Next Article in Journal
Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets
Next Article in Special Issue
Normalized Weighted Bonferroni Harmonic Mean-Based Intuitionistic Fuzzy Operators and Their Application to the Sustainable Selection of Search and Rescue Robots
Previous Article in Journal
An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity
Previous Article in Special Issue
Comparative Evaluation of Sustainable Design Based on Step-Wise Weight Assessment Ratio Analysis (SWARA) and Best Worst Method (BWM) Methods: A Perspective on Household Furnishing Materials
Article Menu
Issue 2 (February) cover image

Export Article

Open AccessArticle
Symmetry 2019, 11(2), 170;

Type-2 Multi-Fuzzy Sets and Their Applications in Decision Making

Department of Computer Science, Heritage Institute of Technology, Kolkata 700107, India
Department of Mathematics, National Institute of Technology, Durgapur 713209, India
Department of Computer Science and Engineering, National Institute of Technology, Durgapur 713209, India
Department of Logistics, University of Defence in Belgrade Military Academy, 11000 Belgrade, Serbia
Author to whom correspondence should be addressed.
Received: 30 December 2018 / Revised: 22 January 2019 / Accepted: 28 January 2019 / Published: 1 February 2019
Full-Text   |   PDF [1112 KB, uploaded 20 February 2019]   |  


In a real-life scenario, it is undoable and unmanageable to solve a decision-making problem with the single stand-alone decision-aid method, expert assessment methodology or deterministic approaches. Such problems are often based on the suggestions or feedback of several experts. Usually, the feedback of these experts are heterogeneous imperfect information collected from various more or less reliable sources. In this paper, we introduce the concept of multi-sets over type-2 fuzzy sets. We have tried to propose an extension of type-1 multi-fuzzy sets into a type-2 multi-fuzzy set (T2MFS). After defining T2MFS, we discuss the algebraic properties of these sets including set-theoretic operations such as complement, union, intersection, and others with examples. Subsequently, we define two distance measures over these sets and illustrate a decision-making problem which uses the idea of type-2 multi-fuzzy sets. Furthermore, an application of a medical diagnosis system based on multi-criteria decision making of T2MFS is illustrated with a real-life case study. View Full-Text
Keywords: Multi-fuzzy sets; type-2 fuzzy sets; type-2 multi-fuzzy sets; distance measures; set operations Multi-fuzzy sets; type-2 fuzzy sets; type-2 multi-fuzzy sets; distance measures; set operations

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

Supplementary material


Share & Cite This Article

MDPI and ACS Style

Kar, M.B.; Roy, B.; Kar, S.; Majumder, S.; Pamucar, D. Type-2 Multi-Fuzzy Sets and Their Applications in Decision Making. Symmetry 2019, 11, 170.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top