Research on fuzzy sets

[EN] This paper explores the advancements in soft sets and their extensions, contributing to the evolving field of soft computing. Soft sets, introduced by Molodtsov, provide a flexible framework for handling uncertainty and imprecision in decision-making processes. The research delves into various extensions of soft sets, including hybrid models with other mathematical structures, enhancing their applicability in diverse domains. Novel methodologies for parameterization and optimization within soft sets are investigated, aiming to improve their efficiency and effectiveness in real-world applications. The study emphasizes the integration of soft sets with machine learning techniques, fostering the development of intelligent systems capable of handling complex and uncertain information. The findings showcase the versatility and potential of soft sets and their extensions, opening new avenues for future research in this dynamic field

Notes on Transformation Techniques for IVIFS: Applications to Aggregation and Decision Making

We delve into the application of two operational transformation techniques that represent a single interval-valued intuitionistic fuzzy number using two intuitionistic fuzzy numbers in a constructive fashion. These techniques are employed to achieve seamless aggregation of interval-valued intuitionistic fuzzy numbers and facilitate multi-attribute decision-making within this framework. The decision-making and prioritization processes rely on comparison laws that consider the score and accuracy of an interval-valued intuitionistic fuzzy number. We illustrate how these parameters can be derived from the analogous proxies associated with the intuitionistic fuzzy numbers that represent it. To wrap up our exploration, we present a comparative study as the culmination of this research endeavor

A systematic literature review of soft set theory

[EN] Soft set theory, initially introduced through the seminal article ‘‘Soft set theory—First results’’ in 1999, has gained considerable attention in the field of mathematical modeling and decision-making. Despite its growing prominence, a comprehensive survey of soft set theory, encompassing its foundational concepts, developments, and applications, is notably absent in the existing literature. We aim to bridge this gap. This survey delves into the basic elements of the theory, including the notion of a soft set, the operations on soft sets, and their semantic interpretations. It describes various generalizations and modifications of soft set theory, such as N-soft sets, fuzzy soft sets, and bipolar soft sets, highlighting their specific characteristics. Furthermore, this work outlines the fundamentals of various extensions of mathematical structures from the perspective of soft set theory. Particularly, we present basic results of soft topology and other algebraic structures such as soft algebras and sigma-algebras. This article examines a selection of notable applications of soft set theory in different fields, including medicine and economics, underscoring its versatile nature. The survey concludes with a discussion on the challenges and future directions in soft set theory, emphasizing the need for further research to enhance its theoretical foundations and broaden its practical applications. Overall, this survey of soft set theory serves as a valuable resource for practitioners, researchers, and students interested in understanding and utilizing this flexible mathematical framework for tackling uncertainty in decision-making processes

A method for temporal fault tree analysis using intuitionistic fuzzy set and expert elicitation

YesTemporal fault trees (TFTs), an extension of classical Boolean fault trees, can model time-dependent failure behaviour of dynamic systems. The methodologies used for quantitative analysis of TFTs include algebraic solutions, Petri nets (PN), and Bayesian networks (BN). In these approaches, precise failure data of components are usually used to calculate the probability of the top event of a TFT. However, it can be problematic to obtain these precise data due to the imprecise and incomplete information about the components of a system. In this paper, we propose a framework that combines intuitionistic fuzzy set theory and expert elicitation to enable quantitative analysis of TFTs of dynamic systems with uncertain data. Experts’ opinions are taken into account to compute the failure probability of the basic events of the TFT as intuitionistic fuzzy numbers. Subsequently, for the algebraic approach, the intuitionistic fuzzy operators for the logic gates of TFT are defined to quantify the TFT. On the other hand, for the quantification of TFTs via PN and BN-based approaches, the intuitionistic fuzzy numbers are defuzzified to be used in these approaches. As a result, the framework can be used with all the currently available TFT analysis approaches. The effectiveness of the proposed framework is illustrated via application to a practical system and through a comparison of the results of each approach.This work was supported in part by the Mobile IOT: Location Aware project (grant no. MMUE/180025) and Indoor Internet of Things (IOT) Tracking Algorithm Development based on Radio Signal Characterisation project (grant no. FRGS/1/2018/TK08/MMU/02/1). This research also received partial support from DEIS H2020 project (grant no. 732242)

Defining Bonferroni means over lattices

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers\u27 preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.<br /

Bipolar querying of valid-time intervals subject to uncertainty

Databases model parts of reality by containing data representing properties of real-world objects or concepts. Often, some of these properties are time-related. Thus, databases often contain data representing time-related information. However, as they may be produced by humans, such data or information may contain imperfections like uncertainties. An important purpose of databases is to allow their data to be queried, to allow access to the information these data represent. Users may do this using queries, in which they describe their preferences concerning the data they are (not) interested in. Because users may have both positive and negative such preferences, they may want to query databases in a bipolar way. Such preferences may also have a temporal nature, but, traditionally, temporal query conditions are handled specifically. In this paper, a novel technique is presented to query a valid-time relation containing uncertain valid-time data in a bipolar way, which allows the query to have a single bipolar temporal query condition

Necessary and possible hesitant fuzzy sets: A novel model for group decision making

We propose an extension of Torra’s notion of hesitant fuzzy set, which appears to be well suited to group decision making. In our model, indecisiveness in judgements is described by two nested hesitant fuzzy sets: the smaller, called necessary, collects membership values determined according to a rigid evaluation, whereas the larger, called possible, comprises socially acceptable membership values. We provide several instances of application of our methodology, and accordingly design suitable individual and group decision procedures. This novel approach displays structural similarities with Atanassov’s intuitionistic fuzzy set theory, but has rather different goals. Our source of inspiration comes from preference theory, where a bi-preference approach has proven to be a useful extension of the classical mono-preference modelization in the fields of decision theory and operations research

Computation of Choquet integral for finite sets: Notes on a ChatGPT-driven experience

The Choquet integral, credited to Gustave Choquet in 1954, initially found its roots in decision making under uncertainty following Schmeidler's pioneering work in this field. Surprisingly, it was not until the 1990s that this integral gained recognition in the realm of multi-criteria decision aid. Nowadays, the Choquet integral boasts numerous generalizations and serves as a focal point for intensive research and development across various domains. Here we share our journey of utilizing ChatGPT as a helpful assistant to delve into the computation of the discrete Choquet integral using Mathematica. Additionally, we have demonstrated our ChatGPT experience by crafting a Beamer presentation with its assistance. The ultimate aim of this exercise is to pave the way for the application of the discrete Choquet integral in the context of N-soft sets

Notes on soft sets and aggregation operators

[EN]Under uncertainty, traditional sets may not be sufficient to represent real-world phenomena, and fuzzy sets can provide a more flexible and natural approach. The concept of fuzzy sets has been widely used in various fields, including artificial intelligence, control theory, decision-making, and pattern recognition. Fuzzy sets can also be combined with other mathematical tools, such as probability theory, to provide a more comprehensive approach to uncertainty management. In these notes, we explore the concept of fuzzy sets under uncertainty, and their applications in various fields. We discuss the fundamental concepts of fuzzy sets, including fuzzy membership functions, fuzzy operations, and fuzzy relations. We also examine different types of uncertainty, including epistemic and aleatory uncertainty, and how fuzzy sets can be used to model and manage uncertainty in these cases

Using fuzzy numbers and OWA operators in the weighted average and its application in decision making

Se presenta un nuevo método para tratar situaciones de incertidumbre en los que se utiliza el operador OWAWA (media ponderada – media ponderada ordenada). A este operador se le denomina operador OWAWA borroso (FOWAWA). Su principal ventaja se encuentra en la posibilidad de representar la información incierta del problema mediante el uso de números borrosos los cuales permiten una mejor representación de la información ya que consideran el mínimo y el máximo resultado posible y la posibilidad de ocurrencia de los valores internos. Se estudian diferentes propiedades y casos particulares de este nuevo modelo. También se analiza la aplicabilidad de este operador y se desarrolla un ejemplo numérico sobre toma de decisiones en la selección de políticas fiscalesWe present a new approach for dealing with an uncertain environment when using the ordered weighted averaging – weighted averaging (OWAWA) operator. We call it the fuzzy OWAWA (FOWAWA) operator. The main advantage of this new aggregation operator is that it is able to represent the uncertain information with fuzzy numbers. Thus, we are able to give more complete information because we can consider the maximum and the minimum of the problem and the internal information between these two results. We study different properties and different particular cases of this approach. We also analyze the applicability of the new model and we develop a numerical example in a decision making problem about selection of fiscal policies