Goldberg, D. E. and Deb, K., 1991, A comparative analysis of selection schemes used in genetic algorithms, Foundations of Genetic Algorithms, G. J. E. Rawlins, ed., pp. 69--93.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that equation (10) admits an analytical solution. Theorem 10 (Convergence) The distribution p(x; t) for proportionate selection is given by p(x; 0)f(x) p(y; 0)f(y) 14) Let M be the set of global optima, then 1=jMj x 2 M (15) Equation (14) was already used by Goldberg and Deb [7] in a di erent context. It enables an exact schema analysis for an ideal genetic algorithm. This is a conceptual algorithm because it needs an exponential amount of computation. But for small problems the increase or decrease of any schema can be exactly computed. In the next section we will ....

D.E. Goldberg and K. Deb. A comparative analysis of selection schemes used in genetic algorithms. In G. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93, Morgan Kaufmann, San Mateo, 1991.

.... location along the chain, and a surface term for each triangle on the surface of the molecule [80] the stochastic reaction kinetics of mutation and fitness proportional selection [19] In order to save computer resources we resort to a somewhat simpler approximate scheme of tournament selection [24] where two individuals in the population are picked at random, their fitness is compared, and the fitter one is replicated. In order to limit the population size, the child organism replaces another randomly picked individual. This reaction scheme in essence reproduces Eigen s quasi species model ....

Goldberg, D. E. and K. Deb: 1991, `A Comparative Analysis of Selection Schemes Used in Genetic Algorithms'. In: G. J. E. Rawlins (ed.): Foundations of Genetic Algorithms. San Mateo, CA, pp. 69--93.

....to implement, computationally efficient, and allows for fine tuning the selective pressure by increasing or decreasing the tournament size q. For an overview of selection methods and a characterization of their selective pressure in terms of numerical measures, the reader should consult [148] [149]. While most of these selection operators have been introduced in the framework of a generational genetic algorithm, they can also be used in combination with the steady state and generation gap methods outlined in section III. The ( evolution strategy uses a deterministic selection scheme. ....

.... the algorithm (e.g. representation independent recombination and mutation operators [170] 171] the requirement that small changes by mutation occur more frequently than large ones [172] 48] and a quantification of the selective pressure imposed by the most commonly used selection operators [149]) Nevertheless, evolutionary algorithms often yield excellent results when applied to complex optimization problems where other methods are either not applicable or turn out be unsatisfactory (a variety of examples can be found in [80] Important practical problem classes where evolutionary ....

D. E. Goldberg and K. Deb, "A comparative analysis of selection schemes used in genetic algorithms," In Rawlins [67], pp. 69--93.

.... to identify the exploiters (selfish agents) We started these experiments with 10 tasks assigned to each agent per evaluation period and increased the Selection of the best candidate from a set of randomly selected candidates is known as tournament selection in the genetic algorithms literature [7]. In our case, the selection pressure is further increased because the candidate set is not chosen randomly but proportionate to the fitness of individuals in the population. Initially we adopted a purely proportionate selection scheme without the tournament selection component. In such cases, ....

K. Deb and D. Goldberg. A comparative analysis of selection schemes used in genetic algorithms. In G. J. Rawlins, editor, Foundations of Genetic Algorithms, pages 69--93, San Mateo, CA, 1991. Morgan Kaufman.

....using the multimodal algorithm. Our first objective is to investigate e#ects of one point crossover on the peakdiscovery capability of the immunity based GA system, if any. Unless stated otherwise, these experiments use an antibody population size of 100, a binary tournament selection scheme [3], one point crossover with various rates, mutation rate of 0.005, and ran for 150 generations. The antigen population is 50 000 . 0 and 50 111 . 1, and both antigens and antibodies are binary strings of length 20. The number of samples, #, is 10, which is 10 of the population size. ....

Goldberg, D. E. and Deb, K.: A Comparative Analysis of Selection Schemes used in Genetic Algorithms. Foundation of Genetic Algorithms (1991) 69-93.

....population (generation t 1) is formed by selecting the ttest individuals. Some members of the new population undergo mutation or recombination. This paper deals with a selection methods that can be used to form a population x t 1 from x t . Among the classical methods found in the literature [9, 15] is tness proportionate selection , sometimes implemented as roulette wheel selection [12] an individual 2 x t is selected for reproduction with probability P [ 2x t f( Another selection method that enjoys large popularity is linear rank selection [14] which de nes the ....

K. Deb and D. E. Goldberg. A comparative analysis of selection schemes used in genetic algorithms. In Gregory J. E. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93, San Mateo, CA, 1991. Morgan Kaufmann.

....and worst member of the population is constant, independent of how many generations have passed. This has the effect of making selective pressure more constant and controlled. Code for linear ranking is given by Whitley [50] Another fast but noisy way to implement ranking is Tournament Selection [19, 17] To construct the intermediate population, select two strings at random and place the best in the intermediate population. In expectation, every string is sampled twice. The best string wins both tournaments and gets 2 copies in the intermediate population. The median string wins one and loses one ....

....is pre assigned according to the position of the individual in the sorted population. This also allows one to prevent duplicates from being introduced into the population. This selection schema also means that the best N 1 solutions are always preserved in a population of size N. Goldberg and Deb [17] have shown that by replacing the worst member of the population, Genitor can generate much higher selective pressure than the canonical genetic algorithm. In practice, steady state genetic algorithms such as Genitor are often better optimizers than the canonical generational genetic algorithm. ....

D. Goldberg and K. Deb. A Comparative Analysis of Selection Schemes Used in Genetic Algorithms. In G. Rawlins, editor, FOGA -1, pages 69--93. Morgan Kaufmann, 1991.

....algorithm. Table 1 summarizes the genetic operators, their 727 application scheme and controlling parameters used in our timetabling problem. Table 1. Genetic operators used in this work. Fitness attribution operator cI I direct evaluation. Selection operator Sr F I binary tournament [6]. Crossover operator uniform crossover [3] crossover probability [0, 1] Mutation operator random mutation with heuristic repair. mutation probability [0, 1] Insertion operator total reinsertion. 4. PROBLEM MAPPING This section present the mapping of the FETP into a constrained GA ....

....to another. The production of new candidate solutions relies on the selection, in the current population, of the fittest candidate solutions and share some of their elements. The whole reproduction process involves a selection operator and a crossover operator. In this work, binary tournament [6] is the application scheme for the selection of the fittest 728 candidate solutions. This scheme randomly select two candidate solutions from the current population and compare their fitness value. The candidate solution with the greatest fitness value wins the tournament. In order to apply ....

[Article contains additional citation context not shown here]

D.E. Goldberg, K. Deb, "A comparative analysis of selection schemes used in genetic algorithms," in G. Rawlins, ed., Foundations of Genetic Algorithms, Morgan Kaufmann, 1991.

....parent. The crossover operation is repeated as often as desired, usually until the new generation is completed. Mutation is carried out by randomly changing the value of a single bit (with small probability) from the bit strings. There are four selection schemes commonly used in genetic algorithms [5]: 1) proportional reproduction or roulette wheel selection, 2) ranking selection, 3) tournament selection and 4) Genitor ( or steady state ) selection. Commonly, some of the best individuals are copied into the next generation population intact. This operation is known as elitism. In the ....

D. Goldberg and K. Deb. A comparative analysis of selection schemes used in genetic algorithms. In G. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93. Morgan Kaufmann, Berlin, 1991.

....the reproducing set of a population. Reproduction: from each reproducing set, apply reproductive strategies to generate a number of new chromosomes. Replacement: replace some or all of the original population with new chromosomes. Selection can use one of the four common selection schema [Goldberg and Deb, 1994]: 1. Proportionate reproduction; 2. Ranking selection; 3. Tournament selection; 4. Genitor (or steady state ) selection; We present here a subset of the first selection scheme which was the approach used in the simple GA, base for this work. Further information for the other schemes can be ....

....2. Ranking selection; 3. Tournament selection; 4. Genitor (or steady state ) selection; We present here a subset of the first selection scheme which was the approach used in the simple GA, base for this work. Further information for the other schemes can be found in the reference [Goldberg and Deb, 1994]. We adopted the model where all strings are able to mate with one another (random mating) The selection strategy of strings for reproduction is based on their fitness values and different methods can be used. The stochastic sampling with replacement (or roulette wheel selection) De Jong, 1975] ....

Goldberg, D. E. and Deb, K.: "A Comparative Analysis of Selection Schemes Used in Genetic Algorithms", Foundations of Genetic Algorithms, pages 69 93. Morgan Kaufmann Publishers, Inc., San Mateo, USA (1994).

....fitnesses of intermediate individuals are regularly spread out. Because of this, the effect of one or two extreme individuals will be negligible, irrespective of how much greater or less their fitnesses are than the rest of the population. C. 2 Tournament Selection In binary tournament selection (Goldberg and Deb, 1991), two individuals are taken at random, and the better individual is selected from the two. If binary tournament selection is being done without replacement, then the two individuals are set aside for the next selection operation, and they are not replaced into the population. Since two individuals ....

Goldberg, D. E. and Deb, K. (1991). A comparative analysis of selection schemes used in genetic algorithms. Foundations of Genetic Algorithms, pages 69--93.

....15 20 25 30 128 256 512 1024 Static Dynamic Execution Time Population Size Time (s) Figure 9. Static versus dynamic implementation of the population in C (Knapsack problem with 20 objects) For example, when selection uses a sorted population (e.g. when applying selection by ranking [4]) it would be better to use some kind of vector operations in order to ease the frequently undertaken sorting steps. See such an scenario implemented in Java in Figure 10 with three different reproductive plans (steady state, generational, and cellular) for solving a difficult deceptive problem ....

Goldberg D.E., Deb K., "A Comparative Analysis of Selection Schemes Used in Genetic Algorithms", in Rawlins G.J.E. (ed.), Foundations of Genetic Algorithms, Morgan Kaufmann, 69-93, 1991.

....nature of multi point crossover may result in a more robust search by encouraging exploration of the search space rather than early convergence to highly fit individuals. For a discussion of # point crossover and generalized multi point crossover schemes see [11] A Tournament selection scheme [8] with a tournament size of 5 is used to select the parents at each generation. In Tournament selection, individuals are chosen randomly from the population and the best individual from this group is selected as a parent. This is repeated until enough parents have been chosen to ....

D.E. Goldberg and K. Deb. A comparative analysis of selection schemes used in genetic algorithms. In Foundations of Genetic Algorithms, pages 69--93. 1991.

....solution. The parameters of selection methods regulate the selection pressure, which in turn determines how fast the algorithms converge. We shall see that the parameters of migration also a ect the selection pressure. The speed of convergence of di erent selection schemes was rst studied by Goldberg and Deb (1991), who introduced the concept of takeover time. The takeover time is the number of generations that selection alone requires to replicate a single individual of the best class until the population is full. The next section of this paper extends Goldberg and Deb s analysis to consider di erent ....

....from migration. Of course, many others have studied and compared di erent selection methods used in EAs (for example, see the papers by B ack (1994) and by Hancock (1997) pressure.tex; 2 11 1999; 15:34; p.4 Migration Policies and Parallel EAs 5 3. Takeover Times This section is based on Goldberg and Deb s (1991) analysis of the takeover times of tournament selection, but similar calculations may be performed for other selection schemes. This section considers a simpli ed population model with only two classes of individuals: good and bad. We may think of the good individuals as representatives of the ....

Goldberg, D. E. and K. Deb: 1991, `A comparative analysis of selection schemes used in genetic algorithms'. Foundations of Genetic Algorithms 1, 69-93. (Also TCGA Report 90007).

....(e.g. 11,7] To achieve this balance, we must understand how selection a ects the composition of the population. Various methods have been used to quantify the e ect of the selection pressure that selection algorithms exert on the population. Goldberg and Deb introduced the takeover time [8], which is the number of generations that the selection algorithm takes to reproduce a single representative of the optimal solution to occupy the entire population. High selection pressures result in short takeover times. Others quantify the selection pressure using the selection intensity ....

....becomes larger with larger tournaments and smaller populations, but in many practical situations the di erences are so small that they may be considered negligible. It is known that the distribution of individuals selected by pairwise tournaments (s = 2) and linear ranking with n = 0 are identical [3,8], and this can be appreciated by comparing gures 5 and 1. 4. Comparing the Selection Methods Although it is convenient to compare selection methods by their takeover time or selection intensity, these may not be suciently complete representations. To understand why and to facilitate ....

D. E. Goldberg and K. Deb. A comparative analysis of selection schemes used in genetic algorithms. Foundations of Genetic Algorithms 1, 69-93, 1991.

....search behavior, leading to good convergence reliability. Characterization of the distinctive features of various selection methods is a prerequisite for effective application of evolutionary algorithms and for the design of new algorithms. Several researchers have studied selection schemes. Goldberg and Deb, 1991, provide an analysis on the convergence time and growth ratio of several selection schemes. Introducing the term takeover time, they considered the sole effect of selection, eliminating the influence of other genetic operators such as crossover and mutation. Bck and Hoffmeister, 1991, studied the ....

....Optimization 64 Figure 4 Evolution of fitness values averaged over 100 runs (N=20) Figure 5 Evolution of fitness values averaged over 100 runs (N=50) 5. Analysis Analysis of selection in evolutionary algorithms can be made at three different levels: the fitness of the best individual (Goldberg and Deb, 1991; Bck, 1996) the average fitness B.T. Zhang J.J. Kim 65 of the population (Mhlenbein and Schlierkamp Voosen, 1993) and the distribution of fitness values in the population (Blickle and Thiele, 1995) In the following, we analyze the experimental results by comparing them with those of ....

[Article contains additional citation context not shown here]

Goldberg D.E. & Deb D. (1991) A comparative analysis of selection schemes used in genetic algorithms, In Rawlins, G.J.E. (ed.) Foundations of Genetic Algorithms, pp. 69-93.

....of o spring produced. Later, De Jong and Sarma (1993) presented additional empirical evidence, and suggested alternative deletion methods to reduce the variance. Whitley (1989) introduced GENITOR, a steady state GA in which the worst individual was deterministically replaced every iteration. Goldberg and Deb (1991) analyzed GENITOR, and they observed that it has a high selective pressure even when the parents were selected randomly. This suggests that the deletion of worst individuals induced a higher selection pressure than the rank based method used to select the parents. This will be quanti ed in section ....

....of I surv (or I s ) is at G = 1=n, and it can be of considerable magnitude. For example, for n = 256, I surv = 2:96, and for n = 1000, I surv = 3:36. So, even if the parents are selected randomly, replacing the worst individuals causes a considerable selection pressure. This is consistent with Goldberg and Deb s (1991) calculations of GENITOR, and Chakraborty et al. s (1996) Markov chains analysis. 1 In his study of ( selection, B ack (1995) shows that for n 50 the approximation is indistinguishable from the real values. 0.2 0.4 0.6 0.8 1 Gap 1 2 3 4 5 I (a) Select best, delete worst 0.2 0.4 ....

Goldberg, D. E., & Deb, K. (1991). A comparative analysis of selection schemes used in genetic algorithms. In Rawlins, G. J. E. (Ed.), Foundations of Genetic Algorithms (pp. 69-93.) San Mateo, CA: Morgan Kaufmann.

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Goldberg, D. E. and Deb, K., 1991, A comparative analysis of selection schemes used in genetic algorithms, Foundations of Genetic Algorithms, G. J. E. Rawlins, ed., pp. 69--93.

....I. INTRODUCTION Genetic algorithms (GAs) traditionally use bit string representation. However, in recent years many researchers have concentrated on using real valued genes in GAs [Michalewicz 94] Surry 96] Eshelman 97] Ono 99] Theoretical studies of real coded GAs have also been performed [Goldberg 91] Eshelman 97] Qi 94] Kita, 99] Higuchi, 00] Previous studies [Tsutsui 99] Higuchi 00] have proposed simplex crossover (SPX) for real coded GAs. SPX has various good characteristics, e.g. it does not depend on a coordinate system, the mean vector of parents and offspring generated with ....

....(p.d.f. p(X,t) of individuals in the population under proportional selection at generation t can be written as X X 1 X 2 SPX X SPX X 1 X 2 X 3 X 3 X X X 1 X 2 SPX X SPX X 1 X 2 X 3 X 3 X Fig. 2. Division of SPX 5 = dX X p X F X p X F t X p t t ) 0 0 (10) Goldberg 91] where p 0 (X) is an initial p.d.f. of individuals and no special bias is assumed to be added by crossover and mutation operators. Normally p 0 (X) is the uniform distribution. From Eq. 10) we can see that the distribution of individuals in a population reflects the fitness landscape ....

Goldberg, D. E., and Deb, K.: A comparative analysis of selection schemes used in genetic algorithms, Foundation of Genetic Algorithms, pp. 69-93 (1991). 12

....iteration Consider the case where the evaluation function F(X) is non negative. Then the probability density function (p.d.f. p(X,t) of individuals in the population under proportional selection at generation t can be written as = dX X p X F X p X F t X p t t ) 0 0 (10) Goldberg 91b] where p 0 (X) is an initial p.d.f. of individuals and no special bias is assumed to be added by crossover and mutation operators. Normally p 0 (X) is the uniform distribution. Then, p(X,t) can be written as = dX X F X F t X p t t ) 11) From Eq. 11) we can see that the ....

Goldberg, D. E., and Deb, K.: A comparative analysis of selection schemes used in genetic algorithms, Foundation of Genetic Algorithms, Morgan Kaufmann Publishers, San Mateo, CA, pp. 69-93 (1991).

....converge solely by genetic drift. 1 Introduction The development of convergence time models for the simple genetic algorithm (SGA) is important for understanding the eciency and scaling capabilities of the algorithm. Some of the rst e orts in this direction were made by Goldberg (1989b) and Goldberg and Deb (1991). Both studies developed estimates of takeover times in genetic algorithms (GAs) focusing on the dynamics of the best individual. A convergence time model was developed by M uhlenbein and Schlierkamp Voosen (1993) using the concept of selection intensity. The validity of this model is limited to ....

Goldberg, D. E., & Deb, K. (1991). A comparative analysis of selection schemes used in genetic algorithms. Foundations of Genetic Algorithms, 1 , 69-93. (Also TCGA Report 90007).

....other available selection schemes can be referenced in Thierens et al. 1994, 1998) The convergence rate of a GA is dependent on the selection scheme used to propagate highly fit individuals. In single objective applications, tournament selection has been the preferred selection scheme because Goldberg and Deb (1991) showed that the method is robust and less prone to premature convergence. In multiobjective applications, stochastic remainder selection is most frequently used for rapid convergence rates because fitness sharing maintains population diversity and prevents premature convergence (Fonseca Fleming ....

Goldberg, D. E., and K. Deb, (1991) A comparative analysis of selection schemes used in genetic algorithms, in Foundations of Genetic Algorithms, pp. 69-93, Morgan Kaufman, San Mateo, CA.

....large and small populations for the available time budget. It gives a new face to the existing population sizing models indicating feasible solution runs for otherwise non optimal population settings. 3. 3 Epoch Duration Convergence (M uhlenbein, 1992; Thierens Goldberg, 1994) and takeover time (Goldberg Deb, 1991) studies are equally important in modeling the continuation operators. The convergence time model gives us an upper bound on the epoch duration or in other words, for a given time budget T , it gives an estimate on the maximum number of generations in a feasible epoch; since a run with population ....

Goldberg, D. E., & Deb, K. (1991). A comparative analysis of selection schemes used in genetic algorithms. Foundations of Genetic Algorithms, 1 , 69-93. (Also TCGA Report 90007).

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Goldberg and Deb, 1991. A comparative analysis of selection schemes used in genetic algorithms. In G. J. E. Rawlins, editor, FOGA, pp 69-93. Morgan Kaufmann.

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D.E. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms, in: G.J.E. Rawlins (Ed.), Foundations of Genetic Algorithms, Morgan Kaufmann, 1991.

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Goldberg D. E. (1991) A comparative analysis of selection schemes used in genetic algorithms. In Gregory Rawlins, editor. Foundations of Genetic Algorithms, pages 69-93, San Mateo, CA: Morgan Kaufmann Publishers.

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Goldberg, D.E. and Deb, K. (1991): A comparative analysis of selection schemes used in genetic algorithms. In: Foundations of Genetic Algorithms, Ed. Gregory J.E. Rawlings (Morgan Kaufmann), pp. 69--93.

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D.E.Goldberg. A comparative analysis of selection schemes used in Genetic Algorithms. In Foundations of Genetic Algorithms, pages 6993, San Mateo, California 1991. Morgan Kaufman Publishers.

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Goldberg and Deb, 1991. A comparative analysis of selection schemes used in genetic algorithms. In G. J. E. Rawlins, editor, FOGA, pp 69-93. Morgan Kaufmann.

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Goldberg, D.E., & Deb, K. 1991. A Comparative Analysis of Selection Schemes Used in Genetic Algorithms. Pages 69-93 of: Rawlins, G. (ed), Foundations of Genetic Algorithms. San Mateo: Morgan-Kaufman.

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D.E. Goldberg and K. Deb. A comparative analysis of selection schemes used in genetic algorithms. In G. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93, Morgan Kaufmann, San Mateo, 1991.

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Goldberg, D.E. & Deb, K. (1991). A comparative analysis of selection schemes used in genetic algorithms. In Rawlins, G. (Ed.), Foundations of Genetic Algorithms (pp. 27 69--93), San Mateo, CA, Morgan-Kaufman.

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D. Goldberg, K. Deb. A comparative analysis of selection scheme used in genetic algorithms. In: Foundations of Genetic Algorithms, Morgan Kaufmann Publishers, 1991, 69-93.

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K. Deb and D. Goldberg. A comparative analysis of selection schemes used in genetic algorithms. In G. J. Rawlins, editor, Foundations of Genetic Algorithms, pages 69--93. Morgan Kaufmann, San Mateo, California, USA, 1991.

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D. E. Goldberg and K. Deb, "A comparative analysis of selection scheme used in genetic algorithms," in Foundations of Genetic Algorithms,G. Rawlins , Ed. San Mateo, CA: Morgan Kaufman, 1991.

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D. E. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms, in FOGA91, vol 1, pp 69-93, 1991, also TCGA Report 90007.

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D. E. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms (1991),in FOGA91, vol. 1, pp 69-93.

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David E. Goldberg and Kalyanmoy Deb. A comparative analysis of selection schemes used in genetic algorithms. In G. Rawlins, editor, Foundations of Genetic Algorithms, pages 69--93, San Mateo, 1991. Morgan Kaufmann.

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D. E. Goldberg and K. Deb, "A comparative analysis of selection schemes used in genetic algorithms," in Foundations of Genetic Algorithms, G. J. E. Rawlins, Ed. 1991, pp. 69--93, Morgan Kaufmann.

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D. E. Goldberg, K. Deb, "A comparative analysis of selection schemes used in genetic algorithms", In G. Rawlins, ed. Foundations of Genetic Algorithms, Morgan Kaufmann, 1991.

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Goldberg, D.E. and Deb, K. A comparative analysis of selection schemes used in genetic algorithms. In Rawlings, G., editor, Foundations of Genetic Algorithms. Morgan Kaufmann, 1991.

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David E. Goldberg. A comparative analysis of selection schemes used in genetic algorithms. In Gregory Rawlins, editor, Foundations of Genetic Algorithms. Morgan Kaufman, 1991. 21

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David E. Goldberg and Kalyanmoy Deb. A comparative analysis of selection schemes used in genetic algorithms. In G. Rawlins, editor, Foundations of Genetic Algorithms, pages 69--93, San Mateo, 1991. Morgan Kaufmann.

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D. E. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms (1991),in FOGA91, vol. 1, pp 69-93.

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D. E. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms, in FOGA91, vol 1, pp 69-93, 1991, also TCGA Report 90007.

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D.E. Goldberg and K. Deb. A Comparative Analysis of Selection Schemes Used in Genetic Algorithms. In G.J.E. Rawlins, editor, Foundations of Genetic Algorithm & Classifier Systems, pages 69--93. Morgan Kaufman, San Mateo, CA, 1991.

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Goldberg, David E. and K. Deb (1991). "A Comparative Analysis of Selection Schemes Used in Genetic Algorithms." In Foundations of Genetic Algorithms. Bruce M. Spatz (Ed.) pp. 69-93. Morgan Kaufmann Publishers, Inc, San Mateo, CA.

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Goldberg D.E., & Deb K. (1991). A comparative analysis of selection schemes used in genetic algorithms. Proceedings of Foundations of Genetic Algorithms FOGA-I. ed. G. Rawlings. pp.69-93. Morgan Kaufmann.

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Goldberg, D.E., Deb, K., A comparative analysis of selection schemes used in genetic algorithms, Foundations of Genetic Algorithms, Morgan Kaufmann, (1991), 69-93.

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D.E. Goldberg. A comparative analysis of selection schemes used in genetic algorithms. In Gregory Rawlins, editor, Foundations of Genetic Algorithms, San Mateo, CA: Morgan Kaufmann Publishers. 1991.

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