Skip to main content
SpringerLink
Log in

On the Optimization of Fuzzy Relation Equations with Continuous t-Norm and with Linear Objective Function

  • Conference paper
Applied Computing (AACC 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3285))

Included in the following conference series:

Abstract

According to [8,12,13,23], the optimization models with a linear objective function subject to fuzzy relation equations is decidable. Algorithms are developed to solve it. In this paper, a complementary problem for the original problem is defined. Due to the structure of the feasible domain and nature of the objective function, individual variable is restricted to become bi-valued. We propose a procedure for separating the decision variables into basic and non-basic variables. An algorithm is proposed to determine the optimal solution. Two examples are considered to explain the procedure.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adamopoulos, G.I., Pappis, C.P.: Some results on the resolution of fuzzy relation equations. Fuzzy Sets and Systems 60, 83–88 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. De Baets, B.: Analytical solution methods for fuzzy relational equations. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets. The Handbook of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  3. Baets, B.D., Kerre, E.: A primer on solving fuzzy relational equations on the unit interval. Internal. J.Uncertain. Fuzziness Knowledge-Based Systems 2, 205–225 (1994)

    Article  Google Scholar 

  4. Bourke, M., Fisher, D.G.: Solution algorithms for fuzzy relational equations with max-product composition. Fuzzy Sets and Systems 94, 61–69 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng, L., Peng, B.: The fuzzy relation equation with union or intersection preserving operator. Fuzzy Sets and Systems 25, 19–204 (1988)

    MathSciNet  Google Scholar 

  6. Chung, F., Lee, T.: A new look at solving a system of fuzzy relational equations. Fuzzy Sets and Systems 88, 343–353 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Czogala, E., Drewniak, J., Pedrycz, W.: fuzzy relation equations on a finite set. Fuzzy Sets and Systems 7, 89–101 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fang, S.C., Li, G.: Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103, 107–113 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Guo, S.Z., Wang, P.Z., Nola, A.D., Sessa, S.: Further contributions to the study of finite relation equations. Fuzzy Sets and Systems 26, 93–104 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gupta, M.M., Qi, J.: Design of fuzzy logic controllers based on generalized T-operators. Fuzzy Sets Syst. 40, 473–489 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Higashi, M., Klir, G.J.: Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems 13, 64–82 (1984)

    Article  MathSciNet  Google Scholar 

  12. Loetamophong, J., Fang, S.C.: An Efficient Solution Procedure for Fuzzy Relation Equations with Max-Product composition. IEEE Tarns. Fuzzy systems 7, 441–445 (1999)

    Article  Google Scholar 

  13. Leotamonphong, J., Fang, S.C.: Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets and Systems 118, 509–517 (2001)

    Article  MathSciNet  Google Scholar 

  14. Li, G., Fang, S.C.: On the resolution of finite fuzzy relation equations, OR Report No.322, North Carolina State University, Raleigh, North Carolina (May 1996)

    Google Scholar 

  15. Lu, J., Fang, S.C.: Solving nonlinear optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 119, 1–20 (2001)

    Article  MathSciNet  Google Scholar 

  16. Di Nola, A.: Relational equations in totally ordered lattices and their complete resolution. J. Math. Anal. Appl. 107, 148–155 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Di Nola, A., Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equation and their Applications to Knowledge Engineering. Kluwer Academic Publisher, Dordrecht (1989)

    Google Scholar 

  18. Di Nola, A., Sessa, S., Pedrycz, W., Pei-Zhuang, W.: Fuzzy relation equations under a class of triangular norms: a survey and new results. Stochastica 8, 99–145 (1984)

    MATH  MathSciNet  Google Scholar 

  19. Pedrycz, W.: An identification algorithm in fuzzy relation systems. Fuzzy Sets Syst. 13, 153–167 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  20. Prevot, M.: Algorithm for the solution of fuzzy relations. Fuzzy Sets and Systems 5, 319–322 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sanchez, E.: Resolution of composite fuzzy relation equations. Information and Control 30, 38–48 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wu, W.: Fuzzy reasoning and fuzzy relational equations. Fuzzy Sets and Systems 20, 67–78 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wu, Y.-K., Guu, S.-M., Liu, J.Y.-C.: An Accelerated Approach for Solving Fuzzy Relation Equations With a Linear Objective Function. IEEE, Trans. Fuzzy Syst. 10(4), 552–558 (2002)

    Article  Google Scholar 

  24. Yager, R.R.: Some procedures for selecting fuzzy set-theoretic operations. Int. J. General Syst. 8, 235–242 (1982)

    Google Scholar 

  25. Zimmermann, H.J.: Fuzzy set theory and its applications. Kluwer Academic Publishers, Boston (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, D.E.I. Dayalbagh, Agra, 282005, India

    Dhaneshwar Pandey

Authors
  1. Dhaneshwar Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, The University of York, YO10 5NG, Heslington, York, UK

    Suresh Manandhar

  2. Advanced Computer Architectures Group, Department of Computer Science, University of York, York, UK

    Jim Austin

  3. Department of Electrical Engineering, Indian Institute of Technology, 400076, Powai, Bombay, India

    Uday Desai

  4. Department of Computer Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-8654, Tokyo, Japan

    Yoshio Oyanagi

  5. Indian Institute of Information Technology, Bangalore, 26/C, Electronics City, Hosur Road, 560100, Bangalore, India

    Asoke K. Talukder

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pandey, D. (2004). On the Optimization of Fuzzy Relation Equations with Continuous t-Norm and with Linear Objective Function. In: Manandhar, S., Austin, J., Desai, U., Oyanagi, Y., Talukder, A.K. (eds) Applied Computing. AACC 2004. Lecture Notes in Computer Science, vol 3285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30176-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-30176-9_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-23659-7

  • Online ISBN: 978-3-540-30176-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation