## A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II (2000)

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@MISC{Deb00afast,

author = {Kalyanmoy Deb and Amrit Pratap and Sameer Agarwal and T. Meyarivan},

title = {A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II},

year = {2000}

}

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### Abstract

Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) O(MN computational complexity (where M is the number of objectives and N is the population size), (ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a non-dominated sorting based multi-objective evolutionary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which alleviates all the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN ) computational complexity is presented. Second, a selection operator is presented which creates a mating pool by combining the parent and child populations and selecting the best (with respect to fitness and spread) N solutions. Simulation results on a number of difficult test problems show that the proposed NSGA-II, in most problems, is able to find much better spread of solutions and better convergence near the true Pareto-optimal front compared to PAES and SPEA - two other elitist multi-objective EAs which pay special attention towards creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems eciently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint non-linear problem, are compared with another constrained multi-objective optimizer and much better performance of NSGA-II is observed. Because of NSGA-II's low computational requirements, elitist approach, parameter-less niching approach, and simple constraint-handling strategy, NSGA-II should find increasing applications in the coming years.

### Citations

1233 |
Multi-objective optimization using evolutionary algorithms
- Deb
- 2001
(Show Context)
Citation Context ...s, it has to be applied many times, hopefully finding a different solution at each simulation run. Over the past decade, a number of multiobjective evolutionary algorithms (MOEAs) have been suggested =-=[1]-=-, [7], [13], Manuscript received August 18, 2000; revised February 5, 2001 and September 7, 2001. The work of K. Deb was supported by the Ministry of Human Resources and Development, India, under the ... |

486 | Genetic algorithms for multiobjective optimization: Formulation discussion and generalization
- Fonseca, Fleming
- 1993
(Show Context)
Citation Context ...as to be applied many times, hopefully finding a different solution at each simulation run. Over the past decade, a number of multi-objective evolutionary algorithms (MOEAs) have been suggested [18], =-=[6]-=-, [11], [24]. The primary reason for this is their ability to find multiple Pareto-optimal solutions in one single simulation run. Since EAs work with a population of solutions, a simple EA can be ext... |

436 | Comparison of multiobjective evolutionary algorithms: Empirical results
- Zitzler, Deb, et al.
- 2000
(Show Context)
Citation Context ...sive algorithm for large population sizes. This large complexity arises because of the complexity involved in the non-dominated sorting procedure in every generation. Lack of elitism: Recent results (=-=[23]-=-, [16]) show clearly that elitism can speed up the performance of the GA significantly, also can help preventing the loss of good solutions once they are found. Need for specifying the sharing paramet... |

384 |
Multiple objective optimization with vector evaluated genetic algorithms
- Schaffer
- 1985
(Show Context)
Citation Context ...cant past studies in this area. Veldhuizen [20] cited a number of test problems which many researchers have used in the past. Of them, we choose four problems, we call them SCH (from Schaffer's study =-=[17]-=-), FON (from Fonseca and Fieming's study [8]), POL (from Poloni's study [14]), and KUR (from Kursawe's study [13]). In 1999, the first author has suggested a systematic way of developing test problems... |

336 |
Messy genetic algorithms: Motivation, analysis and rst results
- Goldberg, Korb, et al.
- 1989
(Show Context)
Citation Context ...ted problem. NSGA-II are able to converge to the Pareto-optimal front (with g(y) = 1 resulting f2 = exp(-f)). This example problem demonstrates that one of the known difficulties (the linkage problem =-=[9]-=-, [10]). of single-objective optimization algorithm can also cause difficulties in a multi-objective problem. However, more systematic studies are needed to amply address the linkage issue in multi-ob... |

334 | An Evolutionary Algorithm for Multiobjective Optimisation: the Strength Pareto Approach. TIK-Report No. 43. Institute für Technische Informatik and Kommunikationsnetze
- Zitzler, Thiele
- 1998
(Show Context)
Citation Context ...ults on a number of difficult test problems, we find that NSGA-II outperforms two other contemporary multi-objective EAs--Pareto-archived evolution strategy (PAES), [12] and strength Pareto EA (SPEA) =-=[22]-=---in terms of finding a diverse set of solutions and in converging near the true Pareto-optimal set. Constrained multi-objective optimization is important from the point of view of practical problem s... |

321 | Multiobjective Evolutionary Algorithm Research: A History and Analysis
- Veldhuizen, Lamont
- 1998
(Show Context)
Citation Context ...lems We first describe the test problems used to compare different multi-objective evolutionary algorithms. Test problems are chosen from a number of significant past studies in this area. Veldhuizen =-=[20]-=- cited a number of test problems which many researchers have used in the past. Of them, we choose four problems, we call them SCH (from Schaffer's study [17]), FON (from Fonseca and Fieming's study [8... |

305 | A niched pareto genetic algorithm for multiobjective optimization
- Horn, Nafpliotis, et al.
- 1994
(Show Context)
Citation Context ... be applied many times, hopefully finding a different solution at each simulation run. Over the past decade, a number of multi-objective evolutionary algorithms (MOEAs) have been suggested [18], [6], =-=[11]-=-, [24]. The primary reason for this is their ability to find multiple Pareto-optimal solutions in one single simulation run. Since EAs work with a population of solutions, a simple EA can be extended ... |

277 |
An investigation of niche and species formation in genetic function optimization
- Deb, Goldberg
- 1989
(Show Context)
Citation Context ... parameter denotes the largest value of that distance metric within which any two solutions share each other’s fitness. This parameter is usually set by the user, although there exist some guidelines =-=[4]-=-. There are two difficulties with this sharing function approach. 1) The performance of the sharing function method in maintaining a spread of solutions depends largely on the chosen value. - - - for ... |

168 | Multi-Objective Optimization and Multiple Constraint Handling with Evolutionary Algorithms – Part I: Application Example
- Fonseca, Fleming
- 1998
(Show Context)
Citation Context ...the concept of sharing. The main problem with sharing is that it requires the specification of a sharing parameter (ashare). Though there has been some work on dynamic sizing of the sharing parameter =-=[8]-=-, a parameter-less diversity preservation mechanism is desirable. In this paper, we address all of these issues and propose an improved version of NSGA, which we call NSGA-II. From the simulation resu... |

166 | Multi-Objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems
- DEB
- 1999
(Show Context)
Citation Context ... (such as O'sbar e needed in the NSGA) is required here. Although the crowding distance is calculated in the objective function space, it can also be implemented in the parameter space, if so desired =-=[2]-=-. However, in all simulations performed in this study, we have used the objective function space niching. IV. SIMULATION RESULTS In this section, we first describe the test problems used to compare th... |

148 | Multiobjective optimization using evolutionary algorithms|A comparative study. In Parallel Problem Solving from Nature (PPSN V
- Zitzler, Thiele
- 1998
(Show Context)
Citation Context ...plied many times, hopefully finding a different solution at each simulation run. Over the past decade, a number of multi-objective evolutionary algorithms (MOEAs) have been suggested [18], [6], [11], =-=[24]-=-. The primary reason for this is their ability to find multiple Pareto-optimal solutions in one single simulation run. Since EAs work with a population of solutions, a simple EA can be extended to mai... |

141 | Simulated binary crossover for continuous search space
- Deb, Agrawal
- 1995
(Show Context)
Citation Context ...on probability of Pm -- 1/n or 1/ (where n is the number of decision variables for real-coded GAs andsis the string length for binary-coded GAs). For NSGA-II (real-coded), we use distribution indices =-=[5]-=- for crossover and mutation operators as /c = 20 and ]m = 20, respectively. The population obtained at the end of 250 generations (the population after elitism mechanism is applied) is used to calcula... |

136 | An efficient constraint handling method for genetic algorithms
- Deb
(Show Context)
Citation Context ... outlines the crowding distance computation procedure of all solutions in an nondominated set Z: crowding-distance-assignment (Z) for each i, set Z[i]distance = 0 for each objective m Z = sort(Z, m) Z=-=[1]-=-istace = Z[l]istace = ec for i = 2 to (l - 1) = + + 1].m - - 1].m) number of solutions in Z initialize distance sort using each objective value so that boundary points are always selected for all othe... |

121 |
The Pareto archived evolution strategy: A new baseline algorithm for multiobjective optimization
- Knowles, Corne
- 1999
(Show Context)
Citation Context ...ll NSGA-II. From the simulation results on a number of difficult test problems, we find that NSGA-II outperforms two other contemporary multi-objective EAs--Pareto-archived evolution strategy (PAES), =-=[12]-=- and strength Pareto EA (SPEA) [22]--in terms of finding a diverse set of solutions and in converging near the true Pareto-optimal set. Constrained multi-objective optimization is important from the p... |

104 | A Variant of Evolution Strategies for Vector Optimization
- Kursawe
- 1991
(Show Context)
Citation Context ...in the past. Of them, we choose four problems, we call them SCH (from Schaffer's study [17]), FON (from Fonseca and Fieming's study [8]), POL (from Poloni's study [14]), and KUR (from Kursawe's study =-=[13]-=-). In 1999, the first author has suggested a systematic way of developing test problems for multi-objective optimization [2]. Zitzler, Deb, and Thiele [23] followed those guidelines and suggested six ... |

101 |
Multi-objective function optimization using nondominated sorting genetic algorithms, Evolutionary Computation
- Srinivas, Deb
(Show Context)
Citation Context ..., it has to be applied many times, hopefully finding a different solution at each simulation run. Over the past decade, a number of multi-objective evolutionary algorithms (MOEAs) have been suggested =-=[18]-=-, [6], [11], [24]. The primary reason for this is their ability to find multiple Pareto-optimal solutions in one single simulation run. Since EAs work with a population of solutions, a simple EA can b... |

87 | On the performance assessment and comparison of stochastic multiobjective optimizers
- Fonseca, Fleming
- 1996
(Show Context)
Citation Context ...f diversity in solutions of the Pareto-optimal set. Clearly, these two tasks cannot be measured with one performance metric adequately. A number of performance metrics have been suggested in the past =-=[7]-=-, [22]. But, here, we define two performance metrics which are more direct in evaluating each of the above two goals in a solution set obtained by a multi-objective optimization algorithm. The first m... |

60 |
Learning gene linkage to efficiently solve problems of bounded difficulty using genetic algorithms
- Harik
- 1997
(Show Context)
Citation Context ...roblem. NSGA-II are able to converge to the Pareto-optimal front (with g(y) = 1 resulting f2 = exp(-f)). This example problem demonstrates that one of the known difficulties (the linkage problem [9], =-=[10]-=-). of single-objective optimization algorithm can also cause difficulties in a multi-objective problem. However, more systematic studies are needed to amply address the linkage issue in multi-objectiv... |

26 | Understanding interactions among genetic algorithm parameters,” in Foundations of Genetic Algorithms
- Deb, Agrawal
- 1998
(Show Context)
Citation Context ...HIS STUDY. ALL OBJECTIVE FUNCTIONS ARE TO BE MINIMIZED. Problem n Variable Objective Optimal Comments bounds functions solutions SCH i [-10 u, 10 u] f(x) = x 2 xs[0, 2] convex f2(x) = (x - 2) 2 FON 3 =-=[-4,4]-=- f(x)-- 1-exp i= (xi- ) x--x2--x3 non-convex f2(x) = 1 - exp Ei= xi +s[-1/v/, 1/v/] roL 2 [-, ] (x) = [1 + (A - B) 2 + (A2 - B2) 2] non-convex, f2(x) = [(x + 3) 2 + (x2 + 1) 2] disconnected A = 0.5 si... |

23 | Multiple objective optimization with vector evaluated genetic algorithms - Schaer - 1985 |

20 | Evolutionary search under partially ordered sets,” Dept - Rudolph - 1999 |

19 |
GA-based decision support system for multicriteria optimization
- Tanaka
- 1995
(Show Context)
Citation Context ...lem SRN was used in the original study of NSGA [20]. Here, the constrained Pareto-optimal set is a subset of the unconstrained Pareto-optimal set. The third problem TNK was suggested by Tanaka et al. =-=[21]-=- and has a discontinuous Pareto-optimal region, falling entirely on the first constraint boundary. In the next section, we show the constrained Pareto-optimal region for each of the above problems. Th... |

18 |
Hybrid GA for multiobjective aerodynamic shape optimization,” in Genetic Algorithms in Engineering and Computer
- Poloni
- 1997
(Show Context)
Citation Context ...ems which many researchers have used in the past. Of them, we choose four problems, we call them SCH (from Schaffer's study [17]), FON (from Fonseca and Fieming's study [8]), POL (from Poloni's study =-=[14]-=-), and KUR (from Kursawe's study [13]). In 1999, the first author has suggested a systematic way of developing test problems for multi-objective optimization [2]. Zitzler, Deb, and Thiele [23] followe... |

12 | Learning gene linkage to e#ciently solve problems of bounded di#culty using genetic algorithms. Doctoral dissertation - Harik - 1999 |

8 |
in press). Multiobjective design optimization by an evolutionary algorithm, Engineering Optimization
- Ray, Kang, et al.
(Show Context)
Citation Context ...change the computational complexity of NSGA-II. The rest of the NSGA-II procedure as described can be used as usual. B. Ray-Kang- Chye ' s Constraint Handling Approach T. Ray, T. Kang, and S. K. Chye =-=[15]-=- suggested a more elaborate constraint handling technique, where constraint violations of all constraints are not simply summed together, instead a non-domination check of constraint violations is als... |

3 |
An evolutionary algorithm for multiobjective optimization
- Ray, Tai, et al.
- 2001
(Show Context)
Citation Context ...domination checks with the constraint-violation values, the proposed approach of this paper is computationally less expensive and is simpler. B. Ray–Tai–Seow’s Constraint-Handling Approach Ray et al. =-=[17]-=- suggested a more elaborate constraint-handling technique, where constraint violations of all constraints are not simply summed together. Instead, a nondomination check of constraint violations is als... |

1 | et al., "CA-based decision support system for multi-criteria optimization - Tanaka - 1995 |

1 | et al., \GA-based decision support system for multi-criteria optimization - Tanaka - 1995 |

1 |
efficient constraint-handling method for genetic algorithms
- “An
- 2000
(Show Context)
Citation Context ...e optimization. VI. CONSTRAINT HANDLING In the past, the first author and his students implemented a penalty-parameterless constraint-handling approach for singleobjective optimization. Those studies =-=[2]-=-, [6] have shown how a tournament selection based algorithm can be used to handle constraints in a population approach much better than a number of other existing constraint-handling approaches. A sim... |

1 |
the performance assessment and comparison of stochastic multiobjective optimizers,” in Parallel Problem Solving from Nature IV
- “On
- 1996
(Show Context)
Citation Context ...et and 2) maintenance of diversity in solutions of the Pareto-optimal set. These two tasks cannot be measured adequately with one performance metric. Many performance metrics have been suggested [1], =-=[8]-=-, [24]. Here, we define two performance metrics that are more direct in evaluating each of the above two goals in a solution set obtained by a multiobjective optimization algorithm. The first metric m... |