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Fuzzy Set Theory (FST) was formalised by Prof. Lofti Zadeh at the University of California in 1965. The significance of fuzzy variables is that they facilitate gradual transition betweenstates and consequently, possess a natural capability to express and deal withobservation and measurement uncertainties.

Traditional variables, which may be referred to as crisp variables do not have this capability. Although the definition of states by crisp sets is mathematically correct, in manycases, it is unrealistic in the face of unavoidable measurement errors. A measurement that falls into a close neighbourhood of each precisely defined border between states of a crisp variable is taken as evidential support for only one of the states, in spite of the inevitable uncertainty involved in decision. The uncertainty reaches its maximum at each border, where any measurement should be regarded as equal evidence for the two states on either side of the border. When dealing with crisp variables, the uncertainty is ignored; the measurement is regarded as evidence for one of the states, the one that includes the border point by virtue of anarbitrary mathematical definition.

Bivalent set theory can be somewhat limiting if we wish to describe a ‘humanistic’ problem mathematically (Zadeh (1987)). For example, Figure 6.2 illustrates bivalent sets tocharacterise the temperature of a room. The limiting feature of bivalent sets is that they are mutually exclusive – it is not possible to have a membership of more than one set. It is not accurate to define a transition from a quantity such as ‘warm’ to ‘hot’. In the real world a smooth (unnoticeable) drift from ‘warm’ to ‘hot’ would occur. The natural phenomenon can be describedmore accurately by FST.

Figure 6.3 shows how the same information can be quantified using fuzzy sets to describe this natural drift. A set, A, with points or objects in some relevant universe, X, is defined as these elements of x that satisfy the membership propertydefined for A. In traditional ‘crisp’ sets theory each element
of x either is or is not an element of A. Elements in a fuzzy set (denoted by –, eg A) can have a continuum of degrees of membership ranging from complete membership to complete non- membership (Zadeh (1987)). /~(x) gives the degree of membership for each element x e X. #(x) is defined on [0,1]. A

membership of 0 means that the value does not belong to the set under consideration. A
membership of 1 would mean full representation of the set under consideration. A membership
somewhere between these two limits indicates the degree of membership. The manner in which
values are assigned to a membership is not fixed and may be established according to the
preference of the person conducting the investigation.

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