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A parallel genetic algorithm with niching technique applied to a nuclear reactor core design optimization problem
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Progress in Nuclear Energy 50 (2008) 740e746 www.elsevier.com/locate/pnucene
A parallel genetic algorithm with niching technique applied to a nuclear reactor core design optimization problem Cla´udio M.N.A. Pereira a,*, Wagner F. Sacco b a
Instituto de Engenharia Nuclear e IEN/CNEN, R. He´lio de Almeida, 75 e Cidade Universita´ria, P.O. Box 68550, 21941-972, Rio de Janeiro, RJ, Brazil b Instituto Polite´cnico, Universidade do Estado do Rio de Janeiro, Depto. de Modelagem Computacional, R. Alberto Rangel s/n, P.O. Box 972285, 28601-970 Nova Friburgo, RJ, Brazil
Abstract Genetic algorithms (GAs) are global optimization techniques inspired by the mechanisms of natural evolution. In many complex problems, GAs have demonstrated to overcome traditional techniques. The key for such success is the efficient exploration of the search space obtained by maintaining diversity during the search process. In standard GAs, the crossover operator is the main responsible for exploration. Its function is to generate new candidate solutions (offspring) by exchanging information between existing ones (parents). In order to provide extra diversity, quite helpful when population is drifting to local optima, mutation operator is used. However, many times it is not able to avoid premature convergence to local optima (due to ‘‘genetic drift’’). More efficient techniques, such as distributed approaches (island GAs) and niching techniques, have been proposed. Island Genetic Algorithms (IGAs) are able to delay efficiently (but not to avoid) the genetic drift. The advantages of this approach are the simplicity of implementation and the efficiency attained by parallel processing. Niching Genetic Algorithms (NGAs), by their turn, prevent genetic drift by the maintenance of subpopulations in the so-called ‘‘niches’’. However, the computational cost is considerably increased in relation to the canonical GA. Motivated by the advantages offered by both techniques, we investigated a hybrid approach e the Niched-Island Genetic Algorithm (NIGA). The NIGA was applied to a nuclear reactor core optimization problem and compared to previously used GA-based techniques. NIGA and NGA obtained the best results. The NIGA achieves results similar to those obtained by the NGA, but in far less processing time. 2008 Elsevier Ltd. All rights reserved. Keywords: Island Genetic Algorithms; Niching methods; Reactor core design
1. Introduction Genetic algorithms (GAs) (Goldberg, 1989) are global optimization techniques inspired by the mechanisms of natural evolution that have demonstrated to perform better and to be more robust than other traditional techniques in many different complex problems. The key for such success is the efficient exploration of the search space obtained by maintaining diversity during the search process. In standard GAs, the crossover operator is the main responsible for exploration. Its function is
* Corresponding author. Tel.: þ55 21 2173 3905. E-mail addresses: [email protected] (C.M.N.A. Pereira), wfsacco@iprj. uerj.br (W.F. Sacco). 0149-1970/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pnucene.2007.12.007
to generate new candidate solutions (offspring) by exchanging information between existing ones (parents). In spite of being suitable for global optimization, the canonical GA is susceptible to being trapped in local optima. This is due to the phenomenon known as ‘‘genetic drift’’ (Goldberg, 1989), where the population loses diversity along the generations and ‘‘converges’’ to a unique point in the search space. In order to provide extra diversity, quite helpful when population is drifting to local optima, the mutation operator is used. Such operator is able to delay the genetic drift; however, it does not avoid it. This drawback has motivated some modifications on the original genetic algorithm (see Mahfoud, 1995, for example). The use of distributed multi-population approaches increases diversity by maintaining isolated subpopulations,
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
which exchange information by means of the migration operator. This technique, known as the Island Genetic Algorithm (IGA, Cantu´-Paz, 2000), generally leads to better results than the traditional GA without, though, avoiding genetic drift. The advantages of this approach are the simplicity of implementation and the efficiency attained by parallel processing. Another efficient approach to maintain diversity is the use of niching techniques. Niching Genetic Algorithms (NGAs, Mahfoud, 1995) prevent genetic drift by the maintenance of subpopulations in the so-called ‘‘niches’’. Results are generally better than those obtained using standard GAs or even IGAs (Sacco et al., 2004, for example), but the computational cost is considerably increased. The computational overhead imposed by the NGA may even limit the scope of its practical application. In fact, in many optimization processes, time-consuming simulators must be used. An example of this situation occurs in the reactor core optimization problem (Pereira et al., 1999) which is attacked here, where thousands of executions of a reactor physics simulator are required. Actually, this sort of problem is better suitable for parallel computation. The reactor core optimization problem has been the focus of many investigations, in which different GA-based techniques have been applied. In the original work (Pereira et al., 1999), a standard GA (SGA) demonstrated to overcome traditional hill-climbing optimization techniques e as expected. To improve search space exploration, the IGA (Pereira and Lapa, 2003) and the NGA (Sacco et al., 2004) models were also investigated, achieving better results than the SGA. The drawback of the NGA is the high computational cost involved, especially in more refined niching techniques. The IGA, by its turn, has proved to be better than the SGA, not only in terms of processing time, but also in terms of the results obtained. To overcome the computational overhead imposed by the use of niching techniques, we propose a hybrid approach e the Niched-Island Genetic Algorithm (NIGA). The motivation is to associate a good diversity maintenance technique (provided by the niching techniques) with a high computational performance (provided by the parallel model). Moreover, the IGA itself may also enhance diversity. Preliminary investigations using fitness sharing (Goldberg and Richardson, 1987), which is the simplest niching method, indicate a slight superiority of the NIGA (Pereira, 2005). In this work, Fuzzy-Clearing (Sacco et al., 2004), a more robust and efficient (but also slower) niching technique is investigated. The proposed NIGA, as well as the necessary background related to the ‘‘island model’’ and niching techniques are described in Section 2. Section 3 describes the reactor core optimization problem. Computational experiments, comparisons and discussions are shown in Section 4 and finally, Section 5 presents the concluding remarks.
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is located in a processor (island) which has its independent evolution process. In order to promote cooperation among the islands, a new operator, called migration is created. According to some predefined strategy, individuals migrate from one island to another. Many topologies for IGAs are possible. In this work we used an eight-island ring topology, as illustrated in Fig. 1. For each island, the searching procedure starts from its own initial population and, for some generations, some part of the search space is explored. As migration occurs, information about different regions of the search space is exchanged between islands, providing a greater populational diversity. Here, the best individual migrates every multiple of 10 generations. According to Cantu´-Paz (2000), IGAs demonstrated to be more robust and efficient than the SGA. If we consider the amount of communication required, the Island GA requires much less communication than other parallel approaches, such as the mastereslave or the cellular model (Cantu´-Paz, 2000). Therefore it is the most indicated model to run on distributed systems. 2.2. Niching techniques In NGAs, the analogy with nature is straightforward, as in an ecosystem there are different subsystems (niches) that contain many diverse species. The number of individuals in a niche is determined by its resources and by the efficiency of each individual in taking profit of these resources. Using this analogy, it is possible to maintain the population diversity in a GA. Each peak of the multi-modal function can be seen as a niche that supports a number of individuals directly proportional to its ‘‘fertility’’, which is measured by the fitness of this peak relatively to the fitness of the other peaks of the domain. The difficulty in implementing niching methods lies in the fact that the peaks are obviously not known beforehand. This complicates the process of populating each niche correctly according to its fitness. Various niching formation methods have been proposed. In a previous work (Pereira, 2005) we used ‘‘fitness sharing’’ (Goldberg and Richardson, 1987), the most widely employed niching paradigm. However, other techniques have demonstrated to be more efficient and robust (Sareni and Kra¨henbu¨hl, 1998). In this article, we employ ‘‘Fuzzy-Clearing’’ (Sacco et al., 2004), which has been
Island 2 Island 1
Island 3
Island 4
Island 8
2. The proposed Niched-Island Genetic Algorithm 2.1. Island topology
Island 7
Island 5 Island 6
The Island Genetic Algorithm (IGA) (Cantu´-Paz, 2000) is a multi-population approach for parallel GAs. Each population
Fig. 1. Island Genetic Algorithm e ring topology.
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
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successful in some real-world optimization problems (Sacco et al., 2006, for example). 2.2.1. Fitness sharing Fitness sharing consists in the reduction of the fitness of an individual proportionally to the number of nearby individuals. The shared fitness fi0 of the i-th individual is given by fi0 ¼
fi ; m0i
ð1Þ
where fi is the original fitness of the i-th individual and m0i ¼
N X
sh dij
ð2Þ
j¼1
is the niche count that takes into account the whole population in relation to the i-th individual, degrading the fitness of this individual according to the nearness of the others. Variables N and dij are, respectively, the population size and the distance between individuals i and j. The function that quantifies this proximity is the following: a d ð3Þ shðdÞ ¼ 1 sshare ; if d < sshare ; 0; otherwise; where sshare and a are user-defined constants. The first constant is of difficult estimation, requiring knowledge of the search space to be properly chosen. Generally it is estimated on a trial and error basis, in this work, it was used sshare ¼ 0.5. The second constant, a, is generally set to 1.0 (Goldberg and Richardson, 1987) so that the function is linear. 2.2.2. Clearing Clearing (Pe´trowski, 1996) lies in the same essence of sharing. As well as sharing, clearing is applied after the fitness evaluation and before selection, using a dissimilarity measure to separate the individuals in subpopulations. In this method each subpopulation contains a dominant individual: the one with the best fitness. The fact that an individual belongs to a subpopulation means that it is at less than a threshold s from this subpopulation’s dominant. But, differently from sharing, in clearing the dominant’s fitness is preserved and all the other individuals have their fitness zeroed (in the case of a maximization problem). In other words, all the resources of the niche are given to a sole individual: the dominant or winner. Pe´trowski generalized the basic clearing algorithm, stating that each niche could have more than a single winner, calling ‘‘niche’s capacity’’ (k) the number of winners. The niche’s capacity can range from 1 to the population size, so that the niching effect can be, respectively, maximized or minimized as convenient. Fig. 2 presents the clearing’s pseudocode (from Pe´trowski, 1996), where P is the population array (n individuals), s is the clearing radius, k is the capacity and nbWinners indicates the number of winners of the subpopulation associated with the current niche.
Function Clearing(σ,κ ) begin sort population P in decreasing fitness order for i = 0 to n-1 if fitness(P[i]) > 0 nbWinners := 1 for j = i+1 to n-1 if fitness(P[j]) > 0 and dist(P[i],P[j]) < σ if nbWinners < κ nbWinners :=nbWinners + 1 else fitness(P[j]) := 0 end if end if end for end if end for end
Fig. 2. Clearing’s pseudocode.
One of the great assets of clearing is that complexity is reduced to O(cn) (being c the number of peaks and n the population size), instead of sharing’s O(n2). Another one is that this method allows dosing the niching effect between the maximum (k ¼ 1) and its absence (k ¼ n). The major weakness of this method is the necessity of estimating s, as Sareni and Kra¨henbu¨hl (1998) pointed out in their article comparing the main niching methods. In spite of this weakness, they considered clearing as the best method overall. 2.2.3. Fuzzy clustering means To group the GA elements in clusters that gather individuals with a high degree of ‘‘natural association’’ is a decisive step toward the formation of niches. Each individual is assigned exclusively to a single cluster by the KMEAN adaptive algorithm (Anderberg, 1975). But a question arises here: can we assure that an individual definitely belongs to the cluster to which it was assigned? The classification scheme of the KMEAN algorithm is deterministic, being far from ideal if we have functions with unknown behavior or even with many adjacent extremes. In other words, the assignment of an individual to a cluster in real-world functions has a high degree of uncertainty. The use of fuzzy logic (Zadeh, 1965) to group the GA individuals in clusters appeared, thus, as a natural step to be taken. So, we applied to the GA the fuzzy class separation algorithm known as Fuzzy Clustering Means (FCM) (Bezdek, 1981). FCM became a popular algorithm exactly because it does not require, at each iteration, the total allocation of an individual to a certain cluster or class (using fuzzy logic terminology). This algorithm borrowed the concept of pertinence from fuzzy logic, which denotes the degree of association of an individual to a given class. In the case of KMEAN, pertinence mik of an individual k to a class i among C classes is given by 1; if dik < dlk ; 1 l c; lsi; mik ¼ ð4Þ 0; otherwise; where dik is the distance of individual k to the centroid of class i. This means that, in KMEAN, the fact that an individual
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
belongs to a given class or cluster excludes its pertinence to all the other classes. In a fuzzy classifier, pertinence mik must satisfy the following conditions: mik ˛½0; 1 for every i and k; 0<
N X
mik < N for every i;
ð5Þ
ð6Þ
mik ¼
1 2 m1 PC dik l¼1 dlk
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ð10Þ
This equation quantifies the pertinence of the k-th individual to the i-th class. Note that it takes into account not only the distance of the k-th individual to the i-th class, but also the distance of this individual to all the other classes. The FCM algorithm is, basically, the following:
k¼1 C X
mik ¼ 1 for every k:
ð7Þ
i¼1
Eq. (7) is called the probabilistic restriction, since it states that the sum of the pertinence of an individual to all the existing classes must total 1 (or 100%). Exactly like in KMEAN (Yin and Germay, 1993), in FCM the processes of representation and assignment are alternated iteratively. (a) Process of representation: a class i is represented by its centroid ci given by (Krishnapuram and Keller, 1993) PN m k¼1 mik $xk ; i ¼ 1; .; C ci ¼ P N m k¼1 mik
ð8Þ
with xk ¼ (xk1, xk2,., xkj). In the equation above, j is the number of variables, xkj is the value of the j-th variable of the k-th individual, xk is the k-th individual and m is the nebulosity degree, that can be varied between 1 and N. Hall et al. (1999) claim that the optimal value of the latter variable is m ¼ 2. Please note that in KMEAN there is no dependency on m, as mik receives either 1 or 0. (b) Process of assignment: each individual belongs to the class with the greatest proximity to it. We decided to express this maximum proximity by the greatest value of pertinence. In other words, an individual k belongs to the class i with the greatest mik. The FCM algorithm implicitly minimizes the total quadratic error of the individuals inside a class, which is given by JðL; UÞ ¼
C X N X
mmik $dik2 ;
ð9Þ
i¼1 k¼1
where L ¼ (c1, c2,., cC) is the array of centroids of all classes and U is the C N matrix that contains all pertinences mik. This matrix is known as the C-partition fuzzy matrix, where C is the number of classes fixed a priori and N is the population size. Deriving (9) in relation to mik and making it equal to zero, respecting constraints (5)e(7), the following equation is achieved:
1. Fix a number of classes C; fix m, 1 < m < N (generally m ¼ 2); 2. Set the iteration counter value to l ¼ 1; 3. Initialize the C-partition fuzzy matrix U(0); 4. Repeat Evaluate ci using Eq. (8); Update the elements of U(l ) using Eq. (10); Increment l; Until (kU(l1) U(l ))k) < 3.
2.2.4. Fuzzy-Clearing As mentioned above, Yin and Germay applied a clustering algorithm to the GA population prior to sharing with a relative success. Based on this and on the results of Sareni and Kra¨henbu¨hl (1998), that showed clearing’s superiority over sharing, we decided to cluster the GA individuals before submitting them to clearing. While in standard clearing an individual dominates those within a range s, in clearing associated with clustering dominance within a cluster was proposed. The adopted clustering method, for the reasons exposed in previous section, is FCM. The pseudocode to be applied after clustering is shown in Fig. 3 (with the modification highlighted). 2.3. The Niched-Island Genetic Algorithm (NIGA) The Niched-Island Genetic Algorithm is a straightforward model. A Niching Genetic Algorithm is implemented in each island and the islands exchange information exactly in Function Fuzzy Clearing(σ,κ) begin sort population P in decreasing fitness order for i = 0 to n-1 if fitness (P[i]) > 0 nbWinners := 1 for j = i+1 to n-1 if fitness(P[j])>0 and (cluster(P[i])=cluster(P[j])) if nbWinners < κ nbWinners := nbWinners + 1 else fitness(P[j]) := 0 end if end if end for end if end for end
Fig. 3. Fuzzy-Clearing’s pseudocode.
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
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the same manner as in the IGA using the canonical GA, as described in Section 2.1. 3. The nuclear core optimization problem As one of the objectives of this work is to compare the GA with niching method (NGA) with the Standard GA (SGA), we address the same problem, described in previous articles (Pereira et al., 1999; Pereira and Lapa, 2003; Sacco et al., 2004), which is briefly described here. Consider a cylindrical three-enrichment-zone PWR, with typical cell composed by moderator (light water), cladding and fuel. Fig. 4 illustrates such reactor. The design parameters that may be changed in the optimization process, as well as their variation ranges are shown in Table 1. The objective of the optimization problem is to minimize the average peak-factor, fp, of the proposed reactor, considering that the reactor must be critical (keff ¼ 1.0 1%) and sub-moderated, providing a given average flux f0. The objective function is written in Eq. (11).
Table 1 Parameters range Parameter
Symbol
Range
Fuel radius (in.) Cladding thickness (in.) Moderator thickness (in.) Enrichment of Zone 1 (%) Enrichment of Zone 2 (%) Enrichment of Zone 3 (%) Fuel material Cladding material
RF DC RE E1 E2 E3 MF MC
0.2e0.5 0.01e0.1 0.01e0.3 2.0e5.0 2.0e5.0 2.0e5.0 {U-Metal or UO2} {Zircaloy-2, Aluminum or Stainless-304}
where Vm is the moderator volume and the min and max subscripts refer to the lower and upper limits of the parameter ranges, given in Table 1. The values of penalty constants r1, r2, r3 were set to 10 as in a previous work (Pereira et al., 1999). The genotype was modeled in a way that each gene comprises a search parameter. A fixed-length binary encoding was used. Dimensions and enrichments were encoded with 7 bits. For fuel and cladding materials 1 and 2 bits have been used, respectively.
8 D0 keff > fp ; Dkeff 0:01; Df 0:01f0 ; > 0; > > DVm > > D0 keff > fp þ r1 $Dkeff ; Dkeff > 0:01; Df 0:01f0 ; > 0; > > DVm > 0 > D keff > > fp þ r2 $Df; Dkeff 0:01; Df > 0:01f0 ; > 0; > DVm > 0 0 > < f þ r $D keff ; Dk 0:01; Df 0:01f ; D keff < 0; p
f¼
3
eff
DVm
0
DVm
0
D keff > fp þ r1 $Dkeff þ r2 $Df; Dkeff > 0:01; Df > 0:01f0 ; > 0; > > DVm > 0 0 > D keff D keff > fp þ r1 $Dkeff þ r3 $ ; Dkeff > 0:01; Df 0:01f0 ; < 0; > > DV DVm m > 0 0 > D keff D keff > > fp þ r2 $Df þ r3 $ ; Dkeff 0:01; Df > 0:01f0 ; < 0; > DV DVm m > > 0 : f þ r $Dk þ r $Df þ r $D keff ; Dk > 0:01; Df > 0:01f ; D0 keff < 0; p
1
eff
2
3
eff
DVm
0
ð11Þ
DVm
4. Computational experiments and results Fuel R1
R2 h
Cladding
R3 Re
Δc Rf
Moderator
a
b
Fig. 4. (a) The reactor and (b) its typical cell.
In order to evaluate the performance and robustness of the proposed NIGA, several experiments using different genetic parameters (random seeds, crossover and mutation rates) were made. In such experiments, the NIGA was compared with other approaches: the SGA, the IGA, and the NGA. To avoid biased results arising from programming features, all approaches were developed using the same basic GA code and the same evaluation function. The Niched-Island Genetic Algorithm and the other GA-based techniques were implemented using the Genetic
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
Search Implementation System (GENESIS) (Greffenstette, 1990). The cell and diffusion equation calculations were performed using Reactor Physics code HAMMER (Suich and Honeck, 1967), which was interfaced to GENESIS. Due to the high processing time required (see Table 2), six experiments of 1000 generations (400,000 executions of the Reactor Physics Code) were performed for each approach. We adopted an eight-island ring topology with 50 individuals in each island for the IGA and the NIGA. A synchronous migration of the best individual at each 10 generations was adopted. For the SGA and the NGA, a single population of 400 individuals was used. For Fuzzy-Clearing, a rule-of-thumb was used to obtain the number of classes or clusters C: C ¼ n½, where n is the number of objects in the data set (McBratney and Moore, 1985). Table 2 shows the results obtained by each approach for different sets of genetic parameters. As expected, the SGA obtained the worst result. By using only crossover and mutation operators the phenomenon of genetic drift seems to be more accentuated. Even when mutation and crossrates are the highest, the GA may drift to a not so good solution, as can be seen in experiment 5. Note that the standard deviation is also the highest one. Better results were obtained by the IGA, which was also the fastest method. Experiments 1, 2 and 5 show comparatively higher fitness values due to premature convergence. The standard deviation is also relatively high. The proposed NIGA, as well as the NGA presented the best results, with the best absolute result obtained by the former. Results obtained by the NGA are comparable with those obtained by the NIGA when considering the average fitness. Such observation may contrast with former work (Pereira, 2005), in which, probably due to the use of a less robust and more parameter-dependent niching technique (the ‘‘fitness sharing’’), results obtained by the NGA were worst than those obtained by the NIGA. Here, may be due to the use of the robust and efficient Fuzzy-Clearing as niching technique, genetic drift could be efficiently avoided in both cases NGA and NIGA. The NIGA was, however, too much faster, with processing time decreased (approximately) linearly with the number of processors.
Table 2 Computational experiments Experiment
Mutation
Crossover
1 0.002 0.6 2 0.002 0.6 3 0.001 0.6 4 0.001 0.6 5 0.01 0.8 6 0.01 0.8 Average Std. Dev. Proc. time for 1000 gens. (h)
SGA
NGA (20 cls)
IGA
NIGA (7 cls)
1.2808 1.2958 1.3173 1.3217 1.3242 1.2801 1.3033 0.0204 23.8
1.2827 1.2800 1.2851 1.2867 1.2812 1.2828 1.2831 0.0025 25.7
1.2920 1.3043 1.2802 1.2803 1.2969 1.2803 1.2890 0.0103 3.0
1.2838 1.2894 1.2899 1.2779 1.2803 1.2814 1.2838 0.0049 3.2
745
5. Conclusions The present work ratified the results obtained by Sacco et al. (2004), which showed the efficiency and robustness of the Fuzzy-Clearing niching method. The limitation for the adoption of this technique, as well as for any niching method, is the necessity of a metric to evaluate the degree of similarity of the individuals. The Fuzzy-Clearing technique has been quite efficient in promoting the populational diversity. In fact, the ability of maintaining diversity through niches formation may be affected by the reduced population in islands (50 individuals). The use of a distributed island topology, however, demonstrated to be able to compensate eventual loss of performance. Besides, the use of this topology reduced considerably the processing time (linearly with the number of processors). One must keep in mind that in most real-world optimization problems, fitness evaluation can be extremely time-consuming. In such cases, the use of a parallel architecture is often a requirement for GA-based optimization. As further development, we will implement an adaptive FCM algorithm, where the number of classes is determined by the algorithm accordingly to the disposition of the data set (Ho¨ppner et al., 1999). Other hybrid approaches should be investigated. Parallel architectures using NGAs with adaptive clustering together with other specialized metaheuristics (for local search within niches) seem to be interesting. Acknowledgements Wagner F. Sacco is supported by FAPERJ (Fundac¸~ao Carlos Chagas Filho de Amparo a` Pesquisa do Estado do Rio de Janeiro) under postdoctoral grant E-26/152.661/2005 (Fixac¸~ao de Pesquisador, Nı´vel 3). Cla´udio Ma´rcio do Nascimento Abreu Pereira is supported by CNPq (National Reasearch Council, Ministry of Science and Technology, Brazil). References Anderberg, M.R., 1975. Clusters Analysis for Applications. Academic Press, New York. Bezdek, J.C., 1981. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York. Cantu´-Paz, E., 2000. Efficient and Accurate Parallel Genetic Algorithms. Kluwer Academic Publishers, Norwell, MA, USA. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, Reading, MA, USA. Goldberg, D.E., Richardson, J., 1987. Genetic algorithms with sharing for multimodal function optimization, genetic algorithms and their applications. In: Proceedings of the Second International Conference on Genetic Algorithms, Hillsdale, New Jersey, USA. p. 41. Greffenstette, J.J., 1990. A User’s Guide to GENESIS, version 5.0. Navy Center for Applied Research in Artificial Intelligence, Washington DC. Hall, L.O., Ozyurt, B., Bezdek, J.C., 1999. Clustering with a genetically optimized approach. IEEE Transactions on Evolutionary Computation 3 (2), 103e112. Ho¨ppner, F., Klawonn, F., Kruse, R., Runkler, T., 1999. Fuzzy Cluster Analysis. John Wiley & Sons, New York.
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Krishnapuram, R., Keller, J.M., 1993. A possibilistic approach to clustering. IEEE Transactions On Fuzzy Systems 1 (2), 98e110. Mahfoud, S.W., 1995. Niching Methods for Genetic Algorithms, PhD thesis, Illinois Genetic Algorithm Laboratory, University of Illinois at UrbanaChampaign, Urbana, IL. McBratney, A.B., Moore, A.W., 1985. Application of fuzzy sets to climatic classification. Agricultural and Forest Meteorology 35, 165. Pereira, C.M.N.A., 2005. A hybrid niched-island genetic algorithm applied to a nuclear core optimization problem. In: International Nuclear Atlantic Conference (INAC), Santos, SP, Brazil. Pereira, C.M.N.A., Schirru, R., Martinez, A.S., 1999. Basic investigations related to genetic algorithms in core designs. Annals of Nuclear Energy 26, 173. Pereira, C.M.N.A., Lapa, C.M.F., 2003. Coarse-grained parallel genetic algorithm applied to a nuclear reactor core design optimization problem. Annals of Nuclear Energy 30, 555. Pe´trowski, A., 1996. A clearing procedure as a niching method for genetic algorithms. In: Proceedings of the 1996 IEEE International Conference on Evolutionary Computation (ICEC’96), Nagoya, Japan. p. 798.
Sacco, W.F., Pereira, C.M.N.A., Machado, M.D., Schirru, R., 2004. The fuzzy clearing approach for a niching genetic algorithm applied to a nuclear reactor core design optimization problem. Annals of Nuclear Energy 31, 55. Sacco, W.F., Lapa, C.M.F., Pereira, C.M.NA., de Oliveira, C.R.E., 2006. A niching genetic algorithm applied to a nuclear power plant auxiliary feedwater system surveillance tests policy optimization. Annals of Nuclear Energy 33, 753e759. Sareni, B., Kra¨henbu¨hl, L., 1998. Fitness sharing and niching methods revisited. IEEE Transactions on Evolutionary Computation 2, 97. Suich, J.E., Honeck, H.C., 1967. The HAMMER System Heterogeneous Analysis by Multigroup Methods of Exponentials and Reactors. Savannah River Laboratory, Aiken, South Carolina. Yin, X., Germay, N., 1993. A fast genetic algorithm with sharing scheme using cluster analysis methods in multimodal function optimization. In: Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms. p. 450. Zadeh, L.A., 1965. Fuzzy sets. Information and Control 8, 338.
Progress in Nuclear Energy 50 (2008) 740e746 www.elsevier.com/locate/pnucene
A parallel genetic algorithm with niching technique applied to a nuclear reactor core design optimization problem Cla´udio M.N.A. Pereira a,*, Wagner F. Sacco b a
Instituto de Engenharia Nuclear e IEN/CNEN, R. He´lio de Almeida, 75 e Cidade Universita´ria, P.O. Box 68550, 21941-972, Rio de Janeiro, RJ, Brazil b Instituto Polite´cnico, Universidade do Estado do Rio de Janeiro, Depto. de Modelagem Computacional, R. Alberto Rangel s/n, P.O. Box 972285, 28601-970 Nova Friburgo, RJ, Brazil
Abstract Genetic algorithms (GAs) are global optimization techniques inspired by the mechanisms of natural evolution. In many complex problems, GAs have demonstrated to overcome traditional techniques. The key for such success is the efficient exploration of the search space obtained by maintaining diversity during the search process. In standard GAs, the crossover operator is the main responsible for exploration. Its function is to generate new candidate solutions (offspring) by exchanging information between existing ones (parents). In order to provide extra diversity, quite helpful when population is drifting to local optima, mutation operator is used. However, many times it is not able to avoid premature convergence to local optima (due to ‘‘genetic drift’’). More efficient techniques, such as distributed approaches (island GAs) and niching techniques, have been proposed. Island Genetic Algorithms (IGAs) are able to delay efficiently (but not to avoid) the genetic drift. The advantages of this approach are the simplicity of implementation and the efficiency attained by parallel processing. Niching Genetic Algorithms (NGAs), by their turn, prevent genetic drift by the maintenance of subpopulations in the so-called ‘‘niches’’. However, the computational cost is considerably increased in relation to the canonical GA. Motivated by the advantages offered by both techniques, we investigated a hybrid approach e the Niched-Island Genetic Algorithm (NIGA). The NIGA was applied to a nuclear reactor core optimization problem and compared to previously used GA-based techniques. NIGA and NGA obtained the best results. The NIGA achieves results similar to those obtained by the NGA, but in far less processing time. 2008 Elsevier Ltd. All rights reserved. Keywords: Island Genetic Algorithms; Niching methods; Reactor core design
1. Introduction Genetic algorithms (GAs) (Goldberg, 1989) are global optimization techniques inspired by the mechanisms of natural evolution that have demonstrated to perform better and to be more robust than other traditional techniques in many different complex problems. The key for such success is the efficient exploration of the search space obtained by maintaining diversity during the search process. In standard GAs, the crossover operator is the main responsible for exploration. Its function is
* Corresponding author. Tel.: þ55 21 2173 3905. E-mail addresses: [email protected] (C.M.N.A. Pereira), wfsacco@iprj. uerj.br (W.F. Sacco). 0149-1970/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pnucene.2007.12.007
to generate new candidate solutions (offspring) by exchanging information between existing ones (parents). In spite of being suitable for global optimization, the canonical GA is susceptible to being trapped in local optima. This is due to the phenomenon known as ‘‘genetic drift’’ (Goldberg, 1989), where the population loses diversity along the generations and ‘‘converges’’ to a unique point in the search space. In order to provide extra diversity, quite helpful when population is drifting to local optima, the mutation operator is used. Such operator is able to delay the genetic drift; however, it does not avoid it. This drawback has motivated some modifications on the original genetic algorithm (see Mahfoud, 1995, for example). The use of distributed multi-population approaches increases diversity by maintaining isolated subpopulations,
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
which exchange information by means of the migration operator. This technique, known as the Island Genetic Algorithm (IGA, Cantu´-Paz, 2000), generally leads to better results than the traditional GA without, though, avoiding genetic drift. The advantages of this approach are the simplicity of implementation and the efficiency attained by parallel processing. Another efficient approach to maintain diversity is the use of niching techniques. Niching Genetic Algorithms (NGAs, Mahfoud, 1995) prevent genetic drift by the maintenance of subpopulations in the so-called ‘‘niches’’. Results are generally better than those obtained using standard GAs or even IGAs (Sacco et al., 2004, for example), but the computational cost is considerably increased. The computational overhead imposed by the NGA may even limit the scope of its practical application. In fact, in many optimization processes, time-consuming simulators must be used. An example of this situation occurs in the reactor core optimization problem (Pereira et al., 1999) which is attacked here, where thousands of executions of a reactor physics simulator are required. Actually, this sort of problem is better suitable for parallel computation. The reactor core optimization problem has been the focus of many investigations, in which different GA-based techniques have been applied. In the original work (Pereira et al., 1999), a standard GA (SGA) demonstrated to overcome traditional hill-climbing optimization techniques e as expected. To improve search space exploration, the IGA (Pereira and Lapa, 2003) and the NGA (Sacco et al., 2004) models were also investigated, achieving better results than the SGA. The drawback of the NGA is the high computational cost involved, especially in more refined niching techniques. The IGA, by its turn, has proved to be better than the SGA, not only in terms of processing time, but also in terms of the results obtained. To overcome the computational overhead imposed by the use of niching techniques, we propose a hybrid approach e the Niched-Island Genetic Algorithm (NIGA). The motivation is to associate a good diversity maintenance technique (provided by the niching techniques) with a high computational performance (provided by the parallel model). Moreover, the IGA itself may also enhance diversity. Preliminary investigations using fitness sharing (Goldberg and Richardson, 1987), which is the simplest niching method, indicate a slight superiority of the NIGA (Pereira, 2005). In this work, Fuzzy-Clearing (Sacco et al., 2004), a more robust and efficient (but also slower) niching technique is investigated. The proposed NIGA, as well as the necessary background related to the ‘‘island model’’ and niching techniques are described in Section 2. Section 3 describes the reactor core optimization problem. Computational experiments, comparisons and discussions are shown in Section 4 and finally, Section 5 presents the concluding remarks.
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is located in a processor (island) which has its independent evolution process. In order to promote cooperation among the islands, a new operator, called migration is created. According to some predefined strategy, individuals migrate from one island to another. Many topologies for IGAs are possible. In this work we used an eight-island ring topology, as illustrated in Fig. 1. For each island, the searching procedure starts from its own initial population and, for some generations, some part of the search space is explored. As migration occurs, information about different regions of the search space is exchanged between islands, providing a greater populational diversity. Here, the best individual migrates every multiple of 10 generations. According to Cantu´-Paz (2000), IGAs demonstrated to be more robust and efficient than the SGA. If we consider the amount of communication required, the Island GA requires much less communication than other parallel approaches, such as the mastereslave or the cellular model (Cantu´-Paz, 2000). Therefore it is the most indicated model to run on distributed systems. 2.2. Niching techniques In NGAs, the analogy with nature is straightforward, as in an ecosystem there are different subsystems (niches) that contain many diverse species. The number of individuals in a niche is determined by its resources and by the efficiency of each individual in taking profit of these resources. Using this analogy, it is possible to maintain the population diversity in a GA. Each peak of the multi-modal function can be seen as a niche that supports a number of individuals directly proportional to its ‘‘fertility’’, which is measured by the fitness of this peak relatively to the fitness of the other peaks of the domain. The difficulty in implementing niching methods lies in the fact that the peaks are obviously not known beforehand. This complicates the process of populating each niche correctly according to its fitness. Various niching formation methods have been proposed. In a previous work (Pereira, 2005) we used ‘‘fitness sharing’’ (Goldberg and Richardson, 1987), the most widely employed niching paradigm. However, other techniques have demonstrated to be more efficient and robust (Sareni and Kra¨henbu¨hl, 1998). In this article, we employ ‘‘Fuzzy-Clearing’’ (Sacco et al., 2004), which has been
Island 2 Island 1
Island 3
Island 4
Island 8
2. The proposed Niched-Island Genetic Algorithm 2.1. Island topology
Island 7
Island 5 Island 6
The Island Genetic Algorithm (IGA) (Cantu´-Paz, 2000) is a multi-population approach for parallel GAs. Each population
Fig. 1. Island Genetic Algorithm e ring topology.
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successful in some real-world optimization problems (Sacco et al., 2006, for example). 2.2.1. Fitness sharing Fitness sharing consists in the reduction of the fitness of an individual proportionally to the number of nearby individuals. The shared fitness fi0 of the i-th individual is given by fi0 ¼
fi ; m0i
ð1Þ
where fi is the original fitness of the i-th individual and m0i ¼
N X
sh dij
ð2Þ
j¼1
is the niche count that takes into account the whole population in relation to the i-th individual, degrading the fitness of this individual according to the nearness of the others. Variables N and dij are, respectively, the population size and the distance between individuals i and j. The function that quantifies this proximity is the following: a d ð3Þ shðdÞ ¼ 1 sshare ; if d < sshare ; 0; otherwise; where sshare and a are user-defined constants. The first constant is of difficult estimation, requiring knowledge of the search space to be properly chosen. Generally it is estimated on a trial and error basis, in this work, it was used sshare ¼ 0.5. The second constant, a, is generally set to 1.0 (Goldberg and Richardson, 1987) so that the function is linear. 2.2.2. Clearing Clearing (Pe´trowski, 1996) lies in the same essence of sharing. As well as sharing, clearing is applied after the fitness evaluation and before selection, using a dissimilarity measure to separate the individuals in subpopulations. In this method each subpopulation contains a dominant individual: the one with the best fitness. The fact that an individual belongs to a subpopulation means that it is at less than a threshold s from this subpopulation’s dominant. But, differently from sharing, in clearing the dominant’s fitness is preserved and all the other individuals have their fitness zeroed (in the case of a maximization problem). In other words, all the resources of the niche are given to a sole individual: the dominant or winner. Pe´trowski generalized the basic clearing algorithm, stating that each niche could have more than a single winner, calling ‘‘niche’s capacity’’ (k) the number of winners. The niche’s capacity can range from 1 to the population size, so that the niching effect can be, respectively, maximized or minimized as convenient. Fig. 2 presents the clearing’s pseudocode (from Pe´trowski, 1996), where P is the population array (n individuals), s is the clearing radius, k is the capacity and nbWinners indicates the number of winners of the subpopulation associated with the current niche.
Function Clearing(σ,κ ) begin sort population P in decreasing fitness order for i = 0 to n-1 if fitness(P[i]) > 0 nbWinners := 1 for j = i+1 to n-1 if fitness(P[j]) > 0 and dist(P[i],P[j]) < σ if nbWinners < κ nbWinners :=nbWinners + 1 else fitness(P[j]) := 0 end if end if end for end if end for end
Fig. 2. Clearing’s pseudocode.
One of the great assets of clearing is that complexity is reduced to O(cn) (being c the number of peaks and n the population size), instead of sharing’s O(n2). Another one is that this method allows dosing the niching effect between the maximum (k ¼ 1) and its absence (k ¼ n). The major weakness of this method is the necessity of estimating s, as Sareni and Kra¨henbu¨hl (1998) pointed out in their article comparing the main niching methods. In spite of this weakness, they considered clearing as the best method overall. 2.2.3. Fuzzy clustering means To group the GA elements in clusters that gather individuals with a high degree of ‘‘natural association’’ is a decisive step toward the formation of niches. Each individual is assigned exclusively to a single cluster by the KMEAN adaptive algorithm (Anderberg, 1975). But a question arises here: can we assure that an individual definitely belongs to the cluster to which it was assigned? The classification scheme of the KMEAN algorithm is deterministic, being far from ideal if we have functions with unknown behavior or even with many adjacent extremes. In other words, the assignment of an individual to a cluster in real-world functions has a high degree of uncertainty. The use of fuzzy logic (Zadeh, 1965) to group the GA individuals in clusters appeared, thus, as a natural step to be taken. So, we applied to the GA the fuzzy class separation algorithm known as Fuzzy Clustering Means (FCM) (Bezdek, 1981). FCM became a popular algorithm exactly because it does not require, at each iteration, the total allocation of an individual to a certain cluster or class (using fuzzy logic terminology). This algorithm borrowed the concept of pertinence from fuzzy logic, which denotes the degree of association of an individual to a given class. In the case of KMEAN, pertinence mik of an individual k to a class i among C classes is given by 1; if dik < dlk ; 1 l c; lsi; mik ¼ ð4Þ 0; otherwise; where dik is the distance of individual k to the centroid of class i. This means that, in KMEAN, the fact that an individual
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belongs to a given class or cluster excludes its pertinence to all the other classes. In a fuzzy classifier, pertinence mik must satisfy the following conditions: mik ˛½0; 1 for every i and k; 0<
N X
mik < N for every i;
ð5Þ
ð6Þ
mik ¼
1 2 m1 PC dik l¼1 dlk
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ð10Þ
This equation quantifies the pertinence of the k-th individual to the i-th class. Note that it takes into account not only the distance of the k-th individual to the i-th class, but also the distance of this individual to all the other classes. The FCM algorithm is, basically, the following:
k¼1 C X
mik ¼ 1 for every k:
ð7Þ
i¼1
Eq. (7) is called the probabilistic restriction, since it states that the sum of the pertinence of an individual to all the existing classes must total 1 (or 100%). Exactly like in KMEAN (Yin and Germay, 1993), in FCM the processes of representation and assignment are alternated iteratively. (a) Process of representation: a class i is represented by its centroid ci given by (Krishnapuram and Keller, 1993) PN m k¼1 mik $xk ; i ¼ 1; .; C ci ¼ P N m k¼1 mik
ð8Þ
with xk ¼ (xk1, xk2,., xkj). In the equation above, j is the number of variables, xkj is the value of the j-th variable of the k-th individual, xk is the k-th individual and m is the nebulosity degree, that can be varied between 1 and N. Hall et al. (1999) claim that the optimal value of the latter variable is m ¼ 2. Please note that in KMEAN there is no dependency on m, as mik receives either 1 or 0. (b) Process of assignment: each individual belongs to the class with the greatest proximity to it. We decided to express this maximum proximity by the greatest value of pertinence. In other words, an individual k belongs to the class i with the greatest mik. The FCM algorithm implicitly minimizes the total quadratic error of the individuals inside a class, which is given by JðL; UÞ ¼
C X N X
mmik $dik2 ;
ð9Þ
i¼1 k¼1
where L ¼ (c1, c2,., cC) is the array of centroids of all classes and U is the C N matrix that contains all pertinences mik. This matrix is known as the C-partition fuzzy matrix, where C is the number of classes fixed a priori and N is the population size. Deriving (9) in relation to mik and making it equal to zero, respecting constraints (5)e(7), the following equation is achieved:
1. Fix a number of classes C; fix m, 1 < m < N (generally m ¼ 2); 2. Set the iteration counter value to l ¼ 1; 3. Initialize the C-partition fuzzy matrix U(0); 4. Repeat Evaluate ci using Eq. (8); Update the elements of U(l ) using Eq. (10); Increment l; Until (kU(l1) U(l ))k) < 3.
2.2.4. Fuzzy-Clearing As mentioned above, Yin and Germay applied a clustering algorithm to the GA population prior to sharing with a relative success. Based on this and on the results of Sareni and Kra¨henbu¨hl (1998), that showed clearing’s superiority over sharing, we decided to cluster the GA individuals before submitting them to clearing. While in standard clearing an individual dominates those within a range s, in clearing associated with clustering dominance within a cluster was proposed. The adopted clustering method, for the reasons exposed in previous section, is FCM. The pseudocode to be applied after clustering is shown in Fig. 3 (with the modification highlighted). 2.3. The Niched-Island Genetic Algorithm (NIGA) The Niched-Island Genetic Algorithm is a straightforward model. A Niching Genetic Algorithm is implemented in each island and the islands exchange information exactly in Function Fuzzy Clearing(σ,κ) begin sort population P in decreasing fitness order for i = 0 to n-1 if fitness (P[i]) > 0 nbWinners := 1 for j = i+1 to n-1 if fitness(P[j])>0 and (cluster(P[i])=cluster(P[j])) if nbWinners < κ nbWinners := nbWinners + 1 else fitness(P[j]) := 0 end if end if end for end if end for end
Fig. 3. Fuzzy-Clearing’s pseudocode.
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the same manner as in the IGA using the canonical GA, as described in Section 2.1. 3. The nuclear core optimization problem As one of the objectives of this work is to compare the GA with niching method (NGA) with the Standard GA (SGA), we address the same problem, described in previous articles (Pereira et al., 1999; Pereira and Lapa, 2003; Sacco et al., 2004), which is briefly described here. Consider a cylindrical three-enrichment-zone PWR, with typical cell composed by moderator (light water), cladding and fuel. Fig. 4 illustrates such reactor. The design parameters that may be changed in the optimization process, as well as their variation ranges are shown in Table 1. The objective of the optimization problem is to minimize the average peak-factor, fp, of the proposed reactor, considering that the reactor must be critical (keff ¼ 1.0 1%) and sub-moderated, providing a given average flux f0. The objective function is written in Eq. (11).
Table 1 Parameters range Parameter
Symbol
Range
Fuel radius (in.) Cladding thickness (in.) Moderator thickness (in.) Enrichment of Zone 1 (%) Enrichment of Zone 2 (%) Enrichment of Zone 3 (%) Fuel material Cladding material
RF DC RE E1 E2 E3 MF MC
0.2e0.5 0.01e0.1 0.01e0.3 2.0e5.0 2.0e5.0 2.0e5.0 {U-Metal or UO2} {Zircaloy-2, Aluminum or Stainless-304}
where Vm is the moderator volume and the min and max subscripts refer to the lower and upper limits of the parameter ranges, given in Table 1. The values of penalty constants r1, r2, r3 were set to 10 as in a previous work (Pereira et al., 1999). The genotype was modeled in a way that each gene comprises a search parameter. A fixed-length binary encoding was used. Dimensions and enrichments were encoded with 7 bits. For fuel and cladding materials 1 and 2 bits have been used, respectively.
8 D0 keff > fp ; Dkeff 0:01; Df 0:01f0 ; > 0; > > DVm > > D0 keff > fp þ r1 $Dkeff ; Dkeff > 0:01; Df 0:01f0 ; > 0; > > DVm > 0 > D keff > > fp þ r2 $Df; Dkeff 0:01; Df > 0:01f0 ; > 0; > DVm > 0 0 > < f þ r $D keff ; Dk 0:01; Df 0:01f ; D keff < 0; p
f¼
3
eff
DVm
0
DVm
0
D keff > fp þ r1 $Dkeff þ r2 $Df; Dkeff > 0:01; Df > 0:01f0 ; > 0; > > DVm > 0 0 > D keff D keff > fp þ r1 $Dkeff þ r3 $ ; Dkeff > 0:01; Df 0:01f0 ; < 0; > > DV DVm m > 0 0 > D keff D keff > > fp þ r2 $Df þ r3 $ ; Dkeff 0:01; Df > 0:01f0 ; < 0; > DV DVm m > > 0 : f þ r $Dk þ r $Df þ r $D keff ; Dk > 0:01; Df > 0:01f ; D0 keff < 0; p
1
eff
2
3
eff
DVm
0
ð11Þ
DVm
4. Computational experiments and results Fuel R1
R2 h
Cladding
R3 Re
Δc Rf
Moderator
a
b
Fig. 4. (a) The reactor and (b) its typical cell.
In order to evaluate the performance and robustness of the proposed NIGA, several experiments using different genetic parameters (random seeds, crossover and mutation rates) were made. In such experiments, the NIGA was compared with other approaches: the SGA, the IGA, and the NGA. To avoid biased results arising from programming features, all approaches were developed using the same basic GA code and the same evaluation function. The Niched-Island Genetic Algorithm and the other GA-based techniques were implemented using the Genetic
C.M.N.A. Pereira, W.F. Sacco / Progress in Nuclear Energy 50 (2008) 740e746
Search Implementation System (GENESIS) (Greffenstette, 1990). The cell and diffusion equation calculations were performed using Reactor Physics code HAMMER (Suich and Honeck, 1967), which was interfaced to GENESIS. Due to the high processing time required (see Table 2), six experiments of 1000 generations (400,000 executions of the Reactor Physics Code) were performed for each approach. We adopted an eight-island ring topology with 50 individuals in each island for the IGA and the NIGA. A synchronous migration of the best individual at each 10 generations was adopted. For the SGA and the NGA, a single population of 400 individuals was used. For Fuzzy-Clearing, a rule-of-thumb was used to obtain the number of classes or clusters C: C ¼ n½, where n is the number of objects in the data set (McBratney and Moore, 1985). Table 2 shows the results obtained by each approach for different sets of genetic parameters. As expected, the SGA obtained the worst result. By using only crossover and mutation operators the phenomenon of genetic drift seems to be more accentuated. Even when mutation and crossrates are the highest, the GA may drift to a not so good solution, as can be seen in experiment 5. Note that the standard deviation is also the highest one. Better results were obtained by the IGA, which was also the fastest method. Experiments 1, 2 and 5 show comparatively higher fitness values due to premature convergence. The standard deviation is also relatively high. The proposed NIGA, as well as the NGA presented the best results, with the best absolute result obtained by the former. Results obtained by the NGA are comparable with those obtained by the NIGA when considering the average fitness. Such observation may contrast with former work (Pereira, 2005), in which, probably due to the use of a less robust and more parameter-dependent niching technique (the ‘‘fitness sharing’’), results obtained by the NGA were worst than those obtained by the NIGA. Here, may be due to the use of the robust and efficient Fuzzy-Clearing as niching technique, genetic drift could be efficiently avoided in both cases NGA and NIGA. The NIGA was, however, too much faster, with processing time decreased (approximately) linearly with the number of processors.
Table 2 Computational experiments Experiment
Mutation
Crossover
1 0.002 0.6 2 0.002 0.6 3 0.001 0.6 4 0.001 0.6 5 0.01 0.8 6 0.01 0.8 Average Std. Dev. Proc. time for 1000 gens. (h)
SGA
NGA (20 cls)
IGA
NIGA (7 cls)
1.2808 1.2958 1.3173 1.3217 1.3242 1.2801 1.3033 0.0204 23.8
1.2827 1.2800 1.2851 1.2867 1.2812 1.2828 1.2831 0.0025 25.7
1.2920 1.3043 1.2802 1.2803 1.2969 1.2803 1.2890 0.0103 3.0
1.2838 1.2894 1.2899 1.2779 1.2803 1.2814 1.2838 0.0049 3.2
745
5. Conclusions The present work ratified the results obtained by Sacco et al. (2004), which showed the efficiency and robustness of the Fuzzy-Clearing niching method. The limitation for the adoption of this technique, as well as for any niching method, is the necessity of a metric to evaluate the degree of similarity of the individuals. The Fuzzy-Clearing technique has been quite efficient in promoting the populational diversity. In fact, the ability of maintaining diversity through niches formation may be affected by the reduced population in islands (50 individuals). The use of a distributed island topology, however, demonstrated to be able to compensate eventual loss of performance. Besides, the use of this topology reduced considerably the processing time (linearly with the number of processors). One must keep in mind that in most real-world optimization problems, fitness evaluation can be extremely time-consuming. In such cases, the use of a parallel architecture is often a requirement for GA-based optimization. As further development, we will implement an adaptive FCM algorithm, where the number of classes is determined by the algorithm accordingly to the disposition of the data set (Ho¨ppner et al., 1999). Other hybrid approaches should be investigated. Parallel architectures using NGAs with adaptive clustering together with other specialized metaheuristics (for local search within niches) seem to be interesting. Acknowledgements Wagner F. Sacco is supported by FAPERJ (Fundac¸~ao Carlos Chagas Filho de Amparo a` Pesquisa do Estado do Rio de Janeiro) under postdoctoral grant E-26/152.661/2005 (Fixac¸~ao de Pesquisador, Nı´vel 3). Cla´udio Ma´rcio do Nascimento Abreu Pereira is supported by CNPq (National Reasearch Council, Ministry of Science and Technology, Brazil). References Anderberg, M.R., 1975. Clusters Analysis for Applications. Academic Press, New York. Bezdek, J.C., 1981. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York. Cantu´-Paz, E., 2000. Efficient and Accurate Parallel Genetic Algorithms. Kluwer Academic Publishers, Norwell, MA, USA. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, Reading, MA, USA. Goldberg, D.E., Richardson, J., 1987. Genetic algorithms with sharing for multimodal function optimization, genetic algorithms and their applications. In: Proceedings of the Second International Conference on Genetic Algorithms, Hillsdale, New Jersey, USA. p. 41. Greffenstette, J.J., 1990. A User’s Guide to GENESIS, version 5.0. Navy Center for Applied Research in Artificial Intelligence, Washington DC. Hall, L.O., Ozyurt, B., Bezdek, J.C., 1999. Clustering with a genetically optimized approach. IEEE Transactions on Evolutionary Computation 3 (2), 103e112. Ho¨ppner, F., Klawonn, F., Kruse, R., Runkler, T., 1999. Fuzzy Cluster Analysis. John Wiley & Sons, New York.
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Krishnapuram, R., Keller, J.M., 1993. A possibilistic approach to clustering. IEEE Transactions On Fuzzy Systems 1 (2), 98e110. Mahfoud, S.W., 1995. Niching Methods for Genetic Algorithms, PhD thesis, Illinois Genetic Algorithm Laboratory, University of Illinois at UrbanaChampaign, Urbana, IL. McBratney, A.B., Moore, A.W., 1985. Application of fuzzy sets to climatic classification. Agricultural and Forest Meteorology 35, 165. Pereira, C.M.N.A., 2005. A hybrid niched-island genetic algorithm applied to a nuclear core optimization problem. In: International Nuclear Atlantic Conference (INAC), Santos, SP, Brazil. Pereira, C.M.N.A., Schirru, R., Martinez, A.S., 1999. Basic investigations related to genetic algorithms in core designs. Annals of Nuclear Energy 26, 173. Pereira, C.M.N.A., Lapa, C.M.F., 2003. Coarse-grained parallel genetic algorithm applied to a nuclear reactor core design optimization problem. Annals of Nuclear Energy 30, 555. Pe´trowski, A., 1996. A clearing procedure as a niching method for genetic algorithms. In: Proceedings of the 1996 IEEE International Conference on Evolutionary Computation (ICEC’96), Nagoya, Japan. p. 798.
Sacco, W.F., Pereira, C.M.N.A., Machado, M.D., Schirru, R., 2004. The fuzzy clearing approach for a niching genetic algorithm applied to a nuclear reactor core design optimization problem. Annals of Nuclear Energy 31, 55. Sacco, W.F., Lapa, C.M.F., Pereira, C.M.NA., de Oliveira, C.R.E., 2006. A niching genetic algorithm applied to a nuclear power plant auxiliary feedwater system surveillance tests policy optimization. Annals of Nuclear Energy 33, 753e759. Sareni, B., Kra¨henbu¨hl, L., 1998. Fitness sharing and niching methods revisited. IEEE Transactions on Evolutionary Computation 2, 97. Suich, J.E., Honeck, H.C., 1967. The HAMMER System Heterogeneous Analysis by Multigroup Methods of Exponentials and Reactors. Savannah River Laboratory, Aiken, South Carolina. Yin, X., Germay, N., 1993. A fast genetic algorithm with sharing scheme using cluster analysis methods in multimodal function optimization. In: Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms. p. 450. Zadeh, L.A., 1965. Fuzzy sets. Information and Control 8, 338.