Fuzzy random programming with equilibrium chance constraints.

*(English)*Zbl 1140.90520Summary: To model fuzzy random decision systems, this paper first defines three kinds of equilibrium chances via fuzzy integrals in the sense of Sugeno. Then a new class of fuzzy random programming problems is presented based on equilibrium chances. Also, some convex theorems about fuzzy random linear programming problems are proved, the results provide us methods to convert primal fuzzy random programming problems to their equivalent stochastic convex programming ones so that both the primal problems and their equivalent problems have the same optimal solutions and the techniques developed for stochastic convex programming can apply. After that, a solution approach, which integrates simulations, neural network and genetic algorithm, is suggested to solve general fuzzy random programming problems. At the end of this paper, three numerical examples are provided. Since the equivalent stochastic programming problems of the three examples are very complex and nonconvex, the techniques of stochastic programming cannot apply. In this paper, we solve them by the proposed hybrid intelligent algorithm. The results show that the algorithm is feasible and effectiveness.

##### MSC:

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90B20 | Traffic problems in operations research |

##### Keywords:

Fuzzy random variable; Equilibrium chance; Fuzzy random programming; Genetic algorithm; Neural network
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\textit{Y.-K. Liu} and \textit{B. Liu}, Inf. Sci. 170, No. 2--4, 363--395 (2005; Zbl 1140.90520)

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##### References:

[1] | Buckley, J.J., Solving possibilistic linear programming problems, Fuzzy sets and systems, 31, 329-341, (1989) · Zbl 0671.90049 |

[2] | Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049 |

[3] | Dubois, D.; Prade, H., Possibility theory, (1988), Plenum New York · Zbl 0645.68108 |

[4] | Colubi, A.; Dominguez-Menchero, J.S.; López-Diaz, M., On the formalization of fuzzy random variables, Information sciences, 133, 3-6, (2001) · Zbl 0988.28008 |

[5] | Grabisch, M.; Murofushi, T.; Sugeno, M., Fuzzy measure of fuzzy event defined by fuzzy integrals, Fuzzy sets and systems, 50, 293-313, (1992) · Zbl 0785.28011 |

[6] | Inuiguchi, M.; Ramı́k, 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, 3-28, (2000) · Zbl 0938.90074 |

[7] | Inuiguchi, M.; Ichihashi, H.; Kume, Y., Modality constrained programming problems: a unified approach to fuzzy mathematical programming problems in the setting of possibility theory, Information sciences, 67, 93-126, (1993) · Zbl 0770.90078 |

[8] | Klir, G.J., On fuzzy-set interpretation of possibility theory, Fuzzy sets and systems, 108, 263-373, (1999) · Zbl 0984.94045 |

[9] | Krätschmer, V., A unified approach to fuzzy random variables, Fuzzy sets and systems, 123, 1-9, (2001) · Zbl 1004.60003 |

[10] | Kibzun, A.I.; Kan, Yu.S., Stochastic programming problems with probability and quantile functions, (1996), Wiley Chichester · Zbl 0927.90089 |

[11] | Kruse, R.; Meye, K.D., Statistics with vague data, (1987), D. Reidel Publishing Company Dordrecht |

[12] | Kwakernaak, H., Fuzzy random variables I: definition and theorems, Information sciences, 15, 1-29, (1978) · Zbl 0438.60004 |

[13] | Lai, Y.-J.; Hwang, C.-L., Fuzzy mathematical programming: methods and applications, (1992), Springer-Verlag Berlin |

[14] | Liu, B., Theory and practice of uncertain programming, (2002), Physica-Verlag Heidelberg · Zbl 1029.90084 |

[15] | Liu, B., Uncertainty theory: an introduction to its axiomatic foundations, (2004), Springer-Verlag Heidelberg · Zbl 1072.28012 |

[16] | Liu, B., Fuzzy random chance-constrained programming, IEEE transactions on fuzzy systems, 9, 713-720, (2001) |

[17] | Liu, B., Fuzzy random dependent-chance programming, IEEE transactions on fuzzy systems, 9, 721-726, (2001) |

[18] | Liu, B.; Iwamura, K., Chance constrained programming with fuzzy parameters, Fuzzy sets and systems, 94, 227-237, (1998) · Zbl 0923.90141 |

[19] | Liu, B.; Iwamura, K., Fuzzy programming with fuzzy decisions and fuzzy simulation-based genetic algorithm, Fuzzy sets and systems, 122, 253-262, (2001) · Zbl 1020.90048 |

[20] | Liu, B.; Liu, Y.-K., Expected value of fuzzy variable and fuzzy expected value models, IEEE transactions on fuzzy systems, 10, 445-450, (2002) |

[21] | Liu, Y.-K.; Liu, B., Fuzzy random variables: a scalar expected value operator, Fuzzy optimization and decision making, 2, 143-160, (2003) |

[22] | López-Diaz, M.; Gil, M.A., Constructive definitions of fuzzy random variables, Statistics and probability letters, 36, 135-143, (1997) · Zbl 0929.60005 |

[23] | Luhandjula, M.K., Fuzziness and randomness in an optimization framework, Fuzzy sets and systems, 77, 291-297, (1996) · Zbl 0869.90081 |

[24] | Luhandjula, M.K.; Gupta, M.M., On fuzzy stochastic optimization, Fuzzy sets and systems, 81, 47-55, (1996) · Zbl 0879.90187 |

[25] | Nahmias, S., Fuzzy variable, Fuzzy sets and systems, 1, 97-101, (1978) |

[26] | Negoita, C.V.; Ralescu, D., Simulation, knowledge-based computing, and fuzzy statistics, (1987), Van Nostrand Reinhold Company New York · Zbl 0683.68097 |

[27] | Prékopa, A., Stochastic programming, (1995), Kluwer Academic Publishers Dordrecht · Zbl 0834.90098 |

[28] | Puri, M.L.; Ralescu, D.A., Fuzzy random variables, Journal of mathematical analysis and applications, 114, 409-422, (1986) · Zbl 0592.60004 |

[29] | M. Sugeno, Theory of fuzzy integral and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974 |

[30] | Sakawa, M., Fuzzy sets and interactive multiobjective optimization, (1993), Plenum New York · Zbl 0842.90070 |

[31] | Stancu-Minasian, I.M., Stochastic programming with multiple objective functions, (1984), D. Reidel Publishing Company Dordrecht · Zbl 0554.90069 |

[32] | Scarselli, F.; Tsoi, A.C., Universal approximation using feedforward neural networks: a survey of some existing methods, and some new results, Neural networks, 11, 15-37, (1998) |

[33] | Tanaka, H.; Guo, P.; Zimmermann, H.-J., Possibility distribution of fuzzy decision variables obtained from possibilistic linear programming problems, Fuzzy sets and systems, 113, 323-332, (2000) · Zbl 0961.90136 |

[34] | Qiao, Z.; Wang, G., On solution and distribution problems of the linear programming with fuzzy random variable coefficients, Fuzzy sets and systems, 58, 155-170, (1993) · Zbl 0813.90127 |

[35] | Wang, G.; Qiao, Z., Linear programming with fuzzy random variable coefficients, Fuzzy sets and systems, 57, 295-311, (1993) · Zbl 0791.90072 |

[36] | Wang, Z.; Klir, J., Fuzzy measure theory, (1992), Plenum New York · Zbl 0812.28010 |

[37] | Yazenin, A.V., On the problem of possibilistic optimization, Fuzzy sets and systems, 81, 133-140, (1996) · Zbl 0877.90087 |

[38] | Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002 |

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