Abstract
The representation theorem for interval type-2 fuzzy sets (IT2 FSs), proposed by Mendel and John, states that an IT2 FS is a combination of all its embedded type-1 (T1) FSs, which can be nonconvex and/or subnormal. These nonconvex and/or subnormal embedded T1 FSs are included in developing many theoretical results for IT2 FSs, including uncertainty measures, the linguistic weighted averages (LWAs), the ordered LWAs (OLWAs), the linguistic weighted power means (LWPMs), etc. However, convex and normal T1 FSs are used in most fuzzy logic applications, particularly computing with words. In this paper, we propose a constrained representation theorem (CRT) for well-shaped IT2 FSs using only its convex and normal embedded T1 FSs, and show that IT2 FSs generated from three word encoding approaches and four computing with words engines (LWAs, OLWAs, LWPMs, and perceptual reasoning) are all well-shaped IT2 FSs. We also compute five constrained uncertainty measures (centroid, cardinality, fuzziness, variance, and skewness) for well-shaped IT2 FSs using the CRT. The CRT and the associated constrained uncertainty measures can be useful in computing with words, IT2 fuzzy logic system design using the principles of uncertainty, and measuring the similarity between two well-shaped IT2 FSs.
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
