Genetic algorithms with shrinking population size


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1 Comput Stat (2010) 25: DOI /s ORIGINAL PAPER Genetic algorithms with shrinking population size Joshua W. Hallam Olcay Akman Füsun Akman Received: 31 July 2008 / Accepted: 7 April 2010 / Published online: 22 April 2010 SpringerVerlag 2010 Abstract A Genetic Algorithm (GA) is an evolutionary computation technique inspired by the principle of biological evolution via natural selection. It employs the fundamental components of evolution, such as selection, mating, and mutation, which continue from generation to generation, creating better solutions as time progresses. Although it is mostly used as an optimization tool, GA enjoys a wide spectrum of applications in diverse fields such as engineering, medicine, and ecology, among others. In this study, we propose three different population size reduction methods for a typical GA optimization, aiming to increase efficiency. Additionally, we compare the accuracy and precision of these methods using Monte Carlo simulations. Keywords Genetic algorithms Population reduction Adaptive Exponential Linear reduction 1 Introduction A genetic algorithm is an optimization technique inspired by biological evolution operating under natural selection. First popularized by Holland (1975) and extensively studied by Goldberg (1989), this technique has been shown to be robust, and capable of dealing with highly multimodal and discontinuous search landscapes, where the traditional optimization techniques fail. Traditional methods such as hillclimbing Supported by program of excellence award from Illinois State University. J. W. Hallam O. Akman (B) F. Akman Department of Mathematics, Illinois State University, Campus Box 4520, Normal, IL , USA
2 692 J. W. Hallam et al. and derivativebased methods are able to find optimal points, but with multimodal landscapes, they may get stuck in local optima, whereas the structure of genetic algorithms help avoid this problem. In genetic algorithms, a group of possible solutions, i.e., a population of chromosomes in genetic algorithm terminology, are evaluated and given a fitness value based on this evaluation. The chromosomes with large fitness values are allowed to mate with other chromosomes, mutate, and move on to the next generation. This process is repeated until either a certain number of generations are reached or there is no change in the best solution found for many generations. At the end of the algorithm, the chromosome with the highest fitness is considered to be the solution. In order to take advantage of the process inspired by evolution and natural selection, chromosomes are encoded using a binary string. Let l denote the length of the string. Typically, if the function being optimized has n independent variables, then l is an integer multiple of n. The binary string is then broken into equal parts of length n, each representing one of the n variables, and converted into a real number based on the range of possible values for the variable. The fitness associated with the chromosome is calculated by evaluating the function being optimized at the n real values for each of the variables. More formally, if f : R n R denotes the function being optimized, and g :{0, 1} l R n gives the transformation from the binary string to the real values, then the fitness of a chromosome is calculated as fitness = f (g(chrom)), and the chromosomes with large fitness are chosen for the next generation. However, this choice is not deterministic. Instead, two chromosomes are selected at random, the one with the higher fitness is chosen, and the other is put back. This technique is called binary tournament. Chromosomes can be chosen more than once in a tournament. The chosen chromosomes are then put in the mating pool. This process continues until the mating pool reaches the size of the population in the next generation. Then two chromosomes are chosen from the pool and mated. This mating is analogous to genetic recombination, in which segments of the code are swapped between the two chromosomes. The number of crossover points is up to the user, but in our work we used three. After the mating occurs, the two new chromosomes are mutated. With a certain small probability, each bit may be changed from 0 to 1 or 1 to 0. This process of crossover and mutation creates two new chromosomes, which will be put into the next generation s population. This continues until all pairs in the mating pool are mated. The process of selection, mating, and mutation continue from generation to generation, creating better solutions as time progresses. In a typical genetic algorithm, the population size remains constant throughout the entire algorithm. The conditions under which the theoretical convergence (when the number of generations tends to infinity) is assured for constant size genetic algorithm are in Bhandari et al. (1996). In this study, we propose a genetic algorithm that reduces population size at every time step. This reduction initially allows for a larger population size. With a larger size, the genetic pool is more expansive, and the algorithm has a better chance of selecting parts of the correct solution early. Additionally, the reduction is controlled by userdefined parameters which do not allow the population to be reduced drastically, to avoid being trapped in local optima. We believe that reducing population size will enable the algorithm to find the correct solution more efficiently.
3 Genetic algorithms with shrinking population size 693 In Sect. 2 we study different methods of population reduction, while in Sect. 3, we examine the efficiency of these methods. Section 4 contains simulation results, followed by a discussion in Sect Methods of population reduction We have developed three different methods of population reduction for genetic algorithms. The first is an adaptive measure and the other two are based on a predetermined pattern. We describe these three methods in detail below. 2.1 Adaptive population reduction Adaptively sizing population is defined as continually changing the population size based on parameters within the algorithm. These changes could include those in average fitness and genetic variance. This method contrasts with predetermined sizing methods, in which the population size at each generation is unaffected by changes in the algorithm. Adaptive measures have been offered by several authors and a review of current methods can be found in Lima and Lobo (2005). Here we present a new method based on the change in best fitness. This approach was used before by Eiben et al. (2004). Their method was to increase the population size if the best fitness increases, decrease the size if there is a short term lack of fitness increase, and increase the population size if no change occurs for a long period of time. This approach may have several problems with it. For example, if the population is increased, then new chromosomes must be created. However, if the chromosomes are just created by cloning existing chromosomes, then there has not been an increase in genetic diversity. In the Eiben et al. study (2004) the best individuals were cloned, which did not increase genetic diversity. It would be more beneficial, in theory, to generate random individuals to simulate gene flow. Another problem is that typically in a genetic algorithm the fitness increases the quickest early in the algorithm, which would imply that the population size grows early in the algorithm. If the individuals are cloned, then the population will lose genetic diversity even faster because of the dominance of the numerous clones with large fitness. It seems, if the population size will likely increase early in the algorithm, that simply starting with a larger randomly generated population and not increasing population size would be better because of higher genetic diversity, and the same amount of computation would be used. Our approach takes the opposite view to Eiben et al. (2004). We believe that as the best fitness increases we may reduce the population size and obtain good results with less computation than a typical genetic algorithm. The only time population size is reduced is when the best fitness increases, and the method never increases the population size. To justify this, suppose we wish to optimize in a multimodal fitness landscape. If we start with a large population, then we can better explore this large landscape. However, as time continues, solutions will aggregate around a certain area in the landscape, and we can reduce the population size. Since the chromosome with the best fitness will be allowed to mate often, the solutions will concentrate around this solution. Thus, the change in best fitness is a good indicator of how well the
4 694 J. W. Hallam et al. algorithm is performing. The initial landscape is typically rugged and we need many chromosomes to explore it, but as the algorithm continues to run we can think of the problem as shrinking in ruggedness since we are concentrating on a smaller section of the landscape. A reduction in ruggedness allows for a smaller population to optimize the problem with the same or better results than a larger population. The small population size, with implementation of elitism, allows genetic drift to finetune the solution without losing the best solution in the process: suppose the population has aggregated in a small partition of the search space such that there are only slight changes in fitness. At this point, it is economical to have a small population, because a chromosome with a small difference in fitness has a better chance to be chosen to participate in a tournament. Although the choice to participate in the tournament is random, with a smaller population, every chromosome has a better chance to be chosen. Thus, those with a slightly better fitness can participate and be chosen for the mating pool. At the same time, this part of the algorithm is merely choosing between solutions which only differ little and it is less important than the phase of the algorithm making large jumps in fitness. We have developed a formula to quantify the amount of reduction. It is based on the idea that the population size should be reduced proportionally to the change in best fitness. Let N t be the population size at generation t. Denote the change in best fitness at generation t by ft best = ( ft 1 best f t 2 best ) / f best t 2. We use the absolute value to deal with fitness values which can be both positive and negative. We then determine a parameter fmax best such that ( ) 1 f best t Nt, if ft best fmax best ( ) N t+1 = 1 f best max Nt, if ft best > fmax best MIN_POPSIZE, if N t+1 is less than MIN_POPSIZE. As seen in Eq. 1, the size of the population of the next generation depends on several factors. The first is the percent change in best fitness ( ft best ). If this is below some threshold value ( fmax best ), then the population of the next generation is reduced by this percent change. If it is more, then the population is reduced by the threshold value. If either of these reductions reduces the size so that it is below the minimum size, then the size of the population is set to be the minimum size. When this type of decrease is used, we implement elitism, allowing the best chromosome to continue to the next generation without change, so that the change in fitness is always nonnegative. Clearly, we have fmax best < 1, since if f best max 1 then the population will be immediately reduced to the minimum size. As a side note, the typical genetic algorithm is a special case of the method we have produced, where fmax best = 0. The determination of minimum population size is arbitrary. However, to avoid the negative effects of extremely small populations, we set MIN_POPSIZE = 20 based on work by Reeves (1993). As can be seen from Eq. 1 and Fig. 1, the shape of the population curve is exponential decrease, followed by a steady section, again followed by an exponential decrease, and this pattern continues. In Fig. 1, the exact solution was not found before the 200th generation for Rastrigin s function and Rosenbrock s valley function. Rastrigin s function is a multimodal function which, in our experiments, (1)
5 Genetic algorithms with shrinking population size 695 Fig. 1 Population size for the adaptive method for the first 200 generations of several functions. Boxed in areas denote the generation where actual solution was found. The exact solution was not found in Rastrigin s function and Rosenbrock s valley function within 200 generations tended to need many generations to converge to the exact solution. Rosenbrock s valley function is unimodal, but very flat, requiring many generations to converge. 2.2 Predetermined exponential decrease Although the adaptive method produces a population curve which has segments of exponential decrease, it requires computing ft best at every generation, as well as the determination of fmax best. We now present a method which requires neither and reduces the population exponentially. Many theoretical results concerning genetic algorithms rest on Holland s Schema Theorem (Holland 1975). A schema can be thought of as parts of solutions (i.e., parts of a binary string) that come together to form more fit chromosomes. Holland was able to show that short schema with high fitness tend to increase in the population exponentially by crossover. This exponential increase brings highly fit segments of strings together to form better and better solutions. The Schema Theorem really underscores the importance of crossover, a characteristic of genetic algorithms that distinguishes it from other methods such as hillclimbing. Based on this theorem, we believe that we can reduce the population size exponentially and obtain results comparable to an algorithm which has no reduction. To perform this reduction, the following formula
6 696 J. W. Hallam et al. is used: N t = (N 0 + α)e c t α, wherec = ln N END α N 0 +α number of generations. (2) Here α is a parameter used to shape the population curve. In all of our studies α = 5 was chosen based on empirical evidence. Also, N END denotes the population size at the end of the algorithm. It is set to be 20, in agreement with the minimum population size used in the adaptive method. 2.3 Predetermined linear decrease It is not possible to predict the shape of the exponential increase of schema without direct and complicated calculations during the algorithm. Therefore, we have also developed a reduction method which is not exponential, but instead, which decreases the population size in a linear trend. This avoids decreasing the population too quickly, but has the benefit of reducing the number of computations needed in a traditional genetic algorithm. The following formula is used to determine the population size at each generation: N t = mt + N 0, where m = N 0 N END number of generations. (3) Again, we set N END = 20. Figure 2 depicts all three methods given above for population reduction and the typical (no reduction) method. 3 Testing reduction methods To determine the effectiveness of the reduction methods, we ran simulations using five sets of problems. Three of the five sets were based on massively multimodal functions and the others on unimodal functions. The three multimodal functions have different properties that make them interesting to compare. Although many real world problems for which genetic algorithms work well are usually multimodal, we chose to use two unimodal functions as well, in order to understand how our new reduction methods are affected by the shape of the fitness landscape. All five of the functions we used can be defined for an arbitrary number of independent variables. In all of the simulations each function had 2 10 variables. Tables 1 and 2 give information about the test functions in which n is the total number of independent variables and x j is the jth independent variable. 3.1 Fixed number of computations Two different types of simulations were run in testing the three new populationreduction techniques. In the first type the number of total computations was fixed.
7 Genetic algorithms with shrinking population size 697 Fig. 2 Population size for the exponential, linear, and adaptive reduction methods and the typical GA (no reduction) Table 1 Test functions and their equations Name Equation Type Sphere model ni=1 xi 2 Unimodal Rastrigin s function 10n + n i=1 (xi 2 10 cos (2π x i )) Multimodal ( Rosenbrock s valley function ni=2 100(x i xi 1 2 )2 + (1 x i 1 ) 2) Unimodal Schwefel s function ni=1 x i (sin x i ) Multimodal ( ni=1 ) x Ackley s path function 20 exp 0.2 i 2 n Multimodal ( ni=1 ) cos (2π x exp i ) n e (A computation refers to a single call to the function being optimized.) Using binary tournament, the number of function calls can be approximated by the number of chromosomes that will continue to the next generation. This is only an approximation, because if a chromosome is chosen more than once for the tournament, then its fitness does not need to be recalculated. After a fixed number of computations were performed, we determined the absolute difference between the best solution given by that run of the algorithm and the actual solution. For each simulation, 100 iterations
8 698 J. W. Hallam et al. Table 2 Test functions and their parameters Name Domain of x i Optimal values x Minimum Sphere model [ 6,6] (0,0,,0) 0 Rastrigin s function [ 6,6] (0,0,,0) 0 Rosenbrock s valley function [ 2.048,2.048] (1,1,,1) 0 Schwefel s function [ 500,500] ( , ,, ) n* Ackley s path function [ ,32.768] (0,0,,0) 0 Table 3 Number of computations allowed based on the number of independent variables Number of variables 2 10, , , , , , , , ,000 Number of computations allowed of the algorithm were run for the three different reduction methods in addition to the traditional genetic algorithm. Using the 100 solutions of each type, we performed t tests to determine if any of the three new methods outperformed the traditional genetic algorithm. Also, we looked at the number of times the algorithm found the correct solution. We used two different types of functions to be optimized, each with a number of variables ranging from 2 to 10. As the number of variables increases, the difficulty in finding the optimal solution increases. In light of this, as the number of variables increased, we allowed more computations to be performed. Table 3 gives the number of computations allowed for a given number of variables. 3.2 Fixed solution with acceptable tolerance With the second type of simulation we performed, the number of computations was not fixed; instead, we let the algorithm run until the solution it gave was in a certain radius about the actual solution. Once the best solution was in the tolerance level, the algorithm terminated, and the number of computations was recorded. Only Rastrigin s function and the sphere model were used for this simulation. Again, 100 iterations of each method were used, and a ttest was performed to determine whether there was a significant difference in the number of computations, required by each method to be within the tolerance level. In the adaptive and typical methods, the choice of number of generations was arbitrary. This was not the case for the exponential and linear decrease,
9 Genetic algorithms with shrinking population size 699 Table 4 Initial population size (number of generations) for fixed number of computations simulation Number of variables Adaptive Exponential Linear Typical (500) 100 (250) 100 (250) 100 (100) (500) 100 (250) 100 (250) 100 (125) (500) 100 (250) 100 (250) 100 (150) (500) 150 (250) 100 (250) 150 (175) (500) 200 (250) 100 (250) 200 (200) (500) 250 (250) 100 (250) 250 (250) (750) 300 (300) 100 (300) 300 (300) (750) 300 (350) 100 (350) 300 (350) (750) 300 (400) 100 (400) 300 (400) Table 5 Initial population size (number of generations) for tolerance study using Rastrigin s function and sphere model respectively Number of variables Adaptive Exponential Linear Typical 2,3,4, ( ) ( ) ( ) ( ) 6,7,8,9, ( ) ( ) ( ) ( ) as their population curves depend on the total number of generations. When choosing the number of generations, we erred on the side of more computation. In effect, we set the number of generations higher than necessary, which caused the population curve to have a less steep decline. This may have skewed results slightly in the direction of overcomputation, but we feel that this outweighed the problem of an algorithm not reaching the tolerance level. 3.3 Simulation parameters In all the simulation runs, the probability of mutation was set at The binary string length was 15 bits per independent variable. The number of crossover points was 3 and each crossover point was chosen so that it occurred at a multiple of 15. For the adaptive reduction method, we set fmax best =.08. For the fixed tolerance level simulation, the tolerance level was.05. Table 4 gives the initial population size and the number of generations for the fixed number of computations simulation. Table 5 gives the same information for the fixed tolerance simulation. In both types of simulations, outliers were removed from the dataset before the ttests were done. 4 Results 4.1 Fixed number of computations It is clear from the mean distance results in Table 6 and Figs. 3 and 4 that the adaptive reduction method performs well for a wide range of functions. It was never
10 700 J. W. Hallam et al. Table 6 Average distance from actual solution after outliers were removed Sphere Adaptive 6.71E E E E E E 7* 2.68E 7* 3.02E 7* 3.35E 7* Exponential 6.71E E E E E E E 6* 1.03E 5* 3.47E 5* Linear 6.71E E E E E 6* 1.33E 5* 8.55E 5* 1.90E 4* 4.05E 4* Typical 6.71E E E E E 7 3.5E E E E 5 Rastrigin s Adaptive 1.33E 5* 2E 5* 2.66E 5* 3.75E 5* 3.25E 1* 4.17E 1* 3.26E 1* 4.47E 1* 6.36E 1* Exponential 3.20E E E E E E E 1* 6.36E 1* 9.83E 1 Linear 3.55E E 1* 5.44E E E 1* 4.56E E Typical 3.84E E E E E E E Rosenbrock s Adaptive 5.30E 3* 1.87E 1* 7.06E 1* 1.59* 2.49* 3.50* Exponential 2.32E 2* 2.97E 1* * 4.27* Linear 2.53E 2* 4.17E * * Typical 1.47E E Schwefel s Adaptive 1.52E 3* 3.45E 2* 6.99E 2* 1.27E 1* 1.64E 1* 2.33E 1* 5.98E 1* 8.53E 1* 1.15* Exponential 8.63E E E E E 1* 2.49E 1* 2.61E 1* 3.33E 1* 4.98E 1* Linear 7.35E E E E E 1* 2.30E 1* 3.09E 1* 4.16E 1* 6.31E 1* Typical 8.21E E E E E E+1 Ackley s Adaptive 4.05E E E E E E 3* 4.46E 3* 4.05E 3* 4.05E 3* Exp 4.05E E E E E E 3* 1.36E 2* 2.00E 2* 3.15E 2* Linear 4.05E E E E E 3* 2.43E 2* 6.95E 2* 9.96E 2* 1.32E 1* Typical 4.05E E E E E E E E E 2 * denotes significantly better than the typical genetic algorithm * denotes significantly worse than the typical genetic algorithm, both at the α =.05 level outperformed by the typical genetic algorithm, and it outperformed the typical genetic algorithm in 34 out of the 45 total categories. This would imply that the adaptive method is robust and could be used on a number of functions with no loss of efficiency and usually with better performance. The predetermined exponential decreasing method was the next best, outperforming 11 times and being outperformed seven times. Finally, the linear reduction method was outperformed 12 times, and it outperformed the typical method nine times. Table 7 gives the fraction of replications that found the correct solution for each function with different number of variables and the average over all variables. In the table, the results for Rosenbrock s valley function were excluded because only once the correct solution was found. Again the results indicate that the adaptive reduction method is the preferred method. It had the largest average frequency of exact hits for three of the four functions considered.
11 Genetic algorithms with shrinking population size 701 Fig. 3 Results for fixed number of computations with 95% CI 4.2 Fixed solution with acceptable tolerance We turn our attention to the simulation in which the number of computations was not fixed, but the tolerance about the solution was fixed (See Table 8 for results). To reiterate, the algorithm ran until it gave a solution that was in the interval of [ 0.05,0.05] around the solution. As with the previous simulation, the adaptive reduction method was the best, finding the neighborhood with a significantly less amount of computation than the typical method in 17 out of the 18 categories. Additionally, the exponential decrease method came in second, outperforming eight times and being outperformed
12 702 J. W. Hallam et al. Fig. 4 Results for fixed number of computations with 95% CI. The legend is the same as in Fig. 3
13 Genetic algorithms with shrinking population size 703 Table 7 The proportion of replications in which the genetic algorithm got the correct solution for different number of variables Average Sphere Adapt Exp Linear Typical Rastrigin s Adapt Exp Linear Typical Schwefel s Adapt Exp Linear Typical Ackley s Adapt Exp Linear Typical The data for Rosenbrock s valley function was omitted because only one of the 3,600 replications found the correct solution four times. The linear decrease method once again did the worst among the reduction methods, outperforming only six times and being outperformed four times. It is interesting to note that with Rastrigin s function and the sphere model, once the algorithm has reached the.05 neighborhood, it is likely that the algorithm will find the correct solution as there is only one minimum in the.05 neighborhood. This is not true for any of the other test functions used in the first simulation. This is why this simulation was run with these two functions. With Rastrigin s function, the average number of computations sometimes exceeded the number of computations allowed for the first type of simulation. In the first simulation, not all the algorithms found the correct solution, supporting the concept that once in the neighborhood the correct solution would be found. 5 Discussion It is clear that the adaptive method was the best method of reduction, outperforming the typical method in almost all the simulations. When we first started the study, we hypothesized that the adaptive method would work well for complex functions. This
14 704 J. W. Hallam et al. Table 8 The average number of computations needed to get into the [.05,.05] neighborhood about the actual solution Sphere Adapt * 1272* 1510* 1736* 1941* 2184* 2401* 2671* Exp * 2163* * 3829* 4487* 5181* Linear * * 4907* 5664* Typical Rastrigin s Adapt 4295* 7810* 13667* 14532* 19673* 23801* 27951* 31944* 39837* Exp * 14127* * * * * Linear * 14747* * * * * Typical * denotes significantly better than the typical genetic algorithm * denotes significantly worse than the typical genetic algorithm, both at the α = 0.05 level is true, but it also did well even for less complex problems. Arguably, the adaptive method should be preferred over a typical genetic algorithm. There are two drawbacks to the adaptive method: one is unique to the adaptive method, and the other is a general problem with population reduction techniques. The first is that the parameter fmax best has to be determined aprioriand usually ad hoc. There are some guidelines that can be given. First, if some information about the search space is known, and the user knows that large jumps in the fitness values are possible, then a lower value should be chosen. This prevents the population from reaching its minimum size too quickly. For example, if the functions f 1 (x) = 100x 2 and f 2 (x) = x 2 were being optimized, we would have f1max best < fbest 2max. Similarly, if the search space is relatively flat, then the parameter should be larger. This is most likely why the adaptive reduction method did not do that well on Rosenbrock s valley function. The search space is quite flat and the changes in fitness are very small, which made the population decrease slowly. In turn, this increased the number of computations quickly from generation to generation, and the algorithm had to terminate at an early generation because it hit the maximum number of computations. The second drawback to the adaptive method, actually to any reduction method, is that usually the initial size is chosen to be larger, and is then reduced to compare it to a typical genetic algorithm with the same number of computations, but with a steady population size. This comparison is fair for a serial implementation of a genetic algorithm, but for a parallel implementation, it may not be. Because most time is spent evaluating the fitness function, parallel implementation can be used to evaluate the chromosomes separately. This can reduce the long computation time caused by the bottleneck of fitness evaluation. However, with a shrinking population that starts with a larger initial size, parallel implementation may not be as beneficial. This is true as long as the number of independent processors is larger than the population size in the typical genetic algorithm. If the number of processors is less than or equal to the minimum number of chromosomes in a shrinking population, then
15 Genetic algorithms with shrinking population size 705 the parallel implementation for a typical genetic algorithm should not be significantly better than that of a population reducing genetic algorithm. The other two methods, exponential and linear decrease, did not do as well as the adaptive reduction method. Possibly this occurred because they reduce the population too slowly, which prevents a large initial size. The large initial size may have been the key factor in the ability of the adaptive method to do so well. Although these tests were done for a genetic algorithm with the same features as a typical genetic algorithm except for the changing population size, there is no reason to believe that the reduction techniques will not work well for genetic algorithms with other modifications. For example, a genetic algorithm with niching may work well with the adaptive reduction technique. In this scenario, each subpopulation could be reduced independently of the other subpopulations, or reduction may depend on the evolution of the entire system. References Bhandari D, Murthy CA, Pal SK (1996) Genetic algorithm with elitist model and its convergence. Intern J Pattern Recognit Artif Intell 10: Eiben AE, Marchiori E, Valkó VA (2004) Evolutionary algorithms with onthefly population size adjustment. In: Yao X et al (ed) Parallel problem solving from nature, PPSN VIII, Lecture notes in computer science, vol Springer, Berlin, pp Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. AddisonWesley, Reading Holland J (1975) Adaptation in natural and artificial systems. The MIT Press, Ann Arbor Lima CF, Lobo FG (2005) A review of adaptive population sizing schemes in genetic algorithms. In: Proceedings of the 2005 workshops on genetic and evolutionary computation. ACM, New York, pp Reeves CR (1993) Using genetic algorithms with small populations. In: Proceedings of the 5th international conference on genetic algorithms. Morgan Kaufmann Publishers, San Francisco, pp 92 97
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