Interpreting the footprint of uncertainty for an interval-valued fuzzy set

In computing with words, words can be modelled by interval-valued (IV) fuzzy sets (FSs). Constructing an footprint of uncertainty (FOU) for an IV FS about a word has been a critical issue. Although a number of methods have been developed, it remains challenging for each people to provide an FOU about his or her word. A primary reason is that FOUs are implicit in indicating the possibilities of the values of the variable. To overcome this challenge, this paper aims to interpret the FOU of an IV FS by revealing the possibilities of the values of its variable. Centroid of an IV FS has been shown to be a measure of the uncertainties inherent in its FOU. The normalized weights of the values of the variable used to compute its centroid, which can be regarded as a type-1 (T1) membership function (MF), are straightforward to show the possibilities of these values. The study can then be performed by revealing the relationship between the FOU and this T1 MF. This T1 FS is called an equivalent T1 (ET1) FS of the IV FS. In this paper, the theory of ET1 FSs will be presented, including how to construct an ET1 FS for an IV FS. Equations relating the lower MF (LMF) and upper MF (UMF) of an IV FS with the MF of its ET1 FS will be established. Using the established equations, properties about the MF of the ET1 FS for an IV FS will be presented to reveal the relationship between its FOU and the MF of its ET1 FS. These properties are helpful for people to relate the FOU of an IV FS with the possibilities of the values of its variable that occur.