Skip to main content
SpringerLink
Log in

The Role of Constrained Fuzzy Arithmetic in Engineering

  • Chapter
Uncertainty Analysis in Engineering and Sciences: Fuzzy Logic, Statistics, and Neural Network Approach

Part of the book series: International Series in Intelligent Technologies ((ISIT,volume 11))

Abstract

When fuzzy arithmetic is employed for dealing with fuzzy systems, which are viewed as systems of linguistic variables, it is essential to take into account all information regarding the linguistic variables involved. It is argued that the standard fuzzy arithmetic does not utilize some of the information available. As a consequence, it may produce results that are more imprecise than necessary or, possibly, even incorrect. Basic ideas of a revised fuzzy arithmetic, which takes all available information into account in terms of relevant requisite constrains, are discussed and illustrated by some examples, focusing particularly on those pertaining to the various areas of engineering.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover 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

  • Alefeld. G. and J. Herzberger [1983], Introduction to Interval Computation, Academic Press. New York.

    Google Scholar 

  • Bowles, J. S. and C. Pelaez [1995], “Application of fuzzy logic to reliability engineering.” Proc. of IEEE, 83(3). pp. 435–449.

    Article  Google Scholar 

  • Buckley. J. J. [1992]. “Solving fuzzy equations.” Fuzzy Sets and Systems, 50(1), pp. 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai. K. Y.. C. Y. Wen and M. L. Zhang [1991]. “Fuzzy reliability modeling of gracefully degradabie computing systems.” Reliability Eng. and Systems Safety, 33, pp. 141–157.

    Article  Google Scholar 

  • Chen, S. [1994], “Fuzzy system reliability analysis using fuzzy number arithmetic operations.” Fuzzy Sets and Systems. 64(1). pp. 31–38.

    Article  MathSciNet  Google Scholar 

  • Chen, S. [1996a], “A new method for evaluating weapon systems using fuzzy set theory.” IEEE Trans. on Systems. Man. and Cybernetics (Part A), 26(4), pp. 493–497.

    Article  Google Scholar 

  • Chen, S. [1996b]. “A fuzzy reasoning approach for rule-based systems based on fuzzy logics.” IEEE Trans, on Systems, Man, and Cybernetics (Part B), 26(5), pp. 769–778.

    Article  Google Scholar 

  • Chen, S. [1996c], “New method for fuzzy system reliability.” Cybernetics and Systems. 27(4), pp. 385–401.

    Article  MATH  Google Scholar 

  • Dijkman, J. G.,. and S. I. De Lange [1983], “Fuzzy numbers.” J. of Math. Analysis and Applications, 92, pp. 301–341.

    Article  MATH  Google Scholar 

  • Dong, W. M, H. C. Shah and F. S. Wong [1985], “Fuzzy computations in risk and decision analysis.” Civil Engineering Systems, 3, pp. 201–208.

    Article  Google Scholar 

  • Dong, W. M. and F. S. Wong [1987], “Fuzzy weighted averages and implementation of the extension principle.” Fuzzy Sets and Systems, 21(2), pp. 183–199.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. [1987], “An application of fuzzy arithmetric to the optimization of industrial machining processes.” Mathematical Modelling, 9(6), pp. 461–475.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and H. Prade [1978], “Operations on fuzzy numbers.” Intern. J. of Systems Science, 9(6), pp. 613–626.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and H. Prade [1979], “Fuzzy real algebra: Some results.” Fuzzy Sets and Systems, 2(4), pp. 327–348.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and H. Prade [1981], “Additions of interactive fuzzy numbers.” IEEE Trans. on Automatic Control, 26, pp. 926–936.

    Article  MathSciNet  Google Scholar 

  • Dubois, D, and H. Prade [1983], “Inverse operations for fuzzy numbers.” In: Sanchez, E., ed., Proc. 1FAC Symp. on Fuzzy Information, Knowledge Representation and Decision Analysis. Pergamon Press, Oxford, pp. 399–404.

    Google Scholar 

  • Person, S. [1996], “Reliable calculation in probabilistic logic: Accounting for small sample size and model uncertainty.” Proc. of 1996 Intern. Multidisciplinary Conf. on Intelligent Systems: A Semiotic Perspective, Gaithersburg, Maryland, Oct. 20-23, pp. 115–121.

    Google Scholar 

  • Hansen. E. R. [1992], Global Optimizaton Using Interval Analysis. Marcel Dekker, New York.

    Google Scholar 

  • Jain, R. [1976], “Tolerance analysis using fuzzy sets.” Intern. J, of Systems Science, 7(12), pp. 1393–1401.

    Article  MATH  Google Scholar 

  • Kaufmann, A. and M. M. Gupta [1985], Introduction to Fuzzy Arithmetic: Theory and Applications. Van Nostrand, New York.

    MATH  Google Scholar 

  • Kearfott, R. B. and V. Kreinovich, eds., [1996], Applications of Interval Computation. Kluwer, Boston.

    Google Scholar 

  • Klir, G. J. [1995], “Principles of uncertainty: What are they? Why do we need them?” Fuzzy Sets and Systems, 74(1), pp. 15–31.

    Article  Google Scholar 

  • Klir, G. J. and B. Yuan [1995], Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, Ml

    MATH  Google Scholar 

  • Kreinovich, V. [1996], “Interval methods in knowledge representation (Abstracts of recent papers).” Intern. J. of Uncertainty, Fuzziness and knowledge-Based Systems, 4(4&5), pp. 391–393,467-490.

    Google Scholar 

  • Law, C. [1996], “Using fuzzy numbers in educational grading systems.” Fuzzy Sets and Systems, 88(3), pp. 311–323.

    Article  Google Scholar 

  • Liou, T. and M. J. Wang [1994], “Subjective assessment of mental workload — A fuzzy linguistic multi-criteria approach.” Fuzzy Sets and Systems, 62(2), pp. 155–165.

    Article  Google Scholar 

  • Mares, M. [1977], “How to handle fuzzy quantities?” Kybernetika, 13(1), pp. 23–40.

    MathSciNet  Google Scholar 

  • Mares. M. [1994], Computation Over Fuzzy Quantities. CRC Press, Boca Ralon. Florida.

    MATH  Google Scholar 

  • Mares. M. and J. Horak [1983], “Fuzzy quantities in networks.” Fuzzy Sets and Systems. 10(2). pp. 123–134.

    Article  MATH  Google Scholar 

  • Mizumoto, M and K. Tanaka [1979]. “Some properties of fuzzy numbers.” In: Gupta. M. M. and R. K.

    Google Scholar 

  • Moore, R. E. [1966]. Interval Analysis. Prentice Hall. Englewood Cliffs. NJ.

    MATH  Google Scholar 

  • Moore. R. E. [1979]. Methods and Applications of Interval Analysis. S1AM. Philadelphia.

    Book  MATH  Google Scholar 

  • Neumaier, A. [1990], Interval Methods for Systems of Equations. Cambridge University Press. Cambridge and New York.

    MATH  Google Scholar 

  • Onisawa, T. and J. Kacprzyk. eds.. [1995], Reliability and Safety Analyses under Fu/.ziness. hysica–Verlag. Heidelberg. Germany.

    Google Scholar 

  • Otto, K. N. and E. K. Antonsson [1991]. “Trade-off strategies in engineering design.” Research in Engineering Design. 3(2). pp. 87–104.

    Article  Google Scholar 

  • Otto. K. N.. A. D. Lewis and E. K. Antonsson [1993]. “Determining optimal points of membership with dependent variables.” Fuzzy Sets and Systems, 60(1). pp. 19–24.

    Article  MathSciNet  Google Scholar 

  • Pan. Y. and G. J. Klir [1997], “Bayesian inference based on interval-valued prior distributions and likelihoods.” Intern. J. of Intelligent and Fuzzy Systems, (in production)

    Google Scholar 

  • Pan, Y.. G. J. Klir and B. Yuan [1996, “Bayesian inference based on fuzzy probabilities.” Proc. Fifth IEEE Intern. Conf. on Fuzzy Systems, New Orleans, Sept. 8-11. pp. 1693–1699.

    Google Scholar 

  • Pan, Y. and B. Yuan [1997], “Bayesian inference of fuzzy probabilities.” Intern. J. of General Systems, 26(in production)

    Google Scholar 

  • Pcdrycz, W. [1994]. “Why triangular membership functions?” Fuzzy Sets and Systems, 64(1). pp. 21–30.

    Article  MathSciNet  Google Scholar 

  • Sanchez. E. [1984]. “Solution of fuzzy equations with extended operations.” Fuzzy Sets and Systems. 12(3). pp. 237–248.

    Article  MathSciNet  MATH  Google Scholar 

  • Sanchez, E. [1993], “Non standard fuzzy arithmetic.” In: Wang, P. Z. and K. F. Loe, eds.. Between Mind and Computer. World Scientific, Singapore, pp. 271–282.

    Google Scholar 

  • Schmucker. K. J. [1984]. Fuzzy Sets. Natural Language Computations, and Risk Analysis. Computer Science Press, Rockville. Md.

    MATH  Google Scholar 

  • Sebastian. H. and E. K. Antonsson, eds., [1996], Fuzzy Sets in Engineering Design and Configuration. Kluwer, Boston.

    MATH  Google Scholar 

  • Singer, D. [1990]. “A fuzzy set approach to fault tree and reliability analysis.” Fuzzy Sets and Systems, 34(2), pp. 145–155.

    Article  Google Scholar 

  • Tee. A. B., M. B. Bowman and K. C. Sinha [1988], “A fuzzy mathematical approach for bridge condition evaluation.” Civil Eng. Systems, 5(1), pp. 17–24.

    Article  Google Scholar 

  • Viertl, R. [1996], Statistical Methods for “Non-Precise Data, CRC Press, Boca Raton, Florida.

    Google Scholar 

  • Warmerdam, J. M and T. L. Jacobs [1994], “Fuzzy set approach for routing and siting hazardous waste operations.” Information Sciences: Applications, 2(1), pp. 1–14.

    Article  MATH  Google Scholar 

  • Wood, K. L., K. N. Otto and E. K. Antonsson [1992], “Engineering design calculations with fuzzy parameters.” Fuzzy Sets and Systems, 52(1), pp. 1–20.

    Article  Google Scholar 

  • Yager. R. R, [1980], “On the lack of inverses in fuzzy arithmetic.” Fuzzy Sets and Systems. 4(1), pp. 73–82.

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, H. Q., H. Yao and D. Jones [1993], “Calculating functions of fuzzy numbers.” Fuzzy Sets and Systems, 55(3), pp. 273–283.

    Article  MathSciNet  Google Scholar 

  • Zadeh, L. A. [1975], “The concept of a linguistic variable and its application to approximate reasoning I, II, III.” Information Sciences, 8, pp. 199–251, 301-357; 9, pp. 43-80.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. George J. Klir
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media New York

About this chapter

Cite this chapter

Klir, G.J. (1998). The Role of Constrained Fuzzy Arithmetic in Engineering. In: Uncertainty Analysis in Engineering and Sciences: Fuzzy Logic, Statistics, and Neural Network Approach . International Series in Intelligent Technologies, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5473-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-5473-8_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7500-5

  • Online ISBN: 978-1-4615-5473-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation