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Multi-strategy monarch butterfly optimization algorithm for discounted {0-1} knapsack problem

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Abstract

As an expanded classical 0-1 knapsack problem (0-1 KP), the discounted {0-1} knapsack problem (DKP) is proposed based on the concept of discount in the commercial world. The DKP contains a set of item groups where each group includes three items, whereas no more than one item in each group can be packed in the knapsack, which makes it more complex and challenging than 0-1 KP. At present, the main two algorithms for solving the DKP include exact algorithms and approximate algorithms. However, there are some topics which need to be further discussed, i.e., the improvement of the solution quality. In this paper, a novel multi-strategy monarch butterfly optimization (MMBO) algorithm for DKP is proposed. In MMBO, two effective strategies, neighborhood mutation with crowding and Gaussian perturbation, are introduced into MMBO. Experimental analyses show that the first strategy can enhance the global search ability, while the second strategy can strengthen local search ability and prevent premature convergence during the evolution process. Based on this, MBO is combined with each strategy, denoted as NCMBO and GMMBO, respectively. We compared MMBO with other six methods, including NCMBO, GMMBO, MBO, FirEGA, SecEGA and elephant herding optimization. The experimental results on three types of large-scale DKP instances show that NCMBO, GMMBO and MMBO are all suitable for solving DKP. In addition, MMBO outperforms other six methods and can achieve a good approximate solution with its approximation ratio close to 1 on almost all the DKP instances.

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References

  1. Cormen TH (2009) Introduction to algorithms. The MIT Press, Cambridge

    MATH  Google Scholar 

  2. Du DZ, Ko KI (2011) Theory of computational complexity. Wiley, New York

    MATH  Google Scholar 

  3. Guldan B (2007) Heuristic and exact algorithms for discounted knapsack problems. Master thesis, University of Erlangen-Nürnberg, Germany

  4. Rong A, Figueira JR, Klamroth K (2012) Dynamic programming based algorithms for the discounted 0–1 knapsack problem. Appl Math Comput 218(12):6921–6933

    MathSciNet  MATH  Google Scholar 

  5. He YC, Wang XZ, Li WB, Zhang XL, Chen YY (2016) Research on genetic algorithms for the discounted {0-1} knapsack problem. Chin J Comput 39(12):2614–2630. doi:10.11897/SP.J.1016.2016.02614

    Article  MathSciNet  Google Scholar 

  6. Rudolph G (1994) Convergence analysis of canonical genetic algorithms. IEEE Trans Neural Netw 5(1):96–101

    Article  Google Scholar 

  7. Golberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison Wesley, New York, p 102

    Google Scholar 

  8. Bellman R (1956) Dynamic programming and Lagrange multipliers. Proc Natl Acad Sci 42(10):767–769

    Article  MathSciNet  Google Scholar 

  9. He YC, Wang XZ, He YL et al (2016) Exact and approximate algorithms for discounted 0–1 knapsack problem. Inf Sci 369:634–647

    Article  MathSciNet  Google Scholar 

  10. Wang G-G, Tan Y (2017) Improving metaheuristic algorithms with information feedback models. IEEE Trans Cybern

  11. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, vol 1, pp 39–43

  12. Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B Cybern 26(1):29–41

    Article  Google Scholar 

  13. Wang G-G, Guo LH, Wang HQ et al (2014) Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Comput Appl 24(3–4):853–871

    Article  Google Scholar 

  14. Wang G-G, Gandomi AH, Alavi AH (2014) Stud krill herd algorithm. Neurocomputing 128:363–370

    Article  Google Scholar 

  15. Wang G-G, Guo L, Gandomi AH et al (2014) Chaotic Krill Herd algorithm. Inf Sci 274(8):17–34

    Article  MathSciNet  Google Scholar 

  16. Wang G-G, Gandomi AH, Alavi AH et al (2014) Hybrid krill herd algorithm with differential evolution for global numerical optimization. Neural Comput Appl 25(2):297–308

    Article  Google Scholar 

  17. Wang G-G, Gandomi AH, Alavi AH (2014) An effective krill herd algorithm with migration operator in biogeography-based optimization. Appl Math Model 38(9-10):2454–2462. doi:10.1016/j.apm.2013.10.052

    Article  MathSciNet  Google Scholar 

  18. Wang G-G, Deb S, Gandomi AH et al (2016) Opposition-based krill herd algorithm with Cauchy mutation and position clamping. Neurocomputing 177:147–157

    Article  Google Scholar 

  19. Wang G-G, Gandomi AH, Yang X-S, Alavi AH (2016) A new hybrid method based on krill herd and cuckoo search for global optimization tasks. Int J Bio Inspir Comput 8(5):286–299. doi:10.1504/IJBIC.2016.10000414

    Article  Google Scholar 

  20. Cui ZH, Fan S, Zeng J et al (2013) Artificial plant optimization algorithm with three-period photosynthesis. Int J Bio Inspir Comput 5(2):133–139

    Article  Google Scholar 

  21. Wang G-G, Deb S, Gao X-Z, Coelho LdS (2016) A new metaheuristic optimization algorithm motivated by elephant herding behavior. Int J Bio Inspir Comput 8(6):394–409

    Article  Google Scholar 

  22. Wang G-G (2016) Moth search algorithm: a bio-inspired metaheuristic algorithm for global optimization problems. Memet Comput 1–14. doi:10.1007/s12293-016-0212-3

    Article  Google Scholar 

  23. Rezoug A, Boughaci D (2016) A self-adaptive harmony search combined with a stochastic local search for the 0-1 multidimensional knapsack problem. Int J Bio Inspir Comput 8(4):234–239

    Article  Google Scholar 

  24. Wang H, Wang W, Sun H et al (2016) Firefly algorithm with random attraction. Int J Bio Inspir Comput 8(1):33–41

    Article  Google Scholar 

  25. Shi Y (2011) Brain storm optimization algorithm. In: International conference in swarm intelligence. Springer, Berlin, pp 303–309

    Google Scholar 

  26. Cheng S, Qin Q, Chen J et al (2016) Brain storm optimization algorithm: a review. Artif Intell Rev 46:445–458. doi:10.1007/s10462-016-9471-0

    Article  Google Scholar 

  27. Cheng S, Shi Y, Qin Q et al (2013) Solution clustering analysis in brain storm optimization algorithm. In: Swarm intelligence (SIS), 2013 IEEE symposium on. IEEE, pp 111–118

  28. Jia Z, Duan H, Shi Y (2016) Hybrid brain storm optimisation and simulated annealing algorithm for continuous optimisation problems. Int J Bio Inspir Comput 8(2):109–121. doi:10.1504/IJBIC.2016.076326

    Article  Google Scholar 

  29. Tan Y, Zhu Y (2010) Fireworks algorithm for optimization. In: International conference in swarm intelligence. Springer, Berlin, pp 355–364

    Google Scholar 

  30. Zheng S, Janecek A, Tan Y (2013) Enhanced fireworks algorithm. In: 2013 IEEE congress on evolutionary computation. IEEE, pp 2069–2077

  31. Cai X, Gao XZ, Xue Y (2016) Improved bat algorithm with optimal forage strategy and random disturbance strategy. Int J Bio Inspir Comput 8(4):205–214

    Article  Google Scholar 

  32. Wang G-G, Deb S, Cui ZH (2015) Monarch butterfly optimization. Neural Comput Appl 1–20. doi:10.1007/s00521-015-1923-y

  33. Wang G-G, Deb S, Zhao X et al (2016) A new monarch butterfly optimization with an improved crossover operator. Oper Res 1–25. doi:10.1007/s12351-016-0251-z

    Article  Google Scholar 

  34. Ghanem WAHM, Jantan A (2016) Hybridizing artificial bee colony with monarch butterfly optimization for numerical optimization problems. Neural Comput Appl 1–19. doi:10.1007/s00521-016-2665-1

    Article  Google Scholar 

  35. Qu BY, Suganthan PN, Liang JJ (2012) Differential evolution with neighborhood mutation for multimodal optimization. IEEE Trans Evol Comput 16(5):601–614

    Article  Google Scholar 

  36. Reynolds AM, Rhodes CJ (2009) The Lévy flight paradigm: random search patterns and mechanisms. Ecology 90(4):877–887

    Article  Google Scholar 

  37. Hinterding R (1995) Gaussian mutation and self-adaption for numeric genetic algorithms. In: IEEE international conference on evolutionary computation, p 384

  38. Higashi N, Iba H (2003) Particle swarm optimization with Gaussian mutation. In: Swarm intelligence symposium, SIS’03. Proceedings of the 2003 IEEE, pp 72–79

  39. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102

    Article  Google Scholar 

  40. He YC, Wang XZ, Kou YZ (2007) A binary differential evolution algorithm with hybrid encoding. J Comput Res Dev 44(9):1476–1484

    Article  Google Scholar 

  41. Feng YH, Wang G-G, Deb S et al (2015) Solving 0-1 knapsack problem by a novel binary monarch butterfly optimization. Neural Comput Appl 1–16. doi:10.1007/s00521-015-2135-1

    Article  Google Scholar 

  42. Feng YH, Yang J, Wu C et al (2016) Solving 0–1 knapsack problems by chaotic monarch butterfly optimization algorithm with Gaussian mutation. Memet Comput 1–16. doi:10.1007/s12293-016-0211-4

    Article  Google Scholar 

  43. Michalewicz Z, Schoenauer M (1996) Evolutionary algorithms for constrained parameter optimization problems. Evol Comput 4(1):1–32

    Article  Google Scholar 

  44. Kort BW, Bertsekas DP (1972) A new penalty function method for constrained minimization. In: Decision and control, 1972 and 11th symposium on adaptive processes. Proceedings of the 1972 IEEE conference on. IEEE, pp 162–166

  45. Di Pillo G, Grippo L (1986) An exact penalty function method with global convergence properties for nonlinear programming problems. Math Program 36(1):1–18

    Article  MathSciNet  Google Scholar 

  46. Chih MC (2015) Self-adaptive check and repair operator-based particle swarm optimization for the multidimensional knapsack problem. Appl Soft Comput 26:378–389

    Article  Google Scholar 

  47. Chu PC, Beasley JE (1998) A genetic algorithm for the multidimensional knapsack problem. J Heuristics 4(1):63–86

    Article  Google Scholar 

  48. He YC, Zhang XL, Li WB et al (2016) Algorithms for randomized time-varying knapsack problems. J Comb Optim 31(1):95–117

    Article  MathSciNet  Google Scholar 

  49. Pisinger D (1995) Algorithms for knapsack problems 31

  50. Alsuwaiyel MH (2009) Algorithms design techniques and analysis. World Scientific Publishing Company, Beijing

    MATH  Google Scholar 

  51. Wilcoxon F (1945) Individual comparisons by ranking methods. Biom Bull 1(6):80–83

    Article  Google Scholar 

  52. Wilcoxon F, Katti SK, Wilcox RA (1970) Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test. Sel Tables Math Stat 1:171–259

    MATH  Google Scholar 

  53. Wang G-G, Deb S, Coelho LDS (2015) Earthworm optimization algorithm: a bio-inspired metaheuristic algorithm for global optimization problems. Int J of Bio Inspir Comput

  54. Zhou A, Qu BY, Li H et al (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evolut Comput 1(1):32–49

    Article  Google Scholar 

  55. Bryden KM, Ashlock DA, Corns S et al (2006) Graph-based evolutionary algorithms. IEEE Trans Evol Comput 10(5):550–567

    Article  Google Scholar 

  56. Li J, Pan Q (2015) Solving the large-scale hybrid flow shop scheduling problem with limited buffers by a hybrid artificial bee colony algorithm. Inf Sci 316:487–502

    Article  Google Scholar 

  57. Han B, Leblet J, Simon G (2010) Hard multidimensional multiple choice knapsack problems, an empirical study. Comput Oper Res 37(1):172–181

    Article  MathSciNet  Google Scholar 

  58. Ren Z, Feng Z, Zhang A (2012) Fusing ant colony optimization with Lagrangian relaxation for the multiple-choice multidimensional knapsack problem. Inf Sci 182(1):15–29

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 61503165), Natural Science Foundation of Jiangsu Province (No. BK20150239), Scientific Research Project Program of Colleges and Universities in Hebei Province (No. ZD2016005), the Open Research Fund of Sichuan Key Laboratory for Nature Gas and Geology (No. 2015trqdz04) and Natural Science Foundation of Hebei Province (No. F2016403055).

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Authors and Affiliations

  1. School of Information Engineering, Hebei GEO University, Shijiazhuang, 050031, China

    Yanhong Feng, Wenbin Li & Ning Li

  2. School of Computer Science and Technology, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, China

    Gai-Ge Wang

  3. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, T6R 2V4, Canada

    Gai-Ge Wang

  4. Institute of Algorithm and Big Data Analysis, Northeast Normal University, Changchun, 130117, China

    Gai-Ge Wang

  5. School of Computer Science and Information Technology, Northeast Normal University, Changchun, 130117, China

    Gai-Ge Wang

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  1. Yanhong Feng
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Correspondence to Gai-Ge Wang.

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Feng, Y., Wang, GG., Li, W. et al. Multi-strategy monarch butterfly optimization algorithm for discounted {0-1} knapsack problem. Neural Comput & Applic 30, 3019–3036 (2018). https://doi.org/10.1007/s00521-017-2903-1

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