Clonal selection algorithm with operator multiplicity - IEEE Xplore

Clonal Selection Algorithm with Operator Multiplicity Adnan Acan Computer Engineering Dept., Eastern Mediterranean University, Gazimagusa, T.R.N.C. Mersin 10, TURKEY Email: [email protected]

Abstract- An artificial immune system using clonal selection principle with multiple hypermutation operators in its implementation is presented. Mutation operators to he used are identified initially. In every mutation operation, the fitness gain achieved by the employed mutation operator is computed and stored. Accordingly, mutation operators are assigned fitness values based on the amount fitness improvements they achieve over a number of previous generations. These fitness values are used to determine operator selection probabilities. This approach is used for the solution of a well-known numerical optimization problem, the frequency assignment problem, for which optimal results are achieved in reasonable computation times even for the very difficult problem instances.

I. INTRODUCTION Artificial immune systems (AIS) are biologically inspired learning and optimization methods which can be used for the solution of many different types of engineering optimization problems. They simulate the behavior of living organisms in protecting themselves against antigens. This system is an example of an evolutionary learning mechanism in which individuals that match better with those within a content addressable memory have higher survival and reproduction capacities. A population of potential solutions is evolved in a direction in which the recognition performance of individual solutions, for the target antigens, is improved over generations. The main mechanism of evolution is affinity maturation through hypermutations that aims to improve the affinity of a potential solution to selected antigens. Hence, the solution space is traversed in a randomized but goal-directed manner ~ 1 PI, . 131, ~41. Artificial immune systems are relatively new in the field of metaheuristics and there is a great deal on the derivation of various algorithmic alternatives for better localization of optimal solutions and application of the algorithms for different classes of problems. Among many different types of AIS implementations, their use for numerical and combinatorial optimization is also studied by several authors. In 1993, Forrest et al. [SI showed that an AIS combined with a GA can be used to evolve a set of antibodies for the purpose of recognizing a set of diverse pathogens. Their algorithm, called the diversity algorithm, was able to maintain diverse subpopulations within a population of individuals and the formation of such populations was similar to fitness sharing in genetic algorithms.

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Hajela et al. [6] extended the diversity algorithm and applied it for the solution of structural and constrained multicriteria optimization problems. Their algorithm was a combination of a genetic algorithm with the diversity algorithm. An initial population is randomly generated and fitness values of all individuals are computed. Also, the degree of constraint violations is evaluated for each individual. Next, subpopulations of feasible and infeasible solutions are formed and their contents are sorted in descending of their fitness values. Then, a number of best feasible solutions are considered as pathogens whereas the infeasible solutions are treated as antibodies. Antibodies are modified by the diversity algorithm. Finally, the designs from the subpopulation of feasible solutions are mixed with multiple copies of the best antibodies to form a population subjected to the standard genetic operations of selection, crossover, and mutation. After evaluating the offspring individuals, the whole process is iterated until a stopping condition is satisfied. The use of AIS for multi-modal function optimization was first addressed by Fukuda et al. [7]. They developed a six-step algorithm. The first step is the recognition of antigens whereas the second step includes the use of memory cells for the production of antibodies (candidate solutions). Memory cells are the individuals that were effective in the past. In the third step, the affinity of the antibodies to the newly entered antigens is computed. Good solutions are saved as memory cells in the fourth step while the remaining solutions are discarded. Good antibodies produce copies of themselves in the fifth step and the number of copies is proportional to the affinity values. In this step, antibodies with extremely high concentrations are suppressed to maintain diversity. Finally, in the sixth step, new solutions are introduced in place of suppressed antibodies. The developed algorithm is used for some numerical optimization problems and proved to be quite efficient for problems with complex landscapes [SI. De Castro and Von Zuben developed a pioneering approach based on theclonal selection algorithm [9], [lo], [ l l ] , [12]. Similar to Fukuda’s work, their algorithm consists of six steps and starts with the generation of a set of candidate solutions. A number of best candidates is selected as the second step. Then, these candidates are cloned with a rate proportional to their goodness. In the fourth step, each clone is subjected hypermutation with a mutation rate inversely proportional to its fitness. The best mutants are used to compose the

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memory set. Finally, a number of low fitness antibodies is replaced by newly generated individuals. Compared to the algorithm of Fukuda et al., the number of control parameters is highly reduced. The efficiency of this algorithm was tested on different classes of problems and, based on its performance, it is demonstrated to be a useful data analysis and function optimization tool. Coello and Cortes developed AIS-based algorithms for multiobjective optimization problems [ 131, [ 141. Their algorithms are based on the clonal selection principle. For all individuals in the population, they are tested for Pareto dominance and feasibility. Next, the population is divided into two subpopulations based on the Pareto dominance criterion. Nondominated or feasible individuals form the antigen subpopulation, while the dominated or infeasible individuals stay in the antibody subpopulation. Antigens are graded as “very good” and “good” by assigning a weight to each antigen. Antigens with higher weights are used as memory cells which are then cloned and mutated as described in [9]. The authors tested their algorithms on several multiobjective optimization problems and demonstrated that the immune system-based approach, named as MISA, shows competitive behavior with respect to other well-known multiobjective optimization algorithms when dealing with unconstrained test functions. However, for constrained test functions the proposed approach does not perform as well as its competitors. In this paper, an AIS using clonal selection principle with multiple mutation operators in its implementation is presented. A number of mutation operators are predefined and labelled. In a hypermutation operation, one mutation operator is selected, among the available ones, for the generation of an offspring. The probability of selection for a mutation operator is adaptively determined based on the cumulative fitness-gain achieved by the mutation operator over a number of previous generations. This way, a performance-based dynamic selection scheme is followed for operator selection. The details of the implementation are presented in the following sections. This paper is organized as follows. Basics of an AIS and the clonal selection algorithm are presented in section 2. The proposed AIS with mutation multiplicity is introduced in section 3. Experimental work and the obtained results for different benchmark problems are demonstrated in section 4. Finally, conclusions and future research directions are specified. 11. AFFINITY MATURATION A N D THECLONAL SELECTION

PRINCIPLE When an antigen enters the body of a living animal, the immune system of the animal attacks the antigen to kill it or to neutralize its effectiveness inside the body. For this purpose, bone marrow produce B cells (lymphocytes) which are able to synthesize antigen-specific antibodies. When invading antigens bind to these antibodies, they stimulate the B cells such that they start to proliferate, through mitosis cell divisions, into antibody secreting cells called plasma cells. In addition to proliferating into plasma cells, B lymphocytes also form a group of long-lived B memory cells. Memory cells circulate

throughout the body and, when exposed to a second antigenic stimulus, they start to proliferate into large lymphocytes c:apable of producing high affinity antibodies for the specific antigen that had stimulated the primary response [21, [9]. Clonal selection and affinity maturation principles are used to develop computer simulation models for the immune system, called the artificial immune systems (AIS). AIS operate on a population of points in search space and, by using the above described generational mechanisms, shows good performance as an optimization algorithm. Some characteristics of the immune system that are important in the development of an AIS can be stated as follows: 1) Immunological Learning: Learning in the immune s:ystem is achieved through two main mechanisms: First, through an expansion process, each antibody is reproduced (cloned) such that the number of copies is proportional to the effectiveness to target antigens. Second, in the expansion process, the reproduced copies are subject to errors where the error rate is inversely proportional to antigenic effectiveness. Those mutated cells with high antigenic affinity are favored to be stored in a repertoire called immunological memory. 2) Immunological Memory: Those antibodies which are proved to be useful in previous antibody-antigen interactions are stored in an antibody repertoire. Cells in this memory are templates that have the best current anti,gen recognition capability. Hence, they are considered as models to be used in the development of new antibodies. The immunological generational mechanism introduces high-rate mutations in the reproduction of these memory cells with the aim of improving the recognition capability of templates. It is easy to see that immunological memory provides main intensification mechanism in immunological search procedure. 3) Immunological Cross-Reaction: The immunological memory is associative in the sense that when an antigen is presented as the input, the memory elements that best match with the input are remembered. This way, the effectiveness of the immune system is enhanced greatly for all new antigens which are structurally similar to the ones in the memory. 4) Afinity Maturation: Antibodies that are found effective in antibody-antigen interactions are stored in immunological memory. Over generational steps, they are replaced by new antibodies with higher affinities and, hence, antibodies present in the memory gain higher recognition capabilities through the evolutionary sdection and reproduction processes. 5 ) Somatic Hypermutations: Over a number of generational steps, it is expected that antibody molecules gain higher antigen recognition capabilities by modifying their binding structures through hypermutations. For this purpose, random changes are introduced into the genomic structures of antibodies during the cloning process. Hypermutation rates are inversely proportional to fitness values (antigenic affinity).

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Immunological Memory Update: After each cloning process, while high-affinity antibodies are selected to enter the immunological repertoire, those cells with low affinities must be removed from the antibody population due to affinity maturation principle. In addition to this, a fraction of antibodies are randomly generated and put into the antibody population for diversification purposes.

111. CLONAL SELECTION W I T H MUTATION MULTIPLICITY

Operator multiplicity in evolutionary algorithms is studied by several authors and significantly better results are obtained compared to conventional single-operator implementations. Among these efforts, the use of multiple crossover operators in genetic algorithms (GAS) was studied in a number of papers. Spears [I61 proposed an adaptive GA which decides between Based on the above described characteristics, the clonal se- 2-point and uniform crossover as it runs. He concluded that, lection algorithm CLONALG was developed by De Castro and his adaptive mechanism works well especially with larger Von Zuben [9]. CLONALG is an algorithmic description of populations. More interestingly, multiplicity feature in GAS the basic mechanisms described above, which makes it possi- was introduced by Esquivel and Gallard such that either mulble to use immunological evolution to be used for engineering tiple crossovers per couple (MCPC) or multiple crossover on applications, such as pattern recognition, machine-learning, multiple parents (MCMP) strategy is used in recombinations multi-modal and multi-objective function optimization. The [17], [IS], [19]. While MCPC aims the exploration of clonal selection algorithms as described in [9] is presented features of promising solutions by multiple application of crossover operators, the objective of MCMP is to balance below: the exploitation and explorations through the use of multiple 1) Generate a set P of candidate solutions. P is composed crossovers and multiple individual samples. Both of these mulof a subset of memory cells M and the remaining tiplicity strategies have been used for the solution of single and multiple objective optimization problems. Their use in parallel subpopulation of antibodies P,. That is, ( P = P,+M). GAS is also described by the same authors. It is demonstrated 2 ) WHILE (NOT SATISFIED) by experimental evaluations that both MCPC and MCMP, 2.1 Select the n hest individuals of the population when combined with fitness proportional crossover selection, P based on an affinity measure. Put these into a provided better results than the conventional single crossover subpopulation (P,,). per couple (SCPC) strategy. Yoon et al. [20] studied the 2.2 Reproduce (clone) the elements of ( P J , giving rise synergy among multiple crossover operators. They started with to a temporary population of clones (C). The clone a number of crossover operators and investigated whether size of an individual in (P,) is an increasing function or not their combinations outperform the sole usage of the of its affinity with the antigens. best crossover operator. They concluded that careful combi2.3 Apply a hypermutation scheme for each individual nation of multiple crossover operators can produce synergy in C, where the hypermutation rate is proportional and multiple crossover combinations and crossover selection to the affinity of the antibody with the antigens. As strategies should be decided in a problem dependent manner. a result, a hypermutated antibody population c ' is Recently, Acan et al. [21] used crossover multiplicity and generated. performance based operator selection schemes in GAS and 2.4 Reselect the improved individuals from c ' to com- demonstrated that their proposed approach outperforms singlepose the memory set M . Some members of P can be operator implementations for the solution of hard numerical replaced by other improved members of c'. and combinatorial optimization problems. 2.5 Replace d antibodies by novel ones (diversity introThe idea behind the proposed approach is to bring the duction). The lower affinity cells have higher proba- advantage of the synergy provided by the use of multiple bilities of being replaced. operators into artificial immune algorithms. In this respect, The application of CLONALG for the solution of a given a number of hypermutation operators are used to change the problem requires an appropriate representation of individual structure of cloned individuals. Each individual is mutated by a solutions such that antibody-antigen interactions can be easily mutation operator which is selected based on its effectiveness measured via complementarity. Each antibody interacts with over a number of previous generations. The fitness gain all antigens that are complements of the antibody within an achieved by each mutation operator is continuously moniaffinity threshold. Accurate measurement of antibody-antigen tored over the whole course of generations and an adaptive affinity is another important component of immune algorithms. performance-based selection scheme is followed for operator Obviously, for different problem classes different affinity mea- selection. An objective function F ( . ) and affinity measure A ( . ) sures may be more appropriate due to the different shape are used in the evaluation of individual solutions. space requirements. However, general distance measures are Mutation operators to be used are predefined and indexed proposed by De Castro [15]. They are the well-known Eu- using a mutation operators' index vector ( M O W ) . After clidean, Manhattan, or Hamming distance between any two n- initializing the population, P(O), and finding the fitness valdimensional vector identities. In case of binary representations, ues of individuals using the fitness function F ( . ) , mutation Hamming distance is the most simple-to-compute and widely operator probabilities to be used are initialized to l / j M O I V \ used distance measure in practice. for all mutation operators. There is a mutation operators' 1911

credit table, MOCT, which is a Kz\MOIV( matrix where rows represent the latest K generations while each column corresponds to a mutation operator. In a mutation operation, the gain achieved by the mutation operator is computed as the fitness improvement between the parent and its offspring. That is, if p is a parent and o is its offspring after mutation, then m a z ( F ( o ) - F ( p ) ,0) is the improvement achieved by the corresponding mutation operator. Over one generation, these improvements are summed up to find the total improvement achieved by a particular mutation operator. MOCT keeps these values for all mutation operators over the latest K generations. Hence, M O C T ( i , j ) is equal to the sum of fitness improvements achieved by the use of mutation operator j in the latest i-th generation. As explained below, the contents of this table are used to compute the mutation operator selection probabilities for a particular hypermutation operation. The modified clonal selection algorithm with mutation operator multiplicity is illustrated below: 1) Generate a set P(0) of candidate solutions. P ( 0 ) is composed of the subset of memory cells M ( 0 ) and the remaining subpopulation of antibodies P,.(0). That is,

+

P(0) = Pr(0) M ( 0 ) . 2) Evaluate the initial population, F ( P ( 0 ) ) . 3 ) MProb(i) = l / l M O I V l , i = l,... , IMOIVI; 4) Initialize M O C T ; 5 ) ITER=I; DONE=FALSE. 6) WHILE (NOT DONE) 6.1 Compute MProb(i),i = 1,.. . , J M O I V / . 6.2 MOCT(l+ mod(ITER, K ) , j ) = 0, j = 1 , " ' ,IMOIVI. 6.3 Select the n best individuals of the population P based o n an affinity measure A ( . ) .Put these into a subpopulation (P,). 6.4 Reproduce (clone) the elements of (Pn), giving rise to a temporary population of clones (C). The clone size of an individual in (P,) is an increasing function of its affinity with the antigens. 6.5 for i=l to /Cl

bilities of being replaced. 6.8 ITER=ITER+I. The multiplicity feature within the proposed AIS strateg;y is demonstrated by mutation selection and operator fitness evaluation procedures. Mutation operator probabilities are nl3t fixed and dynamically changed based on the cumulative fitneirs gain achieved by each mutation operator over a number #of previous generations. In step 7, mutation operators' selection probabilities are computed proportional to their fitness valuas within the corresponding credit assignment tables as,

.. MProb(i) = x M O C T ( j , i ) / x j=1

I=1

MOCT(1,m) ( I ) *=l

which divides the sum of improvements achieved by mutation operator i over the latest K generations by the sum of improvements achieved by all mutation operators over the same period. The selection procedure followed is exactly the method followed in roulette wheel selection. The mutation application probabilities, P,, are restricted to interval [0.05, 1.01. That is, an individual with maximum affinity is mutated using the selected operator with probability 0.05 whereas the one with the smallest affinity will certainly be mutated. These values are experimentally determined and linear scaling is used to find the application probabilities of other individuals. In the initialization of the operator credit table, MOCT, all mutation operators are initially assigned with the same selection probability, MProb(i) = l/IMOIVI, i = 1,. . . , IMOIVI. Then, starting with P(O),K generations are carried out to fill in the tables as explained above. At the beginning of each generation, MOCT(1 mod(lTER,K ) , j ) , j = 1 , . . . , / M O W is set to zero to record the fitness improvements achieved by each mutation operator during the current generation. Since the last K iterations are considered, l + m o d ( I T E R ,K ) gives the row to be updated using offspring individuals within P ( I T E R + 1). Experimentally evaluated parameter values resulting in the best performance are given in the next section.

+

Mutation = SeZect(MOIV, MProb). IV. EXPERIMENTAL WORK Compute A ( C ( i ) ) . A. Experimental Setup 6.5.3 Compute Mutation Probability Pm(A(C(i)). Two test cases are considered for performance evaluation. 6.5.4 Offspring = Mutation(C(i),P m ( i ) ) . The first one includes a number of benchmark problems from 6.5.5 Evaluate Offspring. numerical optimization whereas the second one is a well6.5.6 Fitness-Gain = m a x ( F ( 0 ff s p r i n g ) known instance of combinatorial optimization, the frequency assignment problem (FAP).Due to the nature of the problems F(C(i)),O). used in experiments, CLONALG adapted for optimization 6.5.7 Insert(C*, 0f f s p r i n g ) . problems is considered [lo]. In this respect, antibody afiin6.5.8 Update MOCT(1f Mod(ITER,K ) , ity corresponds to objective function value of the antibody Mutation, FitnessGain). under consideration. Also, since there is no specific antigen 6.6 Reselect the improved individuals from c 'to com- to recognize, the whole population of potential solution!; is pose the memory set M. Some members of P can be considered as the memory set. Correspondingly, all potential replaced by other improved members of C'. solutions (antibodies) within the population are selected for 6.7 Replace d antibodies by novel ones (diversity intro- cloning. The affinity proportionate cloning is implemented so duction). The lower affinity cells have higher proba- that NI 5 Nci 5 Nz, where N I ,Nz are integer constants, 6.5.1 6.5.2

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and N c i is the clone size for a potential solution i. That is, the individual with the worst fitness value has a clone size of N I whereas the best-fitness individual's clone size Nz. In implementations N I is set 0.01* /PI whereas Nz = 0.05*IP/. Considering a moderate-size population, these setting let the worst-fitness individual generate 1 or 2 offspring where the best-fitness individual is allowed to produce 5 to 10 offspring. As explained in the previous section, mutation probability for each individual is inversely proportional to its fitness and is selected within [0.05,1]. In addition to this, elitist selection is employed such that the best 0.02 * /PI individuals are transferred to next generation unmutated. Finally, a number of individuals having the worst fitness values after maturation are replaced by newly generated ones. This number is experimentally determined as 0.01 (PI.In order to make fair comparisons with CLONALG, the same parameters settings, when appropriate, are used in its implementation also. In all experiments, real-valued vectors are used for problem representations. Experiments are carried out using a population size of 200 individuals. Each experiment is performed 10 times. All the tests were run over 1000 generations. Five different mutation operators are used, namely, point-mutation, inversion, non-reciprocal translocation, genesegment swapping, and gene-segment reinitialization. Point mutation and inversion very well-known and widely used in practice. In non-reciprocal translocation, a randomly located and arbitrary-length chromosomal segment is relocated into a randomly selected chromosomal location. In gene-segment swapping, two randomly located equal-length chromosomal segments are swapped and in gene-segment reinitialization a randomly selected chromosomal segment is initialized with randomly chosen genes. Practically, these five mutation operators are the most suitable to implement for the problem under consideration.

B. Performance of the Proposed Approach in Numerical Optimization

Artificial immune systems implemented with clonal selection algorithm, CLONALG, is compared with the proposed AIS strategy which empowers the clonal selection algorithm with mutation multiplicity for the minimization of a number of numerical optimization problems [22]. Each function has 20 variables. The best solutions found using the CLONALG and the proposed AIS strategy are given in Table I, where the proposed approach is named as AISMM (AIS with Mutation Multiplicity). In Table I, benchmark problems Michalewicz, Griewang, Rastrigin, Schwefel, Ackley, and De Jong's step function are indexed from 1 to 6, respectively. The reported global optimum values for these functions can be found in [221. AISMM provided very close to optimal results in all trials and outperformed the CLONALG in the solution of numerical optimization problems in both solution quality and the convergence speed.

TABLE I PERFORMANCE EVALUATION OF CLONALG

AND

AISMM FOR

NUMERICAL OPTIMIZATION.

Funclion

I

B e s t I%und CLONALG 1 BestFaund AlSMM Global MirL 1 ITER I Global Min 1 ITER I 7 I . I 40 1 -9.20 -M~ I. .9h? . ..

,

I

3

1

na--r.

1.0e-4

~~

4

-8264

5 6

1.0~-6 0

I

,tn

I

400

I I

,

1 I

400

. 1.0e-22 0

I I

... IS0 130

TABLE II B E N C H M A RPROBLEM K

INSTANCES FOR THE PHILADELPHIA SYTEM

* CM=Compatibility Matrix

C. Performance of the Proposed Approach in Combinatorial Optimization: The Frequency Assignment Problem The frequency assignment problem (FAP) is defined as the assignment of frequencies within a predefined bandwidth to radio transmitters in a mobile telecommunication network in such a way that certain interference and traffic constraints are all satisfied. FAP is an N€-hard combinatorial optimization problem. In this paper, a well known minimum-order fixed-frequency assignment problem (MO-FFAP), namely the Philadelphia 21-cell MO-FAP [23], [24], is considered and solved using the proposed AIS strategy. This problem is handled by many authors in literature where two different demand vectors and several constraint matrices are used to define its different instances. [23], [241, [251, [261, [271, [28]. The formulation and problem representations of MOFFAP followed in our experiments can be found in [25]. Several commonly handled problem instances of the 21cell Philadelphia system and the corresponding solution lowerbounds determined by analytical methods are listed in Table 11. For these test problem instances, the most difficult ones are the second and sixth problem instances. Comparison of AISMM with some of well-known published results is presented in Table 111. Except the second and the sixth problem problem instances, the optimal solutions are found within a few seconds and within one or two generations. For the second and the sixth problem instances, best results are reported by [24] and the same results with the theoretical lower bounds are also reached by AISMM. Comparative convergence and success rate performances of AISMM and [24] for the second and the sixth problem instances are given in Table IV and Table V. From Tables I11 to V, it is clearly seen that AISMM demonstrated much better performance compared to its well-known competitors in terms of solution quality,

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TABLE 111

approach outperforms the single-operator CLONALG in all trials. Additionally, the AISMM approach exhibits much better performance for both numerical and combinatorial optimization problems. In fact, problems from these classes are purposely chosen to examine this side of the proposed strategy. This work can be further developed to test the effectiveness of mutation multiplicity in negative selection strategy and artificial immune network approaches.

EVALUATION OF AlSMM WITH RESPECT TO PUBLISHED RESULTS NUMBERS CONTAINED I N SQUARE BRACES ARE REFERENCES

REFERENCES [ I ] Takumi K.. Hogeweg, P.: Evolution of the immune repertoire with and TABLE IV OF AlSMM AND CLONALG FOR THE SECOND PROBLEM EVALUATION

[2]

INSTANCE. NUMBERS CONTAINED I N SQUARE BRACES ARE REFERENCES

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[41 151 161

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convergence speed, and success rate

V. CONCLUSIONS AND FUTURE W O R K

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In this paper, a novel CLONALG paradigm employing mutation multiplicity in its implementation is introduced. A quite efficient dynamic operator selection scheme is used where operator selection probabilities are determined based on the fitness gains achieved by each operator over a number of previous generations. The implementation of the proposed approach does not bring a significant complexity to the implementation of the clonal selection algorithm: a small KzlMOIVI matrix is used to keep operator-related fitness gains. The implemented algorithm is used to solve difficult numerical and combinatorial optimization problems and its perfoimance is compared with that of single-operator CLONALG. From the results of case studies, the proposed AISMM

[91

[IO] [Ill [I21 I131

I141 TABLE V OF AISMM A N D CLONALG FOR THE SIXTH PROBLEM EVALUATION INSTANCE. NUMBERS CONTAINED IN SQUARE BRACES A R E REFERENCES

I151 [I61 [I71

[I81

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