For Multi-interval-valued Fuzzy Sets, Centroid Defuzzification Is Equivalent to Defuzzifying Its Interval Hull: A Theorem

In the traditional fuzzy logic, the expert’s degree of certainty in a statement is described either by a number from the interval [0, 1] or by a subinterval of such an interval. To adequately describe the opinion of several experts, researchers proposed to use a union of the corresponding sets – which is, in general, more complex than an interval. In this paper, we prove that for such set-valued fuzzy sets, centroid defuzzification is equivalent to defuzzifying its interval hull. As a consequence of this result, we prove that the centroid defuzzification of a general type-2 fuzzy set can be reduced to the easier-to-compute case when for each x, the corresponding fuzzy degree of membership is convex. 1 Formulation of the Problem Outline of this section. Our main objective is to come up with a centroid defuzzification formula for multi-interval-valued fuzzy sets. Before we start describing our results and algorithms, let us briefly recall why we need centroid defuzzification and why we need multi-interval-valued fuzzy sets. To explain this need: – we will start with the regular fuzzy sets, – then we explain the need for interval-valued fuzzy sets, and – the need for multi-interval-valued fuzzy sets; – finally, we explain the need for centroid defuzzification for all these types of fuzzy sets. Need for interval-valued fuzzy sets: a brief reminder. In the traditional fuzzy logic, an expert describes his or her degree of confidence in different statements by a number from the interval [0, 1]. In particular, for statements like “x is small” corresponding to different values x, the corresponding degree μ(x) form a membership function describing the imprecise (fuzzy) concept like “small”; see, e.g., [1, 6]. In many practical situations, experts are not comfortable describing their degree of confidence by an exact number; they feel more comfortable describing 2 V. Kreinovich and S. Sriboonchitta their degree of confidence by an interval – e.g., by an interval [0.7, 0.8]. In particular, for statements like “x is small”, the corresponding interval-valued degrees of confidence [ μ(x), μ(x) ] form an interval-valued membership function. The intuitive meaning of this membership function is that in principle, we can have many different number-valued membership functions μ(x) as long as μ(x) ∈ [ μ(x), μ(x) ] for every x. Another case when an interval-valued membership function naturally appears is when we ask several experts. For the same value x, different experts give, in general, different degrees of confidence μ1(x), . . . , μn(x). When experts are equally good, there is no reason to select one of these values, it make more sense to consider the interval [ min i μi(x),max i μi(x) ] spanned by these values. This smallest interval containing the values μ1(x), . . . , μn(x) is also known as the interval hull of the corresponding finite set {μ1(x), . . . , μn(x)}. Need for multi-interval-valued fuzzy sets. Once each expert provides his or her degree μi(x) or interval-valued degree [ μ i (x), μi(x) ] , then, instead of taking the interval hull of all these degrees, we can get a more adequate description of the experts’ opinions if we simply take the union of these values and intervals. Such unions are known as multi-intervals. If for each x, the experts’ degrees of confidence in the corresponding statement “x is small” is described by a multiinterval M(x), then we get a multi-interval-valued membership function M(x); see, e.g., [7]. Centroid defuzzification for regular fuzzy sets. In control (or, more generally, decision) applications, when for each possible value x of control, we know the degree μ(x) to which this value is reasonable, we then need to decide which control value c to apply. In fuzzy applications, we usually select the value c for which the weighted mean square deviation from this value is the smallest possible: