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The modes of convergence in the approximation of fuzzy random optimization problems

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Abstract

To develop the approximation approach to fuzzy random optimization problems, it is required to introduce the modes of convergence in fuzzy random theory. For this purpose, this paper first presents several novel convergence concepts for sequences of fuzzy random variables, such as convergence in chance, convergence in distribution and convergence in optimistic value; then deals with the convergence criteria and convergence relations among various types of convergence. Finally, we deal with the convergence theorems for sequences of integrable fuzzy random variables, including dominated convergence theorem and bounded convergence theorem.

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References

  • Chen Y, Liu YK, Chen J (2006) Fuzzy portfolio selection problems based on credibility theory. Lect Notes Artif Intell 3930: 377–386

    Google Scholar 

  • Denneberg D (1994) Nonadditive measure and integral. Kluwer, Dordrecht

    Google Scholar 

  • Feng X, Liu YK (2006) Measurability criteria for fuzzy random vectors. Fuzzy Optim Decis Mak 5(3): 245–253

    Article  MATH  MathSciNet  Google Scholar 

  • Gao J, Liu B (2001) New primitive chance measures of fuzzy random event. Int J Fuzzy Syst 3(4): 527–531

    MathSciNet  Google Scholar 

  • Gao J, Lu M (2005) Fuzzy quadratic minimum spanning tree problem. Appl Math Comput 164(3): 773–788

    Article  MATH  MathSciNet  Google Scholar 

  • Gao J, Liu B (2005) Fuzzy multilevel programming with a hybrid intelligent algorithm. Comput Math Appl 49: 1539–1548

    Article  MATH  MathSciNet  Google Scholar 

  • Gao J (2007) Credibilistic game with fuzzy information. J Uncertain Syst 1(1): 74–80

    Google Scholar 

  • Klir GJ (1999) On fuzzy-set interpretation of possibility theory. Fuzzy Sets Syst 108: 263–373

    Article  MATH  MathSciNet  Google Scholar 

  • Kruse R, Meyer KD (1987) Statistics with vague data. D. Reidel Publishing Company, Dordrecht

    MATH  Google Scholar 

  • Kwakernaak H (1978) Fuzzy random variables–I Definitions and theorems. Inf Sci 15: 1–29

    Article  MATH  MathSciNet  Google Scholar 

  • Liang R, Gao J (2008) Dependent-chance programming models for capital budgeting in fuzzy environments, vol 13, no. 1. Tsinghua Univesity and Technology

  • Liu B (2006) A survey of credibility theory. Fuzzy Optim Decis Mak 5(4): 387–408

    Article  MathSciNet  Google Scholar 

  • Liu B (2007) A survey of entropy of fuzzy variables. J Uncertain Syst 1(1): 4–13

    Google Scholar 

  • Liu B (2002) Theory and practice of uncertain programming. Physica-Verlag, Heidelberg

    MATH  Google Scholar 

  • Liu B (2004) Uncertainty theory. Springer, Berlin

    MATH  Google Scholar 

  • Liu B (2001) Fuzzy random chance-constrained programming. IEEE Trans Fuzzy Syst 9(5): 713–720

    Article  Google Scholar 

  • Liu B (2001) Fuzzy random dependent-chance programming. IEEE Trans Fuzzy Syst 9(5): 721–726

    Article  Google Scholar 

  • Liu B, Liu YK (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans Fuzzy Syst 10(4): 445–450

    Article  Google Scholar 

  • Liu J, Zhu Y (2007) Some inequalites between moments of credibility distributions. J Uncertain Syst 1(2): 137–147

    Google Scholar 

  • Liu YK (2005) Fuzzy programming with recourse. Int J Uncertain Fuzziness Knowl Based Syst 13(4): 381–413

    Article  MATH  Google Scholar 

  • Liu YK (2006) Convergent results about the use of fuzzy simulation in fuzzy optimization problems. IEEE Trans Fuzzy Syst 14(2): 295–304

    Article  Google Scholar 

  • Liu YK, Liu B (2003) Fuzzy random variable: a scalar expected value operator. Fuzzy Optim Decis Mak 2(2): 143–160

    Article  MathSciNet  Google Scholar 

  • Liu YK, Liu B (2003) A class of fuzzy random optimization: expected value models. Inf Sci 155(1–2): 89–102

    Article  MATH  Google Scholar 

  • Liu YK, Liu B (2005) Fuzzy random programming with equilibrium chance constraints. Inf Sci 170(2–4): 363–395

    Article  MATH  Google Scholar 

  • Liu YK, Liu B (2005) On minimum-risk problems in fuzzy random decision systems. Comput Oper Res 32(2): 257–283

    MATH  MathSciNet  Google Scholar 

  • Liu YK, Liu B (2003) Expected value operator of random fuzzy variable and random fuzzy expected value models. Int J Uncertainty Fuzziness Knowl Based Syst 11(2): 195–215

    Article  MATH  Google Scholar 

  • Liu YK, Liu B, Chen Y (2006) The infinite dimensional product possibility space and its applications. Lect Notes Artif Intell 4114: 984–989

    Google Scholar 

  • Liu YK, Gao J (2005) Convergence criteria and convergence relations for sequences of fuzzy random variables. Lect Notes Artif Intell 3613: 321–331

    Google Scholar 

  • Liu YK, Gao J (2007) The independence of fuzzy variables with applications to fuzzy random optimization. Int. J. Uncertainty Fuzziness Knowl Based Syst. 15: Supp.2 1–20

  • Liu YK, Zhu X (2007) Capacitated fuzzy two-stage location-allocation problem. Int J Innov Comput Inf Control 3(4): 987–999

    Google Scholar 

  • Liu YK, Wang S (2006) A credibility approach to the measurability of fuzzy random vectors. J Nat Sci Tech 1(1): 111–118

    Google Scholar 

  • Liu YK, Wang S (2006) Theory of fuzzy random optimization. China Agricultural University Press, Beijing

    Google Scholar 

  • Liu YK, Liu ZQ, Liu Y (2007) Fuzzy optimization problems with critical value-at-risk criteria. Lect Notes Comput Sci 4492: 267–274

    Article  Google Scholar 

  • Wang G, Qiao Z (1993) Linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst 57: 295–311

    Article  MATH  MathSciNet  Google Scholar 

  • Luhandjula MK (2004) Optimisation under hybrid uncertainty. Fuzzy Sets Syst 146: 187–203

    Article  MATH  MathSciNet  Google Scholar 

  • Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1(1): 97–101

    Article  MATH  MathSciNet  Google Scholar 

  • Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114: 409–422

    Article  MATH  MathSciNet  Google Scholar 

  • Wang C, Tang W, Zhao R (2007) Static Bayesian games with finite fuzzy types and the existence of equilibrium. J Uncertain Syst 1(2): 148–160

    MathSciNet  Google Scholar 

  • Wang P (1982) Fuzzy contactability and fuzzy variables. Fuzzy Sets Syst 8: 81–92

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics & Computer Science, Hebei University, 071002, Baoding, Hebei, China

    Yan-Kui Liu

  2. School of Creative Media, City University of Hong Kong, Hong Kong, China

    Zhi-Qiang Liu

  3. School of Information, Renmin University of China, 100872, Beijing, China

    Jinwu Gao

Authors
  1. Yan-Kui Liu
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  2. Zhi-Qiang Liu
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  3. Jinwu Gao
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Correspondence to Yan-Kui Liu.

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Liu, YK., Liu, ZQ. & Gao, J. The modes of convergence in the approximation of fuzzy random optimization problems. Soft Comput 13, 117–125 (2009). https://doi.org/10.1007/s00500-008-0309-9

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