The generalized Heronian mean and geometric Heronian mean operators provide two aggregation operators that consider the interdependent phenomena among the aggregated arguments. In this paper, the generalized Heronian mean operator and geometric Heronian mean operator under the q‐rung orthopair fuzzy sets is studied. First, the q‐rung orthopair fuzzy generalized Heronian mean (q‐ROFGHM) operator, q‐rung orthopair fuzzy geometric Heronian mean (q‐ROFGHM) operator, q‐rung orthopair fuzzy generalized weighted Heronian mean (q‐ROFGWHM) operator, and q‐rung orthopair fuzzy weighted geometric Heronian mean (q‐ROFWGHM) operator are proposed, and some of their desirable properties are investigated in detail. Furthermore, we extend these operators to q‐rung orthopair 2‐tuple linguistic sets (q‐RO2TLSs). Then, an approach to multiple attribute decision making based on q‐ROFGWHM (q‐ROFWGHM) operator is proposed. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Some q‐rung orthopair fuzzy Heronian mean operators in multiple attribute decision making

Pythagorean fuzzy set, generalized by Yager, is a new tool to deal with vagueness considering the membership grade $$\mu $$
 and non-membership $$\nu $$
 satisfying the condition $$\mu ^2+\nu ^2\le 1$$
 . It can be used to characterize the uncertain information more sufficiently and accurately than intuitionistic fuzzy set. Pythagorean fuzzy set has attracted great attention of many scholars that have been extended to new types and these extensions have been used in many areas such as decision making, aggregation operators, and information measures. Because of such a growth, we present an overview on Pythagorean fuzzy set with aim of offering a clear perspective on the different concepts, tools and trends related to their extension. In particular, we provide two novel algorithms in decision making problems under Pythagorean fuzzy environment. It may be served as a foundation for developing more algorithms in decision making.

Pythagorean fuzzy set: state of the art and future directions

This paper presents a new approach for the treatment of uncertainty which is based on interval-valued fuzzy-rough numbers (IVFRN). It is shown that by integrating the rough approach with the traditional fuzzy approach, the subjectivity that exists when defining the borders of fuzzy sets is eliminated. IVFRN make decision making possible using only the internal knowledge in the operative data available to the decision makers. In this way objective uncertainties are used and there is no need to rely on models of assumptions. Instead of different external parameters in the application of IVFRN, the structure of the given data is used. On this basis an original multi-criteria model was developed based on an IVFRN approach. In this multi-criteria model the traditional steps of the BWM (Best–Worst method) and MABAC (Multi-Attributive Border Approximation area Comparison) methods are modified. The model was tested and validated on a study of the optimal selection of fire fighting helicopters. Testing demonstrated that the model based on IVFRN enabled more objective expert evaluation of the criteria in comparison with traditional fuzzy and rough approaches. A sensitivity analysis of the IVFRN BWM-MABAC model was carried out by means of 57 scenarios, the results of which showed a high degree of stability. The results of the IVFRN model were validated by comparing them with the results of the fuzzy and rough extension of the MABAC, COPRAS and VIKOR models.

Modification of the Best–Worst and MABAC methods: A novel approach based on interval-valued fuzzy-rough numbers

The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets

The hesitant fuzzy linguistic term set (HFLTS) has gained great success as it can be used to represent several linguistic terms or comparative linguistic expressions together with some context-free grammars. This new approach has enabled the analysis and computing of linguistic expressions with uncertainties and opened the door for the possibility to develop more comprehensive and powerful decision theories and methods based on linguistic knowledge. Lots of new approaches and proposals for decision-making problems have been proposed to overcome the limitations of previous linguistic decision-making approaches. Now and in the future, decision-making methodologies and algorithms with hesitant fuzzy linguistic models would be a quite promising research line representing a high-quality breakthrough in this topic. To facilitate the study on HFLTS theory, this paper makes a state-of-the-art survey on HFLTSs based on the 134 selected papers from Web of Sciences published from January 2012 to October 2017. We justify the motivation, definitions, operations, comparison methods and measures of HFLTSs. We also summarize the different extensions of HFLTSs. The studies on multiple criteria decision making (MCDM) with HFLTSs in terms of aggregation operators and MCDM methods are clearly reviewed. We also conduce some overviews on decision making with hesitant fuzzy linguistic preference relations. The applications, research challenges and future directions are also given.

Hesitant Fuzzy Linguistic Term Set and Its Application in Decision Making: A State-of-the-Art Survey

https://onlinelibrary.wiley.com/doi/pdf/10.1111/itor.12363

An extended TODIM approach with intuitionistic linguistic numbers

As an extension of type-1 fuzzy sets (T1FSs), interval type-2 fuzzy sets (IT2FSs) can be used to model both extrinsic and intrinsic uncertainties. Based on the likelihood of interval type-2 fuzzy numbers (IT2FNs), this paper proposes a new multi-attributive border approximation area comparison (MABAC) approach to solve multi-criteria decision-making (MCDM) problems. First, an algorithm to decompose IT2FNs into the embedded type-1 fuzzy numbers (T1FNs) is proposed. Second, based on the closeness degree of T1FNs, the likelihood of IT2FNs is defined using the decomposition algorithm, and related properties are discussed. Third, the total ranking of alternatives is obtained using the MABAC approach based on the likelihood of IT2FNs. Finally, a practical example of selecting hotels from a tourism website is presented to verify the validity and feasibility of the proposed approach. A comparative analysis with existing methods is also described.

An Interval Type-2 Fuzzy Likelihood-Based MABAC Approach and Its Application in Selecting Hotels on a Tourism Website

A multi-criteria decision-making model for hotel selection with linguistic distribution assessments

Due to the limitation of knowledge and the vagueness of human being thinking, decision makers prefer to use hesitant fuzzy linguistic sets (HFLSs) to estimate alternatives. Some methods of HFLSs have been researched based on the more familiar means such as the arithmetic mean and the geometric mean; however, Maclaurin symmetric mean (MSM) that can be used to reflect the interrelationships among input arguments have not been applied to solve hesitant fuzzy linguistic multi‐criteria decision‐making problems. In this paper, two hesitant fuzzy linguistic harmonic averaging operators are proposed: the hesitant fuzzy linguistic MSM (HFLMSM) operator and the hesitant fuzzy linguistic weighted MSM (HFLWMSM) operator. Furthermore, an approach based on the HFLWMSM operator is proposed. Finally, to verify the validity and feasibility of the proposed approach, an illustrative example and corresponding comparison analysis are presented in the end.

Hesitant Fuzzy Linguistic Maclaurin Symmetric Mean Operators and their Applications to Multi‐Criteria Decision‐Making Problem

Linguistic hesitant fuzzy sets (LHFSs) are a very useful and appropriate means of expressing preferences of decision-makers; moreover their basic operations and comparison methods have been defined and applied to the solving of MCDM problems. However, there are a number of limitations in the related studies. In this paper, using information from existing studies, several new operations and a new order relationship are defined; moreover four linguistic hesitant fuzzy Heronian mean operators are proposed: the linguistic hesitant fuzzy arithmetic Heronian mean (LHFAHM) operator; the linguistic hesitant fuzzy weighted arithmetic Heronian mean (LHFWAHM) operator; the linguistic hesitant fuzzy geometric Heronian mean (LHFGHM) operator; and the linguistic hesitant fuzzy weighted geometric Heronian mean (LHFWGHM) operator. Furthermore, some useful and desirable properties of these operators are analyzed in some special cases, with respect to the different parameter values in these operators, are discussed. Additionally, an approach based on the LHFWAHM and LHFWGHM operators for solving MCDM problems is proposed. Finally, an illustrative example is provided to verify the validity and feasibility of the proposed approaches, and a comparison analysis is also presented to demonstrate the influences of different parameters on the results of decision-making.

Su-min Yu

Papers

An extended TODIM approach with intuitionistic linguistic numbers

An Interval Type-2 Fuzzy Likelihood-Based MABAC Approach and Its Application in Selecting Hotels on a Tourism Website

A multi-criteria decision-making model for hotel selection with linguistic distribution assessments

Hesitant Fuzzy Linguistic Maclaurin Symmetric Mean Operators and their Applications to Multi‐Criteria Decision‐Making Problem

A Multi-Criteria Decision-Making Method Based on Heronian Mean Operators Under a Linguistic Hesitant Fuzzy Environment