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Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

Two fractal image compression algorithms based on possibility theory are originally presented in this paper. Fuzzy sets are used to represent the edge character of each image block, and two kinds of membership function are designed. A fuzzy integrated judgement model is also proposed. The model generates an accurate value for each edge block, which would be a label during the search process. The edge possibility distribution function and the edge necessity level are designed to control the quantity of the blocks to be searched. Meanwhile the pre-restriction is proposed, the average intensity value at different locations is used to be a necessary condition before the MSE computations. It is shown by our experiments that the encoding times of our two algorithms, compared to that of Jacquin's approach, are reduced to 60%–70% and 10%–20%, respectively.

The goal of this paper is to survey and briefly discuss various rules of conditioning proposed in the framework of possibility theory, as well as various conditional independence relations suggested for these rules. These conditioning rules and conditional independence relations are confronted with formal properties of conditional independence. Special attention is paid to the conditioning in the sense of measure-theoretic approach (based on the notion of a t-norm) recently introduced by de Cooman. It is shown that this approach to conditioning and the related conditional independence notion generalizes and unifies some of the previously suggested rules and conditional independence relations. The properties of the proposed conditional independence relation completely correspond to those possessed by stochastic conditional independence.

Some aspects of the relationship between Goodman and Nguyen's one-point coverage interpretation of a fuzzy set and Zadeh's possibilistic interpretation are discussed. As a result of this, we derive a new interpretation of the strong α-cut of a normalized fuzzy set, namely that of being the most precise set we are sure to contain an unknown parameter with probability greater than or equal to 1-α.

Different authors have observed some relationships between consonant random sets and possibility measures, specially for finite universes. In this paper, we go deeply into this matter and propose several possible definitions for the concept of consonant random set. Three of these conditions are equivalent for finite universes. In that case, the random set considered is associated to a possibility measure if and only if any of them is satisfied. However, in a general context, none of the six definitions here proposed is sufficient for a random set to induce a possibility measure. Moreover, only one of them seems to be necessary.

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.

The scheduling problems with fuzzy processing times and fuzzy due dates are investigated in this paper. The concepts of earliness and tardiness are interpreted by using the concepts of possibility and necessity measures that were developed in fuzzy sets theory. Many types of objective function will be taken into account through the different combinations of possibility and necessity measures. The purpose of this paper is to obtain the optimal schedules based on these objective functions. The genetic algorithm will be invoked to tackle these objective functions. Four numerical examples are also provided and solved by using the commercial software MATLAB.

This note presents the characterization of a fuzzy variable whose possibility distribution is a convex fuzzy set.

In this paper, we propose an interactive method for multiobjective linear programming problems, in which fuzzy coefficients, random variable coefficients and fuzzy random variable coefficients are involved in the objective functions respectively. In the proposed method, it is assumed that the decision maker has a fuzzy goal for each objective function, and such a fuzzy goal can be quantified by eliciting the membership function. Through a possibility measure and a fractile optimization model, the original problem is transformed to the well-defined multiobjective programming problem. Then, a generalized Pareto optimal concept is defined, and an interactive algorithm is proposed to obtain a satisfactory solution from among a generalized Pareto optimal solution set.

Possibility measures have been conceived by Zadeh [1], and developed by, e. g., Dubois, Prade and others, as very simple, if compared with, e.g., probability theory, but still nontrivial and reasonable uncertainty measures. The relatively poor descriptional and operational abilities of possibility measures seem to be closely related to the standard linear ordering relation and the corresponding supremum and infimum operations defined over the unit interval of real numbers. Having discussed, very briefly, the possibilities how to overcome these limitations, we propose and investigate possibility measures taking their values in the well-known Cantor subset of the unit interval but defined with respect to a nonstandard operation of supremum resulting from a nonstandard partial ordering of real numbers from the Cantor set. Such a nonstandard possibility measure is proved to be a sufficient tool to define an infinite sequence of probability measures defined over countable sets, consequently, the distribution of each continuous real-valued random variable defined over a general abstract probability space can be defined by an appropriate nonstandard possibility measure.

The k-order maxitive fuzzy measures on general spaces are discussed. The sum of k_i-order maxitive measures is shown to be a ∑ k_i-order maxitive fuzzy measure. The sum of two maxitive measures (possibility measures) is discussed and completely analyzed.

A fuzzy software reliability model is proposed where the time intervals between the software failures are taken as the fuzzy variables governed by a membership function. The model takes into account the following assumptions: new faults may be introduced into the software during debugging processes, the number of faults removed after a failure may be greater than one, and there is a growth of human experience during debugging. The model can be considered as an extension of the model developed by Cai, Wen and Zhang. An efficient algorithm is presented for estimating parameters of the model. The numerical examples validate the proposed model.

In this paper, new generalizations of the probabilistic averaging operator — Associated Fuzzy Probabilistic Averaging (As-PA and As-FPA) and Immediate Probabilistic Fuzzy Ordered Weighted Averaging (As-IP-OWA and As-IP-FOWA) operators are presented in the environment of fuzzy uncertainty. An uncertainty is presented by associated probabilities of a fuzzy measure. Expert’s evaluations as arguments of the aggregation operators are described by a variable, values of which are compatibility levels on the states of nature defined in positive real or triangular fuzzy numbers (TFNs). Two propositions on the As-FPA operator are proved: (1) The As-FPA operator for the fuzzy measure — capacity of order two coincides with the finite Choquet Averaging (CA) Operator; (2) the As-FPA operator coincides with the FPA operator when a probability measure is used in the role of a fuzzy measure. Analogous propositions for the As-IP-FOWA operator are proved. Some propositions on the connection of the As-FPA and As-IP-FOWA operators are also proved. Information measures — Orness and Divergence for the constructed operators are defined. Some propositions on the connections of these parameters with the corresponding parameters of the finite CA Operator are proved. Two illustrative examples on the applicability of the As-FPA and As-IP-FOWA operators are presented: (1) Several variants of the As-FPA and As-IP-FOWA operators are used for comparison of decision-making results for the problems regarding the fiscal policy of a country; (2) The As-FPA operator is used in the Multi-attribute decision-making (MADM) problem of choosing the best version of the students’ project.

Interval-valued time series (ITS) are interval-valued data that are collected in chronological order. The modeling of ITS is an ongoing issue in domain of time series analysis. This paper presents a new modeling method of ITS based on the synergy of fuzzy set theory and artificial neural networks. The proposed method involves the construction of collection of fuzzy sets describing characteristics of amplitude of ITS, the expression and reconstruction mechanism of ITS and the emergence of model of ITS based on artificial neural network (ANN). The resulting model of ITS not only supports the linguistic output but also the numeric output in interval format. A series of experimental studies is reported for two publicly available financial datasets showing different dynamic characteristics. Experimental results clearly show that the constructed ITS model has the better performance on the linguistic and numeric level.

This chapter shows how Fuzzy Pattern Matching techniques can be applied to the design of a knowledge-based system that can diagnose the most common faults of industrial rotating machinery through the evaluation of vibration data. The system is able to perform the tasks of Fault Detection, Fault Isolation and Fault Identification, and can handle the vagueness implicit in the diagnosis linguistic knowledge, as well as several sources for uncertainty that are generated during the vibration measurement process. Furthermore, a Genetic Algorithms-based module has also been included that, when enough fault data is available, has the ability to adapt some of the system's parameters in order to optimize its diagnosis performance.