Axiomatic Requirements Research Articles

Circular Linear Diophantine Fuzzy Sets and Their Application in Medical Diagnosis

To address the uncertainty and imprecision in a complex system, the concept of the linear Diophantine fuzzy set (LDFS) was introduced, which is a fuzzy set extension that eliminates the constraints of current methodologies and gives the decision-maker complete freedom to select the grades, producing results that are expressive and flexible. A circular intuitionistic fuzzy set (C-IFS) is depicted by a circle with the membership and non-membership values as its center coordinates, addressing the ambiguity regarding membership and non-membership values. In this paper, we propose the notion of a circular linear Diophantine fuzzy set (C-LDFS) as a hybrid structure of both circular intuitionistic fuzzy sets and linear Diophantine fuzzy sets. In addition, some fundamental set operations on C-LDFSs are presented. Furthermore, a similarity metric for C-LDFS is also introduced in the paper. Additionally, we confirm the axiomatic requirement of the similarity measure and highlight a few of its traits. Finally, the proposed similarity metric is implemented in clinical decision-making to illustrate its efficacy.

Some Information Measures for Interval‐Valued Hesitant Fuzzy Sets in Multiple Criteria Decision‐Making

In the fields of information science and artificial intelligence, dealing with uncertainty, fuzziness, and complexity has always been a hot and difficult research topic. Especially in modern society, with the continuous development of science and technology, people are facing more and more complex problems. Interval‐valued hesitant fuzzy set (IVHFS) is an extended form of a fuzzy set. It can more flexibly express and handle uncertainty and fuzziness in decision‐making processes. However, in the practical application of the IVHFS, its information measurement is crucial, which is directly related to the application value of the IVHFS in various fields. Therefore, studying the information measurement of the IVHFS has important theoretical significance and practical value for the fields of information science and artificial intelligence. In spite of significant advances, entropy and similarity as the well‐known information measures for interval‐valued hesitant fuzzy information have not yet been thoroughly researched. In this contribution, we investigate information measures in the IVHFS, including nonprobabilistic entropy, similarity, and cross‐entropy. We first analyze the change law of hesitating uncertainty and fuzzy uncertainty in geometric space, and a nonprobabilistic entropy measurement method and its axiomatic definition for IVHFS are further developed. Then, a novel similarity measurement formula for IVHFS and its axiomatic requirements are proposed on the basis of the two nonfuzzy elements ( and ). Furthermore, the novel similarity measure is used to construct the cross‐entropy measure for IVHFS and its axiomatic requirements based on the association between the similarity and the cross‐entropy. Lastly, a MAGDM method is proposed by using the developed three information measures, and the efficacy of the proposed method is demonstrated by a numerical example of emergency communication support capacity evaluation. Comparative analysis and computational cost analysis are implemented to demonstrate the superiority and validity of the proposed information measures.

Picture Fuzzy Knowledge Measure with Application to MADM

The complementary dual of entropy is termed “knowledge measure” in recent studies concerning fuzzy and intuitionistic fuzzy sets. A picture fuzzy set is an extended and generalized form of fuzzy and intuitionistic fuzzy sets. The broader perspective of the picture fuzzy set inculcated the possibility of the formulation of a picture fuzzy knowledge measure and its potential implications. In this paper, we set up an axiomatic framework for obtaining a complementary dual of the picture fuzzy entropy. Subsequently, we derive two new knowledge measures that strictly follow the axiomatic requirements. Some empirical investigations establish the advantages of our proposed knowledge measure over the existing measures. We also present a novel multiple attribute decision-making (MADM) algorithm, wherein the proposed knowledge measure computes attribute weights and exhibits encouraging performance. The comparative analysis shows the potential implications and advantages of the proposed measures.

Rehabilitating Mean–Variance Portfolio Selection: Theory and Evidence

Recent research has proven that the application of mean–variance portfolio selection is justified if, and only if, asset returns follow a skew-elliptical generalized location and scale (SEGLS) distribution. This irrefutably corrects the widespread fallacy that mean–variance analysis can be used only for portfolios with normally or symmetrically distributed constituents. To make this important finding accessible to a wide range of academics and practitioners, the authors of this article present it in a nontechnical form and additionally highlight that, under the SEGLS distribution and some mild axiomatic requirements, mean–variance analysis and many alternative mean-risk approaches deliver the same optimal portfolios. In a numerical study, they illustrate the key features of the novel SEGLS distribution. In an empirical study, they emphasize its practical relevance by gathering existing and providing new evidence in its favor.

On spherical fuzzy distance measure and TAOV method for decision-making problems with incomplete weight information

A spherical fuzzy set is one of the reliable tools to handle the uncertainties in the data with the help of the three membership degrees: Positive, neutral, and negative. Under this environment, the paper aims to present a novel decision-making algorithm named as MCDM-TAOV with some novel distance measures using matrix norms. In the literature, several kinds of distance measures occur, but they fail to meet certain axioms of the measure. To overwhelm this, the proposed measure satisfies all axiomatic requirements and overcomes the hindrances in the existing distance measures. Several counter-intuitive examples are provided to justify the superiority of the proposed measures. Furthermore, we addressed the total area based on orthogonal vector (TAOV) methodology to solve the MCDM (“multi-criteria decision-making”) problems. A distance-based criteria weight determination method is presented to compute the unknown criteria weight. The approach has been demonstrated by assessing the third-party reverse logistics provider (3PRLP) problem. The practical applicability, comparative analysis and advantages of the study with other decision-making methods are furnished to depict the efficiency of the moulded process.

New distance measure for Fermatean fuzzy sets and its application

As a new extended form of intuitionistic fuzzy sets, Fermatean fuzzy sets are powerful tools for describing vagueness and uncertainty in complex problems. In the method of handling Fermatean fuzzy information, the distance measure is an essential tool to depict the difference between two Fermatean fuzzy sets. However, how to accurately measure the distance between two Fermatean fuzzy sets is still a problem to be solved. In this paper, we devise two novel distance measure methods for Fermatean fuzzy sets. One is the distance measure of Fermatean fuzzy sets based on the Hellinger distance, which is called the FFSH distance. The other is the distance measure of Fermatean fuzzy sets based on the triangular divergence, which is called the FFSTD distance. Then, we prove that the proposed distance measure methods satisfy the axiomatic requirements of the distance function. Afterward, numerical examples are given to reveal that the proposed distance measures are more effective and reasonable than the normalized Euclidean distance measure, which can overcome the counter-intuitive situation. Besides, we utilize the proposed distance measure methods to address the problems of pattern recognition and medical diagnosis under Fermatean fuzzy environment and achieved excellent results. The experimental results illustrate that the proposed distance measure methods can efficiently handle the practical application under Fermatean fuzzy environment, and are more reliable than the normalized Euclidean distance measure.

Process Tomography in General Physical Theories

Process tomography, the experimental characterization of physical processes, is a central task in science and engineering. Here, we investigate the axiomatic requirements that guarantee the in-principle feasibility of process tomography in general physical theories. Specifically, we explore the requirement that process tomography should be achievable with a finite number of auxiliary systems and with a finite number of input states. We show that this requirement is satisfied in every theory equipped with universal extensions, that is, correlated states from which all other correlations can be generated locally with non-zero probability. We show that universal extensions are guaranteed to exist in two cases: (1) theories permitting conclusive state teleportation, and (2) theories satisfying three properties of Causality, Pure Product States, and Purification. In case (2), the existence of universal extensions follows from a symmetry property of Purification, whereby all pure bipartite states with the same marginal on one system are locally interconvertible. Crucially, our results hold even in theories that do not satisfy Local Tomography, the property that the state of any composite system can be identified from the correlations of local measurements. Summarizing, the existence of universal extensions, without any additional requirement of Local Tomography, is a sufficient guarantee for the characterizability of physical processes using a finite number of auxiliary systems and with a finite number of input systems.

Can we trust the standardized mortality ratio? A formal analysis and evaluation based on axiomatic requirements.

BackgroundThe standardized mortality ratio (SMR) is often used to assess and compare hospital performance. While it has been recognized that hospitals may differ in their SMRs due to differences in patient composition, there is a lack of rigorous analysis of this and other—largely unrecognized—properties of the SMR.MethodsThis paper proposes five axiomatic requirements for adequate standardized mortality measures: strict monotonicity (monotone relation to actual mortality rates), case-mix insensitivity (independence of patient composition), scale insensitivity (independence of hospital size), equivalence principle (equal rating of hospitals with equal actual mortality rates in all patient groups), and dominance principle (better rating of unambiguously better performing hospitals). Given these axiomatic requirements, effects of variations in patient composition, hospital size, and actual and expected mortality rates on the SMR were examined using basic algebra and calculus. In this regard, we distinguished between standardization using expected mortality rates derived from a different dataset (external standardization) and standardization based on a dataset including the considered hospitals (internal standardization). The results were illustrated by hypothetical examples.ResultsUnder external standardization, the SMR fulfills the axiomatic requirements of strict monotonicity and scale insensitivity but violates the requirement of case-mix insensitivity, the equivalence principle, and the dominance principle. All axiomatic requirements not fulfilled under external standardization are also not fulfilled under internal standardization. In addition, the SMR under internal standardization is scale sensitive and violates the axiomatic requirement of strict monotonicity.ConclusionsThe SMR fulfills only two (none) out of the five proposed axiomatic requirements under external (internal) standardization. Generally, the SMRs of hospitals are differently affected by variations in case mix and actual and expected mortality rates unless the hospitals are identical in these characteristics. These properties hamper valid assessment and comparison of hospital performance based on the SMR.

Connectedness versus diversification: two sides of the same coin

In the financial framework, the concepts of connectedness and diversification have been introduced and developed respectively in the context of systemic risk and portfolio theory. In this paper we propose a theoretical approach to bring to light the relation between connectedness and diversification. Starting from the respective axiomatic definitions, we prove that a class of proper measures of connectedness verifies, after a suitable functional transformation, the axiomatic requirements for a measure of diversification. The core idea of the paper is that connectedness and diversification are so deeply related that it is possible to pass from one concept to the other. In order to exploit such correspondence, we introduce a function, depending on the classical notion of rank of a matrix, that transforms a suitable proper measure of connectedness in a measure of diversification. We point out general properties of the proposed transformation function and apply it to a selection of measures of connectedness, such as the well-known Variance Inflation Factor.

Entropy Measures for Plithogenic Sets and Applications in Multi-Attribute Decision Making

Plithogenic set is an extension of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic sets, whose elements are characterized by one or more attributes, and each attribute can assume many values. Each attribute has a corresponding degree of appurtenance of the element to the set with respect to the given criteria. In order to obtain a better accuracy and for a more exact exclusion (partial order), a contradiction or dissimilarity degree is defined between each attribute value and the dominant attribute value. In this paper, entropy measures for plithogenic sets have been introduced. The requirements for any function to be an entropy measure of plithogenic sets are outlined in the axiomatic definition of the plithogenic entropy using the axiomatic requirements of neutrosophic entropy. Several new formulae for the entropy measure of plithogenic sets are also introduced. The newly introduced entropy measures are then applied to a multi-attribute decision making problem related to the selection of locations.

On the Individuation of Choice Options

Decision theorists have attempted to accommodate several violations of decision theory’s axiomatic requirements by modifying how agents’ choice options are individuated and formally represented. In recent years, prominent authors have worried that these modifications threaten to trivialize decision theory, make the theory unfalsifiable, impose overdemanding requirements on decision theorists, and hamper decision theory’s internal coherence. In this paper, I draw on leading descriptive and normative works in contemporary decision theory to address these prominent concerns. In doing so, I articulate and assess several different criteria for individuating and formally representing agents’ choice options.

Analysis of One-to-One Matching Mechanisms via SAT Solving: Impossibilities for Universal Axioms

We develop a powerful approach that makes modern SAT solving techniques available as a tool to support the axiomatic analysis of economic matching mechanisms. Our central result is a preservation theorem, establishing sufficient conditions under which the possibility of designing a matching mechanism meeting certain axiomatic requirements for a given number of agents carries over to all scenarios with strictly fewer agents. This allows us to obtain general results about matching by verifying claims for specific instances using a SAT solver. We use our approach to automatically derive elementary proofs for two new impossibility theorems: (i) a strong form of Roth's classical result regarding the impossibility of designing mechanisms that are both stable and strategyproof and (ii) a result establishing the impossibility of guaranteeing stability while also respecting a basic notion of cross-group fairness (so-called gender-indifference).

Some New Trigonometric α-Order Fuzzy Entropies

In this paper, new α-order trigonometric and inverse trigonometric fuzzy entropies are proposed and the fuzzy entropy axiomatic requirements are satisfied for the new fuzzy entropies. A comparison of the new fuzzy entropies is done with several widely used fuzzy entropies in order to find the most fuzziness entropy. The results indicate that the new proposed α-order fuzzy entropy provides larger entropy value than that other fuzzy entropies which are defined.

New Fuzzy Entropy Measure of Order α

In this article, a new fuzzy entropy measure of order α is proposed. The fuzzy entropy axiomatic requirements are discussed for the new measure, and an empirical comparison are made with several entropy measures including Shannon, Rényi and Kapur. It turns out, the proposed measure satisfies all axiomatic and outperform the other entropy measures in terms of the fuzziness degree.Â

Compact representations of temporal databases

This paper investigates the storage compactness of temporal merges. A temporal merge is a joint representation of a set of snapshots of the same database at different points in time. An axiomatic requirement for temporal merges is that each individual snapshot must be derivable from it. We first show that the usual temporal extension of a database is a special kind of temporal merge called a direct merge. For direct merges, finding the most compact representation is easy for separate relations. However, if snapshots feature foreign key dependencies, the representation may contain more tuples than strictly needed. We therefore study inclusion dependencies in temporal databases and show how to use them while maintaining minimality in our representation. Next, we argue that direct merges imply redundancy when they contain non-contiguous, value-equivalent tuples. We therefore introduce a more general kind of temporal merge known as a coherent merge and study its properties in depth. Deriving a minimal coherent merge from a direct one is shown to be NP-hard. However, several greedy algorithms are demonstrated to provide decent results in quadratic time. Next, an incremental algorithm for construction of coherent merges is presented. We provide experimental results on real-life data sets that support the usefulness of the contributions of this paper.

An Uncertainty Measure for Interval-valued Evidences

Interval-valued belief structure (IBS), as an extension of single-valued belief structures in Dempster-Shafer evidence theory, is gradually applied in many fields. An IBS assigns belief degrees to interval numbers rather than precise numbers, thereby it can handle more complex uncertain information. However, how to measure the uncertainty of an IBS is still an open issue. In this paper, a new method based on Deng entropy denoted as UIV is proposed to measure the uncertainty of the IBS. Moreover, it is proved that UIV meets some desirable axiomatic requirements. Numerical examples are shown in the paper to demonstrate the efficiency of UIV by comparing the proposed UIV with existing approaches.

Uncertainty measure for interval-valued belief structures

Interval-valued belief structures, as an extension of belief structures in classical evidence theory, are developed for better exploitation of uncertain and imprecise information. From the point of view of information theory, uncertainty measure of an interval-valued belief structure is critically important for information processing. However, it is still an open issue to measure its uncertainty. Besides discord and non-specificity, which hide in a precise belief structure, we claim that fuzziness is also associated with an interval-valued belief structure. In this paper, axiomatic requirements for uncertainty measure of interval-valued belief structure are defined. Then an uncertainty measure is proposed to measure the information conveyed by interval-valued belief structures. Its properties are mathematically proved. Finally, numerical experiments are employed to illustrate the performance of the proposed uncertainty measure. It is illustrated that the proposed uncertainty measure is sensitive to the change of belief structures, which might have beneficial effects on decision making.

On the axiomatic requirement of range to measure uncertainty

How to measure uncertainty is still an open issue. Probability theory is a primary tool to express the aleatoric uncertainty. The Shannon’s information entropy is an effective measure for the uncertainty in probability theory. Dempster–Shafer theory, an extension of probability theory, has the ability to express the aleatoric and epistemic uncertainty, simultaneously. With respect to such uncertainties in Dempster–Shafer theory, a justifiable uncertainty measure is required to satisfy five axiomatic requirements based on previous studies. In this paper, we show that one of the axiomatic requirements, the requirement of range, is questionable. The correct range of uncertainty should be [0,log22|X|] rather than [0,log2|X|] according to the concept of entropy.

ENTROPY FOR INTUITIONISTIC FUZZY SETS BASED ON DISTANCE AND INTUITIONISTIC INDEX

Many researchers have put forward lots of entropy and distance measures for intuitionistic fuzzy sets (IFSs). This paper firstly reviews some basic concepts for IFSs and several widely used distance measures between IFSs. And then a series of distance measures are presented on the basis of above widely used distance measures. Based on such distance measures and intuitionistic index, a set of entropies for IFSs are proposed and proved to meet the axiomatic requirements given by Szmidt and Kacprzyk in 2001. Besides, two numerical examples are demonstrated to verify the efficiency of the proposed entropies for IFSs. Finally, the paper concludes with suggestions for future research.

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

In preference aggregation a set of individuals express preferences over a set of alternatives, and these preferences have to be aggregated into a collective preference. When preferences are represented as orders, aggregation procedures are called social welfare functions. Classical results in social choice theory state that it is impossible to aggregate the preferences of a set of individuals under different natural sets of axiomatic conditions. We define a first-order language for social welfare functions and we give a complete axiomatisation for this class, without having the number of individuals or alternatives specified in the language. We are able to express classical axiomatic requirements in our first-order language, giving formal axioms for three classical theorems of preference aggregation by Arrow, by Sen, and by Kirman and Sondermann. We explore to what extent such theorems can be formally derived from our axiomatisations, obtaining positive results for Sen’s Theorem and the Kirman-Sondermann Theorem. For the case of Arrow’s Theorem, which does not apply in the case of infinite societies, we have to resort to fixing the number of individuals with an additional axiom. In the long run, we hope that our approach to formalisation can serve as the basis for a fully automated proof of classical and new theorems in social choice theory.