Algorithms for Solving Decision-Making Problem Under Type-2 Fuzzy Sets

Abstract

Multiple attribute decision-making (MADM) is one of the hot topics in the field of the decision-making process to access the best alternative(s) from the feasible ones. In literature, many terms have been used for MADM such as multi-criteria decision analysis (MCDA), multi-objective decision-making (MODM), multi-criteria decision-making (MCDM), etc. and have been frequently used by the researchers to solve real-world decision-making problems. Generally, the MADM issue is explained in the two-stage process: (i) the aggregation of the estimations of criteria for every option (ii) the positioning or ranking between the options.In real decision-making, the decision makers (DMs) need to give their evaluation information of attributes by various types of the evaluation process, such as crisp numbers, interval numbers, fuzzy numbers, and so. However, in many practical cases, because of the increasing uncertainty in the data and various cognition constraints of DMs, it is often difficult for DMs to use real values to express their preferences. To ease with it, a concept of fuzzy set (FS) is introduced by Zadeh in 1965 which adopts the membership degree (MD) to describe the information. After it, various extensions of FSs come into the existence such as intuitionistic fuzzy sets (IFSs), interval-valued IFSs (IVIFSs), Type-2 fuzzy set (T2FSs), Hesitant fuzzy sets (HFSs), and so on, to deal with the uncertain and imprecise information. In the theories of FSs and its extensions, a crisp membership function is assigned to its element. However, in many situations, uncertainty is not probabilities in nature but it is imprecise or vague in nature. To address it, the concept of type-2 fuzzy set (T2FS) was developed by Mendel in 2002, an extension of FS, in which membership values are type-1 FSs on [0,1] is developed. In T2FS, there is an additional membership function which provides an additional degree of freedom to the practices to model the uncertainties and each element is characterized by the degrees of the primary, secondary and a footprint of uncertainty (FOU). In the theories of FSs and its extensions, a crisp membership function is assigned to its element. However, in many situations, uncertainty is not probabilities in nature but it is imprecise or vague in nature. To address it, the concept of type-2 fuzzy set (T2FS) was developed by Mendel in 2002, an extension of FS, in which membership values are type-1 FSs on [0,1] is developed. In T2FS, there is an additional membership function which provides an additional degree of freedom to the practices to model the uncertainties and each element is characterized by the degrees of the primary, secondary and a footprint of uncertainty (FOU).

After this pioneering work, researchers have been engaged in extensions and applications to different disciplines. However, the most important task for the decision-maker is to rank the objects so as to obtain the desired object(s). For this, researchers have made efforts to enrich the concept of information measures as well as aggregation operators in type-2 fuzzy environments. Among these, an aggregation operator is an important part of the decision-making which usually takes the form of mathematical function to aggregate all the input individual data into a single one. However, an information measure such as the distance and similarity measures, complementary to each other, are defined to differentiate between the two or more objects. Thus in order to handle the information in a more accurate and certain manner, there is a need to plan/adopt suitable methodologies to solve the decision-making problems.

The aim of this work is to develop some novel techniques to access the best alternative(s) for the decision makers under the T2FSs and its extensions environment.

Keywords: Type-2 fuzzy and intuitionistic fuzzy sets; Aggregation operators; Triangular interval type-2 (TIT2) intuitionistic fuzzy sets; Symmetric TIT2 intuitionistic fuzzy sets.