## Multi-Objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems (1999)

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Venue: | Evolutionary Computation |

Citations: | 166 - 12 self |

### BibTeX

@ARTICLE{Deb99multi-objectivegenetic,

author = {Kalyanmoy Deb},

title = {Multi-Objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems},

journal = {Evolutionary Computation},

year = {1999},

volume = {7},

pages = {205--230}

}

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### Abstract

In this paper, we study the problem features that may cause a multi-objective genetic algorithm (GA) difficulty in converging to the true Pareto-optimal front. Identification of such features helps us develop difficult test problems for multi-objective optimization. Multi-objective test problems are constructed from single-objective optimization problems, thereby allowing known difficult features of single-objective problems (such as multi-modality, isolation, or deception) to be directly transferred to the corresponding multi-objective problem. In addition, test problems having features specific to multiobjective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multi-objective optimization. Keywords Genetic algorithms, multi-objective optimization, niching, pareto-optimality, problem difficulties, test problems. 1 Introduction After a decade since the pioneering wor...

### Citations

2025 |
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Citation Context ...by one and Go to Step 1. Step 4: All solutions that are not marked ‘dominated’ are non-dominated solutions. A population of solutions can be classified into groups of different non-domination leve=-=ls (Goldberg, 1989-=-). When the above procedure is applied for the first time in a population, the resulting set is the non-dominated set of first (or best) level. In order to have further classifications, these non-domi... |

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Uniform Crossover in Genetic Algorithms
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Citation Context ...tions before they would be able to compare different GA implementations or before they would be able to mimic operators used in single-objective GAs, such as CHC (Eshelman, 1990) or steady-state GAs (=-=Syswerda, 1989-=-). As it is often suggested and used in single-objective GAs, a hybrid strategy of either implementing problem-specific knowledge in GA operators or using a two-stage optimization process of first fin... |

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Citation Context ...ade-offs among the objectives may not have been discovered. In most multi-objective GA implementations, a specific diversity-maintaining operator, such as a niching technique (Deb and Goldberg, 1989; =-=Goldberg and Richardson, 1987-=-), is used to find diverse Paretooptimal solutions. However, the following features of a multi-objective optimization problem may cause multi-objective GAs to have difficulty in maintaining diverse Pa... |

485 | Genetic algorithms for multi-objective optimization: Formulation, discussion and generalization
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Citation Context ...ork with a population of solutions, multiple Pareto-optimal solutions can be found in a GA population in a single simulation run. During the years 1993-95, a number of independent GA implementations (=-=Fonseca and Fleming, 1993-=-; Horn et al., 1994; Srinivas and Deb, 1995) emerged. Later, other researchers successfully used these implementations in various multi-objective optimization applications (Cunha et al., 1997; Eheart ... |

400 | An Overview of Evolutionary Algorithms in Multiobjective Optimization
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Citation Context ...ing multi-objective optimization problems for finding multiple Pareto-optimal solutions. Thus, it is no surprise that a number of different multi-objective GA implementations exist in the literature (=-=Fonseca and Fleming, 1995-=-; Horn et al., 1994; Srinivas and Deb, 1995; Zitzler and Thiele, 1998). Before we discuss the problem features that may cause multi-objective GAs difficulty, let us mention a couple of matters3 that a... |

383 | Multiple objective optimization with vector evaluated genetic algorithms - Schaffer - 1985 |

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335 |
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321 | Multiobjective Evolutionary Algorithm Test Suites - Veldhuizen, Lamont - 1999 |

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278 |
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277 |
An investigation of niche and species formation in genetic function optimization
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- 1989
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Citation Context ...used `1 =10, `2=5, `3=5, `4=5,suchthat` =25. Since the functions are defined with unitation values, we have used genotypic niching with Hamming distance as the distance measure between two solutions (=-=Deb and Goldberg, 1989-=-). Since we expect 11 different function values in f1 (all integers from 1 to 11), we use guidelines suggested in that study and calculate share =9. Figure 6 shows that when a population size of 80 is... |

248 | Genetic Algorithms, Noise, and the Sizing of Populations - Goldberg, Deb, et al. |

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Citation Context ...ul applications are reported, penalty function methods demand an appropriate choice of a penalty parameter for each constraint. Recent suggestions of penalty parameter-less techniques (Deb, in press; =-=Koziel and Michalewicz, 1998-=-) may be worth investigating in the context of multi-objective constrained optimization. 4 A Special Two-Objective Optimization Problem Let us begin our discussion with a simple two-objective optimiza... |

168 | Multi-Objective Optimization and Multiple Constraint Handling with Evolutionary Algorithms – Part I: Application Example - Fonseca, Fleming - 1998 |

148 | Multiobjective optimization using evolutionary algorithms|A comparative study. In Parallel Problem Solving from Nature (PPSN V
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Citation Context ...96). A number of studies have also concentrated in developing new and improved GA implementations (Fonseca and Fleming, 1998; Leung et al., 1998; Kursawe, 1990; Laumanns, Rudolph, and Schwefel, 1998; =-=Zitzler and Thiele, 1998-=-a). Fonseca and Fleming (1995) and Horn (1997) have presented overviews of different multi-objective GA implementations. Recently, van Veldhuizen and Lamont (1998) have made a survey of test problems ... |

129 | Serial and Parallel Genetic Algorithms as Function Optimizers
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Citation Context ...uch a simple bimodal problem. However, a more difficult test problem can be constructed using a standard single-objective multi-modal test problems, such as Rastrigin's function, Schwefel's function (=-=Gordon and Whitley, 1993-=-), and others. 4.2 Deceptive multi-objective optimization problem Next, we shall create a deceptive multi-objective optimization problem, from a deceptive g function. This function is defined over bin... |

115 | Massively multimodality, deception and genetic algorithms
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Citation Context ...not all function values of f 1 are likely to be found, some region in the Pareto-optimal front will be undiscovered. The G-V function for g has a massively multi-modal landscape along with deception (=-=Goldberg, Deb, and Horn, 1992-=-). This function introduces a number of different solutions having the same global optimal g function value. Corresponding to each of these globally optimal solutions for g function, there is one glob... |

105 | Engineering Optimization Methods and Applications - Reklaitis, Ravindran, et al. - 1983 |

101 |
Multi-objective function optimization using nondominated sorting genetic algorithms, Evolutionary Computation
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Citation Context ...areto-optimal solutions can be found in a GA population in a single simulation run. During the years 1993-95, a number of independent GA implementations (Fonseca and Fleming, 1993; Horn et al., 1994; =-=Srinivas and Deb, 1995-=-) emerged. Later, other researchers successfully used these implementations in various multi-objective optimization applications (Cunha et al., 1997; Eheart et al., 1993; Mitra et al., 1998; Parks and... |

98 | Fundamental Principles of Deception in Genetic Search. Foundations of Genetic Algorithms
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Optimization for Engineering Design: Algorithms and Examples
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74 | Deception considered harmful
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Citation Context ...er, we create a multi-modal multi-objective problem and show that a multi-objective GA can get stuck at a local Pareto-optimal front if appropriate GA parameters are not used. Despite some criticism (=-=Grefenstette, 1993-=-), deception, if present in a problem, has been shown to cause GAs to be misled towards deceptive attractors (Goldberg et al., 1989). There is a difference between the difficulties caused by multi-mod... |

72 |
Some experiments in machine learning using vector evaluated genetic algorithm (artificial intelligence, optimization, adaptation, pattern recognition
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- 1984
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Citation Context ...on. In some multi-objective optimization problems, the Pareto-optimal front may not be continuous, instead it may be a collection of discretely spaced continuous sub-regions (Poloni et al., in press; =-=Schaffer, 1984-=-). In such problems, although solutions within each sub-region may be found, competition among these solutions may lead to extinction of some sub-regions. It is also likely that the Pareto-optimal fro... |

70 | Real-coded genetic algorithms with simulated binary crossover: Studies on multi-modal and multi-objective problems
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Citation Context ...in tackling both these problems will largely depend on the constraint-handling technique used. Traditionally, a simple penalty-function based method has been used to penalize each objective function (=-=Deb and Kumar, 1995-=-; Srinivas and Deb, 1995; Weile et al., 1996). Although successful applications are reported, penalty function methods demand an appropriate choice of a penalty parameter for each constraint. Recent s... |

60 |
Learning gene linkage to efficiently solve problems of bounded difficulty using genetic algorithms
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Citation Context ...ys to develop new and improved GAs (such as messy GAs (Goldberg, Korb, and Deb, 1990), Gene expression messy GA (Kargupta, 1996), CHC (Eshelman, 1990), Genitor (Whitley, 1989)), Linkage learning GAs (=-=Harik, 1997-=-), and others. In this paper, we attempt to highlight a number of problem features that may cause a multi-objective GA difficulty. Keeping these properties in mind, we then show procedures of construc... |

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A simulated annealing like convergence theory for the simple genetic algorithm
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Citation Context ... and Miller, 1998; Weile, Michelsson, and Goldberg, 1996). A number of studies have also concentrated in developing new and improved GA implementations (Fonseca and Fleming, 1998; Leung et al., 1998; =-=Rudolph, 1998-=-; Laumanns, Rudolph, and Schwefel, 1998; Zitzler and Thiele, 1998). Fonseca and Fleming (1995) and Horn (1997) have presented overviews of different Currently visiting the Systems Analysis Group, Univ... |

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Citation Context ...Deb, 1995) emerged. Later, other researchers successfully used these implementations in various multi-objective optimization applications (Cunha et al., 1997; Eheart et al., 1993; Mitra et al., 1998; =-=Parks and Miller, 1998-=-; Weile et al., 1996). A number of studies have also concentrated on developing new GA implementations (Kursawe, 1990; Laumanns et al., 1998; Zitzler and Thiele, 1998). Fonseca and Fleming (1995) and ... |

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Citation Context ...ng operators), the presence of which may help us to prove convergence of a GA population to the global Pareto-optimal front. Attempts to some such proofs exist for single-objective GAs (Suzuki, 1993; =-=Rudolph, 1998-=-) and a similar proof may also be attempted for multi-objective GAs. Elitism is an useful and popular mechanism used in single-objective GAs. Elitism ensures that the best solutions in each generation... |

30 |
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- 1993
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Citation Context ...features that may be present in a problem: 1. Multi-modality, 2. Deception, 3. Isolated optimum, and 4. Collateral noise. All the above features are known to cause difficulty in single-objective GAs (=-=Deb et al., 1993-=-) and, when present in a multi-objective problem, may also cause difficulty for a multiobjective GA. In tackling a multi-objective problem having multiple Pareto-optimal fronts, a GA, like many other ... |

26 |
A Markov chain analysis on a genetic algorithm
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Citation Context ...rators), the presence of which may help us to prove convergence of a GA population to the global Pareto-optimal front. Several attempts have been made to achieve such proofs for single-objective GAs (=-=Suzuki, 1993-=-; Rudolph, 1994) and similar attempts may also be made for multi-objective GAs. Elitism is a useful and popular mechanism used in single-objective GAs. Elitism ensures that the best solutions in each ... |

23 |
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Citation Context ...resence of which may help us to prove convergence of a GA population to the global Pareto-optimal front. Several attempts have been made to achieve such proofs for single-objective GAs (Suzuki, 1993; =-=Rudolph, 1994-=-) and similar attempts may also be made for multi-objective GAs. Elitism is a useful and popular mechanism used in single-objective GAs. Elitism ensures that the best solutions in each generation will... |

16 |
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15 |
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Citation Context ... (Fonseca and Fleming, 1993; Horn et al., 1994; Srinivas and Deb, 1995) emerged. Later, other researchers successfully used these implementations in various multi-objective optimization applications (=-=Cunha et al., 1997-=-; Eheart et al., 1993; Mitra et al., 1998; Parks and Miller, 1998; Weile et al., 1996). A number of studies have also concentrated on developing new GA implementations (Kursawe, 1990; Laumanns et al.,... |

11 |
Multiobjective Dynamic Optimization of an Industrial Nylon 6 Semibatch Reactor Using Genetic Algorithm
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- 1998
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Citation Context ... 1994; Srinivas and Deb, 1995) emerged. Later, other researchers successfully used these implementations in various multi-objective optimization applications (Cunha et al., 1997; Eheart et al., 1993; =-=Mitra et al., 1998-=-; Parks and Miller, 1998; Weile et al., 1996). A number of studies have also concentrated on developing new GA implementations (Kursawe, 1990; Laumanns et al., 1998; Zitzler and Thiele, 1998). Fonseca... |

8 |
Sufficient Conditions for Arbitrary Binary Functions
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- 1994
(Show Context)
Citation Context ...n of a substring of length `i is zero and `i, respectively. 9 It can be shown that an equivalent dual maximization function G = `i +1,g(u(`i)) is deceptive according to conditions outlined elsewhere (=-=Deb and Goldberg, 1994-=-). Thus, the above minimization problem is also deceptive. Evolutionary Computation Volume 7, Number 3 215sK. Deb in one subproblem and to the true substring in the two other subproblems. When a suffi... |

2 | Genetic-algorithm-based design of groundwater quality monitoring system
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- 1993
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Citation Context ...g, 1993; Horn et al., 1994; Srinivas and Deb, 1995) emerged. Later, other researchers successfully used these implementations in various multi-objective optimization applications (Cunha et al., 1997; =-=Eheart et al., 1993-=-; Mitra et al., 1998; Parks and Miller, 1998; Weile et al., 1996). A number of studies have also concentrated on developing new GA implementations (Kursawe, 1990; Laumanns et al., 1998; Zitzler and Th... |

1 | A variant of evolution strategies for vector optimization - Kurusawe - 1990 |

1 |
Multiobjective optimization using nondominated sorting annealing genetic algorithms. (Unpublished document
- Leung, Zhu, et al.
- 1998
(Show Context)
Citation Context ...d Gupta, 1998; Parks and Miller, 1998; Weile, Michelsson, and Goldberg, 1996). A number of studies have also concentrated in developing new and improved GA implementations (Fonseca and Fleming, 1998; =-=Leung et al., 1998-=-; Kursawe, 1990; Laumanns, Rudolph, and Schwefel, 1998; Zitzler and Thiele, 1998a). Fonseca and Fleming (1995) and Horn (1997) have presented overviews of different multi-objective GA implementations.... |

1 |
Learning gene linkage to efficiently solve problems of bounded difficulty using genetic algorithms
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