An Improved Immune Genetic Algorithm for Multiobjective Optimization

Guixia He1,Jiaquan Gao1,and Luoke Hu2

1Zhijiang College,Zhejiang University of Technology,Hangzhou310024,China 2College of Mechanical Engineering,Zhejiang University of Technology,

Hangzhou310014,China

hegx_1022@https://www.360docs.net/doc/1819231801.html,

Abstract.The study presents a novel weight-based multiobjective im-

mune genetic algorithm(WBMOIGA),which is an improvement of its

?rst version.In this proposed algorithm,there are distinct characteris-

tics as follows.First,a randomly weighted sum of multiple objectives is

used as a?tness function,and a local search procedure is utilized to facil-

itate the exploitation of the search space.Second,a new mate selection

scheme,called tournament selection algorithm with similar individuals

(TSASI),and a new environmental selection scheme,named truncation

algorithm with similar individuals(TASI),are presented.Third,we also

suggest a new selection scheme to create the new population based on

TASI.Simulation results on three standard problems(ZDT3,VNT,and

BNH)show WBMOIGA can?nd much better spread of solutions and

better convergence near the true Pareto-optimal front compared to the

elitist non-dominated sorting genetic algorithm(NSGA-II).

Keywords:immune genetic algorithm,multiobjective optimization,

similar individuals.

1Introduction

For multiobjective optimization problems(MOOPs),evolutionary algorithms (EAs)seem to be particularly suitable because they process a set of solutions in parallel,possibly exploiting similarities of solutions by recombination.The?rst step towards treating objectives separately in EAs was given by Scha?er(1985) [1].His approach is known as the vector evaluation genetic algorithm(VEGA). Although there are some disadvantages for VEGA,it plays a great role in fa-cilitating the development of multiobjective evolutionary algorithms(MOEAs). Many MOEAs have been presented since1985,such as the weight-based genetic algorithm(WBGA)proposed by Hajela and Lin[2],Ishibuchi and Murata’s ran-dom weight genetic algorithm(RWGA)[3],Srinivas and Deb’s non-dominated sorting genetic algorithm(NSGA)[4],the strength Pareto evolutionary algo-rithm presented by Zitzler et al.[5],the elitist non-dominated sorting genetic algorithm(NSGA-II)by Deb et al.[6],and Knowles’s Pareto archived evolu-tionary strategy(PAES)[7]and so on.

Y.Tan,Y.Shi,and K.C.Tan(Eds.):ICSI2010,Part I,LNCS6145,pp.643–650,2010.

c Springer-Verlag Berlin Heidelberg2010

644G.He,J.Gao,and L.Hu

Among MOEAs,some approaches called the weight-based approaches use the weighted sum of objective function values as the ?tness of a solution.To the weight-based MOEAs such as WBGA and RWGA,their advantages are low computation complexity.However,their main disadvantages are as follows:(1)di?culty in ?nding Pareto-optimal solutions in nonconvex problems,(2)lack of elitism for most cases,and (3)di?culty in generating uniformly distributed Pareto-optimal solutions.At present,we ?nd that the weight-based MOEAs can alleviate or even overcome their disadvantages by improving their mate selection and environmental selection schemes.Thus,we present a novel weight-based MOEA (MOIGA)based on GA and immune theory in the literature[8].Numerical experimental results show that the proposed algorithm can alleviate the above di?culty.

In this paper,we furthermore improve the proposed algorithm with the fol-lowing aspects.(1)For the local search algorithm,the dominated relationship is added to determine whether the current solution v is replaced by a neigh-bourhood solution v ?besides utilizing their ?tness.In particular,the dominated relationship is ?rst considered.(2)TSASI is improved.(3)A new selection opera-tor is presented to create the new https://www.360docs.net/doc/1819231801.html,pared to the random selection approach,the proposed selection operator can better ensure diversity along the current non-dominated front.(4)The improved algorithm gives an approach to handle constrained conditions for MOOPs.

2The Proposed Algorithm

Here,we will present the weight-based MOEA based on an immune operator and GA,which is an improvement of its ?rst version.The main procedures are listed as follows.

(1)A random initial population P 0of size N is created.Set the external archived set Q 0=?and its maximum size is ˉN .Set a counter t =0.(2)Combine P t and Q t and create a set R t =P t ∪Q t .Calculate non-dominated

solutions in R t and then save them to the external archived set Q t +1.If |Q t +1|>ˉN ,then perform TASI to reduce the size of Q t +1to ˉN .If |Q t +1|<ˉN ,then select ˉN ?|Q t +1|solutions from the dominated solutions in R t by using the selection scheme based on TASI and add them to Q t +1.

(3)Repeat the following procedures to select N pairs of parent solutions.

(a)Specify the weight values w 1,w 2,···,w m with a random uniform distri-

bution,where w 1+w 2+···+w m =1,and w i ∈[0,1],i =1,2,···,m .(b)Use TSASI to select a pair of parent solutions.

(4)Apply the simulated binary crossover operator (SBX)[9]to each of the se-

lected N pairs of parent solutions with the prede?ned crossover probability.Two new solutions are generated from each pair of parent solutions.A new solution is selected from the two generated new solutions as the o?spring ac-cording to their ?tness.Then apply the polynomial mutation operator (PB)

[9]to the generated new o?spring with the prede?ned mutation probability,and add them to the population P t +1.

An Improved Immune Genetic Algorithm for Multiobjective Optimization645

(5)Apply the local search algorithm to all N solutions in the population P t+1.

The search direction of the local search for each solution is speci?ed by the weight values in the?tness function by which its parent solutions were selected.The current population is replaced with the N solutions improved by the local search.

(6)If t can be exactly divided by N in(number of inner loops),then apply the

immune operator to the set R t+1=P t+1∪Q t+1.

(7)If t>T(the maximum number of generations),then terminate the algorithm

and output non-dominated solutions in the set R t+1.Otherwise,t=t+1 and return to(2).

In this improved algorithm,we only introduce the modi?ed local search al-gorithm,the modi?ed TSASI,the approach of handling constrained conditions, and a new selection operator.For the other aspects similar to its?rst version, please refer to the original literature[8].

2.1Selection Scheme Based on TASI

A selection operator is important to a MOEA because it is responsible for guiding the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto-optimal front.Here we present a new selection scheme based on TASI in order to preserve diversity of the created external archived set Q t+1. The detailed procedures are listed as follows.

(1)If|Q t+1|>ˉN,copy cells of Q t+1to a setφt+1,and use TASI to reduce

the size ofφt+1toˉN,and then let the external archived set Q t+1=φt+1.

Terminate the procedure.

(2)If|Q t+1|<ˉN,then let a set R t be composed of dominated individuals among

the external archived set Q t and the population P t in the t-th generation.

Next,the set R t is sorted for non-domination.Let us say that the number of non-dominated fronts in R t is K.The maximum number of individuals allowed in the i-th front(i=1,2,...,K)in the new external archived set Q t+1of sizeˉN is calculated according to the following equation.

r i?1,(1)

N i=(ˉN?|Q t+1|)1?r

1?r K

where0

(3)Assume that N t i is the number of individuals of the i-th front in R t.If N t1>

N1(that is,there are more individuals than allowed),we copy individuals of the?rst front in R t to a setφt+1,and use TASI to truncateφt+1until its size is equal to N1,and then add individuals inφt+1to Q t+1.On the other hand,if N t1 N1(that is,there are less or equal number of individuals in R t than allowed),we choose all N t1individuals and count the number of remaining slotsδ1=N1?N t1.The maximum allowed number of individuals in the second front is now increased to N2+δ1.Thereafter,the actual number of solutions N t2present in the second front is counted and is compared with N2as above.This procedure is continued untilˉN?|Q t+1|individuals are selected.

646G.He,J.Gao,and L.Hu

2.2Tournament Selection Algorithm with Similar Individuals

Here we present the tournament selection algorithm with similar individuals (TSASI).The detailed procedures are listed as follows.

(1)For a given weight vector w=[w1,w2,···,w m]T,create two subpopulation

P op1=?and P op2=?.Their maximum sizes are P opsize1and P opsize2, respectively.

(2)Use the binary tournament selection procedure to select an individual from

Q t+1and add it to P op1.

(3)If|P op1|

(4)Similarly,an individual is selected from Q t+1by using the binary tournament

selection procedure,and is added to P op2.

(5)If|P op2|

(6)The average objective vector in P op1is calculated,and then the most dis-

similar solution among P op1from the average objective vector is chosen as Parent A.

(7)Respectively calculate the Euclidean distance in objective space and in de-

cision space from Parent A to any solution in P op2.

(8)Select the most similar solution from P op2as Parent B,whose Euclidean

distance in decision space from Parent A must exceed a given small positive valueδd.

It can be obviously seen that a pair of similar parent individuals,Parent A and Parent B,are chosen using TSASI,and TSASI can overcome the drawback that the same two individuals are chosen as a pair of parent individuals.The similarity in decision space between Parent A and Parent B can be controlled by modifying the parameterδd.

2.3Local Search Algorithm

In this section,we discuss a local search algorithm that is adopted in order to quickly improve the quality of the population.The search direction of the local search for each solution is speci?ed by the weight values in the?tness function. The local search algorithm is similar to that mentioned in the literature[3]. Here we modify it,and add the dominated relationship to determine whether the current solution v is replaced by a neighborhood solution v?besides utilizing their ?tness.Specially,the dominated relationship is?rstly considered.The detailed procedure is as follows.

For each solution v in the population,do

(1)Examine a neighborhood solution v?of the current solution v.

(2)Calculate objective values of v?and v.If v?dominates v,then v=v?.

Otherwise,evaluate the?tness of v?and v.If the?tness of v?is no worse than v,then v=v?.

An Improved Immune Genetic Algorithm for Multiobjective Optimization647

(3)If randomly chosen N neighbor neighborhood solutions of the current solution

v have been already examined(i.e.,if there is no better solution among the examined N neighbor neighborhood solutions of v),then end this procedure.

Otherwise,return to(1).

A neighborhood solution x?of a solution x=(x1,···,x n)is de?ned as x?=(x1+Δx1,x2+Δx2,···,x n+Δx n),whereΔx i(i=1,2,···,n)is a random small number in the interval[?(x U i?x L i),(x U i?x L i)](with uniform probability distribution).x L i and x U i are lower and upper bounds of x i,respec-tively.This algorithm is terminated if no better solution is found among N neighbor neighborhood solutions that are randomly selected from the neighborhood of the current solution.

2.4Constraint Handling

In this study,we mainly adopt the idea of the constrained tournament method to handle constraints.The following constrain-domination conditions are utilized to decide the relationship of domination for any two solutions x p and x q.

De?nition1.(Constrained Domination):A solution x p is said to ‘constrain-dominate’a solution x q(x p?x q),if any of the following conditions are true:

(i)Solution x p is feasible and Solution x q is not.

(ii)Solutions x p and x q are both infeasible,but solution x p has a smaller con-straint violation.

(iii)Solutions x p and x q are both feasible and solution x p dominate solution x q in the usual sense(see De?nition1).

The e?ect of using this constrained-domination principle is that any feasible solution has a better non-domination rank than any infeasible solution.All fea-sible solutions are ranked according to their non-domination level based on the objective function values.However,among two infeasible solutions,the solution with a smaller constraint violation has a better rank.

Based on this constrained-domination principle,the set of non-constrain-dominated solutions can be obtained.In addition,the constrained-domination principle can also be applied to TSASI.However,we modify the binary tourna-ment selection approach used to choose individuals for creating subpopulation in TSASI as follows.

De?nition2.(Binary Constrained Tournament Selection):Given two solutions x p and x q,choose solution x p if any of the following conditions are true: (i)Solution x p belongs to a better non-constrain-dominated set.

(ii)Solutions x p and x q belong to the same non-constrain-dominated set,but solution x p has a better?tness value than solution x q.

By the above de?nitions,it can be seen that this constraint handling strategy does not require any extra computational burden in addition to the constraint violation computations.

648G.He,J.Gao,and L.Hu

3Numerical Experiment

In this section,we compare the performance of WBMOIGA with NSGA-II.All experiments are conducted on an IBM computer,which is equipped with a Pen-tium IV2.8G processor and512MB of internal memory.The operating system is Windows2000server and the programming language is C++.The compiler is Borland C++6.0.

3.1Experimental Setting

The speci?cation of parameters for all two algorithms is listed as follows.For WBMOIGA,the population size N=100,the size of external archived setˉN= 200,P opsize1=P opsize2=15,the crossover possibility p c=0.9,the mutation possibility p m=1/N vars,where N vars is the number of decision variables of the problem,and the reduction rate r=0.95.

For NSGA-II,we maintain the same parameters reported in its original lit-erature[6],which include a population size of100,a crossover rate of0.9,and a mutation rate of1/N vars,where N vars=number of decision variables of the problem.

3.2Performance Measures and Test Problems

Here two di?erent measures(Spacing(S)and Generational Distance(GD)) are chosen[9].In addition,we choose three test problems from a number of signi?cant past studies in this area,which include ZDT3suggested by Zitzler et al.,Viennet’s VNT,and Binh and Korn’s BNH.

3.3Experimental Results and Analysis

For all problems,the comparison of the pareto fronts produced by WBMOIGA and NSGA-II is shown in Figs.1-3,respectively.The values of the two metrics for all algorithms are respectively presented in Table1.

For Figs.1-3,we can see that WBMOIGA shows better behavior than NSGA-II for all test problems.WBMOIGA can not only obtain better Pareto-optimal front,but also have a much better spread of solutions.All these observations are also con?rmed by analyzing the Table1.From Table1,it can be found that WBMOIGA has a smaller value than NSGA-II for each metric.Therefore,we can a?rm that WBMOIGA outperforms NSGA-II for all test problems.

Table1.Results for test problems:mean value and standard deviation(σ)

Test problems Algorithms S(mean)S(σ)GD(mean)GD(σ)

ZDT3WBMOIGA 3.87E-03 1.14E-048.27E-05 1.62E-06 NSGA-II7.83E-037.21E-04 1.26E-04 1.03E-05

VNT WBMOIGA 2.91E-03 1.35E-047.21E-04 4.16E-05 NSGA-II 4.74E-029.63E-03 6.67E-03 5.32E-04

BNH WBMOIGA 3.66E-01 3.27E-02 3.99E-037.00E-04 NSGA-II 6.37E-01 3.29E-027.98E-03 1.47E-03

An Improved Immune Genetic Algorithm for Multiobjective Optimization649

Fig.1.Pareto-optimal front generated by WBMOIGA and NSGA-II for

ZDT3

0.20

f 3

Fig.2.Pareto-optimal front generated by WBMOIGA and NSGA-II for VNT

Fig.3.Pareto-optimal front generated by WBMOIGA and NSGA-II for BNH

4Conclusion

This study improves an immune genetic algorithm suggested in the literature[8]. Numerical results show that the modi?ed algorithm WBMOIGA shows better behavior for all two metrics than NSGA-II for the test problems ZDT3,VNT, and BNH.Therefore,we can a?rm that WBMOIGA outperforms NSGA-II,and can be applied to solve multiobjective optimization problems with constraints or without constraints.

650G.He,J.Gao,and L.Hu

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