Neural Computing and Applications

, Volume 31, Supplement 2, pp 915–929 | Cite as

Multi-objective sine-cosine algorithm (MO-SCA) for multi-objective engineering design problems

  • Mohamed A. TawhidEmail author
  • Vimal Savsani
Original Article


This paper proposes a novel and an effective multi-objective optimization algorithm named multi-objective sine-cosine algorithm (MO-SCA) which is based on the search technique of sine-cosine algorithm (SCA). MO-SCA employs the elitist non-dominated sorting and crowding distance approach for obtaining different non-domination levels and to preserve the diversity among the optimal set of solutions, respectively. The effectiveness of the method is measured by implementing it on multi-objective benchmark problems that have various characteristics of Pareto front such as convex, non-convex and discrete. This proposed algorithm is also checked for the multi-objective engineering design problems with distinctive features. Furthermore, we show the proposed algorithm effectively generates the Pareto front and is easy to implement and algorithmically simple.


Multi-objective optimization Sine-cosine algorithm Multi-objective engineering design problems 



The research of the first author is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). The postdoctoral fellowship of the second author is supported by NSERC.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


  1. 1.
    Aghaei J, Amjady N, Shayanfar HA (2011) Multi-objective electricity market clearing considering dynamic security by lexicographic optimization and augmented epsilon constraint method. Appl Soft Comput 11(4):3846–3858Google Scholar
  2. 2.
    Agrawal S, Dashora Y, Tiwari MK, Son YJ (2008) Interactive particle swarm: a Pareto-adaptive metaheuristic to multiobjective optimization. IEEE Trans Syst Man Cybern A Syst Hum 38(2):258–277Google Scholar
  3. 3.
    Ahrari A, Atai AA (2010) Grenade explosion method—a novel tool for optimization of multimodal functions. Appl Soft Comput 10(4):1132–1140Google Scholar
  4. 4.
    Akbari R, Hedayatzadeh R, Ziarati K, Hassanizadeh B (2012) A multi-objective artificial bee colony algorithm. Swarm Evolut Comput 2:39–52Google Scholar
  5. 5.
    Angus D, Woodward C (2009) Multiple objective ant colony optimisation. Swarm Intell 3(1):69–85Google Scholar
  6. 6.
    Aydin I, Karakose M, Akin E (2011) A multi-objective artificial immune algorithm for parameter optimization in support vector machine. Appl Soft Comput 11(1):120–129Google Scholar
  7. 7.
    Bérubé JF, Gendreau M, Potvin JY (2009) An exact ϵ-constraint method for bi-objective combinatorial optimization problems: application to the Traveling salesman problem with profits. Eur J Oper Res 194(1):39–50MathSciNetzbMATHGoogle Scholar
  8. 8.
    Coello CAC, Lechuga MS (2002) MOPSO: A proposal for multiple objective particle swarm optimization. In: Proceedings of the congress on evolutionary computation (CEC’2002), Honolulu, HI, vol 1, pp 1051–1056 Google Scholar
  9. 9.
    Coello CAC, Van Veldhuizen DA, Lamont GB (2002) Evolutionary algorithms for solving multi-objective problems, vol 242. Kluwer Academic, New YorkzbMATHGoogle Scholar
  10. 10.
    Corne D, Knowles J, Oates M (2000) The Pareto envelope-based selection algorithm for multiobjective optimization. In: Parallel problem solving from nature PPSN VI. Springer, Berlin, Heidelberg, pp 839–848Google Scholar
  11. 11.
    Corne DW, Jerram NR, Knowles JD, Oates MJ (2001) PESA-II: Region-based selection in evolutionary multiobjective optimization. In: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation. Morgan Kaufmann Publishers Inc, pp 283–290Google Scholar
  12. 12.
    Deb K (2001) Multi-objective optimization using evolutionary algorithms, vol 16. Wiley, LondonzbMATHGoogle Scholar
  13. 13.
    Deb K, Agrawal S, Pratab A, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA–II. IEEE Trans Evolut Comput 6:182–197Google Scholar
  14. 14.
    Dorigo M (1992) Optimization, learning and natural algorithms. Ph. D. thesis, Politecnico di MilanoGoogle Scholar
  15. 15.
    Ehrgott M, Gandibleux X (2002) Multiobjective combinatorial optimization—theory, methodology, and applications. In: Ehrgott M, Gandibleux X (eds) Multiple criteria optimization: state of the art annotated bibliographic surveys. Springer, US, pp 369–444Google Scholar
  16. 16.
    Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012) Water cycle algorithm–a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110:151–166Google Scholar
  17. 17.
    Eusuff MM, Lansey KE (2003) Optimization of water distribution network design using the shuffled frog leaping algorithm. J Water Resour Plan Manag 129(3):210–225Google Scholar
  18. 18.
    Farmer JD, Packard NH, Perelson AS (1986) The immune system, adaptation, and machine learning. Physica D 22(1):187–204MathSciNetGoogle Scholar
  19. 19.
    Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3(2):95–99Google Scholar
  20. 20.
    Gong M, Jiao L, Du H, Bo L (2008) Multiobjective immune algorithm with nondominated neighbor-based selection. Evol Comput 16(2):225–255Google Scholar
  21. 21.
    Holland JH (1975) Adaption in natural and artificial systems. The University of Michigan Press, Ann ArborzbMATHGoogle Scholar
  22. 22.
    In: Proceeding of 1st international conference genetic algorithms, pp 93–100Google Scholar
  23. 23.
    Jamuna K, Swarup KS (2012) Multi-objective biogeography based optimization for optimal PMU placement. Appl Soft Comput 12(5):1503–1510Google Scholar
  24. 24.
    Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Global Optim 39(3):459–471MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kaveh A (2017) Charged system search algorithm. In: Advances in metaheuristic algorithms for optimal design of structures. Springer, pp 45–89Google Scholar
  26. 26.
    Kennedy V, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the IEEE international conference on neural networks, pp 1942–1948Google Scholar
  27. 27.
    Khalili-Damghani K, Amiri M (2012) Solving binary-state multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint, multi-start partial bound enumeration algorithm, and DEA. Reliab Eng Syst Saf 103:35–44Google Scholar
  28. 28.
    Knowles J, Corne D (1999) The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation. In: Proceedings of the 1999 congress on evolutionary computation (CEC 1999), vol 1, pp 98–105Google Scholar
  29. 29.
    Krishnanand KR, Panigrahi BK, Rout PK, Mohapatra A (2011) Application of multi-objective teaching-learning-based algorithm to an economic load dispatch problem with incommensurable objectives. In: Panigrahi BK, Suganthan PN, Das S, Satapathy SC (eds) International conference on swarm, evolutionary, and memetic computing. Springer, Berlin, Heidelberg, pp 697–705Google Scholar
  30. 30.
    Laumanns M, Thiele L, Zitzler E (2006) An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur J Oper Res 169(3):932–942MathSciNetzbMATHGoogle Scholar
  31. 31.
    Li H, Zhang Q (2009) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evolut Comput 13(2):284–302Google Scholar
  32. 32.
    Li X, Zhang J, Yin M (2014) Animal migration optimization: an optimization algorithm inspired by animal migration behavior. Neural Comput Appl 24(7–8):1867–1877Google Scholar
  33. 33.
    Mavrotas G (2009) Effective implementation of the ε-constraint method in multi-objective mathematical programming problems. Appl Math Comput 213(2):455–465MathSciNetzbMATHGoogle Scholar
  34. 34.
    Mehrabian AR, Lucas C (2006) A novel numerical optimization algorithm inspired from weed colonization. Ecol Inform 1(4):355–366Google Scholar
  35. 35.
    Merrikh-Bayat F (2015) The runner-root algorithm: a metaheuristic for solving unimodal and multimodal optimization problems inspired by runners and roots of plants in nature. Appl Soft Comput 33:292–303Google Scholar
  36. 36.
    Miettinen K (2012) Nonlinear multiobjective optimization, vol 12. SpringerGoogle Scholar
  37. 37.
    Mirjalili S (2016) SCA: a sine cosine algorithm for solving optimization problems. Knowl Based Syst 96:120–133Google Scholar
  38. 38.
    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61Google Scholar
  39. 39.
    Mondal S, Bhattacharya A, nee Dey SH (2013) Multi-objective economic emission load dispatch solution using gravitational search algorithm and considering wind power penetration. Int J Electr Power Energy Syst 44(1):282–292Google Scholar
  40. 40.
    Moslehi G, Mahnam M (2011) A Pareto approach to multi-objective flexible job-shop scheduling problem using particle swarm optimization and local search. Int J Prod Econ 129(1):14–22Google Scholar
  41. 41.
    Mostaghim S, Teich J (2004) Covering pareto-optimal fronts by subswarms in multi-objective particle swarm optimization. In: Congress on evolutionary computation, CEC2004. IEEE, vol 2, pp 1404–1411Google Scholar
  42. 42.
    Nikoofard AH, Hajimirsadeghi H, Rahimi-Kian A, Lucas C (2012) Multiobjective invasive weed optimization: application to analysis of Pareto improvement models in electricity markets. Appl Soft Comput 12(1):100–112Google Scholar
  43. 43.
    Omkar SN, Senthilnath J, Khandelwal R, Naik GN, Gopalakrishnan S (2011) Artificial Bee Colony (ABC) for multi-objective design optimization of composite structures. Appl Soft Comput 11(1):489–499Google Scholar
  44. 44.
    Passino KM (2002) Biomimicry of bacterial foraging for distributed optimization and control. Control Syst IEEE 22(3):52–67MathSciNetGoogle Scholar
  45. 45.
    Patel VK, Savsani VJ (2016) A multi-objective improved teaching–learning based optimization algorithm (MO-ITLBO). Inf Sci 357:182–200Google Scholar
  46. 46.
    Patel VK, Savsani VJ (2015) Heat transfer search(HTS): a novel optimization algorithm. Inf Sci 324:217–246Google Scholar
  47. 47.
    Patel V, Savsani V (2014) Optimization of a plate-fin heat exchanger design through an improved multi-objective teaching–learning based optimization (MO-ITLBO) algorithm. Chem Eng Res Des 92(11):2371–2382Google Scholar
  48. 48.
    Rao RV, Savsani VJ, Vakharia DP (2011) Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303–315Google Scholar
  49. 49.
    Rao RV, Savsani VJ, Vakharia DP (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems. Inf Sci 183(1):1–15MathSciNetGoogle Scholar
  50. 50.
    Rao SS (2009) Engineering optimization: theory and practice. WileyGoogle Scholar
  51. 51.
    Rashedi E, Nezamabadi-Pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248zbMATHGoogle Scholar
  52. 52.
    Reyes-Sierra M, Coello CC (2006) Multi-objective particle swarm optimizers: a survey of the state-of-the-art. Int J Comput Intell Res 2(3):287–308MathSciNetGoogle Scholar
  53. 53.
    Roy PK, Ghoshal SP, Thakur SS (2010) Biogeography based optimization for multi-constraint optimal power flow with emission and non-smooth cost function. Expert Syst Appl 37(12):8221–8228Google Scholar
  54. 54.
    Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2013) Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems. Appl Soft Comput 13(5):2592–2612Google Scholar
  55. 55.
    Sadollah A, Eskandar H, Kim JH (2015) Water cycle algorithm for solving constrained multi-objective optimization problems. Appl Soft Comput 27:279–298Google Scholar
  56. 56.
    Savsani P, Savsani V (2016) Passing vehicle search (PVS): a novel metaheuristic algorithm. Appl Math Model 40:3951–3978Google Scholar
  57. 57.
    Schaffer JD (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st international conference on genetic algorithms. L. Erlbaum Associates. Inc, pp 93–100 Google Scholar
  58. 58.
    Shareef H, Ibrahim AA, Mutlag AH (2015) Lightning search algorithm. Appl Soft Comput 36:315–333Google Scholar
  59. 59.
    Simon D (2008) Biogeography-based optimization. IEEE Trans Evolut Comput 12(6):702–713Google Scholar
  60. 60.
    Srinivas N, Deb K (1994) Multiobjective optimization using nondominated sorting in genetic algorithms. Evolut Comput 2(3):221–248Google Scholar
  61. 61.
    Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359MathSciNetzbMATHGoogle Scholar
  62. 62.
    Tan KC, Goh CK, Mamun AA, Ei EZ (2008) An evolutionary artificial immune system for multi-objective optimization. Eur J Oper Res 187(2):371–392MathSciNetzbMATHGoogle Scholar
  63. 63.
    Wang Y, Yang Y (2009) Particle swarm optimization with preference order ranking for multi-objective optimization. Inf Sci 179(12):1944–1959MathSciNetGoogle Scholar
  64. 64.
    Yagmahan B, Yenisey MM (2008) Ant colony optimization for multi-objective flow shop scheduling problem. Comput Ind Eng 54(3):411–420Google Scholar
  65. 65.
    Yang XS (2009) Firefly algorithms for multimodal optimization. In: International symposium on stochastic algorithms. Springer, Berlin, Heidelberg, pp 169–178Google Scholar
  66. 66.
    Yang XS (2010) A new metaheuristic bat-inspired algorithm. In: Nature inspired cooperative strategies for optimization (NICSO 2010), pp 65–74Google Scholar
  67. 67.
    Yang XS (2011) Bat algorithm for multi-objective optimisation. Int J Bio Inspir Comput 3(5):267–274Google Scholar
  68. 68.
    Yang XS (2013) Multiobjective firefly algorithm for continuous optimization. Eng Comput 29(2):175–184Google Scholar
  69. 69.
    Yang XS, Deb S (2010) Engineering optimisation by cuckoo search. Int J Math Model Numer Optim 1(4):330–343zbMATHGoogle Scholar
  70. 70.
    Yang XS, Deb S (2013) Multiobjective cuckoo search for design optimization. Comput Oper Res 40(6):1616–1624MathSciNetzbMATHGoogle Scholar
  71. 71.
    Yazdani M, Jolai F (2016) Lion optimization algorithm (LOA): a nature-inspired metaheuristic algorithm. J Comput Des Eng 3(1):24–36Google Scholar
  72. 72.
    Zhang Q, Li H (2007) MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 11(6):712–731Google Scholar
  73. 73.
    Zhang H, Zhu Y, Zou W, Yan X (2012) A hybrid multi-objective artificial bee colony algorithm for burdening optimization of copper strip production. Appl Math Model 36(6):2578–2591zbMATHGoogle Scholar
  74. 74.
    Zhou A, Qu BY, Li H, Zhao SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evolut Comput 1(1):32–49Google Scholar
  75. 75.
    Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evolut Comput 3(4):257–271Google Scholar
  76. 76.
    Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evolut Comput 8(2):173–195Google Scholar

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceThompson Rivers UniversityKamloopsCanada
  2. 2.Department of Mathematics and Computer Science, Faculty of ScienceAlexandria UniversityMoharam Bey, AlexandriaEgypt
  3. 3.Department of Mechanical EngineeringPandit Deendayal Petroleum UniversityGandhinagarIndia

Personalised recommendations