Power Aggregation Operators and Similarity Measures Based on Improved Intuitionistic Hesitant Fuzzy Sets and their Applications to Multiple Attribute Decision Making

Intuitionistic hesitant fuzzy set(IHFS)is amixture of two separated notions called intuitionistic fuzzy set(IFS)and hesitant fuzzy set(HFS),as an important technique to cope with uncertain and awkward information in realistic decision issues.IHFS contains the grades of truth and falsity in the formof the subset of the unit interval.The notion of IHFS was defined by many scholars with different conditions,which contain several weaknesses.Here,keeping in view the problems of already defined IHFSs,we will define IHFS in another way so that it becomes compatible with other existing notions.To examine the interrelationship between any numbers of IHFSs,we combined the notions of power averaging(PA)operators and power geometric(PG)operators with IHFSs to present the idea of intuitionistic hesitant fuzzy PA(IHFPA)operators,intuitionistic hesitant fuzzy PG(IHFPG)operators,intuitionistic hesitant fuzzy power weighted average(IHFPWA)operators,intuitionistic hesitant fuzzy power ordered weighted average(IHFPOWA)operators,intuitionistic hesitant fuzzy power ordered weighted geometric(IHFPOWG)operators,intuitionistic hesitant fuzzy power hybrid average(IHFPHA)operators,intuitionistic hesitant fuzzy power hybrid geometric(IHFPHG)operators and examined as well their fundamental properties.Some special cases of the explored work are also discovered.Additionally,the similarity measures based on IHFSs are presented and their advantages are discussed along examples.Furthermore,we initiated a new approach to multiple attribute decision making(MADM)problem applying suggested operators and a mathematical model is solved to develop an approach and to establish its common sense and adequacy.Advantages,comparative analysis,and graphical representation of the presented work are elaborated to show the reliability and effectiveness of the presented works.

On similarity measures of interval-valued intuitionistic fuzzy sets and their application to pattern recognitions

The concept of the degree of similarity between interval-valued intuitionistic fuzzy sets （IVIFSs） is introduced, and some distance measures between IVIFSs are defined based on the Hamming distance, the normalized Hamming distance, the weighted Hamming distance, the Euclidean distance, the normalized Euclidean distance, and the weighted Euclidean distance, etc. Then, by combining the Hausdorff metric with the Hamming distance, the Euclidean distance and their weighted versions, two other similarity measures between IVIFSs, i. e., the weighted Hamming distance based on the Hausdorff metric and the weighted Euclidean distance based on the Hausdorff metric, are defined, and then some of their properties are studied. Finally, based on these distance measures, some similarity measures between IVIFSs are defined, and the similarity measures are applied to pattern recognitions with interval-valued intuitionistic fuzzy information.

A novel similarity measure between intuitionistic fuzzy sets based on the mid points of transformed triangular fuzzy numbers with applications to pattern recognition and medical diagnosis

Similarity measure is an essential tool to compare and determine the degree of similarity between intuitionistic fuzzy sets (IFSs). In this paper, a new similarity measure between intuitionistic fuzzy sets based on the mid points of transformed triangular fuzzy numbers is proposed. The proposed similarity measure provides reasonable results not only for the sets available in the literature but also gives very reasonable results, especially for fuzzy sets as well as for most intuitionistic fuzzy sets. To provide supportive evidence, the proposed similarity measure is tested on certain sets available in literature and is also applied to pattern recognition and medical diagnosis problems. It is observed that the proposed similarity measure provides a very intuitive quantification.

Fuzzy description logic based on vague sets

To enable the representation and reasoning for fuzzy ontologies with expressive fuzzy knowledge on the semantic web, a new fuzzy extension of description logics called vague ALC which is based on vague sets is presented. The definition of vague set is introduced and then the syntax and semantics of vague ALC are formally defined. The forms of axioms and assertions in the vague ALC knowledge bases are specified. Finally, the tableau algorithm is developed for the reasoning in the vague ALC. The vague ALC based on vague set uses two degrees of membership instead of a single membership degree in the fuzzy sets and is more accurate in representing the imprecision in the degrees of membership. The vague ALC has more expressive power than ALC and can represent fuzzy knowledge and perform reasoning tasks based on them. Therefore, the vague ALC can enable the representation and reasoning for fuzzy ontologies with expressive fuzzy knowledge on the semantic web.

On Similarity Measures and Distance Measures of Fuzzy Soft Sets

In this paper we introduce several new similarity measures and distance measures between fuzzy soft sets, these measures are examined based on the set-theoretic approach and the matching function. Comparative studies of these measures are derived. By introducing two general formulas, we propose a new method to define the similarity measures and the distance measures between two fuzzy soft sets with different parameter sets.

Modelling spatial vagueness based on type-2 fuzzy set

The modelling and formal characterization of spatial vagueness plays an increasingly important role in the imple- mentation of Geographic Information System (GIS). The concepts involved in spatial objects of GIS have been investigated and acknowledged as being vague and ambiguous. Models and methods which describe and handle fuzzy or vague (rather than crisp or determinate) spatial objects, will be more necessary in GIS. This paper proposes a new method for modelling spatial vagueness based on type-2 fuzzy set, which is distinguished from the traditional type-1 fuzzy methods and more suitable for describing and implementing the vague concepts and objects in GIS.

Rough similarity degree and rough close degree in rough fuzzy sets and the applications

Based on rough similarity degree of rough sets and close degree of fuzzy sets, the definitions of rough similarity degree and rough close degree of rough fuzzy sets are given, which can be used to measure the similar degree between two rough fuzzy sets. The properties and theorems are listed. Using the two new measures, the method of clustering in the rough fuzzy system can be obtained. After clustering, the new fuzzy sample can be recognized by the principle of maximal similarity degree.

P-Indeterminate Vector Similarity Measures of Orthopair Neutrosophic Number Sets and Their Decision-MakingMethod with Indeterminate Degrees

In the complexity and indeterminacy of decision making(DM)environments,orthopair neutrosophic number set(ONNS)presented by Ye et al.can be described by the truth and falsity indeterminacy degrees.Then,ONNS demonstrates its advantages in the indeterminate information expression,aggregations,and DM problems with some indeterminate ranges.However,the existing research lacks some similarity measures between ONNSs.They are indispensable mathematical tools and play a crucial role in DM,pattern recognition,and clustering analysis.Thus,it is necessary to propose some similaritymeasures betweenONNSs to supplement the gap.To solve the issue,this study firstly proposes the p-indeterminate cosine measure,p-indeterminate Dice measure,p-indeterminate Jaccard measure of ONNSs(i.e.,the three parameterized indeterminate vector similarity measures of ONNSs)in vector space.Then,a DMmethod based on the parameterized indeterminate vector similarity measures of ONNSs is developed to solve indeterminate multiple attribute DM problems by choosing different indeterminate degrees of the parameter p,such as the small indeterminate degree(p=0)or the moderate indeterminate degree(p=0.5)or the big indeterminate degree(p=1).Lastly,an actual DM example on choosing a suitable logistics supplier is provided to demonstrate the flexibility and practicability of the developed DM approach in indeterminate DM problems.By comparison with existing relative DM methods,the superiority of this study is that the established DMapproach indicates its flexibility and suitability depending on decision makers’indeterminate degrees(decision risks)in ONNS setting.

Cotangent Similarity Measure of Consistent Neutrosophic Sets and Application to Multiple Attribute Decision-Making Problems in Neutrosophic Multi-Valued Setting

A neutrosophic multi-valued set(NMVS)is a crucial representation for true,false,and indeterminate multivalued information.Then,a consistent single-valued neutrosophic set(CSVNS)can effectively reflect the mean and consistency degree of true,false,and indeterminate multi-valued sequences and solve the operational issues between different multi-valued sequence lengths in NMVS.However,there has been no research on consistent single-valued neutrosophic similarity measures in the existing literature.This paper proposes cotangent similarity measures and weighted cotangent similarity measures between CSVNSs based on cotangent function in the neutrosophic multi-valued setting.The cosine similarity measures showthe cosine of the angle between two vectors projected into amultidimensional space,rather than their distance.The cotangent similaritymeasures in this study can alleviate several shortcomings of cosine similarity measures in vector space to a certain extent.Then,a decisionmaking approach is presented in viewof the established cotangent similarity measures in the case of NMVSs.Finally,the developed decision-making approach is applied to selection problems of potential cars.The proposed approach has obtained two different results,which have the same sort sequence as the compared literature.The decision results prove its validity and effectiveness.Meantime,it also provides a new manner for neutrosophic multi-valued decision-making issues.

Novel Distance Measures on Hesitant Fuzzy Sets Based on Equal-Probability Transformation and Their Application in Decision Making on Intersection Traffic Control

The purpose of this study is to reduce the uncertainty in the calculation process on hesitant fuzzy sets(HFSs).The innovation of this study is to unify the cardinal numbers of hesitant fuzzy elements(HFEs)in a special way.Firstly,a probability density function is assigned for any given HFE.Thereafter,equal-probability transformation is introduced to transform HFEs with different cardinal numbers on the condition into the same probability density function.The characteristic of this transformation is that the higher the consistency of the membership degrees in HFEs,the higher the credibility of the mentioned membership degrees is,then,the bigger the probability density values for them are.According to this transformation technique,a set of novel distance measures on HFSs is provided.Finally,an illustrative example of intersection traffic control is introduced to show the usefulness of the given distance measures.The example also shows that this study is a good complement to operation theories on HFSs.

A Knowledge Measure with Parameter of Intuitionistic Fuzzy Sets

A new knowledge measure with parameter of intuitionistic fuzzy sets (IFSs) is presented based on the membership degree and the non-membership degree of IFSs, which complies with the extended form of Szmidt-Kacprzyk axioms for intuitionistic fuzzy entropy. And a sufficient and necessary condition of order property in the Szmidt-Kacprzyk axioms is discussed. Additionally, some numerical examples are given to illustrate the applications of the proposed knowledge measure and some conventional entropies and knowledge measures of IFSs. The experimental results show that the results of the parametric model proposed in this paper are more accurate than those of most of the classic models.

A Synergistic Multi-Attribute Decision-Making Method for Educational Institutions Evaluation Using Similarity Measures of Possibility Pythagorean Fuzzy Hypersoft Sets

Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pPyFHSS)is more flexible in this regard than other theoretical fuzzy set-like models,even though some attempts have been made in the literature to address such uncertainties.This study investigates the elementary notions of pPyFHSS including its set-theoretic operations union,intersection,complement,OR-and AND-operations.Some results related to these operations are also modified for pPyFHSS.Additionally,the similarity measures between pPyFHSSs are formulated with the assistance of numerical examples and results.Lastly,an intelligent decision-assisted mechanism is developed with the proposal of a robust algorithm based on similarity measures for solving multi-attribute decision-making(MADM)problems.A case study that helps the decision-makers assess the best educational institution is discussed to validate the suggested system.The algorithmic results are compared with the most pertinent model to evaluate the adaptability of pPyFHSS,as it generalizes the classical possibility fuzzy set-like theoretical models.Similarly,while considering significant evaluating factors,the flexibility of pPyFHSS is observed through structural comparison.

Information rough communication based on both-branch fuzzy set

The concept of rough communication based on both-branch fuzzy set is proposed, in which the loss of information may exist, for each agent there has a different language and can not provide precise communication to each other. The method of information measure in a rough communication based on both-branch fuzzy set is proposed. By using some concepts, such as ｜α｜-both-branch rough communication cut, the relation theorem between rough communication based on both-branch fuzzy concept and rough communication based on classical concept is obtained. Finally, an example of rough communication based on both-branch fuzzy set is given.

Optimal Route Selection Method Based on Vague Sets

Optimal route selection is an important function of vehicle trac flow guidance system. Its core is to determine the index weight for measuring the route merits and to determine the evaluation method for selecting route. In this paper, subjective weighting method which relies on driver preference is used to determine the weight and the paper proposes the multi-criteria weighted decision method based on vague sets for selecting the optimal route. Examples show that, the usage of vague sets to describe route index value can provide more decision-making information for route selection.

Two-Sided Matching Decision Making with Multi-Attribute Probabilistic Hesitant Fuzzy Sets

In previous research on two-sided matching(TSM)decision,agents’preferences were often given in the form of exact values of ordinal numbers and linguistic phrase term sets.Nowdays,the matching agent cannot perform the exact evaluation in the TSM situations due to the great fuzziness of human thought and the complexity of reality.Probability hesitant fuzzy sets,however,have grown in popularity due to their advantages in communicating complex information.Therefore,this paper develops a TSM decision-making approach with multi-attribute probability hesitant fuzzy sets and unknown attribute weight information.The agent attribute weight vector should be obtained by using the maximum deviation method and Hamming distance.The probabilistic hesitancy fuzzy information matrix of each agent is then arranged to determine the comprehensive evaluation of two matching agent sets.The agent satisfaction degree is calculated using the technique for order preference by similarity to ideal solution(TOPSIS).Additionally,the multi-object programming technique is used to establish a TSM method with the objective of maximizing the agent satisfaction of two-sided agents,and the matching schemes are then established by solving the built model.The study concludes by providing a real-world supply-demand scenario to illustrate the effectiveness of the proposed method.The proposed method is more flexible than prior research since it expresses evaluation information using probability hesitating fuzzy sets and can be used in scenarios when attribute weight information is unclear.

Introduce a Novel PCA Method for Intuitionistic Fuzzy Sets Based on Cross Entropy

In this paper, a new method for Principal Component Analysis in intuitionistic fuzzy situations has been proposed. This approach is based on cross entropy as an information index. This new method is a useful method for data reduction for situations in which data are not exact. The inexactness in the situations assumed here is due to fuzziness and missing data information, so that we have two functions (membership and non-membership). Thus, method proposed here is suitable for Atanasov’s Intuitionistic Fuzzy Sets (A-IFSs) in which we have an uncertainty due to a mixture of fuzziness and missing data information. For the demonstration of the application of the method, we have used an example and have presented a conclusion.

Quasi-cardinality of a Fuzzy Set on a Measurable Universe of Discourse

The concepts of quasi-cardinality, possibility and expectation of a fuzzy set on a measurable set, as a generalization of the quasi-cardinality of a fuzzy set on a finite universe of discourse is proposed. Also some properties of them are given out and thus some relevant results are elucidated.

Modeling Multisource-heterogeneous Information Based on Random Set and Fuzzy Set Theory

This paper presents a new idea, named as modeling multisensor-heterogeneous information, to incorporate the fuzzy logic methodologies with mulitsensor-multitarget system under the framework of random set theory. Firstly, based on strong random set and weak random set, the unified form to describe both data (unambiguous information) and fuzzy evidence (uncertain information) is introduced. Secondly, according to signatures of fuzzy evidence, two Bayesian-markov nonlinear measurement models are proposed to fuse effectively data and fuzzy evidence. Thirdly, by use of "the models-based signature-matching scheme", the operation of the statistics of fuzzy evidence defined as random set can be translated into that of the membership functions of relative point state variables. These works are the basis to construct qualitative measurement models and to fuse data and fuzzy evidence.

A New Matchmaking Algorithm Based on Multi-Level Matching Mechanism Combined with Fuzzy Set

The lack of semantic parts, increasing the number of Web services in the Web, and syntactic-based search operation are the main problems of current Web service technologies, these factors make difficult for clients to find a required web service. This paper shows a matchmaking algorithm to discover Semantic Web Services that are satisfying client requirements. It depends on two factors that distinguish it from any conventional Web service discovery algorithm;the first one is using semantic matching technique to overcome shortcoming of keyword matching techniques, the second one is tying Quality of Service (QoS) metrics of Web Service (WS) with fuzzy words that are used in user’s request. At least fifty percent average gain in search relevancy is obtained when our matchmaking algorithm is applied to WSs that are actually matching the chosen fuzzy semantic theme.