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It provides fuzzy programming approach to solve real-life decision problems in fuzzy environment. Within the framework of credibility theory, it provides a self-contained, comprehensive and up-to-date presentation of fuzzy programming models, algorithms and applications in portfolio analysis.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Credibility Theory

Abstract
The concept of fuzzy set was initialized by Zadeh via membership function in 1965. In order to measure the chance of a fuzzy event occurs, Zadeh proposed the concepts of possibility measure and necessity measure. It is proved that both possibility measure and necessity measure satisfy the properties of normality, nonnegativity and monotonicity. However, neither of them is self-dual. Since the duality is intuitive and important in both theory and practice, Liu and Liu defined a credibility measure as the average value of possibility measure and necessity measure, which was redefined by Li and Liu as a set function satisfying the normality, monotonicity, duality, and maximality axioms. Nowadays, Credibility theory has become a branch of axiomatic mathematics for modeling fuzziness. This chapter mainly introduces some basic concepts and important theorems including credibility measure, fuzzy variable, credibility function, independence, identical distribution, credibility subadditivity theorem, credibility semicontinuous theorem, credibility extension theorem, product credibility theorem, credibility inversion theorem, Zadeh extension theorem, and so on.
Xiang Li

Chapter 2. Credibilistic Programming

Abstract
The decision analysis with fuzzy objective or fuzzy constraints is natural in some real-world applications, and sometimes such analysis seems to be inevitable. Credibilistic programming is a type of mathematical programming for handling the fuzzy decision problems. In the past years, researchers have proposed various efficient modeling approaches including expected value model, chance-constrained programming model, entropy optimization model, cross-entropy minimization model, and regret minimization model. This chapter provides a general description on credibilistic programming. In addition, a brief introduction on genetic algorithm will also be given.
Xiang Li

Chapter 3. Expected Value Model

Abstract
Expected value of a fuzzy variable is the weighted average of all possible values in the sense of credibility measure, which is one of the most well-known credibilistic mappings for ranking fuzzy variables. Based on the concept of expected value, Liu and Liu proposed an expected value model, which had been widely used in many real-life applications. This chapter mainly introduces the concepts of expected value, variance, skewness, moment, as well as the fuzzy simulation technique, expected value model and applications in fuzzy portfolio analysis.
Xiang Li

Chapter 4. Chance-Constrained Programming Model

Abstract
Chance-constrained programming provides a powerful means of modeling decision systems on the assumption that the constraints will hold at least α of time, where α is the confidence level provided as an approximate safety margin by the decision-maker. For fuzzy decision problems, Liu and Iwamura introduced a maximax chance-constrained programming model, and Liu provided a maximin chance-constrained programming model, which respectively maximize the optimistic value and the pessimistic value of the fuzzy objective under certain credibility constraints. Nowadays, fuzzy chance-constrained programming models have been widely used in many real-life applications. This chapter mainly introduces the concepts of optimistic value and pessimistic value, chance-constrained programming models, fuzzy simulation, and applications in fuzzy portfolio analysis.
Xiang Li

Chapter 5. Entropy Optimization Model

Abstract
Fuzzy entropy is used to characterize the uncertainty on the possible values of fuzzy variables, which has been studied by many researchers. Within the framework of credibility theory, Li and Liu presented a Shannon-like entropy for both discrete fuzzy variable and continuous fuzzy variable. Furthermore, Li and Liu proposed the maximum entropy principle, and proved that out of all the credibility functions with fixed expected value and variance, the normal credibility function has the maximum entropy. Based on the concept of fuzzy entropy, Li et al. proposed an entropy optimization model by minimizing the uncertainty of the fuzzy objective under certain expected constraints. This chapter mainly includes the definition of fuzzy entropy, maximum entropy theorems, entropy optimization model and its crisp equivalents, fuzzy simulation, and applications in portfolio selection problem.
Xiang Li

Chapter 6. Cross-Entropy Minimization Model

Abstract
Cross-entropy is used to characterize the divergence between two fuzzy variables. Within the framework of credibility theory, Li and Liu defined the cross-entropy for fuzzy variable by using credibility function, and proposed a fuzzy cross-entropy minimization principle, which tells us that out of all credibility functions satisfying given moment constraints, choose the one that is closest to the given a priori credibility function. Furthermore, Qin et al. proposed a cross-entropy minimization model to study the fuzzy portfolio selection problems. This chapter mainly includes the definition of cross-entropy, minimum cross-entropy principle, cross-entropy minimization model, fuzzy simulation and applications.
Xiang Li

Chapter 7. Regret Minimization Model

Abstract
Distance between fuzzy quantities, used to represent the degree of difference, is a powerful concept in many disciplines of science and engineering. Within the framework of credibility theory, Liu gave an Euclidean distance based on the concept of expected value. Furthermore, Li and Liu proved the triangle inequality and the completeness of the fuzzy metric space. Based on the worst regret criterion, Li et al. proposed a fuzzy regret minimization model to minimize the distance between the fuzzy objective values and the best values. This chapter mainly introduces the concept of distance, regret minimization model, and applications in the portfolio selection problem.
Xiang Li

Backmatter

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