IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 10, NO. 4, AUGUST 2006 405 Efficient Search for Robust Solutions by Means of Evolutionary Algorithms and Fitness Approximation Ingo Paenke, Jürgen Branke, Member, IEEE, and Yaochu Jin, Senior Member, IEEE Abstract—For many real-world optimization problems, the ro- robust solutions is to consider the best worst case performance. bustness of a solution is of great importance in addition to the so- Another definition of robust solutions is to consider a solution’s lution’s quality. By robustness, we mean that small deviations from expected performance over all possible disturbances, which the original design, e.g., due to manufacturing tolerances, should be tolerated without a severe loss of quality. One way to achieve corresponds to a risk-neutral decision maker’s choice. In these that goal is to evaluate each solution under a number of different two definitions for robustness, only one objective is considered, scenarios and use the average solution quality as fitness. However, and we denote such approaches single objective (SO) robustness this approach is often impractical, because the cost for evaluating optimization. However, robustness of solutions might be better each individual several times is unacceptable. In this paper, we defined by considering both the quality and the risk separately, present a new and efficient approach to estimating a solution’s ex- pected quality and variance. We propose to construct local approx- i.e., by converting the problem into a multiobjective problem imate models of the fitness function and then use these approxi- [22]. We denote such approaches as multiobjective (MO) ro- mate models to estimate expected fitness and variance. Based on a bustness optimization. This paper suggests model-based fitness variety of test functions, we demonstrate empirically that our ap- approximation methods that can be employed to improve the proach significantly outperforms the implicit averaging approach, computational efficiency of both SO and MO approaches to as well as the explicit averaging approaches using existing estima- tion techniques reported in the literature. robustness optimization. Disturbances may appear in both environmental variables Index Terms—Evolutionary optimization, fitness approxima- tion, robustness, uncertainty. and design variables. In the following, we focus on robustness against disturbances of design variables, which is important, e.g., in the case of manufacturing tolerances. Formally, if I. INTRODUCTION denotes a design vector (solution) of dimension , and is N MANY real-world optimization scenarios, it is not suf- the fitness (in the context of robustness optimization is I ficient for a solution to be of high quality, but the solution also often called raw fitness, ) of that particular solution, should also be robust. Some examples include the following. then the expected fitness of solution is defined as • In manufacturing, it is usually impossible to produce an item exactly according to the design specifications. In- (1) stead, the design has to allow for manufacturing toler- ances, see, e.g., [2], [14], and [39]. • In scheduling, a schedule should be able to tolerate small where is a disturbance that is distributed according to the prob- deviations from the estimated processing times or be able ability density function . Similarly, the fitness variance of a to accommodate machine breakdowns [17], [23], [32]. solution can be defined as • In circuit design, the circuits should work over a wide range of environmental conditions like different tempera- (2) tures [36]. • In turbine blade design, the turbine should perform well Unfortunately, for reasonably complex problems, (1) and over a range of conditions, e.g., it should work efficiently (2) cannot be computed analytically, usually because is not at different speeds. Similar requirements exist for airfoil known in a closed form. Alternatively, and can be design [31], [40]. estimated by Monte Carlo integration, i.e., by sampling over a There are a number of different possible definitions for robust- number of realizations of . However, each sample corresponds ness (see, e.g., [6, p. 127]). Generally speaking, robustness to one fitness evaluation, and if fitness evaluations are expen- means some degree of insensitivity to small disturbances of sive, this approach is clearly not viable. the environment or the design variables. One definition for Therefore, new approaches are needed which allow to esti- mate a solution’s expected fitness and variance more efficiently. Manuscript received June 16, 2004; revised February 21, 2005 and July 29, In this paper, we propose to use an approximation model to esti- 2005. I. Paenke and J. Branke are with the Institute AIFB, University of Karlsruhe, mate a solution’s robustness. Instead of using the costly raw fit- 76128 Karlsruhe, Germany (e-mail: [email protected]; branke@ ness function in the above mentioned Monte Carlo integration, aifb.uni-karlsruhe.de). we rely on the approximation model for that purpose. In prin- Y. Jin is with the Honda Research Institute Europe, 63073 Offenbach, Germany (e-mail: [email protected]). ciple, this idea could be used in combination with any suitable Digital Object Identifier 10.1109/TEVC.2005.859465 approximation model like artificial neural networks, Kriging 1089-778X/$20.00 © 2006 IEEE 406 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 10, NO. 4, AUGUST 2006 models, or Gaussian processes. In this paper, we use local ap- 1) Variance reduction techniques: Using derandomized proximation models, which have the advantage of being rela- sampling techniques instead of random sampling reduces tively easy to construct and also seem appropriate to approxi- the variance of the estimator, thus allowing a more accu- mate the performance of a solution over a distribution of local rate estimate with fewer samples. In [5] and [26], Latin disturbances. Within that framework, we compare and discuss a Hypercube Sampling is employed (cf. Appendix B), number of alternatives with respect to the approximation model together with the idea to use the same disturbances for all used (interpolation or regression), the complexity of the model individuals in a generation. (linear or quadratic), the number and location of approximation 2) Evaluating important individuals more often: In [4], it models constructed, the sampling method, and how approxima- is suggested to evaluate good individuals more often than tion methods should be exploited for estimation. Empirical re- bad ones, because good individuals are more likely to sur- sults confirm the superiority of our approach to some previous vive, and therefore a more accurate estimate is beneficial. approaches for either SO or MO robustness optimization. In [6], it was proposed that individuals with high fitness Note that we subsequently use the terms raw fitness and variance should be evaluated more often. real fitness. Here, raw fitness is used in contrast to robustness 3) Using other individuals in the neighborhood: Since , whereas real fitness is used in contrast to approxi- promising regions in the search space are sampled several mated fitness. times, it is possible to use information about other indi- This paper is structured as follows. Section II provides a brief viduals in the neighborhood to estimate an individual’s overview of related work. We then introduce the evolutionary expected fitness. In particular, in [4], it is proposed to algorithm (EA) for robustness optimization in Section III. A record the history of an evolution, i.e., to accumulate all short discussion of the approximation techniques used can be individuals of an evolutionary run with corresponding found in Section IV. Then, in Section V, we present our new fitness values in a database, and to use the weighted av- approaches to estimating a solution’s expected fitness and vari- erage fitness of neighboring history individuals. Weights ance. These approaches are evaluated empirically in Section VI are assigned according to the probability distribution based on a variety of benchmark problems. The paper concludes function of the disturbance. We will use this method later with a summary of this paper and some ideas for future work. for comparison and refer to it as weighted history. While all of the above methods explicitly average over a number of fitness evaluations, Tsutsui and Ghosh present in [37] and II. RELATED WORK [38], an idea to simply disturb the phenotypic features before There are a wealth of publications regarding the use of ap- evaluating an individual’s fitness. As the EA is revisiting proximation models to speed up EAs. Feedforward neural net- promising regions of the search space, it implicitly averages works [16], [21], radial basis function networks [33], and poly- over a set of disturbed solutions, which can be seen as an im- nomials [7], [24] have been employed to improve the efficiency plicit averaging approach. Using the schema theorem, Tsutsui of EAs. Besides, estimation of distribution algorithms (EDAs) and Ghosh show that given an infinitely large population size can also be considered as a class of algorithms that approxi- and the proportional selection method, a genetic algorithm mate the fitness landscape implicitly [41]. In the following, we with single disturbed evaluations is actually performing as if will focus on related literature regarding the search for robust it would work on . This implicit averaging has proven solutions. For a general overview on the use of approximation successful for low-dimensional problems. Subsequently, we models in combination with EAs, the reader is referred to [18] refer to this approach as single disturbed. and [19]. B. MO Robustness Optimization As mentioned in the introduction, evolutionary approaches to robustness optimization can be categorized into SO and MO op- In design optimization, several papers have treated the search timization approaches. By far, the majority of research activities for robust optimal solutions as a MO problem, see, e.g., [8] and in this field follows the SO approach. [9]. However, relatively little attention has been paid to evo- lutionary MO search for robust solutions.