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Approaches to Linear Programming Problems with Interactive Fuzzy Numbers

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Fuzzy Optimization

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 254))

Abstract

Fuzzy programming has been developed mainly under the assumption of non-interaction among fuzzy coefficients. However, this assumption is not always suitable in the treatment of real world problems. Several approaches have been proposed to treat the interaction among fuzzy coefficients. In this paper, we review treatments of interaction among fuzzy coefficients in fuzzy linear programming problems. Using a necessity fractile model of a simple linear program with fuzzy coefficients, we will see the differences between non-interactive and interactive problems. We review the five approaches to interactive fuzzy numbers, i.e., weak independent fuzzy numbers, a fuzzy vector with a quadratic membership function, scenario decomposed fuzzy numbers, an oblique fuzzy vector, and a fuzzy polytope.

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References

  1. Hisdal, E.: Conditional possibilities independence and noninteraction. Fuzzy Sets and Systems 1(4), 283–297 (1978)

    Article  MATH  Google Scholar 

  2. Inuiguchi, M.: A necessity measure optimization approach to linear programming problems with oblique fuzzy vectors. Kybernetika 42(4), 441–452 (2006)

    MathSciNet  Google Scholar 

  3. Inuiguchi, M.: Necessity measure optimization in linear programming problems with fuzzy polytopes. Fuzzy Sets and Systems 158, 1882–1891 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Inuiguchi, M., Ramík, J.: Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems 111(1), 3–28 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Inuiguchi, M., Ramík, J., Tanino, T.: Oblique fuzzy vectors and their use in possibilistic linear programming. Fuzzy Sets and Systems 137(1), 123–150 (2003)

    Article  Google Scholar 

  6. Inuiguchi, M., Sakawa, M.: A possibilistic linear program is equivalent to a stochastic linear program in a special case. Fuzzy Sets and Systems 76, 309–318 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Inuiguchi, M., Tanino, T.: Portfolio selection under independent possibilistic information. Fuzzy Sets and Systems 115(1), 83–92 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Inuiguchi, M., Tanino, T.: Possibilistic linear programming with fuzzy if-then rule coefficients. Fuzzy Optimization and Decision Making 1(1), 65–91 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Inuiguchi, M., Tanino, T.: Fuzzy linear programming with interactive uncertain parameters. Reliable Computing 10(5), 357–367 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  11. Rommelfanger, H.: Fuzzy linear programming and applications. European Journal of Operational Research 92(3), 512–527 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rommelfanger, H.: The advantages of fuzzy optimization models in practical use. Fuzzy Optimization and Decision Making 3(4), 295–309 (2004)

    Article  MATH  Google Scholar 

  13. Rommelfanger, H., Kresztfalvi, T.: Multicriteria fuzzy optimization based on Yager’s parameterized t-norm. Foundation of Computing and Decision Sciences 16(2), 99–110 (1991)

    MATH  Google Scholar 

  14. Słowiński, R., Teghem, J. (eds.): Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty. Kluwer Academic Publishers, Dordrecht (1990)

    MATH  Google Scholar 

  15. Stancu-Minasian, I.M.: Stochastic Programming with Multiple Objective Functions. D. Reidel Publishing Company, Dordrecht (1984)

    MATH  Google Scholar 

  16. Tanaka, H., Ishibuchi, H.: Identification of possibilistic linear systems by quadratic membership function. Fuzzy Sets and Systems 41, 145–160 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, Part 1, 2, and 3. Information Sciences 8, 199–249; 8, 301–357; 9, 43–80 (1975)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Osaka, 560-8531, Japan

    Masahiro Inuiguchi

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  1. Masahiro Inuiguchi
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Editors and Affiliations

  1. Department of Mathemaitical and Statistical Sciences, University of Colorado Denver, PO Box 173364, 80217-3364, Colorado, Denver, USA

    Weldon A. Lodwick

  2. Systems Research Institute Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

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Inuiguchi, M. (2010). Approaches to Linear Programming Problems with Interactive Fuzzy Numbers. In: Lodwick, W.A., Kacprzyk, J. (eds) Fuzzy Optimization. Studies in Fuzziness and Soft Computing, vol 254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13935-2_7

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  • DOI: https://doi.org/10.1007/978-3-642-13935-2_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13934-5

  • Online ISBN: 978-3-642-13935-2

  • eBook Packages: EngineeringEngineering (R0)

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