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A new hybrid combinatorial genetic algorithm for multidimensional knapsack problems

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Abstract

Multidimensional knapsack problem (MKP) is known to be a NP-hard problem, more specifically a NP-complete problem, which cannot be resolved in polynomial time up to now. MKP can be applicable in many management, industry and engineering fields, such as cargo loading, capital budgeting and resource allocation, etc. In this article, using a combinational permutation constructed by the convex combinatorial value \(M_j=(1-\lambda ) u_j+ \lambda x^\mathrm{LP}_j\) of both the pseudo-utility ratios of MKP and the optimal solution \(x^\mathrm{LP}\) of relaxed LP, we present a new hybrid combinatorial genetic algorithm (HCGA) to address multidimensional knapsack problems. Comparing to Chu’s GA (J Heuristics 4:63–86, 1998), empirical results show that our new heuristic algorithm HCGA obtains better solutions over 270 standard test problem instances.

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References

  1. Vasquez M, Hao JK (2001) A logic-constrained knapsack formulation and a tabu algorithm for the daily photograph scheduling of an earth observation satellite. Comput Optim Appl 20:137–157

    Article  MathSciNet  MATH  Google Scholar 

  2. Vasquez M, Hao J-K (2001) A hybrid approach for the 0–1 multidimensional knapsack problem. In: Proceedings of the international joint conference on artificial intelligence, pp 328–333

  3. Vasquez M, Vimont Y (2005) Improved results on the 0–1 multidimensional knapsack problem. Eur J Oper Res 165:70–81

    Article  MathSciNet  MATH  Google Scholar 

  4. Hanafi S, Freville A (1998) An efficient tabu search approach for the 0–1 multidimensional knapsack problem. Eur J Oper Res 106(2–3):659–675

    Article  MATH  Google Scholar 

  5. Fréville A (2004) The multidimensional 0–1 knapsack problem: an overview. Eur J Oper Res 155:1–21

    Article  MATH  Google Scholar 

  6. Gottlieb J (2000) On the effectivity of evolutionary algorithms for the multidimensional knapsack problem. Lecture Notes in Computer Science vol 1829, pp 23–37

  7. Raidl GR, Gottlieb J (2005) Empirical analysis of locality, heritability and heuristic bias in evolutionary algorithms: a case study for the multidimensional knapsack problem. Evol Comput J 13(4):441–475

    Article  Google Scholar 

  8. Osorio MA, Glover F, Hammer P (2002) Cutting and surrogate constraint analysis for improved multidimensional knapsack solutions. Ann Oper Res 117(1):71–93

    Article  MathSciNet  MATH  Google Scholar 

  9. Fréville A, Hanafi S (2005) The multidimensional 0–1 knapsack problem bounds and computational aspects. Ann Oper Res 139:195–227

    Article  MathSciNet  MATH  Google Scholar 

  10. Dominique F, Ider T (2004) Global optimization and multi knapsack: a percolation algorithm. Eur J Oper Res 154(1):46–56

    Article  MATH  Google Scholar 

  11. Mahajan R, Chopra S (2012) Analysis of 0/1 knapsack problem using deterministic and probabilistic techniques. In: Second international conference on advanced computing and communication technologies (ACCT), pp 150–155

  12. Khan MHA (2013) An evolutionary algorithm with masked mutation for 0/1 knapsack problem. In: International conference on informatics, Electronics and vision (ICIEV), pp 1–6

  13. Thiongane B, Nagih A, Plateau G (2006) Lagrangean heuristics combined with reoptimization for the 0–1 bidimensional knapsack problem. Discrete Appl Math 154:2200–2211

    Article  MathSciNet  MATH  Google Scholar 

  14. Puchinger J, Raidl GR (2005) Relaxation guided variable neighborhood search. In Proceedings of the XVIII mini EURO conference on VNS, Tenerife

  15. Puchinger J, Raidl G (2008) Bringing order into the neighborhoods: relaxation guided variable neighborhood search. J Heuristics 14(5):457–472

    Article  MATH  Google Scholar 

  16. Løketangen A (2002) Heuristics for 0–1 mixed-integer programming. In: Pardalos PM, Resende MGC (eds) Handbook of applied optimization, Oxford University Press, Oxford, pp 474–477

  17. Løketangen A, Glover F (1998) Solving zero/one mixed integer programming problems usingtabu search. Eur J Oper Res 106:624–658

    Article  Google Scholar 

  18. Fischetti M, Lodi A (2003) Local branching. Mathematical Programming Series B 98:23–47

  19. Danna E, Rothberg E, Le Pape C (2005) Exploring relaxation induced neighborhoods to improve MIP solutions. Math Progr Ser A 102:71–90

    Article  MATH  Google Scholar 

  20. Magazine MJ, Oguz O (1984) A heuristic algorithm for the multidimensional zero-one knapsack problem. Eur J Oper Res 16:319–326

    Article  MathSciNet  MATH  Google Scholar 

  21. Martello S, Toth P (1990) Knapsack problems: algorithms and computer implementations. Wiley, New York

  22. Pirkul H (1987) A heuristic solution procedure for the multiconstrained zero-one knapsack problem. Naval Res Logist 34:161–172

    Article  MATH  Google Scholar 

  23. Volgenant A, Zoon JA (1990) An improved heuristic for multidimensional 0–1 knapsack problems. J Oper Res Soc 41:963–970

    Article  MATH  Google Scholar 

  24. Balas E, Martin CH (1980) Pivot and complement: a heuristic for 0–1 programming. Manag Sci 26(1):86–96

    Article  MathSciNet  MATH  Google Scholar 

  25. Hinterding R (1994) Mapping, order-independent genes and the knapsack problem. In: Fogel DB (ed) Proceedings of the 1st IEEE international conference on evolutionary computation, Orlando, pp 13–17

  26. Khuri S, Bäck T, Heitkötter J (1994) The zero/one multiple knapsack problem and genetic algorithms. In: Proceedings of the 1994 ACM symposium on applied computing, pp 188–193

  27. Olsen AL (1994) Penalty functions and the knapsack problem. In: Fogel DB (ed) Proceedings of the 1st international conference on evolutionary computation, Orlando, pp 559–564

  28. Rudolph G, Sprave J (1996) Significance of locality and selection pressure in the grand deluge evolutionary algorithm. In: Proceedings of the international conference on parallel problem solving from nature IV, pp 686–694

  29. Thiel J, Voss S (1994) Some experiences on solving multiconstraint zero-one knapsack problems with genetic algorithms. INFOR 32:226–242

    MATH  Google Scholar 

  30. Chu PC (1997) A genetic algorithm approach for combinatorial optimization problems, Ph.D. thesis at The Management School, Imperial College of Science, London

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

    Article  MATH  Google Scholar 

  32. Cotta C, Troya JM (1998) A hybrid genetic algorithm for the 0–1 multiple knapsack problem. Artif Neural Nets Genet Algorithms 3:251–255

    Google Scholar 

  33. Sun Y, Wang Z (1994) The genetic algorithm for 0–1 programming with linear constraints. In: Fogel DB (ed) Proceedings of the 1st ICEC94, Orlando, pp 559–564

  34. Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, Berlin

  35. Bäck T, Fogel D, Michalewicz Z (1997) Handbook of evolutionary computation. Oxford University Press, Oxford

  36. Raidl GR (1998) An improved genetic algorithm for the multiconstrained 0–1 knapsack problem. In: Proceedings of the 5th IEEE international conference on evolutionary computation, pp 207–211

  37. Moraga RJ, DePuy GW, Whitehouse GE (2005) Meta-RaPS approach for the 0–1 multidimensional knapsack problem. Comput Ind Eng 48(1):83–96

    Article  Google Scholar 

  38. Bertsimas D, Demir R (2002) An approximate dynamic programming approach to multidimensional knapsack problem. Manag Sci 48(4):550–565

    Article  MATH  Google Scholar 

  39. Li H, Jiao Y-C, Zhang L (2006) Genetic algorithm based on the orthogonal design for multidimensional knapsack problems. Lecture Notes in Computer Science vol 4221, pp 696–705

  40. Alonso C, Caro F, Montana JL (2006) A flipping local search genetic algorithm for the multidimensional 0–1 knapsack problem. Lecture Notes in Artificial Intelligence vol 4177, pp 21–30

  41. Akcay Y, Li HJ, Xu SH (2007) Greedy algorithm for the general multidimensional knapsack problem. Ann Oper Res 150(1):17–29

    Article  MathSciNet  MATH  Google Scholar 

  42. Kong M, Tian P, kao Y (2008) A new ant colony optimization algorithm for the multidimensional knapsack problem. Comput Oper Res 35(8):2672–2683

    Article  MathSciNet  MATH  Google Scholar 

  43. Leguizamon G, Michalewicz Z (1999) A new version of ant system for subset problem. Congr Evol Comput 14:59–64

    Google Scholar 

  44. Fidanova S (2002) Evolutionary algorithm for multidimensional knapsack problem. PPSNVII-Workshop

  45. Alaya I, Solnon C, Ghéira K (2004) Ant algorithm for the multi-dimensional knapsack problem. In: International conference on bioinspired optimization methods and their applications (BIOMA 2004), p 63C72

  46. Hanafi S, Glover F (2007) Exploiting nested inequalities and surrogate constraints. Eur J Oper Res 179(1):50–63

    Article  MathSciNet  MATH  Google Scholar 

  47. Kong M, Tian P (2006) Apply the particle swarm optimization to the multidimensional knapsack problem. Lecture Notes in Computer Science vol 4029, pp 1140–1149

  48. Hembecker F, Lopes H, Godoy W Jr (2007) Particle swarm optimization for the multidimensional knapsack problem. Lecture Notes in Computer Science vol 4431, pp 358–365

  49. Ke L, Feng Z, Ren Z, Wei X (2010) An ant colony optimization approach for the multidimensional knapsack problem. J Heuristics 16(1):65–83

    Article  MATH  Google Scholar 

  50. Hanafi S, Wilbaut C (2008) Scatter search for the 0–1 multidimensional knapsack problem. J Math Modell Algorithms 7(2):143–159

    Article  MathSciNet  MATH  Google Scholar 

  51. Montana J, Alonso C, Cagnoni S, Callau M (2008) Computing surrogate constraints for multidimensional knapsack problems using evolution strategies. Lecture Notes in Computer Science vol 4974, pp 555–564

  52. Marsten RE, Morin TL (1978) A hybrid approach to discrete mathematical programming. Math Progr 14:21–40

    Article  MathSciNet  MATH  Google Scholar 

  53. Gallardo JE, Cotta C, Fernandez AJ (2005) Solving the multidimensional knapsack problem using an evolutionary algorithm hybridized with branch and bound. Lecture Notes in Computer Science vol 3562, pp 21–30

  54. Puchinger J, Raidl G, Pferschy U (2006) The core concept for the multidimensional knapsack problem. Lecture Notes in Computer Science vol 3906, pp 195–208

  55. Balev S, Yanev N, Fréville A, Andonov R (2008) A dynamic programming based reduction procedure for the multidimensional 0–1 knapsack problem. Eur J Oper Res 186(1):63–76

    Article  MATH  Google Scholar 

  56. Boyer V, Elkihel M, Elbaz D (2009) Heuristics for the 0–1 multidimensional knapsack problem. Eur J Oper Res 199(3):658–664

    Article  MathSciNet  MATH  Google Scholar 

  57. Kaparis K, Letchford AN (2008) Local and global lifted cover inequalities for the 0–1 multidimensional knapsack problem. Eur J Oper Res 186(1):91–103

    Article  MathSciNet  MATH  Google Scholar 

  58. Vimont Y, Boussier S, Vasquez M (2008) Reduced costs propagation in an efficient implicit enumeration for the 0–1 multidimensional knapsack problem. J Comb Optim 15(2):165–178

    Article  MathSciNet  MATH  Google Scholar 

  59. Bibliography (2004) For further detail, please visit our website http://www.joics.com

  60. Wang RH (1999) Numerical approximation. Higher Education Press, Beijing

    Google Scholar 

  61. Fusiello A (2000) Uncalibrated euclidean reconstruction: a review. Image Vis Comput 18:555–563

    Article  Google Scholar 

  62. Provot X (1995) Deformation constraints in a mass-spring model to describe rigid cloth behavior. In: Proceedings of graphics interface ’95, pp 147–154

  63. Sun Y (2002) Space deformation with geometric constraint, M.S. Thesis, Department of Applied Mathematics, Dalian University of Technology

  64. Knuth DE (1996) The TEXbook, Addison-Welsey, New York

  65. Ortiz EL (1974) Canonical polynomials in the Lanczos tau-method. In: Scaife B (ed) Studies in numerical analysis. Academic Press, New York, pp 73–93

    Google Scholar 

  66. Egeblad J, Pisinger D (2009) Heuristic approaches for the two- and three-dimensional knapsack packing problem. Comput Oper Res 36:1026–1049

    Article  MathSciNet  MATH  Google Scholar 

  67. Djannaty F, Doostdar S (2008) A hybrid genetic algorithm for the multidimensional knapsack problem. Int J Contemp Math Sci 3(9):443–456

    MathSciNet  MATH  Google Scholar 

  68. Samanta S, Chakraborty S, Acharjee S, Mukherjee A, Dey N (2013) Solving 0/1 knapsack problem using ant weight lifting algorithm. In: IEEE international conference on computational intelligence and computing research (ICCIC), pp 1–5

  69. Htiouech S, Bouamama S, Attia R (2013) Using surrogate information to solve the multidimensional multi-choice knapsack problem. In: IEEE congress on evolutionary computation (CEC), pp 2102–2107

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Acknowledgments

This work is partially supported by the Project of Department of Education of Guangdong Province (No. 2013KJCX0128), the Nature Science Foundation of Guangdong Province (No. 10152104101000004 and No. S2013010013101), and the Foundation of Hanshan Normal University (Grant No. QD20131101). The authors would like to thank the anonymous referees for valuable comments and suggestions, which helped a lot in improving the quality of this paper.

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Authors and Affiliations

  1. Computer Engineering Technical College, Guangdong Institute of Science and Technology, Zhuhai,  519090, China

    Guoming Lai

  2. Department of Mathematics, Hanshan Normal University, Chaozhou , 521041, Guangdong, China

    Dehui Yuan

  3. Department of Computer Science and Engineering, Hanshan Normal University, Chaozhou,  521041, China

    Shenyun Yang

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  1. Guoming Lai
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Correspondence to Shenyun Yang.

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Lai, G., Yuan, D. & Yang, S. A new hybrid combinatorial genetic algorithm for multidimensional knapsack problems. J Supercomput 70, 930–945 (2014). https://doi.org/10.1007/s11227-014-1268-9

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