Skip to main content
SpringerLink
Log in

Toward Eigenvalues and Eigenvectors of Matrices of Z-Numbers

  • Conference paper
  • First Online:
10th International Conference on Theory and Application of Soft Computing, Computing with Words and Perceptions - ICSCCW-2019 (ICSCCW 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1095))

Abstract

Eigenvalues and eigenvectors are widely used in various practical applications in decision making, planning, control and other fields. Particularly, these concepts underlie analysis of consistency of a decision maker’s (DM) knowledge. In real-world problems, DM’s knowledge is naturally characterized by imprecision and partial reliability. This involves combination of fuzzy and probabilistic information. The concept of a Z-number is a formal construct to describe such kind of information. In this paper we initiate study of Z-number valued eigenvalue and eigenvector of matrices, components of which are Z-numbers. A statement of problem and a solution approach for computation of Z-number valued eigensolutions are proposed. An example is provided to prove validity of the proposed approach.

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. Aliev, R.A., Pedrycz, W.: Fundamentals of a fuzzy-logic-based generalized theory of stability. IEEE Trans. Syst. Man, Cybern. Part B (Cybern.) 39(4), 971–988 (2009)

    Article  Google Scholar 

  2. Aliev, R., Tserkovny, A.: Systemic approach to fuzzy logic formalization for approximate reasoning. Inf. Sci. 181(6), 1045–1059 (2011)

    Article  MathSciNet  Google Scholar 

  3. Edelman, A., Rao, N.R.: Random matrix theory. Acta Numer. 14, 233–297 (2005)

    Article  MathSciNet  Google Scholar 

  4. Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  5. Bhowmik, M., Pal, M.: Generalized intuitionistic fuzzy matrices. FarEast J. Math. Sci. 29(3), 533–554 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Gavalec, M.: Computing matrix period in max-min algebra. Discret. Appl. Math. 75, 63–70 (1997)

    Article  MathSciNet  Google Scholar 

  7. Hashimoto, H.: Canonical form of a transitive fuzzy matrix. Fuzzy Sets Syst. 11, 157–162 (1983)

    Article  MathSciNet  Google Scholar 

  8. Thomson, M.G.: Convergence of powers of a fuzzy matrix. J. Math. Anal. Appl. 57, 476–480 (1977)

    Article  MathSciNet  Google Scholar 

  9. Hashimoto, H.: Transitivity of fuzzy matrices under generalized connectedness. Fuzzy Sets Syst. 29(2), 229–234 (1989)

    Article  MathSciNet  Google Scholar 

  10. Hashimoto, H.: Transitivity of generalized fuzzy matrix. Fuzzy Sets Syst. 17, 83–90 (1985)

    Article  Google Scholar 

  11. Hemasinha, R., Pal, N.R., Bezdek, J.C.: Iterates of fuzzy circulant matrices. Fuzzy Sets Syst. 60, 199–206 (1993)

    Article  MathSciNet  Google Scholar 

  12. Khan, S.K., Pal, M.: Interval-valued intuitionistic fuzzy matrices. Notes Intuit. Fuzzy Sets 11(1), 16–27 (2005)

    Google Scholar 

  13. Behera, D., Huang, H.-Z., Chakraverty, S.: Solution of fully fuzzy system of linear equations by linear programming approach. Comput. Model. Eng. Sci. 108(2), 67–87 (2015)

    Google Scholar 

  14. Praščević, N., Praščević, Ž.: Application of fuzzy AHP method based on eigenvalues for decision making in construction industry. Tech. Gaz. 23, 57–64 (2016)

    Google Scholar 

  15. Allahviranloo, T., Salahshour, S., Khezerloo, M.: Maximal- and minimal symmetric solutions of fully fuzzy systems. J. Comput. Appl. Math. 235, 4652–4662 (2011)

    Article  MathSciNet  Google Scholar 

  16. Laarhoven, P.J.M.V., Pedrycz, W.: A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 11, 199–227 (1983)

    Article  MathSciNet  Google Scholar 

  17. Allahviranloo, T., Salahshour, S., Khezerloo, M.: Maximal and minimal symmetric solutions of fully fuzzy systems. J. Comput. Appl. Math. 235, 4652–4662 (2011)

    Article  MathSciNet  Google Scholar 

  18. Behera, D., Chakraverty, S.: Solution method for fuzzy system of linear equations with crisp coefficients. Fuzzy Inf. Eng. 5, 205–219 (2013)

    Article  MathSciNet  Google Scholar 

  19. Behera, D., Chakraverty, S.: New approach to solve fully fuzzy system of linear equations using single and double parametric form of fuzzy numbers. Sadhana 40, 35–49 (2015)

    Article  MathSciNet  Google Scholar 

  20. Ishizaka, A.: Comparison of fuzzy logic, AHP, FAHP and hybrid fuzzy AHP for new supplier selection and its performance analysis. Int. J. Integr. Supply Manag. 9, 1–22 (2014)

    Article  Google Scholar 

  21. Krejcí, J.: Fuzzy eigenvector method for obtaining normalized fuzzy weights from fuzzy pairwise comparison matrices. Fuzzy Sets Syst. 315, 26–43 (2017)

    Article  MathSciNet  Google Scholar 

  22. Zadeh, L.A.: A note on Z-numbers. Inform. Sci. 181, 2923–2932 (2011)

    Article  MathSciNet  Google Scholar 

  23. Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inform. Sci. 290, 134–155 (2015)

    Article  MathSciNet  Google Scholar 

  24. Aliev, R.A., Zeinalova, L.M.: Decision making under Z-information. In: Guo, P., Pedrycz, W. (eds.) Human-Centric Decision-Making Models for Social Sciences (Studies in Computational Intelligence), pp. 233–252. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  25. Aliev, R.A., Huseynov, O.H.: Decision Theory with Imperfect Information. World Scientific, Singapore (2014)

    Book  Google Scholar 

  26. Aliev, R.A., Huseynov, O.H., Aliyev, R.R., Alizadeh, A.V.: The Arithmetic of Z-Numbers: Theory and Applications. World Scientific, Singapore (2015)

    Book  Google Scholar 

  27. Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of continuous Z-numbers. Inform. Sci. 373, 441–460 (2016)

    Article  Google Scholar 

  28. Aliev, R.A., Perdycz, W., Huseynov, O.H.: Functions defined on a set of Z-numbers. Inform. Sci. 423, 353–375 (2018)

    Article  MathSciNet  Google Scholar 

  29. Aliev, R.A., Alizadeh, A.V., Huseynov, O.H., Jabbarova, K.I.: Z-number based linear programming. Int. J. Intell. Syst. 30, 563–589 (2015)

    Article  Google Scholar 

  30. Yager, R.R.: On a view of Zadeh’s Z-numbers. Adv. Comput. Intell. Commun. Comput. Inf. Sci. 299, 90–101 (2012)

    MATH  Google Scholar 

  31. Yager, R.: On Z-valuations using Zadeh’s Z-numbers. Int. J. Intell. Syst. 27, 259–278 (2012)

    Article  Google Scholar 

  32. Kang, B., Deng, Y., Hewage, K., Sadiq, R.: A method of measuring uncertainty for z-number. IEEE Trans. Fuzzy Syst. 27, 731–738 (2018)

    Article  Google Scholar 

  33. Kang, B., Chhipi-Shrestha, G., Deng, Y., Hewage, K., Sadiq, R.: Stable strategies analysis based on the utility of Z-number in the evolutionary games. Appl. Math. Comput. 324, 202–217 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Banerjee, R., Pal, S.K.: Z*-numbers: augmented Z-numbers for machine-subjectivity representation. Inf. Sci. 323, 143–178 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Joint MBA Program, Georgia State University, Atlanta, USA

    R. A. Aliev

  2. Azerbaijan State Oil and Industry University, 20 Azadlig Ave., AZ1010, Baku, Azerbaijan

    R. A. Aliev

  3. Research Laboratory of Intelligent Control and Decision Making Systems in Industry and Economics, Azerbaijan State Oil and Industry University, Baku, Azerbaijan

    O. H. Huseynov

  4. Department of Instrument Making Engineering, Azerbaijan State Oil and Industry University, 20 Azadlig Avenue, AZ1010, Baku, Azerbaijan

    K. R. Aliyeva

Authors
  1. R. A. Aliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. H. Huseynov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. R. Aliyeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. A. Aliev .

Editor information

Editors and Affiliations

  1. Joint MBA Program, Azerbaijan State Oil and Industry University, Baku, Azerbaijan

    Rafik A. Aliev

  2. Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

    Janusz Kacprzyk

  3. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

    Witold Pedrycz

  4. Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, TX, USA

    Mo Jamshidi

  5. Azerbaijan State Oil and Industry University, Baku, Azerbaijan

    Mustafa B. Babanli

  6. Department of Mechatronics, Near East University, Nicosia, Turkey

    Fahreddin M. Sadikoglu

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Aliev, R.A., Huseynov, O.H., Aliyeva, K.R. (2020). Toward Eigenvalues and Eigenvectors of Matrices of Z-Numbers. In: Aliev, R., Kacprzyk, J., Pedrycz, W., Jamshidi, M., Babanli, M., Sadikoglu, F. (eds) 10th International Conference on Theory and Application of Soft Computing, Computing with Words and Perceptions - ICSCCW-2019. ICSCCW 2019. Advances in Intelligent Systems and Computing, vol 1095. Springer, Cham. https://doi.org/10.1007/978-3-030-35249-3_39

Download citation

Publish with us

Policies and ethics

Search

Navigation