Soft Computing

, Volume 21, Issue 16, pp 4677–4691 | Cite as

Improving the multiobjective evolutionary algorithm based on decomposition with new penalty schemes

  • Shengxiang YangEmail author
  • Shouyong Jiang
  • Yong Jiang
Methodologies and Application


It has been increasingly reported that the multiobjective optimization evolutionary algorithm based on decomposition (MOEA/D) is promising for handling multiobjective optimization problems (MOPs). MOEA/D employs scalarizing functions to convert an MOP into a number of single-objective subproblems. Among them, penalty boundary intersection (PBI) is one of the most popular decomposition approaches and has been widely adopted for dealing with MOPs. However, the original PBI uses a constant penalty value for all subproblems and has difficulties in achieving a good distribution and coverage of the Pareto front for some problems. In this paper, we investigate the influence of the penalty factor on PBI, and suggest two new penalty schemes, i.e., adaptive penalty scheme and subproblem-based penalty scheme (SPS), to enhance the spread of Pareto-optimal solutions. The new penalty schemes are examined on several complex MOPs, showing that PBI with the use of them is able to provide a better approximation of the Pareto front than the original one. The SPS is further integrated into two recently developed MOEA/D variants to help balance the population diversity and convergence. Experimental results show that it can significantly enhance the algorithm’s performance.


Multiobjective evolutionary algorithm Decomposition  Penalty boundary intersection Adaptive penalty scheme Subproblem-based penalty scheme 



This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of U.K. under Grant EP/K001310/1 and the National Natural Science Foundation of China under Grant 61273031.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Al Moubayed N, Petrovski A, McCall J (2013) \(D^2MOPSO\): MOPSO based on decomposition and dominance with archiving using crowding distance in objective and solution spaces. Evol Comput 22(1):47–77CrossRefGoogle Scholar
  2. Asafuddoula M, Ray T, Sarker R (2015) A decomposition based evolutionary algorithm for many objective optimization. IEEE Trans Evol Comput 19(3):445–460CrossRefGoogle Scholar
  3. Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76CrossRefGoogle Scholar
  4. Beume N, Naujoks N, Emmerich M (2007) SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur J Oper Res 181(3):1653–1669CrossRefzbMATHGoogle Scholar
  5. Cheng R, Jin Y, Narukawa K (2015) Adaptive reference vector generation for inverse model based evolutionary multiobjective optimization with degenerate and disconnected Pareto fronts. In: Evolutionary Multi-criterion Optimization (EMO 2015), part I, LNCS 9018, pp 127–140Google Scholar
  6. Cheng R, Jin Y, Olhofer M, Sendhoff B (2016) A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput. doi: 10.1109/TEVC.2016.2519378
  7. Chikumbo O, Goodman ED, Deb K (2012) Approximating a multi-dimensional pareto front for a land use management problem: a modified moea with an epigenetic silencing metaphor. In: Proceedings of 2012 IEEE Congress on Evolutionary (CEC 2012), pp 1–9Google Scholar
  8. Deb K, Agrawwal S, Pratap A, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  9. Deb K, Thiele L, Laumanns M, Zitzler E (2005) Scable test problems for evolutionary multi-objective optimization. In: Abraham A, Jain L, Goldberg R (eds) Evolutionary Multiobjective Optimization: Theoretical Advances and Applications. Springer, London, pp 105–145Google Scholar
  10. Deb K, Jain H (2014) An evolutionary many-objective optimization algorithm using reference-point based non-dominated sorting approach, part I: solving problems with box constraints. IEEE Trans Evol Comput 18(4):577–601CrossRefGoogle Scholar
  11. Deb K, Pratap A, Moitra S (2000) Mechanical component design for multiple objectives using elitist non-dominated sorting GA. In: Proceedings of the 6th International Conference on Parallel Problem Solving from Nature (PPSN VI), pp 859–868Google Scholar
  12. Giagkiozis I, Purshouse RC, Fleming PJ (2014) Generalized decomposition and cross entropy methods for many-objective optimization. Inf Sci 282:363–387MathSciNetCrossRefzbMATHGoogle Scholar
  13. Goh C, Tan KC (2007) An investigation on noisy environments in evolutionary multiobjective optimization. IEEE Trans Evol Comput 11(3):354–381CrossRefGoogle Scholar
  14. Gomez RH, Coello Coello CA (2015) Improved metaheuristic based on the r2 indicator for many-objective optimization. In: Proceedings the 2015 Annual Conference on Genetic and Evolutionary Computation (GECCO 15), pp 679–686Google Scholar
  15. Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(2):477–506CrossRefzbMATHGoogle Scholar
  16. Ishibuchi H, Akedo N, Nojima Y (2015) Behavior of multi-objective evolutionary algorithms on many-objective knapsack problems. IEEE Trans Evol Comput 19(2):264–283CrossRefGoogle Scholar
  17. Ishibuchi H, Akedo N, Nojima Y (2013) A study on the specification of a scalarizing function in MOEA/D for many-objective knapsack problems. In: Proceedings of Learning and Intelligence Optimization 7 (LION 7), LNCS 7997, pp 231–246Google Scholar
  18. Ishibuchi H, Tsukamoto N, Hitotsuyanagi Y, Nojima Y (2010) Indicator-based evolutionary algorithm with hypervolume approximation by achievement scalarizing function. In: Proceedings of 12th Annual Conference on Genetic and Evolutionary Computation (GECCO 2010), pp 527–534Google Scholar
  19. Jain H, Deb K (2014) An improved adaptive approach for elitist nondominated sorting genetic algorithm for many-objective optimization. In: Evolutionary Multi-criterion Optimization (EMO 2013), LNCS 7811, pp 307–321Google Scholar
  20. Jia S, Zhu J, Du B, Yue H (2011) Indicator-based particle swarm optimization with local search. In: Proceedings of 2011 7th International Conference on Nature Computation, pp 1180–1184Google Scholar
  21. Jiang S, Yang S (2016) An improved multi-objective optimization evolutionary algorithm based on decomposition for complex Pareto fronts. IEEE Trans Cybern 46(2):421–437CrossRefGoogle Scholar
  22. Knowles JD, Corne DW (1999) The pareto archived evolution strategy: a new baseline algorithm for multiobjective optimisation. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC 1999), pp 98–105Google Scholar
  23. Li H, Zhang Q (2009) Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13(2):284–302CrossRefGoogle Scholar
  24. Li K, Fialho A, Kwong S, Zhang Q (2014a) Adaptive operator selection with bandits for multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 18(1):114–130Google Scholar
  25. Li B, Li J, Tang K, Yao X (2014b) An improved two archive algorithm for many-objective optimization. In: Proceedings of 2014 IEEE Congress on Evolutionary Computation (CEC 2014), pp 2869–2876Google Scholar
  26. Li K, Zhang Q, Kwong S, Li M, Wang R (2014c) Stable matching based selection in evolutionary multiobjective optimization. IEEE Trans Evol Comput 18(6):909–923Google Scholar
  27. Li K, Deb K, Zhang Q, Kwong S (2015a) An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans Evol Comput 19(5):694–716Google Scholar
  28. Li K, Kwong S, Zhang Q, Deb K (2015b) Interrelationship-based selection for decomposition multiobjective optimization. IEEE Trans Cybern 45(10):2076–2088Google Scholar
  29. Liu H, Gu F, Zhang Q (2014) Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems. IEEE Trans Evol Comput 18(3):450–455CrossRefGoogle Scholar
  30. Masoomi Z, Mesgari MS, Hamrah M (2013) Allocation of urban land uses by multi-objective particle swarm optimization algorithm. Int J Geogr Inf Sci 27(3):542–565CrossRefGoogle Scholar
  31. Mendez AM, Coello Coello CA (2015) GD-MOEA: a new multi-objective evolutionary algorithm on the generation distance indicator. In: Evolutionary Multi-criterion Optimization, vol 9018, pp 156–170Google Scholar
  32. Mohammadi A, Omidvar MN, Li X, Deb K (2014) Integrating user preferences and decomposition methods for many-objective optimization. In: Proceedings of 2014 IEEE Congress on Evolutionary Computation (CEC 2014), pp 421–428Google Scholar
  33. Osycza A, Kundu S (1995) A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm. Struct Optim 10(2):94–99CrossRefGoogle Scholar
  34. Pavelski LM, Delgado MR, de Almeida CP, Goncalves RA, Venske SM (2014) ELMOEA/D-DE: extreme learning surrogate models in multi-objective optimization based on decomposition and differential evolution. In: Proceedings of 2014 Brazilian Conference on Intelligent Systems (BRACIS), pp 318–323Google Scholar
  35. Qi T, Ma X, Liu F, Jiao L, Sun J, Wu J (2014) MOEA/D with adaptive weight adjustment. Evol Comput 22(2):231–264CrossRefGoogle Scholar
  36. Reed PM, Hadka D, Herman JD, Kasprzyk JR, Kollat JB (2013) Evolutionary multiobjective in water resources: the past, present, and future. Adv Water Resour 51:438–456Google Scholar
  37. Sato H (2014) Inverted PBI in MOEA/D and its impact on the search performance on multi and many-objective optimization. In: Proceedings of 2014 Conference on Genetic and Evolutionary Computation (GECCO 14), pp 645–652Google Scholar
  38. Wang G, Chen J, Cai T, Xin B (2013) Decomposition-based multi-objective differential evolution particle swarm optimization for the design of tubular permanent magnet linear synchronous motor. Eng Optim 45(9):1107–1127MathSciNetCrossRefGoogle Scholar
  39. Wang L, Zhang Q, Zhou A, Gong M, Jiao L (2015) Constrained subproblems in decomposition based multiobjective evolutionary algorithm. IEEE Trans Evol Comput. doi: 10.1109/TEVC.2015.2457616 Google Scholar
  40. Yang XS, Deb S (2013) Multiobjective cuckoo search for design optimization. Comput Oper Res 40(6):1616–1624MathSciNetCrossRefzbMATHGoogle Scholar
  41. Yuan Y, Xu H, Wang B, Yao X (2015) A new dominance relation based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput. doi: 10.1109/TEVC.2015.2420112 Google Scholar
  42. Yuan Y, Xu H, Wang B (2014) Evolutionary many-objective optimization using ensemble fitness ranking. In: Proceedings of the 2014 Conference on Genetic and Evolutionary Computation (GECCO 2014), pp 669–676Google Scholar
  43. Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731CrossRefGoogle Scholar
  44. Zitzler E, Kunzli S (2004) Indicator-based selection in multiobjective search. In: Proceedings of the 8th International Conference on Parallel Problem Solving from Nature (PPSN VIII), vol 3242, pp 832–842Google Scholar
  45. Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4):257–271CrossRefGoogle Scholar
  46. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195CrossRefGoogle Scholar
  47. Zitzler E, Laumanns M, Thiele L (2002) SPEA2: improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Proceedings of Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001), vol 3242, no 103, pp 95–100Google Scholar
  48. Zitzler E, Thiele L, Laumanns M, Fonseca CM, da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingChina
  2. 2.Centre for Computational Intelligence (CCI), School of Computer Science and InformaticsDe Montfort UniversityLeicesterUK

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