A Novel Discrete Grey Wolf Optimizer for Solving the Bounded Knapsack Problem

  • Zewen Li
  • Yichao HeEmail author
  • Huanzhe Li
  • Ya Li
  • Xiaohu Guo
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 986)


Grey Wolf Optimizer (GWO) is a recently proposed metaheuristic optimizer inspired by the leadership hierarchy and hunting mechanism of grey wolves. In order to solve the bounded knapsack problem by the GWO, a novel Discrete Grey Wolf Optimizer (DGWO) is proposed in this paper. On the basis of DGWO, the crossover strategy of the genetic algorithm is introduced to enhance its local search ability, and infeasible solutions are processed by a Repair and Optimization method based on the greedy strategy, which could not only ensure the effectiveness but also speed up the convergence. Experiment using three kinds of large-scale instances of the bounded knapsack problem is carried out to verify the validity and stability of the DGWO. By comparing and analyzing the results with other well-established algorithms, computational results show that the convergence speed of the DGWO is faster than that of other algorithms, solutions of these instances of the bounded knapsack problem are all well obtained with approximation ratio bound close to 1.


Bounded knapsack problem Grey Wolf Optimizer Genetic algorithm Repair and optimization method 



This work was supported by the Scientific Research Project Program of Colleges and Universities in Hebei Province (ZD2016005), and the Natural Science Foundation of Hebei Province (F2016403055).


  1. 1.
    Mirjalili, S., Mirjalili, S.M., Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)CrossRefGoogle Scholar
  2. 2.
    Masadeh, R., Yassien, E., Alzaqebah, A., et al.: Grey wolf optimization applied to the 0/1 knapsack problem. Int. J. Comput. Appl. 169(5), 11–15 (2017)Google Scholar
  3. 3.
    Sharma, S., Salgotra, R., Singh, U.: An enhanced grey wolf optimizer for numerical optimization. In: International Conference on Innovations in Information, Embedded and Communication Systems, pp. 1–6 (2017)Google Scholar
  4. 4.
    Mirjalili, S.: How effective is the Grey Wolf optimizer in training multi-layer perceptrons. Appl. Intell. 43(1), 150–161 (2015)CrossRefGoogle Scholar
  5. 5.
    Hatta, N.M., Zain, A.M., Sallehuddin, R., et al.: Recent studies on optimisation method of Grey Wolf Optimiser (GWO): a review (2014–2017). Artif. Intell. Rev. May 2018.
  6. 6.
    Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5(2), 266–288 (1957)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gilmore, P.C., Gomory, R.E.: The theory and computation of knapsack functions. Oper. Res. 14(6), 1045–1074 (1966)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wei, S.: A branch and bound method for the multiconstraint zero-one knapsack problem. J. Oper. Res. Soc. 30(4), 369–378 (1979)CrossRefGoogle Scholar
  9. 9.
    Bitran, G.R., Hax, A.C.: Disaggregation and resource allocation using convex knapsack problems with bounded variables. Manag. Sci. 27(4), 431–441 (1981)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pendharkar, P.C., Rodger, J.A.: Information technology capital budgeting using a knapsack problem. Int. Trans. Oper. Res. 13(4), 333–351 (2010)CrossRefGoogle Scholar
  11. 11.
    He, Y.C., Wang, X.Z., Li, W.B., et al.: Research on genetic algorithms for the discounted 0–1 knapsack problem. Chin. J. Comput. 39, 2614–2630 (2016)MathSciNetGoogle Scholar
  12. 12.
    Pisinger, D.: A minimal algorithm for the bounded knapsack problem. In: Balas, E., Clausen, J. (eds.) IPCO 1995. LNCS, vol. 920, pp. 95–109. Springer, Heidelberg (1995). Scholar
  13. 13.
    Wang, X.Z., He, Y.-C.: Evolutionary algorithms for knapsack problems. J. Softw. 28, 1–16 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading (1989)zbMATHGoogle Scholar
  15. 15.
    Kennedy, J., Eberhart, R.: Particle swarm optimization. In: IEEE International Conference on Neural Networks, Proceedings, vol. 4, pp. 1942–1948 (1995)Google Scholar
  16. 16.
    Storn, R., Price, K.: Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Emary, E., Zawbaa, H.M., Hassanien, A.E.: Binary Grey Wolf optimization approaches for feature selection. J. Neurocomputing 172, 371–381 (2016)CrossRefGoogle Scholar
  18. 18.
    Kamboj, V.K., Bath, S.K., Dhillon, J.S.: Solution of non-convex economic load dispatch problem using Grey Wolf optimizer. Neural Comput. Appl. 27(5), 1301–1316 (2016)CrossRefGoogle Scholar
  19. 19.
    Moradi, M., Badri, A., Ghandehari, R.: Non-convex constrained economic dispatch with valve point loading effect using a grey wolf optimizer algorithm. In: 2016 6th Conference on Thermal Power Plants (CTPP), pp 96–104. IEEE (2016)Google Scholar
  20. 20.
    Chandra, M., Agrawal, A., Kishor, A., Niyogi, R.: Web service selection with global constraints using modified Grey Wolf optimizer. In: 2016 International Conference on Advances in Computing, Communications and Informatics (ICACCI), pp 1989–1994. IEEE (2016)Google Scholar
  21. 21.
    He, Y.C., Wang, X.Z., Zhao, S.L., Zhang, X.L.: Design and applications of discrete evolutionary algorithm based on encoding transformation. Ruan Jian Xue Bao/J. Softw. 29(9) (2018). (in Chinese).
  22. 22.
    Michalewicz, Z.: Genetic Algorithm + Data Structure = Evolution Programs, pp. 13–103. Springer, Berlin (1996). Scholar
  23. 23.
    Zou, D.X., Gao, L.Q., Li, S., Wu, J.H.: Solving 0-1 knapsack problem by a novel global harmony search algorithm. Appl. Soft Comput. 11, 1556–1564 (2011). Scholar
  24. 24.
    He, Y.C., Zhang, X.L., Li, X., Wu, W.L., Gao, S.G.: Algorithms for randomized time-varying knapsack problems. J. Comb. Optim. 31(1), 95–117 (2016). Scholar
  25. 25.
    He, Y.C., Wang, X.Z., Li, W.B., Zhao, S.L.: Exact algorithms and evolutionary algorithms for randomized time-varying knapsack problem. Ruan Jian Xue Bao/J. Softw. (2016). (in Chinese with English abstract).,
  26. 26.
    He, Y.C., Song, J.M., Zhang, J.M., et al.: Research on genetic algorithm for solving static and dynamic knapsack problems. Appl. Res. Comput. 32(4), 1011–1015 (2015). (in Chinese)Google Scholar
  27. 27.
    Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1996)zbMATHGoogle Scholar
  28. 28.
    Byrka, J., Li, S., Rybicki, B.: Improved approximation algorithm for k-level uncapacitated facility location problem (with penalties). Theory Comput. Syst. 58, 19–44 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Zewen Li
    • 1
  • Yichao He
    • 1
    Email author
  • Huanzhe Li
    • 1
  • Ya Li
    • 1
  • Xiaohu Guo
    • 1
  1. 1.College of Information and EngineeringHebei GEO UniversityShijiazhuangChina

Personalised recommendations