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.

Interval-valued intuitionistic fuzzy aggregation operators

The notion of the interval-valued intuitionistic fuzzy set （IVIFS） is a generalization of that of the Atanassov＇s intuitionistic fuzzy set. The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. There are various averaging operators defined for IVlFSs. These operators are not monotone with respect to the total order of IVIFS, which is undesirable. This paper shows how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. Moreover, two new aggregation operators based on the t.ukasiewicz triangular norm are proposed, which are monotone with respect to the total order of IVIFS. Finally, an application of the interval-valued intuitionistic fuzzy weighted averaging operator is given to multiple criteria decision making.

Interval-Valued Intuitionistic Fuzzy-Rough Sets

This paper combines interval-valued intuitionistic fuzzy sets and rough sets.It studies rougheness in interval-valued intuitionistic fuzzy sets and proposes one kind of interval-valued intuitionistic fuzzy-rough sets models under the equivalence relation in crisp sets.That extends the classical rough set defined by Pawlak.

Evaluation of Industrial IoT Service Providers with TOPSIS Based on Circular Intuitionistic Fuzzy Sets

Industrial Internet of Things(IIoT)service providers have become increasingly important in the manufacturing industry due to their ability to gather and process vast amounts of data from connected devices,enabling manufacturers to improve operational efficiency,reduce costs,and enhance product quality.These platforms provide manufacturers with real-time visibility into their production processes and supply chains,allowing them to optimize operations and make informed decisions.In addition,IIoT service providers can help manufacturers create new revenue streams through the development of innovative products and services and enable them to leverage the benefits of emerging technologies such as Artificial Intelligence(AI)and machine learning.Overall,the implementation of IIoT platforms in the manufacturing industry is crucial for companies seeking to remain competitive and meet the ever-increasing demands of customers in the digital age.In this study,the evaluation criteria to be considered in the selection of IIoT service provider in small andmedium-sized(SME)manufacturing enterprises will be determined and IIoT service providers alternatives will be evaluated using the technique for order preference by similarity to an ideal solution(TOPSIS)method based on circular intuitionistic fuzzy sets.Based on the assessments conducted in accordance with the literature review and expert consultations,a set of 8 selection criteria has been established.These criteria encompass industry expertise,customer support,flexibility and scalability,security,cost-effectiveness,reliability,data analytics,as well as compatibility and usability.Upon evaluating these criteria,it was observed that the security criterion holds the highest significance,succeeded by cost-effectiveness,data analytics,flexibility and scalability,reliability,and customer support criteria,in descending order of importance.Following the evaluation of seven distinct alternatives against these criteria,it was deduced that the A6 alternative,a German service provider,emerged as the most favorable option.The identical issue was addressed utilizing sensitivity analysis alongside various multi-criteria decision-making(MCDM)methods,and after comprehensive evaluation,the outcomes were assessed.Spearman’s correlation coefficient was computed to ascertain the association between the rankings derived from solving the problem using diverse MCDM methods.

Complex Decision Modeling Framework with Fairly Operators and Quaternion Numbers under Intuitionistic Fuzzy Rough Context

The main goal of informal computing is to overcome the limitations of hypersensitivity to defects and uncertainty while maintaining a balance between high accuracy,accessibility,and cost-effectiveness.This paper investigates the potential applications of intuitionistic fuzzy sets(IFS)with rough sets in the context of sparse data.When it comes to capture uncertain information emanating fromboth upper and lower approximations,these intuitionistic fuzzy rough numbers(IFRNs)are superior to intuitionistic fuzzy sets and pythagorean fuzzy sets,respectively.We use rough sets in conjunction with IFSs to develop several fairly aggregation operators and analyze their underlying properties.We present numerous impartial laws that incorporate the idea of proportionate dispersion in order to ensure that the membership and non-membership activities of IFRNs are treated equally within these principles.These operations lead to the development of the intuitionistic fuzzy rough weighted fairly aggregation operator(IFRWFA)and intuitionistic fuzzy rough ordered weighted fairly aggregation operator(IFRFOWA).These operators successfully adjust to membership and non-membership categories with fairness and subtlety.We highlight the unique qualities of these suggested aggregation operators and investigate their use in the multiattribute decision-making field.We use the intuitionistic fuzzy rough environment’s architecture to create a novel strategy in situation involving several decision-makers and non-weighted data.Additionally,we developed a novel technique by combining the IFSs with quaternion numbers.We establish a unique connection between alternatives and qualities by using intuitionistic fuzzy quaternion numbers(IFQNs).With the help of this framework,we can simulate uncertainty in real-world situations and address a number of decision-making problems.Using the examples we have released,we offer a sophisticated and systematically constructed illustrative scenario that is intricately woven with the complexity ofmedical evaluation in order to thoroughly assess the relevance and efficacy of the suggested methodology.

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

Green supplier selection is an important debate in green supply chain management(GSCM),attracting global attention from scholars,especially companies and policymakers.Companies frequently search for new ideas and strategies to assist them in realizing sustainable development.Because of the speculative character of human opinions,supplier selection frequently includes unreliable data,and the interval-valued Pythagorean fuzzy soft set(IVPFSS)provides an exceptional capacity to cope with excessive fuzziness,inconsistency,and inexactness through the decision-making procedure.The main goal of this study is to come up with new operational laws for interval-valued Pythagorean fuzzy soft numbers(IVPFSNs)and create two interaction operators-the intervalvalued Pythagorean fuzzy soft interaction weighted average(IVPFSIWA)and the interval-valued Pythagorean fuzzy soft interaction weighted geometric(IVPFSIWG)operators,and analyze their properties.These operators are highly advantageous in addressing uncertain problems by considering membership and non-membership values within intervals,providing a superior solution to other methods.Moreover,specialist judgments were calculated by the MCGDM technique,supporting the use of interaction AOs to regulate the interdependence and fundamental partiality of green supplier assessment aspects.Lastly,a statistical clarification of the planned method for green supplier selection is presented.

Research on θ -Intuitionistic Fuzzy Homomorphism Theory

Fuzzy homomorphism is an important research content of fuzzy group theory, different fuzzy mappings will produce different fuzzy homomorphisms. In this paper, the fuzzy homomorphism of groups is generalized. Firstly, the θ-intuitionistic fuzzy mapping is defined, and the θ-intuitionistic fuzzy homomorphism of groups is obtained. The properties of intuitionistic fuzzy subgroups and intuitionistic fuzzy normal subgroups are studied under the θ-intuitionistic fuzzy homomorphism of groups, and the fundamental theorem of θ-intuitionistic fuzzy homomorphism is proved.

Dynamic Analysis of Customer Demand Based on Intuitionistic Fuzzy Number in Product Planning

To address the fuzziness and variability in determining customer demand importance,a dynamic analysis method based on intuitionistic fuzzy numbers is proposed.First,selected customers use intuitionistic fuzzy numbers to represent the importance of each demand.Then,the preference information is aggregated using customer weights and time period weights through the intuitionistic fuzzy ordered weighted average operator,yielding a dynamic vector of the subjective importance of the demand index.Finally,the feasibility of the proposed method is demonstrated through an application example of a vibrating sorting screen.

Target Priority Determination Methods by Interval-Valued Intuitionistic Fuzzy Sets with Unknown Attribute Weights

The paper aims at the problem of multi-targets threat degree being hard to be evaluated accurately in complex air defense battlefield environments. Combined with multi-sensors information fusion and interval-valued intuitionistic fuzzy sets(IVIFS) theories, the target priority determination is studied. The score and accuracy functions of IVIFS are improved with thinking about the hesitating information in order to increase the rationality.Then, the influence factors of target priority and the nonlinear relationship between the influence factors and target priority are analyzed. Next, the algorithms for calculating the factor weights and sensor weights are given. Based on the theory of IVIFS and technique for order preference by similarity to an ideal solution(TOPSIS), two methods of target priority determination based on the IVIFS and TOPSIS are proposed. At last, an application example verifies the effectiveness and flexibility of the proposed algorithms.

NOTES ON “MULTICRITERIA FUZZY DECISION-MAKING METHOD BASED ON A NOVEL ACCURACY FUNCTION UNDER INTERVAL-VALUED INTUITIONISTIC FUZZY ENVIRONMENT”

This note points outs the inappropriateness of an accuracy function introduced by Ye [Ye, J. （2009）. Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Expert Systems with Applications, 36 （3）： 6899-6902] and its misleading use for comparing two interval-valued intuitionistic fuzzy numbers.

A model for selecting green suppliers through interval-valued intuitionistic fuzzy multi criteria decision making models

This study sought to construct a model for selecting green suppliers.The study was conducted in three phases:in phase 1,the literature was reviewed based on the meta-synthesis method and experts’opinions were analyzed to create a model composed of 11 dimensions and 40 criteria for selecting green suppliers.In phase 2,intuitionistic fuzzy best-worst method was used to weight the dimensions and criteria.The most important dimension was“green image”and the most important criteria were the“environmental protection system”and a good“public image.”In phase 3,four suppliers in the field of medical equipment were assessed and ranked.The study proposed a framework for selecting green supplier selection(GSC)and could prove to be more comprehensive than the existing models as it drew on the meta-synthesis method.Another innovation of the study was its reliance on the intuitionistic fuzzy set theory to overcome ambiguities in selecting suppliers.

Aggregation Operators for Interval-Valued Pythagorean Fuzzy Hypersoft Set with Their Application to Solve MCDM Problem

Experts use Pythagorean fuzzy hypersoft sets(PFHSS)in their investigations to resolve the indeterminate and imprecise information in the decision-making process.Aggregation operators(AOs)perform a leading role in perceptivity among two circulations of prospect and pull out concerns from that perception.In this paper,we extend the concept of PFHSS to interval-valued PFHSS(IVPFHSS),which is the generalized form of intervalvalued intuitionistic fuzzy soft set.The IVPFHSS competently deals with uncertain and ambagious information compared to the existing interval-valued Pythagorean fuzzy soft set.It is the most potent method for amplifying fuzzy data in the decision-making(DM)practice.Some operational laws for IVPFHSS have been proposed.Based on offered operational laws,two inventive AOs have been established:interval-valued Pythagorean fuzzy hypersoft weighted average(IVPFHSWA)and interval-valued Pythagorean fuzzy hypersoft weighted geometric(IVPFHSWG)operators with their essential properties.Multi-criteria group decision-making(MCGDM)shows an active part in contracts with the difficulties in industrial enterprise for material selection.But,the prevalent MCGDM approaches consistently carry irreconcilable consequences.Based on the anticipated AOs,a robust MCGDMtechnique is deliberate formaterial selection in industrial enterprises to accommodate this shortcoming.A real-world application of the projectedMCGDMmethod for material selection(MS)of cryogenic storing vessels is presented.The impacts show that the intended model is more effective and reliable in handling imprecise data based on IVPFHSS.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

In this manuscript,our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces.We establish some fixed point theorems in this setting.Also,we plot some graphs of an example of obtained result for better understanding.We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space.Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions.Triangular conorms are known as dual operations of triangular norms.The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.

Business Blockchain Suitability Determinants: Decision-Making through an Intuitionistic Fuzzy Method

Blockchain is one of the innovative and disruptive technologies that has a wide range of applications in multiple industries beyond cryptocurrency.The widespread adoption of blockchain technology in various industries has shown its potential to solve challenging business problems,as well as the possibility to create new business models which can increase a firm’s competitiveness.Due to the novelty of the technology,whereby many companies are still exploring potential use cases,and considering the complexity of blockchain technology,which may require huge changes to a company’s existing systems and processes,it is important for companies to carefully evaluate suitable use cases and determine if blockchain technology is the best solution for their specific needs.This research aims to provide an evaluation framework that determines the important dimensions of blockchain suitability assessment by identifying the key determinants of suitable use cases in a business context.In this paper,a novel approach that utilizes both qualitative(Delphi method)and quantitative(fuzzy set theory)methods has been proposed to objectively account for the uncertainty associated with data collection and the vagueness of subjective judgments.This work started by scanning available literature to identify major suitability dimensions and collected a range of criteria,indicators,and factors that had been previously identified for related purposes.Expert opinions were then gathered using a questionnaire to rank the importance and relevance of these elements to suitability decisions.Subsequently,the data were analyzed and we proceeded to integrate multi-criteria group decision-making(MCGDM)and intuitionistic fuzzy set(IFS)theory.The findings demonstrated a high level of agreement among experts,with the model being extremely sensitive to variances in expert assessments.Furthermore,the results helped to refine and select the most relevant suitability determinants under three important dimensions:functional suitability of the use case,organizational applicability,and ecosystem readiness.

A Novel Incremental Attribute Reduction Algorithm Based on Intuitionistic Fuzzy Partition Distance

Attribute reduction,also known as feature selection,for decision information systems is one of the most pivotal issues in machine learning and data mining.Approaches based on the rough set theory and some extensions were proved to be efficient for dealing with the problemof attribute reduction.Unfortunately,the intuitionistic fuzzy sets based methods have not received much interest,while these methods are well-known as a very powerful approach to noisy decision tables,i.e.,data tables with the low initial classification accuracy.Therefore,this paper provides a novel incremental attribute reductionmethod to dealmore effectivelywith noisy decision tables,especially for highdimensional ones.In particular,we define a new reduct and then design an original attribute reduction method based on the distance measure between two intuitionistic fuzzy partitions.It should be noted that the intuitionistic fuzzypartitiondistance iswell-knownas aneffectivemeasure todetermine important attributes.More interestingly,an incremental formula is also developed to quickly compute the intuitionistic fuzzy partition distance in case when the decision table increases in the number of objects.This formula is then applied to construct an incremental attribute reduction algorithm for handling such dynamic tables.Besides,some experiments are conducted on real datasets to show that our method is far superior to the fuzzy rough set based methods in terms of the size of reduct and the classification accuracy.

Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators

Two interval-valued intuitionistic uncertain linguistic hybrid operators cal ed the induced interval-valued intuitionistic uncertain linguistic hybrid Shapley averaging （I-IIULHSA） operator and the induced interval-valued intuitionistic uncertain linguistic hybrid Shapley geometric （I-IIULHSG） operator are defined. These operators not only reflect the importance of elements and their ordered positions, but also consider the correlations among elements and their ordered positions. Since the fuzzy measures are defined on the power set, it makes the problem exponentially complex. In order to simplify the complexity of solving a fuzzy measure, we further define the induced interval-valued intuitionistic uncertain linguistic hybrid λ-Shapley averaging （I-IIULHλSA） operator and the induced interval-valued intuitionistic uncertain linguistic hybrid λ-Shapley geometric （I-IIULHλSG） operator. Moreover, an approach for multi-attribute group decision making under the interval-valued intuitionistic uncertain linguistic environment is developed. Finally, a numerical example is provided to verify the developed procedure and demonstrate its practicality and feasibility.

Intuitionistic fuzzy C-means clustering algorithms

Intuitionistic fuzzy sets(IFSs) are useful means to describe and deal with vague and uncertain data.An intuitionistic fuzzy C-means algorithm to cluster IFSs is developed.In each stage of the intuitionistic fuzzy C-means method the seeds are modified,and for each IFS a membership degree to each of the clusters is estimated.In the end of the algorithm,all the given IFSs are clustered according to the estimated membership degrees.Furthermore,the algorithm is extended for clustering interval-valued intuitionistic fuzzy sets(IVIFSs).Finally,the developed algorithms are illustrated through conducting experiments on both the real-world and simulated data sets.

Intuitionistic fuzzy hierarchical clustering algorithms

Intuitionistic fuzzy set （IFS） is a set of 2-tuple arguments, each of which is characterized by a membership degree and a nonmembership degree. The generalized form of IFS is interval-valued intuitionistic fuzzy set （IVIFS）, whose components are intervals rather than exact numbers. IFSs and IVIFSs have been found to be very useful to describe vagueness and uncertainty. However, it seems that little attention has been focused on the clustering analysis of IFSs and IVIFSs. An intuitionistic fuzzy hierarchical algorithm is introduced for clustering IFSs, which is based on the traditional hierarchical clustering procedure, the intuitionistic fuzzy aggregation operator, and the basic distance measures between IFSs： the Hamming distance, normalized Hamming, weighted Hamming, the Euclidean distance, the normalized Euclidean distance, and the weighted Euclidean distance. Subsequently, the algorithm is extended for clustering IVIFSs. Finally the algorithm and its extended form are applied to the classifications of building materials and enterprises respectively.

Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems

Intuitionistic trapezoidal fuzzy numbers and their operational laws are defined. Based on these operational laws, some aggregation operators, including intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator are proposed. Expected values, score function, and accuracy function of intuitionitsic trapezoidal fuzzy numbers are defined. Based on these, a kind of intuitionistic trapezoidal fuzzy multi-criteria decision making method is proposed. By using these aggregation operators, criteria values are aggregated and integrated intuitionistic trapezoidal fuzzy numbers of alternatives are attained. By comparing score function and accuracy function values of integrated fuzzy numbers, a ranking of the whole alternative set can be attained. An example is given to show the feasibility and availability of the method.