A Hybrid Multi-objective Immune Algorithm for Numerical

Optimization

Chris S. K. Leung and Henry Y. K. Lau

Department of Industrial & Manufacturing Systems Engineering, The University of Hong Kong, Hong Kong, China

Keywords: Artificial Immune Systems, Artificial Intelligence, Multi-objective Optimization, Hybrid Algorithm,

Genetic Algorithm.

Abstract: With the complexity of real world problems, optimization of these problems often has multiple objectives to

be considered simultaneously. Solving this kind of problems is very difficult because there is no unique

solution, but rather a set of trade-off solutions. Moreover, evaluating all possible solutions requires

tremendous computer resources that normally are not available. Therefore, an efficient optimization

algorithm is developed in this paper to guide the search process to the promising areas of the solution space

for obtaining the optimal solutions in reasonable time, which can aid the decision makers in arriving at an

optimal solution/decision efficiently. In this paper, a hybrid multi-objective immune optimization algorithm

based on the concepts of the biological evolution and the biological immune system including clonal

selection and expansion, affinity maturation, metadynamics, immune suppression and crossover is

developed. Numerical experiments are conducted to assess the performance of the proposed hybrid

algorithm using several benchmark problems. Its performance is measured and compared with other well-

known multi-objective optimization algorithms. The results show that for most cases the proposed hybrid

algorithm outperforms the other benchmarking algorithms especially in terms of solution diversity.

1 INTRODUCTION

In real world, many problems, no matter whether

they are in the domain of engineering, business or

science, can be formulated into different forms of

optimization problems. Most of these problems

normally involve multiple objectives rather than one

single objective, in which some objectives conflict

with others. As such, these problems that require

meeting several objectives simultaneously are called

multi-objective optimization problems. Solving this

kind of problems is never an easy task because

objectives of such problems are often found to be at

least partly non-commensurable and conflicting.

Very often, there is no single best solution to the

multi-objectives optimization problems, but rather a

set of optimal trade-off solutions which cannot be

improved without disadvantaging the optimality of

other objectives. During the solution evaluation

process, a huge number of alternative solutions are

required to be evaluated. However, it is very

difficult to evaluate all possible solutions as this

requires tremendous computer resources that

normally are not available. Thus, an effective and

efficient optimization algorithm is needed to guide

the search process to the promising areas of the

solution space and hence the optimal solutions in

reasonable time.

Over the last decades, different metaheuristic

algorithms have been developed for solving multi-

objective optimization problems. Among the

appealing metaheuristic algorithms, Artificial

Immune Systems (AIS) based on biological immune

system have received special attention among the

research community because the immune system

provides a rich source of stimulation and inspiration

to the research community with their interesting

characteristics: distributed nature, self-organization,

memory and learning capabilities. Motivated by its

great potential for solving multiple-objective

optimization problems, the study reported in this

paper is to develop a hybrid multi-objective

algorithm based on the engineering analogue of the

biological immune system – AIS incorporating some

ideas from GA

for solving multi-objective

optimization problems.

The rest of this paper is organized as follows:

Section 2 gives an overview of related research

Leung, C. and Lau, H.

A Hybrid Multi-objective Immune Algorithm for Numerical Optimization.

DOI: 10.5220/0006014201050114

In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 1: ECTA, pages 105-114

ISBN: 978-989-758-201-1

105

studies and the basis for the design and development

of the proposed algorithm. Section 3 presents the

proposed algorithm and its major features. Section 4

assesses the performance of the algorithm through

the numerical optimization experiments with results

presented and analyzed. Finally, summary and

potential research directions are given in Section 5.

2 IMMUNITY-BASED MULTI-

OBJECTIVE OPTIMIZATION

2.1 Biological Immune System

The biological immune system consists of diverse

sets of specialized cells, molecules, and organs with

a collection of defense mechanisms working

collaboratively. The interactions among various

cells, molecules, and organs result in complicated

immunological behaviors for the purposes of

provoking suitable immune responses, and

defending, recognizing, and memorizing pathogens

for protecting a given host against infections, thus

keeping the host healthy (Goldsby et al., 2003).

Many researchers have successfully developed a

number of powerful multi-objective optimization

algorithms based on the concepts of the biological

immune system. The major inspiration for our

proposed algorithm comes from the clonal selection

principle and the immune network theory. For a

review of immune algorithms, one can refer to

Ataser (2013).

2.1.1 Clonal Selection Principle

Clonal selection principle was developed by Burnet

(1959). It states that only those B-cells capable of

binding with foreign antigens will produce clones

having identical receptors to the original B-cells.

This process is known as clonal expansion. When

the B-cells undergo clonal expansion after binding to

foreign antigens with the help of a second signal

from accessory cells, their average antibody affinity

will increase for the non-self antigens in order to

boost the speed and effectiveness of the immune

response to secondary encounters. Such a process is

known as affinity maturation and results from

somatic hypermutation and selection mechanism.

The hypermutation can change the specificity of

antibodies (cells) by introducing randomness to their

genes, hence introducing diversity into the B-cell

population. Once this process is completed, the B-

cells possessing higher affinity antibodies will be

selected to differentiate into a mature form -

antibody-producing plasma cells, with each

secreting only one type of antibodies. Other than

developing into plasma cells, the activated B-cells

with high affinity are selected to become long-lived

memory B-cells. These cells can be activated by a

very low concentration of the antigen that had

triggered the primary response so that they can be

ready for re-stimulation caused by secondary

antigenic stimulus. Meanwhile, the antibodies of

self-reactive B-cells are given an opportunity to

rearrange their conformation for changing their

specificity through the receptor editing process so as

to prevent them from apoptosis.

Our proposed algorithm mimics the essence of

the clonal selection principle to generate a varied,

enlarged population of antibodies around their

parents based on the corresponding antigenic affinity

through the processes of the cloning and mutation.

In these processes, antibodies perform local

exploitation in different directions along the

objective space, while the receptor editing process

performs global exploration through the whole

search space. At the end of each generation, the

population will return to its original size with elitist

antibodies having better affinity. This principle

ensures the selection pressure is only placed on good

individuals evolving towards the optimal solution set

with reduced redundant search as well as strikes a

balance between exploitation and exploration for

assuring the achievement of a good result.

2.1.2 Immune Network Theory

The immune network theory describes the behavior

of one of the key working principles of the adaptive

immune system, which was mainly developed by

Jerne (1974). The theory explains the properties of

the immune system including immunological

memory and tolerance through the existence of a

mutually reinforcing network of B-cells that have

variable region, i.e. idiotope and paratope. These

variable regions bind not only to antigens, but also

to other variable regions in the system. The

interactions between B-cells result in stimulation on

the B-cells with a paratope that has recognized an

antigen. However, suppression can also result from

the interactions between B-cells where an anti-

idiotypic antibody is involved, hence bringing about

a regulatory mechanism.

Our proposed algorithm bases on the immune

network theory to introduce a suppression operator

to the antibody population after the cloning and

mutation processes for avoiding antibody

redundancy and maintaining the population diversity

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

106

so as to acquire the uniformly distributed Pareto

front. To achieve this, the affinity among all

antibodies is determined in order to determine

whether to retain or discard individual antibody.

2.2 Multi-objective Optimization

Algorithms

Finding the solutions to the multi-objective

optimization problems has long been a challenge to

researchers because both the Pareto optimality and

the diversity of the generated solutions must be

simultaneously addressed. Unlike solutions in single

objective optimization problems, which can easily

be compared according to the value of the objective

function, solutions in multi-objective problems

cannot directly be compared with each other unless

employing classical techniques, such as, weighted

objective aggregation methods and constraint

approaches. As such, the multi-objective

optimization problems are simplified and solved by

converting the multi-objective problems into the

single objective problems. Although these

approaches are simple, they have some drawbacks

for solving multi-objective problems such as the

limited spread of non-dominated solutions, the lack

of the ability of capturing the characteristics of all

objectives, the lack of the ability of generating

concave and nonconvex portions of the Pareto front,

and their high dependence on user experiences and

preference information.

Due to the complexities of real world problems

and the limitations of these classical methods,

modern evolutionary optimization algorithms such

as GA, ES, AIS, etc. incorporating the concept of

Pareto optimality with capability of generating

diversified solutions have been proposed and have

become popular. The reason of employing the

concept of Pareto optimality is that it can facilitate

the determination of the relative strength between

candidate solutions based on their fitness value

without converting the multiple-objective problems

into the single objective problems, thus resulting in a

set of non-dominated solutions or Pareto optimal

solutions. They are said to be globally optimal to

multi-objective optimization problems because no

improvement in any one objective can be obtained

without sacrificing the optimality of other

objectives.

During the past few decades, evolutionary

algorithms have received great interest and a

significant number of publications have been done in

multi-objective optimization domain since the first

multi-objective evolutionary algorithm has been

developed by Schaffer (1985). These algorithms

have been proved to be effective ways for solving

multi-objective optimization problems by finding the

approximated Pareto front, including NSGA-II (Deb

et al., 2000), SPEA2 (Zitzler et al., 2001), PESA-II

(Corne et al., 2001), micro-GA2 (Pulido and Coello

Coello, 2003), omni-aiNet (Coelho and Von Zuben,

2006), NNIA (Gong et al., 2008), omni-AIOS

(Zhang, 2011), etc. For a comprehensive review of

multi-objective evolutionary algorithms, one can

refer to Deb (2001) and Coello Coello (2007).

Recently, the hybridization of AIS with other

evolutionary algorithms has increasingly become a

prevalent trend. For example, Luh et al. (2003)

introduced an algorithm called Multi-objective

Immune Algorithm (MOIA), which is devised based

on the features of the biological immune system and

gene evolution for efficiently solving multi-objective

optimization problems. Cutello et al. (2006)

extended the algorithm called Pareto Archived

Evolution Strategy (PAES) (Knowles and Corne,

1999) with a different representation based on

immune inspired computing principles in order to

devise a modified version of PAES denoted by I-

PAES. This algorithm is applied to a multi-objective

Protein Structure Prediction (PSP) problem. Wong et

al. (2009) developed an immunity-based hybrid EA

called Hybrid Artificial Immune Systems (HAIS) for

solving constrained multi-objective global container

repositioning problems. Qiu and Lau (2014)

proposed a new AIS-based hybrid algorithm which

hybridizes two AIS theories: clonal selection

principle and immune network theory with particle

swarm optimization (PSO) theory for solving static

job shop scheduling problems with the objective of

makespan minimization.

Based on these studies, it is true to point out that

complementarily combing the various optimization

techniques can usually offer better performance in

terms of convergence, computational efficiency,

diversity and solution quality than individually

employing them by overcoming their own

weaknesses. In our proposed multi-objective

optimization immune algorithm, other than immune

operators, a crossover operator of GAs is adopted to

enhance the performance in terms of diversity and

convergence (Coello Coello et al., 2007).

3 OPTIMIZATION ALGORITHM

DESIGN AND DEVELOPMENT

In this section, an innovative hybrid multi-objective

A Hybrid Multi-objective Immune Algorithm for Numerical Optimization

107

optimizer – Suppression-controlled Multi-objective

Immune Algorithm (SCMIA) based on the clonal

selection principle and immune network theory as

well as incorporated the ideas from GA is

developed. The mapping between the biological

immune system and the proposed artificial one is

given in Table 1.

Table 1: Mapping between the biological immune system

and SCMIA.

Biological

Immune

System

SCMIA

Antigen (Ag) Objective function to be optimized

Antibody (Ab) Candidate solution (a set of

decision variables) to be optimized

Ag-Ab affinity Fitness value of each candidate

solution evaluated based on Pareto

dominance

Ab-Ab affinity Crowding-distance working as a

measure of population diversity

Immune

suppression

Mechanism to control the number

of nearby candidate solutions based

on similarity among candidate

solutions in both the objective

space and decision variable space

Memory cell Current best non-dominated

solution

The proposed SCMIA comprises five immune

operators: cloning, hypermutation, suppression,

selection & receptor editing, and memory updating,

and one genetic operator: crossover. Each of them

takes responsibility for different tasks for the

purpose of finding uniformly distributed Pareto

front. The cloning operator generates a number of

copies to explore the solution space where better

individuals are given more chances for being cloned.

The hypermutation operator works on the clones to

bring variation to the clone population, hoping for

producing better offspring and increasing population

diversity. The crossover operator is used to enhance

the diversity of the clone population and the

convergence of the algorithm by avoiding getting

trapped into local optima. The suppression operator

works on the whole population including the

mutated clones and parent cells to eliminate similar

individuals in order to avoid a particular search

space being over exploited. The selection & receptor

editing operator works like a director to guide the

search towards the promising regions of a given

fitness landscape by selecting the best antibodies to

form the next generation and allowing the genes of

the less-fit to be randomly restructured for changing

their specificity through the receptor editing process.

The memory updating operator works as an elitist

mechanism for helping preserve the best solutions

that represent the Pareto front found over the search

process. The proposed algorithm is conducted by

applying these heuristic and stochastic operators on

the antibody population for balancing both the local

and global search capabilities. SCMIA is indeed a

specific multi-objective algorithm, but its basic

structure can be considered a generic framework for

multi-objective optimization which can be

implemented in different ways according to the

problem at hand. Details of the proposed algorithm

are discussed below and the block diagram showing

the computational steps for the proposed SCMIA is

presented in Figure

Figure 1: Computational steps for the proposed SCMIA.

Step 1: A random uniformly distributed antibody

population Ab(t) = {ab

: i = 1, 2, …, N} is initially

generated, where t is the iteration counter, N is the

size of the population, and ab

= {x

: j = 1, 2, …, m}

is a candidate solution containing m decision

variables to the fitness function. Other than this

online population, an external memory population

P(t-1) with the size of N

for storing the non-

dominated solutions is also created and initialized to

be empty.

Step 2: The values of objective functions of each

antibody ab

are evaluated. In this way, a fitness

Yes

Clone

Population

Mutation

Cloning

Initial

Population

Initialization

Fitness Assignment

Non-dominated

Neighbor-based Selection

Active

Population

Mutated

Population

Termination

Condition

End

Suppression

Fitness Assignment

Crossover

Mature

Population

New

Population

Selection & Receptor Editing

Memory Updating

New

Memory Set

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

108

array ab(f) storing the values of all objective

functions for each parent can be determined. And

then the Pareto dominance relation of each antibody

is determined by comparing their fitness values with

respect to each objective. Through this comparison,

each of them is assigned an antigenic Ab-Ag affinity

called Pareto fitness pf

. The Pareto fitness of each

antibody is computed as follows:

= D

(1)

where D

is the number of antibodies that

dominates the antibody ab

. It is noted that when the

fitness of an antibody pf is equal to 0, this antibody

is considered a non-dominated solution as it is not

dominated by any other antibodies in the population.

Step 3: Based on the idea proposed by Gong in

NNIA (Gong et al., 2008), only non-dominated

antibodies are selected to form an active parent

population A(t) with the size of N

for undergoing

cloning, crossover and hypermutation. If the number

of non-dominated antibodies is smaller than N

, all

non-dominated antibodies are selected to form A(t).

However, if the number of non-dominated

antibodies is larger than N

, an antibody density

measure called crowding-distance (Deb et al., 2002)

analogous with Ab-Ab affinity in biological immune

system is employed. The crowding-distance for a

non-dominated antibody is computed as follows:

(



∗

) =

∑







∗













∗









(





)(





)





(2)

where 

(





∗

)

is the crowding-distance of the i-

th non-dominated antibody ab

, (





) and

(





) are the maximum and minimum fitness

values of the j-th objective, 



∗





 and 



∗







are the fitness values of the nearest neighboring

antibodies from both sides in terms of the fitness.

With this measure, N

antibodies with a larger

crowding-distance value are selected to form A(t) in

order to enhance the population diversity. It is worth

emphasizing that this approach guides the search

paying more attention to the less-crowded regions in

the current Pareto front at each generation.

Step 4: Cloning operator enlarges the population

by generating a number of copies of each antibody

in A(t) and the number of copies is directly

proportional to its Ab-Ab affinity, thus forming a

clone population C(t). Hence the size of the

population now is N + N

and N

is obtained by:

round(c

max

×(



∗

))

(3)

= ∑c

(4)

where N

is the total number of copies produced,

is the clone size for the antibody, c

max

is the pre-

defined maximum clone size of each antibody, and

round() is an operator for rounding its argument to

the closest integer. Clearly, the higher the Ab-Ab

affinity an antibody has, the more the number of

copies it can generate.

Step 5: The hypermutation operator induces

multi-point mutations to the clones. The mutation

depends on the Ab-Ab affinity of their active

parents. The reason to take account of the Ab-Ab

affinity is to maintain the population diversity and

prevent the crowding of antibodies. The clones are

mutated proportionally as follows:

α = 

×(

∗

)

(5)

(t) = C(t) + α × R

(6)

where α represents the mutation rate inversely

proportional to the Ab-Ab affinity (

∗

), is an

exponential coefficient controlling the decay of α, R

∈ [-1, 1] is a m-dimensional random vector

obtained with uniform distribution, and C

(t) is the

mutated clone population.

Step 6: A modified single point crossover

operator works on the mutated clone population to

generate a mature clone population C

(t) with the

size of N

. With this crossover operator, each

offspring is generated by randomly selecting a single

crossover point on a clone and then swapping its

content beyond that point with that of an active

parent antibody randomly selected from A(t). The

diversity can be further enhanced through the

crossover operation while the quick convergence can

be ensured because some good genes from the active

parent can be passed to the offspring.

Step 7: Objective functions of each mature clone

are evaluated. And then a combined population is

formed by combining both the parent cells and their

mature clones for fitness assignment. For each cell

in the active parent population and mature clone

population, the Pareto dominance relation is

determined and the Pareto fitness value 





assigned. As such, the Pareto fitness being assigned

to each cell is dependent on the performance of all

other cells in the combined population.

Step 8: Suppression operator is introduced and

works on each cell in the combined population A(t)

∪ C

(t) to avoid antibody redundancy and maintain

the population diversity so as to acquire the

uniformly distributed Pareto front based on the idea

of immune network theory (Jerne, 1974). To achieve

this, the antibody similarity among all antibodies has

to be determined. Different from other AIS-based

multi-objective optimization algorithms, the

A Hybrid Multi-objective Immune Algorithm for Numerical Optimization

109

similarity among antibodies in this algorithm is

determined in terms of both the objective space and

the decision variable space so as to determine

whether to retain or discard individual antibody. The

suppression operation has two phases: in 1

phase,

the suppression will be applied to all antibodies and

the similarity between two antibodies is defined as

follows:

dO(ab

, ab

)

= 









−









 ≤ 



(7)

where dO(ab

, ab

)

is the distance between

antibodies ab

and ab

in terms of j-th objective and





refers to the threshold value for j-th objective. In

this phase, if the distances for all objectives between

two cells are smaller than the thresholds, the two

cells are said to be similar and hence the cell with

poorer Pareto fitness will be suppressed and

eliminated from the population. This procedure is

repeated until all antibodies in the combined

population are compared in order to ensure the

population diversity.

In 2

phase, the suppression will only be applied

to the similarity between non-dominated cells and

dominated cells and the similarity between two

antibodies is defined as follows:

dV(ab

, ab

) =



∑

[







−













]



≤ ε

(8)

where dV(ab

, ab

) is the Euclidean distance in

decision variable space between the two antibodies

(dominated cell) and ab

(non-dominated cell)

and  refers to the threshold value for the decision

space. In this phase, if the distance between two

antibodies is smaller than the thresholds in decision

variable space, they are said to be similar and hence

the dominated cell will be suppressed and eliminated

from the population. This procedure is repeated until

all antibodies between non-dominated and

dominated are compared in order to avoid redundant

search. Eventually, surviving populations A

(t) ∪

(t) are obtained and then enter into the selection &

receptor editing process and memory updating

process simultaneously.

To enhance the population diversity and facilitate

the search of uniformly distributed non-dominated

solutions along the Pareto front of a given problem,

the threshold values for the decision variable space

and the objective space are dynamically calculated

according to the maximum and minimum values

found so far, hence adapting to the new values that

appear in the population.

Step 9A: An evolutionary selection operator is

used to select all non-dominated antibodies with

respect to the Pareto fitness from the surviving

populations to form a new population Ab(t+1) with

the size of N for the next generation. If Ab(t+1) is

not full, dominated antibodies with a better Pareto

fitness are selected and some genes of these

antibodies are then randomly selected to be replaced

by randomly generated genes. These restructured

antibodies are finally added to the new population

until the population is full in order to further

enhance the population diversity. This process

actually mimics the process of receptor editing in the

biological immune system. However, if the number

of non-dominated antibodies found exceeds the

population limit, only N non-dominated antibodies

with higher Ab-Ab affinity are selected.

Step 9B: The memory set P(t) is updated and

used to store all the non-dominated solutions from

the surviving populations for the replacement of the

previous memory set P(t-1). These best solutions are

non-dominated with regard to both the antibodies in

the current generation and the antibodies that tried to

enter the memory set in previous generations.

Step 10: The termination function returns True if

an optimal Pareto front is found, i.e., no significant

changes (change within an acceptable range, η) on

performance metrics of the memory set over

successive iterations, term_max. The optimization

process will also terminate if the predetermined

maximum number of iterations T

max

is performed. If

these conditions are not satisfied Steps 3 to 9 are

repeated until one of the predetermined termination

conditions is met.

4 NUMERICAL EXPERIMENTS

In this benchmarking study, a set of experiments

based on several multi-objective numerical

optimization problems was performed to benchmark

the proposed algorithm with other well-known

multi-objective optimization algorithms, that is, two

immune algorithms – MISA (Coello Coello and

Cortés, 2005) and NNIA (Gong et al., 2008) and two

other evolutionary algorithms – NSGA-II (Deb et

al., 2000) and SPEA2 (Zitzler et al., 2001). All these

experiments were conducted using a computer with

Xeon E5-2620 2 GHz CPU with 2 GB RAM and the

Excel with VBA was used as an implementation

platform.

4.1 Test Problems for Multi-objective

Optimization

Several numerical functions with different

characteristics and degrees of complexity reported in

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

110

the literature are selected to validate SCMIA. The

test functions employed in this study are taken from

three sources, including the traditional test problems

used in early multi-objective optimization studies,

namely SCH proposed by Schaffer (Schaffer, 1984)

and FON proposed by Fonseca & Fleming (Fonseca

and Fleming, 1995), as well as the ZDT test suite

proposed by Zitzler et al. (Zitzler et al., 2000). In

this study, ZDT5 is not selected for the

benchmarking study largely because it is formulated

based on binary coding, which is different from our

study with the focus on real coding.

4.2 Performance Metrics

Three performance metrics are adopted to examine

the quality of solution set in terms of the optimality

and diversity in order to provide a quantitative

comparison of the results, including 1) Error Ratio

(ER) (Van Veldhuizen, 1999), 2) Spacing (S)

(Schott, 1995), and 3) Inverted Generational

Distance (IGD) (D. A. Van Veldhuizen and G. B.

Lamont, 1998).

4.3 Experimental Setup

To conduct the experiments, the true Pareto front for

each test problem is required. The true Pareto fronts

of the test problems are generated by enumeration.

However, since infinite number of solutions to be

generated along the true Pareto fronts is impossible,

a large number of random solutions, that is, 10,000

solutions are generated for representing the true

Pareto fronts. In this research, the decision variables

of all algorithms are real coded despite some of them

originally are binary coded. Since the experimental

results may be sensitive to runtime parameters of

SCMIA, the runtime parameters are manually tuned

based on preliminary sensitivity analysis. Three

parameters, namely active population size,

maximum clone size and mutation factor, are chosen

for the sensitivity analysis. Based on the results of

the sensitivity analysis, the parameters of SCMIA

were set as follows: Initial population size, N = 100;

Size of active population, 



= 40; Size of the

memory population, 



= 100; Maximum number

of clone for each cell, max_clone = 20; Exponential

distribution coefficient, ρ

= 0.05. To allow a fair

comparison among the algorithms compared, the

parameters of the benchmarking algorithms were set

with same values and the values suggested by the

researchers in their original papers as follows:

MISA: Initial population size, N = 100; Size of

clone population, 



= 600; Size of the memory

population, 



= 100; Number of grid subdivisions,

subd_size = 25; Initial mutation rate, ω = 0.6 (it

decreases linearly over time until reaching the rate

of 1/m, where m is the number of decision

variables.). NNIA: Size of dominant population, 



= 100; Size of active population, 



= 20; Size of

clone population, 



= 100; Crossover probability,





= 0.9; Mutation probability, 



= 1/m;

Distribution indexes for crossover and mutation

operators, 



and 



= 20. NSGA-II: Initial

population size, N = 100; Crossover probability, 



= 0.9; Mutation probability, 



= 1/m; Distribution

indexes for crossover and mutation operators, 



and





= 20. SPEA2: Initial population size, N = 100;

Archive size, 



= 100; Crossover probability, 



0.9; Mutation probability, 



= 1/m; Distribution

indexes for crossover and mutation operators, 



and





= 20.

4.4 Experimental Results and Analysis

For test problems, the same parameters and

hardware configurations are used with 100

generations over 15 trials being performed. The

results of comparing the proposed algorithm with the

benchmarking algorithms are shown in the following

tables.

Table 2: Spacing (S).

SCMIA MISA NNIA NSGA-

SPEA2

Mean

(Standard Deviation)

FON 1.28E-02

(1.47E-

03)

1.60E-02

(4.23E-

03)

1.80E-02

(1.44E-

03)

1.49E-02

(3.53E-

03)

1.84E-02

(1.07E-

02)

SCH 9.95E-02

(1.42E-

01)

2.23E-02

(2.07E-

02)

1.15E-01

(2.29E-

02)

1.02E-01

(2.57E-

02)

2.31E-01

(7.42E-

02)

ZDT

1.41E-02

(4.16E-

03)

8.99E-02

(6.48E-

02)

1.76E-02

(1.52E-

03)

2.46E-02

(4.61E-

03)

1.18E-01

(9.79E-

02)

ZDT

1.12E-02

(1.94E-

03)

2.24E-02

(2.48E-

02)

2.03E-02

(2.58E-

03)

2.44E-02

(5.02E-

03)

1.42E-01

(9.82E-

02)

ZDT

2.40E-02

(7.42E-

03)

1.24E-01

(7.10E-

02)

3.02E-02

(5.80E-

03)

2.64E-02

(5.78E-

03)

1.26E-01

(8.21E-

02)

ZDT

2.74E-02

(2.19E-

02)

0.27

(0.32)

1.73E-02

(1.25E-

03)

2.59E-01

(4.30E-

01)

3.05E-01

(1.28E-

01)

ZDT

3.22E-02

(3.51E-

02)

3.41E-02

(5.00E-

03)

1.39E-02

(1.64E-

03)

2.26E-02

(1.86E-

02)

8.28E-02

(2.57E-

02)

A Hybrid Multi-objective Immune Algorithm for Numerical Optimization

111

Table 3: Error Ratio (ER).

SCMIA MISA NNIA NSGA-

SPEA2

Mean

(Standard Deviation)

FON 2.08E-01

(1.17E-

01)

0.00

(0.00)

8.87E-02

(3.02E-

02)

8.00E-03

(7.75E-

03)

4.28E-01

(2.33E-

01)

SCH 7.33E-03

(1.28E-

02)

0.00

(0.00)

0.00

(0.00)

0.00

(0.00)

3.73E-02

(4.37E-

02)

ZDT

8.00E-03

(6.76E-

03)

1.00

(0.00)

1.33E-03

(3.52E-

03)

2.00E-03

(5.61E-

03)

1.00

(0.00)

ZDT

7.33E-03

(4.58E-

03)

1.00

(0.00)

0.00

(0.00)

2.00E-02

(4.34E-

02)

1.00

(0.00)

ZDT

1.43E-02

(7.64E-

03)

1.00

(0.00)

0.00

(0.00)

7.33E-03

(1.16E-

02)

1.00

(0.00)

ZDT

2.53E-02

(1.81E-

02)

1.00

(0.00)

0.00

(0.00)

2.59E-01

(4.30E-

01)

1.00

(0.00)

ZDT

0.00

(0.00)

1.00

(0.00)

0.00

(0.00)

1.00

(0.00)

1.00

(0.00)

Table 4: Inverted Generational Distance (IGD).

SCMIA MISA NNIA NSGA-

SPEA2

Mean

(Standard Deviation)

FON 1.28E-03

(4.59E-

04)

1.15E-04

(1.54E-

05)

7.18E-04

(1.10E-

04)

3.37E-04

(5.17E-

05)

7.52E-03

(1.36E-

02)

SCH 1.39E-03

(4.32E-

03)

5.59E-05

(8.09E-

06)

5.49E-05

(1.55E-

05)

5.54E-05

(1.08E-

05)

3.70E-03

(6.26E-

03)

ZDT

5.43E-04

(3.90E-

04)

3.65E-02

(5.44E-

03)

2.06E-04

(9.30E-

05)

4.14E-04

(2.37E-

04)

6.47E-02

(1.65E-

02)

ZDT

4.03E-04

(2.90E-

04)

6.19E-02

(1.63E-

02)

4.17E-05

(3.59E-

05)

9.37E-04

(7.99E-

04)

9.19E-02

(1.40E-

02)

ZDT

4.81E-04

(1.62E-

04)

3.86E-02

(7.73E-

03)

1.14E-04

(5.80E-

05)

2.95E-04

(1.25E-

04)

5.50E-02

(1.70E-

02)

ZDT

7.45E-03

(4.37E-

03)

3.45

(1.08)

7.45E-05

(5.40E-

05)

1.14E-03

(1.41E-

03)

4.41E-01

(1.42E-

01)

ZDT

1.09E-02

(5.20E-

03)

4.05E-01

(1.13E-

02)

4.43E-05

(1.24E-

05)

1.17E-01

(2.60E-

02)

3.68E-01

(7.66E-

02)

Firstly, we compare the results of the mean and

standard deviation of the three metrics, namely, ER,

S and IGD over 15 trials obtained by the proposed

algorithm - SCMIA with that of the other immune

algorithms, namely, MISA and NNIA. From the

above tables, we found that SCMIA generally is able

to provide a similar result as other immune

algorithms do, which is close to the true Pareto front

true

and in some cases SCMIA can even

outperform them. As for the traditional test problems

(FON and SCH), MISA, NNIA and NSGA-II can

generate slightly better results in the proximity

aspect by achieving better performances in terms of

ER and IGD. For the ZDT test suite, SCMIA and

NNIA can generate much better results, which

completely outperform the results generated by

MISA in terms of the proximity and diversity with

much lower values in the three metrics in all of the

five ZDT test problems. By comparing SCMIA with

NNIA, SCMIA performs better in diversity aspect

with smaller S values in three (ZDT 1, 2 and 3) of

the five ZDT test problems while NNIA performs

better in proximity aspect with lower ER in four

(ZDT 1, 2, 3 and 4) of the five ZDT test problems

and the standard deviation of zero in ER indicates

NNIA can consistently achieve the best performance

in all trials. With regard to the convergence rate,

MISA is the worst one among these three

algorithms, which is revealed through the much

higher ER and IGD in all of the five ZDT test

problems. The standard deviation of zero in ER

indicates MISA consistently cannot converge to the

true Pareto front within 100 generations in all trials.

In conclusion, it is shown that although NNIA is the

best one in the proximity aspect, SCMIA is able to

achieve better results in the diversity aspect among

these three immune algorithms.

Secondly, we compare the results obtained by

SCMIA with that of the other evolutionary

algorithms, namely, NSGA-II and SPEA2. For the

traditional test problems, NSGA-II can generate the

best results in the proximity aspect because it

achieves the best performance in terms of ER and

IGD for the two traditional test problems, whereas

SCMIA and SPEA2 obtain the similar results in the

proximity aspect. With respect to the diversity,

SCMIA achieves the best performance in terms of S

for the traditional test problems. For the ZDT test

suite, SCMIA can generate the best results in terms

of the diversity with much lower values in S in

almost all of the five ZDT test problems except

ZDT6. In terms of the proximity, SCMIA can also

outperform NSGA-II and SPEA2 in three (ZDT 2, 4

and 6) of the five ZDT test problems because

SCMIA has lower values in ER and IGD. With

respect to the convergence rate, SPEA2 is the worst

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

112

one among these three algorithms, which is revealed

through the very high ER in all of the five ZDT test

problems. The standard deviation of zero in ER

indicates SPEA2 consistently cannot converge to the

true Pareto front within 100 generations in all trials.

In conclusion, it is shown that SCMIA outperforms

the other evolutionary algorithms for most of the

benchmark test problems especially in the diversity

aspect.

Several important points can be concluded based

on the results. Firstly, the proposed SCMIA provides

comparable results regarding the other four

algorithms against which it is compared. Although it

does not always provide the best performance in

terms of the three metrics adopted, it is able to

generate reasonably good approximations of the true

Pareto front of each test problem under

investigation, including those with a convex, a

nonconvex or a disconnected Pareto front. Also, it is

generally shown to outperform MISA and SPEA2

with the quality of solution being similar to NNIA

and NSGA-II in approximating the true Pareto front

in terms of the proximity, diversity and convergence

in almost all test problems. Finally, SCMIA clearly

performs better than other benchmarking algorithms

in the diversity aspect. This is largely attributed to

the operators employed in the algorithm, including

selection, cloning, hypermutation, crossover, and

suppression. The selection operator, cloning operator

and hypermutation operator incorporate the

crowding-distance as a measure to select antibodies

for undergoing the subsequent evolutionary

processes, generate a number of copies to explore

the solution space especially the less-crowded

regions, and bring variation to the clone population

respectively, in order to produce better offspring and

increasing population diversity. The diversity is

further enhanced through the crossover operation

while the quick convergence can be ensured by

preventing from being trapped into local optima

because some good genes from the active parent can

be passed to the offspring while bad genes would

have a chance to be replaced with better genes

through hypermutation. The suppression operator

helps reduce antibody redundancy, hence

significantly minimizes the number of unnecessary

searches and increases the population diversity.

5 CONCLUSION AND FUTURE

WORK

This research develops a hybrid immune algorithm -

SCMIA for solving multi-objective optimization

problems. The results show that SCMIA is able to

generate a well-distributed set of solutions while it

represents good approximation to the true Pareto-

optimal set for most of the benchmark problems.

Such satisfactory results are largely attributed to the

characteristics of the algorithm, namely, distributed

nature, self-organization, specificity, memory and

learning capabilities from AIS as well as the

complementary effect from crossover operation of

GA to the hypermutation operation in AIS due to

their different style of solution space traversal.

Future research could extend this approach to

solve real world complex business problems with

real world dynamics and to solve large scale

problems with a large number of parameters,

operators and equipment involved in order to

establish the practical value of the algorithm in

multi-objective optimization context.

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