Uncertainty in engineering analysis usually pertains to stochastic uncertainty, i.e.,variance in product or process parameters characterized by probability (uncertainty in truth). Methods for calculating under stochastic uncertainty are well documented. It has been proposed by the authors that other forms of uncertainty exist in engineering design. Imprecision, or the concept of uncertainty in choice, is one such form. This paper considers real-time techniques for calculating with imprecise parameters. These techniques utilize interval mathematics and the notion of α-cuts from the fuzzy calculus. The extremes or anomalies of the techniques are also investigated, particularly the evaluation of singular or multi-valued functions. It will be shown that realistic engineering functions can be used in imprecision calculations, with reasonable computational performance.

This article deal with simulation of approximate models of dynamic systems. We propose an approach that is appropriate when the uncertainty intrinsic in some models cannot be reduced by traditional identification techniques, due to the impossibility of gathering experimental data about the system itself. The article presents a methodology for qualitative modeling and simulation of approximately known systems. The proposed solution is based on the Fuzzy Sets theory, extending the power of traditional numerical-logical methods. We have implemented a fuzzy simulator that integrates a fuzzy, qualitative approach and traditional, quantitative methods.

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The formal concept of logical equivalence in fuzzy logic, while theoretically sound, seems impractical. The misinterpretation of this concept has led to some pessimistic conclusions. Motivated by practical interpretation of truth values for fuzzy propositions, we take the class (lattice) of all sub-intervals of the unit interval [0,1] as the truth value space for fuzzy logic, subsuming the traditional class of numerical truth values from [0,1]. The associated concept of logical equivalence is stronger than the traditional one. Technically, we are dealing with much smaller set of pairs of equivalent formulas, so that we are able to check equivalence algorithmically. The checking is done by showing that our strong equivalence notion coincides with the equivalence in logic programming.

A fuzzy set can be represented by a family of crisp sets using its α-level sets, whereas a rough set can be represented by three crisp sets. Based on such representations, this paper examines some fundamental issues involved in the combination of rough-set and fuzzy-set models. The rough-fuzzy-set and fuzzy-rough-set models are analyzed, with emphasis on their structures in terms of crisp sets. A rough fuzzy set is a pair of fuzzy sets resulting from the approximation of a fuzzy set in a crisp approximation space, and a fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a crisp set in a fuzzy approximation space. The approximation of a fuzzy set in a fuzzy approximation space leads to a more general framework. The results may be interpreted in three different ways.

This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (non-fuzzy) case.

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This paper presents a multiple criteria decision aiding approach in order to build a ranking on a set of alternatives. The partial evaluations of the alternatives on the points of view can be fuzzy numbers. The aggregation is performed through the use of a fuzzy extension of the Choquet integral. We detail how to assess the parameters of the aggregator by using alternatives which are well-known to the decision maker, and which originate from his domain of expertise.

A new technique is presented to identify intervals for parameters and initial conditions for nonlinear dynamic systems based on an imprecise mathematical model and measurements of system variables. This technique employs a fuzz interval qualitative simulator for interval dynamical models and a qualitative description of measured signals, the episodes. These are based on the sign of the variable and of its first and second derivative. Episodes describe in a very simple and intuitive form the dynamic behaviour of the measured variable. By combining this qualitative with numerical information, a structure and parameter identification can be done in an very intuitive and explainable way. This technique is used within a tool, TAM-C, to model and assess chemical processes involving exothermic chemical reactions. Two applications of this technique are given: modelling of a chemical reaction system and safety assessment of an exothermic chemical process. The examples are based on real industrial and laboratory data.

This paper presents a software tool (Qua.Si. III) for the simulation of continuous dynamical systems whose parameters and/or initial conditions are modelled by fuzzy distributions. The tool, which is the last version of the qualitative simulator Qua.Si. (Bonarini, Bontempi, 1994a), implements a new approach to the numerical integration of fuzzy dynamical systems, where the problem of propagating a fuzzy distribution in the phase space is solved as a problem of constrained multivariable optimisation. Numerical simulations of two fuzzy dynamical systems, are also reported. 1. Introduction A continuous dynamical system is traditionally intended as the formalisation of a phenomenon by a set of variables, named the state, and a set of deterministic differential equations, named the model. To simulate the system behaviour means to integrate the differential model. Exact analytical solutions can be obtained only in the simplest cases: indeed, several numerical methods (e.g. Runge Kutta) have...