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Volumetric fuzzy set and its application in optimization problems
Author(s) -
Arman Hosein
Publication year - 2021
Publication title -
international journal of intelligent systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.291
H-Index - 87
eISSN - 1098-111X
pISSN - 0884-8173
DOI - 10.1002/int.22429
Subject(s) - mathematics , degree (music) , euclidean distance , membership function , point (geometry) , fuzzy set , set (abstract data type) , euclidean geometry , plane (geometry) , fuzzy logic , coordinate system , algorithm , mathematical optimization , computer science , artificial intelligence , geometry , programming language , physics , acoustics
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.