Intuitionistic Fuzzy Sets, Theory and Applications
Chapters (4)
A new operator defined over an interval-valued intuitionistic fuzzy set is defined. It gives as a result an intuitionistic fuzzy set. Some properties of the new operator are studied.
The definition of intuitionistic fuzzy sets will serve as a basis for further definitions of the elements of the intuitionistic fuzzy logics (IFLs). Here we shall present basic elements of:
Intuitionistic Fuzzy Propositional Calculus (IFPC),
Intuitionistic Fuzzy Predicate Logic (IFPL),
Intuitionistic Fuzzy Modal Calculus (IFMC), and
Temporal Intuitionistic Fuzzy Logic (TIFL).
The intuitionistic fuzzy sets can be used everywhere the ordinary fuzzy sets can be applied. On the other hand, this is not always necessary.
The basic problems related to intuitionistic set theory (IFS) were discussed. Constructing an axiomatic system, and developing efficient algorithms for the construction of degree of membership and non-membership are the most common problems related to an IFS. Research of the possible geometrical interpretations of IFSs and their classification were also discussed. Particularization of the possible operations and relations over IFS and integration of their algebraic properties are some other major problems.
... Atanassov [14] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs (IFGs). Articles [8,9,14,35] motivated us to analyze balanced IFGs and their properties. ...
... Atanassov [14] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs (IFGs). Articles [8,9,14,35] motivated us to analyze balanced IFGs and their properties. We modify balanced IFGs notion to semi-IFGs and study their properties. ...
... [14] An intuitionistic fuzzy graph (simply, IFG) is of the formG : (V; E) where (i) V = f 0 ; 1 ; :::::::; n g such that 1 : V ! [0; 1] and 1 : V ! ...
In this paper, we introduce the relatively new notions of semi-fuzzy graph and balanced semi-fuzzy graph. We study several operations on these graphs such as complement, union, join, composition, direct product, semi-strong product and strong product. In addition, we provide some classes of balanced semi-fuzzy graphs. Similar work is also done for Intuitionistic semi-fuzzy graphs.
... The concept of IFSs though resourceful could not model situation in which the summation of non-membership degree and membership degree is at least one. Hence, intuitionistic fuzzy set of the second type (IFSST) [17] was proposed. IFSST is popularly known as Pythagorean fuzzy sets (PFSs) [18], [19]. ...
... Similarly, using (13)(14)(15)(16)(17)(18) and (20), we get the similarities in Table II. ...
... Similarly, the similarities in Table IV are gotten using (13)(14)(15)(16)(17)(18) and (20). The results in Table IV show that the new approach of similarity satisfies Definition 7 unlike Ψ 1 , Ψ 2 and Ψ 5 with a better performance. ...
... Definition 0.17. [2,3] "An intuitionistic fuzzy set (IFS) A in X can be represented as an object of the form ...
... Remark 0.18. [2,3] (i) "When µ A (x) + ν A (x) = 1, ∀x ∈ X. Then A is called a fuzzy set. ...
The investigation of mathematics underlines accuracy, precision, and flawlessness, yet in numerous genuine circumstances, individuals face equivocalness, ambiguity, imprecision, and so forth. Intuitionistic fuzzy set theory, rough set theory, and soft set theory are three noble techniques in mathematics that are utilised for decision-making in vague and uncertain information systems. Intuitionistic fuzzy algebra-based math plays a huge part in the current era of mathematical research, and it deals with the algebraic concepts and models of intuitionistic fuzzy sets. The investigation of different ordered algebraic structures, like lattice-ordered groups, Riesz spaces, etc., is of great importance in algebra. The theory of lattice-ordered G-modules is very useful in the study of lattice-ordered groups and similar algebraic structures. In this article, the theories of intuitionistic fuzzy sets and lattice-ordered G-modules are synchronised in a reasonable way to develop a novel concept in mathematics, i.e., intuitionistic fuzzy lattice-ordered G-modules, which would pave the way for new researchers in intuitionistic fuzzy mathematics to explore much more in this field.
... (a) Interval type 2 fuzzy numbers (Mendel 2017) which are characterized by two membership functions, and (b) The intuitionistic fuzzy sets introducing another degree of freedom into a set description (membership function, non-membership function, and intuitionistic index foundation) (Atanassov 1999). ...
... Definition 7 Let a non-empty set is given X. An IFS from the set X may be presented in the following way (Atanassov 1999): where: ...
The Delphi method has been widely utilized in a number of scientific domains to aid forecasting and decision-making. Created by Dalkey and Helmer (Manag Sci 9:458–467, 1963), the method was developed to forecast future scenarios based on the consensus of expert opinions collected through several iterations of structured questionnaires. Although the approach was adopted in various disciplines, several shortcomings of the method, such as lack of consistent standards for result interpretation, limited generalization, the long-time needed for implementation, and susceptibility to the subjective interpretation of the qualitative results, have been identified over time. The purpose of the study is to review the available scientific literature on the application of a Fuzzy Delphi method in the marketing domain, in order to measure the interest in the method, identify the studies that have utilized the fuzzy Delphi method for marketing related topics, and determine the areas of marketing where the method has been applied so far. Fuzzy Delphi method has found the most extensive application in operation management, decision sciences, and business intelligence and has not been widely adopted as a method of choice in marketing studies.
... Regarding step (i), in [52], the contributors gave the name "de-i-fuzzification" for the scheme for generating a convenient fuzzy set out of an IFS. Moreover, they suggested utilizing the operator presented in [53]: ...
Convolutional Neural Networks (CNNs) are a kind of artificial neural network designed to extract features and find out patterns for tasks such as segmentation, recognizing objects, and drawing up classification. Within a CNNs architecture, pooling operations are used until the number of parameters and the computational complexity are reduced. Numerous papers have focused on investigating the impact of pooling on the performance of Convolutional Neural Networks (CNNs), leading to the development of various pooling models. Recently, a fuzzy pooling operation based on type-1 fuzzy sets was introduced to cope with the local imprecision of the feature maps. However, in fuzzy set theory, it is not always accurate to assume that the degree of non-membership of an element in a fuzzy set is simply the complement of the degree of membership. This is due to the potential existence of a hesitation degree, which implies a certain level of uncertainty. To overcome this limitation, intuitionistic fuzzy sets (IFS) were introduced to incorporate the concept of a degree of hesitation. In this paper, we introduce a novel pooling operation based on intuitionistic fuzzy sets to incorporate the degree of hesitation heretofore neglected by a fuzzy pooling operation based on classical fuzzy sets, and we investigate its performance in the context of image classification. Intuitionistic pooling is performed in four steps: bifuzzification (by the transformation of data through the use of membership and non-membership maps), first aggregation (through the transformation of the IFS into a standard fuzzy set, second aggregation (through the transformation and use of a sum operator), and the defuzzification of feature map neighborhoods by using a max operator. IFS pooling is used for the construction of an intuitionistic pooling layer that can be applied as a drop-in replacement for the current, fuzzy (type-1) and crisp, pooling layers of CNN architectures. Various experiments involving multiple datasets demonstrate that an IFS-based pooling can enhance the classification performance of a CNN. A benchmarking study reveals that this significantly outperforms even the most recent pooling models, especially in stochastic environments.
... It is a very significant and interesting expanded form of the FS with its best applicability. In different areas, IFSs are applied such as DM, medical diagnosis, and optimization (Atanassov, 1999;Kozae et al., 2020). In IFSs, it is necessary that the sum of membership grade and non-membership grade should be 1. ...
... However there were limitations of FS too. To overcome the limitations Atanassov [2,3,10] further generalised FS theory and launched the concept of Intuitionstic FS (IFS) theory. Here degree of neutrality was not taken into consideration. ...
Picture fuzzy set (PFS) is a recent advancement tool to deal with vulnerability. It is an immediate expansion of intuitionistic fuzzy set that can display vulnerability in such circumstances including more responses of these kinds: indeed, decline, no. In this manuscript the idea of Picture fuzzy normed linear space (PFNLS) is discussed for the first time. Naturally PFNLS is an hybrid concept of PFS and normed linear space. Also Convergence in PFNLS are shown. Later on Completeness property on PFNLS are explored. Finally boundedness of Cauchy sequence in PFNLS is analysed.
... Definition 1. An intuitionistic fuzzy set (IFS) S in X is defined by (Atanassov, 1999), as an object of the following form as, ...
The choice of similarity measure (SM) plays an important role in distinguishing between objects. Similarity measure of Pythagorean fuzzy sets (PFSs) is very useful and effective in discriminating between different Pythagorean fuzzy sets. Therefore, in this paper, we suggest a new similarity measure for PFSs based on converting the PFSs into their lower, upper and middle fuzzy sets (FSs) to calculate their degree of similarity. We construct an axiomatic definition for a new SM of PFSs. Furthermore, we put forward a new way to express the similarity measure of PFSs to show its competency, reliability and applicability. For establishing reasonability and usefulness of the proposed methods, we present several practical examples related to pattern recognition and multicriteria decision making problems. Finally, we construct an algorithm for Portuguese of interactive and multiple attributes decision making (TODIM) method based on the proposed similarity measures, for handling complex multicriteria decision making problems related to day to day life. Our final results show that the suggested method is reasonable, reliable and useful in managing different complex decision making problems in the context of Pythagorean fuzzy sets as the domain.
... Te membership function was insufcient to explain exactly the complexity of object features, leading to the suggestion of a nonmembership function with fuzzy sets (FS). Te extension of FS, which is called intuitionistic fuzzy sets (IFS), was introduced [6][7][8]. Later, the notion of m-polar interval-valued intuitionistic fuzzy graphs was introduced [9]. Also, several types of arcs in the interval-valued intuitionistic (S, T)-fuzzy graphs and their properties were studied [10,11]. ...
Neutrosophic graphs are used to model inconsistent information and imprecise data about any real-life problem. It is regarded as a generalization of intuitionistic fuzzy graphs. Since interval-valued neutrosophic sets are more accurate, compatible, and flexible than single neutrosophic sets, interval-valued neutrosophic graphs (IVNGs) were defined. The interval-valued neutrosophic graph is a fundamental issue in graph theory that has wide applications in the real world. Also, problems may arise when partial ignorance exists in the datasets of membership [0, 1], and then, the concept of IVNG is crucial to represent the problems. Line graphs of neutrosophic graphs are significant due to their ability to represent and analyze uncertain or indeterminate information about edge relationships and complex networks in graphs. However, there is a research gap on the line graph of interval-valued neutrosophic graphs. In this paper, we introduce the theory of an interval-valued neutrosophic line graph (IVNLG) and its application. In line with that, some mathematical properties such as weak vertex isomorphism, weak edge isomorphism, effective edge, and other properties of IVNLGs are proposed. In addition, we defined the vertex degree of IVNLG with some properties, and by presenting several theorems and propositions, the relationship between fuzzy graph extensions and IVNLGs was explored. Finally, an overview of the algorithm used to solve the problems and the practical application of the introduced graphs were provided.
... Mathew and Sunitha (2010) investigated node and arc connectivity in FGs. Atanassov (1999) introduced the idea of intuitionistic fuzzy graphs (IFGs). Chakraborty and Mahapatra (2020) provided notes on IFGs. ...
Incidence graphs are an effective to model interconnected networks with additional vertex-edge interactions. They are widely used to establish modes of operation and controllers to illustrate the influence of one factor on another. The purpose of this paper is to present the concept of directed Pythagorean fuzzy incidence graphs. Physical problems that Pythagorean fuzzy incidence graphs cannot effectively illustrate can be modeled by restricting the flow direction because the interactions in these graphs are not symmetric. We discuss the connectivity of directed Pythagorean fuzzy incidence graphs differently by emphasizing legal and illegal network flows. Furthermore, we introduce the concept of legal and illegal flow reduction vertices, edges, and pairs in directed Pythagorean fuzzy incidence graphs and give some significant results. We study the types of legal edges in directed Pythagorean fuzzy incidence graphs and establish some results about legal strong edges. Moreover, we discuss the application of legal fuzzy incidence trees in decision-making, that is, to select the optimal location of the electronic toll collection system on one-way toll roads to maximize toll revenue. Additionally, we provide an algorithm to understand the methods we use in our application. Finally, we compare the proposed method with an existing model that includes numerical results and logical arguments to demonstrate its feasibility and applicability.
... Intuitionistic fuzzy sets was proposed by [5,6] as a generalization of fuzzy set theory by [50]. In this section, we provide basic useful background on intuitionistic fuzzy sets that we later use in our proposal of fuzzy composite metrics. ...
The performance assessment of companies in terms of sustainability requires to find a balance between multiple and possibly conflicting criteria. We here rely on composite metrics to rank a set of companies within an industry considering environmental, social and corporate governance criteria. To this end, we connect intuitionistic fuzzy sets and composite programming to propose novel composite metrics. These metrics allow to integrate important environmental, social and governance principles with the gradual membership functions of fuzzy set theory. The main result of this paper is a sustainability assessment method to rank companies within a given industry. In addition to consider multiple objectives, this method integrates two important social principles such as maximum utility and fairness. A real-world example is provided to describe the application of our sustainability assessment method within the motor industry. A further contribution of this paper is a multicriteria generalization of the concept of magnitude of a fuzzy number.
... Definition 1. Assume X is a given complete set; set S is said to be an Intuitionistic Fuzzy Set (IFS) if it can be expressed by the following formula [51]: ...
The extension of intuitionistic fuzzy sets (IFS) to Pythagorean fuzzy sets (PFS) is a significant advancement, addressing the inherent limitations of IFS. This study introduces a novel entropy measure specifically designed for Pythagorean fuzzy sets, establishing its axiomatic definition and presenting key properties. Decision making guided by entropy is advantageous, as it effectively mitigates ambiguity with increasing entropy values. Furthermore, a numerical example is provided to facilitate a comparative assessment of our newly introduced entropy measure in contrast to existing PFS entropy measures. The validation of our findings is achieved through the application of the COPRAS method, which determines decision outcomes based on a multitude of influencing factors. Notably, the determination of weights in this method is underpinned by the utilization of our innovative entropy measure.
... Sometimes we need to work with intuitionistic fuzzy events. An intuitionistic fuzzy event is an intuitionistic fuzzy set A = (µ A , ν A ) such that µ A , ν A : Ω → [0, 1] are S-measurable (see [2,3,8]). The family of all IF-events on (Ω, S) will be denoted by F. ...
The aim of this contribution is to formulate some definitions of almost uniformly convergence for a sequence of observables in the MV-algebra of the intuitionistic fuzzy sets. We define a partial binary operation ⊖ called difference on MV-algebra of intuitionistic fuzzy sets. As an illustration of the use the almost uniformly convergence we prove a variation of Egorov’s theorem for the observables in MV-algebra of intuitionistic fuzzy sets.
... The authors in [3] developed the concepts of fuzzy graph and fuzzy hypergraph. As a generalization of fuzzy sets, Atanassov [1] introduced the concepts of intuitionistic fuzzy set in 1983. In [7], regular fuzzy graphs and totally regular fuzzy graphs are compared through various examples. ...
A graph in which the edge can connect more than two vertices is called a Hypergraph. A ''k''-partite hypergraph is a hypergraph whose vertices can be split into ''k'' different independent sets. In this paper, regular, totally regular, totally irregular, totally neighborly irregular Intuitionistic Fuzzy ''k''-Partite Hypergraphs (IF''k''-PHGs) are defined. Also order and size along with the properties of regular and totally regular IF''k''-PHGs are discussed. It has been proved that the size S(\mathscr{H}) of a ''r''-regular IF''k''-PHG is \frac{tr}{2} where t=\left|V\right| . The dual IF''k''-PHG has also been defined with example.
Picture fuzzy set (PFS) concepts are modified versions of fuzzy sets. Picture fuzzy sets cover all aspects of portraying human opinion accurately. In this paper, we create Muirhead mean (MM) operators employing arithmetic operations modelled by Hamacher t-norm (TN) and t-conorm (TCN) using picture fuzzy information. These operators are known as picture fuzzy. Hamacher MM (PFHMM) and picture fuzzy Hamacher weighted MM (PFHWMM). Combining Hamacher t-norm and t-conorm arithmetic with the MM operator allows for flexible aggregation and consideration of attribute interrelationships. Also, MM is a generalization of commonly used aggregation operators, including arithmetic mean (AM), geometric mean (GM), Bonferroni mean (BM), and Maclaurin symmetric mean (MSM). The paper discusses some desirable properties and exceptional cases of proposed operators. The study also examines the Multiple Attribute Decision Making (MADM) technique using the suggested PFHWMM operator under the system of PFS information. A MADM problem is about allocating healthcare resources during a pandemic to test how well the suggested operators and methods work. To demonstrate the superiority of the currently proposed methods, we conducted a comprehensive comparative analysis to contrast the results of these approaches with the prevailing theories in the literature.
This paper proposes a novel approach to rank popular digital payment apps in India based on multiple criteria, such as user experience, security, and overall industry impact. To achieve this, we adopt multiple criteria decision-making (MCDM) methodologies, using q-rung orthopair fuzzy sets (q-ROFS) as the input range. The use of q-ROFS enhances the flexibility, resilience, and appropriateness of our model, surpassing intuitionistic fuzzy sets (IFS) and Pythagorean fuzzy sets (PFS). Incorporating the social network aspect, we account for trust relationships between experts, influencing decision-making. The weights assigned to each expert are determined through the Jaccard similarity measure. To further enhance our model's robustness, we adopt the entropy weight method (EWM) for determining criteria weights. The paper leverages the Digital India program, a flagship initiative of the Indian government, aimed at transforming India into a digitally empowered society and knowledge economy. With a significant increase in players in the UPI (Unified Payment Interface) market, it is imperative to have a transparent and useful ranking system to address intense competition. A comprehensive case study validates the proposed approach, evaluating the effectiveness of digital payment apps in India.
As the global economy continues to grow, the need for transportation also grows. Transportation researchers are developing new methods for integrating new technologies into existing transportation systems and for addressing the associated challenges. This paper addressed bi-objective fixed charge solid transportation problem that consider two objectives minimizing the total transportation cost, including fixed and variable costs, and minimizing the total transportation time. It is a challenging optimization problem, as it is difficult to find a solution that simultaneously minimizes both objectives. Additionally, the problem with bi-objective fixed-charge solid transportation problem (BOFCSTP) under uncertainty with neutrosophic concept is presented here. This problem constructed with all the parameters such as cost, fixed-charge, source availability and requirements as neutrosophic values. Neutrosophic sets are efficiently handling the indeterminacy and imprecise data in many fields and single-valued neutrosophic sets are extension and simpler form of NS. Further, to convert the neutrosophic values to crisp values a ranking function is used. To solve the considered BOFCSTP different approaches are employed namely, neutrosophic linear programming, neutrosophic goal programming, fuzzy goal programming to get the compromised solution to the problem. Additionally, a real-life problem is given with numerical example and the results compared with the different approaches.
Interval-valued fuzzy sets are the generalization of classical fuzzy sets. The assumption behind the theoretical interpretation of interval-valued fuzzy sets is that each element has exactly one real-valued truth membership degree from an interval. Information and knowledge measures play a major part in the interval-valued fuzzy set theory. This manuscript’s main objective is to investigate the information and knowledge measures in an interval-valued fuzzy context. A knowledge measure for interval-valued fuzzy sets is proposed axiomatically in this manuscript. The effectiveness and consistency of the proposed knowledge measure are demonstrated by numerical examples for structured linguistic comparison, ambiguity, and criteria weights computation in the interval-valued fuzzy context. An accuracy measure in interval-valued fuzzy-context is developed using the proposed knowledge measure. Apart from that, a similarity and a dissimilarity measure in an interval-valued fuzzy context are proposed. The suggested accuracy, similarity, and dissimilarity measures are used to solve cluster analysis and pattern detection issues. Additionally, a case study on the damage caused by floods and heavy rainfall in India between 2012 and 2021 is discussed, and the data obtained from this study are used to create clusters using the suggested accuracy measure. Furthermore, the accuracy, similarity, and dissimilarity measures that have been proposed, are used to address the pattern detection problems.
The present study focuses on the development of a novel distance measure for q-rung orthopair fuzzy sets in cognitive decision-making problems. The q-rung orthopair fuzzy set is an extension of the intuitionistic fuzzy set that allows for precise manipulation of ambiguous environments. By combining the concept of distance measure with q-rung orthopair fuzzy sets, the study aims to address the shortcomings of previous distance measures and provide a more accurate evaluation of q-rung orthopair fuzzy inherent information. The new distance measure is designed to calculate the differences between different pairs of q-rung orthopair fuzzy sets. The study illustrates the advantages of the proposed distance measure through meticulous numerical examples and examines its general axioms. The comparative study reveals that many established distance measures yield identical values for different pairs of q-rung orthopair fuzzy sets, which makes them unsuitable for accurately representing q-rung orthopair fuzzy inherent information. Additionally, the study introduces a new algorithm for solving fuzzy transportation problem using the proposed distance measure. A numerical example is discussed to validate the feasibility and applicability of the algorithm in real-life scenarios. The significance of the proposed method is evaluated through a comparative study, degree of hesitancy analysis, statistical tests, result analysis, and computational complexity assessment. The study thoroughly discusses these aspects to demonstrate the effectiveness of the proposed method. Finally, the study concludes by summarizing the findings and discussing future research directions for further improvement in the field of fuzzy transportation problem. Overall, the present study contributes to the field of cognitive decision-making by introducing a novel distance measure for q-rung orthopair fuzzy sets and demonstrating its advantages through various analyses and examples.
This paper uses the Aczel-Alsina t-norm and t-conorm to make several new linguistic interval-valued intuitionistic fuzzy aggregation operators. First, we devised some rules for how linguistic interval-valued intuitionistic fuzzy numbers should work. Then, using these rules as a guide, we created a set of operators, such as linguistic interval-valued intuitionistic fuzzy Aczel-Alsina weighted averaging (LIVIFAAWA) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina weighted geometric (LIVIFAAWG) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina ordered weighted averaging (LIVIFAAOWA) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina ordered weighted geometric (LIVIFAAOWG) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina hybrid weighted averaging (LIVIFAAHWA) operator and linguistic interval-valued intuitionistic fuzzy Aczel-Alsina hybrid weighted geometric (LIVIFAAHWG) operators are created. Several desirable qualities of the newly created operators are thoroughly studied. Moreover, a multi-criteria group decision-making (MCGDM) method is proposed based on the developed operators. The proposed operators are then applied to real-world decision-making situations to demonstrate their applicability and validity to the reader. Finally, the suggested model is contrasted with the currently employed method of operation.
Topological indices (TIs) are numerical structures that are associated with a graph to identify its topology. TIs are highly popular in the literature with a wide range of applications from chemistry to economics. However, TIs have limitations in representating complex relations within the graphs creating some uncertainities. Fuzzy graph (FG) and intuitionistic fuzzy graph (IFG) are introduced to overcome these uncertainities. While a FG a describes degree of membership of an object in a graph, IFG delineate information on membership or nonmembership under uncertainity. This study aims to introduce novel TIs such as the general second Zagreb index, the Sombor index of the third version, and the Sombor index of the fourth version in the IFG framework in order to improve practicality of FG and IFG applications. Some properties of the proposed indices and their upper bounds are provided as well. Proposed TIs are applied to an internet routing network as a case study. Results of the study show that adding more internet routers in the network can increase internet speed and the strength of the entire system. Finally, comparative studies for the Sombor index of the third version and the Sombor index of the fourth version are also revealed.
Complex geometries, fine details, and various designs that are difficult to create using traditional methods can easily be turned into a tangible object with Three-Dimensional (3D) printers. 3D printers have advantages such as providing design flexibility, obtaining prototypes in the shortest possible time, allowing for personalization, and reducing waste through the use of advanced technology. These advantages emphasize the significance of 3D printers in a sustainable production model. The widespread usage of 3D printers leads to increased efficiency and cost reduction in production. When the literature is examined, it is observed that there are limited studies on the evaluation of supplier performances for company using 3D printers. The aim of this study is to address 3D printers, which are highly significant for sustainable production, and to reveal the criteria that companies utilizing these printers need to consider for determining their suppliers. As a result of the literature review and expert interviews, a model has been developed that gathers the criteria to be considered for supplier selection, which is an important cost factor for companies involved in designing and producing 3D printers under five main and 18 sub-criteria. The importance weights of the criteria have been determined using the Interval Valued Pythagorean Fuzzy Analytic Hierarchy Process (IVPF-AHP) method, and the most suitable supplier among alternative suppliers has been selected using the Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method. Finally, the supplier scores have been statistically analyzed to show the validation of the results of the proposed method. According to the results, it has been concluded that for company using 3D printers, quality and technical service criteria are more important in the supplier selection. Additionally, cost of the material/equipment, product price and easy maintenance criteria also play a critical role in the supplier selection of 3D printer.
The concept of a single-valued neutrosophic graph SVNS-G is recently studied, the bond between neutrosophic graph N-G and neutrosophic topological graph NT-G was my goal in this research, I try to find a relation on the vertices of the SVN-G to structure NT-G add some theorems and corollaries.
Entropy is an important tool of information measurement in the fuzzy set and its inference. The research on information measurement based on interval-valued Pythagorean fuzzy sets mostly involves the distance formula for interval-valued Pythagorean fuzzy numbers, but seldom involves the measurement of fuzziness. In view of this situation, we have aimed to propose new interval-valued Pythagorean fuzzy entropy and weighted exponential entropy schemes. Based on the interval-valued Pythagorean fuzzy weighted averaging operator, a strategy based on weighted exponential entropy is proposed to solve the multi-criteria group decision-making (MCGDM) problem in the interval-valued Pythagorean environment. Two examples illustrate that this paper provides a feasible new method to solve the MCGDM problem in an interval-valued Pythagorean fuzzy (IVPF) environment. Finally, by comparing with the existing methods, it is concluded that the entropy measure of IVPF schemes and the corresponding MCGDM method can select the optimal solution of the practical problem more precisely and accurately. Therefore, the comparative analysis shows that the proposed measurement method has the characteristics of flexibility and universality.
The ambiguous information in multi-criteria decision-making (MCDM) and the vagueness of decision-makers for qualitative judgments necessitate accurate tools to overcome uncertainties and generate reliable solutions. As one of the latest and most powerful MCDM methods for obtaining criteria weight, the best–worst method (BWM) has been developed. Compared to other MCDM methods, such as the analytic hierarchy process, the BWM requires fewer pairwise comparisons and produces more consistent results. Consequently, the main objective of this study is to develop an extension of BWM using spherical fuzzy sets (SFS) to address MCDM problems under uncertain conditions. Hesitancy, non-membership, and membership degrees are three-dimensional functions included in the SFS. The presence of three defined degrees allows decision-makers to express their judgments more accurately. An optimization model based on nonlinear constraints is used to determine optimal spherical fuzzy weight coefficients (SF-BWM). Additionally, a consistency ratio is proposed for the SF-BWM to assess the reliability of the proposed method in comparison to other versions of BWM. SF-BWM is examined using two numerical decision-making problems. The results show that the proposed method based on the SF-BWM provided the criteria weights with the same priority as the BWM and fuzzy BWM. However, there are differences in the criteria weight values based on the SF-BWM that indicate the accuracy and reliability of the obtained results. The main advantage of using SF-BWM is providing a better consistency ratio. Based on the comparative analysis, the consistency ratio obtained for SF-BWM is threefold better than the BWM and fuzzy BWM methods, which leads to more accurate results than BWM and fuzzy BWM.
A huge factor in the success of a franchise business is the optimal selection of franchisees to represent the brand to customers. The dynamism in market parameters predetermines the use of Atanasov’s concepts of intuitionistic fuzzy logic and index matrices as a modeling and problem-solving tool for optimal business franchisee selection. In our previous work [34], we have introduced an index matrix model for a temporal intuitionistic fuzzy algorithm for optimal franchise candidate selection under uncertainty (OTIFAFr) and a software program for its execution. In this paper, which is an extension of [34], we will further outline a practical framework for fast-food restaurant chains for the process of franchisee selection from a pool of potential candidates applying the OTIFAFr utility. An example of choosing a franchisee for the McDonald’s restaurant chain is considered in this work.
This chapter presents an exploration of foundational concepts in the domain of uncertainty modeling through different types of fuzzy sets such as type-2 fuzzy sets, triangular interval type-2 fuzzy sets, simplified neutrosophic sets, and triangular fuzzy hesitant fuzzy sets. Algebraic operations, including addition, subtraction, multiplication, and division, are defined for all the given fuzzy sets extensions. The concepts of score and deviation functions are defined also defined to facilitate comparisons and ordering among sets. The extensions of fuzzy sets introduced in this chapter provide preliminaries to the methods applied in the upcoming sections and are presented in the order they appear in the chapters for the reader's comprehension.
The determination of attribute weights in solving multi-attribute decision-making (MADM) problems is crucial and significantly impacts the results. Many researchers have highlighted the effectiveness of deriving attribute weights objectively based on the assessments provided by decision-makers for MADM problems. One approach involves using entropy measures to determine weights based on the given ratings. This paper introduces a novel intuitionistic fuzzy entropy measure that takes the form of an exponential function. This new entropy measure is combined with the TOPSIS method to propose a new decision-making method for solving MADM problems. The proposed method does not require attribute weights, thereby eliminating the need for their determination.
Fuzzy rough entropy established in the notion of fuzzy rough set theory, which has been effectively and efficiently applied for feature selection to handle the uncertainty in real-valued datasets. Further, Fuzzy rough mutual information has been presented by integrating information entropy with fuzzy rough set to measure the importance of features. However, none of the methods till date can handle noise, uncertainty and vagueness simultaneously due to both judgement and identification, which lead to degrade the overall performances of the learning algorithms with the increment in the number of mixed valued conditional features. In the current study, these issues are tackled by presenting a novel intuitionistic fuzzy (IF) assisted mutual information concept along with IF granular structure. Initially, a hybrid IF similarity relation is introduced. Based on this relation, an IF granular structure is introduced. Then, IF rough conditional and joint entropies are established. Further, mutual information based on these concepts are discussed. Next, mathematical theorems are proved to demonstrate the validity of the given notions. Thereafter, significance of the features subset is computed by using this mutual information, and corresponding feature selection is suggested to delete the irrelevant and redundant features. The current approach effectively handles noise and subsequent uncertainty in both nominal and mixed data (including both nominal and category variables). Moreover, comprehensive experimental performances are evaluated on real-valued benchmark datasets to demonstrate the practical validation and effectiveness of the addressed technique. Finally, an application of the proposed method is exhibited to improve the prediction of phospholipidosis positive molecules. RF(h2o) produces the most effective results till date based on our proposed methodology with sensitivity, accuracy, specificity, MCC, and AUC of 86.7%, 90.1%, 93.0% , 0.808, and 0.922 respectively.
The concept of a neutrosophic set is an extension of a fuzzy set that uses indeterminacy. Similarly, an intuitionistic set has an extension, which is known as a single-valued neutrosophic set. The extension of intuitionistic fuzzy graphs and fuzzy graphs is single-valued neutrosophic fuzzy graphs (SVNF-graphs), which is the new component of graph theory. These versions of graph theory play an important role in many real-world problems, like medical diagnoses, law, engineering, finance, and industry. SVNF-graphs play an important role in linguistics, genetics, networking, sociology, computer technology, economics, and communication. The topological graph parameter gives a real number to the associated graph. There are numerous topological graph parameters proposed in the literature. In topological graph parameters, some uncertainty exists. Rosenfeld, Atanssov, and Smarandache introduced the concepts of a fuzzy graph, intuitionistic fuzzy graph (IFG), and SVNF-graph to overcome these uncertainties. SVNF-graph, IFG, and FG have vital roles in solving world-life problems. In this research work, we proposed a pythonic environment for the single-value neutrosophic fuzzy topological graph parameters. We introduced for the very first time some SVNF-graph parameters, like the Sombor graph parameter: the third and fourth versions of the SVNF-Sombor graph parameters for the SVNF-graph framework. Also, we have proved some characteristics and bounds of these topological graph parameters. We have discussed the social media application for the SVNF-Sombor graph parameter and its third and fourth versions. Under consideration application, We have shown that deleting a person (vertex) in the network can increase or decrease the chances of sending friend requests to other people of artificial intelligence.
The idea of probabilistic q-rung orthopair linguistic neutrosophic (P-QROLN) is one of the very few reliable tools in computational intelligence. This paper explores a significant breakthrough in nanotechnology, highlighting the introduction of nanoparticles with unique properties and applications that have transformed various industries. However, the complex nature of nanomaterials makes it challenging to select the most suitable nanoparticles for specific industrial needs. In this context, this research facilitate the evaluation of different nanoparticles in industrial applications. The proposed framework harnesses the power of neutrosophic logic to handle uncertainties and imprecise information inherent in nanoparticle selection. By integrating P-QROLN with AO, a comprehensive and flexible methodology is developed for assessing and ranking nanoparticles according to their suitability for specific industrial purposes. This research contributes to the advancement of nanoparticle selection techniques, offering industries a valuable tool for enhancing their product development processes and optimizing performance while minimizing risks. The effectiveness of the proposed framework are demonstrated through a real-world case study, highlighting its potential to revolutionize nanoparticle selection in HVAC (Heating, Ventilation, and Air Conditioning) industry. Finally, this study is crucial to enhance nanoparticle selection in industries, offering a sophisticated framework probabilistic q-rung orthopair linguistic neutrosophic quantification with an aggregation operator to meet the increasing demand for precise and informed decision-making.
The purpose of this paper is to introduce the notion of intuitionistic fuzzy neutronsophic soft ideal in Intuitionistic fuzzy neutronsophic soft set theory. The concept of intuitionistic fuzzy neutrosophic soft local function is also introduced. These concepts are discussed with a view to find new intuitionistic fuzzy neutronsophic soft topologies from the original one. The basic structure, respecially a basic for such generated Intuitionistic fuzzy neutronsophic soft topologies also studied here. Finally, the notion of compatibility of intuitionistic fuzzy neutronsophic soft ideals with Intuitionistic fuzzy neutrosophic soft topologies is introduced and some equivalent conditions concerning, this topic are established here.
Intuitionistic fuzzy sets aims at taking the hesitancy of an expert into account in assigning a membership degree or a non-membership degree. The direct assignment of decimal numbers for membership and non-membership degrees of an element in intuitionistic fuzzy sets is not practical. Besides, the assigned degrees are generally composed of one digit or at most two digits after dot. This problem has not been addressed as much as it deserves in the literature. The hypothesis of the paper is that the determination of proportional relationships between membership and non-membership degrees is more appropriate than the direct assignment to obtain the degrees. Proportional intuitionistic fuzzy (PIF) sets require only the proportion relations between an intuitionistic fuzzy set’s parameters. The accuracy of the results obtained with multi-criteria decision-making models definitely depends on how accurately the membership degrees are determined. In this paper, we extend Combinative distance-based assessment (CODAS) method by using single-valued proportional intuitionistic fuzzy sets. We compare the proposed PIF CODAS method with ordinary fuzzy CODAS method. A cloud service provider selection problem is handled to show the validity of the proposed PIF CODAS method. Additionally, a comparative analysis and a sensitivity analysis together with a discussion are presented.
The essential extension to fuzzy sets is known as intuitionistic fuzzy sets (or Atanassov intuitionistic fuzzy sets, or IFS sets), and was developed by Krassimir Atanassov in 1983 [1]. Atanassov considered not only the degree of membership but also the degree of non-membership for any element belonging to such sets.
To determine the optimal value and solution for balanced and unbalanced intuitionistic fuzzy transportation problems (UBIFTPs), two different approaches are proposed in this chapter. Additionally, the parameters of the proposed problems are considered to be triangular intuitionistic fuzzy numbers (TIFNs). Two efficient methods, namely method-I (named linear programming method) and method-II (named PSK method), are presented. Both of them are used to solve the proposed problems. The ideas of these two methods are illustrated with the help of simple examples, and their relevant computer programming is presented. From software such as RStudio, LINGO, RGui, and MATLAB, the obtained solutions for the proposed problems are compared and analyzed with the solutions obtained by the proposed methodologies and already existing methodologies. The unique or superior results, discussions, advantages, and disadvantages of the proposed methods are all presented. Finally, the author's future work is mentioned.
The concept of interval-valued picture fuzzy sets (IVPFSs) is the most generalized form of fuzzy sets (FSs) and is proven a useful tool to manipulate complications that arise due to incomplete information more effectively. One of the most powerful feature of IVPFSs is that it allocates the membership, non membership and neutral membership values as intervals to any element of the given data. Due to this, IVPFSs play a key role to deal uncertain data with multiple attributes. In this study, we introduce the notion of interval-valued picture fuzzy hypergraphs (IVPFHGs) which is the combination of both IVPFSs and hypergraphs and provide its application in decision making. We describe several types of IVPFHGs such as partial, simple, support, support simple, elementary IVPFHGs etc. We also initiate the concepts of dual of IVPFHGs. Moreover, \(([\iota , \kappa ], [\lambda , \epsilon ], [\rho , \nu ])\)-level cuts of IVPFHGs are also addressed. We present a comparative analysis of our newly established terms with those existing in the literature and elaborate the superiority of IVPFHGs over the other existing fuzzy hypergraphs structures. Finally, we provide an application of IVPFHGs with algorithm and flowchart towards decision making.
A (p, q)-fuzzy set is the resulting structure when the fuzzy membership and non-membership values are bounded by a general nonlinear relation. This set enhances the range of depicting uncertain information by making the feasible region larger, and thereby widening the purview of decision-making. Data aggregation is crucial in optimal decision-making. This article attempts to formulate aggregation operators for (p, q)-fuzzy sets using additive generators of strict t-norms and strict t-conorms. The utility of these operators is showcased by examining a decision-making problem, where the best decision is obtained by ranking the alternatives based on their score values. A comparative study is also carried out using some existing aggregation operators to test the validity and effectiveness of the introduced operators.
In order to handle incomplete, ambiguous, and inconsistent data, the concept of a neutrosophic set (NS) has gained immense popularity. Due to its independent indeterminacy component, it has been shown to be a successful tactic and expanded in literature by numerous experts. The primary aim of the current study is to discuss the generality of the NS and its development processes within the studied field. According to the requirements in various sectors, there have been numerous techniques and improvements toward neutrosophic perception during these periods. To understand the most current developments in the neutrosophic approach, a review of the last ten years literature has been conducted in the current study. To comprehend the advantages and disadvantages of various approaches, tabular and graphical comparisons are included. Based on the outcomes of the review papers, managerial implications have been investigated. For further comprehension, the research gap between existing methods and corresponding future approach is also emphasized.
The circular Pythagorean fuzzy (Cir-PyF) sets (Cir-PyFs) not only contain the membership and non-membership grades, but also have the radius around the circle of each element. Cir-PyFSs can cope with many real-life problems. In this paper, we consider the Hamacher t-norm and t-conorm operational laws for any two Cir-PyF numbers (Cir-PyFNs) and describe their exceptional cases such as algebraic and Einstein operational laws. Furthermore, the Cir-PyF Hamacher averaging (Cir-PyFHA) operator, Cir-PyF Hamacher ordered averaging (Cir-PyFHOA) operator, Cir-PyF Hamacher geometric (Cir-PyFHG) operator, and Cir-PyF Hamacher ordered geometric (Cir-PyFHOG) operator are proposed. Some properties for the above operators are also discussed in detail. Moreover, to select the simplest and best procedure for evaluating the source of gold in mines, we illustrate the application of the multi-attribute decision-making (MADM) technique based on the derived operators. Finally, we demonstrate some examples for comparing the proposed operators with some existing methods to expand the attraction of the proposed way.
The combination of MCDM and fuzzy sets offers new potential ways to solve the challenges posed by complex image contents, such as selecting the optimal segmentation algorithm or evaluating the segmentation quality based on various parameters. Since no single segmentation algorithm can achieve the best results on satellite image datasets, it is essential to determine the most appropriate segmentation algorithm for each satellite image, the content of which can be characterized by relevant visual features. In this research, we proposed a set of visual criteria representing the fundamental aspects of satellite image segmentation. The segmentation algorithms chosen for testing were evaluated for their performance against each criterion. We introduced a new method to create a decision matrix for each image using fuzzy fusion, which combines the image content vector and the evaluation matrix of the studied segmentation algorithms. An extension of the Preference Ranking Organization Method Enrichment Evaluation (PROMETHEE) using intuitive fuzzy sets (IFSs) was applied to solve this problem. The results acquired by the proposed methodology were validated by comparing them with those obtained in expert ratings and by performing a sensitivity analysis.
Average edge connectivity is a fundamental metric in classical and fuzzy graph theory. It is a key parameter in evaluating the reliability of a network. Fuzzy average
edge connectivity more accurately describes the overall stability of links in the network by providing a more precise measure of the connectivity of the fuzzy graphs. It is particularly significant in situations where connectivity strength and dependability possess a degree of fuzziness, including communication networks with variable signal intensities or transit systems with unpredictable travel durations. In this article, we describe the types of edges in Pythagorean fuzzy graphs and the ideas of the Pythagorean fuzzy trees and cycles. We define the Pythagorean fuzzy average edge connectivity of Pythagorean fuzzy graphs using the concept of minimal Pythagorean fuzzy local edge cut. We provide the bounds for the Pythagorean fuzzy average edge connectivity of Pythagorean fuzzy graphs and strong Pythagorean fuzzy graphs. We discuss the Pythagorean fuzzy average edge connectivity of edge-deleted Pythagorean fuzzy subgraphs of Pythagorean fuzzy graphs. Moreover, we determine the effect of the different types of edges on the Pythagorean fuzzy average edge connectivity. We establish some results on Pythagorean fuzzy average edge connectivity. Furthermore, we design some algorithms to compute the Pythagorean fuzzy average edge connectivity of particular Pythagorean fuzzy graphs as complete Pythagorean fuzzy graphs and
saturated Pythagorean fuzzy cycles. We provide the application of Pythagorean fuzzy average edge connectivity in human trafficking. Finally, we compare the suggested
approach with an existing model to illustrate its viability and applicability.
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