Subsethood Measure Research Articles

Novel Subsethood Measures for Totally DependentNeutrosophic Sets and Their Usage in Multiple Attribute Decision-Making

Novel Subsethood Measures for Totally DependentNeutrosophic Sets and Their Usage in Multiple Attribute Decision-Making

Modeling relationships in three-way conflict analysis with subsethood measures

The relations on agents and issues are essential topics in three-way conflict analysis. While there is a general agreement on the formulation and understanding of relations regarding individual agents and issues, different opinions have been proposed when it involves agent or issue sets. On the one hand, these opinions provide diverse perspectives on interpreting the relations, enriching approaches to other essential topics in conflict analysis, such as finding a feasible strategy for a group of agents. On the other hand, a general and unified framework still needs to be developed in order to investigate the relations systematically and uniformly. Based on the quantitative subsethood measures, this work presents such a framework and conducts a comprehensive and systematic study of the relations in conflict analysis. We pay particular attention to the agent relations with respect to a set of issues and the relations between individual agents and issue sets, which are in the spotlight of the existing works. For each of them, we present a general formulation of the relations using subsethood measures and show that a few popular existing formulations could be explained and formulated under the presented formulation.

An interval type-2 fuzzy axiomatic design method: A case study for evaluating blockchain deployment projects in supply chain

This study is concerned with the development of the axiomatic design (AD) method under an interval type-2 fuzzy (IT2F) environment and its application in evaluating blockchain deployment projects in supply chains. Blockchain is a transformative technology that has received significant attention recently. Blockchain technology can process various business transactions by offering a reliable and decentralized infrastructure. Supply chain management is an important application area of blockchains due to its desirable properties, including data security, extended visibility, product traceability, digitalization, and disintermediation. Since blockchain technologies are in their infancy, adopting them to supply chains requires proper design methodologies. Fuzzy AD offers valuable computational mechanisms to evaluate design options in the presence of functional requirements. However, extending AD to different fuzzy extensions is not an easy task, and area-based calculations hinder its widespread applicability. In this study, an IT2F-AD method is developed based on the concept of fuzzy subsethood. The potential of the fuzzy subsethood measure as the main computation engine within type-1 and IT2F-AD is demonstrated. Finally, an integrated multiple criteria decision-making (MCDM) model is proposed by using IT2F Best-Worst Method (IT2F-BWM) and IT2F-AD. The proposed model is used to prioritize blockchain deployment projects in a real-life case study.

Subsethood measure for picture fuzzy sets and its applications on multicriteria decision making

Picture fuzzy set is a direct generalization of intuitionistic fuzzy set and is therefore more capable of dealing with uncertainty while working on real life problems. The concept of inclusion is a subject that is frequently studied in family of fuzzy sets and has many applications in real life problems. Therefore, in this work, the measuring degree of inclusion between picture fuzzy sets is introduced. For this purpose, firstly axioms for subsethood measure are given and then a subsethood measure based on a distance measure for picture fuzzy sets is proposed. Then, a numerical example is provided to illustrate the applicability and usefulness of the presented measure. Finally, results are compared with the existing methods and aggregation operator to show validity of subsethood measure for PFS.

A method of constructing fuzzy implications from the FIφ-construction

In the recent years, several new construction methods of fuzzy implications have been proposed. However, these construction methods actually care about that the new implication could preserve more properties. In this paper, we introduce a new method for constructing fuzzy implications based on an aggregation function with F (1, 0) =1, a fuzzy implication I and a non-decreasing function φ, called FIφ-construction. Specifically, some logical properties of fuzzy implications preserved by this construction are studied. Moreover, it is studied how to use the FIφ-construction to produce a new implication satisfying a specific property. Furthermore, we produce two new subclasses of fuzzy implications such as UIφ-implications and GpIφ-implications by this method and discuss some additional properties. Finally, we provide a way to generate fuzzy subsethood measures by means of FIφ-implications.

A novel risk ranking method based on the single valued neutrosophic set

<p style='text-indent:20px;'>Risk assessment is a key issue in the process of product design and manufacturing. Traditionally risk assessment uses the risk priority number (RPN) method to rank the extent of a threat. However, this simultaneously includes quantitative and qualitative evaluation factors in the process of risk assessment. Moreover, the information provided by different experts for evaluation factors contain ambiguous, incomplete and inconsistent information. These problems lead to more difficulty for risk assessment, and cannot be effectively solved by the traditional RPN method. To solve some limits of the traditional risk analysis method, this paper integrates the single valued neutrosophic set and subsethood measure method to rank the extent of the threat. For missing or incomplete information in the information aggregation process, the minimum, averaging and maximum operators are used to perform data imputation to avoid the distortion of decision results. Finally, a numerical example of high-dose-rate (HDR) brachytherapy treatments is provided to demonstrate the effectiveness and feasibility of the proposed method, and a comparative analysis with some other existing methods is given.</p>

Interval Subsethood Measures with Respect to Uncertainty for the Interval-Valued Fuzzy Setting

In this paper, the problem of measuring the degree of subsethood in the interval-valued fuzzy setting is addressed. Taking into account the widths of the intervals, two types of interval subsethood measures are proposed. Additionally, their relation and main properties are studied. These developments are made both with respect to the regular partial order of intervals and with respect to admissible orders. Finally, some construction methods of the introduced interval subsethood measures with the use interval-valued aggregation functions are examined.

Three-way decision on two universes

Rough set models on two universes are important generalizations of Pawlak’s classical model. In most of the two-universe models, the upper and lower approximations are constructed based on inclusion relation. We generalize inclusion relation to the general evaluation function and define the models of three-way decision on two universes. As an important class of evaluation functions, subsethood measures are considered. We compare our models with other five existing two-universe models in rough set theory and point out that the model of three-way decision on two-universe unifies the five two-universe models. Besides, properties of the two-universe model of three-way decision are also given. More importantly, we propose an approach to computing the pair of thresholds α and β. Our approach is based on the maximum value of the accuracy measure with respect to tri-partitions of a universe induced by all thresholds.

On the properties of subsethood measures

This paper explores the properties of subsethood measures, including boundary conditions, monotonicity, duality, and additivity. These properties can be used as candidates of axioms to define specific subsethood measures. The standard set theory introduces the set-inclusion relation with a qualitative nature, that is, a set is either a subset of the other or not. As a quantitative generalization, a subsethood measure considers the degree of inclusion. It should keep essential characteristics of the qualitative set-inclusion and satisfy generalized properties with the qualitative set-inclusion relation as a special case. A systematic study of the relationships between the qualitative and quantitative frameworks results in the four types of properties of subsethood measures presented in this paper. A combination of different types of properties helps in constructing an appropriate subsethood measure in real-world applications.

Similarity measures, penalty functions, and fuzzy entropy from new fuzzy subsethood measures

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.

A study of (T,N)-implications and its use to construct a new class of fuzzy subsethood measure

It is not hard to see that implications can be obtained by means of other operators. To name a few, we can mention (S,N)-, R- and QL-implications, which has been widely investigated. The aim of this work is to study a fuzzy implication called (T,N)-implication, obtained from the composition of a fuzzy negation and a t-norm. It is also discussed the conditions under which such functions preserve the main properties of fuzzy implications, such that some are related to the laws of contraposition. We also prove under what conditions our implication satisfies the following iterated boolean-like law in fuzzy logic: I(x,I(x,y))=I(x,y) for all x and y in the unit interval. Finally, we provide a way to construct fuzzy subsethood measures by means of (T,N)-implications.

Interval Type-2 Mutual Subsethood Fuzzy Neural Inference System (IT2MSFuNIS)

This paper presents an interval type-2 mutual subsethood fuzzy neural inference system (IT2MSFuNIS). A mutual subsethood measure between two interval type-2 fuzzy sets (IT2 FS) has been derived and has been used in determining the similarity between the IT2 FS inputs and IT2 FS antecedents. The consequent weights are taken to be interval sets. The inputs to the system are fuzzified into IT2 FSs with Gaussian primary membership function having fixed center and uncertain variance. Aggregation of type-2 mutual subsethood based activation spreads is performed using product operator. The output is obtained using simplified type-reduction followed by defuzzification. The system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space. The mathematical modeling and empirical studies of IT2MSFuNIS bring forth its efficacy in problems pertaining to function approximation, time-series prediction, control, and classification. Comparisons with other type-1 and type-2 neuro-fuzzy systems verify that IT2MSFuNIS compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy. Empirical studies indicate the intelligent decision making capability of the proposed model. The main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between IT2 FSs and to find out its applicability in diverse application domains. The improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of IT2 FSs. The integration of mutual subsethood with interval type-2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well-known problems reported in the literature.

Measurement of general granules

The measurement of granules is often considered the most foundational problem in granular computing and it has been studied in many fields but in relative isolation. This paper attempts to explore the measurement for general granules, whether they are granules based on equivalence or general relations, whether they are macrocosmic or microcosmic granules, and whether they are granules on a finite or an infinite set. For a finite set we discuss the measurement of granules corresponding to equivalence relations and introduce definitions of granularity, fineness and granular entropy. In order to distinguish those granules with the same granularity, fineness and granular entropy we present conditional granularity, conditional fineness and conditional granular entropy and obtain the necessary and sufficient condition which can judge one granule as coarser or finer than another. While the conditional granularity is a subsethood measure which satisfies monotonic properties. What’s more, we introduce relation matrix to measure the granules, and it not only can be easily generalized to measure the granules on general relations but also connects granular computing to graph theory. For an infinite set we can define different granularities in different questions, but we suggest the basic idea of granularity, that is, the granularity of a granule is the sum of the relative information contents of its atomic granules.

A u-map representation of general type-2 fuzzy sets via concepts from activation detection: Application to constructing type-2 fuzzy set measures

Following the introduction of type-2 fuzzy sets (T2FS) by Zadeh in 1975, the theory of T2FS has been further developed by a series of notable researchers, and it has been widely used in a variety of applications nowadays. While recent theoretical developments have led to simplifications in the theory of T2FS, it can be argued that theoretical difficulties concerning these developments still persist, inhibiting T2FS from reaching its full potential. With this in mind, in this article, we introduce a concept termed the u-map representation of type-2 fuzzy sets, a simple yet powerful tool for T2FS specialists to further develop T2FS theories, and for non-specialists to adapt T2FS for their applications. With the u-map representation, we show that it is extremely simple to translate manipulations involving type-1 fuzzy sets into corresponding manipulations involving type-2 fuzzy sets. This means that a measure for type-1 fuzzy sets could be converted into a corresponding measure for type-2 fuzzy sets in a straightforward and theoretically satisfying manner, overcoming some of the key theoretical difficulties and restrictions of previous approaches. Moreover, we describe a foundational mental model underlying the u-map representation. Such a foundational model empowers one to check the reasonableness of various definitions, propositions, and computation results associated with measures constructed under the u-map representation. We illustrate the utility of the u-map representation and its foundational mental model via the constructions and the interpretations of a T2FS subsethood measure, a T2FS entropy measure, and a T2FS relative entropy measure.

An algorithmic study of relative cardinalities for interval-valued fuzzy sets

The main topic of this paper is the notion of relative cardinality for interval-valued fuzzy sets – its definition, properties and computation. First we define relative cardinality for interval-valued fuzzy sets following the concept of uncertainty modelling given by Mendel's Wavy-Slice Representation Theorem. We expand on previous approaches by considering relative cardinality based on different t-norms and scalar cardinalities and we initiate an investigation of its properties and possible applications. Drawing on the Nguyen–Kreinovich and Karnik–Mendel algorithms, we propose efficient algorithms to compute relative cardinality depending on a chosen t-norm. This seems to be the first such broad and consistent analysis to have been made of relative cardinality for interval-valued fuzzy sets. As a promising application we consider using interval-valued relative cardinality to construct the family of parameterised subsethood measures.

Subsethood measure for single valued neutrosophic sets

The main aim of this paper is to introduce a neurosophic subsethood measure for single valued neutrosophic sets. For this purpose, we first introduce a system of axioms for subsethood measure of single valued neutrosophic sets. Then we give a simple subsethood measure based to distance measure. Fin ally, to show effectiveness of intended subsethood measure, an application is presented in multicriteria decision making problem and the obtained results are discussed. Though having a simple measure for calculation, the subsethood measure presents a new and useful approach to deal with neutrosophic information.

Subsethood measures for interval-valued fuzzy sets based on the aggregation of interval fuzzy implications

The connection between fuzzy subsethood measures and fuzzy implications straightforwardly follows from the definition of inclusion for (crisp) sets. In accordance with this connection we introduce a new constructive method for (weak) fuzzy subsethood measures for interval-valued fuzzy sets based on the aggregation of fuzzy interval implications. For this purpose we generalize aggregation functions and propose a technique for aggregation of intervals. The benefit of our approach lies in the wide range of obtained fuzzy subsethood measures for interval-valued fuzzy sets. Moreover, the method gives the possibility to construct new fuzzy subsethood measures with properties that are appropriate to the situation under investigation. Another advantage of the method is its clear semantic interpretation.

Quantitative rough sets based on subsethood measures

Subsethood measures, also known as set-inclusion measures, inclusion degrees, rough inclusions, and rough-inclusion functions, are generalizations of the set-inclusion relation for representing graded inclusion. This paper proposes a framework of quantitative rough sets based on subsethood measures. A specific quantitative rough set model is defined by a particular class of subsethood measures satisfying a set of axioms. Consequently, the framework enables us to classify and unify existing generalized rough set models (e.g., decision-theoretic rough sets, probabilistic rough sets, and variable precision rough sets), to investigate limitations of existing models, and to develop new models. Various models of quantitative rough sets are constructed from different classes of subsethood measures. Since subsethood measures play a fundamental role in the proposed framework, we review existing methods and introduce new methods for constructing and interpreting subsethood measures.

A note on entropy subsethood relationship

We comment on subsethood measure defined by Kosko and Young and give some new aspects of these measures. Finally we would like to discard the entropy subsethood relationship established by the authirs. We present some properties of subsethood measure from set theoretic approach and also from axiomatic approach with the expectation that these would help in removing the shortcomings that currently exist in these definitions.

Inclusion and subsethood measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets

The main aim of this paper is to propose new subsethood measures for continuous, general type-2 fuzzy sets. For this purpose, we introduce inclusions and subsethood measures for interval-valued fuzzy sets first. Then, using an α-plane representation for type-2 fuzzy sets, we extend these inclusions and subsethood measures to general type-2 fuzzy sets. Subsethood measures for interval-valued fuzzy sets (hence, also for type-2 fuzzy sets) rely on already known subsethood measures for ordinary fuzzy sets. We focus on a special subsethood measure for ordinary fuzzy sets, based on α-cut representation, and show, how to compute subsethood measures for continuous type-2 fuzzy sets with no need for discretizing of universe. This is a very interesting and useful property of proposed subsethood measures, which is one of the reason, why our approach has less computational demand than the others.