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A k-means binarization framework applied to multidimensional knapsack problem

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Abstract

The multidimensional knapsack problem (MKP) is one of the widely known integer programming problems. The MKP has received significant attention from the operational research community for its large number of applications. Solving this NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. In this paper we present a k-means transition ranking (KMTR) framework to solve the MKP. This framework has the property to binarize continuous population-based metaheuristics using a data mining k-means technique. In particular we binarize a Cuckoo Search and Black Hole metaheuristics. These techniques were chosen by the difference between their iteration mechanisms. We provide necessary experiments to investigate the role of key ingredients of the framework. Finally to demonstrate the efficiency of our proposal, MKP benchmark instances of the literature show that KMTR competes with the state-of-the-art algorithms.

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Notes

  1. OR-Library: http://www.brunel.ac.uk/mastjjb/jeb/orlib/mknapinfo.html.

References

  1. Akhlaghi M, Emami F, Nozhat N (2014) Binary tlbo algorithm assisted for designing plasmonic nano bi-pyramids-based absorption coefficient. J Mod Opt 61(13):1092–1096

    Article  Google Scholar 

  2. Albo Y, Lanir J, Bak Px, Rafaeli S (2016) Off the radar Comparative evaluation of radial visualization solutions for composite indicators. IEEE Trans Vis Comput Graph 22(1):569–578

    Article  Google Scholar 

  3. Alegrıa J, Túpac Y (2014) A generalized quantum-inspired evolutionary algorithm for combinatorial optimization problems XXXII international conference of the Chilean computer science society SCCC, November, pp 11–15

    Google Scholar 

  4. Alvarez MJ, Shen Y, Giorgi FM, Lachmann A, Belinda Ding B, Hilda Ye B, Califano A (2016) Functional characterization of somatic mutations in cancer using network-based inference of protein activity. Nat Genet

  5. Balas E, Zemel E (1980) An algorithm for large zero-one knapsack problems. Oper Res 28(5):1130–1154

    Article  MathSciNet  MATH  Google Scholar 

  6. Bansal JC, Deep K (2012) A modified binary particle swarm optimization for knapsack problems. Appl Math Comput 218(22):11042–11061

    MathSciNet  MATH  Google Scholar 

  7. Baykasoġlu A, Ozsoydan FB (2014) An improved firefly algorithm for solving dynamic multidimensional knapsack problems. Expert Syst Appl 41(8):3712–3725

    Article  Google Scholar 

  8. Beasley JE (1990) Or-library: distributing test problems by electronic mail. J Oper Res Soc 41(11):1069–1072

    Article  Google Scholar 

  9. Bhattacharjee KK, Sarmah SP (2016) Modified swarm intelligence based techniques for the knapsack problem. Appl Intell, 1–22

  10. Chajakis E, Guignard M (1992) A model for delivery of groceries in vehicle with multiple compartments and lagrangean approximation schemes Proceedings of congreso latino ibero-americano de investigación de operaciones e ingeniería de sistemas

    Google Scholar 

  11. Chandrasekaran K, Simon SP (2012) Network and reliability constrained unit commitment problem using binary real coded firefly algorithm. Int J Electr Power Energy Syst 43(1):921–932

    Article  Google Scholar 

  12. Changdar C, Mahapatra GS, Pal RK (2013) An ant colony optimization approach for binary knapsack problem under fuzziness. Appl Math Comput 223:243–253

    MathSciNet  MATH  Google Scholar 

  13. Chen E, Li J, Liu X (2011) In search of the essential binary discrete particle swarm. Appl Soft Comput 11(3):3260–3269

    Article  Google Scholar 

  14. Chih M (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 

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

    Article  MATH  Google Scholar 

  16. Crawford B, Soto R, Cuesta R, Olivares-Suárez M, Johnson F, Olguin E (2014) Two swarm intelligence algorithms for the set covering problem 2014 9th international conference on software engineering and applications (ICSOFT-EA), pp 60–69

    Google Scholar 

  17. Crawford B, Soto R, Olivares-Suarez M, Palma W, Paredes F, Olguin E, Norero E (2014) A binary coded firefly algorithm that solves the set covering problem. Romanian J Inf Sci Technol 17(3):252–264

    Google Scholar 

  18. Dey S, Bhattacharyya S, Maulik U (2015) New quantum inspired meta-heuristic techniques for multi-level colour image thresholding. Appl Soft Comput 888:999

    Google Scholar 

  19. Drake JH, Özcan E, Burke EK (2016) A case study of controlling crossover in a selection hyper-heuristic framework using the multidimensional knapsack problem. Evol Comput 24(1):113–141

    Article  Google Scholar 

  20. Fayard D, Plateau G (1982) An algorithm for the solution of the 0–1 knapsack problem. Computing 28 (3):269–287

    Article  MATH  Google Scholar 

  21. García J, Crawford B, Soto R, García P (2017) A multi dynamic binary black hole algorithm applied to set covering problem International conference on harmony search algorithm. Springer, pp 42–51

  22. Gavish B, Pirkul H (1982) Allocation of databases and processors in a distributed computing system. Manag Distributed Data Process 31:215–231

    Google Scholar 

  23. Geem Z, Kim J, Loganathan GV (2001) A new heuristic optimization algorithm: harmony search. Simulation 76(2):60–68

    Article  Google Scholar 

  24. Gherboudj A, Layeb A, Chikhi S (2012) Solving 0-1 knapsack problems by a discrete binary version of cuckoo search algorithm. Int J Bio-Inspired Comput 4(4):229–236

    Article  Google Scholar 

  25. Gilmore PC, Gomory RE (1966) The theory and computation of knapsack functions. Oper Res 14 (6):1045–1074

    Article  MathSciNet  MATH  Google Scholar 

  26. Gong T, Tuson AL (2007) Differential evolution for binary encoding Soft computing in industrial applications. Springer, pp 251–262

  27. Haddar B, Khemakhem M, Hanafi S, Wilbaut C (2016) A hybrid quantum particle swarm optimization for the multidimensional knapsack problem. Eng Appl Artif Intell 55:1–13

    Article  Google Scholar 

  28. Hamilton R, Fuller J, Baldwin K, Vespa P, Xiao H, Bergsneider M (2016) Relative position of the third characteristic peak of the intracranial pressure pulse waveform morphology differentiates normal-pressure hydrocephalus shunt responders and nonresponders Intracranial pressure and brain monitoring XV. Springer, pp 339–345

  29. Hatamlou A (2013) Black hole: A new heuristic optimization approach for data clustering. Inf Sci 222:175–184

    Article  MathSciNet  Google Scholar 

  30. Hota AR, Pat A (2010) An adaptive quantum-inspired differential evolution algorithm for 0–1 knapsack problem 2010 2nd world congress on nature and biologically inspired computing (naBIC), pp 703–708

    Chapter  Google Scholar 

  31. Ibrahim AA, Mohamed A, Shareef H, Ghoshal SP (2011) An effective power quality monitor placement method utilizing quantum-inspired particle swarm optimization 2011 international conference on electrical engineering and informatics (ICEEI), pp 1–6

    Google Scholar 

  32. Ionita-Laza I, McCallum K, Bin X, Buxbaum JD (2016) A spectral approach integrating functional genomic annotations for coding and noncoding variants. Nat Genet 48(2):214–220

    Article  Google Scholar 

  33. Kennedy JJ, Eberhart R (1997) A discrete binary version of the particle swarm algorithm. IEEE 4105:4104–4108

    Google Scholar 

  34. Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. Technical report, Technical report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department

  35. Khalil T, Youseef H, Aziz M (2006) A binary particle swarm optimization for optimal placement and sizing of capacitor banks in radial distribution feeders with distorted substation voltages. In: Proceedings of AIML international conference, pp 137–143

  36. Kong X, Gao L, Ouyang H, Li S (2015) Solving large-scale multidimensional knapsack problems with a new binary harmony search algorithm. Comput Oper Res 63:7–22

    Article  MathSciNet  MATH  Google Scholar 

  37. Kotthoff L (2014) Algorithm selection for combinatorial search problems. Surv AI Mag 35(3):48–60

    Google Scholar 

  38. Layeb A (2011) A novel quantum inspired cuckoo search for knapsack problems. Int J Bio-Inspired Comput 3(5):297–305

    Article  Google Scholar 

  39. Layeb A (2013) A hybrid quantum inspired harmony search algorithm for 0–1 optimization problems. J Comput Appl Math 253:14–25

    Article  MathSciNet  MATH  Google Scholar 

  40. Layeb A, Boussalia SR (2012) A novel quantum inspired cuckoo search algorithm for bin packing problem. Int J Inf Technol Comput Sci (IJITCS) 4(5):58

    Google Scholar 

  41. Leonard BJ, Engelbrecht AP, Cleghorn CW (2015) Critical considerations on angle modulated particle swarm optimisers. Swarm Intell 9(4):291–314

    Article  Google Scholar 

  42. Li X-L, Shao Z-J, Qian J-X (2002) An optimizing method based on autonomous animats: fish-swarm algorithm. Syst Eng Theory Pract 22(11):32–38

    Google Scholar 

  43. Liu J, Changzhi W, Cao J, Wang X, Teo KL (2016) A binary differential search algorithm for the 0–1 multidimensional knapsack problem. Appl Math Model

  44. Liu W, Liu L, Cartes D (2007) Angle modulated particle swarm optimization based defensive islanding of large scale power systems. In: IEEE power engineering society conference and exposition in Africa, pp 1–8

  45. Long Q, Changzhi W, Huang T, Wang X (2015) A genetic algorithm for unconstrained multi-objective optimization. Swarm Evol Comput 22:1–14

    Article  Google Scholar 

  46. Martello S, Toth P (1988) A new algorithm for the 0-1 knapsack problem. Manag Sci 34(5):633–644

    Article  MathSciNet  MATH  Google Scholar 

  47. McMillan C, Plaine DR (1973) Resource allocation via 0–1 programming. Decis Sci 4:119–132

    Article  Google Scholar 

  48. (2014). IBM IBM ILOG CPLEX Optimizer. http://www.ibm.com/software/commerce/optimization/cplex-optimizer/. Cited on, page 1

  49. Özcan E, Baṡaran C (2009) A case study of memetic algorithms for constraint optimization. Soft Comput 13(8-9):871–882

    Article  Google Scholar 

  50. Palit S, Sinha SN, Molla MA, Khanra A, Kule M (2011) A cryptanalytic attack on the knapsack cryptosystem using binary firefly algorithm International conference on computer and communication technology (ICCCT), vol 2, pp 428–432

  51. Pampara G (2012) Angle modulated population based algorithms to solve binary problems. Phd thesis, University of Pretoria, Pretoria

  52. Pan W-T (2012) A new fruit fly optimization algorithm: taking the financial distress model as an example. Knowl-Based Syst 26:69–74

    Article  Google Scholar 

  53. Pandiri V, Singh A (2016) Swarm intelligence approaches for multidepot salesmen problems with load balancing. Appl Intell 44(4):849–861

    Article  Google Scholar 

  54. Petersen CC (1967) Computational experience with variants of the balas algorithm applied to the selection of r&d projects. Manag Sci 13(9):736–750

    Article  Google Scholar 

  55. Pirkul H (1987) A heuristic solution procedure for the multiconstraint zero? One knapsack problem. Nav Res Logist 34(2):161–172

    Article  MathSciNet  MATH  Google Scholar 

  56. Raidl GR (1998) An improved genetic algorithm for the multiconstrained 0-1 knapsack problem The 1998 IEEE international conference on evolutionary computation proceedings, 1998. IEEE World Congress on Computational Intelligence, pp 207–211

    Google Scholar 

  57. Rajalakshmi N, Padma Subramanian D, Thamizhavel K (2015) Performance enhancement of radial distributed system with distributed generators by reconfiguration using binary firefly algorithm. J Inst Eng (India): B 96(1):91–99

    Google Scholar 

  58. Rashedi E, Nezamabadi-Pour H, Saryazdi S (2009) Gsa: a gravitational search algorithm. Inf Sci 179 (13):2232–2248

    Article  MATH  Google Scholar 

  59. Robinson D (2005) Reliability analysis of bulk power systems using swarm intelligence. In: IEEE, pp 96–102

  60. Saremi S, Mirjalili S, Lewis A (2015) How important is a transfer function in discrete heuristic algorithms. Neural Comput Applic 26(3):625–640

    Article  Google Scholar 

  61. Shi Y et al (2001) Particle swarm optimization: developments, applications and resources Proceedings of the 2001 congress on evolutionary computation, 2001, vol 1, pp 81–86

  62. Shih W (1979) A branch and bound method for the multiconstraint zero-one knapsack problem. J Oper Res Soc 30(4):369–378

    Article  MATH  Google Scholar 

  63. Shuyuan Y, Min W, Licheng J (2004) A quantum particle swarm optimization. IEEE Congress Evol Comput 1:19–23

    Google Scholar 

  64. Simon J, Apte A, Regnier E (2016) An application of the multiple knapsack problem: The self-sufficient marine. Eur J Oper Res

  65. Soto R, Crawford B, Olivares R, Barraza J, Johnson F, Paredes F (2015) A binary cuckoo search algorithm for solving the set covering problem Bioinspired computation in artificial systems. Springer, pp 88–97

  66. Swagatam D, Rohan M, Rupam K (2013) Multi-user detection in multi-carrier cdma wireless broadband system using a binary adaptive differential evolution algorithm. In: Proceedings of the 15th annual conference on genetic and evolutionary computation, GECCO, pp 1245–1252

  67. Thaker NG et al (2016) Radar charts show value of prostate cancer treatment options. Pharmaco Econ Outcomes News 762:33–24

  68. Totonchi A, Reza M (2008) Magnetic optimization algorithms, a new synthesis IEEE international conference on evolutionary computations

    Google Scholar 

  69. Wang I, Zhang Y, Zhou Y (2008) Discrete quantum-behaved particle swarm optimization based on estimation of distribution for combinatorial optimization. In: IEEE evolutionary computation, pp 897–904

  70. Wang L, Zheng X-L, Wang S-Y (2013) A novel binary fruit fly optimization algorithm for solving the multidimensional knapsack problem. Knowl-Based Syst 48:17–23

    Article  Google Scholar 

  71. Weingartner HM (1963) Mathematical programming and the analysis of capital budgeting problems. Markham Pub. Co.

  72. Weingartner MH, Ness DN (1967) Methods for the solution of the multidimensional 0/1 knapsack problem. Oper Res 15(1):83–103

    Article  Google Scholar 

  73. Yang X-S (2009) Firefly algorithms for multimodal optimization International symposium on stochastic algorithms. Springer, pp 169–178

  74. Yang X-S (2010) A new metaheuristic bat-inspired algorithm Nature inspired cooperative strategies for optimization (NICSO 2010). Springer, pp 65–74

  75. Yang X-S, Deb S (2009) Cuckoo search via lévy flights World congress on nature & biologically inspired computing, 2009. naBIC 2009, pp 210–214

    Chapter  Google Scholar 

  76. Yang Y, Yi M, Yang P, Jiang Y (2013) The unit commitment problem based on an improved firefly and particle swarm optimization hybrid algorithm Chinese automation congress (CAC), 2013, pp 718–722

    Chapter  Google Scholar 

  77. Zakaria D, Chaker D (2015) Binary bat algorithm: on the efficiency of mapping functions when handling binary problems using continuous-variable-based metaheuristics. Comput Sci Appl 456:3–14

    Google Scholar 

  78. Zhang B, Pan Q-K, Zhang X-L, Duan P-Y (2015) An effective hybrid harmony search-based algorithm for solving multidimensional knapsack problems. Appl Soft Comput 29: 288–297

    Article  Google Scholar 

  79. Zhang G (2011) Quantum-inspired evolutionary algorithms: a survey and empirical study. J Heuristics 17 (3):303–351

    Article  MATH  Google Scholar 

  80. Zhang X, Changzhi W, Li J, Wang X, Yang Z, Lee J-M, Jung K-H (2016) Binary artificial algae algorithm for multidimensional knapsack problems. Appl Soft Comput 43:583–595

    Article  Google Scholar 

  81. Zhao J, Sun J, Wenbo X (2005) A binary quantum-behaved particle swarm optimization algorithm with cooperative approach. Int J Comput Sci 10(2):112–118

    Google Scholar 

  82. Zhifeng W, Houkuan H, Xiang11 Z (2008) A binary-encoding differential evolution algorithm for agent coalition. J Comput Res Dev 5:019

  83. Zhou Y, Bao Z, Luo Q, Zhang S (2016) A complex-valued encoding wind driven optimization for the 0-1 knapsack problem. Appl Intell, 1–19

  84. Zhou Y, Chen X, Zhou G (2016) An improved monkey algorithm for a 0-1 knapsack problem. Appl Soft Comput 38:817–830

    Article  Google Scholar 

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Acknowledgments

Broderick Crawford is supported by grant CONICYT/FONDECYT/REGULAR 1171243, Ricardo Soto is supported by Grant CONICYT /FONDECYT /REGULAR /1160455, and José García is supported by INF-PUCV 2016.

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Authors and Affiliations

  1. Telefónica I+D, Av. Manuel Montt 1404, Third Floor, Providencia, Santiago, Chile

    José García

  2. Pontificia Universidad Católica de Valparaíso, 2362807, Valparaíso, Chile

    José García, Broderick Crawford & Ricardo Soto

  3. Universidad Técnica Federico Santa María, Valparaíso, Chile

    Carlos Castro

  4. Escuela de Ingeniería Industrial, Universidad Diego Portales, Santiago, Chile

    Fernando Paredes

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  1. José García
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Correspondence to José García.

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García, J., Crawford, B., Soto, R. et al. A k-means binarization framework applied to multidimensional knapsack problem. Appl Intell 48, 357–380 (2018). https://doi.org/10.1007/s10489-017-0972-6

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