Genetic Algorithms, Selection Schemes, and the Varying Eects of Noise. IlliGAL Report No November Department of General Engineering

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1 Genetic Algorithms, Selection Schemes, and the Varying Eects of Noise Brad L. Miller Det. of Comuter Science University of Illinois at Urbana-Chamaign David E. Goldberg Det. of General Engineering University of Illinois at Urbana-Chamaign IlliGAL Reort No November 1995 Deartment of General Engineering University of Illinois at Urbana-Chamaign 117 Transortation Building 104 South Mathews Avenue Urbana, IL 61801

2 Genetic Algorithms, Selection Schemes, and the Varying Eects of Noise Brad L. Miller Det. of Comuter Science University of Illinois at Urbana/Chamaign David E. Goldberg Det. of General Engineering University of Illinois at Urbana/Chamaign November 14, 1995 Abstract This aer analyzes the eect of noise on dierent selection mechanisms for genetic algorithms. Models for several selection scheme are develoed that successfully redict the convergence characteristics of genetic algorithms within noisy environments. The selection schemes modeled in this aer include roortionate selection, tournament selection, - selection, and linear ranking selection. These models are shown to accurately redict the convergence rate of genetic algorithms under a wide range of noise levels. 1 Introduction Selection schemes rimarily determine the convergence characteristics of genetic algorithms (GAs). Good rogress has been made in develoing models for several dierent selection schemes that successfully redict the convergence characteristics of a GA within a deterministic (noiseless) environment. However, these models are not designed for noisy environments, where tness functions only aroximately measure the true tness of individuals. This aer seeks to model the convergence characteristics of several selection schemes for noisy environments. Convergence models of dierent selection schemes was rst broached in Goldberg (1989), and later exanded in Goldberg and Deb (1991). Muhlenbein and Schlierkam-Voosen (1993) introduced the use of selection intensity for convergence analysis of genetic algorithm selection schemes. The convergence characteristics of several dierent selection schemes have recently been successfully modeled for deterministic environments. In Thierens and Goldberg (1994), convergence models for deterministic environments were develoed for several selection schemes, including roortionate selection, binary tournament selection, and truncation selection. Both Back (1995) and Miller and Goldberg (1995) alied order statistics to extend Thierens and Goldberg's tournament selection model to handle tournament sizes larger than two. Back also used order statistics to develo a model for - selection (Back, 1995). A convergence model for linear ranking was resented by Blickle and Thiele (1995). The urose of this aer is to further our understanding of how selection ressure works, with secial attention on how noise alters the eects of selection ressure. Convergence models are then develoed that utilize our new understanding of how noise aects selection ressure. Convergence models for noisy environments are develoed for several selection schemes, including tournament selection, - selection, linear ranking, and stochastic universal selection. These models accurately redict the convergence of GAs, and are veried for a wide range of noise levels using the onemax domain. Section 2 rovides the basic background needed to understand this aer, including a discussion of selection scheme tyes, selection intensity, noise denitions, and the onemax domain. Section 3 exands the selection intensity equation to handle noise, and adats the noisy selection intensity model to the onemax domain. This forms the basis of the convergence models for both roortionate-based and ordinal-based selection. Section 1

3 2 4 discusses the exerimental methodology used, and resents the results of exeriments comaring actual convergence erformance of several selection schemes with redicted erformance. Lastly, section 5 resents some general conclusions from this research. 2 Background In this section, a brief overview of the basic background information needed to understand this aer is given. The role of selection schemes in GAs is discussed, and a general selection intensity model is resented. Next, noise and noisy tness functions are exlained. The last subsection deals with the onemax domain, as well as using the general selection intensity model to redict erformance in the deterministic onemax domain. 2.1 Selection Schemes Genetic algorithms use a selection scheme to select individuals from the oulation to insert into a mating ool. Individuals from the mating ool are used by a recombination oerator to generate new osring, with the resulting osring forming the basis of the next generation. As the individuals in the mating ool ass their genes on to the next generation, it is desirable that the mating ool be comrised of \good" individuals. A selection scheme in GAs is simly a rocess that favors the selection of better individuals in the oulation for the mating ool. The selection ressure is the degree to which the better individuals are favored: the higher the selection ressure, the more the better individuals are favored. This selection ressure drives the GA to imrove the oulation tness over succeeding generations. The convergence rate of a GA is largely determined by the magnitude of the selection ressure, with higher selection ressures resulting in higher convergence rates. Genetic algorithms are able to to identify otimal or near-otimal solutions under a wide range of selection ressure (Goldberg, Deb, & Thierens, 1993). However, if the selection ressure is too low, the convergence rate will be slow, and the GA will unnecessarily take longer to nd the otimal solution. If the selection ressure is too high, there is an increased chance of the GA rematurely converging to an incorrect (sub-otimal) solution. In addition to roviding selection ressure, selection schemes should also reserve oulation diversity, as this hels avoid remature convergence. There are two basic tyes of selection schemes commonly used today: roortionate selection and ordinalbased selection. Proortionate-based selection selects individuals based uon their tness values relative to the tness of the other individuals in the oulation. Some common roortionate-based selection schemes are roortionate selection (Holland, 1975), stochastic remainder selection (Booker, 1982; Brindle, 1981), and stochastic universal selection (Baker, 1987; Grefenstette & Baker, 1989). Ordinal-based selection schemes select individuals not uon their tness, but based uon their rank within the oulation. This entails that the selection ressure is indeendent of the tness distribution of the oulation, and is solely based uon the relative ordering (ranking) of the oulation. Some common ordinal-based selection schemes are tournament selection (Brindle, 1981; Goldberg, Korb, & Deb, 1989), - selection (Schwefel, 1981), truncation selection (Muhlenbein & Schlierkam-Voosen, 1993), and linear ranking (Baker, 1985; Baker, 1987; Grefenstette & Baker, 1989). Ordinal selection schemes are normally referred over roortional selection schemes for a variety of reasons, the most rominent of which is the scaling roblem (Forrest, 1985; Goldberg, 1989; Whitley, 1989). A selection scheme is said to be scale invariant if multilying every individuals' tness by a constant does not change the selection ressure. A selection scheme is said to be translation invariant if adding a constant to every individuals' tness does not change the selection ressure. Proortionate selection methods are normally scale invariant, but translation variant. Ordinal-based selection schemes are translation and scale invariant. 2.2 Selection Intensity The selection intensity, I, measures the magnitude of the selection ressure rovided by a selection scheme. The selection intensity of genetic algorithms, as dened by Muhlenbein and Schlierkam-Voosen (1993), is the exected average tness of a oulation after selection is erformed uon a oulation whose tness is distributed according to the unit normal distribution N(0; 1). If the selection intensity I of a selection scheme is known, and the oulation tness at generation t is distributed N( t ; 2 t ), the exected mean tness of a oulation after selection can be determined: t+1 = t + t I: (1)

4 3 An imortant assumtion of this model is that oulation tness is normally distributed before selection. In ractice, this is true or aroximately true for many domains, as recombination and mutation oerators have a normalizing eect on the oulation tness distribution. Table 1 gives the selection intensity for several common selection schemes. Back (1995) and Miller and Goldberg (1995) indeendently alied order statistics to derive the selection intensity for tournament selection. The order statistics are for the unit normal distribution N(0; 1); thus i:j reresents the exected value of the ith biggest samle out of a samle of size j drawn from the unit normal distribution. The maximal order statistic s:s determines the selection ressure of a tournament of size s. The study by Back (1995) also derives the selection intensity for - selection. In - selection, the best individuals are selected out of a random samle of individuals. The selection ressure is simly the average of the to th order statistics of samle size. The selection intensity of linear ranking is given by Blickle and Thiele (1995), where n + denotes the number of desired coies of the best individual. Linear ranking selects each individual in the oulation with a robability linearly roortional to the rank of the individual. Imlicit in the selection intensity value for linear ranking is that 1 n + 2, and n + + n? = 2, where n? is the number of desired coies of the worst individual. Muhlenbein and Schlierkam-Voosen (1993) derived the selection intensity for roortionate selection, which directly deends on the current mean t and standard deviation t of the oulation in generation t. Proortionate selection selects individuals for the mating ool with a robability directly roortional to the individuals' tness. The selection intensity equation for roortionate selection is used in this aer to redict the erformance of stochastic universal selection, one of a handful of dierent roortionate selection schemes. The selection intensity of roortionate selection is unique in that it is the only one that is sensitive to the current oulation distribution. Selection Scheme Parameters Selection Intensity I Tournament Selection s s:s 1 X - Selection ; i=?+1 Linear Ranking n + (n +? 1) 1 Proortionate Selection t ; t t = t i: Table 1: Selection Intensity of Common Selection Schemes. 2.3 Noise While there are many dierent denitions of noise, in this aer we are concerned with the factors that revent the accurate evaluation of the tness of individuals. This noise results in the tness functions being inaccurate, so the tness function in turn are referred to as noisy tness functions. There are many factors that may necessitate the use of noisy tness functions. In some domains, there may be no known tness function that can accurately assess an individual's tness, so an aroximate (noisy) tness function must be used. In domains where comutational seed is aramount, fast but noisy tness functions may be referred over slow but accurate tness functions. Noisy information can also negatively aect the accuracy of a tness evaluation. Information noise can come from a variety of sources, including noisy data, knowledge uncertainty, samling error, sensor inut, and human error. As the selection rocess is based uon tness values, noisy tness functions cause the selection rocess itself to be noisy. The noisy tness value of an individual can be viewed as the sum of the real tness of the individual lus a random noise comonent. In this aer, we assume that the noise comonent is randomly drawn from an unbiased (mean of zero) normal distribution. The assumtion of an unbiased, normally distributed noise source is true or aroximately true in many noisy domains, and allows the eects of noise to be more easily modeled. 2.4 Onemax Domain This aer uses the onemax domain, also known as the counting ones or bit-counting roblem, to verify the accuracy of the selection scheme models. The ability to accurately redict the erformance of a selection

5 4 scheme oerating within the onemax domain demonstrates a basic understanding of the underlying selection mechanism. This section reviews the onemax domain, and derives the convergence model for the onemax domain. Within the onemax domain, the real tness of an individual is simly the number of one bits in the chromosome. The otimal chromosome is simly the chromosome containing all one bits. The oulation mean tness and variance for the onemax domain are given by t = l t and t 2 = l t (1? t ); resectively (Muhlenbein & Schlierkam-Voosen, 1993; Thierens & Goldberg, 1994), where l is the chromosome length and t is the ercentage of correct alleles in the oulation at generation t. Given that the general selection intensity equation 1 can be reresented as t+1? t = t I; the rate of change of the ercentage of correct alleles can be determined by t+1? t = I t (1? t ): l Aroximating the above dierence equation with a dierential equation yields d t (1? t ): (2) dt = I l For a randomly initialized oulation, 0 = 0:5 is a reasonable aroximation for the initial ercentage of correct alleles. Using this, Equation 2 can be solved exactly to yield t = 1 2 (1 + sin( l I t)): (3) The time until convergence t conv for 0 = 0:5 can also be solved to yield t conv = l 2I : (4) The above derivation holds for cases where the selection intensity I is indeendent of the ercentage of correct alleles t. However, for roortional selection, the selection intensity is directly deendent on t, so the derivation is dierent. For roortionate selection (Thierens & Goldberg, 1994), t+1? t = 1 l ti; = 2 t l t ; = l t(1? t ) l 2 t ; = 1 l (1? t): Aroximating the above with the dierential d dt, and again assuming that 0 = 0:5, gives t = 1? 0:5e? t l (5) for the secial case of roortionate selection. It is imortant to note that unlike the ordinal-based selection mechanisms, roortionate selection, as shown by equation 5, will never reach absolute convergence ( = 1). Thierens and Goldberg (1994) calculated the amount of time until the oulation convergences to within an arbitrary amount of = 1: t conv =?l ln(2):

6 5 For the uroses of this aer, is chosen to be = 1, so the converge time for roortionate selection is 2l calculated by t conv = l ln(l): (6) 3 Noise and Selection Intensity This section extends the selection intensity equation, given by equation 1, to accurately redict the selection ressure in the resence of noise. This section generalizes the derivation of the noisy tournament selection model resented in Miller and Goldberg (1995) to work for all selection scheme models that are based uon selection intensity. This section then derives the corresonding convergence models for the onemax environment. 3.1 Noisy Selection Intensity Model Derivation The model derivation in this section has three major stes. First, the relationshi between an individual's noisy tness and true tness values is determined, so that the exected true tness value of an individual can be estimated from the noisy tness evaluation. Next, the relationshi between true and noisy tness is extended to handle subsets of individuals, so that the mean true tness of a subset of the oulation can be estimated from the mean noisy tness of the subset. Lastly, we use the general selection intensity equation, equation 1, to estimate the mean noisy tness value of the mating ool, where the mating ool was selected based on the noisy tness values. This mean noisy tness value is then lugged into the formula found in the second ste to estimate the mean true tness of the mating ool. The selection ressure, based on the exected mean true tness value of the mating ool, is thus determined. The result is a redictive model for selection schemes that can handle varying noise levels. In a noisy environment, the noisy tness f 0 of an individual is given by f 0 = f + noise, where f is the real tness of the individual, and noise is the noise inherent in the tness function evaluation. The real tness of the oulation F is assumed to be normally distributed N( F;t ; 2 F;t). This section further assumes that the noise is unbiased and normally distributed N(0; 2 N). This facilitates modeling the eects of the noise, and is a reasonable assumtion for many domains. Using these assumtions, along with the additive roerty of normal distributions, gives that the noisy tness F 0 of the oulation is normally distributed N( F;t ; 2 F;t + 2 N ). Although the real tness value for an individual is unknown, the exected value of the real tness can be determined from the individual's noisy tness value, which is generated by a noisy tness function evaluation. As both the true tness and the noisy tness are normally distributed, the bivariate normal distribution can be used to obtain the exected true tness value of F for a given noisy tness value f 0 of F 0. For normal random variables X and Y, the bivariate normal distribution states that the exected value of Y for a secic value x of X is XY E(Y jx) = Y + XY Y X (x? X ); where XY is the correlation coecient for X and Y. The correlation coecient XY can be calculated by = XY X Y, where XY is the covariance of X and Y. The covariance between F and F 0 is simly F, 2 thus E(F jf 0 ) = F + 2 F F F 0 F F 0 = F + 2 F 2 F 0 (f 0? F 0); = F + 2 F 2 F + 2 N (f 0? F 0); (f 0? F 0): (7) As the above formula is linear, the exected value of F for any subset R of the oulation can be calculated using equation 7, with f 0 set to the noisy tness mean R of the subset. Of course, the subset we are interested in is the mating ool selected by the noisy selection rocess. The general selection intensity equation, equation 1, can be used to obtain the exected mean noisy tness of the mating ool when the selection rocess is itself based uon noisy tness values. The exected noisy tness mean of the mating ool subset can be calculated

7 6 using equation 1, as the mean F 0 ;t and variance F 0 ;t of the noisy oulation are known: F 0 ;t+1 = F 0 ;t + F 0 ;ti; q = F 0 ;t F;t 2 N I: Setting f 0 to F 0 ;t+1 in equation 7 roduces the exected true tness value of the mating ool: E(F t+1 j F 0 ;t+1) = F;t+1 ; = F;t + = F;t + 2 F;t 2 F;t + 2 N q ( F 0 ;t + 2 F;t + 2 N I? F ;t); 0 2 F;t I: (8) 2 F;t + 2 N As exected, equation 8 reduces to equation 1, the formula for the deterministic case, when the noise variance 2 N equals zero. Equation 8 is signicant in that it extends the basic selection intensity convergence model to handle noise. By doing so, accurate convergence rate rediction in the resence of noise is now ossible for selection schemes that can be modeled using the selection intensity aroach. An interesting ramication of equation 8 is that roortionate selection is noise invariant: increased noise levels do not aect the selection ressure of genetic algorithms using roortionate selection! Plugging in I = F 0 ;t from Table 1 to equation 8, and realizing that F 0 ;t = F;t and F 0 ;t = 2 F 0 F;t + 2 N for unbiased ;t normally distributed noise, yields F;t F;t+1 = F;t + F;t ; (9) F;t which is identical to the deterministic equation of selection ressure for roortionate selection. 3.2 Selection Intensity Model for Noisy Onemax Environment In subsection 2.4, the basic convergence model for the deterministic domain was derived. This section derives the converge model for the noisy onemax domain. Using the same aroach outlined in subsection 2.4, along with equation 8, it is easy to see that t+1? t = I 2 F;t : l 2 F;t + 2 N t (1? t ) = I : (10) l t (1? t ) + 2 N Aroximating the above dierence equation with a dierential equation yields d = dt I t (1? t ) l t (1? t ) + 2 N : (11) Although equation 11 is integrable, it does not reduce to convenient form in the general case for t ; however, it can be easily solved numerically for t, and for the noiseless case ( N = 0) t can be solved exactly (see equation 3). While equation 11 is not directly solvable for t, it can be solved for t as a function of : t() = 1 I " l arctan N log? 1 l(2? 1) 2 2 N + l (1? )?l? 2 2 N + l? 2 N 2 N + l (1? )! 2 2 N + l + 2 N 2 N + l (1? ) + (12)! + c # :

8 7 For binary alleles, at time t = 0 we can assume that half of the alleles are initially correct = 0:5. Using this to solve for c in equation 12 gives that c = 0. We are articularly interested in the time t conv it takes for all alleles to converge ( = 1). For the deterministic case, equation 12 reduces to t conv = l 2I ; (13) which is equivalent to the result obtained in equation 4. While t conv can not be solved exactly for the noisy onemax environment, it can be easily solved numerically. Useful aroximations for small, medium, and large levels of noise are derived from equation 12 in Miller and Goldberg (1995), where the convergence aroximations derived for tournament selection can be generalized to the selection intensity model by simly substituting I for s:s. These aroximation for small, medium, and large amounts of noise can be used to quickly estimate the convergence time of GAs in noisy environments, and are resented in table 2. Noise Case Noise Level t conv Aroximation h 1 l i Small N 0 l arctan I 2 N + 2 N log(2 N ) h 1 l Medium N F l arctan I 2 N + 2 N log( 2 N ) Large N 1 1 I h l 2 N + N log(l? 1) Table 2: Time of Convergence Aroximations for Noisy Onemax Domain. i i 4 Exerimental Validation In this section we verify the accuracy of the noisy selection models derived above for four dierent selection schemes: tournament selection, - selection, linear ranking, and stochastic universal selection. The exeriments are run using the noisy onemax domain, where the tness function noise is simulated by adding a random noisy value to the true tness of each individual. The noise is randomly drawn from an unbiased normal distribution, where the variance of the normal distribution is given for each exeriment. The exerimental results of running GAs with the dierent selection schemes in various noise levels is then comared to the results redicted by the models derived above. 4.1 Methodology For each dierent selection scheme, the GA is run 10 times in the noisy onemax domain for each noise level, and the results are averaged. The results are then lotted against redicted erformance for each noise level. For the onemax domain, the chromosome length is l = 100, and the oulation size is set according to Goldberg, Deb, and Clark (1992), which works out to N = 8( 2 + F 2 N ). For a binomially distributed oulation, the initial oulation variance is l(1? ), and the initial the ercentage of correct alleles in the oulation is assumed to be 0:5. Thus the oulation sizing equation works out to be N = 8( N ). To isolate the eects of selection, no mutation mechanism is used. The crossover mechanism is uniform crossover (Syswerda, 1989). As done in Thierens and Goldberg (1994) and Miller and Goldberg (1995), two rounds of crossover are erformed in each generation. The mating ool is selected by the selection scheme, crossover is used to roduce a new ool of osring, and then crossover is run again on the new osring to roduce the next generation. The creates a new generation that is more normally distributed than if one crossover er generation is used. This makes the selection model assumtion of a normally distributed oulation more accurate. However, Miller and Goldberg (1995) shows that for tournament selection, two crossovers er generation only has a small eect in the convergence erformance of the GA comared to one crossover er generation. The tested noise levels are taken to be 0, 0:5, 1, 2, and 4 times the initial function variance of 2 = 25. Thus the noise levels used in verifying the selection mechanisms are F 2 N = f0; 12:5; 25; 50; 100g. In each GA run, the run is stoed only after the oulation has fully converged.

9 8 4.2 Results This subsection details the results of our exeriments. The exerimental results are lotted along with the redicted erformance for each selection scheme in gure 1. In each lot, the dotted line is the redicted erformance, and the solid lines are the exerimental erformance (avg. of 10 runs). In most cases, the model accurately redicts the exerimental erformance, so that the redicted erformance (dashed line) is mostly obscured by the exerimental results (solid line). For the ordinal-based selection mechanisms (tournament selection, - selection, and linear ranking), there are ve sets of lines corresonding to the ve dierent noise levels. The lines, from left to right, corresond to the ve dierent noises levels of 2 N = f0; 12:5; 25; 50; 100g. Equations 3 and 11 are used to redict the erformance of the ordinal based selection schemes in the deterministic and noisy onemax domains resectively. The selection intensity for each of the ordinal selection schemes is calculated according to table 1, and is described in more detail below. The results for roortionate selection are resented in the stochastic universal selection below Tournament Selection For tournament selection, a tournament size of s = 2 was used, with a corresonding selection intensity (from table 1) of I = 2:2 = 0:5642: The maximal order statistic value was obtained from Harter (1970). The results, lotted in gure 1a, demonstrate that the convergence models were very accurate at redicting the convergence of tournament selection Selection For - selection, = 4 and = 8 was used. The corresonding selection intensity was calculating using table 1, along with the order statistic values obtained from Harter (1970). I = 1 4 ( 5:8 + 6:8 + 7:8 + 8:8 ) = 1 (0: : : :4236) 4 = 0:7253: The results, lotted in gure 1b, demonstrate that the convergence models were very accurate at redicting the convergence of - selection. Note that this exeriment had the same = ratio value of 0.5 as the binary tournament selection exeriment, yet had a higher selection ressure. The tradeo of eectively having higher values for and for the same ratio is that the resulting selected oulation has a lower variance. In general, selection mechanisms should maximize the resulting oulation variance for a given level of selection ressure in order to avoid remature convergence. This entails that for - selection, if there are several ossible and combinations that roduce a desired selection ressure, the combination with the smallest should be referred in ordered to maximize oulation diversity Linear Ranking For linear ranking selection, the number of desired coies of the best individual n + was set to 2, which made n? = 0. The corresonding selection intensity (from table 1) is I = 1 = 0:5642; which is equivalent to the binary tournament selection case. The results, lotted in gure 1c, demonstrate that the convergence models were very accurate at redicting the convergence of linear ranking.

10 9 100 Tournament Selection: s=2 100 Mu-Lambda Selection: mu=4, lambda=8 % Convergence Noise Var: 0,12.5,25,50,100 Predictive Model Exerimental Avg Generations % Convergence Noise Var: 0,12.5,25,50,100 Predictive Model Exerimental Avg Generations (a) Tournament Selection (b) - Selection 100 Linear Ranking Selection: n+ = Stochastic Universal Selection: c=0 % Convergence Noise Var: 0,12.5,25,50,100 Predictive Model Exerimental Avg Generations % Convergence Noise Var: 0,12.5,25,50,100 Predictive Model Exerimental Avg Generations (c) Linear Ranking Selection (d) - Selection Figure 1: Exerimental Results for the Onemax Domain Stochastic Universal Selection As equation 9 demonstrated that the convergence equation for noisy environments was equivalent to the deterministic case for roortionate selection, equation 5 was used to redict the erformance for stochastic universal selection at all noise levels. The redictive accuracy of our model for stochastic universal selection, equation 5, is shown in gure 1d. As the model is noise invariant, the runs at the ve dierent noise level roduced roughly the same lot, and therefore overlay each other in the gure. This accounts for the one dark line, which is also the redicted erformance for stochastic universal selection. This bears out the model's assumtion of noise invariance. Note that the number of generations until convergence is much larger than the rank-based selection schemes. While gure 1d demonstrates that our model accurately redicts the selection ressure of tness roortionate selection, it does not show that the absolute convergence times dier for dierent noise levels and oulation sizes. While the number of generation until the oulation reached 99.5% convergence is roughly equivalent for all ve noise levels, the time for absolute convergence ranged from 790 generation to 1420 generations. This can be exlained by the following: as the oulation aroaches near convergence for tness roortionate selection schemes, the resulting selection ressure becomes negligible. At this oint, the eect of genetic drift dominates the convergence rocess. The time until convergence under genetic drift is inuenced both by the noise and size of the oulation. However, this ending genetic drift is not free to let the oulation drift any which way, as the further away the oulation drifts from the otimum, the more selection ressure is alied. So in eect, there is a one-way genetic drift tendency of the oulation towards the otimum.

11 10 Accurate modeling of this one-way genetic drift in the resence of noise is an interesting toic that merits further research. 5 Conclusions Now that GAs are increasingly being used by industry, the eects of noise on erformance are becoming more imortant. This aer's main contribution is that it demonstrates how tness function noise will aect the convergence of GAs utilizing several common selection schemes. A noisy selection intensity model is derived that accurately redicts the erformance of several selection schemes in noisy environments, including tournament selection, - selection, linear ranking and stochastic universal selection. Furthermore, any selection scheme model based uon the deterministic selection intensity model can be similarly adated to handle noise. This noisy selection intensity model develoed in this aer has several immediate ractical alications. It can redict the convergence rate of a GA within a noisy environment, which is critical for time-sensitive alications. The model can be used to redict the solutions quality after a certain number of generations, and therefore be used to determine aroriate stoing criteria for noisy environments. For noisy tness functions where the noise is due to samling error, the model can be used to determine an otimal samle size for the tness function. An otimum samle size will maximize the erformance of a GA within a xed comutational time. This research also has several long term ramications. The basic aroach of this aer can be used to study the delaying eects of noise on other selection schemes. It may also be ossible to aly the same aroach to redict convergence delays resulting from noise inherent within other GA oerators, such as recombination or mutation. 6 Acknowledgments This work was suorted under NASA Grant No. NGT 9-4. This eort was also sonsored by the Air Force Oce of Scientic Research, Air Force Materiel Command, USAF, under grant numbers F and F The U.S. Government is authorized to reroduce and distribute rerints for Governmental uroses notwithstanding any coyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interreted as necessarily reresenting the ocial olicies or endorsements, either exressed or imlied, of the Air Force Oce of Scientic Research or the U.S. Government. References Back, T. (1995). Generalized convergence models for tournament- and (,)-selection. In Eschelman, L. (Ed.), Proceedings of the Sixth International Conference on Genetic Algorithms (. 2{8). San Francisco, CA: Morgan Kaufmann. Baker, J. E. (1985). Adative selection methods for genetic algorithms. In Grefenstette, J. J. (Ed.), Proceedings of an International Conference on Genetic Algorithms and Their Alications (. 101{111). Hillsdale, NJ: Lawrence Erlbaum Associates. Baker, J. E. (1987). Reducing bias and ineciency in the selection algorithm. In Grefenstette, J. J. (Ed.), Proceedings of the Second International Conference on Genetic Algorithms (. 14{21). Hillsdale, NJ: Lawrence Erlbaum Associates. Blickle, T., & Thiele, L. (1995). A comarison of selection schemes used in genetic algorithms (Technical Reort No. 11). Gloriastrasse 35, CH-8092 Zurich: Swiss Federal Institute of Technology (ETH) Zurich, Comuter Engineering and Communications Networks Lab (TIK). Booker, L. B. (1982). Intelligent behavior as an adatation to the task environment. Dissertations Abstracts International, 43 (2), 469B. (University Microlms No ). Brindle, A. (1981). Genetic algorithms for function otimization. Unublished doctoral dissertation, University of Alberta, Edmonton, Canada.

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