Skip to main content
Springer Nature Link
Log in

Chance-constrained programming with fuzzy stochastic coefficients

  • Published:
Fuzzy Optimization and Decision Making Aims and scope Submit manuscript

Abstract

We consider fuzzy stochastic programming problems with a crisp objective function and linear constraints whose coefficients are fuzzy random variables, in particular of type L-R. To solve this type of problems, we formulate deterministic counterparts of chance-constrained programming with fuzzy stochastic coefficients, by combining constraints on probability of satisfying constraints, as well as their possibility and necessity. We discuss the possible indices for comparing fuzzy quantities by putting together interval orders and statistical preference. We study the convexity of the set of feasible solutions under various assumptions. We also consider the case where fuzzy intervals are viewed as consonant random intervals. The particular cases of type L-R fuzzy Gaussian and discrete random variables are detailed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Aiche, F. (1995). Sur l’optimisation floue stochastique, Dissertation (pp. 97–115). Algeria: University of Tizi-ouzou.

  • Aiche, F., & Dubois, D. (2010). An extension of stochastic dominance to fuzzy random variables. In E. Hüllermeier et al. (Eds.), International conference on information processing and management of uncertainty in knowledge-based systems (IPMU 2010), LNAI 6178 (pp. 159–168), Dortmund (Germany): Springer.

  • Ammar E. E. (2009) On fuzzy random multiobjective quadratic programming. European Journal of Operational Research 193: 329–341

    Article  MathSciNet  MATH  Google Scholar 

  • Baas S. M., Kwakernaak H. (1977) Rating and ranking of multiple aspect alternatives using fuzzy sets. Automatica 13: 47–58

    Article  MathSciNet  MATH  Google Scholar 

  • Chakraborty D., Rao K. R., Tiwari R. N. (1994) Interactive decision making in mixed (fuzzy and stochastic) environment. European Journal of Operational Research 31: 89–107

    MathSciNet  MATH  Google Scholar 

  • Chanas S., Delgado M., Verdegay J. L., Vila M. A. (1993) Ranking fuzzy real intervals in the setting of random sets. Information Sciences 69: 201–217

    Article  MathSciNet  MATH  Google Scholar 

  • Chanas S., Nowakowski M. (1988) Single value simulation of fuzzy variable. Fuzzy Sets and Systems, 25: 43–57

    Article  MathSciNet  MATH  Google Scholar 

  • Chanas S., Zielinski P. (1999) Ranking fuzzy intervals in the setting of random sets-further results. Information Sciences 117: 191–200

    Article  MathSciNet  MATH  Google Scholar 

  • Charnes A., Cooper W. W. (1959) Chance-constrained programming. Management Science 6: 73–79

    Article  MathSciNet  MATH  Google Scholar 

  • Couso I., Dubois D. (2009) On the variability of the concept of variance for fuzzy random variables. IEEE Transactions on Fuzzy Systems 17: 1070–1080

    Article  Google Scholar 

  • Couso I., Sánchez L. (2011) Upper and lower probabilities induced by a fuzzy random variable. Fuzzy Sets and Systems 165(1): 1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Dantzig G. B. (1955) Linear programming under under uncertainty. Management Sciences 1: 3–4

    Article  MathSciNet  Google Scholar 

  • David H. (1963) The method of paired comparisons, Griffins statistical monographs & courses (Vol. 12). Charles Griffin & D. Ltd., London

  • de Campos L., Gonzalez Munoz A. (1989) A subjective approach for ranking fuzzy numbers. Fuzzy Sets & Systems 29: 145–153

    Article  MATH  Google Scholar 

  • Destercke, S., Dubois, D., & Chojnacki, E. (2008). Unifying practical uncertainty representations Part I: Generalized p-boxes. International Journal of Approximate Reasoning, 49, 649–663; Part II: Clouds. 49, 664–677.

    Google Scholar 

  • Dubois, D. (1987). Linear programming with fuzzy data. In J. C. Bezdek (Ed.), Analysis of Fuzzy Information, volume III, Application in Engineering and Sciences (pp. 241–263). Boca Raton, FL: CRC Press.

  • Dubois, D., Kerre, E., Mesiar, R., & Prade, H. (2000). Fuzzy interval analysis. In: Dubois, D., & Prade, H. (Eds.), Fundamentals of fuzzy sets, The Handbooks of Fuzzy Sets Series (pp. 483–581). Boston, MA: Kluwer.

  • Dubois D., Prade H. (1983) Ranking fuzzy numbers in the setting of possibility theory. Information Sciences 30: 183–225

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D., Prade H. (1987a) Fuzzy numbers an overview. In: Bezdek J. C. (Ed.), Analysis of fuzzy information, Vol. 2. CRC Press, Boca Raton, pp 3–39

    Google Scholar 

  • Dubois D., Prade H. (1987b) The mean value of a fuzzy number. Fuzzy Sets & Systems 24: 279–300

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D., Prade H. (1988) Possibility theory. Plenum Press, New York

    Book  MATH  Google Scholar 

  • Dubois D., Prade H., Sabbadin R. (2001) Decision theoretic foundations of qualitative possibility theory. European Journal of Operational Research 128: 459–478

    Article  MathSciNet  MATH  Google Scholar 

  • Fortemps P., Roubens M. (1996) Ranking and defuzzification methods based on area compensation. Fuzzy Sets & Systems 82: 319–330

    Article  MathSciNet  MATH  Google Scholar 

  • Fishburn P. (1987) Interval orderings. Wiley, New York

    Google Scholar 

  • Inuiguchi, M. (2007). On possibilistic/fuzzy optimization. In Foundations of fuzzy logic and soft computing, IFSA 2007, LNCS 4529 (pp. 351–360), Cancun, Mexico: Springer.

  • Inuiguchi M., Ichihashi H., Kume Y. (1992) Relationships between modality constrained programming problems and various fuzzy mathematical programming problems. Fuzzy Sets and Systems 49: 243–259

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Iskander M. G. (2005) A suggested approach for possibility and necessity dominance indices in stochastic fuzzy linear programming. Applied Mathematics Letters 18: 395–399

    Article  MathSciNet  MATH  Google Scholar 

  • Kall D. (1978) Stochastic linear programming. Springer, Berlin, pp 79–92

    Google Scholar 

  • Katagiri H., Sakawa M., Ishii H. (2004) Fuzzy random bottleneck spanning tree problems using possibility and necessity measures. European Journal of Operational Research 152: 88–95

    Article  MathSciNet  MATH  Google Scholar 

  • Katagiri H., Sakawa M., Kato K., Nishizaki I. (2008) Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability. European Journal of Operational Research 188: 330–339

    Article  MathSciNet  Google Scholar 

  • Kruse H., Meyer K. D. (1987) Statistics with vague data. D. Riedel, Dordrecht

    Book  MATH  Google Scholar 

  • Kwakernaak H. (1978) Fuzzy random variables I. Information Sciences 15: 1–29

    Article  MathSciNet  MATH  Google Scholar 

  • Li J., Xu J., Gen M. (2006) A class of multiobjective linear programming model with fuzzy random coefficients. Mathematical and Computer Modelling 44: 1097–1113

    Article  MathSciNet  MATH  Google Scholar 

  • Liou, T., & Wang, J. (1992). Ranking fuzzy numbers with integral value. Fuzzy Sets & Systems, 50, 247–255.

    Google Scholar 

  • Luhandjula M. K. (1996) Fuzziness and randomness in an optimization framework. Fuzzy Sets & Systems 77: 291–297

    Article  MathSciNet  MATH  Google Scholar 

  • Luhandjula M. K. (2004) Optimization under hybrid uncertainty. Fuzzy Sets & Systems 146: 187–203

    Article  MathSciNet  MATH  Google Scholar 

  • Luhandjula, M. K. (2006). Fuzzy stochastic linear programming: Survey and future research directionsfla. European Journal of Operational Research, 174, 1353–1367.

    Google Scholar 

  • Luhandjula, M. K., & Gupta M. M. (1996). On fuzzy stochastic optimization. Fuzzy Sets and Systems, 81, 47–55.

    Google Scholar 

  • Luhandjula M. K., Joubert J. W. (2010) On some optimisation models in fuzzy-stochastic environment. European Journal of Operational Research 207: 1433–1441

    Article  MathSciNet  MATH  Google Scholar 

  • Ogura Y., Li S. M., Ralescu D. A. (2001) Set Defuzzification and Choquet Integral. International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems 9: 1–12

    MathSciNet  MATH  Google Scholar 

  • Puri M., Ralescu D. (1986) Fuzzy random variables. Journal of the Mathematics and Applications 1114: 409–420

    Article  MathSciNet  Google Scholar 

  • Qiao Z., Wang W. (1993) On solution and distribution problem of the linear programming with fuzzy random variable coefficients. Fuzzy Sets & Systems 58: 155–170

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao Z., Zhang Y., Wang W. (1994) On fuzzy random linear programming. Fuzzy Sets & Systems 65: 31649

    MathSciNet  Google Scholar 

  • Shapiro A. F. (2009) Fuzzy random variables. Insurance: Mathematics and Economics 44: 307–314

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X., Kerre E. (2001) Reasonable properties for ordering of fuzzy quantities. Fuzzy Sets & Systems 118: 375–406

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R. R. (1978). Ranking fuzzy subsets over the unit interval. In Proceedings of the IEEE international conference on decision and control (pp. 1435–1437).

  • Yager R. R. (1980) On choosing between fuzzy subsets. Kybernetes 9: 151–154

    Article  MATH  Google Scholar 

  • Yager R. R. (1993) Criteria for evaluating fuzzy ranking methods of the unit interval. Information Sciences 24: 139–157

    Google Scholar 

  • Yazini A. V. (1987) Fuzzy and stochastic programming. Fuzzy Sets & Systems 22: 171–188

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Mouloud Mammeri, BP 17 RP, Tizi-ouzou, Algeria

    Farid Aiche

  2. Faculté de Mathématiques, USTHB, LAID3, BP 32 EL, 16311, Alia, Alger, Algeria

    Moncef Abbas

  3. IRIT, CNRS and University of Toulouse, 118, Route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois

Authors
  1. Farid Aiche
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Moncef Abbas
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Didier Dubois
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Didier Dubois.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aiche, F., Abbas, M. & Dubois, D. Chance-constrained programming with fuzzy stochastic coefficients. Fuzzy Optim Decis Making 12, 125–152 (2013). https://doi.org/10.1007/s10700-012-9151-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10700-012-9151-8

Keywords