## Optimal Models and Methods with Fuzzy QuantitiesI originated a submission titled “Fuzzy Geometric Programming” for the Proceeding of the Second International Fuzzy Systems Association (IFSA) Congress (Tokyo) in 1987, and later published through rigorous selection in Fuzzy Sets and Systems. In 1989, I brought up“Study on non-distinct se- regression forecast model” for discussion by using Zadeh's theory on fuzzy sets. From then on, I have done researches on an optimal model with fuzzy informationquantities. In the book,Iregardthe modelwith fuzzy quantities, including fuzzy coe?cients and fuzzy variables, as a main line, introducing the molding of various problems and their practical examples, completely and clearly, in some ?elds. Many of my papers are indexed in SCI (Science Citation Index), EI (Engineering Index) and ISTP (Index to Scienti?c & Technical Proceedings), commented or extracted in American Mathematical Reviews and Zentralblatt Math. The researching and writing have been funded by the National Na- ral Science Foundation of China for three times (1997, 2003, 2008). At the same time, it is supported by the Science and Technology Project of - nan Province, the Science Research Foundation of Changsha Electric Power University,“211 Project”Foundation of Shantou University and Li Ka-Shing Science Development Foundation of Shantou University, and Scienti?c - search Foundation of Guangzhou University. The research project won the ThirdAwardofGuangdongScienceandTechnologyAwardedbytheGove- ment of Guangdong Province (2005) and Third Award of Excellent Papers in Natural Science by it (2003), successively. The book contains ten chapters as follows: Chapter 1. Prepare Knowledge; Chapter2. RegressionandSelf-regressionModelswith FuzzyCoe?cients; Chapter 3. |

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### Contents

Prepare Knowledge | 1 |

Regression and Selfregression Models with Fuzzy Coefficients | 33 |

Regression and Selfregression Models with Fuzzy Variables | 63 |

Fuzzy InputOutput Model | 95 |

Fuzzy Cluster Analysis and Fuzzy Recognition | 117 |

Fuzzy Linear Programming | 138 |

Fuzzy Geometric Programming | 193 |

Fuzzy Relative Equation and Its Optimizing | 254 |

Interval and Fuzzy Differential Equations | 293 |

Interval and Fuzzy Functional and their Variation | 327 |

363 | |

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algorithm antinomy antinomy appears Cartesian product convex function convex fuzzy convex fuzzy-valued function convex set corresponding Definition denotes determined dual programming equivalent Euler equations Example exists fc=i flat fuzzy follows forecast formula function at fuzzy fuzzy coefficients fuzzy constraint fuzzy function fuzzy linear programming fuzzy numbers fuzzy optimal solution fuzzy points fuzzy posynomial geometric fuzzy set fuzzy variables fuzzy-valued functional G X(A gi(x given cone index go(x greatest solution hence input-output model interval functional interval numbers Lagrange problem Lemma linear regression membership degree membership function method Min-Max minimum solution model with T-fuzzy objective function obtain operation optimal value optimum posynomial geometric programming programming with fuzzy Proof Proposition prove real number regression model respectively satisfies self-regression model solution set solve Step Suppose T-fuzzy data theorem holds transitive closure trapezoidal fuzzy unique variable vector variation