Types of Membership Functions

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In principle a membership function associated with a fuzzy set A depends not only on the concept to be represented, but also on the context in which it is used. The graphs of thefunctions may have very different shapes and may have some specific properties (e.g.continuity).

Whether a particular shape is suitable or not can be determined only in the application context (Klir and Yuan (1995)). In many practical instances, fuzzy sets can berepresented explicitly by families of parameterised functions, the most common being: where m is a modal value and a and b denote lower and upper bounds, respectively, for non-zero values of .4(x).

Sometimes it is more convenient to use the notation explicitly highlightingthemembership function parameters, in this case it is given by: Any fuzzy set can be regarded as a family of fuzzy sets. This is the essence of an identity principle known also as the representation theorem. To explain this construction, it is required to define the notion of an acut of a fuzzy set. The a -cut of A, denoted by A~, is a set consisting of those elements in theuniverse X whose membership values exceed the threshold level a.

This is formally represented by: Conversely, any fuzzy set can be “reconstructed” from a family of nested sets (assuming that they satisfy the constraint of consistency: if al > a2 then Aa~ c Aa2). This theorem’s importance lies in its underscoring of the very nature of the generalisation provided by fuzzy sets. Furthermore, the theorem implies that problems formulated in the framework of fuzzy sets (such as risk and reliability analysis) can be solved by transforming these fuzzy sets into their corresponding families of nested a-cuts and determining solutions to each using standard, non- fuzzy techniques.

Subsequently, all the partial results derived in this way can be merged, reconstructing a solution to the problem in its original formulation based on fuzzy sets. By Since FST was proposed almost four decades ago, it has found many useful applications. The linguistic approach based on fuzzy sets has given very good results for modelling qualitativeinformation. It has been widely used in different fields, for example, informationretrieval (Bordogna and Pasi (1993)), clinical diagnosis (Degani and Bortolan (1988)), marketing (Yager et al. (1994)), risk modelling in software development (Lee (1996a), Lee (1996b)), technology transfer strategy selection (Chang and Chen (1994)), education (Law (1996)), decision making (Bordogna et al. (1997)), environmental engineering (Deshpande (1999)), and many more.

A review by Maiers and Sherif in 1985, covered over 450 papers addressing FST application in areas of automation control, decision making, biology and medicine, economics and the environment (Maiers and Sherif (1985)). The use of FST in system safety and reliability analyses could prove to be a useful tool, as these analyses often require the use of subjective judgement and uncertain data. By allowing imprecision and approximate analysis, FST helps to restore integrity to reliability analyses by allowing uncertainty and not forcing precision where it is not possible.

However, the theory can be difficult to use directly. The use of linguistic variables allows a flexible modelling of imprecise data and information. A linguistic variable differs from a numerical variable in that its values are not numbers but words or sentences in a natural or artificial language. Since words in general are less precise than numbers, the concept of a linguistic variable serves the purpose ofproviding ameans of approximate characterisation of phenomena, which are too complex or ill defined to be amenable to description in conventional quantitative terms (Schmucker (1984)).

More specifically, fuzzy sets, which represent the restriction associated with the values of linguisticvariable, may be viewed as summaries of various sub-classes of elements in auniverse of discourse (a universe of discourse is the range of all possible values for an input to a fuzzy system). This is analogous to the role played by words and sentences in anaturallanguage.

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