Trapezoidal fuzzy numbers

In this chapter, the due times of different production orders are represented as trapezoidal fuzzy numbers (TrFN) with the following definition ... 

In the context of interest here, namely that of fault tree analysis, Lindley Singpurwalla (1986) present a formal probabilistic (Bayesian) procedure for the use of expert opinions, assuming expert input in the form of mean and standard deviation of lognormally distributed failure rates. In Tanaka et al. (1983), Liang Wang (1991) and Huang et al. (2001), basic event probabilities (chances) are treated as trapezoidal fuzzy numbers and the extension principle is applied to compute the probability (chance) of occurrence of the top event. In order to deal with repeated basic events in fault tree analysis. Soman Misra (1993) provide a simple method for fuzzy fault tree analysis based on the a-cut method, also known as resolution identity. Another approach to fuzzy fault tree analysis based on the treatment of the system state as a fuzzy variable has been proposed by Huang et al. (2004). 

Up till now we have dealt with (fuzzy) set-theoretic operations. Really it is more important in computational chemistry to compute with numbers, in particular with fuzzy numbers. For our purposes a fuzzy number is an element of the real axis E that has to satisfy the following conditions (i) there is only one xq, the mean value of A, with ix(xo) = I, (ii) /z is piecewise continuous, and (iii) /z(x) < /z(xo) is monotonically increasing and /z(x) > ix xo) is monotonically decreasing. The extension of the principle to fuzzy points, fuzzified functions and fuzzy functions is explained by Bandemer and Otto in a chemical context. A further extension of fuzzy numbers are flat fuzzy numbers that can model a fuzzy interval, e.g., by a trapezoidal membership function. 

As an example, several possible membership functions that are reasonable for defining the set of real numbers that are close to 3 are shown in Fig. 1. Which of these functions captures best the concept close to 3 depends on the context within which the concept is applied. It turns out, however, that most current applications of fuzzy set theory are not overly sensitive to changes in shapes of the membership functions employed. Since triangular shapes (function A in Fig. 1) and trapezoidal shapes (Fig. 

In such rules, X and Y and Z are linguistic variables whose values, e.g., small, medium, and large, are words rather than numbers. In effect, the values of linguistic variables are labels for fuzzy sets. It is understood that the membership functions of these sets must be specified in context. Usually, the membership functions are assumed to be triangular or trapezoidal. 

Figure 6.18 shows the membership function of the riskiness of an event on an arbitrary scale, which would later be used to defuzzify the fuzzy conclusion and rank the risk according to a priority number. The membership function used is a triangular function. Unlike the trapezoidal function, the membership value of 1 in the triangular function is limited to (mly one value of the variable on the x-axis. 

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