Skip to main content
SpringerLink
Log in

An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem

  • Mathematics
  • Published:
Vestnik St. Petersburg University: Mathematics Aims and scope Submit manuscript

Abstract

An extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied. It is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components. The minimax problem of location of a single facility (the Rawls problem) on a plane with rectilinear distance is considered. For this problem, we give the corresponding representation in terms of idempotent algebra and suggest a new algebraic solution, which is based on the results of investigation of the extremal property of eigenvalue and reduces to finding the eigenvalue and eigenvectors of a certain matrix.

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. F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems (Wiley, Chichester, 1992).

    MATH  Google Scholar 

  2. V. N. Kolokoltsov and V. P. Maslov, Idempotent Analysis and Applications (Fizmatlit, Moscow, 1994; Kluwer, Dordrecht, 1997).

    Google Scholar 

  3. R. A. Cuninghame-Green, “Minimax algebra and applications,” in Advances in Imaging and Electron Physics, Vol. 90, Ed. by P. W. Hawkes (Academic, San Diego, 1994), pp. 1–121.

    Google Scholar 

  4. J. S. Golan, Semirings and Affine Wquations over them: Theory and Applications (Kluwer, Dordrecht, 2003).

    Google Scholar 

  5. B. Heidergott, G. J. Olsder, and J. van der Woude, Max Plus at Work. Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications (Princeton University Press, Princeton, 2005).

    Google Scholar 

  6. N. K. Krivulin, Methods of Idempotent Algebra in Problems of Modeling and Analysis of Complex Systems (St. Petersb. Univ., St. Petersburg, 2009).

    Google Scholar 

  7. P. Butkovi, Max-linear Systems: Theory and Algorithms (Springer, London, 2010).

    Book  Google Scholar 

  8. R. A. Cuninghame-Green, “Minimax Algebra and Applications,” Fuzzy Set. Syst. 41, 251–267 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Zimmermann, “Disjunctive Optimization, Max-separable Problems and Extremal Algebras,” Theoret. Comput. Sci., 293(1), 45–54 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Tharwat and K. Zimmermann, “One Class of Separable Optimization Problems: Solution Method, Application,” Optimization, 59(5), 619–625 (2008).

    Article  MathSciNet  Google Scholar 

  11. N. K. Krivulin, “Matrix Eigenvalues and Eigenvectors in Idempotent Algebra,” Vestn. St. Petersb. Univ., Ser. 1, Issue 2, 29–40 (2006).

  12. N. K. Krivulin, “Evaluation of the Mean Growth Rate of the State Vector in a Linear Dynamic Stochastic System in Idempotent Algebra,” Vestn. St. Petersb. Univ., Ser. 1, Issue 2, 45–54 (2005).

  13. J. Elzinga and D. W. Hearn, “Geometrical Solutions for Some Minimax Location Problems,” Transport. Sci., 6(4), 379–394 (1972).

    Article  MathSciNet  Google Scholar 

  14. R. L. Francis, “A Geometrical Solution Procedure for a Rectilinear Distance Minimax Location Problem,” IIE Trans. 4(4), 328–332 (1972).

    MathSciNet  Google Scholar 

  15. E. Moradi and M. Bidkhori, “Single Facility Location Problem,” in Facility Location: Concepts, Models, Algorithms and Case Studies, Ed. by R. Z. Farahani and M. Hekmatfar (Physica-Verlag, Heidelberg, 2009), pp. 37–68.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

    N. K. Krivulin

Authors
  1. N. K. Krivulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. K. Krivulin.

Additional information

Original Russian Text © N.K. Krivulin, 2011, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2011, No. 4, pp. 42–51.

About this article

Cite this article

Krivulin, N.K. An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem. Vestnik St.Petersb. Univ.Math. 44, 272–281 (2011). https://doi.org/10.3103/S1063454111040078

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1063454111040078

Keywords

Search

Navigation