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Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 1999
"... Evolutionary algorithms (EA’s) are often wellsuited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a singl ..."
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Cited by 781 (22 self)
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Evolutionary algorithms (EA’s) are often wellsuited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EA’s are compared quantitatively where an extended 0/1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the Strength Pareto EA (SPEA), that combines several features of previous multiobjective EA’s in a unique manner. It is characterized by a) storing nondominated solutions externally in a second, continuously updated population, b) evaluating an individual’s fitness dependent on the number of external nondominated points that dominate it, c) preserving population diversity using the Pareto dominance relationship, and d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proofofprinciple results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware–software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Paretooptimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EA’s on the 0/1 knapsack problem.
SPEA2: Improving the Strength Pareto Evolutionary Algorithm
, 2001
"... The Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999) is a relatively recent technique for finding or approximating the Paretooptimal set for multiobjective optimization problems. In different studies (Zitzler and Thiele 1999; Zitzler, Deb, and Thiele 2000) SPEA has shown very ..."
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Cited by 673 (19 self)
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The Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999) is a relatively recent technique for finding or approximating the Paretooptimal set for multiobjective optimization problems. In different studies (Zitzler and Thiele 1999; Zitzler, Deb, and Thiele 2000) SPEA has shown very good performance in comparison to other multiobjective evolutionary algorithms, and therefore it has been a point of reference in various recent investigations, e.g., (Corne, Knowles, and Oates 2000). Furthermore, it has been used in different applications, e.g., (Lahanas, Milickovic, Baltas, and Zamboglou 2001). In this paper, an improved version, namely SPEA2, is proposed, which incorporates in contrast to its predecessor a finegrained fitness assignment strategy, a density estimation technique, and an enhanced archive truncation method. The comparison of SPEA2 with SPEA and two other modern elitist methods, PESA and NSGAII, on different test problems yields promising results. 1
Comparison of Multiobjective Evolutionary Algorithms: Empirical Results
, 2000
"... In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in conver ..."
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Cited by 605 (39 self)
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In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Paretooptimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search.
An Overview of Evolutionary Algorithms in Multiobjective Optimization
 Evolutionary Computation
, 1995
"... The application of evolutionary algorithms (EAs) in multiobjective optimization is currently receiving growing interest from researchers with various backgrounds. Most research in this area has understandably concentrated on the selection stage of EAs, due to the need to integrate vectorial performa ..."
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Cited by 487 (13 self)
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The application of evolutionary algorithms (EAs) in multiobjective optimization is currently receiving growing interest from researchers with various backgrounds. Most research in this area has understandably concentrated on the selection stage of EAs, due to the need to integrate vectorial performance measures with the inherently scalar way in which EAs reward individual performance, i.e., number of offspring. In this review, current multiobjective evolutionary approaches are discussed, ranging from the conventional analytical aggregation of the different objectives into a single function to a number of populationbased approaches and the more recent ranking schemes based on the definition of Paretooptimality. The sensitivity of different methods to
Evolutionary Algorithms for Multiobjective Optimization
, 2002
"... Multiple, often conflicting objectives arise naturally in most realworld optimization scenarios. As evolutionary algorithms possess several characteristics due to which they are well suited to this type of problem, evolutionbased methods have been used for multiobjective optimization for more than ..."
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Cited by 436 (14 self)
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Multiple, often conflicting objectives arise naturally in most realworld optimization scenarios. As evolutionary algorithms possess several characteristics due to which they are well suited to this type of problem, evolutionbased methods have been used for multiobjective optimization for more than a decade. Meanwhile evolutionary multiobjective optimization has become established as a separate subdiscipline combining the fields of evolutionary computation and classical multiple criteria decision making. In this paper, the basic principles of evolutionary multiobjective optimization are discussed from an algorithm design perspective. The focus is on the major issues such as fitness assignment, diversity preservation, and elitism in general rather than on particular algorithms. Different techniques to implement these strongly related concepts will be discussed, and further important aspects such as constraint handling and preference articulation are treated as well. Finally, two applications will presented and some recent trends in the field will be outlined.
Multiobjective Evolutionary Algorithms: Analyzing the StateoftheArt
, 2000
"... Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mideighties in an attempt to stochastically solve problems of this generic class. During the past decade, ..."
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Cited by 424 (7 self)
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Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mideighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of c...
Approximating the nondominated front using the Pareto Archived Evolution Strategy
 EVOLUTIONARY COMPUTATION
, 2000
"... We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its ..."
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Cited by 324 (21 self)
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We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some realworld applications when local search seems superior to or competitive with populationbased methods. We introduce (1 + λ) and (μ  λ) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks.
A Comprehensive Survey of EvolutionaryBased Multiobjective Optimization Techniques
 Knowledge and Information Systems
, 1998
"... . This paper presents a critical review of the most important evolutionarybased multiobjective optimization techniques developed over the years, emphasizing the importance of analyzing their Operations Research roots as a way to motivate the development of new approaches that exploit the search cap ..."
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Cited by 286 (22 self)
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. This paper presents a critical review of the most important evolutionarybased multiobjective optimization techniques developed over the years, emphasizing the importance of analyzing their Operations Research roots as a way to motivate the development of new approaches that exploit the search capabilities of evolutionary algorithms. Each technique is briefly described mentioning its advantages and disadvantages, their degree of applicability and some of their known applications. Finally, the future trends in this discipline and some of the open areas of research are also addressed. Keywords: multiobjective optimization, multicriteria optimization, vector optimization, genetic algorithms, evolutionary algorithms, artificial intelligence. 1 Introduction Since the pioneer work of Rosenberg in the late 60s regarding the possibility of using geneticbased search to deal with multiple objectives, this new area of research (now called evolutionary multiobjective optimization) has grown c...
Multiobjective Optimization Using Evolutionary Algorithms  A Comparative Case Study
, 1998
"... . Since 1985 various evolutionary approaches to multiobjective optimization have been developed, capable of searching for multiple solutions concurrently in a single run. But the few comparative studies of different methods available to date are mostly qualitative and restricted to two approaches. I ..."
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Cited by 222 (12 self)
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. Since 1985 various evolutionary approaches to multiobjective optimization have been developed, capable of searching for multiple solutions concurrently in a single run. But the few comparative studies of different methods available to date are mostly qualitative and restricted to two approaches. In this paper an extensive, quantitative comparison is presented, applying four multiobjective evolutionary algorithms to an extended 0/1 knapsack problem. 1 Introduction Many realworld problems involve simultaneous optimization of several incommensurable and often competing objectives. Usually, there is no single optimal solution, but rather a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior to them when all objectives are considered. They are known as Paretooptimal solutions. Mathematically, the concept of Paretooptimality can be defined as follows: Let us consider, without loss of generality, a multio...
Combining convergence and diversity in evolutionary multiobjective optimization
 Evolutionary Computation
, 2002
"... Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to �nd a number of Paretooptimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms c ..."
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Cited by 156 (15 self)
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Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to �nd a number of Paretooptimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Paretooptimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Paretooptimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept ofdominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modi�cations to the baseline algorithm are also suggested. The concept ofdominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.