�

Abstract—In this paper, a novel algorithm based on membrane systems theory is proposed for solving multi-objective optimization problems. The proposed algorithm is composed mainly by some elements, such as symbol-objects, multi-sets, regions, rules and so on, which are inspired by the structure and functioning of the living cell. In the inner regions of membrane systems, a symbol-object represents a candidate solution of the multi-objective optimization problem. Thereafter, some symbol-objects may construct a multi-set which is sent into the appointed region by the specific rule. In addition, some evolution rules are employed to evolve the multi-set in the inner region of elementary. Moreover, the diversity of the proposed algorithm is improved by the maintaining strategy and the rules of division and dissolution. Comparative study shows that the proposed method performs better in four performance metrics when solving these benchmark problems compared with three the state-of-art algorithms. Sensitivity analysis indicates that it could quickly obtain the approximate Pareto front and satisfy the requirement of diversity of Pareto front. So, it is feasible and effective to solve multi-objective optimization problems.

I. INTRODUCTION

large number of optimization problems exist in the domain of scientific research and engineering

application. These problems are usually solved by searching their maximum or minimum value under certain constrained conditions. If the number of objective function is more than or equal two, it is called multi-objective problems (MOPs). In recent years, many scholars have carried out a considerable of research work in the domain of multi-objective optimization, including the traditional mathematical methods and population-based evolutionary algorithms. Compared with the traditional mathematical methods, the population-based evolutionary algorithms have obtained better results on high-dimensional, nonlinear and discontinuous problems.

As a new branch of natural computing, membrane systems are proposed by Pãun who is inspired by biological cells[1]. With the deep study on membrane systems, they have become to focus on multi-disciplinary such as the current computer science, biology and artificial intelligence. Recently, in the aspect of single-objective optimization problems, a great success has been achieved. Nishida[3] first proposed a novel membrane algorithm to solve the traveling salesman problem. Huang[4] presented an optimization algorithm for large

Manuscript received June 30, 2011. This work was supported by the project (61074096) of the National Nature Science Foundation of China

Chuang Liu, Min Han and Xinzhe Wang are with the Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, 116023 Dalian, China. (email: chuang.liu@mail.dlut.edu.cn, minhan@dlut.edu.cn ).

feasible space and parameters problems, which is inspired by Membrane Computing. Zhang[5] introduces an evolutionary algorithm which is on the basic of the quantum-inspired evolutionary approach and membrane systems and solve a well-known combinatorial optimization problem, the knapsack problem.

However, it is rare that the membrane systems are used to solve MOPs. Huang[6] first proposed a new multi-objective optimization algorithm, named PMOA, which is based on membrane systems. It selected an important objective and optimized first, then optimized other objective. But, this algorithm could not effectively share the information among objectives. Also, the selection of the important objective directly affects the final Pareto fronts. Zhang[7] proposed a multi-objective membrane algorithm based on membrane systems and quantum-inspired evolutionary algorithms, called MOMA to solve multi-objective knapsack problems. However, he does not take into account the rule of division and dissolution, which maintain the diversity of symbol objects.

In this paper, we proposed a novel evolutionary multi-objective optimization algorithm based on the theory of membrane computing, named as multi-objective optimization membrane computing (MOMC). The main idea of this algorithm is described as follows. The concepts of membrane systems, such as symbol-objects, multi-sets, regions, rules and so on, are introduced to the proposed algorithm. In the inner regions of membrane systems, a symbol-object represents a candidate solution of MOP. Next, some symbol-objects may construct a multi-set which is sent into the appointed region by the communication rule. In addition, evolving rules in the inner regions of membrane are employed to obtain the non-inferior solutions for MOPs. The diversity of solutions is implemented by the maintaining strategy and the rules of division and dissolution in the inner regions of membrane. Finally, non-dominated solutions from the inner regions of elementary are released atomically to the inner region of skin membrane in each cycle. These solutions constitute the final approximate Pareto front for MOPs.

The remainder of this paper is organized as follows: Section II presents a brief review on Membrane computing and describes its relevant function. In Section III, the details of the proposed algorithm are elaborated. Comprehensive study and experimental results are discussed in Section IV, and finally, Section V provides concluding remarks of the study.

A Multi-Objective Evolutionary Algorithm based on Membrane Systems

Chuang Liu, Min Han, Senior Member, IEEE, and Xin-zhe WANG

A

103

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE

II. MEMBRANE COMPUTING (MC)

Membrane computing is a branch of natural computing, which imitated the structure and functioning of the living cell and abstracted distributed parallel computing models, called membrane systems or P systems. A membrane structure is a hierarchically arranged set of membranes, as shown in Fig. 1. Next, a basic structure of the membrane system is described

briefly on the degree of n . Equation (2) shows.

1( , , , , , , )nV T w w R�� � � (2)

Where:

(1) V is an alphabet; its elements are called

symbol-objects;

(2) T V� T is the output alphabet;

(3) � is a membrane structure of degree n;

(4) *w Vi � 1 i n� � are symbol-objects from V*

representing multi-sets over V associated with the regions 1,

2, ..., n of �(5) iR , 1 i n� � are finite sets of evolution rules over

V associated with the regions 1, 2, ..., n of �

Fig.1 shows the generic structure of P systems. The description of some concepts in Fig. 1 is elaborated in the following sections. We distinguish the external membrane (usually called the skin membrane) and several internal membranes (corresponding to the membranes and the elementary). If a membrane without any other membrane inside, it is said to be elementary. Each membrane determines a compartment, also called a region. In the basic variant of P systems, each region contains a multi-set of symbol-objects. The symbol-objects evolve by means of evolution rules, which are also localized, associated with the regions of the membrane structure.

III. MULTI-OBJECTIVE MEMBRANE COMPUTATION

In this section, a method based on membrane computation theory will be introduced for dealing with MOPs. In addition, the proposed method incorporates several strategies to maintain the diversity of non-dominated solutions and to enhance the searching ability for solutions. Finally, the algorithm is detailed in the following section.

A. Algorithm Overview

As described in Section I, the main motivation of this paper is to develop a novel evolutionary multi-objective optimization algorithm, named MOMC, which is inspired by molecular model based on membrane computation concepts and theory. If an evolutionary multi-objective algorithm is suitable to solve the MOPs, it should have the characteristics of good diversity of non-inferior solutions and fast convergence to the known Pareto front. To achieve this goal, a maintenance strategy is introduced for the implementation of the diversity of non-inferior solutions, and an update strategy is proposed for the global optimum solutions. The maintenance process adopts non-dominated sorting strategy and crowing distance strategy from NSGAII[10]. It is effective to enhance the diversity of non-inferior solutions. And the update strategy constructs an archiving solution set for finding global optimal solution. It makes the algorithm quickly converge to the known Pareto front. The flow chart of MOMC is shown in Fig. 2.

B. Description of MOMC

In order to clearly illustrate the proposed method, a pseudo-code is given in Fig. 3. It describes the evolution steps of MOMC. The algorithm of MOMC can be described

unambiguously as follows

Step1 Initialization. At first, the basic parameters of the

proposed algorithm are initialized. Then, under the constraints, symbol-objects are generated randomly in the inner regions of skin membrane. Meanwhile, their encoding format employs the decimal-real code. The generated detail of a symbol-object is described by the following formula:

()is min+(max - min) rand� (3)

Where, 1 i N� � " "is denotes the i -th symbol-object

"min" denotes the minimum of solutions; "max" denotes the

Fig. 1. The generic structure of P systems.

The dissolve of Elementary

Initialize the set of objects in Skin Membrane

Elementary Elementary…

store non-inferior solution in Skin Membrane

Termination conditions

In Skin Membrane, execution division rules and allocation symbolobjects into elementary

End

Start

Fig. 2. The flow chart of MOMC.

104

maximum of solutions; " "()rand is a random function to

generate a number between 0 and 1. Finally, those symbol-objects are combined into M multi-sets. The length of each multi-set is N/M. Their form is described as:

( : / )kset s j j N M� (4)

Where /j j N M� 1,k M� � and 0j �" "kset denotes the k-th multi-set; " "s denotes the set of

symbol objects from the Equation(3); k denotes the total of

symbol objects.

Step2 Division. After the initialization operation, the

division rule is utilized in the regions of skin membrane. It may generate M elementary, which have the ability to solve multi-object optimization problems. Then, initialized multi-sets are sent to the inner regions of M elementary by commutation rules.

Step3 Evolution. We are inspired by genetic algorithms

during the whole evolving process. Some operating rules in elementary are proposed to solve MOPs. The operating rules are related on selection, rewrite and mutation. We will present clearly the role of each operator rule in the inner regions of elementary with the following description.

1) Selection_ruleIt is based on crowding distance mechanism, which finds a

winner by comparing the measurement of crowing distance of two symbol-objects. At first, two symbol-objects in an inner of elementary are randomly selected. Secondly, one is selected according to the value of their crowding distance. Finally, one with the small measurement of crowing distance is voted as a winner. It may be described as follows:

1 1_ 2 _

2

1_

2 _

( ()).

( ()).

dis dis

k

dis k

dis k

s s ss

s otherwise

s set rand crowdDistance

s set rand crowdDistance

���� ���

�

(5)

Where, " ( ())"kset rand denotes a symbol-object which is

selected randomly from the k-th symbol-object.

" "crowdDistance denotes the crowd distance of a symbol

object. 1_" "diss , 2 _" "diss are the crowding distance of

symbol-objects, respectively. If the value of 1_" "diss is

smaller than 2 _" "diss , the symbol-object 1" "s is selected.

Otherwise, the symbol-object 2" "s is selected.

2) Rewrite_rule It is an operator rule of rewriting the real numbers based on

the theory of simulation binary coding. It is realized by the idea of simulated binary crossover [11]. Under its role, an offspring symbol-object encoded by the real number has a better performance than its parent. It can improve the algorithm in the aspect of exploiting the capability of the known solution space.

3) Mutation_rule It is proposed to improve the diversity of solutions and

prevent some local optimal solutions obtained in membrane systems. We take the idea of the polynomial mutation from genetic algorithms. It effectively enhances the algorithm capability of exploring in unknown solution space. Its form is described as follows:

1/( 1)

1/ ( 1)

( )

(2 ) 1 , 0.5

1 2 (1 ) ,

m

i i U Lt t t t t

tt m

t

s s s s

r rt

r otherwise

�

�

�

�

� �

� � ��� � ���

(6)

Where " "Uts denotes the maximum solution of

symbol-objects; " "Lts denotes the minimum of

symbol-objects; " "its denotes the solution of the ith

symbol-objects; " "m� is a coefficient; " "tr is a random

number.

Step4 Dissolution. After all evolving operator rules in

elementary are executed, dissolution rule would be invoked. It may dissolve the corresponding elementary. Then, the non-dominated symbol-objects in the inner region of elementary are automatically released into the inner regions of skin membrane. It would improve the capability of the algorithm to explore and exploit the global unknown solution space. Therefore, it may effectively enhance the diversity of solutions for MOPs. Finally, until each of the elementary has been dissolved, this operator is terminated.

Step5 Archive. In the region of skin membrane, an

archive is implemented to save any good symbol-objects found from elementary during the search process. At each iteration count, new symbol-objects are compared with respect to any symbol-objects in the archive. If new symbol-objects are not dominated by any symbol-objects in the archive, they will be inserted into the archive. Otherwise, the symbol-objects dominated will be deleted from the archive. When the size of the archive exceeds the threshold, the mechanism of the crowding distance will be employed to remove the symbol-objects with the minimum of crowding distance. Finally, the archive is updated and the symbol-objects in next generation are formed. It could improve the convergence speed to the known Pareto front and increase the number of excellent non-inferior solutions.

Step6 Iteration. If the termination condition is not met,

Fig. 3. The pseudo-code of MOMC.

105

step2 will be sequentially executed. Eventually, the iteration is broken. Once the search process is terminated, multi-sets in the inner region of skin membrane become the final approximation Pareto front of MOPs.

IV. EXPERIMENTS

In this section, we present several test functions and performance metrics for MOPs, which are adopted to compare the performance of the proposed method with other state-of-art algorithms.

The experiments are run under the environment of Intel Pentium M 2.66 GHz core and 512MB memory. In table I, we can clearly see the parameters, which are about the proposed algorithm and the state-of-art algorithm such as SPEA2 [12], PAES [13] and NSGAII [10]. Furthermore, the proposed algorithm is initialized according to the formula:

,

0 1 1 2 2 10 10 0

1 10

,1 ,10 ,

' '

, 1 , 2 , 1 , 2

'

, ,

{ |1 10,1 10},

[ [ ] ,[ ] , ,[ ] ] ,

, , ,

{ , ,

,

,

[ ] ,

[ ] }

i j

i i i j

i j i j i j i j

i j i j

i j

i j

s i j

s s

s s s

s s s s

s s

s s

s s

� � � ������

���� � ��� ��� ��

���

���

(7)

where, " "� denotes a membrane system; ," "i js denotes

the j-th symbol-object in the i-th multi-set;

0 1 1 2 2 10 10 0"[ [ ] ,[ ] , ,[ ] ] "� denotes the skin membrane which is

marked as zero and contains ten elementary; 1 10" , , "s s�denotes multi-sets labeled from one to ten, which are

combined into symbol-objects. The rule of

,1 ,10 ," , "i i i js s s�� denotes a selection_rule from many

symbol-objects. The rule of ' '

, 1 , 2 , 1 , 2" "i j i j i j i js s s s� denotes a

rewrite_rule between two symbol objects. The rule of '

, ," "i j i js s� denotes a mutation_rule. The rule of

" [ ] "i js s� denotes a division operator. The rule of

"[ ] "i js s� denotes a dissolution operator.

A. Test Functions

Simulation experiments are designed to verify the effectiveness of the proposed algorithm. We select the

benchmark two-dimensional problems such as ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6[14]. The characteristic of their Pareto front is the convex, non-continuous, uneven distribution, multi-model and so on, so they are widely used to test evolutionary multi-objective algorithm.

From Fig. 4 to Fig. 8, we can see simulation results including ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6. As shown in figures, the approximate Pareto fronts are obtained by the proposed algorithm.

ZDT1 of the multi-objective test problem has a convex Pareto front. As presented in Fig. 4, the proposed algorithm has a good diversity and distribution on this problem, and the result is efficiently approximate to the known Pareto front.

ZDT2 of the multi-objective test problem has a non-convex Pareto front. In Fig. 5, the simulation result shows that the proposed algorithm makes an efficient approximation to the

known Pareto front. Furthermore, its non-inferior solutions have a good diversity and distribution.

Fig. 4. Pareto fronts produced by the MOMC and the known Pareto fronts for ZDT1.

Fig. 5. Pareto fronts produced by the MOMC and the known Pareto fronts for ZDT2.

TABLE IPARAMETERS SETTING OF MULTI-OBJECTIVE ALGORITHMS

MOMC SPEA2[12] PAES[13] NSGA-II[10]

Population size - 100 100 100Object size 100 - - -

Archive size 100 100 100 100Crossover rate - 0.9 - 0.9

Mutation rate - 1/ NumberOfVariables 1/ NumberOfVariables 1/ NumberOfVariables

Max Evaluation 20000 20000 20000 20000

Elementary 10 - - -

106

ZDT3 of the multi-objective test problem has a non-convex and discontinuous Pareto front. As shown in Fig. 6, the proposed algorithm maintains a uniform distribution and is approximate to the known Pareto front effectively. In addition, its non-inferior solutions have a good diversity and

distribution.

For ZDT4 of the multi-objective test problem, it is very difficult to find the global optimal Pareto fronts, because there are some local optimal Pareto fronts. From the Fig. 7, we can see that the proposed algorithm is effectively out of the local optimal solutions and obtain the global optimal solutions. It is an effective approximation to the known Pareto front. Moreover, its non-inferior solutions have a good diversity and distribution.

At last, ZDT6 of the multi-objective test problem has a non-convex Pareto front, and the more the proposed

algorithm is approximate to the known Pareto front, the smaller the density of the solution is. In Fig. 8, an efficient approximation to the known Pareto front is achieved. In addition, its non-inferior solutions have a good diversity and distribution.

B. Performance Analysis

From Fig. 4 to Fig. 8, the simulation results show that the proposed algorithm is effective to solve the ZDT test problems. To further test the performances of the proposed algorithm, some performance metrics are employed to assess the capability of the MOMC and the state-of-art algorithms compared.

Four different qualitatively performance metrics, including

Error Rate (ER) [15] General Distance (GD) [15]

Diversity [10] and MaxSpread (MS) [16], have been widely used in the study of the multi-objective evolutionary algorithm. Meanwhile, they are capable of evaluating non-dominated solutions in several nontrivial issues. On this count, they are employed to test the quality of algorithms.

From Table II to Table IV, the mean and standard deviation of the state-of-art algorithms and the proposed algorithm are

given. The upper part in the cell denotes the value of mean while the lower part in the cell denotes the value of standard deviation on the corresponding indicators.

The metric of ER [15] indicates the percentage of solutions obtained which are not members of the known Pareto front. If the value of ER equals zero, it indicates that all elements found are in the Pareto optimal set. Table.II shows the value of ER, which the proposed method performs satisfactorily in some of the test functions. The proposed method has better results than PAES. Furthermore, it has equivalent results to NSGAII and SPEA2.

The metric of GD [15] is a way to estimate the Euclidean distance between the non-dominated solution found so far and its nearest solutions in the known Pareto front. If the value of GD equals zero, it indicates the distance is zero between the current obtained Pareto front and the know Pareto front. As we can see clearly from the Table.III, the

Fig. 6. Pareto fronts produced by the MOMC and the known Pareto fronts for ZDT3.

Fig. 7. Pareto fronts produced by the MOMC and the known Pareto fronts for ZDT4.

Fig. 8. Pareto fronts produced by the MOMC and the known Pareto fronts for ZDT6.

TABLE IICOMPARISON OF DIFFERENT ALGORITHMS ON ER MERIC

MOMC SPEA2[12] PAES[13] NSGA-II[10]

ZDT10.7918

0.0284

0.7963

0.0263

0.8007

0.0366

0.7882

0.0293

ZDT20.7736

0.0262

0.7848

0.0285

0.7812

0.0381

0.7850

0.0290

ZDT30.9196

0.0180

0.9243

0.0175

0.9139

0.0272

0.9195

0.0189

ZDT40.7951

0.0254

0.7921

0.0258

0.7905

0.0490

0.7967

0.0271

ZDT60.8937

0.0187

0.8964

0.0209

0.8661

0.0413

0.8962

0.0182

107

proposed method performs better results than the state-of-the-art algorithms such as NSGAII, SPEA2 AND PAES. The results could indicate that the Pareto solutions obtained by the proposed algorithm are more closely to the known Pareto front.

The metric of Diversity [10] measures the extent of spread achieved among the solutions obtained. A small diversity represents the projections of the solutions are uniform on the objective space. If the value of Diversity equals zero, it indicates an ideal distribution or spread. From Table IV, we can draw a conclusion that the proposed algorithm has an equivalent result to NSGAII, and it abtained better results than PAES, but it shows an inferior result compared with SPEA2.

The metric of maximum spread [16] are evaluations for

diversity and expansibility. It takes a larger value, which implies a better spread of solutions. The value is close to one means the best spread. From Table.V, we can draw the conclusion that the proposed algorithm performs better results than the other related state-of-the-art algorithms. The results could explain the Pareto fronts obtained by the proposed algorithm equal the maximum expansibility of known Pareto fronts.

In summary, the experiment results show that the proposed method is competitive in terms of performance in both qualitative and quantitative measures for the selected test functions. From Fig. 4 to Fig. 8, the proposed method has the ability to find pareto fronts, which is approximate to the known Pareto Front in each test function. From the above data table, we can see clearly the proposed algorithm is effective through comparing its average and variance with the state-of-the-art algorithms. In addition, on the metrics of GD, MS, and ER, the proposed method has better value than SPEA2 to solve ZDT problems, but its performance about the metrics of spread is worse than SPEA2. This illustrates the diversity of the proposed algorithm need to be improved further. Finally, as a whole, it has a low standard deviation. It is convincing proof that the proposed method is quite robust and stable. Moreover, the proposed algorithm can improve the search ability and distribution to solve MOPs.

V. CONCLUSIONS

In this paper, based on the theory of membrane computation, we proposed a novel evolutionary multi-objective algorithm, which employs elementary to solve MOPs. Some rules, including rewrite_rue, mutation_rule, division, dissolution and so on, are utilized in order to implement the diversity of solutions. The archive maintenance makes it quickly converge to the known Pareto front. The simulation results showed that it not only satisfies the diversity of the non-dominated solutions obtained, but also attains more well distribution. In addition, compared with the state-of-art algorithms based on the population, the proposed algorithm is competitive in terms of performance for the selected test functions. So, it has obvious more

TABLE IIICOMPARISON OF DIFFERENT ALGORITHMS ON GD MERIC

MOMC SPEA2[12] PAES[13] NSGAII[10]

ZDT1

0.1796

e-3

0.0395

0.2248

e-3

0.0284

0.0017

0.0061

0.2216

e-3

0.0419

ZDT2

0.1232

e-3

0.0578

0.1763

e-3

0.0555

0.0025

0.0083

0.1841

e-3

0.0899

ZDT3

0.2072

e-3

0.0192

0.2308

e-3

0.0163

0.0008

0.0026

0.2128

e-3

0.0140

ZDT4

0.1353

e-3

0.0478

0.7173

e-3

0.5647

0.1300

0.2073

0.5319

e-3

0.4454

ZDT6

0.5460

e-3

0.0353

0.0017

0.0002

0.0085

0.0199

0.0010

0.0001

TABLE IVCOMPARISON OF DIFFERENT ALGORITHMS ON DIVERSITY MERIC

MOMC SPEA2[12] PAES[13] NSGA-II[10]

ZDT10.3660

0.0310

0.1496

0.0135

0.7673

0.1067

0.3696

0.0302

ZDT20.3716

0.0287

0.1715

0.0465

0.7953

0.1264

0.3713

0.0384

ZDT3 0.7536

0.0152

0.7101

0.0055

1.0489

0.0783

0.7480

0.0128

ZDT4 0.4612

0.0395

0.2873

0.1276

1.2292

0.1445

0.3976

0.0593

ZDT6 0.4097

0.0364

0.2246

0.0242

0.9091

0.2306

0.3606

0.0324

TABLE VCOMPARISON OF DIFFERENT ALGORITHMS ON MS MERIC

MOMC SPEA2[12] PAES[13] NSGA-II[10]

ZDT11.0000

0.0000

0.9997

0.0003

0.9705

0.0330

0.9999

0.0001

ZDT21.0000

0.0000

0.9960

0.0148

0.9880

0.0190

0.9999

0.0001

ZDT30.9999

0.0001

0.9976

0.0153

0.8716

0.0906

0.9988

0.0109

ZDT41.0000

0.0000

0.9549

0.0527

0.9750

0.0423

0.9985

0.0065

ZDT61.0001

0.0000

1.0000

0.0002

0.9982

0.0030

1.0001

0.0001

108

algorithms.

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advantages than some kinds of population-based evolutionary