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Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment
Abstract
The interval-valued intuitionistic fuzzy weighted arithmetic average operator, the interval-valued intuitionistic fuzzy weighted geometric average operator, and an accuracy function of interval-valued intuitionistic fuzzy value are introduced in this paper. A novel accuracy function for interval-valued intuitionistic fuzzy sets (IVIFSs) is proposed by taking into account the unknown degree (hesitancy degree) of IVIFSs to overcome the situation of difficult decision of existing accuracy functions to the alternatives in some cases. To identify the best alternative in multicriteria decision-making problems, a multicriteria fuzzy decision-making method is established in which criterion values for alternatives are IVIFSs. We utilize the interval-valued intuitionistic fuzzy weighted aggregation operators to aggregate the interval-valued intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the accuracy degree of the aggregated the interval-valued intuitionistic fuzzy information corresponding to the new accuracy function. Finally, an illustrative example is given to verify the developed approach.
... For aggregating intuitionistic fuzzy numbers, Xu and Yager [9] presented a geometric aggregation operator. In addition, related to IFS, some authors proposed several other tools to handle vague and imprecise information whereby two or more sources of vagueness appear simultaneously [10][11][12][13][14]. The notion of SR-Fuzzy Sets was first introduced by Al-Shami et al. [15] as a generalization of FS. ...
... We now utilize this information by applying the existing aggregation techniques to it. To check the application and effectiveness of the proposed method, we compare our approach to the IFAAWA operator [49], IFAAWG operator [49], IFAAWOA operator [49], IFAAOWG operator [49], IFWG operator [10] and IFWA operator [10]. The ranking results are shown in Table 10. ...
... We now utilize this information by applying the existing aggregation techniques to it. To check the application and effectiveness of the proposed method, we compare our approach to the IFAAWA operator [49], IFAAWG operator [49], IFAAWOA operator [49], IFAAOWG operator [49], IFWG operator [10] and IFWA operator [10]. The ranking results are shown in Table 10. ...
The complex interval-valued intuitionistic fuzzy set (CIIFS), as an extension of the complex intuitionistic fuzzy set (CIFS), is one of the better tools for dealing with time-periodic information in a single set of two-dimensional information. So, this paper makes some new operators based on Aczel-Alsina’s t-norm and t-conorm in the CIIFS environment. First, some Aczel-Alsina t-norm and t-conorm operational laws are defined for complex interval-valued intuitionistic fuzzy numbers (CIIFNs). Then, using the proposed operational laws, Aczel-Alsina The operators Complex interval-valued intuitionistic Aczel-Alsina weighted average (CIIFAAWA), Complex interval-valued intuitionistic Aczel-Alsina weighted geometric (CIIFAAWG), Complex interval-valued intuitionistic Aczel-Alsina ordered weighted average (CIIFAAOWA), and Complex interval-valued intuitionistic Aczel-Alsina ordered weighted geometric (CIIFAAOWG) are developed. Further, some properties of the proposed operators are discussed and investigated in detail. A group decision-making method is also suggested based on developed operators in the CIIFS environment. The proposed operators are more adaptable and give more accurate results than existing ones. Furthermore, a multi-criteria decision-making technique is developed based on the proposed operators. A decision-making problem for healthcare facilities in public hospitals is provided to demonstrate the benefits and applicability of the proposed method. Finally, a comparative study is also investigated using the existing methods.
... Information aggregation models play a crucial role in group decision-making environments, and the study of information aggregation operators has been active in recent years. To meet the needs of various information processing requirements, a variety of types of information fusion aggregation operators have been developed, including arithmetic averaging operator [11][12][13][14], ordered weighted averaging operator [11,14,15], hybrid weighted averaging operator [11,14], weighted geometric operator [11][12][13]16], ordered weighted geometric operator [11,16], hybrid weighted geometric operator [11,16], Bonferroni mean operator [17][18][19], Choquet integral operator [20,21], Heronian mean operator [22,23], etc. However, all of the above aggregation operators are based on a combination of algebraic products and algebraic sums. ...
... Information aggregation models play a crucial role in group decision-making environments, and the study of information aggregation operators has been active in recent years. To meet the needs of various information processing requirements, a variety of types of information fusion aggregation operators have been developed, including arithmetic averaging operator [11][12][13][14], ordered weighted averaging operator [11,14,15], hybrid weighted averaging operator [11,14], weighted geometric operator [11][12][13]16], ordered weighted geometric operator [11,16], hybrid weighted geometric operator [11,16], Bonferroni mean operator [17][18][19], Choquet integral operator [20,21], Heronian mean operator [22,23], etc. However, all of the above aggregation operators are based on a combination of algebraic products and algebraic sums. ...
... A critical issue that requires discussion is how to aggregate this decision-making information, to obtain a comprehensive expression matrix and ultimately, arrive at the target decision outcome. In the process of information aggregation, a majority of studies adopt the arithmetic average operator [11,13,14] to aggregate the opinions of multiple experts. However, the arithmetic averaging operator compensates for differences, resulting in a bias in the decision outcome. ...
Dynamic multi-attribute group decision-making (DMAGDM) is a widespread practice in which evaluations are provided by multiple decision-makers at various times and early evaluations impact later evaluations. Additionally, attributes and alternatives can be added or removed over time. An R-numbers DMAGDM method is developed based on the advantages of R-numbers in capturing risks. This paper introduces the R-numbers Einstein weighted averaging (RNEWA) operator and R-numbers weighted Einstein geometric (RNEWG) operator, which are distinct from conventional algebraic operations, and examines their properties. Moreover, an expert weight determination model is constructed using the similarity measure of R-numbers. The attribute weight determination model in the R-numbers environment is also proposed with the method based on the criteria removal effects method (MEREC). A static rating calculation model, which utilizes the combination compromise solution (CoCoSo) method in the R-numbers environment, is built using the RNEWA operator and RNEWG operator. Furthermore, a new dynamic rating calculation model is proposed which does not require storage of all decision information over time. Finally, the applicability and effectiveness of the R-numbers DMAGDM method is demonstrated through a case study on supply chain risk assessment of manufacturing enterprises.
... Therefore, many scholars put forward new sort functions from different perspectives. Ye [23] proposed the accurate function from the perspective of hesitation in 2009. Nayagam and Sivaraman et al. [24,25], Gao [26], Kang [27] and Wang [28,29] proposed some different sort functions. ...
... After pointing out the limitations of Xu [14], Ye [23] proposed the following accurate function from the perspective of hesitation in 2009. ...
... In 2011, Nayagam and Sivaraman et al. [24,25] proposed two accurate functions based on Xu [14] and Ye [23]. ...
The classical Analytic Hierarchy Process (AHP) requires an exact value to compare the relative importance of two attributes, but experts often can not obtain an accurate assessment of every attribute in the decision-making process, there are always some uncertainty and hesitation. Compared with classical AHP, our new defined interval-valued intuitionistic fuzzy AHP has accurately descripted the vagueness and uncertainty. In decision matrix, the real numbers are substituted by fuzzy numbers. In addition, each expert will make different evaluations according to different experiences for each attribute in the subjective weighting method, which neglects objective factors and then generates some deviations in some cases. This paper provides two ways to make up for this disadvantage. On the one hand, by combining the interval-valued intuitionistic fuzzy AHP with entropy weight, an improved combination weighting method is proposed, which can overcome the limitations of unilateral weighted method only considering the objective or subjective factors. On the other hand, a new score function is presented by adjusting the parameters, which can overcome the invalidity of some existing score functions. In theory, some theorems and properties for the new score functions are given with strictly mathematical proof to validate its rationality and effectiveness. In application, a novel fuzzy portfolio is proposed based on the improved combination weighted method and new score function. A numerical example shows that these results of our new score function are consistent with those of most existing score functions, which verifies that our model is feasible and effective.
... Let This implies that M = N. This example shows that none of these methods [1,2,6] are better to each other; 9. ...
... Comparison of GIS score function with Novel accuracy score function M in Ye [2]: Using example 2.1. in [6], Garg has shown the inconsistency of Ye's [2] method. ...
... Comparison of GIS score function with Novel accuracy score function M in Ye [2]: Using example 2.1. in [6], Garg has shown the inconsistency of Ye's [2] method. However, his method also fails to compare arbitrary IVIFNs which we can be see from the following example. ...
Ranking of interval-valued intuitionistic fuzzy numbers (IVIFNs) is an important task for solving real-life Decision-Making problems. It is a potential area of research that has attracted the researchers working in fuzzy mathematics. Researchers worldwide are looking for a unique ranking principle that can be used to discriminate any two arbitrary IVIFNs. Various ranking functions on the set of IVIFNs have been proposed. However, every method has some drawbacks in ranking arbitrary IVIFNs due to the partial ordering. This paper introduces a new ranking principle for comparing two arbitrary IVIFNs by defining a new score function based on the non-membership value of IVIFNs. In this paper, firstly, the limitations of a few well-known and existing ranking methods for IVIFNs have been discussed. Secondly, a new non-membership score on the class of IVIFNs has been introduced. Thirdly, the superiority of the proposed score function in ranking arbitrary IVIFNs over the existing methods has been demonstrated. Finally, the proposed non-membership score function has been utilized in interval-valued intuitionistic fuzzy TOPSIS (IVIF-TOPSIS) using numerical examples.
... (iii) Novel Accuracy function (Ye, 2009): ...
... To say the least, an IVIFN and its complement will bear the same entropy value and hence cannot be differentiated. Chen et al. (2012) proposed a ranking method to overcome the drawback of the accuracy function of Ye (2009), and later (Wu & Chiclana, 2014) presented a risk attitudinal ranking method for IVIFNs based on novel score and accuracy expected functions. Garg (2016a) introduced an improvised score function for IVIFSs incorporating a weighted hesitancy factor: ...
With the increase of fall incidents in emergency cases and expenditures in the healthcare sector, fall prevention has become a very important study. The aim of this paper is to develop a ranking system to capture the risky gaits to aid in the reduction of fall incidents. With this, an approach driven framework under the interval-valued intuitionistic fuzzy environment is developed for decision making. The proposed framework is validated by experimental data and is seen to be effective and reliable. It is applied to risky gaits and a ranking is obtained based on the mindset one chooses. Technically, we also extended the entropy measure to incorporate the hesitancy factor to interval-valued region where a new measuring function is introduced. We believe that the proposed system will provide a better support for rehabilitation arrangements.
... In recent times, Xu [23] applied the score function and accuracy function to compare interval-valued intuitionistic fuzzy numbers (IVIFNs). Subsequently, Ye [25], Nayagam et al. [17], Sahin [19], Zhang and Xu [30] and Nayagam et al. [16] provided some additional accuracy functions to compare IVIFNs and developed multicriteria fuzzy decision-making techniques. It is well known that these approaches cannot identify the difference between two arbitrary IVIFNs in some situations due to the particular properties of intervals. ...
... Methodologies that rank any two IVIFNs have been studied by many researchers, such as Xu [23], Ye [25], Nayagam et al. [17], Sahin [19], Zhang and Xu [30] and Nayagam et al. [16]. Following this purpose, they have introduced various novel accuracy functions. ...
This study is inspired by those of Huang et al. (Soft Comput. 25, 2513--2520, 2021) and Wang et al. (Inf. Sci. 179, 3026--3040, 2009) in which some ranking techniques for interval-valued intuitionistic fuzzy numbers (IVIFNs) were introduced. In this study, we prove that the space of all IVIFNs with the relation in the method for comparing any two IVIFNs based on a score function and three types of entropy functions is a complete chain and obtain that this relation is an admissible order. Moreover, we demonstrate that IVIFNs are complete chains to the relation in the comparison method for IVIFNs on the basis of score, accuracy, membership uncertainty index, and hesitation uncertainty index functions.
... The IVIFS presented here was developed by Ye [25] and is an extension of the intuitionistic fuzzy set (IFS) originally proposed by Atanassov [26]. IVIFS, which has multidimensional mathematical functions, is used for addressing decision-making problems with several uncertainties, falsity, hesitation, and periodicity. ...
Railway infrastructure is generally classified as either fixed or movable infrastructure assets. Failure in any of the assets could lead to the complete shutdown and disruption of the entire system, economic loss, inconvenience to passengers and the train operating company(s), and can sometimes result in death or injury in the event of the derailment of the rolling stock. Considering the importance of the railway infrastructure assets, it is only necessary to continuously explore their behavior, reliability, and safety. In this paper, a proactive multi-criteria decision-making model that is based on an interval-valued intuitionistic fuzzy set and some reliability quantitative parameters has been proposed for the evaluation of the reliability of the infrastructure assets. Results from the evaluation show that the failure mode ‘Broken and defective rails’ has the most risk and reliability concerns. Hence, priority should be given to the failure mode to avoid a total system collapse.
... adapted from the work of Ye[48], featuring four companies in which one can investTo find a decision-making solution using interval neutrosophic values (INVs) and criterion weights, ranking alternatives8 Peng, Juan-juan, 2015, International Journal of General Systems[43] Simplified Neutrosophic Sets and Their Applications in Multi-Criteria Group Decision-Making Problems Synthetic data adapted from the work of Gallego-Lupianez[49], featuring four companies in which one can invest and three criteria to decide uponTo identify issues in a set of certain numbers using simplified neutrosophic sets ( ...
In recent years, neutrosophic theory has garnered increasing attention within scholarly circles due to its applicability in various domains. Within these domains, the field of decision-making has derived significant advantages from the progressions in neutrosophic theory. Notably, neutrosophic theory has made substantial contributions by advancing and offering a range of aggregation operators and information measures specifically designed for enhancing decision-making processes. In this context, this study aims to conduct a comprehensive bibliometric analysis of the current research landscape in the field of neutrosophic theory, with a specific focus on understanding its applications and development trends. Our analysis reveals that the scientific literature addresses neutrosophic theory in a diverse range of applications. This examination encompasses a scrutiny of key contributors, affiliated academic institutions, influential publications, and noteworthy journals within the neutrosophic domain. To achieve this, we have curated a dataset comprising scholarly papers retrieved from Clarivate Analytics’ Web of Science Core Collection database, employing keywords closely aligned with neutrosophic theory and its applications, spanning a specified timeframe starting from the year in which the first paper on neutrosophic theory was published, namely, from 2005 until 2022. Our findings underscore sustained and robust scholarly interest in neutrosophic theory, characterized by a considerable high annual growth rate of 43.74% during the specified period. Additionally, our investigation delves into the identification and analysis of pivotal keywords and emerging trends, shedding light on prominent research trajectories within this domain. Furthermore, we elucidate collaborative networks among authors, their academic affiliations, and the global distribution across diverse countries and territories, providing valuable insights into the worldwide proliferation of neutrosophic research and applications. Employing n-gram analysis techniques across titles, keywords, abstracts, and keyword-plus fields unveils a multitude of applications where neutrosophic theory plays a central role. The analysis culminates in a review of globally cited documents and a comprehensive discussion highlighting the significance of neutrosophic theory in contemporary research and problem-solving contexts.
... An IFS contains both the MG and NMG denoted by m and f , respectively, with the restriction 0 m þ f 1. Keeping in the view that IFS as a generalization of FS many researchers did a lot of work on the structure. For further study we may refer to Das [10], Lin [11], Xu [12], Liu [13], Xu [14], and Ye [15][16][17]. An IFS has not been able to deliver in some situations and fails. ...
The basic aim of this study is to introduce an innovative concept of bipolar T-spherical fuzzy set (BTSFS) which is a hybrid structure of bipolar fuzzy set (BFS) and T-spherical fuzzy Set (TSFS). BTSFS is a unique type of fuzzy set that works with bipolarity of all membership grades (MGs), that is, truth MG, abstained MG and non-membership grade (NMG). This paper scrutinizes a few basic characteristics and essential operations of BTSFSs. The proposed results are compared with the existing fuzzy structures and also the importance of the proposed work is discussed. Further, the idea of new distance measures of BTSFSs has been presented to investigate the geometrical properties of BPFSs. An example related to pattern recognition (PR) based on new distance measures have been examined in the article. Furthermore, some new similarity measures (SMs) based on BTSFSs such as Dice SM and weighted Dice SM have been initiated. We have also studied the propositions related to proposed SMs. A multi-criteria decision-making (MCDM) method based on proposed SMs for BSTSFSs has also been presented in this article. Then, in the end, to check capability and viability, a practical demonstration of the planned approach has also been explained.
... Some similarity measures for IFSs are developed in Deli and Çağman (2016). IFS is widely used in MADM Das et al., (2016); Lin et al., (2007) ;Ye, (2009) and(2010) ;Liu, & Wang, (2007) ;. ...
In this work, ambiguity and unclarity are coped with the effective tools of picture fuzzy sets (PFSs), especially where the conditions demand simulation of various dimensions for evaluation, for example decision making. PFS requires operators to measure the coordination of two PFSs. As far as this paper is concerned, we bring new operators to PFSs with an application, validating this as the generalization of the concept of Fuzzy Sets (FSs) and Intuitionistic Fuzzy Sets (IFSs). The hybrid structure of PFSs has been incorporated with other operators to develop picture fuzzy Dombi Hamy mean (PFDHM) operator, picture fuzzy weighted Dombi Hamy mean (PFWDHM) operator, picture fuzzy Dombi dual Hamy mean (PFDDHM) operator, and PF weighted Dombi dual Hamy mean (PFWDDHM) operator. Further, the properties such as Idempotency, Monotonicity, Boundedness, and Commutativity related to each proposed operator have been discussed. By using these operators the multiple attribute group decision-making (MAGDM) methods are proposed. Moreover, we have explained the application by providing an example of a car supplier. The results are concluded by selecting the best car on the basis of attributes such as quality, production, service efficiency, and risk factors using operators defined on PFSs. A comparative study is also conducted to study the significance of the developed work.
... Yager [7], Xu and Yager [8], Xu [9] respectively introduced the OWA operator, IFHG operator, IFOWG operator, IFWG operator, IFHA operator, IFOWA operator, and IFWA operator, and presented their advantages in our daily life problems. Ye [10,11] presented the notion of accuracy under environments, such that intuitionistic fuzzy numbers and intervalvalued intuitionistic fuzzy numbers. Wang and Liu [12,13] and Zhao and Wei [14] presented numerous new methods using Einstein's operation laws, namely IFEWG operator, IFEOWG operator, IFEWA operator, IFEOWA operator, IFEHA operator and IFEHG operators and their structural properties and applications. ...
The objective of this paper is to introduce some new logarithm operational laws for intuitionistic fuzzy sets. Some structure properties have been developed and based on these, various aggregation operators, namely confidence logarithmic intuitionistic fuzzy Einstein weighted geometric (CLIFEWG) operator, confidence logarithmic intuitionistic fuzzy Einstein ordered weighted geometric (CLIFEOWG) operator, confidence logarithmic intuitionistic fuzzy Einstein hybrid geometric (CLIFEHG) operator, confidence logarithmic intuitionistic fuzzy Einstein weighted averaging (CLIFEWA) operator, confidence logarithmic intuitionistic fuzzy Einstein ordered weighted averaging (CLIFEOWA) operator, confidence logarithmic intuitionistic fuzzy Einstein hybrid averaging (CLIFEHA) operator have been presented. To show the validity and the superiority of the proposed operators, we compared these methods with the existing methods and concluded from the comparison and sensitivity analysis our proposed techniques are more effective.
... Step 4: Evaluate the weights ξ it (i = 1, 2, 3, 4, t = 1, 2, 3, 4, 5) by applying Equation (17) Step 5: Utilize the IHCPWA operators and IHCPWG operators given in Equations (18) and (19), respectively, to obtain all IHCEs J it (i = 1, 2, 3, 4, t = 1, 2, 3, 4, 5) of the emerging enterprises H i . The aggregating values are reflected in Table 3. Step 6: By Equation (20), the values obtained by applying the score function are produced in Table 4. Step 7: Agreeing to the score values given in Table 4 and evaluating the formula of score functions, the rank of the emerging technology companies is stated in Table 5. Recall that this idea " > " indicates "preference." ...
The intuitionistic hesitant fuzzy set (IHFS) is an enriched version of hesitant fuzzy sets (HFSs) that can cover both fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). By assigning membership and non-membership grades as subsets of [0, 1], the IHFS can model and handle situations more proficiently. Another related theory is the theory of set pair analysis (SPA), which considers both certainties and uncertainties as a cohesive system and represents them from three aspects: identity, discrepancy, and contrary. In this article, we explore the suitability of combining the IHFS and SPA theories in multi-attribute decision making (MADM) and present the hybrid model named intuitionistic hesitant fuzzy connection number set (IHCS). To facilitate the design of a novel MADM algorithm, we first develop several averaging and geometric aggregation operators on IHCS. Finally, we highlight the benefits of our proposed work, including a comparative examination of the recommended models with a few current models to demonstrate the practicality of an ideal decision in practice. Additionally, we provide a graphical interpretation of the devised attempt to exhibit the consistency and efficiency of our approach.
... 1527 to rank the interval valued intuitionistic fuzzy numbers (IVIFNs) in the decision making process [5,7,8,9]; and (iii) methods of entropy measurement, which prioritise data discrimination by assessing uncertain information [10,11]. ...
In this research, a unique approach for determining unknown decision making weights utilising the sumudu transform and the integro-differential equation is provided, and the weights obtained are used in calculations to choose the best alternative. In addition, for the optimal solution, we calculate the prospect projection of each alternative and rank all of the alternatives based on the values of the prospect projection. Finally, to define the methodology, a numerical example is provided.
... From Table 5 it is seen that A 2 is the best alternative for case 1, case 2, case 3, case 4 and case 6 whereas A 4 is foremost alternative for case 5. These outcomes obtained from our suggested method are compliance with the works mentioned in [31,39]. ...
Fermatean fuzzy sets (FFSs), an orthopair fuzzy set proposed by Senapati and Yager (Journal of Ambient Intelligence and Humanized Computing 11:663–674, 2020, [24]), can handle the situation with ambiguous and incomplete information in a more effective manner than the Pythagorean fuzzy sets presented by Yager (Pythagorean fuzzy subsets, 2013 Joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), pp 57–61, 2013, [2]) and the intuitionistic fuzzy sets presented by Atanassov (Fuzzy Sets and Systems 20:87–96, 1986, [3]). Sergi et al. (Journal of Intelligent & Fuzzy Systems 42:365–376, 2022, [40]) initiated interval-valued Fermatean fuzzy sets (IVFFSs) and established IVFFS ordering, as well as some mathematical operations. In addition, Jeevraj (Expert Systems with Applications 185:1–20, 2021, [37]) introduced the concept of a score and accuracy function for IVFFSs. The main objective of this chapter is to suggest some score functions for acceptable ranking of IVFFSs, as well as the interval-valued Fermatean fuzzy TOPSIS method for solving multi-criteria decision making problems. Here, we have proposed six new variants of score functions for effective ranking of interval-valued Fermatean fuzzy sets. Depending on various types of score functions, we have used a TOPSIS-based multi-criteria decision making (MCDM) problem in which decision makers’ (DMs’) preference knowledge is summarized in the pattern of interval-valued Fermatean fuzzy sets. To illustrate the usefulness of the proposed method, a computational paradigm has been considered. Finally, concluding remarks and future scope of research have been mentioned.
KeywordsInterval-valued Fermatean fuzzy setFermatean fuzzy setTOPSISMulti-criteria decision makingScore function
... Xu [37] has introduced two other score functions (score and accuracy) on the set of IVIFNs and compared any two arbitrary IVIFNs by using them. Jun Ye [16] has proposed a new novel accuracy score function M for comparing arbitrary IVIFNs and utilized novel accuracy score function in solving Multi-criteria Decision-Making (MCDM) problems. Nayagam and Geetha [19] have introduced another accuracy function by including a new parameter d (represents Decision maker confidence), which has performed better in ranking IVIFNs. ...
Conventional trapezoidal intuitionistic fuzzy numbers (CTrIFNs) are used in the literature to handle many real-life problems with imprecise information. However, the CTrIFNs are not the real generalization of interval-valued intuitionistic fuzzy numbers (IVIFNs) and triangular intuitionistic fuzzy numbers (TIFNs). This study discusses the non-conventional trapezoidal intuitionistic fuzzy numbers called trapezoidal-valued intuitionistic fuzzy numbers (TrVIFNs) that generalize (contain) all real-valued intuitionistic fuzzy numbers, IVIFNs and TIFNs. Various researchers worldwide are looking for a complete ranking principle on the set of TrVIFNs. However, none of them yields a complete ranking. The paper’s main aim is to propose a new complete ranking principle on the class of TrVIFNs. The complete ranking principle on the set of TrVIFNs makes the decision-making algorithm more robust. To achieve the main aim, firstly, we introduce eight different score functions on the set of TrVIFNs and study their mathematical properties. Secondly, we present a new ranking principle by considering all eight score functions in a linear order. Further, we prove (mathematically) that the introduced ranking principle defines complete ranking on the set of TrVIFNs. Thirdly, we show the significance of the proposed method compared with the existing methods on various classes of fuzzy and intuitionistic fuzzy numbers. Finally, we implement a case application of the proposed complete ranking principle for evaluating the social sustainability performance of Indian firms.
... Liu (2011, 2012), explored the notion of many operators such as IFEWG operator, IFEOWG operator, IFEWA operator, IFEOWA operator based on Einstein operations and applied these operators on decision-making problems. Ye (2007Ye ( , 2009, using IFS and developed the notion of a new accuracy degree for interval-valued and developed MAGDM based on vague sets. To develop the decisionmaking process in daily life problems. ...
The focus of our this paper is to develop a series of Einstein hybrid aggregation operators using confidence level, such as confidence intuitionistic fuzzy Einstein hybrid averaging operator, confidence intuitionistic fuzzy Einstein hybrid geometric operator, generalized confidence intuitionistic fuzzy Einstein hybrid averaging operator and generalized confidence intuitionistic fuzzy Einstein hybrid geometric operator. The main advantage of the new operators is that these operators not only provide information to the experts of the problems, but these methods also consider the degrees of the experts of the problems that they are familiar with the selection option. The new approaches provide more general, more accurate and precise results as compared to the existing methods. Finally, the proposed operators have been applied to decision-making problems to show the validity, practicality and effectiveness of the new approach.
... In this section, we will compare the aggregated results achieved by using IVIFDWA and IVIFDWG operators with various other IVIF information based on AOs. For this purpose, we applied averaging and geometric AOs [45] of the IVIFs framework, the Hamy mean operators of IVIFSs [46], and IVIF Bonferroni mean operators [47] on the IVIF information given in Table 1, and the results are given in Table 7, below. ...
Dombi t-norm (DTN) and t-conorm (TCN) are among the most effective triangular norms in fuzzy systems for aggregation purposes. The environment of interval-valued intuitionistic fuzzy (IVIF) set gives us precision in expressing uncertain information by using a membership grade (MG) and non-membership grade (NMG) in the form of closed subintervals of 0, 1. The goal of this paper is to introduce DTN-based aggregation operators (AOs) for IVIF numbers (IVIFNs) and study their performance in the evaluation of the worth of energy recourses to be opted in Pakistan to deal with the energy crises situation. We first introduced some DTN and TCN-based operations for IVIFNs and developed two new AOs known as IVIF Dombi weighted averaging (IVIFDWA) and IVIF Dombi weighted geometric (IVIFDWG) operators. The validity and fitness of the proposed operators are tested. A case study is presented where the energy resources of Pakistan are discussed and the problem of the selection of sustainable energy resources in the context of Pakistan is investigated. The sensitivity analysis of the proposed IVIFDWA and IVIFDWG operators is studied and a comparative analysis of the current work with previous studies is established.
... The accuracy function M defined on IVIFNs [18] is extended to the TraIFNs: Let ...
The aim of this paper proposes an improved accuracy function (AF) on the collection of trapezoidal intuitionistic fuzzy numbers (TraIFNs) and makes comparisons with existing functions. A trapezoidal intuitionistic fuzzy TOPSIS method based on the proposed AF is applied to rank the alternatives, a trapezoidal intuitionistic fuzzy index matrix based on the proposed accuracy function is applied to rank the alternatives, and the most suitable one(s) can be chosen in the decision-making process.KeywordsIntuitionistic fuzzy valuesTrapezoidal intuitionistic fuzzy numbersImproved accuracy functionTOPSIS methodTrapezoidal intuitionistic fuzzy index matrixMulti-criteria decision-making
... However, interval-valued intuitionistic fuzzy sets are among the most known extensions of fuzzy sets. It provides the limits of belongingness and non-belongingness for the elements to obtain the vague sets with the limited numbers (Ye, 2009). Earlier version of this approach is presented by Atanassov in 1980s (Atanassov, 1986) and interval-valued intuitionistic fuzzy sets are applied for different kinds of decision-making problems gradually (Atanassov, 1994;Atanassov & Gargov, 1989). ...
Serious environmental problems occur as a result of the unconscious production and use of energy. Hence, there is a strong need to develop an effective energy culture to provide sustainability for the energy investments. This study aims to identify the influence directions of dimensions and criteria for energy culture in emerging economies for sustainable energy investments. For this purpose, a multi-criteria decision-making model is proposed based on 2-tuple linguistic values and interval-valued intuitionistic fuzzy sets. The weighting results of energy culture are also evaluated for E7 economies. The findings demonstrate that energy companies should consider investing in technologies, assets, and objects which enable to reproduce energy usage. Furthermore, the analysis revealed that there is a misalignment of practices with material culture and norms what creates the potential for behavior change of consumers. Finally, the potential of bigger countries for investment opportunities in sustainable energies has been disclosed by ranking emerging economies. The results let us provide insights into development of energy investments by considering strategic priorities of energy culture in emerging economies. The novelty of this paper is to define a set of criteria and dimensions for the strategic priorities of energy culture in emerging economies and construct a novel decision-making technique with DEMATEL and TOPSIS based on 2-tuple and interval-valued intuitionistic fuzzy sets.
... Liu (2011, 2012) explored the notion of several operators, such as the IFEWG operator, the IFEOWG operator, the IFEWA operator, the IFEOWA operator based on Einstein operations and applied these operators on decision making problems. Ye (2007Ye ( , 2009 using IFS and developed the notion of a new accuracy degree for intervalvalued and developed MAGDM based on vague sets. Zhao and Wei (2013) familiarized the concept of some Einstein hybrid operators, namely the IFEHA operator, the IFEHG operator and their weight and advantages in real-life problems. ...
In this paper, a logarithmic operational law for intuitionistic fuzzy numbers is defined, in which the based ℓ is a real number such that ℓ∈ (0 , 1) with condition ℓ≠ 1. Some properties of logarithmic operational laws have been studied and based on these, several Einstein averaging and Einstein geometric operators, namely logarithmic intuitionistic fuzzy Einstein weighted averaging (LIFEWA) operator, logarithmic intuitionistic fuzzy Einstein ordered weighted averaging (LIFEOWA) operator, logarithmic intuitionistic fuzzy Einstein hybrid averaging (LIFEHA) operator, logarithmic intuitionistic fuzzy Einstein weighted geometric (LIFEWG) operator, logarithmic intuitionistic fuzzy Einstein ordered weighted geometric (LIFEOWG) operator, and logarithmic intuitionistic fuzzy Einstein hybrid geometric (LIFEHG) operator have been introduced, which can overcome the weaknesses of algebraic operators. Furthermore, based on the proposed operators a multi-attribute group decision-making problem is established under logarithmic operational laws. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
... PFSs and IVPFSs have attracted the attention of many researchers within a very short period of time. There are several methods in the field of PFS to solve real-life multi-criteria, decision-making problems (Ye, 2009;Zhang et al., 2014;Zhang, 2016;Gou et al., 2016;Geng et al., 2017;Jing et al., 2017;Li et al., 2018). Several researchers have also proposed real-life applications under a Pythagorean fuzzy environment. ...
The intention of this paper is to propose some similarity measures between Fermatean fuzzy sets (FFSs). Firstly, we propose some score based similarity measures for finding similarity measures of FFSs and also propose score based cosine similarity measures between FFSs. Furthermore, we introduce three newly scored functions for effective uses of Fermatean fuzzy sets and discuss some relevant properties of cosine similarity measure. Fermatean fuzzy sets introduced by Senapati and Yager can manipulate uncertain information more easily in the process of multi-criteria decision making (MCDM) and group decision making. Here, we investigate score based similarity measures of Fermatean fuzzy sets and scout the uses of FFSs in pattern recognition. Based on different types of similarity measures a pattern recognition problem viz. personnel appointment is presented to describe the use of FFSs and its similarity measure as well as scores. The counterfeit results show that the proposed method is more malleable than the existing method(s). Finally, concluding remarks and the scope of future research of the proposed approach are given.
... Liu (2011, 2012) explored the notion of several operators, such as the IFEWG operator, the IFEOWG operator, the IFEWA operator, the IFEOWA operator based on Einstein operations and applied these operators on decision making problems. Ye (2007Ye ( , 2009 using IFS and developed the notion of a new accuracy degree for interval-valued and developed MAGDM based on vague sets. Zhao and Wei (2013) familiarized the concept of some Einstein hybrid operators namely, the IFEHA operator, the IFEHG operator and their weight and advantages in real life problems. ...
In this paper, a logarithmic operational law for intuitionistic fuzzy numbers is defined, in which the based1 is a real number such that1 ∈(0,1) with condition1 ≠ 1. Some properties of logarithmic operational laws have been studied and based on these, several Einstein averaging and Einstein geometric operators namely, logarithmic intuitionistic fuzzy Einstein weighted averaging (LIFEWA) operator, logarithmic intuitionistic fuzzy Einstein ordered weighted averaging (LIFEOWA) operator, logarithmic intuitionistic fuzzy Einstein hybrid averaging (LIFEHA) operator, logarithmic intuitionistic fuzzy Einstein weighted geometric (LIFEWG) operator, logarithmic intuitionistic fuzzy Einstein ordered weighted geometric (LIFEOWG) operator, and logarithmic intuitionistic fuzzy Einstein hybrid geometric (LIFEHG) operator have been introduced, which can overcome the weaknesses of algebraic operators. Furthermore, based on the proposed operators a multi-attribute group decision-making problem is established under logarithmic operational laws. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.
... The concept of fuzzy sets has been thoroughly investigated and is widely accepted in the field of decision-making. To date, fuzzy sets have been extended to intuitionistic fuzzy set (IFS) (Atanassov 1986), interval-valued IFS (Ye 2009), hesitant fuzzy set (HFS) (Torra 2010) and many more. However, fuzzy sets cannot handle the indeterminacy elements in the real-world problems. ...
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.
... • If ( 1 ) < ( 2 ), then 1 < 2 . Ye [52] offered a new formulation of the accuracy function (i.e., the novel accuracy function) that allows comparison among IVIFS values. (19) where ( ) ∈ [−1, +1]. ...
The recovery efforts of the tourism and hospitality sector are compromised by the emergence of COVID-19 variants that can escape vaccines. Thus, maintaining non-pharmaceutical measures amidst massive vaccine rollouts is still relevant. The previous works which categorize tourist sites and restaurants according to the perceived degree of tourists' and customers’ exposure to COVID-19 are deemed relevant for sectoral recovery. Due to the subjectivity of predetermining categories, along with the failure of capturing vagueness and uncertainty in the evaluation process, this work explores the use k-means clustering with dataset values expressed as interval-valued intuitionistic fuzzy sets. In addition, the proposed method allows for the incorporation of criteria (or attribute) weights into the dataset, often not considered in traditional k-means clustering but relevant in clustering problems with attributes having varying priorities. Two previously reported case studies were analyzed to demonstrate the proposed approach, and comparative and sensitivity analyses were performed. Results show that the priorities of the criteria in evaluating tourist sites remain the same. However, in evaluating restaurants, customers put emphasis on the physical characteristics of the restaurants. The proposed approach assigns 12, 15, and eight sites to the “low exposure”, “moderate exposure”, and “high exposure” cluster, respectively, each with distinct characteristics. On the other hand, 16 restaurants are assigned “low exposure”, 16 to “moderate exposure”, and eight to “high exposure” clusters, also with distinct characteristics. The characteristics described in the clusters offer meaningful insights for sectoral recovery efforts. Findings also show that the proposed approach is robust to small parameter changes. Although idiosyncrasies exist in the results of both case studies, considering the characteristics of the resulting clusters, tourists or customers could evaluate any tourist site or restaurant according to their perceived exposure to COVID-19.
... However, intuitionistic fuzzy sets are a well-known extension of fuzzy sets that have, to date, once again become a hot topic in the decision-making area [54,55]. They have several advantages, for example, they provide the extreme limits for the belongingness and non-belongingness elements and construct the vague sets more accurately [56]. Atanassov [57] introduced the intuitionistic sets in the 1980s as a generalized form of fuzzy sets and developed IVIFSs [58,59]. ...
The purpose of this study is to analyze new service development competencies based on house of quality (HoQ) in the healthcare industry by using a hybrid interval-valued intuitionistic decision-making approach with fuzzy linguistic information modelled by hesitant fuzzy linguistic term sets (HFLTSs) and linguistic 2-tuples. The novelty of the manuscript is to identify a list of criteria, dimensions, and alternatives for HoQ-based new service development used for the investment decisions in healthcare industry. Thus, a hybrid model with HFLTSs has been proposed, considering 2-tuple linguistic information, Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) and a defuzzification process with an attitudinal expected score function. Hence, a DEMATEL extension dealing with HFLTSs, linguistic 2-tuple and IVIFS has been developed and used to weigh the criteria and dimensions of customer needs. Similarly, a TOPSIS extension dealing with the same type of information is introduced and applied in order to rank the technical requirements, the so-called “new service development competencies in healthcare industry”. The results show which are the most and least important dimensions for patients’ needs, and also point out which dimensions can be applied more effectively to satisfy the needs of service demanders. The findings obtained could be generalized to a country level analysis, and other extensions of multi-criteria decision-making methods could be applied to future studies.
... Because of Xu's ranking method's flaws, many scholars have done a lot of research on the score function and the accuracy function. For example, to overcome the drawback of Xu's accuracy function, Ye [11] proposed an improved accuracy as follows: However, the proposed techniques for ranking IVIFNs using a score function, the scope of solutions is limited. For this reason, Yao [13] proposed a new score function to overcome this problem. ...
In this paper, an improved multiple attribute decision making (MADM) method based on the proposed novel score function and accuracy function of interval-valued intuitionistic fuzzy numbers (IVIFNs) is proposed to aggregate large-scale data. The attribute values in the decision matrices provided by each decision-maker (DM), which are characterized by interval numbers. First, a transformation matrix is introduced to define the concepts of satisfactory set, un- satisfactory set and uncertainty set of alternatives. An approach is then developed for aggregating attribute values into IVIFNs, and we will obtain the collective evaluation of each alternative. Next, using the interval-valued intuitionistic fuzzy weighted averaging operator, the collective attribute values characterized by IVIFNs are aggregated to get the overall evaluation of alternatives. The score function and accuracy function are applied to calculate the score degree and the rank of each alternative. Finally, a large-scale example is given to verify the validity of the reported method.
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.
Atanassov’s intuitionistic fuzzy sets play a vital role in handling vagueness and uncertainty successfully as they associate an additional non-membership degree to each element of the universal set. Interval-valued intuitionistic fuzzy (IVIF) sets are more generalized quantities in this direction as they use intervals (⊂[0, 1]) to define the membership and non-membership grades of each element of the set. However, without defining basic arithmetic operations on the IVIF values/sets, the study is incomplete. This paper broadly examines the division and subtraction operations over any arbitrary IVIF values/sets using the concept of Hamming distance between them. A deterministic linear optimization model is proposed and solved in order to obtain the complete expressions for these operations. Next, various fundamental properties and relationships are examined and proved for the proposed arithmetic operations over IVIF values/sets. Furthermore, an example of a multi-criteria decision-making problem under uncertain conditions is studied and analyzed using the developed arithmetic operations on IVIF numbers.
As China’s economic and social development enters a new stage, the role of innovation and entrepreneurship is becoming increasingly prominent, and its importance is also being emphasized. As the main force for future employment and national economic construction, college students naturally become the new force for innovation and entrepreneurship. Therefore, it is imperative for universities to carry out in-depth innovation and entrepreneurship education (IEE) for college students. Then, with the continuous development of social development needs and the professional growth needs of college students, the “innovation and entrepreneurship” education for college students should also be adjusted in a timely manner in terms of educational concepts, models, and methods. The IEE environment evaluation in universities under the background of “ Double Innovation” is looked as multiple attribute decision-making (MADM). In this paper, the information entropy model is employed to calculate the objective weight of the evaluation attribute. Then, interval-valued intuitionistic fuzzy Combined Compromise Solution (IVIF-CoCoSo) is built based on the Hamming distance and Euclid distance to cope with MADM under interval-valued intuitionistic fuzzy sets (IVIFSs). The new MADM method is proposed for IEE environment evaluation in universities under the background of “ Double Innovation”. Finally, the IVIF-CoCoSo approach is compared with existing methods to verify the effectiveness of IVIF-CoCoSo algorithm. The main contributions of this constructed paper are: (1) the IVIF-CoCoSo method is built based on the Hamming distance and Euclid distance. (2) the information entropy model is employed to calculate the objective weight of the evaluation attribute. (3) The new MADM method is proposed for IEE environment evaluation in universities under the background of “ Double Innovation” based on IVIF-CoCoSo. (4) The IVIF-CoCoSo model is compared with existing methods to verify the effectiveness of the IVIF-CoCoSo algorithm.
The ambiguous sets have recently been proposed to represent inherent uncertainties, imprecision, and vague information. In this study, we generalize the concept of the ambiguous set to a single-valued ambiguous number (SVAN). We propose many comparison functions and operational laws for SVANs. This study also introduces three new aggregation operators, the ambiguous weighted geometric operator (AWGO), the ambiguous ordered weighted geometric operator (AOWGO), and the ambiguous hybrid geometric operator (AHGO), for aggregating the input information in the form of SVANs. We discuss some significant properties of the operators along with numerical examples. We develop a method based on the AWGO and AHGO for solving multi-criteria group decision-making (MCGDM) problems. Finally, a real-time example of a MCGDM problem associated with the selection of cars by customers is presented, and its solution is recommended using the proposed method.
Let = (U, V) be a Single valued Neutrosophic graph. A subset ∈ () is a said to be score equitable set if the score value of any two nodes in S differ by at most one. That is, | ()-()| ≤ 1, , ∊. If e is an edge with end vertices u and v and score of u is greater than or equal to score of v then we say u strongly dominates v. If every vertex of V − S is strongly influenced by some vertex of S then S is called strong score set of G. The minimum cardinality of a strong dominating set is called the strong score number of G. The equitable integrity of Single valued Neutrosophic graph G which is defined as E () = {| | + (−): is a score equitable set in }, where (−) denotes the order of the largest component in −. The strong integrity of Single valued Neutrosophic graph G which is defined as S () = {| | + (−): is a strong score set in }. In this paper, we study the concepts of equitable integrity and strong equitable integrity in different classes of regular Neutrosophic graphs and discussed the upper and lower bounds.
Ambiguous set has recently been proposed to represent inherent uncertainties, imprecision, and vague information. In this study, we generalize the concept of ambiguous set to a single-valued ambiguous number (SVAN). We propose many comparison functions and operational laws for SVANs. This study also introduces three new aggregation operators, such as the ambiguous weighted geometric operator (AWGO), the ambiguous ordered weighted geometric operator (AOWGO), and the ambiguous hybrid geometric operator (AHGO), which are used to aggregate input information in the form of SVANs. We discuss some significant properties of the operators along with numerical examples. Based on the AWGO and AHGO, we develop a method for solving the multi-criteria group decision-making (MCGDM) problem. Finally, a real-time example of an MCGDM problem associated with the selection of cars by customers is presented, and its solution is recommended using the proposed method.
In this manuscript, we firstly proposed a novel concept of complex interval-valued q-rung orthopair linguistic (CIVq-ROL) information by integrating the concepts of complex interval-valued q-rung orthopair fuzzy (CIVq-ROF) information and linguistic set (LS), and it is more generalized than most existing sets, and it is very feasible and valuable for depicting awkward and unreliable information in a difficult situation. Further, we developed some new operational laws of the CIVq-ROF information based on algebraic t-norm and t-conorm. It is also clear that it is a very challenging task to fuse the collection of data into a singleton set in information fusion and decision making, and the Heronian mean (HM) operator is an important aggregation operator, which can not only achieve the aggregation function from the collection of data into a singleton one, but also consider the relationship between any two data. Therefore, based on the CIVq-ROL information and HM operator, we proposed the CIVq-ROL Heronian mean (CIVq-ROLHM), CIVq-ROL weighted HM (CIVq-ROLWHM), CIVq-ROL geometric HM (CIVq-ROLGHM), and CIVq-ROL weighted geometric HM (CIVq-ROLWGHM) operators, and then, the various desirable properties and specific cases of them are also investigated. Furthermore, we developed a multi-attribute decision-making (MADM) procedure for evaluating the finest business in the country from the collection of four different types of electronics-related businesses based on the proposed operators. Finally, various examples are given to show the application of the invented techniques, and a comparative analysis is carried out for the parameters to show the advantages of the proposed approaches.
Assessment specialists (experts) are sometimes expected to provide two types of information: knowledge of rating domains and the performance of rating objects (called confidence levels). Unfortunately, the results of previous information aggregation studies cannot be properly used to combine the two categories of data covered above. Additionally, a significant range of symmetric/asymmetric events and structures are frequently included in the implementation process or practical use of fuzzy systems. The primary goal of the current study was to use cubic Fermatean fuzzy set features to address such situations. To deal with the ambiguous information of the aggregated arguments, we defined information aggregation operators with confidence degrees. Two of the aggregation operators we initially proposed were the confidence cubic Fermatean fuzzy weighted averaging (CCFFWA) operator and the confidence cubic Fermatean fuzzy weighted geometric (CCFFWG) operator. They were used as a framework to create an MCDM process, which was supported by an example to show how effective and applicable it is. The comparison of computed results was carried out with the help of existing approaches.
Multi-criteria decision-making (MCDM) approaches are effective and reliable to solve problems with uncertain conditions. The Cubic Pythagorean fuzzy set (CPFS) is the newest model of fuzzy set theory. The aim of this article is to develop cubic Pythagorean fuzzy Combinative Distance-based assessment (PCF-CODAS) by combining CODAS with CPFS, to solve MCDM problems. This method employs the Euclidean distance (ℰ𝒟̅̅̅̅) as the major measure and Taxicab distance (𝒯𝒟̅̅̅̅) as the secondary measure to determine the desirability of the alternative. These distances are calculated with the negative-ideal solution. Furthermore, an example is provided to assess the practicality of the proposed methodology. We also provided a comprehensive sensitivity study to test the CPF-CODAS results and compare them with some existing MCDM methods. These evaluations reveal that the proposed methodology is efficient and the results are stable.
Recently the useful redefinition of the intuitionistic and interval-valued intuitionistic fuzzy set theories (A−IFS, A−IVIFS) based on the Dempster-Shafer theory of evidence (DST) was developed. It was named the “Belief-Plausibility” (BP) approach as only the basic mathematical tools of the DST were used for its foundation. In this paper, a new approach to compare the real, interval and fuzzy-valued intuitionistic fuzzy and BP numbers is proposed. Among known approaches to compare intuitionistic fuzzy objects, most justified and used in applications are those based on the so-called score and accuracy functions. The observed multiplicity of these functions treatments and modifications indicates that the source of such a divergence is the impossibility to solve this problem exclusively in the framework of the A−IFS. Therefore, to develop the reliable methods for the considered types of comparison, a redefinition of this problem based on the BP approach is proposed. This redefinition allows us to develop new two and three criteria methods for comparison of real, interval and fuzzy-valued intuitionistic fuzzy and BP numbers, based on the local criteria with transparent interpretation in the framework of the BP approach.
Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can hold much more information and can be used to describe both an interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) at the same time to handle data uncertainties. Based on it, the present study is classified into three phases. The first phase is to modify the existing operational laws and aggregation operators (AOs) in the article presented by Abbas et al. (Journal of Intelligent & Fuzzy Systems, vol. 37, no. 1, pp. 1529–1544, (2019)). The main objective of improved operational laws is to eliminate the flows and ambiguities in existing AOs. Secondly, based on these laws, various AOs to aggregate the information are acquired along with their requisite properties and relations. Lastly, an approach for interpreting the multi-attribute decision-making (MCDM) problem based on the stated operators is given and illustrated with an example. Some of the existing models are used to perform a comprehensive comparative analysis to demonstrate their impacts.
Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can hold much more information and can be used to describe both an interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) at the same time to handle data uncertainties. Based on it, the present study is classified into three phases. The first phase is to modify the existing operational laws and aggregation operators (AOs) in the article presented by
In this paper, a new type of IVIFN is defined from TraIFN. Some operations are defined by using the concept of arithmetic operations on intervals. Later, the main theme of the paper is elaborated on that is how the operation defined over the new type of IVIFNs has an impact on existing ranking functions. It is a seed for several developments in the field of MCDM, fuzzy multi-objective LPP, and in various fields of fuzzy theory. Finally, an application of one of the ranking methods is discussed.
Let = (U, V) be a Single valued Neutrosophic graph. A subset ∈ () is a said to be score equitable set if the score value of any two nodes in S differ by at most one. That is, | ()-()| ≤ 1, , ∊. If e is an edge with end vertices u and v and score of u is greater than or equal to score of v then we say u strongly dominates v. If every vertex of V − S is strongly influenced by some vertex of S then S is called strong score set of G. The minimum cardinality of a strong dominating set is called the strong score number of G. The equitable integrity of Single valued Neutrosophic graph G which is defined as E () = {| | + (−): is a score equitable set in }, where (−) denotes the order of the largest component in −. The strong integrity of Single valued Neutrosophic graph G which is defined as S () = {| | + (−): is a strong score set in }. In this paper, we study the concepts of equitable integrity and strong equitable integrity in different classes of regular Neutrosophic graphs and discussed the upper and lower bounds.
Graph pattern matching (GPM) in big graph has been widely used in decision making, such as expert finding, social group discovery, etc. However, these existing works consider neither the preference of the decision maker (DM), nor the subjectivity of constraints during the process of GPM. Therefore, this paper proposes an interval-valued intuitionistic fuzzy decision (IVIFD) with graph pattern in big graph. As traditionally IVIFD can maximally reduce the uncertainty of decision making and it is only valid for small datasets, which makes it impossible to be applied in big graph. In this paper, GPM is adopted to prune the searching space, which makes it possible to process IVIFD under the preference of the DM later. Technically, firstly, each DM selects the preferred vertices and/or edges in big graph, and the interval-valued intuitionistic fuzzy preference (IVIFP) is calculated and used to form the contextual constraints to conduct GPM. Secondly, a probability-certainty density function is introduced to capture the subjective probability of the contextual preference of subgraphs via the bijection from rating space of the context to preference space of the context, which leads to an interval-valued intuitionistic fuzzy set (IVIFS). In addition to the IVIFS, the IVIFD is made through interval-valued intuitionistic fuzzy cross entropy and grey relation degree. Moreover, the weight problem between different contexts is taken into account and handled respectively as three cases. Finally, numerical experiments and perturbation analysis validate the effectiveness and stability of our proposed method, and verify its necessity and efficiency through ablation experiments.
The ranking of interval-valued intuitionistic fuzzy values (IVIFVs) has an important role in real-life decision-making problems. Even though there are many approaches related to the ranking methods of the IVIFVs, some of them have some shortcomings. In this study, the disadvantages of the existing ranking functions of the IVIFVs are discussed. It is revealed that both the most popular ranking functions and their recently improved versions may lead to unacceptable results. Furthermore, in this article, a new score function is offered to cope with the shortcomings of the ranking functions. The importance and performance of this new score function are proven with the help of many examples. Also, the decision-making algorithm adapted by integrating the new function is presented to illustrate the applicability of the new score function in the decision-making problems.
In this paper the concepts of weighted arithmetic and geometric operators on IVPFSS are defined. Using both the arithmetic and geometric operators accuracy function is defined and the level of accuracy is calculated. The IVPFSS weighted aggregation operators are used to aggregate the IVPFSS information corresponding to each alternative. The alternatives are then ranked based on the values of the accuracy function. An algorithm based on this multi criteria decision-making method is developed. To illustrate this MCDM method five candidates (alternatives) are selected for a poetry competition and their poetic skills are taken as the parameters. The candidate whose accuracy function value is maximum is chosen as the winner. Comparison of accuracy function vales shows that the maximum values corresponding to the alternative A5 were on par with each other. However, the ranking of the other alternatives differ.
Samarandache introduced the neutrosophic set concept in 1995 to solve uncertainty data. After that, Maji introduced the concept of neutrosophic soft set by combining the neutrosophic sets and soft sets as a general mathematical tool. Due to some structural deficiencies in Maji’s study, Ozturk et al. re-defined some operations on neutrosophic soft sets. The neutrosophic soft mapping was firstly defined by Alkhazaleh et al. However, Alkhazaleh’s study does not provide some basic classical mapping properties and it has structural problems. For that reason, we propose a new approach to neutrosophic soft mapping. In this study, we re-define neutrosophic soft mappings and studied the images and inverse images of neutrosophic soft sets. Further, we investigate the basic operations and other related properties of neutrosophic soft mappings. Finally, the neutrosophic soft mapping have been applied to application in decision making.
In this chapter we especially report on the concept of single-valued spherical hesitant neutrosophic number (SVSHNN for short) induced by single-valued hesitant neutrosophic set and single-valued spherical neutrosophic set. Next, using Dombi operations, we construct some salient operations coined as single-valued spherical hesitant neutrosophic Dombi operations between the SVSHNNs. We study in detail the key features of these proposed operations. We also pursue here that these proposed operations invoke single-valued spherical hesitant neutrosophic Dombi weighted arithmetic aggregation operators (SVSHNDWAA for short) and single-valued spherical hesitant neutrosophic Dombi weighted geometric aggregation operators (SVSHNDWGA for short), to aggregate the single-valued spherical hesitant neutrosophic information. In addition, most importantly, we provide the score of SVSHNNs to ranking the SVSHNNs. We develop a multi-attribute decision-making approach in the perspectives of single-valued spherical hesitant neutrosophic weighted aggregation operators and score of SVSHNNs in which the evaluation values of alternatives on the attribute are expressed in terms of SVSHNNs and the ranking for those alternatives is made subject to the score values of SVSHNNs to select the best suited one. Last but not the least, we make use of this proposed approach into the Bulbul disaster assessment and conform of its proficiency.
Suppose that all of C∞ functions f1,…, fk have the zero property. We give a necessary and sufficient condition for their product to have the same property This is a generalization of Bochnak’s result ([1]).
A definition of the concept ‘intuitionistic fuzzy set’ (IFS) is given, the latter being a generalization of the concept ‘fuzzy set’ and an example is described. Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
The basic definitions of the concept of interval-valued intuitionistic fuzzy set and of the operations, relations and operators over it are given. Some of the most important applications are described. Ideas for future development of the theory of interval-valued intuitionistic fuzzy sets are discussed.
Methods for aggregating interval-valued intuitionistic fuzzy information are investigated. Some operational laws of interval-valued intuitionistic fuzzy numbers are defined. Based on these operational laws, some aggregation operators, including interval-valued intuitionistic fuzzy weighted arithmetic aggregation operator and interval-valued intuitionistic fuzzy weighted geometric aggregation operator, are proposed. The score function and accuracy function of interval-valued intuitionistic fuzzy number are defined, and based on these two functions, a method for ranking interval-valued intuitionistic fuzzy numbers is presented. Finally, an approach for decision making with interval-valued intuitionistic fuzzy information is developed, and a practical example is provided to illustrate the developed approach.
In this article, the ordered weighted aggregation operator and hybrid aggregation operator are developed for aggregating interval-valued intuitionistic preference information. Interval-valued intuitionistic judgment matrix and its score matrix and accuracy matrix are defined. Some of their desirable properties are investigated in detail. The relationships among interval-valued intuitionistic judgment matrix, intuitionistic judgment matrix, and complement judgment matrix, are discussed. On the basis of the arithmetic aggregation operator and hybrid aggregation operator, an approach to group decision making with interval-valued intuitionistic judgment matrices is given. Finally, a practical example is provided to verify the effectiveness of the developed approach.
A generalization of the notion of intuitionistic fuzzy set is given in the spirit of ordinary interval valued fuzzy sets. The new notion is called interval valued intuitionistic fuzzy set (IVIFS). Here we present the basic preliminaries of IVIFS theory.
The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.
A vague set is a set of objects, each of which has a grade of
membership whose value is a continuous subinterval of [0,1]. Such a set
is characterized by a truth-membership function and a false-membership
function. The notion of inclusion, union, intersection, and complement
are extended to such sets, and various properties of vague sets are
established. Finally, convex vague sets are introduced
This paper introduces and discusses a method of approximate inference which operates on the extension of the concept of a fuzzy set by the concept of an interval-valued fuzzy set. This method allows a formal, fuzzy representation to be built for verbal decision algorithms. Furthermore, it can have an effective computer representation. An example showing how this method operates is provided.
Different operators are defined over the interval valued intuitionistic fuzzy sets and their basic properties are studied.
We recapitulate the definition given by Atanassov (1983) of intuitionistic fuzzy sets as well as the definition of vague sets given by Gau and Byehrer (1993) and see that both definitions coincide.
A study on the steps to follow in linguistic decision analysis is presented in a context of multi-criteria/multi-person decision making. Three steps are established for solving a multi-criteria decision making problem under linguistic information: (i) the choice of the linguistic term set with its semantic in order to express the linguistic performance values according to all the criteria, (ii) the choice of the aggregation operator of linguistic information in order to aggregate the linguistic performance values, and (iii) the choice of the best alternatives, which is made up by two phases: (a) the aggregation of linguistic information for obtaining a collective linguistic performance value on the alternatives, and (b) the exploitation of the collective linguistic performance value in order to establish a rank ordering among the alternatives for choosing the best alternatives. Finally, an example is shown.
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Interval valued fuzzy sets are proposed for the representation of combined concepts based on normal forms where linguistic connectives as well as variables are assumed to be fuzzy.It is shown that the proposed representation (1) exists for certain families of the conjugate pairs of t-norms and t-conorms; and (2) resolves some of the difficulties associated with particular interpretations of conjunction, disjuntion, and implication in fuzzy set theories.
An improved method is presented, which provides improved score functions to measure the degree of suitability of each of a set of alternatives, with respect to a set of criteria presented with vague values. The improved algorithm for score functions is introduced by taking into account the effect of an unknown degree (hesitancy degree) of the vague values on the degree of suitability to which each alternative satisfies the decision-maker’s requirement. The meaning of the proposed function is more transparent than that of other existing functions, which are not reasonable in some cases. The proposed function illustrates that it has stronger discrimination in comparison with previous functions. The applicability of this improved multicriteria fuzzy decision-making approach is also demonstrated by means of examples. The improved method can be used to rank the decision alternatives according to the decision criteria. The functions proposed in this paper can provide a more useful technique than previous functions, in order to efficiently help the decision-maker.
An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators.
A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems
- M B Gorzaleczany
Gorzaleczany, M. B. (1987). A method of inference in approximate reasoning based
on interval-valued fuzzy sets. Fuzzy Sets and Systems, 21, 1-17.
Vague sets are intuitionistic fuzzy sets
- Bustince
A method of inference in approximate reasoning based on interval-valued fuzzy sets
- Gorzaleczany