutions are fluctuated outwards or towards the best solution using a mathematical model based on sine and cosine functions. Lion Optimization Algorithm (LOA), proposed by Yazdani and Jolai [17], is inspired by simulation of the solitary and cooperative behaviors of lions. Finally, Whale Optimization Algorithm (WOA), introduced by Mirjalili and Lewis [18], is inspired by the bubble-net hunting strategy of humpback whales. Such meta-heuristic optimization algorithms have been applied in the various engineering fields, and its performance was also verified by various engineering design problems.

Years of on-site experience in ship design are very important in order to dominate the technical market in advance under competitiveness. The application of Common Structural Rules (CSR) [52] proposed by International Association of Classification Societies (IACS) was expected to standardize existing technology and infrastructure competitiveness. The reason is that it is necessary to design according to regulations pre-defined, not by the experience-based, the required design know-how would not be needed. However, in actual, more competitiveness was required in the background of engineering knowledge than in the past. Furthermore, in the case of the midship section, which is the target of this research, if the number of members is increased, design parameters and constraint conditions are inevitably increased. Therefore, an efficient optimization algorithm is required to satisfy the CSR standard and obtain global optimum solution. In order to consider the design problem, it is required as follows:

- To define the design variables

4 - To set the constraint conditions - To operate the objective function

These problems can be determined naturally by the adoption of CSR. Thus the problem that designers must consider is to efficiently find global optimum solution solutions based on these conditions. For instance, in order to optimize the midship section area for bulk carrier of the Kamsarmax class, it must be consider 69 design variables and 475 constraint conditions at least. In order to manage the various conditions and search a global optimum solution, an efficient optimization algorithm is required. Furthermore, an effective designer-oriented integrated system to operate the optimization algorithm should be constructed.

1.2 Objectives

The objectives of this research are as follows:



To introduce a novel metaheuristic optimization algorithm named Insect Antennae Algorithm (IAA) that can be applied to various optimization problems as well as engineering design problems efficiently.



To develop an Optimal Design Support System (ODSS) for bulk carriers based on common structural rules that designers can use when to minimize the midship section area



To enhance the performance of ODSS such as computing time, exploration ability to find a global optimum solution, enhanced usability, precise definition of design variables and introduce a penalty method for efficient management of constraint conditions.

1.3 Outline

This dissertation consists of five chapters as follows:

Chapter 1, “Introduction”, presents a background, motivation, objectives of research

and composition.

5

Chapter 2, “Insect Antennae Algorithm”, introduces a novel metaheuristic optimization algorithm, named Insect Antennae Algorithm, inspired by antennae of insects, and shows the robustness and efficiency of the proposed algorithm via well- known benchmark functions and constrained functions.

Chapter 3, “Engineering Design with Insect Antennae Algorithm”, verifies the applicability of the Insect Antennae Algorithm via well-known engineering design problems.

Chapter 4, “Development of Optimal Design Support System for Midship Section Optimization of Bulk Carriers”, introduces an Optimal Design Support System (ODSS) to minimize the midship section area based on Common Structural Rules of International Association of Classification Societies (IACS). In order to enhance the performance of ODSS, optimization algorithm hybridization, parallel computing techniques, several preconditions are presented. Also, the numerical results obtained by ODSS with standard Kamsarmax are mentioned.

Chapter 5, “Conclusion”, presents the results obtained in this research.

6

CHAPTER 2. INSECT ANTENNAE ALGORITHM

2.1 Introduction

In this chapter, a new metaheuristic algorithm named Insect Antennae Algorithm (IAA) is introduced for solving optimization problems. The proposed IAA is inspired by the antennae of insects. In IAA, an insect is simplified as a body and two antennae, and the body and antennae are discretized as three robots. A detailed explanation of the IAA is presented in Section 2. The proposed algorithm was validated with the following nine well-known benchmark functions: Ackley, Drop-Wave, Goldstein and Price, Griewank, Rastrigin, Rosenbrock, Schaffer, Schwefel, and Shubert. The global optimal solutions were obtained very fast and with high accuracy. In order to verify the performance of the IAA under constrained conditions, the IAA was evaluated using four constrained functions. All of the constrained conditions of each function were satisfied very precisely, and the best solutions obtained by the IAA were outstanding when compared with other state-of-the-art optimization algorithms. Finally, in order to confirm the practical applicability of the IAA to the engineering fields, six well- known engineering design problems (the tension/compression spring design, welded beam design, pressure vessel design, speed reducer design, stepped cantilever beam design, and 15-bar planar truss design) were considered. From the results of the comparison of the IAA to others algorithms, it was clear that the IAA outperforms the state-of-the-art algorithms.

The rest of this chapter is organized as follows: After the introduction, Section 2.2

presents the IAA in detail. Section 2.3 describes the performance of the IAA via

benchmark functions and compares the efficiency of the IAA with the state-of-the-art

algorithms via solving the constrained functions. The applicability of the IAA shows via

solving the well-known engineering design problems in Section 2.4. Finally, the

summary is provided in Section 2.5.

7

In this section, the inspiration and implementation of the proposed IAA is presented firstly, then the search strategy, organization, and parameters using in the IAA are described in detail.

2.2.1 Inspiration and Implementation

The antennae of the insects in the nature are classified into various shapes and functions, and they are complex sensory organs that perform olfaction, audition, and gustation as well as tactile stimulation. Such functions are employed to find the location of the opposite sex, prey, and enemy.

The insects decide the movement direction using the surrounding information collected through their antennae and move to the desired final location based on an iterative information collection process. Most of the insects have two antennae as shown in Figure 2.1 (a), and an insect can be simplified as a robot with their own two antennae as shown in Figure 2.1 (b). As shown in Figure 2.1 (c), a robot has randomly select two robots as own antennae among the others robots. In the IAA, the randomly selected antennae method was adopted in order to reduce the number of evaluations of the objective function and improve the searching efficiency.

An insect can be expressed using three robots as shown in Figure 2.1 (c), and a robot have a position information in the solution domain, and can be carried out a body role or an antenna role for the others robots. It means that n insects can be represented by n robots.

The search strategy based on the robot swarm is described in the next section.

8

(a) (b) (c)

Figure 2.1 Discretization of a body and two antennae of an insect: (a) natural antennae of the insects (e.g., longicorn beetles); (b) robot(R1) with their own two antennae; and (c) robot(R1) with randomly selected two antennae.

2.2.2 Search Strategy

The population-based optimization algorithms such as genetic algorithm, particle swarm optimization, ant colony optimization, firefly algorithm, bat algorithm, lion optimization algorithm, whale optimization algorithm, and so on, consist of their own population, which is inspired from natural creatures. Depending on the type of algorithm, the population can be expressed as DNA, birds, insects, or mammals.

The proposed IAA is also one of the population-based optimization algorithms, and the appropriate number of insects (population) is required in order to find the best solution. An insect is discretized by using three robots, and each robot performs a body role or an antenna role. One robot is sequentially selected, and two robots are randomly selected.

There are three hypotheses, which are used in the IAA as the primary conception.

First, a local best solution desires the migration to the global best solution. This hypothesis is adopted to calculate the direction vector (V1).

Second, two antennae desire to find the best solution of surrounding by using the

peripheral information. This hypothesis is applied to calculate the direction vectors

(V2 and V3).

9

Figure 2.2 Pseudo code of the Insect Antennae Algorithm.

Finally, the next position of a body is expressed by the sum of the three direction vectors. This hypothesis is used to calculate the direction vector (PX).

Figure 2.2 presents the pseudo code of the IAA. The IAA consists of four procedures:

initialize all robots, calculate the objective function, update the position of robots, and update the best solutions. Figure 2.3 illustrates how to migrate the robots to the next position. The detailed explanation of each line is as follows:

Line 1: The position (Robot*.X) of each robot is determined based on the range of design variables.

Line 2: The fitness (Robot*.F) of each robot is calculated using the objective function.

Line 3: The best solution (Robot*.bX and Robot*.bF) of robots and the global best solution (gX and gF) are updated.

Lines 1 to 3 are carried out before entering into the main loop of iteration.

Line 5: The Attraction Rate (AR) is selected randomly in the new calculated range of attraction rate (AR*), which is adjusted based on the number of iterations as follows:

   

 

min max

* * * *

min min max max min max

n N ,

1 , 2

1, 2, ,

.

con con

Rate

Rate AR AR Rate Rate AR AR Rate

AR AR Rate AR AR Rate AR AR AR



     

    

(2.1)

10

where n is the current iteration number, N is the maximum number of iterations, ARcon is the convergence value of attraction rate, ARmin is the initial ARmin, ARmax is the initial ARmax, and AR*min and AR*max are the newly calculated range of the attraction rate. The new range of AR is updated for each iteration.

Line 7: The ith robot and two randomly selected robots are ranked according to their fitness (Robot*.F). The ranked robots are expressed as R1, R2, and R3. It is means that a robot has the best solution among the three robots. For instance, as you can see in Figure 2.3 (b) and (c), R1, R2 and R3 correspond to 7, 4 and 5, respectively.

Line 8: The four direction vectors are calculated based on the three hypotheses mentioned previous as follows:

1

1 2

1 3

1

.

. .

. .

. .

R

R R

R R

R

Robot

Robot Robot Robot Robot

Robot

 

 

 

   

V1 gX bX

V2 bX X

V3 bX X

PX X V1 V2 V3

(2.2) Line 9: The position of the ith robot is updated with the new calculated range of AR from Line 5.

. . ( . )

i i i

Robot XRobot X PXRobot X AR

(2.3)

Line 10: The fitness (Roboti.F) of the ith robot is calculated.

Line 11: The best solution (Roboti.bX and Roboti.bF) of the ith robot and global best solution (gX and gF) is updated, if there is a better solution.

Lines 7 to 11 are performed iteratively for updating the information of robots.

Figure 2.3 can be explained in detail as follows:

(a) We suppose that there are ten robots, and an optimal solution, gX, which is

obtained until current stage.

11

(b) The routine is started from P1 to P10, and the current stage is fifth. The P5 is sequentially (automatically) selected and P4 and P7 are randomly selected. Three robots are prepared to operate the IAA algorithm.

(c) In order to rank the three robots, the best solution, bX, which is obtained until current stage of each robot, is used. In this case, P7 have a best solution among the three robots, and P4 have secondly best solution. Therefore, the P7 will be R1, P4 will be R2, and P5 will be R3. R1 will operate a body role, and R2 and R3 will operate the antennae roles.

(d) The two hypotheses are adapted to calculate the three vectors, V1, V2, and V3. V1 can be calculated by first hypothesis: a local best solution desires to migrate to the global best solution. V2 and V3 can be calculated by second hypothesis: two antennae desire to find the best solution of surrounding by using the peripheral information.

(e) The third hypothesis is adapted to calculate the new position of robot. A new position, PX, can be obtained the sum of three vectors at the R1.

(f) This stage is fifth. Therefore P5 (R3) will be moved to the new position, PX. The direction vector, Q, can be calculated from PX to P5.

(g) The actual direction vector, v, can be calculated by applying the Attraction Rate (AR). In order to define the AR, a real number will be randomly selected between 0.4 and 0.6.

(h) Finally, the P5 will be moved to the new position, P5’.

12

(a) (b)

(c) (d)

13

(e) (f)

(g) (h)

Figure 2.3 Movement procedure of the ith robot: (a) initial status; (b) select two

robots randomly; (c) rank three robots by using fitness of each robot; (d) calculate the

three vectors V*; (e) calculate a new position PX; (f) calculate a vector Q; (g) calculate

a velocity V using attraction rate; and (h) update the position of the ith robot.

14 2.2.3 Organization of Antennae and Body

Most of the insects consist of a body and a two antennae system. However, it is necessary to ensure the efficiency of the one body and two antennae system.

Accordingly, five cases are considered as shown in Figure 2.4. Case A is the one body system without any antenna, Case B is the one body with one antenna system, Case C is the one body with two antennae system, Case D is the one body with three antennae system, and, finally, Case E is the one body with four antennae system.

(a) (b) (c) (d) (e)

Figure 2.4 Number of antennae for a body: (a) a body without antenna; (b) a body with an antenna; (c) a body with two antennae; (d) a body with three antennae; and (e) a body with four antennae.

In order to evaluate each case, a 10-dimensional sphere function is adopted. The number of robots and the number of iterations are set to 100 and 1,000, respectively.

Each case is performed 1,000 times, and 1,000 of best solutions are obtained. The mathematical formulation is as follows:

2

1

( )

n

i i

f x x



 

(2.4) where

Figure 2.5 illustrates the 1,000 best solutions obtained by each case. As can be seen in

Figure 2.5 (a), Case A shows a very low level of searching ability and convergence. As

shown in Figure 2.5 (b), Case B shows a higher level of searching ability than Case A

15

and it can find the exact solution. However, there is no robustness. For Case C, as shown in Figure 2.5 (c), all of the solutions are reached to the exact solution with very high accuracy and precision (smaller than 1.0E-42). It means that Case C has sufficient searching ability and robustness as an optimization algorithm. In contrast, Case D and Case E show very low searching ability and robustness less than Case A. From the results, we can know that too few or too many antennae are inefficient to find the global optimum.

For these reasons, the Case C (one body with two antennae) was adopted in the IAA.

The Fitness shown in Figure 2.5 means a real number, which is a solution of Equation 2.4.

(a)

(b)

0 200 400 600 800 1000

0 6 12 18 24 30

Fitness

No. of solution

Case A: 1 Body

[x100]

Exact Solution IAA

0 200 400 600 800 1000

0 6 12 18 24 30

Fitness

No. of solution

Case B: 1 Body 1 Antenna

[x100]

Exact Solution IAA

16 (c)

(d)

(e)

0 200 400 600 800 1000

0 6 12 18 24 30

Fitness

No. of solution

Case C: 1 Body 2 Antennae

[x100]

Exact Solution IAA

0 200 400 600 800 1000

0 6 12 18 24 30

Fitness

No. of solution

Case D: 1 Body 3 Antennae

[x100]

Exact Solution IAA

0 200 400 600 800 1000

0 20 40 60 80 100

Fitness

No. of solution

Case E: 1 Body 4 Antennae

[x100]

Exact Solution IAA

17 (f)

Figure 2.5 Convergence trend according to the number of antennas: (a) a body without antenna; (b) a body with one antenna; (c) a body with two antennae; (d) a body with three antennae; (e) a body with four antennae; and (f) comparison of average and standard deviation for each case.

2.2.4 Parameters

The proposed IAA has five parameters, which are required for the user decision: the number of iterations, the number of robots, the range of the attraction rate (minimum value and maximum value), and the convergence value. The design variables and the range of the variables are determined by the design problem considered. For example, Figure 2.6 shows the parameter values for solving the welded beam design problem, and this problem has four design variables. As shown in Figure 2.6, the number of iterations, the number of robots and the number of variables are set to 400, 40, and 4, respectively. The range of the attraction rate is set to 0.4 to 0.6 and the convergence value is set to 0.5. Because the range of the attraction rate and the convergence value were determined based on the statistical results of tests, it is not necessary to set other values in almost all of the cases. Therefore, in the IAA, only two parameters (the number of iterations and the number of robots) need to be determined. However, In general, the number of robots can be set to 10 times the number of variables, and the number of iterations can be set to 100 times the number of variables by experience.

1 2 3 4 5

0 12 24 36 48 60

0 4 8 12 16 20

Average

Case

Average and SD of Fitness

SD Average

A B C D E

[x100] [x100]

SD

18

This setting showed very good efficiency and robustness in all of problems mentioned in this paper and other many problems used to test the IAA even though not mentioned.

In order to deal with an unknown design problem, the number of variables and the range of each variable should be determined by designer based on the experimental experience and theoretical background, and it is very important. In the IAA, it is very important to define the appropriate attraction rates because they affect directly to the results. The explanation for the range of the attraction rate and the convergence value is discussed in detail in the following section.

Figure 2.6 Example of parameter settings.

2.2.5 Attraction Rate and Convergence Value

The robots are migrated by using the direction vector (PX) and the attraction rate (AR)

as explained in Section 2.2.2. The AR is selected randomly in the chosen range of

attraction rate. The reason for the adoption of the ranged attraction rate is in order to

give a chance to escape from the local minima via the allowance of the diversity of

migration during the search. In order to decide an appropriate range for the attraction

rate, it is necessary to investigate what value is available and effective. First, the

19

effectiveness of the attraction rate is analyzed with 13 values: 0.1, 0.2, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.8, and 0.9. The 10-dimensional sphere function is employed as an objective function. The number of robots and the number of iterations are set to 100 and 1,000, respectively. Each case is performed 1,000 times, and the best 1,000 solutions are obtained. Figure 2.7 shows the convergence trend according to the 13 fixed attraction rates. As can be shown from the results, Cases F, G, and H perfectly reached the global best solution, and their attraction rates are 0.45, 0.5, and 0.55, respectively.

The Fitness shown in Figure 2.7 means a real number, which is a solution of Equation 2.4.

(a)

(b)

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case B: AR = 0.10

Exact Solution IAA

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case D: AR = 0.20

Exact Solution IAA

20 (c)

(d)

(e)

0 200 400 600 800 1000

0 25 50 75 100

No. of solution

Case F: AR = 0.30

Fitness

Exact Solution IAA

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case G: AR = 0.35

Exact Solution IAA

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case H: AR = 0.40

Exact Solution IAA

21 (f)

(g)

(h)

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case I: AR = 0.45

Exact Solution IAA

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case J: AR = 0.50

Exact Solution IAA

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case K: AR = 0.55

Exact Solution IAA

22 (i)

(j)

(k)

0 200 400 600 800 1000

0 25 50 75 100

Fitness

No. of solution

Case L: AR = 0.60

Exact Solution IAA

0 200 400 600 800 1000

0 50 100 150 200

Fitness

No. of solution

Case M: AR = 0.65

Exact Solution IAA [x100]

0 200 400 600 800 1000

0 50 100 150 200

Fitness

No. of solution

Case N: AR = 0.70

Exact Solution IAA [X100]

23 (l)

(m)

(n)

0 200 400 600 800 1000

0 50 100 150 200

Fitness

No. of solution

Case P: AR = 0.80

Exact Solution IAA [X100]

0 200 400 600 800 1000

0 50 100 150 200

Fitness

No. of solution

Case R: AR = 0.90

Exact Solution IAA [X100]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0

5 10 15 20

Average & SD

Case

Average and SD of Fitness

SD Average

A B C D E F GH I J K L M

24 (o)

Figure 2.7 Convergence trend according to the fixed attraction rates: (a), (b), (c), (d), and (e) show the results of evaluation with attraction rates of 0.1, 0.2, 0.3, 0.35, and 0.4, respectively; (f), (g), and (h) indicate that the global solution is searched successfully with 0.45, 0.5, and 0.55; (i), (j), (k), (l), and (m) show the results of evaluation with attraction rates of 0.6, 0.65, 0.7, 0.8, and 0.9, respectively; (n) and (o) show average and standard deviation graphs scaled for case A to case H and case I to case M, respectively.

In order to decide an appropriate range of attraction rate, seven cases (0.3-0.45, 0.35- 0.5, 0.4-0.55, 0.45-0.55, 0.45-0.6, 0.5-0.65, and 0.55-0.7), are considered based on the test results of fixed attraction rate. The 20-dimension sphere function is adopted as an objective function. The number of robots and the number of iterations are set to 200 and 1,000, respectively. Each case is also performed 1,000 times, and 1,000 of the best solutions are gathered. Figure 2.8 illustrates the convergence trend in accordance with the seven attraction rates. As can be seen in Figure 2.8, the best solutions are obtained perfectly in Cases C, D, and E, and the corresponding ranges of attraction rate are 0.4-0.55, 0.45-0.55, and 0.45-0.6.

In order to ensure that a global best solution is found without being trapped in the local minima, it is important to use a search domain that is as wide as possible until the convergence is guaranteed. Therefore, the minimum value and the maximum value of the range of attraction rate are set to 0.4 and 0.6, respectively.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0

5 10 15 20

Average & SD

Case

Average and SD of Fitness

SD Average

A B C D E F GH I J K L M

[X1000]

25

It has been confirmed that the range of attraction rate is very important, as shown in Figure 2.8. Therefore, the convergence value of attraction rate will affect the final best solution.

The Fitness shown in Figure 2.8 means a real number, which is a solution of Equation 2.4.

(a)

(b)

0 200 400 600 800 1000

0 250 500 750 1000

Fitness

No. of solution

Case A: AR = [0.30, 0.45]

Exact Solution IAA

0 200 400 600 800 1000

0 250 500 750 1000

Fitness

No. of solution

Case B: AR = [0.35, 0.50]

Exact Solution IAA

26 (c)

(d)

(e)

0 200 400 600 800 1000

0 250 500 750 1000

Fitness

No. of solution

Case C: AR = [0.40, 0.55]

Exact Solution IAA

0 200 400 600 800 1000

0 250 500 750 1000

Fitness

No. of solution

Case D: AR = [0.45, 0.55]

Exact Solution IAA

0 200 400 600 800 1000

0 250 500 750 1000

Fitness

No. of solution

Case E: AR = [0.45, 0.60]

Exact Solution IAA

27 (f)

(g)

(h)

0 200 400 600 800 1000

0 250 500 750 1000

Fitness

No. of solution

Case F: AR = [0.50, 0.65]

Exact Solution IAA

0 200 400 600 800 1000

0 50 100 150 200

Fitness

No. of solution

Case G: AR = [0.55, 0.70]

Exact Solution IAA [x100]

1 2 3 4 5 6 7

0 60 120 180 240

Average & SD

Case

Average and SD of Fitness

SD Average

A B C D E F G

28 (i)

Figure 2.8 Convergence trend according to the range of attraction rates: (a) and (b) are needed enough convergence process; (c), (d) and (e) are reached to the exact solution;

and (f) and (g) are shown early convergence to the local optima; (h) and (i) show average and standard deviation graphs scaled for case A to case E and case F to case G, respectively.

Figure 2.9 shows the fitness history of the 20-dimensional sphere function according to the three reduction methods of the ranged attraction rate. Figure 2.9 (a) is a fixed value method, and it does not convergence. Figure 2.9 (b) is a linearly reduced method, and it converges to 0.5. Finally, Figure 2.9 (c) is a stepped value method, and it is also converges to 0.5. As shown in Figure 2.9, the linearly reduced method does not provide dramatically good results compared with other methods, but it is sure to provide better results. Therefore, the linearly reduced type was adopted in the IAA.

The Fitness shown in Figure 2.9 means a logged real number, which is a solution of Equation 2.4.

1 2 3 4 5 6 7

0 30 60 90 120

Average & SD

Case

Average and SD of Fitness

SD Average

A B C D E F G

[x100]

29

(a) (b)

(c)

Figure 2.9 Convergence trend according to the reduction types of ranged attraction rate: (a) fixed ranged attraction rate, (b) linearly reduced ranged attraction rate; and (c) stepped ranged attraction rate.

0 500 1000 1500 2000

−80

−60

−40

−20 0

0.4 0.5 0.6

Fitness

Iteration Sphere Function (20d)

Attraction Rate

Log10

0 500 1000 1500 2000

−80

−60

−40

−20 0

0.4 0.5 0.6

Fitness

Iteration Sphere Function (20d)

Attraction Rate

Log10

0 500 1000 1500 2000

−80

−60

−40

−20 0

0.4 0.5 0.6

Fitness

Iteration Sphere Function (20d)

Attraction Rate

Log10

30

In this section, nine mathematical benchmark functions were tested to check the basic performance of the IAA, and four constrained functions were adopted to confirm the convergence and effectiveness of the IAA under constraint conditions. The results were compared with several well-known state-of-the-art algorithms.

The IAA has been implemented using the C++ programming language, and the parameters for the IAA were set as follows: the number of iterations is 200, the number of robots is 20, the number of variables is 2, the range of the attraction rate is [0.4, 0.6], the convergence value of the attraction rate is 0.5, and the range of the variables is set according to each function.

2.3.1 Benchmark Functions

The landscape of nine functions is shown in Figure 2.10.

The features of each function are as follows:

2.3.1.1 Ackley Function

This function [7] is widely used for testing optimization algorithms, and it has numerous local minima. The optimization algorithm, based on the hill-climbing method, will be trapped in one of its many local minima. The global minimum is

( *) 0

f x 

at

.

2

1 1

1 1 1

cos( 2 )

( ) 20 20

n n

i i

i x i x

n n

f x e e e

  



 

    _(2.5)

where

and

. 2.3.1.2 Drop-Wave Function

This function [20] has multimodal features and is highly complex. The global minimum

is located at

with

.

31

 

 

2 2

1 2

2 2

1 2

1 cos 12 ( )

2 0.5

x x f x

x x

 

   

(2.6) where

.

2.3.1.3 Goldstein and Price Function

This function [21] has several local minima. The global minimum is at

with

( *) 3 f x 

.

   

 

   

 

2 2 2

1 2 1 1 2 1 2 2

2 2 2

1 2 1 1 2 1 2 2

( ) 1 1 19 14 3 14 6 3

30 2 3 18 32 12 48 36 27

f x x x x x x x x x

x x x x x x x x

        

        ^(2.7)

where

.

2.3.1.4 Griewank Function

This function [22] has many widespread local minima, which are regularly distributed.

The global minimum is at

with

.

2

1 1

( ) 1 1 cos

4000

n n

i i

i i

f x x x

i

 

    

 

 

 

(2.8)

where

and

. 2.3.1.5 Rastrigin Function

This function [7] has several local minima. It is highly multimodal, but locations of minima are regularly distributed. The global minimum is at

with

.

 





1

( ) 10 10 cos 2

n

i i

i

f x n x  x



   

(2.9)

where

and

.

32 2.3.1.6 Rosenbrock Function

This function [23] is a popular test problem for gradient-based optimization algorithms.

It is a unimodal type. However, even though this valley is easy to find, convergence to the minimum is difficult. The global minimum is at

with

.

   

 

1 2 2 2

1 1

( ) 1 100

n

i i i

i

f x x x x







    

(2.10)

where

and

. 2.3.1.7 Schaffer Function

This function has numerous minima. The global minimum is at

with

.

 

 

2 2 2

1 2

2 2 2

1 2

sin 0.5

( ) 0.5

1 0.001 x x f x

x x

 

 

 

 

  _(2.11)

where

. 2.3.1.8 Schwefel Function

This function [25] is non-convex, multimodal, and additively separable. The global minimum is

at

.

 

1

( ) 418.982887 sin

n

i i

i

f x n x x



  

(2.12)

where

and

. 2.3.1.9 Shubert Function

This function [24] has 18 global minima in the below search domain. The value of global minima is

.

 

     

5 5

1 2

1 1

( ) cos 1 cos 1

i i

f x i i x i i i x i

 

          

     (2.13)