Fuzzy sets tools

Fuzzy logie was first proposed by Zadeh (1965) and is based on the eoneept of fuzzy sets. Fuzzy set theory provides a means for representing uneertainty. In general, probability theory is the primary tool for analysing uneertainty, and assumes that the... 

One of the most important tools in applications of fuzzy set theory is the concept of linguistic variables (LV). These are groups of fuzzy sets with (partially) overlapping membership functions over a common (crisp) basic variable x. To represent several classes within an LV, the membership functions should cover all the relevant definition space of the basic variable x. The overlaps of these functions define the fuzziness. A linguistic variable L, classified by n fuzzy sets Aj, can be defined as... 

Fuzzy sets serve as mathematical tools for the description of problems where the classification criteria are not clearly defined. In ordinary set theory a point p is either an element or not an element of a set A, i.e., the membership function of a point p can have values of either 1 or 0. By contrast, in fuzzy set theory the membership function of a point p is regarded as the "degree of belonging" to the fuzzy set A, and this membership function can take any value from the interval [0,1]. In some sense, fuzzy set theory can be regarded as probability theory turned inside out, where events are considered with "a posteriory" certainty after they have occurred, but some uncertainty is associated with Judging and classifying these events. 

As discussed in Section 6.7, fuzzy sets are tools for treating classification problems with imprecisely defined classification criteria [382-385]. We recall that for an ordinary set A, a point p either does or does not belong to the set A. This can be... 

Whereas both the external f -deficiency measure SRyc(A) and the R-deficiency measure ARyB(A) are mathematically correct and valid tools for comparisons of the degrees of symmetry deficiencies and their correlations with unusual reactivities of functional groups, the actual physical existence of fuzzy sets involved in the determination of the internal -deficiency measure SRyB(A) nevertheless suggests that this measure is likely to provide more reliable correlations with variations in the chemical reactivity of functional groups. 

A mathematical tool which has gained importance in environmental assessments is fuzzy logic (Adriaenssens et al 2004 Hollert et al., 2002 Silvert, 1997 Tran et al 2002). Fuzzy sets which describe different categories, such as low contamination and moderate contamination (Figure 8.2.2), can overlap and hence enable the membership... 

Application of Fuzzy Set Theory in Chemistry Tools for Dealing with Fuzzy Sets Related Articles References... 

The next few steps are very similar to those required in any software project. One of the first stages is the clear definition of the knowledge domain. It must be clear which problems the expert system must solve. It is at this stage not the intention to define how this can be done. Clarity and specificity must be the major guides here. Fuzziness at this stage will, more than in classical software projects, have to be paid for later when different interpretations cause misunderstandings. Equally important is the clear definition of the end user(s). An expert system set up as decision support tool for professionals is totally different from an expert system that can be used as a training support for less professional people. 

Virtual screening on a 2-D basis is far more efficient in terms of computing time, although the information content of the resulting virtual hits is far less sophisticated than for a 3-D pharmacophore search. It is in this setting that combinatorial chemistry approaches are applied most effectively. The many interactions possible with a compound library based on virtual hits compensates for the inherent fuzziness of any prediction tool. Examples have been described where the 2-D structure of a... 

David J. Soderberg, BP Chemicals You mentioned a very coherent and concise set of tools that are used portfolio management, fuzzy front end, and stage gate. Those work very well within a business unit context. They re very focused and allow you to allocate resources. 

Fuzzy clustering is an important tool in identifying the strueture in data. In general, a fuzzy clustering algorithm with objective function can be formulated as follows suppose X= V y=l, 2,. .., N is a finite set of feature vectors, where V= the number of measurements and = the dimension of the input variables x = In addition,/ = (F i,/ 2, ---,... 

An important tool for interrelating fuzzy problems and crisp problems is the a-cut Ga(oi) = x fj-A(x) = a of a fuzzy subset A of a set X, defined as the crisp set of all those points x of X with membership function tXA(x) equal to a. A set where /xa(x) is greater than or equal to the value a is Da oi) = (x maU) > These two choices for Qf-cuts are analogous to the electron density level sets and the density domains used in the topological analysis of molecular shapes. " ... 

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