Skip to main content
Springer Nature Link
Log in

Non-dominated Sorting Based Multi/Many-Objective Optimization: Two Decades of Research and Application

  • Chapter
  • First Online:
Multi-Objective Optimization

  • 1958 Accesses

Abstract

For more than two decades, non-dominated sorting has been a cornerstone in most successful multi/many-objective optimization algorithms. In this chapter, we discuss the effect of non-dominated sorting in multi- and many-objective scenarios. Thereafter, we present some of the most widely used optimization algorithms involving non-dominated sorting, where we discuss their extent and ubiquity across many scientific disciplines. Finally, we go over some of the state-of-the-art combinations of non-dominated sorting with other optimization techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    For simplicity, we only consider strong domination. However, according to the definition, domination can be weak as well. In our example, weak domination is when two cars have exactly the same price but one of them is more comfortable than the other. The more comfortable car is said to weakly dominate the other.

  2. 2.

    Unlike the car example, throughout the rest of this chapter, we assume a minimization context unless otherwise stated.

  3. 3.

    The terms “solution”, “individual” and “points” are used interchangeably throughout this chapter.

  4. 4.

    It is worth noting that some of the later ones were shown to solve up to three objectives as well.

  5. 5.

    Although the original study used the notion of a “reference point”, here we will mostly use the notion of a reference direction, which is the vector connecting the ideal point to the “reference point”. We found this notion more conceivable and avoids the confusion that may arise between the two terms “point” (an actual solution) and “reference point” (a hypothetical point in the objective space).

  6. 6.

    Other than the standard evolutionary parameters, e.g., population size, number of Solution evaluations(SEs)/generations, recombination/mutation probability, etc.

  7. 7.

    One of the most important resources in optimization is the number of function evaluations (FEs) consumed to reach a solution. In a multi-objective optimization scenario, we use the term solution evaluation (SE) instead, as evaluating a single solution involves evaluating more than one function.

References

  • S.F. Adra, P.J. Fleming, Diversity management in evolutionary many-objective optimization. IEEE Trans. Evol. Comput. 15(2), 183–195 (2011)

    Article  Google Scholar 

  • H. Aguirre, K. Tanaka, Many-objective optimization by space partitioning and adaptive \(\varepsilon \)-ranking on mnk-landscapes, in Evolutionary Multi-criterion Optimization (Springer, 2009), pp. 407–422

    Google Scholar 

  • R. Arora, S. Kaushik, R. Kumar, R. Arora, Multi-objective thermo-economic optimization of solar parabolic dish stirling heat engine with regenerative losses using nsga-ii and decision making. Int. J. Electr. Power Energy Syst. 74, 25–35 (2016)

    Article  Google Scholar 

  • M. Asafuddoula, T. Ray, R. Sarker, A decomposition-based evolutionary algorithm for many objective optimization. IEEE Trans. Evol. Comput. 19(3), 445–460 (2015)

    Article  Google Scholar 

  • R. Bolaños, M. Echeverry, J. Escobar, A multiobjective non-dominated sorting genetic algorithm (nsga-ii) for the multiple traveling salesman problem. Decis. Sci. Lett. 4(4), 559–568 (2015)

    Article  Google Scholar 

  • C.A.C. Coello, D.A. Van Veldhuizen, G.B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, vol. 242 (Springer, 2002)

    Google Scholar 

  • I. Das, J.E. Dennis, Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998)

    Article  MathSciNet  Google Scholar 

  • K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms vol.16 (Wiley, 2001)

    Google Scholar 

  • K. Deb, R.B. Agrawal, Simulated binary crossover for continuous search space. Complex Syst. 9(2), 115–148 (1995)

    MathSciNet  MATH  Google Scholar 

  • K. Deb, D.E. Goldberg (1989) An investigation of niche and species formation in genetic function optimization. in Proceedings of the Third International Conference on Genetic Algorithms, pp. 42–50

    Google Scholar 

  • K. Deb, H. Jain, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part i: solving problems with box constraints. IEEE Transa. Evol. Computat. 18(4), 577–601 (2014)

    Article  Google Scholar 

  • K. Deb, S. Tiwari, Omni-optimizer: a generic evolutionary algorithm for single and multi-objective optimization. Eur. J. Oper. Res. 185(3), 1062–1087 (2008)

    Article  MathSciNet  Google Scholar 

  • K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  • K. Deb, L. Thiele, M. Laumanns, E. Zitzler (2005) Scalable test problems for evolutionary multiobjective optimization, in Evolutionary multiobjective optimization theoretical advances and applications, pp. 105–145

    Google Scholar 

  • K.A. DeJong, An analysis of the behavior of a class of genetic adaptive systems. Ph.D. Thesis, Ann Arbor, MI, University of Michigan, dissertation Abstracts International 36(10), 5140B (University Microfilms No. 76-9381)

    Google Scholar 

  • S. Dhanalakshmi, S. Kannan, K. Mahadevan, S. Baskar, Application of modified nsga-ii algorithm to combined economic and emission dispatch problem. Int. J. Electr. Power Energy Syst. 33(4), 992–1002 (2011)

    Article  Google Scholar 

  • E. Fallah-Mehdipour, O.B. Haddad, M.M.R. Tabari, M.A. Mariño, Extraction of decision alternatives in construction management projects: application and adaptation of nsga-ii and mopso. Expert Syst. Appl. 39(3), 2794–2803 (2012)

    Article  Google Scholar 

  • C.M. Fonseca, P.J. Fleming et al., Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. ICGA Citeseer 93, 416–423 (1993)

    Google Scholar 

  • D. Gale, L.S. Shapley, College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)

    Article  MathSciNet  Google Scholar 

  • M. Garza-Fabre, G.T. Pulido, C.A.C. Coello, Ranking methods for many-objective optimization, in MICAI 2009: Advances in Artificial Intelligence (Springer, 2009), pp. 633–645

    Google Scholar 

  • C.K. Goh, K.C. Tan, Evolutionary multi-objective optimization in uncertain environments (Issues and Algorithms, Studies in Computational Intelligence, 2009), p. 186

    MATH  Google Scholar 

  • D.E. Goldberg, J. Richardson, Genetic algorithms with sharing for multimodal function optimization, in Genetic Algorithms and Their Applications: Proceedings of the Second International Conference on Genetic Algorithms (1987), pp. 41–49

    Google Scholar 

  • D. Hadka, P. Reed, Borg: An auto-adaptive many-objective evolutionary computing framework. Evol. Comput. 21(2), 231–259 (2013)

    Article  Google Scholar 

  • J. Horn, N. Nafpliotis, D.E. Goldberg, A niched pareto genetic algorithm for multiobjective optimization, in 1994 Proceedings of the First IEEE Conference on Evolutionary Computation, IEEE World Congress on Computational Intelligence (IEEE, 1994) pp. 82–87

    Google Scholar 

  • A. Ibrahim, S. Rahnamayan, M.V. Martin, Deb K Elitensga-iii: an improved evolutionary many-objective optimization algorithm, in 2016 IEEE Congress on Evolutionary Computation (CEC) (IEEE, 2016), pp. 973–982

    Google Scholar 

  • H. Ishibuchi, N. Tsukamoto, Y. Nojima, Evolutionary many-objective optimization: a short review, in IEEE Congress on Evolutionary Computation, Citeseer (2008) pp. 2419–2426

    Google Scholar 

  • H. Jain, K. Deb, An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part ii: handling constraints and extending to an adaptive approach. IEEE Trans. Evol. Comput. 18(4), 602–622 (2014)

    Article  Google Scholar 

  • S. Jeyadevi, S. Baskar, C. Babulal, M.W. Iruthayarajan, Solving multiobjective optimal reactive power dispatch using modified nsga-ii. Int. J. Electr. Power Energy Syst. 33(2), 219–228 (2011)

    Article  Google Scholar 

  • V. Khare, X. Yao, K. Deb, Performance scaling of multi-objective evolutionary algorithms, in Evolutionary Multi-Criterion Optimization (Springer, 2003) pp. 376–390

    Google Scholar 

  • J. Knowles, D. Corne, The pareto archived evolution strategy: a new baseline algorithm for pareto multiobjective optimisation, in Proceedings of the 1999 Congress on Evolutionary Computation, CEC 99, vol. 1 (IEEE, 1999)

    Google Scholar 

  • J. Knowles, D. Corne, K. Deb, Multiobjective Problem Solving from Nature: from Concepts to Applications (Springer Science & Business Media, 2007)

    Google Scholar 

  • M. Köppen, K. Yoshida, Substitute distance assignments in nsga-ii for handling many-objective optimization problems, in Evolutionary Multi-Criterion Optimization (Springer, 2007), pp. 727–741

    Google Scholar 

  • K. Li, Q. Zhang, S. Kwong, M. Li, R. Wang, Stable matching-based selection in evolutionary multiobjective optimization. IEEE Trans. Evol. Comput. 18(6), 909–923 (2014)

    Article  Google Scholar 

  • K. Li, K. Deb, Q. Zhang, S. Kwong, An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans. Evol. Comput. 19(5), 694–716 (2015)

    Article  Google Scholar 

  • M. Li, J. Zheng, K. Li, Q. Yuan, R. Shen, Enhancing diversity for average ranking method in evolutionary many-objective optimization, in Parallel Problem Solving from Nature, PPSN XI (2010), pp. 647–656

    Google Scholar 

  • H.L. Liu, F. Gu, Q. Zhang, Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems. IEEE Trans. Evol. Comput. 18(3), 450–455 (2014)

    Article  Google Scholar 

  • M.W. Mkaouer, M. Kessentini, S. Bechikh, K. Deb, M. Ó Cinnéide, High dimensional search-based software engineering: finding tradeoffs among 15 objectives for automating software refactoring using nsga-iii, in Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation (ACM, 2014), pp. 1263–1270

    Google Scholar 

  • W. Mkaouer, M. Kessentini, A. Shaout, P. Koligheu, S. Bechikh, K. Deb, A. Ouni, Many-objective software remodularization using nsga-iii. ACM Trans. Softw. Eng. Methodol (TOSEM) 24(3), 17 (2015)

    Article  Google Scholar 

  • P. Murugan, S. Kannan, S. Baskar, Application of nsga-ii algorithm to single-objective transmission constrained generation expansion planning. IEEE Trans. Power Syst. 24(4), 1790–1797 (2009)

    Article  Google Scholar 

  • R.C. Purshouse, P.J. Fleming, On the evolutionary optimization of many conflicting objectives. IEEE Trans. Evol. Comput. 11(6), 770–784 (2007)

    Article  Google Scholar 

  • H. Sato, H.E. Aguirre, K. Tanaka, Pareto partial dominance moea and hybrid archiving strategy included cdas in many-objective optimization, in 2010 IEEE Congress on Evolutionary Computation (CEC) (IEEE, 2010), pp. 1–8

    Google Scholar 

  • J.D. Schaffer, Multiple objective optimization with vector evaluated genetic algorithms, in Proceedings of the 1st International Conference on Genetic Algorithms (L. Erlbaum Associates Inc., 1985), pp. 93–100

    Google Scholar 

  • H. Seada, K. Deb, A unified evolutionary optimization procedure for single, multiple, and many objectives. IEEE Trans. Evol. Comput. 20(3), 358–369 (2016)

    Article  Google Scholar 

  • H. Seada, M. Abouhawwash, K. Deb, Towards a better balance of diversity and convergence in nsga-iii: first results, International Conference on Evolutionary Multi-Criterion Optimization (Springer, Cham, 2017), pp. 545–559

    Google Scholar 

  • H. Seada, K. Deb, M. Abouhawwash, Multi-phase balance of diversity and convergence in multiobjective optimization. Technical Report COIN Report Number 2018001, 3247 EB, (Michigan State University, East Lansing, Michigan, 48824, USA, 2018)

    Google Scholar 

  • D. Sharma, K. Deb, N. Kishore, Towards generating diverse topologies of path tracing compliant mechanisms using a local search based multi-objective genetic algorithm procedure, in 2008 IEEE Congress on Evolutionary Computation, CEC 2008 (IEEE World Congress on Computational Intelligence) (IEEE, 2008) pp. 2004–2011

    Google Scholar 

  • K. Sindhya, K. Miettinen, K. Deb, A hybrid framework for evolutionary multi-objective optimization. IEEE Trans. Evol. Comput. 17(4), 495–511 (2013)

    Article  Google Scholar 

  • H.K. Singh, A. Isaacs, T. Ray, A pareto corner search evolutionary algorithm and dimensionality reduction in many-objective optimization problems. IEEE Trans. Evol. Comput. 15(4), 539–556 (2011)

    Article  Google Scholar 

  • N. Srinivas, K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol. Comput. 2(3), 221–248 (1994)

    Article  Google Scholar 

  • A.K.A. Talukder, K. Deb, S. Rahnamayan, Injection of extreme points in evolutionary multiobjective optimization algorithms, in International Conference on Evolutionary Multi-Criterion Optimization (Springer, 2017), pp. 590–605

    Google Scholar 

  • K.C. Tan, E.F. Khor, T.H. Lee, Multiobjective Evolutionary Algorithms And Applications (Springer Science & Business Media, 2006)

    Google Scholar 

  • M. Tavana, Z. Li, M. Mobin, M. Komaki, E. Teymourian, Multi-objective control chart design optimization using nsga-iii and mopso enhanced with dea and topsis. Expert Syst. Appl. 50, 17–39 (2016)

    Article  Google Scholar 

  • Z. Wang, Q. Zhang, M. Gong, A. Zhou, A replacement strategy for balancing convergence and diversity in moea/d, in 2014 IEEE Congress on Evolutionary Computation (CEC) (IEEE, 2014), pp. 2132–2139

    Google Scholar 

  • U.K. Wickramasinghe, X. Li, A distance metric for evolutionary many-objective optimization algorithms using user-preferences, in AI 2009: Advances in Artificial Intelligence (Springer, 2009), pp. 443–453

    Google Scholar 

  • X. Yuan, H. Tian, Y. Yuan, Y. Huang, R.M. Ikram, An extended nsga-iii for solution multi-objective hydro-thermal-wind scheduling considering wind power cost. Energy Convers. Manage. 96, 568–578 (2015)

    Article  Google Scholar 

  • Y. Yuan, H. Xu, B. Wang, X. Yao, A new dominance relation-based evolutionary algorithm for many-objective optimization. IEEE Trans. Evol. Comput. 20(1), 16–37 (2016a)

    Article  Google Scholar 

  • Y. Yuan, H. Xu, B. Wang, B. Zhang, X. Yao, Balancing convergence and diversity in decomposition-based many-objective optimizers. IEEE Trans. Evol. Comput. 20(2), 180–198 (2016b)

    Article  Google Scholar 

  • Q. Zhang, H. Li, Moea/d: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)

    Google Scholar 

  • X. Zhang, Y. Tian, Y. Jin, A knee point-driven evolutionary algorithm for many-objective optimization. IEEE Trans. Evol. Comput. 19(6), 761–776 (2015)

    Article  Google Scholar 

  • Y. Zhu, J. Liang, J. Chen, Z. Ming, An improved nsga-iii algorithm for feature selection used in intrusion detection. Knowl.-Based Syst. 116, 74–85 (2017)

    Article  Google Scholar 

  • E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)

    Article  Google Scholar 

  • E. Zitzler, K. Deb, L. Thiele, Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8(2), 173–195 (2000)

    Article  Google Scholar 

  • E. Zitzler, M. Laumanns, L. Thiele, Spea2: Improving the Strength Pareto Evolutionary Algorithm (2001)

    Google Scholar 

  • X. Zou, Y. Chen, M. Liu, L. Kang, A new evolutionary algorithm for solving many-objective optimization problems. IEEE Trans. Syst. Man Cybern. Part B Cybern. 38(5), 1402–1412 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brown University, 70 Ship, Providence, RI, 02903, USA

    Haitham Seada

  2. Michigan State University, 428 South Shaw Lane, 2320 EB, East Lansing, MI, 48824, USA

    Kalyanmoy Deb

Authors
  1. Haitham Seada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kalyanmoy Deb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kalyanmoy Deb .

Editor information

Editors and Affiliations

  1. University of Kalyani, Kalyani, West Bengal, India

    Jyotsna K. Mandal

  2. Assam University, Silchar, Assam, India

    Somnath Mukhopadhyay

  3. Visva Bharati University, Bolpur, Santiniketan, West Bengal, India

    Paramartha Dutta

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Seada, H., Deb, K. (2018). Non-dominated Sorting Based Multi/Many-Objective Optimization: Two Decades of Research and Application. In: Mandal, J., Mukhopadhyay, S., Dutta, P. (eds) Multi-Objective Optimization. Springer, Singapore. https://doi.org/10.1007/978-981-13-1471-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-981-13-1471-1_1

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-13-1470-4

  • Online ISBN: 978-981-13-1471-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics