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Optimal Elevator Group Control by Evolution Strategies

Thomas Beielstein1 , Claus-Peter Ewald2 , and Sandor Markon3

Universtit¨t Dortmund, D-44221 Dortmund, Germany a Thomas.Beielstein@udo.edu, WWW home page: http://ls11-www.cs.uni-dortmund.de/people/tom/index.html 2 NuTech Solutions GmbH, Martin-Schmeisser Weg 15, D-44227 Dortmund, Germany Ewald@nutechsolutions.de, WWW home page: http://www.nutechsolutions.de 3 FUJITEC Co.Ltd. World Headquarters, 28-10, Shoh 1-chome, Osaka, Japan markon@rd.fujitec.co.jp, WWW home page: http://www.fujitec.com

1

Abstract. E?cient elevator group control is important for the operation of large buildings. Recent developments in this ?eld include the use of fuzzy logic and neural networks. This paper summarizes the development of an evolution strategy (ES) that is capable of optimizing the neuro-controller of an elevator group controller. It extends the results that were based on a simpli?ed elevator group controller simulator. A threshold selection technique is presented as a method to cope with noisy ?tness function values during the optimization run. Experimental design techniques are used to analyze ?rst experimental results.

1

Introduction

Elevators play an important role in today’s urban life. The elevator supervisory group control (ESGC) problem is related to many other stochastic tra?c control problems, especially with respect to the complex behavior and to many di?culties in analysis, design, simulation, and control [Bar86,MN02]. The ESGC problem has been studied for a long time: ?rst approaches were mainly based on analytical approaches derived from queuing theory, in the last decades arti?cial intelligence techniques such as fuzzy logic (FL), neural networks (NN), and evolutionary algorithms (EA) were introduced, whereas today hybrid techniques, that combine the best methods from the di?erent worlds, enable improvements. Therefore, the present era of optimization could be classi?ed as the era of computational intelligence (CI) methods [SWW02,BPM03]. CI techniques might be useful as quick development techniques to create a new generation of self-adaptive ESGC systems that can handle high maximum tra?c situations. In the following we will consider an ESGC system that is based on a neural network to control the elevators. Some of the NN connection weights can be modi?ed, so that di?erent weight settings and their in?uence on the ESGC performance can be tested. Let x denote one weight con?guration. We can de?ne

2 the optimal weight con?guration as x? = arg min f (x), where the performance measure f () to be minimized is de?ned later. The determination of an optimal weight setting x? is di?cult, since it is not trivial 1. to ?nd an e?cient strategy that modi?es the weights without generating too many infeasible solutions, and 2. to judge the performance or ?tness f (x) of one ESGC con?guration. The performance of one speci?c weight setting x is based on simulations of speci?c tra?c situations, which lead automatically to stochastically disturbed ? (noisy) ?tness function values f (x). The rest of this article deals mainly with the second problem, especially with problems related to the comparison of two noisy ?tness function values. Before we discuss a technique that might be able to improve the comparison of stochastically disturbed values in section 3, we introduce one concrete variant of the ESGC problem in the next section. The applicability of the comparison technique to the ESGC problem is demonstrated in section 4, whereas section 5 gives a summary and an outlook.

2

The Elevator Supervisory Group Control Problem

In this section, we will consider the elevator supervisory group control problem [Bar86,SC99,MN02]. An elevator group controller assigns elevators to service calls. An optimal control strategy is a precondition to minimize service times and to maximize the elevator group capacity. Depending on current tra?c loads, several heuristics for reasonable control strategies do exist, but which in general lead to suboptimal controllers. These heuristics have been implemented by Fujitec, one of the world’s leading elevator manufacturers, using a fuzzy control approach. In order to improve the generalization of the resulting controllers, Fujitec developed a neural network based controller, which is trained by use of a set of the aforementioned fuzzy controllers, each representing control strategies for di?erent tra?c situations. This leads to robust and reasonable, but not necessarily optimal, control strategies [Mar95]. Here we will be concerned with ?nding optimal control strategies for destination call based ESGC systems. In contrast to traditional elevators, where customers only press a button to request up or down service and choose the exact destination from inside the elevator car, a destination call system lets the customer choose the desired destination at a terminal before entering the elevator car. This provides more exact information to the group controller, and allows higher e?ciency by grouping of passengers into elevators according to their destinations; but also limits the freedom of decision. Once a customer is assigned to a car and the car number appears on the terminal, the customer moves away from the terminal, which makes it very inconvenient to reassign his call to another car later. The concrete control strategy of the neural network is determined by the network structure and neural weights. While the network structure as well as E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

3

Fig. 1. The architecture of an elevator supervisory group control system [Sii97].

many of the weights are ?xed, some of the weights on the output layer, which have a major impact on the controller’s performance, are variable and therefore subject to optimization. Thus, an algorithm is searched for to optimize the variable weights of the neural controller. The controller’s performance can be computed by the help of a discrete-event based elevator group simulator. Fig.2 illustrates how the output from the simulator is visualized. Unfortunately, the resulting response surface shows some characteristics which makes the identi?cation of globally optimal weights di?cult if not impossible. The topology of the ?tness function can be characterized as follows: – – – – – – highly nonlinear, highly multi-modal, varying fractal dimensions depending on the position inside the search space, randomly disturbed due to the nondeterminism of service calls, dynamically changing with respect to tra?c loads, local measures such as gradients can not be derived analytically.

Furthermore, the maximum number of ?tness function evaluations is limited to the order of magnitude 104 , due to the computational e?ort for single simulator calls. Consequently, e?cient robust methods from the domain of black-box optimization are required where evolutionary computation is one of the most promising approaches. Therefore, we have chosen an evolution strategy to perform the optimization [BFM00,Bey01]. The objective function for this study is the average waiting time of all passengers served during a simulated elevator movement of two hours. Further experiments will be performed in future to compare this objective function to more complex ones, which e.g. take into account the maximum waiting time as well. There are three main tra?c patterns occurring during a day: up-peak (morning E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

4

Fig. 2. Visualization of the output from the elevator group simulator.

rush hour), two-way (less intense, balanced tra?c during the day), and downpeak tra?c (rush hour at closing time). These patterns make up the simulation time to one third each, which forces the resulting controller to cope with di?erent tra?c situations. As described above, the objective function values are massively disturbed by noise, since the simulator calculates di?erent waiting times on the same objective variables (network weights), which stems from changing passenger distributions generated by the simulator. For the comparison of di?erent ES parameter settings the ?nal best individuals produced by the ES were assigned handling capacities at 30, 35, and 40 seconds. A handling capacity of n passengers per hour at 30s means that the elevator system is able to serve a maximum of n passengers per hour without exceeding an average waiting time of 30s. These values were created by running the simulator with altering random seeds and increasing passenger loads using the network weights of the best individuals found in each optimization run. Finally, to enable the deployment of our standard evaluation process, we needed a single ?gure to minimize. Therefore, the handling capacities were averaged and then subtracted from 3000 pass./h. The latter value was empirically chosen as an upper bound for the given scenario. The resulting ?tness function is shown in (9) and is called ‘inverse handling capacity’ in the following.

3

3.1

Evolution Strategies and Threshold Selection

Experimental Noise and Evolution Strategies

In this section we discuss the problem of comparing two solutions, when the available information (the measured data) is disturbed by noise. A bad solution E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

5 might appear better than a good one due to the noise. Since minor di?erences are unimportant in this situation, it might be useful to select only much better values. A threshold value τ can be used to formulate the linguistic expression ‘much better’ mathematically: ? ? x is much better (here: greater) than y ? f (x) > f (y) + τ. (1)

Since comparisons play an important role in the selection process of evolutionary algorithms, we will use the term threshold selection (TS) in the rest of this article. TS can be generalized to many other stochastic search techniques such as particle swarm optimizers or genetic algorithms. We will consider evolution strategies in this article only. The goal of the ES is to ?nd a vector x? , for which holds: f (x? ) ≤ f (x) ?x ∈ D, (2)

where the vector x represents a set of (object) parameters, and D is some ndimensional search space. An ES individual is usually de?ned as the set of object parameters x, strategy parameters s, and its ?tness value f (x) [BS02]. In the following we will consider ?tness function values obtained from computer simulation experiments: in this case, the value of the ?tness function depends on the seed that is used to initialize the random stream of the simulator. The exact ?tness function value f (x) is replaced by the noisy ?tness function ? value f (x) = f (x) + . In the theoretical analysis we assume normal-distributed noise, that is ? N (?, σ 2 ). It is crucial for any stochastic search algorithm to decide with a certain amount of con?dence whether one individual has a better ?tness function value than its competitor. This problem will be referred to as the selection problem in the following [Rud98,AB01]. Reevaluation of the ?tness function can increase this con?dence, but this simple averaging technique is not applicable to many real-world optimization problems, i.e. if the evaluations are too costly. 3.2 Statistical Hypothesis Testing

Threshold selection belongs to a class of statistical methods that can reduce the number of reevaluations. It is directly connected to classical statistical hypothesis testing [BM02]. Let x and y denote two object parameter vectors, with y being proposed as a ’better’ (greater) one, to replace the existing x. Using statistical testing, the selection problem can be stated as testing the null hypothesis H0 : f (x) ≤ f (y) against the one-sided alternative hypothesis H1 : f (x) > f (y). Whether the null hypothesis is accepted or not depends on the speci?cation of a critical region for the test and on the de?nition of an appropriate test statistic. Two kinds of errors may occur in testing hypothesis: if the null hypothesis is rejected when it is true, an alpha error has been made. If the null hypothesis is not rejected when it is false, a beta error has been made. Consider a maximization problem: ? the threshold rejection probability Pτ is de?ned as the conditional probability, E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

6

Fig. 3. Hypothesis testing to compare models: we are testing whether threshold selection has any signi?cance in the model. The Normal QQ-plot, the box-plot, and the interaction-plots lead to the conclusion that threshold selection has a signi?cant e?ect. Y denotes the ?tness function values that are based on the inverse handling capacities: smaller values are better, see (9). Threshold, population size ? and selective pressure (o?spring-parent ratio ν) are modi?ed according to the values in Tab. 3. The function α(t) as de?ned in (6) was used to determine the threshold value.

E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

7 that a worse candidate y has a better noisy ?tness value than the ?tness value of candidate x by at least a threshold τ :

? Pτ := P {f (y) ≤ f (x) + τ | f (y) ≤ f (x)}, n i=1

(3)

? where f (x) := f (xi )/n denotes the sample average of the noisy val? ues [BM02]. Pτ and the alpha error are complementary probabilities:

? Pτ = 1 ? α.

(4)

(4) reveals, how TS and hypothesis testing are connected: maximization of the threshold rejection probability, an important task in ES based optimization, is equivalent to the minimization of the alpha error. Therefore, we provide techniques from statistical hypothesis testing to improve the behavior of ES in noisy environments. Our implementation of the TS method is based on the common practice in hypothesis testing to specify a value of the probability of an alpha error, the so called signi?cance level of the test. In the ?rst phase of the search process the explorative character is enforced, whereas in the second phase the main focus lies on the exploitive character. Thus, a technique that is similar to simulated annealing is used to modify the signi?cance level of the test during the search process. In the following paragraph, we will describe the TS implementation in detail. 3.3 Implementation Details

The TS implementation presented here is related to doing a hypothesis test to see whether the measured di?erence in the expectations of the ?tness function values is signi?cantly di?erent from zero. The test result (either a ‘reject’ or ‘failto-reject’ recommendation) determines the decision whether to reject or accept a new individual during the selection process of an ES. In the following, we give some recommendations for the concrete implementation. We have chosen a parametric approach, although non-parametric approaches might be applicable in this context too. Let Yi1 , Yi2 , . . . Yin (i = 1, 2) be a sample of n independent and identically distributed (i.i.d.) measured ?tness function values. The n di?erences Zi = Y1j ?Y2j are also i.i.d. random variables (r.v.) with sample mean Z and sample variance S 2 (n). Consider the following test on means ?1 and ?2 of normal distributions: if H0 : ?1 ? ?2 ≤ 0 is tested against H1 : ?1 ? ?2 > 0, we have the test statistic t0 = z ? (?1 ? ?2 ) s 2/n , (5)

with sample standard deviation S (small letters denote realizations of the corresponding r.v.). The null hypothesis H0 : ?1 ? ?2 ≤ 0 would be rejected if t0 > t2n?2,1?α , where t0 > t2n?2,1?α is the upper α percentage point of the t distribution with 2n ? 2 degrees of freedom. Summarizing the methods discussed so far, we recommend the following recipe: E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

8 1. Select an alpha value. The α error is reduced during the search process, e.g. the function α(t) = (1 ? t/tmax )/2, (6) with tmax the maximum number of iterations and t ∈ {0, tmax }, the actual iteration number, can been used. Evaluate the parent and the o?spring candidate n times. To reduce the computational e?ort, this sampling can be performed every k generations only. Determine the sample variance. Determine the threshold value τ = t2n?2,1?α · s · 2/n A new individual is accepted, if its (perturbed) ?tness value plus τ is smaller than the (perturbed) ?tness value of the parent.

2.

3. 4. 5.

A ?rst analysis of threshold selection can be found in [MAB+ 01,BM02]. In the following section we will present some experimental results that are based on the introduced ideas.

4

4.1

Threshold Selection in the Context of ESGC Optimization

Statistical experimental design

The following experiments have been set up to answer the question: how can TS improve the optimization process if only stochastically perturbed ?tness function values (e.g. from simulation runs) can be obtained? Therefore, we simulated alternative ES parameter con?gurations and examined their results. We wanted to ?nd out if TS has any e?ect on the performance of an ES, and if there are any interactions between TS and other exogenous parameters such as population size or selective pressure. A description of the experimental design (DoE) methods we used is omitted here. [Kle87,LK00] give excellent introductions into design of experiments, the applicability of DoE to evolutionary algorithms is shown in [Bei03]. The following vector notation provides a very compact description of evolution strategies [BS02,BM02]. Consider the following parameter vector of an ES parameter design: pES = (?, ν, S, nσ , τ0 , τi , ρ, R1 , R2 , r0 ) , (7)

where ν := λ/? de?nes the o?spring-parent ratio, S ∈ {C, P } de?nes the selection scheme resulting in a comma or plus strategy. The representation of the selection parameter S can be generalized by introducing the parameter κ that de?nes the maximum age (in generations) of an individual. If κ is set to 1, we obtain the comma-strategy, if κ equals +∞, we model the plus-strategy. The mixing number ρ de?nes the size of the parent family that is chosen from the parent pool of size ? to create λ o?springs. We consider global intermediate GI, global discrete GD, local intermediate LI, and local discrete LD recombination. E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

9 R1 and R2 de?ne the recombination operator for object resp. strategy variables, and r0 is the random seed. This representation will be used throughout the rest of this article and is summarized in Tab. 1. Typical settings are: pES = √ √ 5, 7, 1, 1, 1/ 2 D, 1/ 2D, 5, GD, GI, 0 . (8)

Our experiment with the elevator group simulator involved three factors. A 23 full factorial design has been selected to compare the in?uence of di?erent population sizes, o?spring-parent ratios and selection schemes. This experimental design leads to eight di?erent con?gurations of the factors that are shown in Tab.1. The encoding of the problem parameters is shown in Tab.2. Tab.3 displays the parameter settings that were used during the simulation runs: the population size was set to 5 and 20, whereas the parent-o?spring ratio was set to 2 and 5. This gives four di?erent (parent, o?spring) combinations: (5,20), (5,25), (20,40), and (20,100). Combined with two di?erent selection schemes we obtain 8 di?erent run con?gurations. Each run con?guration was repeated 10 times. 4.2 Analysis

Although a batch job processing system, that enables a parallel execution of simulation runs, was used to run the experiments, more than a full week of round-the-clock computing was required to perform the experiments. Finally 80 con?gurations have been tested (10 repeats of 8 di?erent factor settings). The simulated inverse handling capacities can be summarized as follows: Min. Mean : 850.0 :1003.1 1st Qu.:916.7 3rd Qu.:1083.3 Median :1033.3 Max. :1116.7

A closer look at the in?uence of TS on the inverse handling capacities reveals that ES runs with TS have a lower mean (931.2) than simulations that used a plus selection strategy (1075.0). As introduced in section 2, the ?tness function reads (minimization task): g(x) = 3000.0 ? f pES (x), (9)

where the setting from Tab. 3 was used, f pES is the averaged handling capacity (pass./h), and x is a 36 dimensional vector that speci?es the NN weights. The minimum ?tness function value (850.0) has been found by TS. Performing a t-test (the null hypothesis ‘the true di?erence in means is not greater than 0’ that is tested against the ‘alternative hypothesis: the true di?erence in means is greater than 0’) leads to the p-value smaller than 2.2e ? 16. An interaction plot plots the mean of the ?tness function value for two-way combinations of factors, e.g. population size and selective strength. Thus, it can illustrate possible interactions between factors. Considering the interaction plots in Fig. 3, we can conclude that the application of threshold selection improves the E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

10

Table 1. DoE parameter for ES. Symbol Parameter ? ν = λ/? nσ τ0 τi κ ρ R1 R2 number of parent individuals o?spring-parent ratio number of standard deviations global mutation parameter individual mutation parameter age mixing number recombination operator for object variables recombination operator for strategy variables Recommended Values 10 . . . 100 1 . . . 10 1...D √ 1/√ 2 D 1/ 2D 1...∞ ? {discrete} {intermediate}

Table 2. DoE parameter for the ?tness function. Symbol Parameter f D Nexp Ntot σ ?tness function, optimization problem dimension of f number of experiments for each scenario total number of ?tness function evaluations noise level Values ESGC problem, minimization, see (9) 36 10 5 · 103 unknown

Table 3. ES parameter designs.

ES ESGC-Design Model (D = 36) Variable Low High (?1) (+1) ? 5 20 ν 2 5 κ +∞ Threshold Selection The following values remain unchanged: nσ 1 √ τ0 1/√ 2 D τ1 1/ 2D ρ ? R1 GD R2 GI

E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

11 behavior of the evolution strategy in any case. This improvement is independent from other parameter settings of the underlying evolution strategy. Thus, there is no hint that the results were caused by interactions between other factors.

5

Summary and Outlook

The elevator supervisory group control problem was introduced in the ?rst part of this paper. Evolution strategies were characterized as e?cient optimization techniques: they can be applied to optimize the performance of an NN-based elevator supervisory group controller. The implementation of a threshold selection operator for evolution strategies and its application to a complex real-world optimization problem has been shown. Experimental design methods have been used to set up the experiments and to perform the data analysis. The obtained results gave ?rst hints that threshold selection might be able to improve the performance of evolution strategies if the ?tness function values are stochastically disturbed. The TS operator was able to improve the average passenger handling capacities of an elevator supervisory group control problem. Future research on the thresholding mechanism will investigate the following topics: – Developing an improved sampling mechanism to reduce the number of additional ?tness function evaluations. – Introducing a self-adaptation mechanism for τ during the search process. – Combining TS with other statistical methods, e.g. Staggge’s e?ciently averaging method [Sta98]. Acknowledgments. T. Beielstein’s research was supported by the DFG as a part of the collaborative research center ‘Computational Intelligence’ (531). R, a language for data analysis and graphics was used to compute the statistical analysis [IG96].

References

[AB01] Dirk V. Arnold and Hans-Georg Beyer. Investigation of the (?, λ)-ES in the presence of noise. In J.-H. Kim, B.-T. Zhang, G. Fogel, and I. Kuscu, editors, Proc. 2001 Congress on Evolutionary Computation (CEC’01), Seoul, pages 332–339, Piscataway NJ, 2001. IEEE Press. G. Barney. Elevator Tra?c Analysis, Design and Control. Cambridg U.P., 1986. T. Beielstein. Tuning evolutionary algorithms. Technical Report 148/03, Universit¨t Dortmund, 2003. a Hans-Georg Beyer. The Theory of Evolution Strategies. Natural Computing Series. Springer, Heidelberg, 2001. Th. B¨ck, D. B. Fogel, and Z. Michalewicz, editors. Evolutionary Computaa tion 1 – Basic Algorithms and Operators. Institute of Physics Publ., Bristol, 2000.

[Bar86] [Bei03] [Bey01] [BFM00]

E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

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[BM02] T. Beielstein and S. Markon. Threshold selection, hypothesis tests, and DOE methods. In David B. Fogel, Mohamed A. El-Sharkawi, Xin Yao, Garry Greenwood, Hitoshi Iba, Paul Marrow, and Mark Shackleton, editors, Proceedings of the 2002 Congress on Evolutionary Computation CEC2002, pages 777–782. IEEE Press, 2002. [BPM03] T. Beielstein, M. Preuss, and S. Markon. A parallel approach to elevator optimization based on soft computing. Technical Report 147/03, Universit¨t a Dortmund, 2003. [BS02] Hans-Georg Beyer and Hans-Paul Schwefel. Evolution strategies – A comprehensive introduction. Natural Computing, 1:3–52, 2002. [IG96] R. Ihaka and R. Gentleman. R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5(3):299–314, 1996. [Kle87] J. Kleijnen. Statistical Tools for Simulation Practitioners. Marcel Dekker, New York, 1987. [LK00] Averill M. Law and W. David Kelton. Simulation Modelling and Analysis. McGraw-Hill Series in Industrial Egineering and Management Science. McGraw-Hill, New York, 3 edition, 2000. [MAB+ 01] Sandor Markon, Dirk V. Arnold, Thomas B¨ck, Thomas Beielstein, and a Hans-Georg Beyer. Thresholding – a selection operator for noisy ES. In J.-H. Kim, B.-T. Zhang, G. Fogel, and I. Kuscu, editors, Proc. 2001 Congress on Evolutionary Computation (CEC’01), pages 465–472, Seoul, Korea, May 27– 30, 2001. IEEE Press, Piscataway NJ. [Mar95] Sandor Markon. Studies on Applications of Neural Networks in the Elevator System. PhD thesis, Kyoto University, 1995. [MN02] S. Markon and Y. Nishikawa. On the analysis and optimization of dynamic cellular automata with application to elevator control. In The 10th Japanese-German Seminar, Nonlinear Problems in Dynamical Systems, Theory and Applications. Noto Royal Hotel, Hakui, Ishikawa, Japan, September 2002. [Rud98] G¨nter Rudolph. On risky methods for local selection under noise. In u A. E. Eiben, Th. B¨ck, M. Schoenauer, and H.-P. Schwefel, editors, Parallel a Problem Solving from Nature – PPSN V, Fifth Int’l Conf., Amsterdam, The Netherlands, September 27–30, 1998, Proc., volume 1498 of Lecture Notes in Computer Science, pages 169–177. Springer, Berlin, 1998. [SC99] A.T. So and W.L. Chan. Intelligent Building Systems. Kluwer A.P., 1999. [Sii97] M. L. Siikonen. Planning and Control Models for Elevators in High-Rise Buildings. PhD thesis, Helsinki Unverstity of Technology, Systems Analysis Laboratory, October 1997. [Sta98] Peter Stagge. Averaging e?ciently in the presence of noise. In A.Eiben, editor, Parallel Problem Solving from Nature, PPSN V, pages 188–197, Berlin, 1998. Springer-Verlag. [SWW02] H.-P. Schwefel, I. Wegener, and K. Weinert, editors. Advances in Computational Intelligence – Theory and Practice. Natural Computing Series. Springer, Berlin, 2002.

E. Cant?-Paz et al. (Eds.): GECCO 2003, LNCS 2724, pp. 1963–1974, 2003. u c Springer-Verlag Berlin Heidelberg 2003

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