Extension Of Fuzzy Sets Research Articles

Interval-Valued and Circular Intuitionistic Fuzzy Present Worth Analyses

Present worth (PW) analysis is an important technique in engineering economics for investment analysis. The values of PW analysis parameters such as interest rate, first cost, salvage value and annual cash flow are generally estimated including some degree of uncertainty. In order to capture the vagueness in these parameters, fuzzy sets are often used in the literature. In this study, we introduce interval-valued intuitionistic fuzzy PW analysis and circular intuitionistic fuzzy PW analysis in order to handle the impreciseness in the estimation of PW analysis parameters. Circular intuitionistic fuzzy sets are the latest extension of intuitionistic fuzzy sets defining the uncertainty of membership and non-membership degrees through a circle whose radius is r. Thus, we develop new fuzzy extensions of PW analysis including the uncertainty of membership functions. The methods are given step by step and an application for water treatment device purchasing at a local municipality is illustrated in order to show their applicability. In addition, a multi-parameter sensitivity analysis is given. Finally, discussions and suggestions for future research are given in conclusion section.

Pythagorean Fuzzy Linguistic Power Generalized Maclaurin Symmetric Mean Operators and Their Application in Multiple Attribute Group Decision-Making

As an extension of Pythagorean fuzzy sets and linguistic term sets, Pythagorean fuzzy linguistic sets (PFSs) are powerful to describe decision-making information quantificational and qualitatively, which have received much scholars’ attention. The purpose of this paper is to propose a new multiple attribute group decision-making (MAGDM) approach with Pythagorean fuzzy linguistic (PFL) information. To this end, we firstly analyze the drawbacks of existing operations of PFL numbers and propose new operational rules based on linguistic scale function. The power average (PA) operator is famous for its capacity of reducing the negative influence of unreasonable evaluation values provided by prejudiced decision makers on the decision results. The generalized Maclaurin symmetric mean (GMSM) can not only capture the interrelationship among multiple inputs but also manipulate the effect of related properties by adjusting the parameters. When considering aggregation operators of PFL numbers, we combine PA with GMSM and propose the PFL power generalized Maclaurin symmetric and the PFL power generalized weighted Maclaurin symmetric operators. We also study important properties and special cases of these operators. We continue to investigate MAGDM problems with PFL decision information and propose a novel method to determine the optimal alternative. Finally, we conduct numerical examples to demonstrate the effectiveness of our proposed method. We also attempt to illustrate the advantages and superiorities of the proposed method via comparative analysis.

N,m-Rung Orthopair Fuzzy Sets With Applications to Multicriteria Decision Making

A q-rung orthopair fuzzy set is one of the effective generalizations of fuzzy set for dealing with uncertainties in information. Under this environment, in this study, we define a new type of extensions of fuzzy sets called n,m-rung orthopair fuzzy sets and investigate their relationship with Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets. The n,m-rung orthopair fuzzy sets can supply with more doubtful circumstances than Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets because of their larger range of depicting the membership grades. There is a symmetry between the values of this membership function and non-membership function. Here, any power function scales are utilized to widen the scope of the decision-making problems. In addition, the novel notion of an n,m-rung orthopair fuzzy set through double universes is more flexible when debating the symmetry between two or more objects that are better than the diffusing concept of an n-rung orthopair fuzzy set, as well as m-rung orthopair fuzzy set. The main advantage of n,m-rung orthopair fuzzy sets is that it can describe more uncertainties than Fermatean fuzzy sets, which can be applie in many decision-making problems. Then, we discover the essential set of operations for the n,m-rung orthopair fuzzy sets along with their several properties. Finally, we introduce a new operator, namely, n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) over n,m-rung orthopair fuzzy sets and apply this operator to the MADM problems for evaluation of alternatives with n,m-rung orthopair fuzzy information.

N-Cylindrical Fuzzy Neutrosophic Sets

In this paper, we introduce a new type of fuzzy Neutrosophic set called n-Cylindrical fuzzy Neutrosophic set (n-CyFNS), with I as independent neutrosophic component. The n-CyFNS can be claimed as the largest extension of fuzzy sets. In n-CyFNS, the degree of positive, neutral and negative membership functions are satisfying the condition, 0≤ βA(x) ≤1 and 0≤ αA n(x) + γAn(x) ≤ 1, 1, is an integer. Also the distance between two n- CyFNS and its properties are also defined. Along with basic operations on n- CyFNSs, we put forward two concepts-Neutrosophic affinity degree Neutrosophic similarity index which is used to compare and correlate n-CyFNSs respectively. A comparison is made in the n-CyFNS environment using the existing correlation measures to check its reliability.

Complex‐Valued Migrativity of Complex Fuzzy Operations

Complex fuzzy sets (CFSs), as an important extension of fuzzy sets, have been investigated in the literature. Operators of CFSs are of high importance. In addition, α−migrativity for various fuzzy operations on [0, 1] has been well discussed, where α is a real number and α ∈ [0,1]. Thus, this paper studies α−migrativity for binary functions on the unit circle of the complex plane O, where α is a complex number and α ∈ O. In particular, we show that a binary function is α−migrativity for all α ∈ O if and only if it is α−migrativity for all , where is the boundary point subset of O. Finally, we discuss the relationship between migrativity and rotational invariance of binary operators on O.

Fourth party logistics firm assessment using a novel neutrosophic MCDM

The fourth party logistics (4PL) is an combiner that designs and implements the holistic supply chain solutions by using skills, knowledge, technology and resources of the service provider and its customer. A 4PL provider is also a technological service provider with eligible intellectual capital and the sufficient computer/software infrastructure. Defining the most appropriate 4PL service provider from the alternatives is not easy for companies, the solution can be addressed within the framework of the Multi-Criteria Decision Making (MCDM) problem, and subjective and uncertain data are required for this solution. “Fuzzy set theory” is a helpful tool for dealing with such subjectivity and uncertainty. In recent times, extensions of fuzzy sets have been evolved to address and describe the subjectivities and uncertainties more widely. Neutrosophic sets are one of the extensions of fuzzy sets, and unlike other extensions, they use the independent indeterminacy-membership function, thereby extracting important information and improving the accuracy of the decision-making process. A neutronophic MCDM method was proposed for the assessment of 4PL providers’ performance. In the application part of the study, neutrosophic language scale was used by three experts to evaluate the performance of 4PL providers. Then the closeness coefficient of each alternative was computed and sequenced in descending order. We also presented a comparative analysis with neutrosphic TOPSIS method. The results determined that the proposed neutrosophic MCDM method could be used in the performance evaluation of 4PL providers and similar problems.

A Novel spherical fuzzy CRITIC method and its application to prioritization of supplier selection criteria

Linguistic terms are quite suitable to make evaluations in multiple criteria decision making problems since humans prefer them rather than sharp evaluations. When linguistic evaluations are used in the decision matrix instead of exact numerical values, fuzzy set theory can capture the vagueness in the linguistic evaluations. Ordinary fuzzy sets have been extended to many new types of fuzzy sets such as intuitionistic fuzzy sets, neutrosophic sets, spherical fuzzy sets and picture fuzzy sets. Spherical fuzzy sets are an extension of picture fuzzy sets whose squared sum of their parameters is at most equal to one. This paper develops a novel spherical fuzzy CRiteria Importance Through Intercriteria Correlation (CRITIC) method and applies it for prioritizing supplier selection criteria. Supplier selection is one of the most critical aspects of any organization since any mistake in this process may cause poor supplier performance and inefficiencies in the business processes. Supplier selection is a multi-criteria decision making problem involving several conflicting criteria and alternatives. A numerical illustration of the proposed method is also given for this problem.

Circular intuitionistic fuzzy TOPSIS method: pandemic hospital location selection

A circular intuitionistic fuzzy set (CIFS) recently introduced by Atanassov as a new extension of intuitionistic fuzzy sets is represented by a circle whose radius is r and whose center is composed of membership and non-membership degrees. The idea is similar to type-2 fuzzy sets, which are based on the fuzziness of membership functions with a third dimension. CIFSs help us define membership functions more flexibly, taking into account the vagueness in membership and non-membership degrees. In this study, TOPSIS, which is a multi-criteria decision-making (MCDM) method, is developed under circular intuitionistic fuzzy environment. The proposed CIF-TOPSIS method is applied to determine the most appropriate pandemic hospital location selection problem. Then, a sensitivity analysis based on criteria weights and the weight of the decision maker’s optimistic and pessimistic attitudes are conducted to check the robustness of the decisions given by the proposed approach. A comparative analysis with the single-valued intuitionistic fuzzy TOPSIS, Pythagorean fuzzy TOPSIS, picture fuzzy TOPSIS methods is also performed to verify the developed approach and to demonstrate its effectiveness.

Process design and capability analysis using penthagorean fuzzy sets: surgical mask production machines comparison

Process capability analysis (PCA) is a tool for measuring a process’s ability to meet specification limits (SLs), which the customers define. Process capability indices (PCIs) are used for establishing a relationship between SLs and the considered process’s ability to meet these limits as an index. PCA compares the output of a process with the SLs through these capability indices. If the customers’ needs contain vague or imprecise terms, the classical methods are inadequate to solve the problem. In such cases, the information can be processed by the fuzzy set theory. Recently, ordinary fuzzy sets have been extended to several new types of fuzzy sets such as intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and spherical fuzzy sets. In this paper, a new extension of intuitionistic fuzzy sets, which is called penthagorean fuzzy sets, is proposed, and penthagorean fuzzy PCIs are developed. The design of production processes for COVID-19 has gained tremendous importance today. Surgical mask production and design have been chosen as the application area of the penthagorean fuzzy PCIs developed in this paper. PCA of the two machines used in surgical mask production has been handled under the penthagorean fuzzy environment.

Fermatean fuzzy Heronian mean operators and MEREC‐based additive ratio assessment method: An application to food waste treatment technology selection

Uncertainty is often occurred in real-life decision-making problems due to the lack of complete information, imprecise data, and the vagueness of decision making experts in qualitative judgment, thus, the crisp values of criteria may be insufficient to handle such types of complex real situations. As the extension of fuzzy set, intuitionistic fuzzy set and Pythagorean fuzzy set, the Fermatean Fuzzy Set (FFS) has been demonstrated as a powerful tool to handle the uncertainty arisen in practical decision-making problems. Thus, this study aims to introduce an integrated Fermatean fuzzy information-based decision-making method by combining method based on the removal effects of criteria (MEREC) and additive ratio assessment (ARAS) methods with the application in a food waste treatment technology selection problem. By using Fermatean fuzzy numbers, the suggested approach successfully handle the qualitative data and uncertain information that often occur in practical situations. This study consists of four phases. First, entropy measure is developed for FFS and further utilized for determining the experts’ weights. Second, some Fermatean fuzzy Heronian mean operators and their properties are introduced to aggregate the Fermatean fuzzy information. These operators can provide us a valuable means to handle practical multicriteria decision-making problems on FFSs context. Third, an extended MEREC technique is originated to assess objective criteria weights within FFS context. Fourth, an integrated ARAS method is introduced with the combination of proposed entropy measure, generalized weighted Fermatean fuzzy Heronian mean operator and MEREC technique to evaluate and rank the alternatives. To confirm the reasonableness and practicality of the proposed methodology, an empirical case study of food waste treatment technology selection is discussed on FFSs settings. Further, a comparison with extant models and a sensitivity investigation are performed to confirm the validity and robustness of the obtained outcomes.

Multi-criteria decision making and pattern recognition based on similarity measures for Fermatean fuzzy sets

Higher-order fuzzy decision-making methods have become powerful tools to support decision-makers in solving their problems effectively by reflecting uncertainty in calculations better than crisp sets in the last 3 decades. Fermatean fuzzy set proposed by Senapati and Yager, which can easily process uncertain information in decision making, pattern recognition, medical diagnosis et al., is extension of intuitionistic fuzzy set and Pythagorean fuzzy set by relaxing the restraint conditions of the support for degrees and support against degrees. In this paper, we focus on the similarity measures of Fermatean fuzzy sets. The definitions of the Fermatean fuzzy sets similarity measures and its weighted similarity measures on discrete and continuous universes are given in turn. Then, the basic properties of the presented similarity measures are discussed. Afterward, a decision-making process under the Fermatean fuzzy environment based on TOPSIS method is established, and a new method based on the proposed Fermatean fuzzy sets similarity measures is designed to solve the problems of medical diagnosis. Ultimately, an interpretative multi-criteria decision making example and two medical diagnosis examples are provided to demonstrate the viability and effectiveness of the proposed method. Through comparing the different methods in the multi-criteria decision making and the medical diagnosis application, it is found that the new method is as efficient as the other methods. These results illustrate that the proposed method is practical in dealing with the decision making problems and medical diagnosis problems.

Aggregation Operators in Hesitant Fuzzy Set for Decision Making

Uncertainty is prevalent in a wide range of real-world issues. The fuzzy sets, vague sets or intuitionistic fuzzy sets are widely used in recent years for decision making and various analysis where uncertainty is predominant. An extension of fuzzy sets is Hesitant Fuzzy Sets, which deals with ambiguous situations that arise when determining an element's membership degree in a set. Researchers have defined various ideas, extensions, aggregation operators, and measurements to deal with reluctant information as a result of this new approach. Machine leaning algorithms are also exploiting hesitant fuzzy sets for better decision making.

Shortest path algorithm of a network via spherical fuzzy digraphs

Many extension and generalization of fuzzy sets have been introduced and studied in the literature. Picture fuzzy set acquired more concentration in the domain of decision making, as many real time circumstances might have choices of more than one and researchers are looking for optimum choice/decision. Spherical fuzzy digraph is a generalization of intuitionistic fuzzy set and fuzzy graph. In this paper, we redefine some preliminary operations of Spherical fuzzy graph and it is referred as spherical fuzzy digraph. We discuss some arithmetic operations and relations among spherical fuzzy digraph. We further proposed a method to solve a shortest path problem using score function.

Choquet integral-based intuitionistic fuzzy arithmetic aggregation operators in multi-criteria decision-making

Intuitionistic fuzzy set (IFS), a classical extension of fuzzy set, is a powerful tool to describe fuzzy information in multi-criteria decision-making (MCDM) because of its membership and non-membership functions. Intuitionistic fuzzy aggregation operators are the standard mathematical tools for the combination of several inputs with respect to criteria into one unique output. The traditional intuitionistic fuzzy aggregation operator-based methods have been criticized because of reasons that include disregard for comprehensive correlative relationships of criteria and ignorance for combination of fuzzy measures of criteria and weights of positions. In this paper, the effectiveness of traditional aggregation operator-based MCDM techniques is improved by a novel method to select the best alternative(s) under the intuitionistic fuzzy environment. We begin by defining the Choquet integral-based intuitionistic fuzzy arithmetic aggregation (CIIFAA) operator, the remarkable properties of which are proved in details. Further, the Choquet integral based intuitionistic fuzzy hybrid arithmetic aggregation (CIIFHAA) operator is defined as well as the proofs of its properties. The CIIFHAA operator can not only capture the comprehensive correlative relationships of criteria in a simpler manner, but also combine the weights of positions. Then, we put forward a MCDM method based on CIIFHAA operator with intuitionistic fuzzy evaluations. An illustrative example is solved, the best alternative and the ranking of alternatives are obtained, which can verify the feasibility of the proposed method. Finally, two comparisons are conducted to illustrate the stability and advantages of the proposed method.

The cross-entropy and improved distance measures for complex q-rung orthopair hesitant fuzzy sets and their applications in multi-criteria decision-making

The complex q-rung orthopair fuzzy set (Cq-ROFS) is the extension of complex Pythagorean fuzzy set (CPFS) in which the sum of the q-power of the real part (imaginary part) of the support for and the q-power of the real part (imaginary part) of the support against is limited by one; however, it is difficult to express the hesitant information. In this study, the conception of complex q-rung orthopair hesitant fuzzy set (Cq-ROHFS) by combining the Cq-ROFS and hesitant fuzzy set (HFS) is proposed, and its properties are discussed, obviously, Cq-ROHFS can reflect the uncertainties in structure and in detailed evaluations. Further, some distance measures (DMs) and cross-entropy measures (CEMs) are developed based on complex multiple fuzzy sets. Moreover, these proposed measures are utilized to solve a multi-criteria decision-making problem based on TOPSIS (technique for order preference by similarity to ideal solution) method. Then, the advantages and superiority of the proposed measures are explained by the experimental results and comparisons with some existing methods.

Selecting Personnel with the Weighted Cross-Entropy TOPSIS of Hesitant Picture Fuzzy Linguistic Sets

Personnel selection is a key important role for the human resource department of organization, and hesitant picture fuzzy linguistic sets (HPFLSs) elaborated the advantages of both hesitant linguistic set and picture fuzzy set, which is more flexible and effective to solve the decision-making problems of personnel selection than other extension of fuzzy linguistic sets (FLSs). Cross-entropy, as effective measurement tools, is wildly used under fuzzy multicriteria decision-making (FMCDM) environment; thus, in order to elaborate the advantages of both cross-entropy and HPFLSs under FMCDM environment, the cross-entropy definition of HPFLSs is firstly given in this paper. Meanwhile, several novel cross-entropy measures between two HPFLSs are introduced, and their related properties are proved. Then, an approach based on the weighted cross-entropy measures and TOPSIS under hesitant picture fuzzy linguistic environment is proposed. Finally, the proposed method is applied to the real personnel’s selection, and the ranking results show that the proposed methods are practical and effective.

A New Technique for solving Picture Fuzzy Differential Equation

Picture Fuzzy set (PFS) is an extension of fuzzy set (FS) and intuitionistic fuzzy set (IFS) that can model the uncertainty by integrating the concept of positive, negative and neutral membership degree of an element. In this paper, the solution of Picture Fuzzy ordinary differential equation of first order by means of picture fuzzy number is exemplified and intend to define the picture fuzzy number for (∝, δ, β)-cut. Finally, we illustrate the numerical example for drug distribution in human body for different drug levels is discussed for determining its effectiveness and practicality of the first order differential equation involving picture fuzzy numbers.

Novel Technique for Group Decision-Making under Fuzzy Parameterized q -Rung Orthopair Fuzzy Soft Expert Framework

The q -rung orthopair fuzzy sets and their hybrid models are capable of dealing with uncertain situations very effectively than the theories of intuitionistic and Pythagorean fuzzy sets and thus have numerous decision-making applications in daily life, while the fuzzy parameterized soft set theory has its impact on different decision-making scenarios. Motivated by these facts, in this research article, these theories are combined to form a new structure named fuzzy parameterized q -rung orthopair fuzzy soft expert sets ( FP q ROFSESs) for dealing with more generalized information. The developed model is an efficient extension of fuzzy parameterized intuitionistic fuzzy soft expert sets. Some of its basic notions, including subset, complement, OR operation, AND operation, intersection, and union are studied and illustrated via examples. Moreover, to show the applicability and efficiency of the developed model, two real-life applications are solved under the FP q ROFSES approach, which is supported by an algorithm, the first application is about selecting an appropriate site for a cafe outlet, and the second application is about selecting the Best News Channel for an award. At last, a comparison of the initiated model with some existing approaches is presented to verify its advantages over them.

A Bonferroni mean considering Shapley fuzzy measure under hesitant bipolar-valued neutrosophic set environment for an investment decision.

Bonferroni mean (BM) operators have been established as a powerful tool for handling the interrelationship between the input arguments under various decision-making information. However, the existing BM operators do not take into account the overall interaction among decision makers or criteria. To overcome this limitation, this study considers the Shapley fuzzy measure (SFM) with the normalized weighted BM (NWBM) operator under a neutrosophic environment. In addition, the current research ignores the bipolarity and hesitancy during decision elicitations, resulting in the imprecise decision results. In this paper, the hesitant bipolar-valued neutrosophic set (HBNS) which is the extension of hesitant fuzzy set and bipolar neutrosophic set is employed. The main focus of this paper is in the development of an aggregation operator for HBNS. Based on the literature review, we would like to fill in the gaps by developing a hesitant bipolar-valued neutrosophic Shapley NWBM (HBN-SNWBM) operator where the overall interaction among decision makers can be considered. Besides that, a three-phase decision making framework is also proposed to show the applicability of the proposed aggregation operator to the real-world decision problems. The HBN-SNWBM operator and the decision making framework are applied to two examples of investment selection where evaluations are implemented using the proposed aggregations that based upon hesitant bipolar-valued neutrosophic sets. In the first example, it is found that a weapon company is the best alternative for investment followed by a food company. Sensitivity of parameters of the aggregation operator is also analysed and it is found that the ranking results are consistent despite of different parameter values used. This verifies the insensitivity of p,q parameters in the developed aggregation operator. The proposed decision making framework and hesitant bipolar-valued neutrosophic sets would be a great significance for the practical implementation of the aggregation operators.

New Generalized Intuitionistic Fuzzy Divergence Measure with Applications to Multi-Attribute Decision Making and Pattern Recognition

Background: The notion of fuzzy set was introduced by Zadeh. After that, many researchers extended the concept of fuzzy sets in different ways. Atanassov introduced the concept of intuitionistic fuzzy sets as an extension of fuzzy sets. This concept is applied in many fields such as bio-informatics, image processing, decision making, feature selection, pattern recognition, etc. Objectives: The prime objective of this paper is to introduce a new generalized intuitionistic fuzzy divergence measure with proof of its validity and discussions on its elegant properties. Applications of the proposed divergence measure in multi-attribute decision making and pattern recognition are also discussed with some numerical illustrations. Further, the proposed divergence measure is compared with other methods for solving MADM and pattern recognition problems which exist in the literature. Methods: The divergence measure method is used to measure the divergence between two given sets. In addition, the results of the other existing measures are also given to compare with the proposed measure. Results: It was observed that the proposed divergence measure found much better results in comparison with the other existing methods. Conclusion: A new divergence measure for intuitionistic fuzzy sets is introduced with some of its properties. Applications of the proposed divergence measure to pattern recognition and MADM are illustrated through examples. The comparison of the proposed method with the existing methods shows the legacy of the results of the proposed method. It is concluded that the proposed divergence measure is effective for solving real-world problems related to MADM and pattern recognition.