Multi-Objective Optimization using Genetic Algorithms

Postadress: Besöksadress: Telefon:

Box 1026 Gjuterigatan 5 036-10 10 00 (vx)

551 11 Jönköping

Multi-Objective Optimization

using Genetic Algorithms

Kaveh Amouzgar

THESIS WORK 2012

PRODUCT DEVELOPMENT AND MATERIALS

ENGINEERING

Postadress: Besöksadress: Telefon:

Box 1026 Gjuterigatan 5 036-10 10 00 (vx)

551 11 Jönköping

Multi-Objective Optimization

using Genetic Algorithms

Kaveh Amouzgar

This thesis work has been carried out at the School of Engineering in

Jönköping in the subject area Product Development and Materials Engineering.

The work is a part of the master’s degree.

The authors take full responsibility for opinions, conclusions and findings

presented.

Supervisor: Niclas Strömberg

Scope: 30 ECTS credits

Date: 2012-05-30

This thesis has been prepared using LATEX.

Abstract

In this thesis, the basic principles and concepts of single and multi-objective Ge-

netic Algorithms (GA) are reviewed. Two algorithms, one for single objective and

the other for multi-objective problems, which are believed to be more efficient,

are described in details. The algorithms are coded with MATLAB and applied

on several test functions. The results are compared with the existing solutions

in literatures and shows promising results. Obtained pareto-fronts are exactly

similar to the true pareto-fronts with a good spread of solution throughout the

optimal region. Constraint handling techniques are studied and applied in the two

algorithms. Constrained benchmarks are optimized and the outcomes show the

ability of algorithm in maintaining solutions in the entire pareto-optimal region.

In the end, a hybrid method based on the combination of the two algorithms is

introduced and the performance is discussed. It is concluded that no significant

strength is observed within the approach and more research is required on this

topic. For further investigation on the performance of the proposed techniques,

implementation on real-world engineering applications are recommended.

Keywords

Single Objective Optimization, Multi-objective Optimization, Constraint Han-

dling, Hybrid Optimization, Evolutionary Algorithm, Genetic Algorithm, Pareto-

Front, Domination.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Purpose and aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theoretical background 3

2.1 What is Optimization? . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Single-Objective Optimization . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Evolutionary Method . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 Genetic Algorithm Concept . . . . . . . . . . . . . . . . . . 4

2.2.3 Genetic Algorithm Principles . . . . . . . . . . . . . . . . . 5

2.2.4 Real Parameter GA . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.5 Generalization Generation Gap Algorithm (G3) . . . . . . . 7

2.2.6 Parent-Centric Recombination Operator (PCX) . . . . . . . 8

2.2.7 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Multi-objective Optimization . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Multi-Objective Optimization Formulation . . . . . . . . . . 10

2.3.2 Multi-Objective Optimization Definitions . . . . . . . . . . . 11

2.3.3 Approaches Towards Non-Dominated Set . . . . . . . . . . . 14

2.3.4 Approaches Towards Multi-Objective Optimization . . . . . 14

2.3.5 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . 15

2.3.6 MOEA Techniques . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.7 Comparison of MOEAs . . . . . . . . . . . . . . . . . . . . . 17

2.3.8 SPEA2: Improved Strength Pareto Evolutionary Algorithm 18

2.3.9 Overall SPEA2 Algorithm . . . . . . . . . . . . . . . . . . . 18

2.3.10 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Hybrid Multi-Objective Optimization Approach . . . . . . . . . . . 23

i

3 Implementation 24

3.1 Single Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Unconstrained Test Functions . . . . . . . . . . . . . . . . . 24

3.1.2 Constrained Test Functions . . . . . . . . . . . . . . . . . . 26

3.2 Multi objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Unconstrained Test Functions . . . . . . . . . . . . . . . . . 29

3.2.2 Constrained Test Functions . . . . . . . . . . . . . . . . . . 32

3.3 Hybrid Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Test Results 36

4.1 Single Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Unconstrained Functions . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Constrained Functions . . . . . . . . . . . . . . . . . . . . . 38

4.2 Multi-Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Unconstrained Functions . . . . . . . . . . . . . . . . . . . . 40

4.2.2 Constrained Functions . . . . . . . . . . . . . . . . . . . . . 51

4.3 Hybrid Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Conclusion 54

6 Bibliography 55

A Hand Calculation of G3 Algorithm with Constraints 59

B Hand Calculation of SPEA2 Algorithm 63

B.1 Constraint Handling Method of SPEA2 Algorithm . . . . . . . . . . 70

ii

List of Figures

1 Trade-off curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Min-Min pareto-front . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 The welded beam problem. . . . . . . . . . . . . . . . . . . . . . . . 28

4 Tension/compression string problem. . . . . . . . . . . . . . . . . . 29

5 Convergence of Schwefel’s function . . . . . . . . . . . . . . . . . . 37

6 Convergence of Rosenbrock function . . . . . . . . . . . . . . . . . . 37

7 Convergence of Test Function 1 . . . . . . . . . . . . . . . . . . . . 38

8 Convergence of welded beam problem . . . . . . . . . . . . . . . . . 38

9 Convergence of Tension/Compression Spring . . . . . . . . . . . . . 38

10 Pareto-front of Exercise 14, single objective . . . . . . . . . . . . . . 40

11 Pareto-front of Exercise 14, multi objective . . . . . . . . . . . . . . 40

12 Pareto-front of Kursawe test function . . . . . . . . . . . . . . . . . 41

13 Pareto-front of ZDT1 test function . . . . . . . . . . . . . . . . . . 41





18 Scatter-plot matrix of Kursawe test function . . . . . . . . . . . . . 45

19 Scatter-plot matrix of ZDT1 test function . . . . . . . . . . . . . . 46





24 Pareto-front of BNH test function . . . . . . . . . . . . . . . . . . . 51

iii

25 Pareto-front of OSY test function . . . . . . . . . . . . . . . . . . . 51

26 Pareto-front of SRN test function . . . . . . . . . . . . . . . . . . . 51

27 Pareto-front of TNK test function . . . . . . . . . . . . . . . . . . . 51

28 ZDT1 Hybrid and random archive population . . . . . . . . . . . . 52



31 Min-Ex pareto-front and initial solutions . . . . . . . . . . . . . . . 63

32 Constrained Min-Ex pareto-front, feasible region and initial solutions 70

List of Tables

1 Results of unconstrained test functions, single objective. . . . . . . 36

2 Comparison of welded beam results . . . . . . . . . . . . . . . . . . 38

3 Comparison of tension/compression spring results . . . . . . . . . . 39

4 Pre-defined parameters of unconstrained SPEA2 . . . . . . . . . . . 41

5 Pre-defined parameters of constrained SPEA2 . . . . . . . . . . . . 51

6 Random initial solutions for G3 algorithm hand calculation example 60

7 Current and external initial random population of SPEA2 . . . . . 63

8 Fitness assignment procedure of SPEA2 . . . . . . . . . . . . . . . 65

9 Constraint handling data of SPEA2 . . . . . . . . . . . . . . . . . . 71

iv

1 Introduction

Although substantial amount of search in optimization is conducted with regards

to single objective problems, optimization problems with multi conflicting objec-

tives are inevitable in many topics specially engineering applications. Two main

methods have been proposed by scientist for solving multi-objective optimization

problems: 1) Classical method, 2) Evolutionary algorithms. Classical methods are

able to reach one optimal solution at each run, while evolutionary algorithms are

based on a population of solutions which will hopefully lead to a number of opti-

mal solutions at every generation. The evolutionary algorithm method which had

shown benefits over the classical approach can be categorized in several categories.

Genetic Algorithm is one of the methods that mimic the evolution of genes and

chromosomes.

1.1 Background

Previous works by Beasley and Bull (1993); Coello (2007); Deb (1995, 2001, 2002,

2004); Deb et al. (2001); Fonseca and Fleming (1993); Haupt et al. (2004); Kim

et al. (2004); Kukkonen (2006); Man et al. (1996); Zitzler and Thiele (1998);

Zitzler et al. (2001) on the theory, concepts and algorithms of single and multi-

objective optimization using evolutionary algorithms.

Previous studies by Deb (2000); Deb et al. (2002); Jimnez et al. (1999); Kuri-

Morales and A.Gutierrez-Garcia (2002); Mezura-Montes et al. (2003); T. Ray

(2001) on constraint handling methods.

Test functions and their comparison has been studied by (Binh and Korn, 1997);

Deb (1991, 1999); Gamot and Mesa (2008); Kuri-Morales and A.Gutierrez-Garcia

(2002); Kursawe (1991); Osyczka and Kundu (1995); Srinivas and Deb (1994);

Tanaka and Watanabe (1995) and Zitzler et al. (2000).

1.2 Purpose and aims

The aim is to develop a fast and efficient multi-objective optimization technique

by using GA (Genetic Algorithm) method, in order to solve multi-objective opti-

mization problems with constraints. Several benchmarks are to be optimized with

the developed algorithm. Also a hybrid approach will be suggested and further

studied. MATLAB shall be used to code the algorithm.

1

1.3 Delimitations

Genetic Algorithm is the only method used in developing the technique. Other

evolutionary methods like Evolution strategies, Evolutionary programming and

Genetic Programming are not considered in the thesis.

1.4 Outline

The organization of the thesis is in five different sections:

Section 1: Introductory, background and purpose is described.

Section 2: General theory of optimization, single and multi-objective optimiza-

tion is explained. Evolutionary algorithms especially Genetic Algorithms are

discussed in details and two algorithms are suggested. Constraint handling

techniques and a hybrid method are theoretically defined.

Section 3: A short description on implementation of algorithms and a number

of benchmarks will be presented in this section.

Section 4: The results obtained from the benchmarks are illustrated and com-

pared with references.

Section 5: The conclusion of the performed work will be summarized.

2

2 Theoretical background

2.1 What is Optimization?

Optimization is a process of making things better. Life is full of optimization

problems which all of us are solving many of them each day in our life. Which

route is closer to school? Which bread is better to buy having the lowest price while

giving the required energy? Optimization is fine-tuning the inputs of a process,

function or device to find the maximum or minimum output(s). The inputs are

the variables, the process or function is called objective function, cost function or

Fitness value (function) and the output(s) is fitness or cost (Haupt et al., 2004).

In the thesis minimization of cost is tackled, in functions which maximum of cost

is required, by slapping a minus in front of objective function, the output will be

minimized. Therefore all the problems and functions in the thesis are addressed

as minimization problem.

When only one objective function involves in the problem, it is called single-

objective optimization, however in most real world problems more than one objec-

tive function is required to be optimized, and therefore these problems are named

multi-objective optimization.

Deb (2001) classified optimization solving methods into following two major cat-

egories:

• Classical methods

• Evolutionary methods

The classical methods commonly use a single random solution, updated in every

iteration by a deterministic procedure to find the optimal solution. These methods

are classified into two distinct groups: direct methods where only the objective

function and the constraints value are used to find the optimum and gradient-

based methods whereas the first and second derivative of objective function and/or

constraints are applied to find the search direction and optimal solution (Deb,

2001).

3

2.2 Single-Objective Optimization

2.2.1 Evolutionary Method

This method was inspired by the evolutionary process of human being and the

interests for imitating living being is increasing since 1960′s. Evolutionary method

mimics the evolution principle of nature which results in a stochastic search and

optimization algorithm. It also can out pace the classical method in many ways

(Gen and Cheng, 1997).

Evolutionary method (algorithm) uses an initial population of random solutions in

each iteration, instead of using a single solution as in classical method. This initial

population is updated in each generation to finally converge to a single optimal

solution. Having a population of optimum solution in a single simulation run,

is a unique characteristic of the method in solving multi-objective optimization

problems (Deb, 2001).

Gen and Cheng (1997) divides the method into three main types: genetic al-

gorithm, evolutionary programming and evolutionary strategy while Deb (2001)

describes an additional type to the three above-mentioned: genetic programming.

2.2.2 Genetic Algorithm Concept

Many real-world optimization problems are extremely difficult and complex in

terms of number of variables, nature of the objective function, many local optimal,

continuous or discrete search space, required computation time and resources, etc.

in various domains including service, commerce and engineering.

Genetic algorithm was first introduced by John Holland (1975) in 1970′s, whereas

one of his students David Goldberg had an important contribution in popularizing

this method in his dissertation by solving a complex problem (Haupt et al., 2004).

Genetic algorithm is an inspiration of the selection process of nature, where in

a competition the stronger individuals will survive (Man et al., 1996). In nature

each member of a population competes for food, water and territory, also strive

for attracting a mate is another aspect of nature. It is obvious the stronger in-

dividuals have a better chance for reproduction and creating offspring, while the

poor performers have less offspring or even non. Consequently the gen of the

strong or fit individuals will increase in the population. Offspring created by two

4

fit individual (parents) has a potential to have a better fitness compared to both

parents called super-fit offspring. By this principle the initial population evolves

to a better suited population to their environment in each generation (Beasley and

Bull, 1993).

2.2.3 Genetic Algorithm Principles

As mentioned, in genetic algorithm unlike other classical methods, a population

of random solution is selected. Each solution of the problem is represented as a

set of parameters which are known as genes. Joining genes create a binary bit

string of values, denoting each member of population referred as chromosome. A

chromosome evolves through iterations, called generation (Gen and Cheng, 1997).

After representation a fitness or objective function is required. Also during the run

a selection of parents for reproduction and recombination for creating offspring is

essential. These aspects are called GA′s operators (Beasley and Bull, 1993).

A selection or reproduction operator during reproduction phase of GA selects par-

ents from population which they create offspring by recombination comprising

next generation. The main objective of selection operator is to keep and du-

plicate the fit solutions and eliminate the poor chromosomes, while keeping the

size of population constant. Deb (2001) describes some schemes for achieving the

above objective: tournament selection, proportionate selection, ranking selection,

roulette wheel selection (RWS) and stochastic universal selection (SUS). It is ob-

vious this operator cannot create new chromosomes to the initial population, it

only make copies of good solutions.

In reproduction phase the two parents nominated by selection operator recombine

to create one or more offspring with crossover or mutation operators.

They are number of different crossover operators in literature but the main concept

is selecting two strings of solution (chromosomes) from the mating pool of selection

operator and exchanging some portion of these two strings from a random selected

point(s). Single point cross over is one basic type of this operator for binary GA

(Deb, 2001).

A mutation operator is applied to individual solutions after cross over operator

which a gene(s) is randomly changed in a string with a small probability to create

a new chromosome. The aim of this operator is to maintain the diversity of the

5

population and increase the possibility of not losing any potential solution and find

the global optimal, while cross over operator is a technique of rapid exploration

of search space (Beasley and Bull, 1993).

To sum up, the selection operator selects and maintains the good solutions; while

crossover recombines the fit solutions to create a fitter offspring and mutation

operator randomly alter a gene or genes in a string to hopefully find a better

string (Deb, 2001).

2.2.4 Real Parameter GA

There are some difficulties in binary-coded GAs, including inability to solve the

problems where the values of variables have continuous search space or when the

required precision is high. According to Deb (2001) hamming cliffs related to

certain strings (01111 or 11110) is one of the difficulties where altering to a near

neighbour string requires changes in many genes. He also claims necessity of large

strings (chromosomes with many genes) in order to fulfil a necessary precision

which in result increases the size of population, as another struggle for binary

GAs. Therefore using floating point numbers to represent the variables in most

problems is more logical which requires less storage than binary coded strings. In

addition, since there is no need for decoding the chromosomes before evaluation

of objective function in selection phase the real parameter GA (in some literature

called continuous GA) is inherently faster than binary GA (Haupt et al., 2004).

Since the real value of parameters are directly used to find the fitness value in

selection operator and there is no decoding to a string in real parameter GAs, this

operator does not alter with binary GA selection operators and the same operators

can be used in real parameter GAs. On the other hand, since the cross over and

mutation operators used in binary GAs are based on strings and alteration in

genes (bits), new cross over and mutation operators shall be defined for this type

of GA.

Deb (2001) outlines some real parameter crossover operators such as linear crossover,

naive crossover, blend crossover (BLX), simulated binary crossover (SBX), fuzzy

recombination operator, unimodal normally distributed crossover (UNDX), sim-

plex crossover (SPX), fuzzy connectives based crossover and unfair average crossover.

Other cross over operators including parent centric crossover (PCX), modified

PCX (mPCX) are recommended in literatures (Deb and Joshi, 2001).

6

Since in real parameter crossover operator two or more parents directly recombine

to create on or more offspring and it has the same concept as mutation operator,

a question comes up: Is there a good reason for using a mutation operator along

with crossover operator? The debate still remains, however Deb (2001) argues, the

different between these two operators is in the number of parent solutions selected

for perturbation. He claims if offspring is created from one parent the operator

is mutation while offspring created from more than one parent is crossover. He

also mentions some common mutation operators in his book: Random mutation,

non-uniform mutation, normally distributed mutation and polynomial mutation.

2.2.5 Generalization Generation Gap Algorithm (G3)

Deb (2002) proposes a population-based, four steps, real-parameter optimization

algorithm-generator called Generalization Generation Gap (G3) model. In the

same paper performance of the G3 algorithm is studied on three commonly used

test problems and is compared with a number of evolutionary and classical opti-

mization algorithms, also Deb (2004) performs a systematic parametric study on

G3 model, both of these studies concludes to out performance of the algorithm

to a number of existing classical and evolutionary algorithms. The G3 algorithm

is coded in MATLAB for solving single-objective optimization problems in the

thesis.

Generalization generation gap algorithm is modified steady-state GA to make it

computationally faster, in which in every iteration only two new solutions are

updated in the GA population. This model preserves elite solutions from previ-

ous generation (Deb, 2002). Four plans are used which are Selection Plan (SP),

Generation Plan (GP), Replacement Plan (RP) and Update Plan (UP).

The steps in algorithm are as follows:

Step 1 (SP): From the population B (set B) of size N, the best parent and µ−1

other parents are randomly selected. These µ solutions create Set P.

Step 2 (GP): λ offspring are created from µ parents in set P, with using any

recombination operator, which creates Set C.

Step 3 (RP): r random parents are chosen from set B, which creates Set R.

Step 4 (UP): r random parent chosen in step 3 (RP) are replaced with r best

solutions from the combined set C ∪R (set RC), in set B.

7

Several parametric studies such as Deb (2001, 2002, 2004); Kita (2001), compare

the performance of G3 model to other evolutionary algorithms, and in all of the

studies G3 model has shown a better performance and robustness. Also using

different recombination operators has been examined and the overall result shows

faster computation time and lower number of evaluation required to meet a desired

accuracy of a parent centric recombination operator (PCX) proposed by Deb et al.

(2001), which will be briefly described.

2.2.6 Parent-Centric Recombination Operator (PCX)

Deb et al. (2001) suggests a variation operator (combination of crossover and mu-

tation operator) for this algorithm, called parent centric recombination operator

(PCX). A parent centric operator ensures identically of population mean of the

total offspring population to that of the parent population while mean centric

operators preserve the mean between the participating parents and resulting off-

spring. The paper states the benefit of parent centric recombination operators

over mean centric operator, as the parents are selected from the fittest solution

in selection plan and in most real parameter optimization problems it is assumed

that the solutions near the parents can be the potential good solutions. Therefore,

creating new solutions close to parents as how it is in PCX is a steady and reliable

search technique.

The mean vector ~g of the chosen parents is computed. For each offspring, one

parent ~x(p) is chosen with equal probability. The direction vector ~d = ~g − ~x(p) is

calculated. Thereafter from each of the µ−1 parents perpendicular distance Di to

the line ~d(p) are computed and their average D is found. The offspring is created

as follows:

~y = ~x(p) + wζ ~d(p) +

µ∑i=1,i 6=p

wηD~e(i)

where ~e(i) are the µ− 1 orthonormal bases that span the subspace perpendicular

to ~d(p). Parameters wζ and wη are zero-mean normally distributed variables with

variance w2ζ and w2

η, respectively.

8

2.2.7 Constraint Handling

Most existing constraint handling methods in literatures are classified in five cat-

egories which Deb (2001) describes them briefly:

• Method based on preserving feasibility of solutions.

• Method based on penalty functions.

• Methods biasing feasible over infeasible solutions.

• Methods based on decoders.

• Hybrid methods.

In the thesis, the method based on penalty function is used for single-objective

optimization.

Penalty function method transforms a constrained optimization problem to an

unconstrained problem usually by using an additive penalty term or penalty mul-

tiplier. Penalty method can also be categorized in seven different type:

• Death Penalty

• Static Penalties

• Dynamic Penalties

• Annealing Penalties

• Adaptive Penalties

• Segregated GA

In the Static Penalty method which is implemented in this section, the penalty

parameters do not change within generations and is only applied to infeasible

solutions.

There are number of approaches in this method suggested by authors but Morales

et al. (1997) penalizes the objective function of infeasible solutions by using the

information on the number of violated constraints. His approach is formulated as

follows:

9

F (x) =

f(x), if xisfeasible,

K −∑s

i=1

[Km

], otherwise.

where s is the number of non-violated constraints and m is the total number of

constraints. K is a large positive constant. Morales et al. (1997) uses 1× 109 for

this constant which should be large enough to assign a bigger fitness to infeasible

solution compared to feasible individual. A simple single-objective constrained

optimization problem is solved for one generation by using a step by step hand

calculation of G3 algorithm in appendix A.

2.3 Multi-objective Optimization

In real world applications, most of the optimization problems involve more than

one objective to be optimized. The objectives in most of engineering problems are

often conflicting, i.e., maximize performance, minimize cost, maximize reliability,

etc. In the case, one extreme solution would not satisfy both objective functions

and the optimal solution of one objective will not necessary be the best solution for

other objective(s). Therefore different solutions will produce trade-offs between

different objectives and a set of solutions is required to represent the optimal

solutions of all objectives.

Figure 1 shows the trade-off curve of decision making involved in buying a house

problem.

The trade-off curve reveals that considering the extreme optimal of one objective

(price) requires a compromise in other objective (house area). However there

exists number of trade-off solutions between the two extreme optimal, that each

are better with regards to one objective.

2.3.1 Multi-Objective Optimization Formulation

Basically a multi-objective optimization problem has more than one objective

function, in engineering problems usually two objectives, to be optimized. In

the thesis, minimization problems with only two objectives investigated, while

maximization problems are transformed to minimizing optimization types.

The multi-objective optimization problems may also have one or more constraints

10

Figure 1: Trade-off solutions illustrated for a house-buying decision-making

including inequality, equality and/or variable bounds to be satisfied. However

in real engineering applications usually more than one constraint is involved in

the problem. A general formulation of a multi-objective optimization problem is

defined as follows:

Minimize/Maximize fm(x), m = 1, 2, ...,M ;

Subject to gj(x) ≤ 0, j = 1, 2, ..., J ;

hk(x) = 0, k = 1, 2, ..., K;

x(L)i ≤ xi ≤ x

(U)i , i = 1, 2, ..., n.

2.3.2 Multi-Objective Optimization Definitions

In order to fully understand multi-objective optimization problems (MOOP), al-

gorithms and concepts some definitions must be clarified.

• Decision variable and objective space: The variable bounds of an opti-

mization problem restrict each decision variable to a lower and upper limit

which institutes a space called decision variable space.

In multi-objective optimization values of objective functions create a mutli-

dimensional space called objective space. Each decision variable on variable

11

space corresponds to a point in objective space.

• Feasible and infeasible solutions: A solution that satisfies all the con-

straints (inequality and equality) and variable bounds is referred to as a

feasible solution. On the other hand, a solution that does not satisfy all

constraints and variable bounds is called an infeasible solution.

• Ideal objective vector: If x∗(i) is a vector of variables that optimizes

(minimize or maximize) the ith objective in a multi-objective optimization

problem with M conflicting objectives:

∃x∗(i) ∈ Ω, x∗(i) =[x∗(i)1 , x

∗(i)2 , ..., x

∗(i)M

]T: fi(x

∗(i)) = OPTfi(x).

Then, the vector

z∗ = f ∗ = [f ∗1 , f∗2 , ..., f

∗M ]T

where f ∗M is the optimum of the M th objective function, is ideal for a multi-

objective optimization problem and the point in <n which determines this

vector is the ideal solution, therefore called the ideal objective vector. Gen-

erally, ideal objective vector is a solution that does not exist. The reason

is that the optimal solution of each objective in a MOP is not necessary

the same solution. However, if the optimal solutions of all objectives are

identical the ideal vector is feasible.

• Utopian objective vector: A vector that all of its components are marginally

smaller (in case of minimization MOP) than that of the ideal objective vector

is called utopian objective vector. In other words:

∀ i = 1, 2, ...,M : z∗∗i = z∗i − εi , εi > 0.

• Linear and non- linear MOOP: If all objectives and constraints are linear

the problem is named a linear optimization problem (MOLP). In contrast, if

one or more of the objectives and/or constrains are non-linear the problem

in non-linear MOOP (Deb, 2001).

• Convex and Non-convex MOOP: The problem is convex if all objective

functions and feasible region are convex. Therefore a MOLP problem is

convex (Deb, 2001).

Convexity is an important issue in MOOPs, where in non-convex problems

the solutions obtained from a preference-based approach will not cover the

12

non-convex part of the trade-off curve. Moreover many of the existing algo-

rithms can only be used for convex problems. Convexity can be defined on

both of spaces (objective and decision variable space). A problem can have

a convex objective space while the decision variable space is non-convex.

• Domination (dominated, dominating and non-dominated): Most

of real world applications consist of conflicting objectives. Optimizing a

solution with respect to one objective will not result in an optimal solution

regarding the other objective(s). For a M objective MOP, the operator /

between two solutions i and j as i/j is translated as solution i is better than

solution j on a particular objective. Also, i. j means that solution i is worse

than solution j on this objective. Therefore, if the MOP is a minimization

case, the operator / denotes < and vice versa. Now a general definition of

domination for both minimization and maximization MOP can be made:

A feasible solution x(1) is said to dominate another feasible solution x(2) (or

mathematically x(1) x(2)), if and only if:

1. The solution x(1) is no worse than x(2) with respect to all objectives

value, or fj(x(1)) 7 fj(x

(2)) for all j = 1, 2, ...,M .

2. The solution x(1) is strictly better than x(2) in at least one objective

value, or fj(x(1)) C fj(x

(2)) for at least one j ∈ 1, 2, ...,M.

Therefore solution x(1) dominates solution x(2), solution x(1) is non-dominated

by solution x(2) or solution x(2) is dominated by solution x(1).

• Pareto- optimal set (non-dominated set): A solution is pareto-optimal

if it is not dominated by any other solution in decision variable space. The

pareto-optimal is the best known (optimal) solution with respect to all objec-

tives and cannot be improved in any objective without worsening in another

objective. The set of all feasible solutions that are non-dominated by any

other solution is called the pareto-optimal or non-dominated set. If the non

dominated set is within the entire feasible search space, it is called globally

pareto-optimal set. In other words, for a given MOP, the pareto-optimal set,

P∗, is defined as:

P ∗ = x ∈ Ω | ¬∃x′ ∈ Ω F (x′) F (x).

• Pareto-front: The values of objective functions related to each solution of

a pareto-optimal set in objective space is called pareto-front. In other words,

13

Figure 2: Pareto-front of a Min–Min problem

for a given MOP, F (x), and pareto-optimal set, P ∗, the pareto-front, PF ∗

is given by:

PF ∗ := u = F (x) |x ∈ P ∗.

Figure 2 illustrates a typical pareto-front of a two objective minimizing type

optimization problem in objective space.

Since the concept of domination enables comparison of solutions with respect

to multi-objectives, most of multi-objective optimization algorithms practice

this concept to obtain the non-dominated set of solutions, consequently the

pareto-front.

2.3.3 Approaches Towards Non-Dominated Set

They are several methods and algorithms towards finding the non-dominated set

of solutions from a given population in an optimization problem. Deb (2001)

describes three of the most common methods in his book from a naive and slow

to an efficient and fast approach.

Approach 1: Naive and slow

Approach 2: Continuously updated

Approach 3: Kung et al.s efficient method

Approach 3 has the least computational complexity among the three and according

to Kung and Luccio (1975) is the most efficient method. In all methods the concept

of domination is used to compare the solution with respect to different objective

functions.

14

2.3.4 Approaches Towards Multi-Objective Optimization

Extensive studies have been conducted in multi-objective optimization algorithms.

But most of the researches avoid the complexity in the true multi-objective opti-

mization problem by transforming the problem into single- objective optimization

with the use of some user defined parameters.

Deb classifies the approaches towards solving multi-objective optimization in two

groups.

• Ideal multi-objective optimization, where a set of solutions in form of a

trade-off curve is obtained and the desired solution is selected according to

some higher level information of problem.

• Preference based multi-objective optimization, which by using the higher

level information a preference vector transforms the multi-objective prob-

lem to a single-objective optimization. The optimal solution is obtained by

solving the single-objective problem.

The ideal approach is less subjective compared to preference based approach,

where analysis of non-technical, qualitative and experimental information is re-

quired to find the preference vector. Therefore the second approach will not be

further discussed in the thesis.

In absence of higher level information in an optimization problem within ideal

approach none of the pareto-optimal solutions is preferred over others. Therefore

in the ideal approach the main objective is to converge to a set of solution as

close as possible to true pareto-optimal set, which is the common objective of

all optimization tasks. However, diversity in the obtained pareto-optimal set is

the second objective specific to multi-objective problems. With a more divers set

of solutions that covers all parts of pareto-front in objective space, the decision

making process at the next level using the higher level information is easier. Since

two spaces are involved in MOOP, diversity of solutions in both decision and

objective space is defined. Solutions with a large Euclidean distance in variable

and objective space are referred as divers set of solutions in variable and objective

space, respectively. The diversity in the two spaces are often Symmetric, however

in complex and non-linear problems this property may not be true. Hence, Deb

(2001) assumes that there are two goals in multi-objective optimization:

15

1. To find a set of non-dominated solutions with the least distance to pareto-

optimal set.

2. To have maximum diversity in the non-dominated set of solutions.

Recall from section 2.1, that classifies optimization solving methods into; classi-

cal and evolutionary method , the classification is also valid for multi-objective

optimization problems.

In the classical method objectives are transformed to one objective function by

means of different techniques. The easiest and probably most common is the

weighted sum method which the objectives are scalarized to one objective by

multiplying the sum of objectives to a weight vector (Deb, 2001). Other techniques

are such as considering all objectives except one as constraints and limiting them

by a user defined value (ε− constraint) (Haimes and A., 1971). Deb (2001) very

well presents some of the most important classical methods in one chapter of the

book.

2.3.5 Evolutionary Algorithms

The characteristic of evolutionary methods which use a population of solutions

that evolve in each generation is well suited for multi-objective optimization prob-

lems. Since one of the main goals of MOOP solvers is to find a set of non-dominated

solutions with the minimum distance to pareto-front, evolutionary algorithms can

generate a set of non-dominated solutions in each generation.

Requirement of little prior knowledge from the problem , less vulnerability to shape

and continuity of pareto-front, easy implementation, robustness and the ability to

be carried out in parallel are some of the advantages of evolutionary algorithms

listed in Goldberg (2005).

The first goal in multi-objective optimization is achieved by a proper fitness as-

signment strategy and a careful reproduction operator. Diversity in the pareto-set

can be obtained by designing a suitable selection operator. Preserving the elitism

during generations shall be carefully considered in evolutionary algorithms. Elite-

preserving operators, as Deb (2001) names them, are introduced to directly carry

over the elit solutions to the next generation.

Coello (2007) presents the basic concepts and approaches of multi-objective opti-

mization evolutionary algorithms. The book further explores some hybrid methods

16

and introduces the test functions and there analysis. Various applications of multi-

objective evolutionary algorithms (MOEA) are also discussed in the book. Deb

(2001) is another comprehensive source of different MOEAs. The book divides the

evolutionary algorithms into non-elitist and elitist algorithms.

2.3.6 MOEA Techniques

All researchers are agreed upon that the invention of first MOEA is devoted to

David Schaffer with his Vector Evaluation Genetic Algorithm (VEGA) in the mid-

1980s, aimed at solving optimization problems in machine learning.

Deb (2001) and Coello (2007) both name various MOEAs which shows the differ-

ence in the frame work and their operators as follows:

• Vector Evaluated GA (VEGA)

• Vector Optimized Evolution Strategy (VOES)

• Weight Based GA (WBGA)

• Multiple Objective GA (MOGA)

• Niched Pareto GA (NPGA, NPGA2)

• Non-dominated Sorting GA (NSGA,NSGA-II)

• Distance-Based Pareto GA (DPGA)

• Thermodynamical GA (TDGA)

• Strength Pareto Evolutionary Algorithm (SPEA, SPEA2)

• Multi-Objective Messy GA (MOMGA-I, II, III)

• Pareto Archived Evolution Strategy (PAES)

• Pareto Enveloped Based Selection Algorithm (PESA, PESA II)

• Micro GA-MOEA (µGA, µGA2)

Coello (2007) describes the concept of each EA along with an illustration of algo-

rithm and short notes on advantages and disadvantages. At the end he summarizes

all EAs in a table. While Deb (2001) devotes two complete chapter of the book

17

to fully define the concept and principle of each EA by step-by step description of

algorithm, hand calculation, discussion on advantages and short comings, calcu-

lating the computational complexity and simulating an identical test problem for

all algorithms.

2.3.7 Comparison of MOEAs

Since there exist several MOEAs, a question of which algorithm has the best

performance is a common question among scientist and researchers. In order to

settle to an answer several test problems has been designed and various amount of

researches is carried out. In Deb’s book, a few significant studies on comparison

of EAs are discussed. (Deb, 2001)

Konak et al. (2006) demonstrates the advantages and disadvantages of most well-

known EAs in a table.

However the most representative, discussed and compared evolutionary algorithms

are Non-dominated Sorting GA (NSGA-II) (Deb et al., 2002), Strength Pareto

Evolutionary Algorithm (SPEA, SPEA2) (Zitzler and Thiele, 1998; Zitzler et al.,

2001), Pareto archived Evolution Strategy (PAES)(Knowles, 1999, 2000) , and

Pareto Enveloped Based Selection Algorithm (PESA, PESA II) (Corne and Knowles,

2000; Corne et al., 2001).

Extensive comparison studies and numerical simulation on various test problems

shows a better overall behavior of NSGA-II and SPEA2 compared to other al-

gorithms. In cases where more than two objectives are present SPEA2 seems to

indicate some advantages over NSGA-II. Strength Pareto Evolutionary Algorithm

(SPEA2) is comprehensively described in next section. Also SPEA2 is coded and

implemented on a number of test functions.

2.3.8 SPEA2: Improved Strength Pareto Evolutionary Algorithm

Zitzler et al. (2001) improves the original SPEA (Zitzler and Thiele, 1998), and

addresses some potential weaknesses of SPEA.

SPEA2 uses an initial population and an archive (external set). At the start,

random initial and archive population with fixed sizes are generated. The fitness

value of each individual in the initial population and archive is calculated per

iteration. Next, all non-dominated solutions of initial and external population are

18

copied to the external set of next iteration (new archive). With the environmental

selection procedure the size of the archive is set to a predefined limit. After wards,

mating pool is filled with the solutions resulted from performing binary tournament

selection on the new archive set. Finally, cross-over and mutation operators are

applied to the mating pool and the new initial population is generated. If any of

the stopping criteria is satisfied the non-dominated individuals in the new archive

forms the pareto-optimal set.

Kim et al. (2004) adds two new mechanisms to SPEA2 in order to improve the

searching ability of algorithm. The SPEA2 + algorithm, as it is named, uses a

more effective crossover (Neighborhood Crossover) and new archive mechanism to

diversify the solutions in both objective and variable spaces.

Kukkonen (2006) introduces a pruning method, which can be used to improve the

performance of SPEA2. The idea of pruning is to reduce the size of a set of non-

dominated solution to a pre-defined limit, while the maximum possible diversity

is encountered.

2.3.9 Overall SPEA2 Algorithm

The overall algorithm of SPEA2 is as follows:(Zitzler et al., 2001)

Algorithm (SPEA2 Main Loop)

Input : N (population size)

N (archive size)

T (maximum number of generations)

Output: A (non-dominated set)

Step 1: Initialization: an initial population P0 and archive (external set) P 0 is

generated. Set t = 0.

Step 2: Fitness assignment : Fitness values of individuals in Pt and P t are calcu-

lated. (Fitness Assignment section)

Step 3: Environmental selection: All non-dominated individuals in Pt and P t

shall be copied to P t+1. If size of P t+1 exceeds N , reduction of P t+1 is

achieved by means of the truncation operator, otherwise P t+1 is filled with

19

dominated individuals in Pt and P t, if size of P t+1 is less than N . (Environ-

mental Selection)

Step 4: Termination: If t > T or another stopping criterion is satisfied then, the

non-dominated individuals in P t+1 creates the output set A.

Step 5: Mating Selection: Binary tournament selection with replacement is per-

formed on P t+1 in order to fill the mating pool.

Step 6: Variation: Recombination and mutation operators shall be applied to

the mating pool and the resulting population is set to Pt+1 . Increment

generation counter (t = t+ 1) and go to Step 2.

Fitness Assignment

Each individual i in the archive P t and the population Pt is assigned a strength

value S(i), representing the number of solutions it dominates:

S(i) = |j|j ∈ Pt + P t ∧ i j||,

the raw fitness R(i) of an individual i is calculated:

R(i) =∑

j∈Pt+P t,ji

S(j).

The density estimation technique is adopted from the k-th nearest neighbor method

(Silverman 1986), where the density at any point is a (decreasing) function of the

distance to the k-th nearest data point. In SPEA2 the inverse of the distance to

the k-th nearest neighbor is considered as the density measurement. The density

D(i) corresponding to i is defined by:

D(i) =1

σki + 2,

where,

k =√N +N.

20

and σki is the distance of solution i to the k-th nearest neighbour. Finally, the

fitness of an individual i is calculated by adding D(i) to the raw fitness value

R(i):

F (i) = R(i) +D(i)

Environmental Selection

The first step is to copy all non-dominated individuals, i.e., the ones with fitness

value lower than one, from archive and population to the external set of the next

generation:

P t+1 = i|i ∈ Pt + P t ∧ F (i) < 1.

If the size of non-dominated solutions is exactly the same as archive size (|P t+1| =N) the environmental selection step is completed. Otherwise, there can be two

situations:

• The archive is too small (|P t+1| < N): The best N − |P t+1| dominated

individuals in the previous external set and population are copied to the

new archive. This can be achieved by sorting the multi-set P t + P t+1 from

lowest to highest fitness values and copy the first N − |P t+1| individuals i

with F (i) > 1 from the sorted list to P t+1 .

• The archive is too large (|P t+1| > N): In this case, an archive truncation

procedure is invoked which iteratively removes individuals from P t+1 until

(|P t+1| = N) . Here, at each iteration the solution i is chosen for removal

for which i ≤d j for all j ∈ P t+1 with:

i ≤d j :⇔ ∀ 0 < k < |P t+1| : σkj = σki ∨

∃ 0 < k < |P t+1| :[(∀ 0 < l < k : σli = σlj

)∧ σki < σkj

],

where σki denotes the distance of i to a user-predefined (k-th) nearest neigh-

bor in P t+1 . In other words, at each stage removed solution will be the one

with the least distance to the k-th neighbor; if there is more than one solu-

tion with the same distance the judgement will be upon the second smallest

distance and so forth.

21

In appendix B, hand calculation and step by step simulation of a simple

example minimization problem is fully described. This will help on better

understanding of algorithm and the working principle of each step.

2.3.10 Constraint Handling

Handling constraints within MOEAs is an essential issue which must be consid-

ered carefully, especially when dealing with real world engineering applications

where constraints are always involved. Constraints can be in form of equality or

inequality. Another classification of constraints are hard and soft constraints. A

hard constraint is a must to be satisfied, while on the other hand, a soft one can

be relaxed in order to accept a solution (Coello, 2007; Deb, 2000). Normally only

inequality constraints are handled in MOEAs, however equality constraints can

be easily transformed to inequality using:

|h(x)| − ε 6 0

where h(x) = 0 is the equality constraint and ε is very small value.

Constraints divide the decision space into two separate parts: feasible and infea-

sible regions. A solution in the feasible region of search space satisfies all the

constraints and it is called a feasible solution, otherwise the solution is infeasible.

The most popular and common way of handling constraints is the penalty function

method. However sensitivity of penalty method to the penalty parameter is a

drawback in this method.

In addition to penalty method, Jimnez et al. (1999) proposed a systematic con-

straint handling procedure. Two other method which are more credited and elab-

orated are the Ray-Tai- Seows constraint handling approach (T. Ray, 2001) and

the Deb et al. (2002) proposed constraint handling method, which is implemented

in NSGA II algorithm.

In Debs method a binary tournament selection operator is used for any two so-

lutions selected from the population. Therefore in presence of constraints three

scenarios will occur: 1) Both solutions are feasible; 2) One is feasible and the other

is infeasible; 3) Both solutions are infeasible. In the method for each scenario fol-

lowing rule is applied:

22

• Scenario 1) the solution with better objective function is selected (Crowded

comparison).

• Scenario 2) the feasible solution will win the tournament.

• Scenario 3) the solution with less constraint violation is selected.

Deb modifies the definition of domination as solution i constraint dominates solu-

tion j, if any of the following conditions is true:

1. Solution i is feasible and j not.

2. Solution i and j are both infeasible, but solution i has a smaller overall

constrained violation.

3. Solution i and j are both feasible and solution i dominates solution j.

In the SPEA2 algorithm proposed for the thesis, binary tournament is applied to

the archive population (P t+1), which holds the non-dominated individuals to create

the mating pool. Afterwards the genetic operators are used to generate the child

from the mating pool which is the initial population of next generation (Pt+1). In

constrained problems the modified definition of domination is implemented and the

non-dominated solutions are selected according to constraint domination concept.

Appendix B.1 simulates the principle and procedures of constraint domination

concept on a simple problem.

23

2.4 Hybrid Multi-Objective Optimization Approach

In real world engineering problems there is no prior knowledge on the true global

pareto-front. Although Evolutionary algorithms have shown a good convergence

in benchmarks, hybrid methods have been proposed to ensure the convergence of

an algorithm to the true pareto-front. Several hybridization techniques (combining

an MOEA with other methods) are discussed in literatures.

Coello (2007) comprehensively deliberate the use of local search and co-evolutionary

techniques as a hybrid method in a complete chapter of his book. He specifies local

search decision space approaches such as depth-first search (hill-climbing), simu-

lated annealing and Tabu search for consideration in hybridization.

Deb (2001) also argues the use of local search techniques with an MOEA. Accord-

ing to Goldberg, the best way to achieve convergence to the exact pareto-front is

implementing the local search techniques on the solutions obtained from an EA.

However Deb proposes two other ways to use local search techniques; 1) during

EA generations, 2) at the end of an EA run.

Here a new method of hybridization is introduced and tested on benchmarks to

investigate the performance of the technique. A combination of single and multi-

objective optimization evolutionary algorithms discussed in previous subsections

are applied to obtain the global optimal solutions. The archive population in

SPEA2, which holds the non-dominated solutions of each generation, is created

using the single-objective genetic algorithm optimization method introduced in

earlier sections called G3 algorithm.

First, the objectives are transformed to a single objective function by using the

weighted sum method. A number of random weights equal to size of population

are multiplied to each objective to scalarize the objective function. Then, every

scalarized function is optimized with G3 single objective GA. After finding the

optimal solution for each weighted function, the required initial population is

obtained. Finally, the multi-objective algorithm (SPEA2) is used to optimize the

function. Therefore, the hybridizing technique is applied before EA generations

to create the required initial archive population.

24

3 Implementation

All the algorithms in the thesis are coded with MATLAB. Several benchmarks are

encompassed and solved with the coded algorithms to ensure the accuracy and

efficiency of algorithm.

3.1 Single Objective

3.1.1 Unconstrained Test Functions

Sphere Function

fSphere(x) =n∑i=1

x2i

has a global minimum of 0 at x∗ = (0, 0)T .

Ellipsoidal Function

The behaviour of the algorithms for a poorly scaled objective function is discussed

using the following objective function:

fEllipsoid(x) = ax21 +

n∑i=1

x2i

where a is a positive parameter. If a = 1, the function has a valley structure. In

this experiment, a = 0.01 is used. The global minimum is 0 at x∗i = 0, i = 1, ..., n.

Schwefel’s Function

fSchwefel(x) =n∑i=1

(i∑

j=1

xi

)2

the global minimum is 0 at xi = 0.

25

Goldstein-Price Function

The Goldstein-Price function is given by:

fGoldstein(x1, x2) = (1 + (x1 + x2 + 1)2)(19− 14x1 + 3x21 − 14x2 + 6x1x2 + 3x2

2))

×(30 + (2x1 − 3x2)2(18− 32x1 + 12x21 + 48x2 − 36x1x2 + 27x2

2))

has a global minimum of 3 at x∗ = (0,−1)T . The typical search range is −2 ≤xi ≤ 2, i = 1, 2.

Rosenbrock Function

This function is used to discuss the behaviour of the algorithms for functions

having complex non-separable structure, such as a curved, deep valley, given by

fRosenbrock(x) =n∑i=2

(100(x21 − xi)2 + (1− xi)2),

and has a global minimum of 0 at x∗i = 1. The typical search range is −5.12 ≤xi ≤ 5.12, i = 1, ..., n.

Colville Function

The Colville function is defined as

fColville(x1, x2, x3, x4) = 100(x2 − x21)2 + (1− x1)2 + 90(x4 − x2

3)2 + (1− x3)2

+10.1((x2 − 1)2 + (x4 − 1)2) + 19.8(x2 − 1)(x4 − 1).

The search range is −10 ≤ xi ≤ 10 and the global minimum of f ∗Colville = 0 at

x∗i = 1, i = 1, ..., 4.

Considering the results of systematic studies on parameters of G3 algorithm (Deb,

2004), in all above cases, a population size of N = 100, a parent size µ = 3, number

of offspring λ = 2 and r = 2 (Step 2 and 3) are used. For the PCX different values

of ση and σζ is implemented.

In addition, for Spherical, Ellipsoidal, Schwefel and Rosenbrock functions two cases

are considered for initial population:

26

• Normal Case: the distribution of initial population is surrounding the

optimal solution. The population is generated by uniform random numbers

in the region below:

−1 < xi < 1, i = 1, ..., n

• Offset Case: the distribution of initial population is faraway from the op-

timal solution.The population is generated by uniform random numbers in

the region below:

−10 < xi < −5, i = 1, ..., n

3.1.2 Constrained Test Functions

Three constrained optimization problems, two of them real engineering problems,

is used to evaluate the performance of the G3 algorithm and selected constraint

handling method.

Test Function 1

This test problem is a two dimensional constrained optimization problem:

Minimize f(x) = (x21 + x2 − 11)2 + (x1 + x2

2 − 7)2,

Subject to g1(x) = 4.84− (x1 − 0.05)2 − (x2 − 2.5)2 ≥ 0,

g2(x) = x21 + (x2 − 2.5)2 − 4.84 ≥ 0,

0 ≤ x1, x2 ≤ 6.

The optimal solution is x∗ = (2.246826, 2.381865) with a function value equal to

f ∗ = 13.59085.

Welded Beam Design

The objective is to minimize the cost of the welded beam subject to the constraints

on shear stress (τ), bending stress in the beam (σ), bucking load on the bar (Pc),

end deflection of the beam (δ), and side constraints. The problem has four design

variables h(x1), l(x2), t(x3), b(x4) and five inequality constraints as follows:

27

Minimize f(x) = 1.1047x21x2 + 0.04811x3x4(14.0 + x2),

Subject to g1(x) = τMAX − τ(x) ≥ 0,

g2(x) = σMAX − σ(x) ≥ 0,

g3(x) = x4 − x1 ≥ 0,

g4(x) = Pc(x)− P ≥ 0,

g5(x) = δMAX − δ(x) ≥ 0,

0.1 ≤ x1, x4 ≤ 2,

0.1 ≤ x2, x3 ≤ 10

where

τ(x) =

√((τ ′(x))2 + (τ ”(x))2 + x2τ

′(x)τ ”(x))/√

0.25 [x22 + (x1 + x3)2],

σ(x) =6PL

x23x4

,

δ(x) =4PL3

Ex23x4

,

Pc(x) =4.013E

√x23x

64

36

L2

[1− x3

2L

√E

4G

],

where

τ′(x) =

P√2x1x2

, τ ”(x) =MR

J

′,

M = P[L+

x2

2

], R =

√x2

2

4+ (

x1 + x3

2)2.

P = 6000 lb, L = 14 in, E = 30× 106 psi, G = 12× 106 psi,

τMAX = 13600 psi, σMAX = 30000 psi, δMAX = 0.25 in.

28

The optimized solution reported in literature (Deb, 1991) is x = (0.2489, 6.1730, 8.1789, 0.2533)

with f = 2.43 using binary GA.

Figure 3: The welded beam problem.

Minimization of the Weight of a Tension/Compression Spring

The problem consists of minimizing the weight of a tension/compression spring

subject to constraints on minimum deflection, shear stress, surge frequency, limits

on outside diameter and on design variables. The design variables are the mean

coil diameter D(x2), the wire diameter d(x1) and the number of active coils N(x3).

The problem can be expressed as follows:

Minimize f(x) = x21x2(x3 + 2),

Subject to g1(x) =(x3

2x3)

(71785x41)− 1,

g2(x) = 1− 4 ∗ x22 − x1 ∗ x2

12566 ∗ x31 ∗ (x2 − x1)

− 1

5108 ∗ x21

,

g3(x) =140.45 ∗ x1

xx3 ∗ x22

− 1,

g4(x) = 1− x1) + x2

1.5− 1,

0.05 ≤ x1 ≤ 2,

0.25 ≤ x2 ≤ 1.3,

2 ≤ x2 ≤ 15.

29

The best optimal solution obtained by using static penalty function is f ∗ =

0.012729 and the lowest optimal found by (Mezura-Montes et al., 2003) is f ∗ =

0.012688 using an approach based on Evolution Strategy.

Figure 4: Tension/compression string problem.

3.2 Multi objective

Similar to single objective, two sets of test functions, one for unconstrained and

the other for constrained, are utilized to assess the performance of SPEA2 and the

proposed constrained handling method. The Algorithm is coded with MATLAB.

3.2.1 Unconstrained Test Functions

Exercise 14

A non-convex function presented in Stromberg (2011) with one variable.

Minimize

f1(x) = 1− 2x+ x2,

f2(x) =√x,

0 ≤ x ≤ 1.

The pareto-front is obtained by using two different methods; 1) Single objective

G3 algorithm (the function is transformed to single-objective by weighted sum

method), 2) Multi-objective SPEA2 algorithm.

30

Kursawe’s Test Function

Kursawe (1991) used a complicated two-objective, three variable function with a

non-convex and disconnected pareto-optimal set.

KUR:

Minimize F = (f1(x), f2(x)) ,

f1(x) =∑n−1

i=1

(−10e−0.2

√x2i+x

2i+1

),

f2(x) =∑n

i=1

(|xi|0.8 + 5 sin3

i

).

−5 ≤ xi ≤ 5, i = 1, 2, 3.

Deb (2001) illustrates the pareto-front of KUR function in figure 201. Three

distinct disconnected regions create the pareto-front of the problem. Also figure

202 of the same book shows the pareto-optimal solutions in decision space.

Zitzler-Deb-Thiele’s Test Functions

Zitzler et al. (2000) introduced six set of multi-objective problems (ZDT1 to ZDT6)

which are based on a unique structure with different level of difficulties. The

functions have two objective with the aim of minimization:

ZDT:


f1(x),

f2(x) = g(x)h(f1(x), g(x)).

In the thesis five of the six test problems (ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6)

are implemented with SPEA2 algorithm.

ZDT1 Test Function

ZDT1 is a function with 30 variables and convex pareto-optimal set as follows:

ZDT1:


f1(x) = x1,

f2(x) = g(x)(

1−√f1/g(x)

),

g(x) = 1 + 9∑n

i=2

xin− 1

.

31

All the variables are limited between 0 and 1. Figure 213 of Deb (2001) shows

the search space and pareto-front in objective space. This is the easiest among all

ZDT’s and the only difficulty is the large number of variables.

ZDT2 Test Function

Another 30-variable test function with a non-convex pareto-front:

ZDT2:


f1(x) = x1,

f2(x) = g(x)(1− (x1/g(x))2) ,

g(x) = 1 +9

n− 1

∑ni=2 xi.

The range for all the variables is [0, 1]. Pareto-front and the search region in

objective space is shown in figure 214 of Deb (2001). Non-convexity of pareto-

optimal set is the only difficulty of this problem.

ZDT3 Test Function

ZDT3 problem with 30 variables, has a number of disconnected pareto-optimal

sets:

ZDT3:


f1(x) = x1,

f2(x) = g(x)(

1−√f1/g(x)

)− (f1/g(x)) sin (10πf1) ,

g(x) = 1 + 9∑n

i=2

xin− 1

.

All the variables are limited within [0, 1]. Finding all the discontinuous pareto-

optimal regions with a good diversity of non-dominated solutions may be difficult

for an MOEA. Deb (2001) show the search space and pareto-front in figure 215.

32

ZDT4 Test Function

This is a 10 variable problem with a convex pareto-front:

ZDT4:


f1(x) = x1,

f2(x) = g(x)(

1−√x1/g(x)

),

g(x) = 1 + 10 (n− 1) +∑n

i=2 (x2i − 10 cos (4πxi)) .

All the variables except x1, which lies in the range [0, 1], are limited within −5

and 5. Large number of multiple local pareto-fronts, shown in figure 216 of Deb

(2001), will create a difficult convergence to global pareto-front for an MOEA.

ZDT6 Test Function

This is a problem with 10 variables and a non-convex pareto-optimal set:

ZDT6:


f1(x) = 1− exp(−4x1) sin6(6πx1),

f2(x) = g(x)(1− (f1/g(x))2) ,

g(x) = 1 + 9

(∑ni=2

xin− 1

)1/4

.

All the variables lie in the range [0, 1]. Non-convexity of pareto-front, coupled with

adverse density solutions across the front, may rise some difficulty in convergence.

Figure 218 in Deb (2001) shows the pareto-optimal region for this problem.

3.2.2 Constrained Test Functions

The presence of constraints may cause hurdles for an MOEA to converge to the true

and global pareto-front, also maintaining diversity in the non-dominated solutions

may be another problem. A number of common test problems used in literatures,

are presented in this section and implemented in the SPEA2 code.

33

Binh and Korn Test Function

Binh and Korn (1997) introduced a problem with two-variable as follows:

BNH:

Minimize f1(x) = 4x21 + 4x2

2,

Minimize f2(x) = (x1 − 5)2 + (x2 − 5)2,

subject to C1(x) = (x1 − 5)2 + x22 ≤ 25,

C2(x) = (x1 − 8)2 + (x2 + 3)2 ≥ 7.7,

0 ≤ x1 ≤ 5,

0 ≤ x2 ≤ 3.

Deb (2001) illustrates the decision variable and objective space of the problem in

figures 219 and 220. In BNH problem, constraints will not add any difficulty to

the unconstrained problem.

Osysczka and Kundu Test Function

Osyczka and Kundu (1995) used the following six variable test function:

OSY:

Minimize f1(x) = −[25(x1 − 2)2 + (x2 − 2)2 + (x3 − 2)2 + (x4 − 2)2

+(x5 − 2)2],

Minimize f2(x) = x21 + x2

2 + x23 + x2

4 + x25 + x2

6,

subject to C1(x) = x1 + x2 − 2 ≥ 0,

C2(x) = 6− x1 − x2 ≥ 0,

C3(x) = 2 + x1 − x2 ≥ 0,

C4(x) = 2− x1 + 3x2 ≥ 0,

C5(x) = 4− (x3 − 3)2 − x4 ≥ 0,

C6(x) = (x5 − 3)2 + x6 − 4 ≥ 0,

0 ≤ x1, x2, x6 ≤ 10,

1 ≤ x3, x5 ≤ 5, 0 ≤ x4 ≤ 6.

The pareto-front as shown in figure 221 of Deb (2001), is a line connecting some

parts of five different region. Since the algorithm should maintain the solutions

within intersections of constraint boundaries, this is a difficult problem to solve.

34

Srinivas and Deb Test Function

Srinivas and Deb (1994) suggested the following problem:

SRN:

Minimize f1(x) = 2 + (x1 − 2)2 + (x2 − 1)2,

Minimize f2(x) = 9x1 − (x2 − 1)2,

subject to C1(x) = x21 + x2

2 ≤ 225,

C2(x) = x1 − 3x2 + 10 ≤ 0,

−20 ≤ x1, x2 ≤ 20.

Since the constraints eliminate some parts of the original pareto-front, difficulties

may arise in solving the problem. Figures 222 and 223 in Deb (2001) shows the

corresponding pareto-front of feasible decision variable and objective space.

Tanaka Test Function

Tanaka and Watanabe (1995) proposed a two variable test function as follows:

TNK:

Minimize f1(x) = x1,


subject to C1(x) = x21 + x2

2 − 1− 0.1 cos

(16 arctan

x1

x2

)≥ 0,

C2(x) = (x1 − 0.5)2 + (x2 − 0.5)2 ≤ 0.5,

0 ≤ x1, x2 ≤ π.

The pareto solutions lie on a surface which is non-linear. Therefore optimization

algorithms may face some difficulties in finding a diverse set of feasible pareto

solutions. A figure of feasible decision variable spaces for the problem can be seen

in Deb (2001). (Figure 224)

35

3.3 Hybrid Approach

In order to test the performance of the proposed hybrid method, the archive pop-

ulation generated from the hybrid technique (weighted sum single objective G3

algorithm) is compared with the random archive population for different test prob-

lems. It is obvious that the test functions with non-convex pareto-fronts are not

suitable for the technique, since the weighted sum method is used to transform

the objectives into one objective. Therefore, the random and hybrid archive pop-

ulation are plotted in objective space for ZDT1 and ZDT3 test functions. Also

despite the non-convex property of ZDT6 problem, comparison of archive popu-

lation has also been applied to this problem to study the performance of hybrid

method on non-convex benchmarks.

36

4 Test Results

4.1 Single Objective

4.1.1 Unconstrained Functions

To examine the behaviour of algorithm and code, evaluation in two and multi-

dimensional search space is carried out for some of the test functions as blow:

• Spherical: n = 2, 4

• Ellipsoidal: n = 2, 4

• Schwefel: n = 2, 10, 15

• Rosenbrock: n = 2, 5

Goldstein function is by default in two dimensional search space and Colville is a

four variable function.

Table 1: Results of unconstrained test functions, single objective.

Function Initial Number of Number of Variance from ση,Population Variable Evaluation Global Optimum σζ

Sphere Normal 2 98 7.32× 10−6 0.1Sphere Normal 2 67 5.47× 10−6 0.4Sphere Normal 4 279 3.54× 10−6 0.4Sphere Offset 2 261 3.65× 10−6 0.1

Ellipsoidal Normal 2 99 3.75× 10−6 0.1Ellipsoidal Offset 2 405 3.65× 10−6 0.1Ellipsoidal Offset 4 553 8.10× 10−6 0.4Schwefel Normal 2 553 8.10× 10−6 0.1Schwefel Offset 10 553 8.10× 10−6 0.3Schwefel Offset 15 6434 7.76× 10−6 0.3

Rosenbrock Normal 2 146 3.55× 10−6 0.1Rosenbrock Offset 5 2200 3.9308× 10−6 0.5Rosenbrock Offset 5 5095 3.55× 10−6 0.9Goldstein - 2 211 9.46× 10−6 0.1Colville - 4 no result no result 0.1Colville - 4 1055 9.46× 10−6 0.3Colville - 4 1468 8.09× 10−6 0.6

37

The experiment for each function runs until the best objective function of the pop-

ulation reaches a minimum difference of 10−5 from the optimal solution. Number

of Generation (evaluation) and best fitness are shown in table 1.

The result shows acceptable behaviour of algorithm for two dimensional search

space with ση = 0.1 and σζ = 0.1 , but when the number of variable increases

or functions are more complex such as Rosenbrock, the algorithm converges in

local optima or the global optima is obtained with high number of generations.

Therefore by increasing the variance of zero-mean normally distributed variables

in PCX operator better results are obtained as it can be seen in table of results

the Schwefel’s function with 15 variable has reached the required variance from

global optimal. Convergence of the best individual obtained from some of the test

functions during generations can be seen in the figures 5 and 6.

Figure 5: Convergence of Variance from optimal solution for Schwefel’s functionwith 15 variables and optimal f ∗ = 0

Figure 6: Convergence of Variance from optimal solution for Rosenbrock functionwith 5 variables and optimal f ∗ = 0

38

4.1.2 Constrained Functions

The Penalty method used for constrained test functions shows a good behaviour.

All three functions reached the optimal solution reported in literature, in ad-

dition the optimal solution found by the algorithm in this thesis with related

constraint handling method is better than some other approaches used in liter-

ature. In the first test function the optimal solution of 13.590842 is found at

x∗ = (2.246818, 2.381735) which is better than the optimal found at literature

(Deb, 2000) with the value f ∗ = 13.59085. Furthermore, figures 7, 8 and 9 illus-

trate the convergence of results for the three constrained test functions.

Tables 2 and 3 compare the results of different methods for welded beam de-

sign problem and the minimization of the weight of a tension/compression spring,

which shows the out-performance of the approach used here to some methods.

Figure 7: Convergence of objective function for Test Function 1 with obtainedoptimal of f ∗ = 13.590842

Figure 8: Convergence of objective function for welded beam problem with ob-tained optimal of f ∗ = 1.834756

39

Table 2: Comparison of the results of different methods for welded beam designproblem.

Method f ∗(x)This Thesis 1.83475678

Coello (self-adaptive penalty approach) 1.74830941Arora (constraint correction at constant cost) 2.43311600

He and Wang (CPSO) 1.728024Ragsdell and Phillips (Geometric programming) 2.385937

Deb (GA) 2.433116Coello and Montes (feasibility-based tournament selection) 1.728226

Ebehart (modified PSO) 1.72485512

Figure 9: Convergence of objective function for Tension/Compression Spring withobtained optimal of f ∗ = 0.012710175

Table 3: Comparison of the results of different methods for the minimization ofthe weight of a tension/compression spring.

Method f ∗(x)This Thesis 0.012710175

Coello (self-adaptive penalty approach) 0.01270478Arora (constraint correction at constant cost) 0.12730274

He and Wang (CPSO) 0.0126747Belegundu (numerical optimization technique) 0.0128334

Coello and Montes (feasibility-based tournament selection) 0.0126810Ebehart (modified PSO) 0.01266614

40

4.2 Multi-Objective

SPEA2 algorithm coded with MATLAB is used to solve multi-objective test func-

tions. Simulated Binary Crossover (SBX) and Polynomial Mutation (Deb, 2001)

are implemented in the step 6 (Variation) of SPEA2 algorithm as recombination

and mutation operators.

4.2.1 Unconstrained Functions

In exercise 14 test function, a weight vector with 20 weight factors within the range

[0, 1] with step length of 0.05 is used to create 20 single objective functions and

each function is optimized separately to obtain the pareto-front shown in figure

10 . In multi-objective method, the initial and archive set with population of 30

individuals after 100 generations result in the pareto-front (figure 11).

Figure 10: Pareto-front of Exercise 14 using weighted single objective algorithm

Figure 11: Pareto-front of Exercise 14 using weighted single objective algorithm

41

Since the function is non-convex the pareto-front obtained from single objective

method does not cover the non-convex parts of pareto-optimal set. In the other

hand, the pareto-front of multi-objective method clearly illustrates all parts of

pareto-optimal region. Thus, an important drawback of single-objective weighted

sum method for solving multi-objective optimization problems is the weakness in

non-convex problems.

Table 4 shows the defined parameters of SPEA2 algorithm, such as size of initial

and archive population and number of generations, for the other test functions

from Kursawe to ZDT6.

Table 4: Pre-defined parameters of SPEA2 algorithm for unconstrained multi-objective test functions

Test Function Initial Population Archive Population Number ofSize (N) Size (N) Generations

Kursawe 50 50 100ZDT1 50 50 400ZDT2 50 50 400ZDT3 50 50 250ZDT4 100 100 250ZDT6 100 100 250

Figures 12, 13, 14, 15, 16 and 17 shows the non-dominated solutions obtained

from SPEA2 algorithm for problems Kursawe, ZDT1, ZDT2, ZDT3, ZDT4 and

ZDT6. Comparing the figures to the true pareto-fronts illustrated in literature

(Deb, 2001), confirms the excellent performance of SPEA2 algorithm in finding

the pareto-front of problems with upto 30 variables.

Figure 12: Pareto-front of Kursawe test function

42

Figure 13: Pareto-front of ZDT1 test function



Scatter-Plot Matrix Method for Representation of Non-Dominated So-

lutions

Throughout this thesis, all the multi-objective test functions contain two objec-

tives, thus the performance of an algorithm can be measured and illustrated with

representing the non-dominated solutions in a two-dimensional objective space

43

plot. However, illustrating the non-dominated solutions in a multi-objective prob-

lem with more than two objectives can be a difficult task. Even the 3D plot for

three objective problems, which each axes represents one objective, is confusing

and unhelpful.

There are number of methods for presenting problems with more than two ob-

jectives in literatures. Scatter-plot matrix is one way, which Meisel (1973) and

Cleveland (1994) suggest to plot all (M2 ) pairs of plots among the M objective

functions. Therefore, the non-dominated solutions of a problem with three ob-

jectives will be illustrated with 6 plots in a 5 × 5 matrix. Each diagonal plot is

used to mark the axis for the matching off diagonal plots. In this method the

non-dominated solutions in each pair of objective spaces are shown twice with the

difference in the axis marked for each objective.



44

The scatter plot matrix can also be used for comparison of two different algorithms

on an identical problem. The upper diagonal plots shows the non-dominated

solutions of one algorithm and lower diagonal plot is utilized to illustrate the

corresponding solutions of other algorithm.

Furthermore, in engineering applications the relation of variables with objective

functions and the non-dominated solutions in variable space is an imperative issue.

Investigating the variations of each variable of non-dominated solutions and the

effect of the variations to objective functions and other variables can be very helpful

in better understanding the optimized problem. Here, the scatter-plot matrix is

used to show these variations and their affects. For this purpose Kursawe, ZDT1,

ZDT2, ZDT3, ZDT4 and ZDT6 are optimized with three variables by using SPEA2

algorithm. In figures 18, 19, 20, 21, 22 and 23 each variable and the two objective

functions are marked in the diagonal plots of a 5×5 matrix for mentioned problems.

The off-diagonal plots clearly illustrate the non-dominated solutions in objective

and variable space.

45

Fig

ure

18:

Sca

tter

-plo

tm

atri

xof

Kurs

awe

test

funct

ion

46

Fig

ure

19:

Sca

tter

-plo

tm

atri

xof

ZD

T1

test

funct

ion

47

Fig

ure

20:

Sca

tter

-plo

tm

atri

xof

ZD

T2

test

funct

ion

48

Fig

ure

21:

Sca

tter

-plo

tm

atri

xof

ZD

T3

test

funct

ion

49

Fig

ure

22:

Sca

tter

-plo

tm

atri

xof

ZD

T4

test

funct

ion

50

Fig

ure

23:

Sca

tter

-plo

tm

atri

xof

ZD

T6

test

funct

ion

51

4.2.2 Constrained Functions

The constrained test functions are optimized by SPEA2 algorithm with the pre-

defined parameters shown in table 5. The non-dominated solutions obtained for

BNH, OSY, SRN and TNK problems are illustrated respectively in figures 24, 25,

26 and 27.

Table 5: Pre-defined parameters of SPEA2 algorithm for constrained multi-objective test functions

Test Function Initial Population Archive Population Number ofSize (N) Size (N) Generations

BNH 30 30 100OSY 30 30 600SRN 30 30 100TNK 30 30 100

Figure 24: Pareto-front of BNH test function

Figure 25: Pareto-front of OSY test function

52

Comparing the figures with the true pareto-fronts reported in literatures, proves

the good performance of algorithm in converging to optimal results with a good

diversity of solutions.

Figure 26: Pareto-front of SRN test function

Figure 27: Pareto-front of TNK test function

53

4.3 Hybrid Approach

Figures 28 and 29 illustrate the comparison of the two archive population for ZDT1

and ZDT3 test functions respectively.

Assessing the plots with the existing true pareto-optimal fronts in the literatures,

shows that the hybrid approach generates a population near the actual pareto-

front.

However, figure 30 which plots the two population of ZDT6 problem confirms the

fact that convexity of objective function has an important influence in diversity

and closeness of population to pareto-front.

Furthermore, the obtained hybrid populations are the outcome of a single objective

GA with relatively high number of generations. Consequently, the proposed hybrid

approach do not show any improvement in overall computation time of test func-

tions. However the number of generations to reach near the actual pareto-front,

accordingly the computation time in the multi-objective part of the algorithm

decreases.

Figure 28: Hybrid (left) and random (right) initial archive population for ZDT1.


54

More precise and reliable judgement can be made only after conduction of an

extensive research on convergence of optimization problems and introducing a

proper metric to compare the two approaches in a more scientific way.

Also, parameter setting in the hybrid method will have an important effect on

computation time. There exist a large number of parameters including size of

population in each algorithm, number of generations for single objective algorithm

and size of different sets used in the algorithm, which will have a great impact

in computation time. Nevertheless, creating a predefined archive population may

enhance convergence to the true pareto-front in an EA.


55

5 Conclusion

After implementing the proposed algorithm for single objective optimization test

functions, it was concluded that the approach showed a good performance in con-

verging to the true optimal solution. However parameter setting in problems with

higher number of variables is crucial. The penalty method used for constrained

handling managed to find the optimal solution for all three test functions. Also,

better behaviour was observed in comparison to some of the other techniques.

The SPEA2 algorithm, for multi-objective optimization problems, was applied on

several benchmarks and the obtained pareto-fronts were completely similar to the

fronts reported in literatures. Furthermore, the diversity and spread of solutions

along the pareto-optimal region appeared to be equally distanced and the non-

dominated solutions were uniformly distributed in all parts of pareto-front. The

constraint handling approach performed well on all test functions and the pareto-

fronts were exactly comparable to the true fronts illustrated in references. It is

recommended to extend the research on a real-world engineering application and

problems with more than two objectives with the aim of assessing the performance

of algorithm in different situations.

The scatter-plot matrix method for illustrating the non-dominated solutions, could

be very supportive in studying real world engineering problems, where understand-

ing the relations between variables and objectives are crucial.

The suggested hybrid approach did not show any advantages in overall computa-

tion time, and in some problems it can be considered as a weakness regarding this

issue. There is an essential need of comprehensive studies related to convergence

of optimization problems, comparison metrics and different ways of combining

single and multi-objective methods in order to conclude in a more precise and

scientific manner. It is believed that the hybrid method may improve the ability

of algorithm in finding the global optimal solutions.

56

6 Bibliography

Beasley, D. and Bull, D. R. (1993). An Overview of Genetic Algorithms : Part 1

, Fundamentals 1 Introduction 2 Basic Principles. Building, pages 1–16.

Binh, T. and Korn, U. (1997). MOBES: A multiobjective evolution strategy

for constrained optimization problems. The Third International Conference on

Genetic, 1(1).

Cleveland, W. S. (1994). The Elements of Graphing Data. Murry Hill,NJ: ATI&T

Bell Laboratories.

Coello, Carlos A., L. G. B. V. D. A. (2007). Evolutionary Algorithms for Solving

Multi-Objective Problems. Genetic and Evolutionary Computation. Springer

Science + Business Media, LLC, second edition edition.

Corne, D., Jerram, N., and Knowles, J. (2001). PESA-II: Region-based selection

in evolutionary multiobjective optimization. Genetic and Evolutionary.

Corne, D. and Knowles, J. (2000). The Pareto envelope-based selection algo-

rithm for multiobjective optimization. Problem Solving from Nature PPSN VI,

(Mcdm).

Deb, K. (1991). Optimal design of a welded beam structure via genetic algorithm.

AIAA.

Deb, K. (1995). Real-coded Genetic Algorithms with Simulated Binary Crossover

: Studies on Multimodal and Multiobjective Problems. Writing, 9:431–454.

Deb, K. (1999). Multi-objective genetic algorithms: problem difficulties and con-

struction of test problems. Evolutionary computation, 7(3):205–30.

Deb, K. (2000). An efficient constraint handling method for genetic algorithms.

Computer methods in applied mechanics and engineering, 186(2-4):311–338.

Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms.

John Wiley & Sons Ltd, The Atrium,Southern Gate, Chichester, West Sussex

PO19 8SQ, England.

Deb, K. (2002). A Computationally Effcient Evolutionary Algorithmfor Real-

Parameter Optimization. Evolutionary Computation, 10(4):371–395.

Deb, K. (2004). A population-based algorithm-generator for real-parameter opti-

mization. Soft Computing, 9(4):236–253.

57

Deb, K., Joshi, D., and Anand, A. (2001). Real-coded evolutionary algorithms with

parent-centric recombination. Technical report, KanGAL Report No. 2001003,

Kanpur Genetic Algorithms Laboratory (KanGAL), Indian Institute of Tech-

nology, Kanpur.

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). A fast and elitist

multiobjective genetic algorithm: Nsga-ii. IEEE Transactions on Evolutionary

Computation, 6(2):182–197.

Fonseca, C. and Fleming, P. (1993). Genetic algorithms for multiobjective op-

timization: Formulation, discussion and generalization. conference on genetic

algorithms, (July).

Gamot, R. and Mesa, A. (2008). Particle Swarm OptimizationTabu Search Ap-

proach to Constrained Engineering Optimization Problems. WSEAS Transac-

tions on Mathematics, 7(11):666–675.

Gen, M. and Cheng, R. (1997). Genetic algorithms and engineering design. New

York: Wiley.

Goldberg, Robert (ed.), A. A. e. J. L. (2005). Evolutionary Multiobjective Op-

timization: Theoretical Advances and Applications. Springer London Ltd, 1st

edition. edition 2005 edition.

Haimes, Y. Y., L. L. S. and A., W. D. (1971). On a bicriterion formulation of

the problems of integrated system identification and system optimization. IEEE

Transactions on Systems, Man and Cybernetics, 1(3)():296–297.

Haupt, R. L., Haupt, S. E., and Wiley, A. J. (2004). ALGORITHMS PRACTICAL

GENETIC ALGORITHMS. John Wiley & Sons, Inc., Hoboken, New Jersey.

Jimnez, F., Verdegay, J. L., and G’omez-skarmeta, A. F. (1999). Evolutionary

techniques for constrained multiobjective optimization problems.

Kim, M., Hiroyasu, T., Miki, M., and Watanabe, S. (2004). Spea2+: Improving

the performance of the strength pareto evolutionary algorithm 2. 3242:742–751.

Kita, H. (2001). A comparison study of self-adaptation in evolution strategies and

real-coded genetic algorithms. Evolutionary computation, 9(2):223–41.

Knowles, J. (1999). The pareto archived evolution strategy: A new baseline algo-

rithm for pareto multiobjective optimisation. Evolutionary Computation, 1999.

CEC.

58

Knowles, J. (2000). Approximating the nondominated front using the Pareto

archived evolution strategy. Evolutionary computation, pages 1–35.

Konak, a., Coit, D., and Smith, a. (2006). Multi-objective optimization us-

ing genetic algorithms: A tutorial. Reliability Engineering & System Safety,

91(9):992–1007.

Kukkonen, S. (2006). A fast and effective method for pruning of non-dominated

solutions in many-objective problems. Parallel Problem Solving from Nature-

PPSN IX, pages 1–20.

Kung, H. and Luccio, F. (1975). On finding the maxima of a set of vectors. Journal

of the ACM (JACM), (4):469–476.

Kuri-Morales, A. and A.Gutierrez-Garcia, J. (2002). Penalty function methods for

constrained optimization with genetic algorithms: A statistical analysis. MICAI

2002: Advances in Artificial Intelligence, pages 187–200.

Kursawe, F. (1991). A variant of evolution strategies for vector optimization.

Parallel Problem Solving from Nature.

Man, K. F., Tang, K. S., and Kwong, S. (1996). Genetic Algorithms : Concepts

and Applications. October, 43(5).

Meisel, W. L. (1973). Tradeoff decision in multiple criteria decision making. Mul-

tiple Criteria Decision Making, pages 461–476.

Mezura-Montes, E., Coello Coello, C., and Landa-Becerra, R. (2003). Engineer-

ing optimization using simple evolutionary algorithm. In Tools with Artificial

Intelligence, 2003. Proceedings. 15th IEEE International Conference on, pages

149–156. IEEE.

Morales, A. K., Quezada, C. V., Investigacion, C. D. E., Computacion, E. N.,

Batiz, J. D. D., and Lindavista, C. (1997). A Universal Eclectic Genetic Con-

strained Optimization Algorithm for Constrained Optimization. Optimization,

pages 2–6.

Osyczka, A. and Kundu, S. (1995). A new method to solve generalized multicri-

teria optimization problems using the simple genetic algorithm. Structural and

Multidisciplinary Optimization, 99(Goldberg 1989):94–99.

Srinivas, N. and Deb, K. (1994). Multi-objective function optimization using non-

dominated sorting genetic algorithms. Evolutionary Computation, 2(3):221–248.

59

Stromberg, N. (2011). Nonlinear FEA abd Design Optimization for Mechanical

Engineers. Jonkoping University, 2nd edition.

T. Ray, K. Tai, a. C. S. (2001). An evolutionary algorithm for multiobjective

optimization. Engineering and Optimization, 33(3):399–424.

Tanaka, M. and Watanabe, H. (1995). GA-based decision support system for

multicriteria optimization. Systems, Man and.

Zitzler, E., Deb, K., and Thiele, L. (2000). Comparison of multiobjective evolu-

tionary algorithms: Empirical results. Evolutionary computation, 8(2):173–195.

Zitzler, E., Laumanns, M., and Thiele, L. (2001). SPEA2: Improving the strength

Pareto evolutionary algorithm. Computer Engineering, pages 1–21.

Zitzler, E. and Thiele, L. (1998). An evolutionary algorithm for multiobjective

optimization: The strength pareto approach. Technical report, Zurich, Switzer-

land: Computer Engineering and Network Laboratory (TIK), Swiss Federal

Institute of Technology (ETH).

60

Appendices

A Hand Calculation of G3 Algorithm with Con-

straints

A simple real world single objective with constraint optimization problem is pre-

sented and solved by the proposed G3 algorithm and related constraint handling

method for one generation run.

A car spare part manufacturing company manufactures disk brakes and brake

pads. A disk brake takes 8 hours to manufacture and 2 hours to finish and pack.

A brake pad takes 2 hours to manufacture and 1 hour to finish and pack. The

maximum number of labour-hours per day is 400 for the manufacturing process

and 120 for the finishing and packing process. If the profit on a disk brake is

90Euro and the profit on a brake pad is 25Euro, how many disk brakes and brake

pads should be made each day to maximize the profit (assuming that all of the

disk brakes and brake pads can be sold)?

Therefore the objective is to maximize the profit by maximizing:

Profit = 990x1 + 25x2

Where x1 is the number of disk brakes and x2 is the number of brake pads. The ob-

jective can be transformed to a minimization problem by multiplying the objective

function by −1. The constraints are labour hours for each product:

8x1 + 2x2 ≤ 400

2x1 + x2 ≤ 120

The overall optimization problem is formulated by:

Spare part company:

Minimize −90x1 − 25x2,

subject to 8x1 + 2x2 − 400 ≤ 0,

2x1 + x2 − 120 ≤ 0,

x1, x2 ≥ 0.

The G3 algorithm has four steps (plans): 1) Selection plan, 2) Generation Plan,

61

3) Replacement plan, 4) Update plan. A population of random solutions named

set B, with 10 individuals is created. Table 6 shows set B and the initial random

solutions. Other pre-defined parameters are:

µ = 3, λ = 2, r = 2.

Table 6: Random initial solutions for hand calculation of G3 algorithm exampleproblem (profit of spare part manufacturing company)

Solution x1 x2 Fitness1 64.56 20.46 109

2 77.39 28.45 109

3 26.70 63.96 −4002.474 19.05 11.10 −1992.255 97.83 8.99 109

6 80.16 74.53 109

7 33.28 34.16 −3849.328 33.97 62.01 5× 108

9 21.43 48.04 −3129.6910 68.42 75.20 109

Step 1: The best solution in set B and µ − 1 other random solutions create set

P . First, the best solution, with regards to its fitness, is chosen. Thus, the

fitness (value of objective function) of each individual in set B has to be

calculated. Since a constraint is involved, the fitness is assigned according

to the proposed constraint handling method:

Step C1: The feasibility or infeasibility of each solution is inspected. Feasi-

ble solutions are 3, 4, 7, 9 and infeasible solutions are 1, 2, 5, 6, 8, 10.

Step C2: Number of non-violated constraints of each infeasible individual

is counted.

Step C3: Fitness of feasible solutions is the value of objective function. For

example solution 3 is feasible, therefore:

F (3) = −4002.47

Step C4: Fitness of infeasible solutions are calculated by:

F (x) = K −s∑i=1

[K

m

],

62

where K is a large enough pre-defined penalty factor (K = 109), s

is the number of constraints (s = 2) and m is the number of non-

violated constraints. Solution 8, is infeasible and only satisfies the first

constraint, therefore:

m = 1, F (8) = 5× 108.

By comparing the fitnesses, solution 3 has the minimum fitness value.

We assume that solutions 2 and 5 are the other µ−1 random solutions.

Therefore, set P = 2, 3, 5 is created.

Step 2: λ (λ = 2) offspring are created from the chosen three parents in set

P with PCX crossover.

Step PCX1: Mean vector of the three parents are calculated:

~g = [67.31, 33.80]

Step PCX2: For each offspring one parent is randomly selected from

set P . For instance, the first randomly selected parent is solution

2. The direction vector is obtained:

~d = ~g − ~x = [10.08,−5.35]

Step PCX3: The µ − 1 perpendicular distances from the two other

parents in set P to the direction vector (~d) is calculated and their

average is found:

D = [7.62, 7.62], D = 7.62

Step PCX4: The µ − 1 orthonormal basis that span the subspace

perpendicular to ~d is obtained:

~e = [0.47, 0.88]

Step PCX5: Zero-mean normally distributed variables with variance

63

w2ζ = 0.5 and w2

η = 0.51 are:

wη = 0.17, wζ = 0.17.

Step PCX6: The offspring is created by:

~y = ~x(p) + wζ ~d(p) +

µ∑i=1,i 6=p

wηD~e(i)

Therefore corresponding offspring to the first parent (solution 2)

is:

Parent1 = [77.39, 28.45], Offspring1 = [81.28, 27.84].

By assuming that the second random parent is solution 5, the re-

sulted offspring is:

Parent2 = [97.83, 8.99], Offspring2 = [100.67, 1.41].

The two (λ) offspring create set C = Offspring1,Offspring2.

Step 3: Solutions 2 and 7 (r solutions) are randomly selected from set B

to create set R = 2, 7.

Step 4: Set R ∪ C = 2, 7,Offspring1,Offspring2 is generated.

The solutions are arranged in ascending order with respect to their

fitness2, therefore the arranged set is:

R ∪ Csorted = Offspring2, 2, 7,Offspring1

The two randomly selected solutions in step 3, solutions 2 and 7, are re-

placed with the first r (r = 2) solutions from R∪Csorted, in set B. There-

fore set B is modified to Bnew = 1, offspring2, 3, 4, 5, 6, 2, 8, 9, 10 and

the first generation is completed. Next generation starts with the new

set Bnew, from step 1.

1The value of variance is selected according to the desired distance of offspring from parent.In other words, higher values of variances increases the distance of offspring from parent, whereasa small variance creates an offspring close to the parent.

2The fitness of the two newly created offspring is calculated according to the same proceduredescribed in steps C1 to C3 considering the feasibility or infeasibility of offspring.

64

B Hand Calculation of SPEA2 Algorithm

A simple minimization type optimization example problem is defined. Sim-

ulation of the steps in SPEA2 and hand calculation of one generation is

described in this appendix.

A two-objective with two variable minimization problem introduced by Deb

(2001) is chosen to illustrate the function of SPEA2.

Min-Ex:


Minimize f2(x) =x2 + 1

x1

,

subject to 0.1 ≤ x1 ≤ 1,

0 ≤ x2 ≤ 5.

This problem unlike the simple look, has two conflicting objective which

create a convex pareto-front as shown in figure 31. The search space is also

illustrated in the figure.

Figure 31: Min-Ex pareto-front and initial solutions

The SPEA2 algorithm has two initial sets; one the initial population and the

other the external set which holds the non-dominated solutions. In order to

show the working principle of algorithm, six random solutions in the search

space is chosen for the first set and three random solutions for the archive set,

as it is done in the code. These solutions and their corresponding objective

function values are listed in table 7.

Step 1: The two initial random populations are created. The solution in

65

Table 7: Current and external initial random population of SPEA2 with theirobjective function values

Initial population Pt External population PtSolution x1 x2 f1 f2 Solution x1 x2 f1 f2

1 0.31 0.89 0.31 6.10 a 0.27 0.87 0.27 6.932 0.43 1.92 0.43 6.79 b 0.79 2.14 0.79 3.973 0.22 0.56 0.22 7.09 c 0.58 1.62 0.58 4.524 0.59 3.63 0.59 7.855 0.66 1.41 0.66 3.656 0.83 2.51 0.83 4.23

the two sets are P0 = 1, 2, 3, 4, 5, 6 and P0 = a, b, c, where N = 6,

N = 3 and t = 0. In this example, we will run the algorithm for one

generations (T = 2).

Step 2: In this section fitness is assigned for all solutions in P0 + P0. For

this purpose:

Step F1: For each solution the number of individuals it dominates is

calculated as strength of that solution S(i). For example, since so-

lution 1 has lower values of objective functions (in both objectives)

compared to solution 2 and 4, therefore; solution 1 dominates solu-

tions 2 and 4 or solutions 2 and 4 are dominated by solution 1. On

the other hand, neither solution 1 dominates solution 3 (value of

first objective for solution 3 is lower than solution 1) nor solution

3 dominates solution 1 (value of second objective for solution 1 is

lower than solution 3). The strength values of all solutions and the

individuals that each solution dominate are shown in table 8.

Step F2: The raw fitness of each solution, that is the sum of strength

values (S(i)) of all solutions that dominate solution i, is calculated.

In other words, since solution 1 is not dominated by any other

solution, the raw fitness of solution 1 is 0. Solution 4 is dominated

by solutions 1, 2, 3, a, c and the strength value of these solutions

are respectively 2, 1, 1, 1, 1; therefore the raw fitness of solution

1 is R(1) = 2 + 1 + 1 + 1 + 1 = 6. Raw fitness of all individuals in

P0 + P0 is also listed in table 8.

Step F3: Density of each solution, with predefined value ofK =√N +N =

√6 + 3 = 3 is calculated. First, the normalized Euclidean distance

of every solution from all solutions in objective space shall be com-

66

puted:

dij =

√√√√ |M |∑k=1

(f

(i)k − f

(j)k

fmaxk − fmink

)2

where, M is number of objective functions and fmaxk , fmink are the

upper and lower values of each objective function (fmin = [0.1, 0]

and fmax = [1, 59]).3 For instance, resultant distances of solution

1 are:

d12 = 0.0179, d13 = 0.0103, d14 = 0.0977, d15 = 0.1530,

d16 = 0.3348, d1a = 0.0022, d1b = 0.2857, d1c = 0.0907.

Next, these distances are all arranged in ascending order. The

kth distance, which is the distance of solution 1 from the 3rd (kth)

nearest solution, is considered as σ31 (σki ) which is d13 = 0.0103.

Finally, the Density of solution 1 is obtained by:

D(1) =1

σ31 + 2

= 0.4974

Step F4: Fitness value of each individual is calculated. Fitness of

solution 1 is:

F (1) = R(1) +D(1) = 0 + 0.4974 = 0.4974

Density and fitness values of all solutions are listed in the two last

columns of table 8.

Step 3: All non-dominated solutions of P0+P0, solutions with fitness values

smaller than one (F (i) < 1), are copied to P1. From table 8, we can

observe that solutions 1, 3, 5, a, c have fitness values of lower than

one, therefore size of non-dominated solutions is 5. Since it is more

than the predefined size of archive set (N = 3), we need to use the

environmental selection procedure (truncation) to reduce the size of P1

to 3. In other words, two of these solutions must be eliminated from

P1.

3Here the minimum and maximum of each objective is obtained by first, calculating a sampleof random solutions and then use the corresponding upper and lower objective values as initialbounds. If in any generation, the limits are changed and lower or higher bounds are found, thesample minimum and maximum values will be replace with the new values.

67

Table 8: Fitness assignment procedure of current and external set of SPEA2

Initial population PtSolution S(i) Dominated Solutions R(i) F (i)

1 2 2, 4 0 0.49742 1 4 2 2.49283 1 4 0 0.49744 0 6 6.4975 2 6, 8 0 0.49726 0 3 3.4912

External population PtSolution S(i) Dominated Solutions R(i) F (i)

a 1 4 0 0.4992b 1 6 2 2.4948c 1 4 0 0.4980

Step E1: First, the normalized Euclidean distance of each solution in

P1 from other solutions in that set is calculated. In order to reduce

the computation time, the distances obtained in step F3 can be

used here, where for solution 1:

d11 = 0, d13 = 0.0103, d15 = 0.1530, d1a = 0.0022, d1c = 0.0907.

Step E2: After sorting the distances in increasing order, the kth near-

est solution (here we set k = 2) to solution 1 is solution a with

d1a = 0.0022. The same procedure is done for all other solutions.

With comparing the 2nd (kth) nearest solution to all solutions in

P1, we can conclude that solutions a and c should be eliminated.

They have the lowest distance to their 2nd nearest solution; there-

fore they are removed from the archive set to improve the diversity

of non-dominated solutions.

Step 4: Since the stopping criteria which is the number of generations is

not met, we will continue to step 5.

Step 5: In this step the three individuals in P1 will participate in a binary

tournament selection with replacement. Each solution will participate

in two tournaments and the better solution in matter of lower fitness will

win the tournament and be placed in the mating pool. Since solution

5 has the best fitness among the three, it will win both tournaments,

thereby creating two copies of it in the mating pool. Therefore the

mating pool is filled with solutions 5, 5, 3.

68

Step 6: The variation step creates child population from the parent popu-

lation of mating pool. Here we used cross-over probability equal to 0.9

and mutation probability of 0.1.

Step V1: Two parents are randomly selected; we assume that solution

5 is the first parent and 3 is the second one.

Step V2: Two children are created from the parents by using SBX

cross-over and pre-defined value of ηc = 2 as follows:

Step SBX1: A random number ui ∈ [0, 1] is chosen. For example

u1 = 0.7577.

Step SBX2: βqi is calculated by:

βqi =

(2ui)

1

ηc + 1 , ifui ≤ 0.5;(1

2(1− ui)

) 1

ηc + 1, otherwise.

Since, u1 = 0.7577 > 0.5 therefore:

βq1 = 1.2732

Step SBX3: The two offspring are computed by:

x(1,t+1)i = 0.5[(1 + βqi)x

(1,t)i + (1− βqi)x

(2,t)i ],

x(2,t+1)i = 0.5[(1− βqi)x

(1,t)i + (1 + βqi)x

(2,t)i ].

here, x(1,t)1 = 0.66 and x

(2,t)1 = 0.22 therefore the children are:

x(1,t+1)1 = 0.7201, x

(2,t+1)1 = 0.1599.

The same procedure with a random u2 = 0.72 is done for the second

variable and the resulted children are:

x(1,t+1)2 = 1.5157, x

(2,t+1)2 = 0.4543.

The two parents and their new solutions (offspring) are:

parent1 = [0.66, 1.41], offspring1 = [0.7201, 1.5157],

parent2 = [0.22, 0.56], offspring2 = [0.1599, 0.4543].

69

Step V3: By keeping in mind the mutation probability, mutation of

a randomly selected parent is achieved with polynomial mutation

operator and predefined parameter ηm = 2 :

Step PBX1: A random number ri ∈ [0, 1] is chosen. For example

ri = 0.7.

Step PBX2: Parameter δi is calculated:

δi =

(2ri)1

ηm+1 − 1, if ri < 0.5,(1− [2(1− ri)]

1ηm+1

), if ri ≥ 0.5.

Since r1 = 0.7 > 0.5 therefore; δ1 = 1− [2(1−0.7)]1

2+1 = 0.1566

Step PBX3: The new mutated child is computed with:

y(1,t+1)i = x

(1,t+1)i + δi.

with randomly selecting solution 5 as parent the child will be

y(1,t+1)1 = 0.66 + 0.1566 = 0.8166. The same procedure is ap-

plied to obtain the second variable of mutated parent.

Step V3: After three solutions have been generated the parents are

replaced by the new children.

Step 7: The new generated solutions are set to P2. Counter for number

of generations is incremented (t = 1) and the next generation will

be started from step 2. After the stopping criteria is met, the non-

dominated solutions in Pt create the pareto-optimal solutions of prob-

lem.

70

B.1 Constraint Handling Method of SPEA2 Algorithm

The same simple minimizing optimization problem used for hand calculation of

SPEA2 algorithm is used to simulate the working principle of suggested constraint

handling method. However, two constraints are added to the problem:

Min-Ex:


Minimize f2(x) =x2 + 1

x1

,

subject to x2 + 9x1 ≥ 6,

−x2 + 9x1 ≥ 1,

0.1 ≤ x1 ≤ 1,

0 ≤ x2 ≤ 5.

Constraints divide the decision and objective search space into two regions. A part

of the unconstrained pareto-optimal front is not feasible and a new pareto-front

will emerge. The new constrained pareto-front is convex. The same two initial

sets, six solutions for the first set and three solutions for the external set (table 7),

are chosen. Figure 32 illustrates the constrained pareto-front and the solutions in

objective space.

Figure 32: Constrained Min-Ex pareto-front, feasible region and initial solutions

The only difference of constrained SPEA2 algorithm with the normal algorithm

is the definition of domination concept. Therefore, a step by step simulation of

constraint domination concept is described by using the initial chosen population.

71

Step CD1: Feasibility or infeasibility of every solution in current and external

set is examined.

Step CD2: The three scenarios explained in constraint handling method are in-

vestigated for each pair of solutions:

Scenario 1(Both solutions feasible): Solutions 4 and c are both feasible,

therefore by considering the domination concept described in previous

appendix (solution c has a lower fitness compared to solution 4) solution

c constraint dominates solution 4.

Scenario 2 (One solution feasible): For instance, solution 1 is infeasible

while solution 5 is feasible, thus solution 5 constraint dominates solution

1.

Scenario 3 (Both solutions infeasible): Solutions 2 and 3 are both in-

feasible. In this case, the constraint violation of each solution is cal-

culated and the one with lower violation wins the tournament. Here,

Solution 2 with a violation value of 0.0356 constraint dominates solution

3 with violation value of 0.5767.

Now by knowing the number of individuals each solution constraint domi-

nates, fitness assignment of SPEA2 algorithm can be accomplished. Table 9

shows feasibility, constraint violation and constraint dominated individuals

of each solution. Other steps are similar to the non-constrained optimization

problem illustrated in appendix B.

Table 9: Constraint handling data for each solution of SPEA2

Initial population PtSolution Feasibility Constraint violation Dominated Solutions

1 Infeasible 0.3867 3, a2 Infeasible 0.0356 1, 3, a3 Infeasible 0.5767 −4 Feasible 0 1, 2, 3, a5 Feasible 0 1, 2, 3, 6, a, b6 Feasible 0 1, 2, 3, a

External population PtSolution Feasibility Constraint violation Dominated Solutions

a Infeasible 0.4500 3b Feasible 0 1, 2, 3, 6, ac Feasible 0 1, 2, 3, 4, a

72

Documents

Multi-Objective Optimization using Genetic Algorithms