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.

Fuzzy sets that have been presented so far are generally related to one-coordinate variables. However, some variables are identified by two coordinates, such as a location on a plane. In this case, the membership degree of each point in a fuzzy set is determined by considering the simultaneous values of both coordinates. In this paper, we develop the classical fuzzy set for two-coordinate variables; we call it the volumetric fuzzy set (VFS) due to the fact that a point on a plane along with its membership degree forms a corresponding point in a three-dimensional coordinate system. In this study, we present a specific type of VFS in which there is only one ideal point with the highest membership degree, and the membership degrees of other points are inversely correlated with their distances from this ideal point. For this purpose, we apply three types of distances which are Euclidean, Manhattan, and Chebyshev distances. The membership function extracted based on Euclidean distance may have a common volumetric shape, a cone shape, for example, described in detail in this paper; while those extracted based on the other types of distance usually have unusual volumetric shapes. This study also shows the application of the VFS in optimization problems. For this purpose, three location optimization problems are developed based on different types of volumetric membership functions and then solved using the max–min approach. To illustrate the proposed volumetric optimization models, three numerical examples are used and their results are compared.

There are simple approximated methods to extract the local weights from a pairwise comparison matrix which are row sums, inverse of column sums, arithmetic mean, and geometric mean. In this paper, first, we extend these methods to fuzzy analytic hierarchy process (FAHP) to extract the local weights as fuzzy numbers (FNs). Then, these weights are defuzzified using the center of gravity (COG) method. We also propose an approach to integrate different local weights obtained from different approximated methods to achieve a unified local weight. Moreover, this study proposes a novel and simple approach in which uncommon FNs are indirectly defuzzified based on COG method. This helps extend the multi-attribute decision-making methods to uncommon FNs. To illustrate the applicability of the proposed approaches, three numerical examples are given and their results are compared with some well-known FAHP methods in the literature.