Baoding Liu

Expected value of fuzzy variable and fuzzy expected value models

This paper will present a novel concept of expected values of fuzzy variables, which is essentially a type of Choquet integral and coincides with that of random variables. In order to calculate the expected value of general fuzzy variable, a fuzzy simulation technique is also designed. Finally, we construct a spectrum of fuzzy expected value models, and integrate fuzzy simulation, neural network, and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models.

Chance constrained programming with fuzzy parameters

This paper extends chance constrained programming from stochastic to fuzzy environments. Analogous to stochastic programming, some crisp equivalents of chance constraints in fuzzy environments are presented. We also propose a technique of fuzzy simulation for the chance constraints which are usually hard to be converted to their crisp equivalents. Finally, a fuzzy simulation based genetic algorithm is designed for solving this kind of problems and some numerical examples are discussed.

A survey of credibility theory

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.

Expected value operator of random fuzzy variable and random fuzzy expected value models

Random fuzzy variable is a mapping from a possibility space to a collection of random variables. This paper first presents a new definition of the expected value operator of a random fuzzy variable, and proves the linearity of the operator. Then, a random fuzzy simulation approach, which combines fuzzy simulation and random simulation, is designed to estimate the expected value of a random fuzzy variable. Based on the new expected value operator, three types of random fuzzy expected value models are presented to model decision systems where fuzziness and randomness appear simultaneously. In addition, random fuzzy simulation, neural networks and genetic algorithm are integrated to produce a hybrid intelligent algorithm for solving those random fuzzy expected valued models. Finally, three numerical examples are provided to illustrate the feasibility and the effectiveness of the proposed algorithm.

Existence and uniqueness theorem for uncertain differential equations

Canonical process is a Lipschitz continuous uncertain process with stationary and independent increments, and uncertain differential equation is a type of differential equations driven by canonical process. This paper presents some methods to solve linear uncertain differential equations, and proves an existence and uniqueness theorem of solution for uncertain differential equation under Lipschitz condition and linear growth condition.

Fuzzy random chance-constrained programming

By fuzzy random programming, we mean the optimization theory dealing with fuzzy random decision problems. This paper presents a new concept of chance of fuzzy random events, and constructs a general framework of fuzzy random chance-constrained programming. We also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming. To speed up the process of handling uncertain functions, we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation. Finally, we integrate the fuzzy random simulation, neural network, and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples.

Fuzzy Random Variables: A Scalar Expected Value Operator

Fuzzy random variable has been defined in several ways in literature. This paper presents a new definition of fuzzy random variable, and gives a novel definition of scalar expected value operator for fuzzy random variables. Some properties concerning the measurability of fuzzy random variable are also discussed. In addition, the concept of independent and identically distributed fuzzy random variables is introduced. Finally, a type of law of large numbers is proved.

A note on chance constrained programming with fuzzy coefficients

This paper deals with nonlinear chance constrained programming as well as multiobjective case and goal programming with fuzzy coefficients occurring in not only constraints but also objectives. We also present a fuzzy simulation technique for handling fuzzy objective constraints and fuzzy goal constraints. Finally, a fuzzy simulation based genetic algorithm is employed to solve a numerical example.

A model for portfolio selection with order of expected returns

Abstract This paper proposes a new model for portfolio selection in which the expected returns of securities are considered as variables rather than as the arithmetic means of securities. A genetic algorithm is designed to solve the optimization problem which is difficult to solve with the existing traditional algorithms due to its nonconcavity and special structure. We illustrate the new model by a numerical example and compare the results with those derived from the traditional model of Markowitz. Scope and purpose Portfolio selection, originally articulated by Markowitz, has been one of the most important research fields in modern finance. Several new models and extensions such as the inclusion of transaction costs and taxes have been proposed to improve the performance of portfolio investment. All those models and extensions have advantages and disadvantages in both theory and practical applications. The purpose of this paper is to describe the return and risk of a portfolio more accurately. On the basis of an order of expected returns of securities, we propose a new model for portfolio selection.

A SUFFICIENT AND NECESSARY CONDITION FOR CREDIBILITY MEASURES

Possibility measures and credibility measures are widely used in fuzzy set theory. Compared with possibility measures, the advantage of credibility measures is the self-duality property. This paper gives a relation between possibility measures and credibility measures, and proves a sufficient and necessary condition for credibility measures. Finally, the credibility extension theorem is shown.