T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems

As the extension of fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, and picture fuzzy set, the spherical fuzzy set is characterized by three functions expressing the positive-membership degree, the neutral-membership degree and the negative-membership degree which the sum squares of them is equal or less than 1. In this work, we shall present some novel Dice similarity measures of spherical fuzzy sets and the generalized Dice similarity measures of spherical fuzzy sets and indicates that the Dice similarity measures and asymmetric measures (projection measures) are the special cases of the generalized Dice similarity measures in some parameter values. Then, we propose the generalized Dice similarity measures-based multiple attribute group decision making models with spherical fuzzy information. Then, we apply the generalized Dice similarity measures between spherical fuzzy sets to multiple attribute group decision making. Finally, an illustrative example is given to demonstrate the efficiency of the similarity measures for selecting the desirable ERP system.

The existing concepts of picture fuzzy sets (PFS), spherical fuzzy sets (SFSs), T-spherical fuzzy sets (T-SFSs) and neutrosophic sets (NSs) have numerous applications in decision-making problems, but they have various strict limitations for their satisfaction, dissatisfaction, abstain or refusal grades. To relax these strict constraints, we introduce the concept of spherical linear Diophantine fuzzy sets (SLDFSs) with the inclusion of reference or control parameters. A SLDFS with parameterizations process is very helpful for modeling uncertainties in the multi-criteria decision making (MCDM) process. SLDFSs can classify a physical system with the help of reference parameters. We discuss various real-life applications of SLDFSs towards digital image processing, network systems, vote casting, electrical engineering, medication, and selection of optimal choice. We show some drawbacks of operations of picture fuzzy sets and their corresponding aggregation operators. Some new operations on picture fuzzy sets are also introduced. Some fundamental operations on SLDFSs and different types of score functions of spherical linear Diophantine fuzzy numbers (SLDFNs) are proposed. New aggregation operators named spherical linear Diophantine fuzzy weighted geometric aggregation (SLDFWGA) and spherical linear Diophantine fuzzy weighted average aggregation (SLDFWAA) operators are developed for a robust MCDM approach. An application of the proposed methodology with SLDF information is illustrated. The comparison analysis of the final ranking is also given to demonstrate the validity, feasibility, and efficiency of the proposed MCDM approach.

The extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and neutrosophic sets (NS), whose membership functions are based on three dimensions, aim at collecting experts’ judgments more informatively and explicitly. In the literature, generalized three-dimensional spherical fuzzy sets have been developed by Kutlu Gündoğdu and Kahraman [1], including their arithmetic operations, aggregation operators, and defuzzfication operations. Spherical Fuzzy Sets (SFS) are a new extension of Intuitionistic, Pythagorean and Neutrosophic Fuzzy sets. In this paper, our aim is to employ SF-VIKOR method to waste management problems. We handle a waste disposal site selection problem with five alternatives and four criteria in order to demonstrate the performance of the proposed SF-VIKOR method.

Design for six sigma (DFSS) is a structured approach for designing new products, processes and/or services or redesigning the existing ones. It builds quality into the product during the development and ensures the customer expectations are understood, met, or exceed. In DFSS, variation is aimed to be minimized by applying the techniques of the six sigma approach. Some of these techniques are process control and process capability analysis. Vagueness and impreciseness in these analyses such as the statistical distribution of the considered process can be handled by the fuzzy set theory. Recently, ordinary fuzzy sets have been extended to several new types of fuzzy sets such as Pythagorean fuzzy sets, spherical fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. In this paper, we propose a new extension of intuitionistic fuzzy sets, which is called Penthagorean fuzzy sets and employ it in the process capability analysis of a production process. Penthagorean fuzzy process capability indices are developed and applied to an Autonomous Underwater Vehicle (AUV) design example.

The extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and neutrosophic sets (NS), whose membership functions are based on three dimensions, aim to describe expert’s judgments more informatively and explicitly. Introduction of generalized three dimensional spherical fuzzy sets (SFS) including some essential differences from the other fuzzy sets is presented in the literature with their arithmetic, aggregation, and defuzzfication operations [1]. This study summarizes the previously introduced spherical fuzzy sets and as an application spherical fuzzy CODAS method will be applied to the warehouse location selection problem.

In addition to the well-known fuzzy sets, a novel type of fuzzy sets called as spherical fuzzy set (SFS) is recently introduced in the literature. SFS is the generalized structure over existing structures of fuzzy sets (intuitionistic fuzzy sets, Pythagorean fuzzy sets, and neutrosophic sets) based on three dimensions (truth, falsehood, and indeterminacy) to provide a wider choice for decision makers (DMs). Although the SFS is introduced recently, the topic attracts the attention of academicians at a remarkable rate. Hence, a comprehensive literature review of recent and state-of-the-art papers is vital to draw a framework of the past, and to shed light on future directions. To do so, a systematic review methodology is followed and 43 academic papers include SFS are selected and reviewed. The papers are then analyzed and categorized based on the titles and abstracts to construct a useful foundation of past research. Finally, trends and gaps in the literature are identified to clarify and to suggest future research opportunities in fuzzy logic area.

This paper deals with uncertainty, asymmetric information, and risk modelling in a complex power system. The uncertainty is managed by using probability and decision theory methods. Multiple-criteria decision making (MCDM) is a very effective and well-known tool to investigate fuzzy information more effectively. However, the selection of houses cannot be done by utilizing symmetry information, because enterprises do not have complete information, so asymmetric information should be used when selecting enterprises. In this paper, the notion of soft set (SftS) and interval-valued T-spherical fuzzy set (IVT-SFS) are combined to produce a new and more effective notion called interval-valued T-spherical fuzzy soft set (IVT−SFSftS). It is a more general concept and provides more space and options to decision makers (DMs) for making their decision in the field of fuzzy set theory. Moreover, some average aggregation operators like interval-valued T-spherical fuzzy soft weighted average (IVT−SFSftWA) operator, interval-valued T-spherical fuzzy soft ordered weighted average (IVT−SFSftOWA) operator, and interval-valued T-spherical fuzzy soft hybrid average (IVT−SFSftHA) operators are explored. Furthermore, the properties of these operators are discussed in detail. An algorithm is developed and an application example is proposed to show the validity of the present work. This manuscript shows how to make a decision when there is asymmetric information about an enterprise. Further, in comparative analysis, the established work is compared with another existing method to show the advantages of the present work.

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

Pythagorean fuzzy set (PFS), proposed by Yager (2013), is a generalization of the notion of Atanassov's intuitionistic fuzzy set, which has received more and more attention. In this paper, first, we define the weighted Minkowski distance with interval-valued PFSs. Second, inspired by the idea of the Pythagorean fuzzy linguistic variables, we define a new fuzzy variable called interval-valued Pythagorean fuzzy linguistic variable set (IVPFLVS), and the operational laws, score function, accuracy function, comparison rules, and distance measures of the IVPFLVS are defined. Third, some aggregation operators are presented for aggregating the interval-valued Pythagorean fuzzy linguistic information such as the interval-valued Pythagorean fuzzy linguistic weighted averaging (IVPFLWA), interval-valued Pythagorean fuzzy linguistic ordered weighted averaging (IVPFLOWA) , interval-valued Pythagorean fuzzy linguistic hybrid averaging, and generalized interval-valued Pythagorean fuzzy linguistic ordered weighted average operators. Fourth, some desirable properties of the IVPFLWA and IVPFLOWA operators, such as monotonicity, commutativity, and idempotency, are discussed. Finally, based on the IVPFLWA or interval-valued Pythagorean fuzzy linguistic geometric weighted operator, a practical example is provided to illustrate the application of the proposed approach and demonstrate its practicality and effectiveness.

The notions of fuzzy set (FS) and intuitionistic fuzzy set (IFS) make a major contribution to dealing with practical situations in an indeterminate and imprecise framework, but there are some limitations. Pythagorean fuzzy set (PFS) is an extended form of the IFS, in which degree of truthness and degree of falsity meet the condition 0≤Θ˘2(x)+K2(x)≤1. Another extension of PFS is a q´-rung orthopair fuzzy set (q´-ROFS), in which truthness degree and falsity degree meet the condition 0≤Θ˘q´(x)+Kq´(x)≤1,(q´≥1), so they can characterize the scope of imprecise information in more comprehensive way. q´-ROFS theory is superior to FS, IFS, and PFS theory with distinguished characteristics. This study develops a few aggregation operators (AOs) for the fusion of q´-ROF information and introduces a new approach to decision-making based on the proposed operators. In the framework of this investigation, the idea of a generalized parameter is integrated into the q´-ROFS theory and different generalized q´-ROF geometric aggregation operators are presented. Subsequently, the AOs are extended to a “group-based generalized parameter”, with the perception of different specialists/decision makers. We developed q´-ROF geometric aggregation operator under generalized parameter and q´-ROF geometric aggregation operator under group-based generalized parameter. Increased water requirements, in parallel with water scarcity, force water utilities in developing countries to follow complex operating techniques for the distribution of the available amounts of water. Reducing water losses from water supply systems can help to bridge the gap between supply and demand. Finally, a decision-making approach based on the proposed operator is being built to solve the problems under the q´-ROF environment. An illustrative example related to water loss management has been given to show the validity of the developed method. Comparison analysis between the proposed and the existing operators have been performed in term of counter-intuitive cases for showing the liability and dominance of proposed techniques to the existing one is also considered.

T-spherical fuzzy set (TSFS) is a generalized version of the spherical fuzzy set (SFS) and picture fuzzy set (PFS) to manage awkward and unpredictable information in realistic decision issues. TSFS deals with yes, abstinence, no, and refusal type of fuzzy information. This manuscript aims to observe the drawbacks of some existing dice similarity measures (DSMs) and to propose some new DSMs in the environment of TSFSs. The validation of the new DSMs is proved. The defined DSMs are further extended to introduce some generalized DSMs (GDSMs) and their special cases are studied. Additionally, the TOPSIS method using the entropy measures (EMs) based on TSFSs is also explored and verified with the help of some examples. The proposed new GDSMs and TOPSIS method are applied to the problem of building material recognition, medical diagnosis, clustering, and the results obtained are investigated. A comparison of the new theory is established where the advancement of the proposed DSMs is elaborated under some conditions. The advantages of the new DSMs and the drawbacks of the previous DSMs of IFSs, PyFSs, and PFSs have been studied because of their applicability. The article is comprehensively summarized, and some possible future directions are stated.

In a fuzzy set (FS), there is a concept of alpha-cuts of the FS for alpha in [0,1]. Further, this concept was extended into (alpha,delta)-cuts in an intuitionistic fuzzy set (IFS) for delta in [0,1]. One of the expansions of FS and IFS is the picture fuzzy set (PFS). Hence, the concept of (alpha,delta)-cuts was developed into (alpha,delta,beta)-cuts in a PFS where beta is an element of [0,1]. Since a picture fuzzy graph (PFG) consists of picture fuzzy vertex or edge sets or both of them, we have an idea to construct the notion of the (alpha,delta,beta)-cuts in a PFG. The steps used in this paper are developing theories and algorithms. The objectives in this research are to construct the concept of (alpha,delta,beta)-cuts in picture fuzzy graphs (PFGs), to construct the (alpha,delta,beta)-cuts coloring of PFGs, and to design an algorithm for finding the cut chromatic numbers of PFGs. The first result is a definition of the (alpha,delta,beta)-cut in picture fuzzy graphs (PFGs) where (alpha,delta,beta) are elements of a level set of the PFGs. Further, some properties of the cuts are proved. The second result is a concept of PFG coloring and the chromatic number of PFG based on the cuts. The third result is an algorithm to find the cuts and the chromatic numbers of PFGs. Finally, an evaluation of the algorithm is done through Matlab programming. This research could be used to solve some problems related to theories and applications of PFGs.

Picture fuzzy set (PFS) is a direct generalization of the fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs) and is quite powerful in modelling the situations that involve more answers of the type yes, no, abstain, and refuse. In this study, we introduce a novel picture fuzzy (PF) similarity measure on the basis of direct operation on the function of membership, the function of non-membership, the function of neutrality, the function of refusal, and the upper bound of the function of membership of two PFSs instead on the basis of distance measure or the association between the functions of membership, non-membership, and neutrality. The comparison of the proposed PF similarity measure with the existing PF similarity measures reveals that it does not give unreasonable results and also overcomes the drawbacks of the existing PF similarity measures. The application of the proposed measure in pattern recognition is also discussed. Moreover, we also introduce a new multi-attribute decision-making (MADM) method using the proposed PF similarity measure that overcomes a major drawback of the technique for order preference by similarity to ideal solution (TOPSIS). Finally, we contrast the performance of the proposed MADM method with several existing MADM methods in the PF environment.

Multi-attribute group decision-making based on q-rung orthopair fuzzy Aczel–Alsina power aggregation operators

The researcher has been facing problems while handling imprecise and vague information, i.e., the problems of networking, decision-making, etc. For encountering such complicated data, the notion of fuzzy sets (FS) has been considered an influential tool. The notion was extended to its generalizations by a number of researchers in different ways which helps to understand and assess even more complex issues. This article characterizes imprecision with four kinds of values of membership. In this work, we aim to define and examine cubic picture fuzzy sets and give an application on averaging aggregation operators. We first introduce the notion of a cubic picture fuzzy set, which is a pair of interval-valued picture fuzzy set and a picture fuzzy set by giving examples. Then, we define two kinds of ordering on these sets and also discuss some set-theoretical properties. Moreover, we introduce three kinds of averaging aggregation operators based on cubic picture fuzzy sets and, at the end, we illustrate the results with a decision-making problem by using one of the provided aggregation operators.