## Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach (1999)

### Cached

### Download Links

- [www.tik.ee.ethz.ch]
- [www.das.ufsc.br]
- [www.jeo.org]
- DBLP

### Other Repositories/Bibliography

Citations: | 560 - 19 self |

### BibTeX

@MISC{Zitzler99multiobjectiveevolutionary,

author = {Eckart Zitzler and Lothar Thiele},

title = {Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

Evolutionary algorithms (EAs) 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 EAs 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 EAs 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 domina...

### Citations

8169 |
Genetic Algorithms
- Goldberg
- 1989
(Show Context)
Citation Context ...ording to one of the objectives [9], [10]. Other non-Pareto algorithms use multiple linear combinations of the objectives in parallel [11], [12]. Pareto-based fitness assignment was first proposed in =-=[13]-=-. All approaches of this type explicitly use Pareto dominance in order to determine the reproduction probability of each individual. While non-Pareto EA’s are often sensitive to the nonconvexity of Pa... |

703 | No free lunch theorems for optimization
- Wolpert, Macready
- 1997
(Show Context)
Citation Context ...search and optimization might be a problem area where EA’s do better than other blind search strategies [1], [2]. Although this statement must be qualified with regard to the “no free lunch” theorems =-=[3]-=-, up to now there are few if any alternatives to EA-based multiobjective optimization [4]. Since the mid-1980’s, there has been a growing interest in solving multicriteria optimization problems using ... |

570 |
Knapsack problems. Algorithms and computer implementation
- Martello, Toth
- 1990
(Show Context)
Citation Context .... The task is to find a subset of items which maximizes the total of the profits in the subset, yet all selected items fit into the knapsack, i.e., the total weight does not exceed the given capacity =-=[31]-=-. This single-objective problem can be extended directly to the multiobjective case by allowing an arbitrary number of knapsacks. Formally, the multiobjective 0/1 knapsack problem considered here is d... |

537 |
Genetic Algorithms with sharing for multimodal function optimization
- Goldberg, Richardson
- 1987
(Show Context)
Citation Context ...t from premature convergence), but in addition niching techniques are characterized by their capability of promoting the formulation and maintenance of stable subpopulations (niches). Fitness sharing =-=[17]-=- is used most frequently, which is a niching technique based on the idea that individuals in a particular niche have to share the available resources. The more individuals are located in the neighborh... |

485 | Genetic algorithms for multi-objective optimization: Formulation, discussion and generalization
- Fonseca, Fleming
- 1993
(Show Context)
Citation Context ...convexity of Pareto-optimal sets, this is not the case for Pareto-based EA’s [1]. Finally, some multiobjective EA’s also make use of combinations of the presented fitness assignment strategies (e.g., =-=[14]-=-, [15]). C. Multimodal Optimization and Preservation of Diversity When we consider the case of finding a set of nondominated solutions rather than a single-point solution, multiobjective EA’s have to ... |

434 | Comparison of Multiobjective Evolutionary Algorithms
- Zitzler, Deb, et al.
- 2000
(Show Context)
Citation Context ...esponding medians of the other multiobjective EA’s by more than ten quartile deviations. Although the Pareto-optimal fronts of the test problems considered here are all convex, we have shown recently =-=[45]-=- that SPEA also has advantages over the other EA’s for different types of problems (e.g., nonconvex functions). • As Fig. 1 indicates, SPEA can find solutions that are closer to the Pareto-optimal fro... |

404 | Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation
- Srinivas, Deb
- 1994
(Show Context)
Citation Context ...timization, Fonseca and Fleming [1] categorize 1 The definitions and terms presented in this section correspond to the mathematical formulations most widespread in multiobjective EA literature (e.g., =-=[6]-=-, [1]). For more detailed information, we refer to [7] and [8]. (1) (2) several multicriteria EA’s and compare different fitness assignment strategies. In particular, they distinguish plain aggregatin... |

400 | An Overview of Evolutionary Algorithms in Multiobjective Optimization
- Fonseca, Flemming
- 1995
(Show Context)
Citation Context ...oiting similarities of solutions by recombination. Some researchers suggest that multiobjective search and optimization might be a problem area where EA’s do better than other blind search strategies =-=[1]-=-, [2]. Although this statement must be qualified with regard to the “no free lunch” theorems [3], up to now there are few if any alternatives to EA-based multiobjective optimization [4]. Since the mid... |

383 |
Multiple objective optimization with vector evaluated genetic algorithms
- Schaffer
- 1985
(Show Context)
Citation Context ...ng is incorporated to preserve diversity in the population. B. A Simple Test Function: Schaffer’s A very simple test function for multiobjective optimizers is the well-known function used by Schaffer =-=[44]-=-. It is defined as follows: minimize where Obviously, the Pareto-optimal points are located in the range . Outside this interval, as well as are increasing, while within the interval, there is a trade... |

361 | Mulitple Criteria Optimization: Theory, Computation, and Application - Steuer - 1986 |

351 |
Graphical methods for data analysis
- Chambers, Cleveland, et al.
- 1983
(Show Context)
Citation Context ...method are connected by dashed lines and RAND is not included in the figure. Note that SPEA and SP-S are described later. , values per test problem according to the 30 runs performed. Here, box plots =-=[35]-=- are used to visualize the distribution of these samples. A box plot consists of a box summarizing 50% of the data. The upper and lower ends of the box are the upper and lower quartiles, while a thick... |

304 | A Niched Pareto Genetic Algorithm for Multiobjective Optimization
- Horn, Nafpliotis, et al.
- 1994
(Show Context)
Citation Context ...edsum aggregation (e.g., [12]). We have chosen HLGA to represent this class of multiobjective EA’s. 3) Niched Pareto Genetic Algorithm: The niched Pareto genetic algorithm (NPGA) proposed in [18] and =-=[26]-=- combines tournament selection and the concept of Pareto dominance. Two competing individuals and a comparison set of other individuals are picked at random from the population; the size of the compar... |

277 |
An investigation of niche and species formation in genetic function optimization
- Deb, Goldberg
- 1989
(Show Context)
Citation Context ...OMINATION PRESSURE (tdom) be dependent on the complexity of the test problem, as can be seen in Table I: the more knapsacks and items involved, the greater the value for . Following the guidelines in =-=[34]-=-, the niche radius was calculated based on normalized distance, assuming the formation of ten (15 and 20, respectively) independent niches in the case of two (three and four, respectively) knapsacks. ... |

205 | Niching Methods for Genetic Algorithms
- Mahfoud
- 1995
(Show Context)
Citation Context ...rform a multimodal search that samples the Pareto-optimal set uniformly. Unfortunately, a simple (elitist) EA tends to converge toward a single solution and often loses solutions due to three effects =-=[16]-=-: selection pressure, selection noise, and operator disruption. To overcome this problem, several methods have been developed that can be divided into niching techniques and nonniching techniques [16]... |

148 | Multiobjective optimization using evolutionary algorithms|A comparative study. In Parallel Problem Solving from Nature (PPSN V
- Zitzler, Thiele
- 1998
(Show Context)
Citation Context ... a few algorithms. Therefore, extensive quantitative comparisons are needed in order to assess the performance of the EA’s in a greater context. Previous effort in this direction has been reported in =-=[5]-=-. In the present study, we provide a comparison of five multicriteria EA’s, four previously known and one new, by solving a multiobjective 0/1 knapsack problem. Thereby, two complementary quantitative... |

128 |
A multi-objective genetic local search algorithm and its application to flowshop scheduling
- Ishibuchi, Murata
- 1998
(Show Context)
Citation Context ...iques in order to find multiple Pareto-optimal solutions in parallel. On one hand, similarly to other multiobjective EA’s, it: • stores the nondominated solutions found so far externally (e.g., [10], =-=[12]-=-, [19]); • uses the concept of Pareto dominance in order to assign scalar fitness values to individuals; • performs clustering to reduce the number of nondominated solutions stored without destroying ... |

124 | Multiobjective optimization using the niched Pareto genetic algorithm”, IlliGAL
- Horn, Nafpliotis
- 1993
(Show Context)
Citation Context ... sharing and phenotypic sharing; phenotypic sharing can be performed on the decision vectors or the objective vectors. Currently, most multiobjective EA’s implement fitness sharing (e.g., [11], [14], =-=[18]-=-, [6], [15], [19], [20]). Among the nonniching techniques, restricted mating is the most common in multicriteria function optimization. Basically, two individuals are allowed to mate only if they are ... |

107 | Finding Multimodal Solutions Using Restricted Tournament Selection
- Harik
- 1995
(Show Context)
Citation Context ...he number of population members that dominate it. For the purpose of a diverse population, he incorporated a niching technique which has been rather seldom used: restricted tournament selection (RTS) =-=[48]-=-. RTS is a special binary tournament selection for steady-state EA’s where two individuals hold tournament with the most similar individual of a randomly chosen group; winners replace inferior individ... |

104 | A Variant of Evolution Strategies for Vector Optimization
- Kursawe
- 1991
(Show Context)
Citation Context ...election criterion during the reproduction phase, the search is guided in several directions at the same time. Often, fractions of the mating pool are selected according to one of the objectives [9], =-=[10]-=-. Other non-Pareto algorithms use multiple linear combinations of the objectives in parallel [11], [12]. Pareto-based fitness assignment was first proposed in [13]. All approaches of this type explici... |

100 | Searching for diverse, cooperative populations with genetic algorithms
- Smith, Forrest, et al.
- 1993
(Show Context)
Citation Context ... symbiotic evolution can speed up the search process. In [37]–[40], a similar concept was applied to immune system models where two cooperative populations were used to maintain population diversity; =-=[39]-=- reported that this method has emergent properties that are similar to fitness sharing.266 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 3, NO. 4, NOVEMBER 1999 2) Reducing the Pareto Set by Cl... |

90 | System-Level Synthesis Using Evolutionary Algorithms
- Blickle, Teich, et al.
- 1998
(Show Context)
Citation Context ...pplication to System-Level Synthesis The third application is a larger problem in the domain of computer engineering that is concerned with computer-based system-level synthesis. Blickle et al. [24], =-=[46]-=-, [47] have presented an evolutionary approach to this problem which we use as the basis for the SPEA implementation. 1) Problem Description: In [47], system-level synthesis is considered as the probl... |

89 |
Genetic search strategies in multicriterion optimal design
- Hajela, Lin
- 1992
(Show Context)
Citation Context ...he same time. Often, fractions of the mating pool are selected according to one of the objectives [9], [10]. Other non-Pareto algorithms use multiple linear combinations of the objectives in parallel =-=[11]-=-, [12]. Pareto-based fitness assignment was first proposed in [13]. All approaches of this type explicitly use Pareto dominance in order to determine the reproduction probability of each individual. W... |

72 | Using genetic algorithms to explore pattern recognition in the immune system
- Forrest, Javornik, et al.
- 1993
(Show Context)
Citation Context ...wo interacting populations has been inspired by [37]–[41]. Paredis [41] studied the use of cooperating populations in EA’s and showed that symbiotic evolution can speed up the search process. In [37]–=-=[40]-=-, a similar concept was applied to immune system models where two cooperative populations were used to maintain population diversity; [39] reported that this method has emergent properties that are si... |

68 | A comparison of selection schemes used in evolutionary algorithms
- Blickle, Thiele
- 1997
(Show Context)
Citation Context ...plied. But the influence of the selection scheme on the outcome of an EA cannot be neglected, e.g., fitness proportionate selection, which is used in VEGA, is well known to have serious disadvantages =-=[32]-=-. In order to guarantee a fair comparison, all EA’s considered were implemented with the same selection scheme: binary tournament selection with replacement. This selection method turned out to be sup... |

68 |
Compaction of symbolic layout using genetic algorithms
- Fourman
- 1985
(Show Context)
Citation Context ...the selection criterion during the reproduction phase, the search is guided in several directions at the same time. Often, fractions of the mating pool are selected according to one of the objectives =-=[9]-=-, [10]. Other non-Pareto algorithms use multiple linear combinations of the objectives in parallel [11], [12]. Pareto-based fitness assignment was first proposed in [13]. All approaches of this type e... |

64 | The zero/one multiple knapsack problem and genetic algorithms
- Khuri, Bäck, et al.
- 1994
(Show Context)
Citation Context ...actical relevance it has been subject to several investigations in various fields. In particular, there are some publications in the domain of evolutionary computation related to the knapsack problem =-=[27]-=-–[29], even in conjunction with multiobjective optimization [30].260 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 3, NO. 4, NOVEMBER 1999 1) Formulation as Multiobjective Optimization Problem:... |

44 |
Tournament selection, niching, and the preservation of diversity
- Oei, Goldberg, et al.
- 1991
(Show Context)
Citation Context ...ing selection on our test problems—that has been confirmed experimentally. Unfortunately, a conventional combination of fitness sharing and tournament selection may lead to chaotic behavior of the EA =-=[33]-=-. Therefore, both NSGA and HLGA were implemented using a slightly modified version of sharing, called continuously updated sharing, which was proposed by the same researchers. Thereby, the partly fill... |

43 | Multi-objective optimization by genetic algorithms: a review. Evol. Comput - Tamaki, Kita, et al. - 1996 |

37 |
The Symbiotic Evolution of Solutions and Their Representations
- Paredis
- 1995
(Show Context)
Citation Context ...ces the niching capability as we will discuss in the next section. Furthermore, it has to be mentioned that this kind of fitness assignment using two interacting populations has been inspired by [37]–=-=[41]-=-. Paredis [41] studied the use of cooperating populations in EA’s and showed that symbiotic evolution can speed up the search process. In [37]–[40], a similar concept was applied to immune system mode... |

34 |
Reducing the size of the nondominated set: pruning by clustering
- Morse
- 1980
(Show Context)
Citation Context ...while maintaining its characteristics might be necessary or even mandatory. A method that has been applied to this problem successfully and studied extensively in the same context is cluster analysis =-=[42]-=-, [43]. In general, cluster analysis partitions a collection of elements into groups of relatively homogeneous elements, where . The average linkage method [42], a clustering approach that has proven ... |

29 |
Theory of Evolutionary Algorithms and Application to System Synthesis
- Blickle
- 1996
(Show Context)
Citation Context ...hing methods like crowding [22] and its derivatives as well as nonniching techniques as isolation by distance [23] have never been applied to EA’s with multiple objectives (an exception is offered in =-=[24]-=-, cf. Section IV-D). D. Four Population-Based Approaches In the following we present the multiobjective EA’s applied to the knapsack problem in our comparison. For a thorough discussion of other evolu... |

29 |
Genetic algorithms and the immune system
- Forrest, Perelson
- 1991
(Show Context)
Citation Context ...fluences the niching capability as we will discuss in the next section. Furthermore, it has to be mentioned that this kind of fitness assignment using two interacting populations has been inspired by =-=[37]-=-–[41]. Paredis [41] studied the use of cooperating populations in EA’s and showed that symbiotic evolution can speed up the search process. In [37]–[40], a similar concept was applied to immune system... |

26 | Population diversity in an immune system model: Implications for genetic search. in Foundations of Genetic Algorithms - Smith, Forrest - 1993 |

24 | An interactive fuzzy satisficing method for multi objective nonconvex programming problems with fuzzy numbers through coevolutionary genetic algorithms
- Sakawa, Yauchi
(Show Context)
Citation Context ...in various fields. In particular, there are some publications in the domain of evolutionary computation related to the knapsack problem [27]–[29], even in conjunction with multiobjective optimization =-=[30]-=-.260 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 3, NO. 4, NOVEMBER 1999 1) Formulation as Multiobjective Optimization Problem: Generally, a 0/1 knapsack problem consists of a set of items, w... |

22 | A non generational genetic algorithm for multiobjective optimization
- Valenzuela-Rendón, Uresti-Charre
(Show Context)
Citation Context ...d for reproduction. Horn and Nafpliotis [18], [26] used phenotypic sharing on the objective vectors. This algorithm seems to be widespread and has been often taken as reference in recent publications =-=[2]-=-, [21], [20], hence, it is also examined here. 4) Nondominated Sorting Genetic Algorithm: Srinivas and Deb [6] also developed an approach based on [13], called nondominated sorting genetic algorithm (... |

17 | A multiple criteria genetic algorithm for containership loading - Todd, Sen - 1997 |

15 |
Use of genetic algorithms in multicriteria optimization to solve industrial problems
- Cunha, Oliveira, et al.
- 1997
(Show Context)
Citation Context ... sharing; phenotypic sharing can be performed on the decision vectors or the objective vectors. Currently, most multiobjective EA’s implement fitness sharing (e.g., [11], [14], [18], [6], [15], [19], =-=[20]-=-). Among the nonniching techniques, restricted mating is the most common in multicriteria function optimization. Basically, two individuals are allowed to mate only if they are within a certain distan... |

15 | The Neighborhood Constraint Method: A Genetic Algorithm-based Multi-objective Optimization Technique - Loughlin, Ranjithan, et al. - 1997 |

13 |
F1.9 multicriterion decision making. Handbook of Evolutionary Computation 97/1
- Horn
- 1997
(Show Context)
Citation Context ...rch strategies [1], [2]. Although this statement must be qualified with regard to the “no free lunch” theorems [3], up to now there are few if any alternatives to EA-based multiobjective optimization =-=[4]-=-. Since the mid-1980’s, there has been a growing interest in solving multicriteria optimization problems using evolutionary approaches. In the meantime, several multiobjective EA’s are available that ... |

11 | Fitness functions for multiple objective optimization problems: Combining preferences with pareto rankings - Greenwood, Hu, et al. - 1996 |

8 |
Multiobjective Optimization: Behavioral and Computational Considerations
- Ringuest
- 1992
(Show Context)
Citation Context ...ns and terms presented in this section correspond to the mathematical formulations most widespread in multiobjective EA literature (e.g., [6], [1]). For more detailed information, we refer to [7] and =-=[8]-=-. (1) (2) several multicriteria EA’s and compare different fitness assignment strategies. In particular, they distinguish plain aggregating approaches, population-based non-Pareto approaches, and Pare... |

7 | Perelson, "Searching for diverse, cooperative populations with genetic algorithms - Smith, Forrest, et al. - 1992 |

7 |
Reducing the Pareto optimal set in multicriteria optimization, Eng Optim 8
- Rosenman, Gero
- 1985
(Show Context)
Citation Context ...maintaining its characteristics might be necessary or even mandatory. A method that has been applied to this problem successfully and studied extensively in the same context is cluster analysis [42], =-=[43]-=-. In general, cluster analysis partitions a collection of elements into groups of relatively homogeneous elements, where . The average linkage method [42], a clustering approach that has proven to per... |

6 |
Genetic algorithms for the 0/1 knapsack problem
- Michalewicz, Arabas
- 1994
(Show Context)
Citation Context ...the results concerning the former type of knapsack capacity in the following. 3) Implementation: Concerning the chromosome coding as well as the constraint handling, we drew upon results published in =-=[28]-=-, which examined EA’s with different representation mappings and constraint handling techniques on the (singleobjective) 0/1 knapsack problem. Concluding from the experiments in [28], penalty function... |

6 |
Solving large knapsack problems with a genetic algorithm
- Spillman
(Show Context)
Citation Context ...al relevance it has been subject to several investigations in various fields. In particular, there are some publications in the domain of evolutionary computation related to the knapsack problem [27]–=-=[29]-=-, even in conjunction with multiobjective optimization [30].260 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 3, NO. 4, NOVEMBER 1999 1) Formulation as Multiobjective Optimization Problem: Gene... |

6 |
Niche and species formation in genetic algorithms,” in Practical Handbook of Genetic Algorithms
- Ryan
- 1995
(Show Context)
Citation Context ... in the field of multiobjective EA’s (e.g., [11], [14], [21]). To our knowledge, other niching methods like crowding [22] and its derivatives as well as nonniching techniques as isolation by distance =-=[23]-=- have never been applied to EA’s with multiple objectives (an exception is offered in [24], cf. Section IV-D). D. Four Population-Based Approaches In the following we present the multiobjective EA’s a... |

3 | Multiobjective Optimization - Ringuest - 1992 |

2 |
An analysis of the bevavior of a class of genetic adaptive systems
- Jong
- 1975
(Show Context)
Citation Context ...performance. Nevertheless, as mentioned in [1], it does not appear to be widespread in the field of multiobjective EA’s (e.g., [11], [14], [21]). To our knowledge, other niching methods like crowding =-=[22]-=- and its derivatives as well as nonniching techniques as isolation by distance [23] have never been applied to EA’s with multiple objectives (an exception is offered in [24], cf. Section IV-D). D. Fou... |

1 | F1.9 multicriteria decision making," in Handbook of Evolutionary Computation - Horn - 1997 |

1 | Niche and species formation in genetic algorithms," in Practical Handbook of Genetic Algorithms - Ryan - 1995 |