A simple noniterative method to accurately calculate the centroid of an interval type‐2 fuzzy set

Abstract

An interval type-2 fuzzy set (IT2FS) contains the infinite number of embedded membership functions (MFs) defined as type-1 fuzzy sets. The center of gravity (COG) of an IT2FS is obtained by integrating the centroids of all these MFs. However, obtaining the COG of an IT2FS in a continuous domain is impossible because the number of embedded MFs is infinite and, in a discrete domain, comes with exponential time complexity. Therefore, different algorithms were proposed to find the center of centroids (COCs) instead of the COG. These algorithms do not necessarily obtain the exact value of COC; they compete to find a more accurate result in less time. In this study, we propose a simple noniterative method to obtain the COG of IT2FSs. This method only considers the MFs which form the boundary values, that is, the lower MF (LMF) and upper MF (UMF), and obtains their centroids separately. We show that the COG of an IT2FS is equal to the weighted sum of the centroids of its LMF and UMF if the relative areas of LMF and UMF are considered their corresponding weights. We call this method the Indirect COG (ICOG) method because it indirectly obtains the COG of an IT2FS. This method finds the accurate COG of an IT2FS by considering only two MFs instead of infinite MFs. It can also obtain the accurate COG of an uncommon polygonal IT2FS schematically. We use numerical examples to illustrate that the ICOG obtains the accurate COG of an IT2FS very fast in a nonexponential time. We also develop a new triangular interval type-2 fuzzy analytical hierarchy process that uses the ICOG method to accurately defuzzify the IT2 fuzzy weights.