Skip to main content
SpringerLink
Log in

Methods of Tropical Optimization in Rating Alternatives Based on Pairwise Comparisons

  • Conference paper
  • First Online:
Operations Research Proceedings 2016

Part of the book series: Operations Research Proceedings ((ORP))

Abstract

We apply methods of tropical optimization to handle problems of rating alternatives on the basis of the log-Chebyshev approximation of pairwise comparison matrices. We derive a direct solution in a closed form, and investigate the obtained solution when it is not unique. Provided the approximation problem yields a set of score vectors, rather than a unique (up to a constant factor) one, we find those vectors in the set, which least and most differentiate between the alternatives with the highest and lowest scores, and thus can be representative of the entire solution.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barzilai, J.: Deriving weights from pairwise comparison matrices. J. Oper. Res. Soc. 48(12), 1226–1232 (1997)

    Article  Google Scholar 

  2. Chu, M.T.: On the optimal consistent approximation to pairwise comparison matrices. Linear Algebra Appl. 272(1–3), 155–168 (1998)

    Article  Google Scholar 

  3. Farkas, A., Lancaster, P., Rózsa, P.: Consistency adjustments for pairwise comparison matrices. Numer. Linear Algebra Appl. 10(8), 689–700 (2003)

    Article  Google Scholar 

  4. Golan, J.S.: Semirings and Affine Equations Over Them, Mathematics and Its Applications, vol. 556. Springer, New York (2003)

    Book  Google Scholar 

  5. González-Pachón, J., Rodríguez-Galiano, M.I., Romero, C.: Transitive approximation to pairwise comparison matrices by using interval goal programming. J. Oper. Res. Soc. 54(5), 532–538 (2003)

    Article  Google Scholar 

  6. Heidergott, B., Olsder, G.J., van der Woude, J.: Max Plus at Work. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2006)

    Google Scholar 

  7. Krivulin, N.: Extremal properties of tropical eigenvalues and solutions to tropical optimization problems. Linear Algebra Appl. 468, 211–232 (2015)

    Article  Google Scholar 

  8. Krivulin, N.: Rating alternatives from pairwise comparisons by solving tropical optimization problems. In: Tang, Z., Du, J., Yin, S., He, L., Li, R. (eds.) 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), pp. 162–167. IEEE (2015)

    Google Scholar 

  9. Krivulin, N.: Solving a tropical optimization problem via matrix sparsification. In: Kahl, W., Winter, M., Oliveira, J.N. (eds) Relational and Algebraic Methods in Computer Science, Lecture Notes in Computer Science, vol. 9348, pp. 326–343. Springer, Cham (2015)

    Google Scholar 

  10. Krivulin, N.: A maximization problem in tropical mathematics: a complete solution and application examples. Informatica 27(3), 587–606 (2016)

    Article  Google Scholar 

  11. Krivulin, N.: Using tropical optimization techniques to evaluate alternatives via pairwise comparisons. In: Gebremedhin, A.H., Boman, E.G., Ucar, B. (eds.) 2016 Proceedings of the 7th SIAM Workshop on Combinatorial Scientific Computing, pp. 62–72. SIAM, Philadelphia (2016)

    Google Scholar 

  12. Saaty, T.L.: On the measurement of intangibles: a principal eigenvector approach to relative measurement derived from paired comparisons. Not. Am. Math. Soc. 60(2), 192–208 (2013)

    Article  Google Scholar 

  13. Saaty, T.L., Vargas, L.G.: Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Math. Model. 5(5), 309–324 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Russian Foundation for Humanities (grant No. 16-02-00059). The author is very grateful to the referees for their valuable comments and suggestions, which have been incorporated into the revised version of the manuscript.

Author information

Authors and Affiliations

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

    Nikolai Krivulin

Authors
  1. Nikolai Krivulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolai Krivulin .

Editor information

Editors and Affiliations

  1. Institute of Computer Science, Helmut Schmidt University, Institute of Computer Science, Hamburg, Germany

    Andreas Fink

  2. Applied Mathematics, Helmut Schmidt University, Applied Mathematics, Hamburg, Germany

    Armin Fügenschuh

  3. Institute of Logistics and Organization, Helmut Schmidt University, Institute of Logistics and Organization, Hamburg, Germany

    Martin Josef Geiger

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Cite this paper

Krivulin, N. (2018). Methods of Tropical Optimization in Rating Alternatives Based on Pairwise Comparisons. In: Fink, A., Fügenschuh, A., Geiger, M. (eds) Operations Research Proceedings 2016. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-55702-1_13

Download citation

Publish with us

Policies and ethics

Search

Navigation