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Scalarization of the fuzzy optimization problems

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Abstract

Scalarization of the fuzzy optimization problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Two solution concepts are proposed by considering two convex cones. The set of all fuzzy numbers can be embedded into a normed space. This motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to solve the fuzzy optimization problems. By applying scalarization to the optimization problem with fuzzy coefficients, we obtain its corresponding scalar optimization problem. Finally, we show that the optimal solution of its corresponding scalar optimization problem is the optimal solution of the original fuzzy optimization problem.

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References

  • R. E. Bellman L. A. Zadeh (1970) ArticleTitleDecision making in a fuzzy environment Management Science 17 141–164 Occurrence Handle301613 Occurrence Handle10.1287/mnsc.17.4.B141

    Article  MathSciNet  Google Scholar 

  • M. Delgado J. Kacprzyk J.-L. Verdegay M.A. Vila (Eds) (1994) Fuzzy optimization: Recent advances Physica-Verlag New York Occurrence Handle0812.00011

    MATH  Google Scholar 

  • D. P. Filev R. R. Yager (1993) ArticleTitleAn adaptive approach to defuzzification based on level sets Fuzzy Sets and Systems 54 355–360 Occurrence Handle1215580 Occurrence Handle10.1016/0165-0114(93)90383-S

    Article  MathSciNet  Google Scholar 

  • P. Fortemps M. Roubens (1996) ArticleTitleRanking and defuzzification methods based on area compensation Fuzzy Sets and Systems 82 319–330 Occurrence Handle0886.94025 Occurrence Handle1409703 Occurrence Handle10.1016/0165-0114(95)00273-1

    Article  MATH  MathSciNet  Google Scholar 

  • M. Inuiguchi (2004) Multiple objective linear programming with fuzzy coefficient J. Figueira S. Greco M. Ehrgot (Eds) Multiple criteria decision analysis: State of the art surveys Springer-Verlag New York 723–760

    Google Scholar 

  • M. Inuiguchi T. Tanino (1999) ArticleTitleModality goal programming models with interactive fuzzy numbers Proceedings of the eighth international fuzzy systems association world congress 1 519–523

    Google Scholar 

  • J. Jahn (1986) Mathematical vector optimization in partially ordered linear spaces Verlag Peter Lang GmbH Frankfurt am Main Occurrence Handle0578.90048

    MATH  Google Scholar 

  • O. Kaleva (1990) ArticleTitleThe calculus of fuzzy valued functions Applied Mathematics Letters 3 55–59 Occurrence Handle0708.26015 Occurrence Handle1052249 Occurrence Handle10.1016/0893-9659(90)90014-3

    Article  MATH  MathSciNet  Google Scholar 

  • Y.-J. Lai C.-L. Hwang (1992) Fuzzy mathematical programming: Methods and applications. Lecture Notes in Economics and Mathematical Systems NumberInSeriesVol. 394 Springer-Verlag New York

    Google Scholar 

  • Y.-J. Lai C.-L. Hwang (1994) Fuzzy multiple objective decision making: Methods and applications, Lecture Notes in Economics and Mathematical Systems NumberInSeriesVol. 404 Springer-Verlag New York

    Google Scholar 

  • M. Ma A. Kandel M. Friedman (2000) ArticleTitleA new approach for defuzzification Fuzzy Sets and Systems 111 351–356 Occurrence Handle0968.93046 Occurrence Handle1748552 Occurrence Handle10.1016/S0165-0114(98)00176-6

    Article  MATH  MathSciNet  Google Scholar 

  • C. V. Negoita D. A. Ralescu (1975) Applications of fuzzy sets to systems analysis Wiley NY Occurrence Handle0326.94002

    MATH  Google Scholar 

  • A. V. Patel B. M. Mohan (2002) ArticleTitleSome numerical aspects of center of area defuzzification method Fuzzy Sets and Systems 132 401–409 Occurrence Handle1010.93523 Occurrence Handle1936874 Occurrence Handle10.1016/S0165-0114(02)00107-0

    Article  MATH  MathSciNet  Google Scholar 

  • M. L. Puri D. A. Ralescu (1983) ArticleTitleDifferentials of fuzzy functions Journal of Mathematical Analysis and Applications 91 552–558 Occurrence Handle0528.54009 Occurrence Handle690888 Occurrence Handle10.1016/0022-247X(83)90169-5

    Article  MATH  MathSciNet  Google Scholar 

  • I. Ramik M. Vlach (2002) Generalized concavity in fuzzy optimization and decision analysis Kluwer Academic Publishers Boston Occurrence Handle1059.90152

    MATH  Google Scholar 

  • Slowiński, R. (Ed.) (1998). Fuzzy sets in decision analysis, operations research and statistics, Kluwer Academic Publishers.

  • H.-C. Wu (2003) ArticleTitleFuzzy optimization problems based on ordering cones Fuzzy Optimization and Decision Making 2 13–29 Occurrence Handle1967361 Occurrence Handle10.1023/A:1022896029935

    Article  MathSciNet  Google Scholar 

  • H.-C. Wu (2004) ArticleTitleAn (α ,β)-optimal solution concept in fuzzy optimization problems Optimization 53 203–221 Occurrence Handle1144.90528 Occurrence Handle2058298 Occurrence Handle10.1080/02331930410001699928

    Article  MATH  MathSciNet  Google Scholar 

  • Zadeh, L. A. (1975). The concept of linguistic variable and its application to approximate reasoning I, II and III. Information Sciences, 8, 199–249; 8, 301–357; 9, 43–80.

    Google Scholar 

  • Zimmermann H.-J. (1996). Fuzzy set theory – and its applications. (3rd edn). Kluwer Academic Publishers.

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  1. Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, 802, Taiwan

    Hsien-Chung Wu

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  1. Hsien-Chung Wu
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Correspondence to Hsien-Chung Wu.

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Wu, HC. Scalarization of the fuzzy optimization problems. Fuzzy Optim Decis Making 5, 331–353 (2006). https://doi.org/10.1007/s10700-006-0019-7

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  • DOI: https://doi.org/10.1007/s10700-006-0019-7

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