Soft Computing

, Volume 22, Issue 24, pp 8273–8288 | Cite as

Multi-objective imperialistic competitive algorithm with multiple non-dominated sets for the solution of global optimization problems

  • Zhavat Sherinov
  • Ahmet Ünveren
Methodologies and Application


In this paper, we propose a multi-objective imperialistic competitive algorithm (MOICA) for solving global multi-objective optimization problems. The MOICA is a modified and improved multi-objective version of the single-objective imperialistic competitive algorithm previously proposed by Atashpaz-Gargari and Lucas (IEEE Congr Evolut Comput 7:4661–4666. doi: 10.1109/CEC.2007.4425083, 2007). The presented algorithm utilizes the metaphor of imperialism to solve optimization problems. Accordingly, the individuals in a population are referred to as countries, of which there are two types—colonies and imperialists. The MOICA incorporates competition between empires and their colonies for the solution of multi-objective problems. To this end, it employs several non-dominated solution sets, whereby each set is referred to as a local non-dominated solution (LNDS) set. All imperialists in an empire are considered non-dominated solutions, whereas all colonies are considered dominated solutions. In addition to LNDS sets, there is one global non-dominated solution (GNDS) set, which is created from the LNDS sets of all empires. There are two primary operators in the proposed algorithm, i.e., assimilation and revolution, which use the GNDS and LNDS sets, respectively. The significance of this study lies in a notable feature of the proposed algorithm, which is that no special parameter is used for diversity preservation. This enables the algorithm to prevent extra computation to maintain the spread of solutions. Simulations and experimental results on multi-objective benchmark problems show that the MOICA is more efficient compared to a few existing major multi-objective optimization algorithms because it produces better results for several test problems.


Multi-objective metaheuristics Imperialistic competitive algorithm Multiple non-dominated sets Global optimization 



This study was funded by Eastern Mediterranean University (02).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Atashpaz-Gargari E, Lucas C (2007) Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition. IEEE Congr Evolut Comput 7:4661–4666. doi: 10.1109/CEC.2007.4425083 CrossRefGoogle Scholar
  2. Deb K (2001) Multiobjective optimization using evolutionary algorithms. Wiley, ChichesterzbMATHGoogle Scholar
  3. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evolut Comput 6(2):182–197. doi: 10.1109/4235.996017 CrossRefGoogle Scholar
  4. Dorigo M, Blum C (2005) Ant colony optimization theory: a survey. Theor Comp Sci 344:243–278. doi: 10.1016/j.tcs.2005.05.020 MathSciNetCrossRefzbMATHGoogle Scholar
  5. Doush IA, Bataineh MQ (2015) Hybedrized NSGA-II and MOEA/D with harmony search algorithm to solve multi-objective optimization problems. In: Arik S, Huang T, Lai W, Liu Q (eds) Neural information processing. Springer, Switzerland, pp 606–614CrossRefGoogle Scholar
  6. Duan H, Xu C, Liu S, Shao S (2010) Template matching using chaotic imperialist competitive algorithm. Pattern Recogn Lett 31:1868–1875. doi: 10.1016/j.patrec.2009.12.005 CrossRefGoogle Scholar
  7. Ebrahimzadeh A, Addeh J, Rahmani Z (2012) Control chart pattern recognition using K-MICA clustering and neural networks. ISA Trans 51(1):111–119. doi: 10.1016/j.isatra.2011.08.005 CrossRefGoogle Scholar
  8. Eiben AE, Smit SK (2011) Evolutionary algorithm parameters and methods to tune them. In: Hamadi Y, Monfoy E, Saubion F (eds) Autonomous search. Springer, Berlin, pp 15–36CrossRefGoogle Scholar
  9. Fonseca CM, Fleming PJ (1993) Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. In: Forrest S (ed) Proceedings of the 5th international conference on genetic algorithms. Morgan Kauffman Publishers, San Mateo, CA, pp 416–423Google Scholar
  10. Fonseca CM, Fleming PJ (1998) Multiobjective optimization and multiple constraint handling with evolutionary algorithms—part II: application example. IEEE Trans Syst Man Cybern (A) 28:38–47. doi: 10.1109/3468.650319 CrossRefGoogle Scholar
  11. Goudarzi M, Vahidi B, Naghizadeh RA (2013) Optimum reactive power compensation in distribution networks using imperialistic competitive algorithm. Sci Int (Lahore) 25(1):27–31Google Scholar
  12. Horn J, Nafploitis N, Goldberg DE (1994) A niched Pareto genetic algorithm for multiobjective optimization. In: Michalewicz Z (ed) Proceedings of the 1st IEEE conference on evolution computer. IEEE Press, Piscataway, NJ, pp 82–87Google Scholar
  13. Jordehi AR (2016) Optimal allocation of FACTS devices for static security enhancement in power systems via imperialistic competitive algorithm (ICA). Appl Soft Comput 48:317–328. doi: 10.1016/j.asoc.2016.07.014 CrossRefGoogle Scholar
  14. Kashani AR, Gandomi AH, Mousavi M (2014) Imperialistic competitive algorithm: a metaheuristic algorithm for locating the critical slip surface in 2-dimensional soil slopes. Geosci Front 7(1):83–89. doi: 10.1016/j.gsf.2014.11.005 CrossRefGoogle Scholar
  15. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the IEEE international conference on neural network IV 1942–1948Google Scholar
  16. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680. doi: 10.1126/science.220.4598.671 MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kursawe F (1990) A variant of evolution strategies for vector optimization. In: Schwefel H-P, Manner R (eds) Parallel problem solving from nature. Springer, Berlin, pp 193–197Google Scholar
  18. Mitchell M (1999) An introduction to genetic algorithms. MIT Press, CambridgezbMATHGoogle Scholar
  19. Nazari-Shirkouhi S, Eivazy H, Ghodsi R, Rezaie K, Atashpaz-Gargari E (2010) Solving the integrated product mix-outsourcing problem by a novel meta-heuristic algorithm: imperialist competitive algorithm. Expert Syst Appl 37:7615–7626. doi: 10.1016/j.eswa.2010.04.081 CrossRefGoogle Scholar
  20. Niknam T, Taherian FE, Pourjafarian N, Rousta A (2011) An efficient hybrid algorithm based on modified imperialist competitive algorithm and \(k\)-means for data clustering. Eng Appl Artif Intell 24(2):306–317. doi: 10.1016/j.engappai.2010.10.001 CrossRefGoogle Scholar
  21. Qi Y, Ma X, Liu F, Jiao L, Sun J, Wu J (2014) MOEA/D with adaptive weight adjustment. Evol Comput 22(2):231–264. doi: 10.1162/EVCO_a_00109 CrossRefGoogle Scholar
  22. Razmjooy N, Mousavi BS, Soleymani F (2013) A hybrid neural network imperialist competitive algorithm for skin color segmentation. Math Comput Model 57:848–856. doi: 10.1016/j.mcm.2012.09.013 CrossRefGoogle Scholar
  23. Schaffer JD (1987) Multiple objective optimization with vector evaluated genetic algorithms. In: Grefensttete JJ (ed) Proceedings of the 1st international conference on genetic algorithms. Lawrence Erlbaum, Hillsdale, NJ, pp 93–100Google Scholar
  24. Seyedmohsen H, Abdullah AK (2014) A survey on the imperialist competitive algorithm metaheuristic: implementation in engineering domain and directions for future research. Appl Soft Comput 24:1078–1094. doi: 10.1016/j.asoc.2014.08.024 CrossRefGoogle Scholar
  25. Sherinov Z, Unveren A, Acan A (2011) An evolutionary multi-objective modeling and solution approach for fuzzy vehicle routing problem. In: International symposium on innovations in intelligent systems and applications (INISTA), pp 450–454. doi: 10.1109/INISTA.2011.5946143
  26. Srinivas N, Deb K (1995) Multiobjective function optimization using nondominated sorting genetic algorithms. Evolut Comput 2(3):221–248. doi: 10.1162/evco.1994.2.3.221 CrossRefGoogle Scholar
  27. Van Veldhuizen DA, Lamont GB (1998) Multiobjective evolutionary algorithm research: a history and analysis. Technical Report TR-98-03, Department of Electrical and Computer Engineering, Graduate School of Engineering, Air Force Institute of Technology (AFIT), Wright-Patterson AFB, OHGoogle Scholar
  28. Vedadi M, Vahidi B, Hosseinian SH (2015) An imperialist competitive algorithm maximum power point tracker for photovoltaic string operating under partially shaded conditions. Sci Int (Lahore) 27(5):4023–4033Google Scholar
  29. Xiang Y, Zhou Y (2015) A dynamic multi-colony artificial bee colony algorithm for multi-objective optimization. Appl Soft Comput 35:766–785. doi: 10.1016/j.asoc.2015.06.033 CrossRefGoogle Scholar
  30. Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S (2009) Multiobjective optimization test instances for the CEC 2009 special session and competition. Technical Report CES-487, University of Essex, Essex, UKGoogle Scholar
  31. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evolut Comput 8(2):173–195. doi: 10.1162/106365600568202 CrossRefGoogle Scholar
  32. Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithms: a comparative case study. In: Eiben AE, Bäck T, Shoenauer M, Schwefel HP (eds) Parallel problem solving from nature. Springer, Berlin, pp 292–301Google Scholar
  33. Zitzler E, Thiele L, Laumanns M, Fonseca CM, Grunert da Fonseca V (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evolut Comput 7(2):117–132. doi: 10.1109/TEVC.2003.810758 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Computer Engineering DepartmentEastern Mediterranean UniversityMağusa, Mersin 10Turkey

Personalised recommendations