Skip to main content
SpringerLink
Log in

An Approximate Algorithm for Lexicographic Search in Multiple Orders for the Solution of the Multidimensional Boolean Knapsack Problem

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

A new scheme of approximate lexicographic search is proposed for the solution of the multidimensional Boolean knapsack problem. The main idea of the algorithm is to gradually define the lexicographic order (ordering of variables) in which “qualitative” solutions of the problem belong to a direct two-sided lexicographic constraint whose upper bound is the lexicographic maximum of the set of feasible solutions of the problem in this order. Since the search for “qualitative” solutions in each order is carried out on a bounded lexicographic interval, the proposed algorithm is called a bounded lexicographic search. The quality of the approximate method of bounded lexicographic search is investigated by solving test tasks from the well-known Beasley and Glover–Kochenberger sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. V. Sergienko and V. P. Shylo, Discrete Optimization Problems: Challenges, Solution Technics, Analysis [in Russian], Naukova Dumka, Kyiv (2003).

    Google Scholar 

  2. A. Fréville, “The multidimensional 0-1 knapsack problem: An overview,” European Journal of Operational Research, Vol. 155, 1–21 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  3. P. C. Chu and J. E. Beasley, “A genetic algorithm for the multidimensional knapsack problem,” Journal of Heuristics, Vol. 4, 63–86 (1998).

    Article  MATH  Google Scholar 

  4. F. Glover and G. A. Kochenberger, “Critical event tabu search for multidimensional knapsack problems,” in: Metaheuristics: The Theory and Applications, Academic Publishers, Kluwer (1996), pp. 407–427.

  5. V. P. Shilo, “Solution of multidimensional knapsack problems by the method of global equilibrium search,” in: Theory of Optimal Solutions [in Russian], V. M. Glushkov Inst. Cybern. of NASU, Kyiv (2000), pp. 10–13.

  6. M. Vasquez and J.-K. Hao, “A hybrid approach for the 0-1 multidimensional knapsack problem,” in: Proc. 17th Intern. Joint Conf. on Artificial Intelligence, Seattle (2001), pp. 328–333.

  7. M. Vasquez and Y. Vimont, “Improved results on the 0-1 multidimensional knapsack problem,” European Journal of Operational Research, Vol. 165, 70–81 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  8. Q. Yuan and Z. X. Yang, “A weight-coded evolutionary algorithm for the multidimensional knapsack problem,” Advances in Pure Mathematics, Vol. 6, 659–675 (2016).

    Article  Google Scholar 

  9. V. Boyer, M. Elkihel, and D. El Baz, “Heuristics for the 0-1 multidimensional knapsack problem,” European Journal of Operational Research, Vol. 199, 658–664 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. V. Chupov, “New approaches to solving discrete programming problems on the basis of lexicographic search,” Cybernetics and Systems Analysis, Vol. 52, No. 4, 536–545 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  11. S. V. Chupov, “The stochastic algorithm of lexicographic search of discrete programming problem,” Uzhhorod University Scientific Herald, Ser. Math. and Inform., Iss. 1 (28), 150–159 (2016).

  12. J. E. Beasley, “OR-library: Distributing test problems by electronic mail,” Journal of Operational Research Society, Vol. 41, 1069–1072 (1990).

    Article  Google Scholar 

  13. S. V. Chupov, “Stochastic algorithm for lexicographic search for the optimal solution of the Boolean knapsack problem,” in: Proc. Vth Intern. Sci.-Pract. Conf. “Mathematics. Information Technologies. Education,” Vezha-Print, Lutsk (2016), pp. 51–52.

  14. S. V. Chupov, “Modifications of the algorithm for searching for the lexicographic maximum of a set,” Uzhhorod University Scientific Herald, Ser. Math. and Inform., Iss. 2 (27), 168–173 (2015).

Download references

Author information

Authors and Affiliations

  1. Uzhhorod National University, Uzhhorod, Ukraine

    S. V. Chupov

Authors
  1. S. V. Chupov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Chupov.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2018, pp. 56–69.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chupov, S.V. An Approximate Algorithm for Lexicographic Search in Multiple Orders for the Solution of the Multidimensional Boolean Knapsack Problem. Cybern Syst Anal 54, 563–575 (2018). https://doi.org/10.1007/s10559-018-0057-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-018-0057-5

Keywords

Search

Navigation