Iranian journal of fuzzy systems
Volume:20 Issue: 3, May-Jun 2023

In this note, we consider the problem of computing a scalar-valued, and also a fuzzy-valued integral over fuzzy spatial domains, which is useful, for instance, to calculate  fuzzy areas described by a certain property, allowing the density function to be variable, or evaluate magnitudes in the case where the field function is fuzzily known.

Attitudinal Choquet integral (ACI) is a recent aggregation operator thatconsiders in the aggregation process the criteria interaction and the DM's attitude, both of which arespecific to the decision-maker. However, this capability comes at the cost of increasedcomplexity that hinders its applicability in big data analytics.To address the same, in this paper, we explore some heuristics-based forms of the ACI operator, so as to somehow overcome its complexity.We devise new and efficient forms of $\mathcal{ACI}$, and test their validityin the real world datasets, against the backdrop of preference learning.

In this paper, we have proved Diaz-Metcalf inequality for fuzzy integrals. More precisely:\\If $f, g: [0, 1]\to\mathbb{R}$ are continuous and strictly increasing functions, then the fuzzy integral inequality$$ - \hspace{-1em} \int_0^1 f^s d\mu\cdot  - \hspace{-1em} \int_0^1  g^sd\mu\le  - \hspace{-1em} \int_0^1\left(f\cdot g\right)^sd\mu,$$holds, where $s>1$ and $\mu$ is  the Lebesgue measure on $\mathbb{R}$. In addition, we have shown this inequality for pseudo-integrals.

In this paper, we consider a generalized fuzzy differential system (GFDS) consisting of a fuzzy Caputo fractional differential inclusion    combined with a fuzzy mixed quasivariational inequality. The GFDS has been known as a framework of fuzzy fractional differential quasivariational inequalities involving Caputo fractional derivatives. First, we verify the existence of solutions for the fuzzy mixed quasivariational inequality by using the Kakutani-Fan-Glicksberg fixed point theorem. Then, the existence of mild solutions for the GFDS is also obtained under some mild conditions. Finally, the upper semicontinuity of the solution mapping to the GFDS provided in the case of the perturbed    parameters is discussed.

K-Nearest Neighbors (KNN) is a classification algorithm based on supervised machine learning, which works according to a voting system. The performance of the KNN algorithm depends on different factors, such as unbalanced distribution of classes, the scalability problem, and considering equal values for all training samples. Regarding the importance of the KNN algorithm, different improved versions of this algorithm are introduced, such as fuzzy KNN, weighted KNN, and KNN with variable neighbors. In this paper, a weighted KNN based on Whale Optimization Algorithm is proposed for the objective of increasing the level of detection accuracy. The proposed algorithm devotes a weight to each training sample of every feature by employing the WOA to explore the optimized weight matrix. The algorithm is implemented and experimented on five standard datasets. The evaluation results prove that the proposed algorithm performs better than both weighted KNN based on the Genetic Algorithm (GA) and the classic KNN algorithm.

If there is a fuzzy relation $\widetilde{R}$ between two spaces $U$ and $V$, the fuzzy approximations in both spaces based on $\widetilde{R}$ are widely studied, and they basically reflect only the influence from one space to another. In this paper, on each space of $U$ and $V$, two new fuzzy relations are derived from $\widetilde{R}$, a positive low-value relation and a conservative high-value relation, to reflect the interaction and feedback between the two spaces. So, the fuzzy approximations based on them can reflect the combination of the action and the reaction from one space to another. Therefore, two spaces $U$ and $V$ are closely accompanied, and $(U, V, \widetilde{R})$ is a whole, so it is called a fuzzy accompanied approximation space (FAAS). In an FAAS, the properties of the fuzzy approximation models on each space are studied, the relationships between fuzzy approximation models of two spaces are researched, and examples to show how the approximation operator models in the FAAS to solve practical problems from multiple perspectives are also illustrated. More importantly, when the fuzzy relation $\widetilde{R}$ is a binary relation $R$ or the two spaces are the same, the special cases of FAAS are investigated and some important new models and new results are obtained, which add new ideas and methods to the current research.

As a proper generalization of the ordinal sum t-norm construction on bounded lattices proposed in [E. A\c{s}{\i}c{\i}, R. Mesiar, New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice, International Journal of General Systems, {\bf 49}(2) (2020), 143-160], the present paper studies a new extension of a triangular norm on a subinterval $[0,\alpha]$ via an interior operator to the underlying entire bounded lattice, where the necessary and sufficient conditions under which the constructed operation is again a t-norm are given. By comparing the graphic structures of two t-norms on a common bounded lattice which are constructed in different ways, it is shown that the new method in this paper is essentially different from the ones existing in the literature. As an end, this new construction is generalized to construct ordinal sums of finitely many t-norms by recursion on bounded lattices. The dual results for ordinal sum construction of t-conorms via closure operators on bounded lattices are also presented.

A new method for solving a fuzzy linear system (FLS), $A\tilde X=\tilde Y$, where the coefficient matrix $A$ is an arbitrary real matrix is obtained. A necessary and sufficient condition for the ${\cal R}$-consistency of the associated system of linear equations is obtained, related to itsrepresentative solutions.  Moreover, the general form of representative solutions of such linear systems is presented. The straightforward method for solving $m\times n$ FLS based on an arbitrary $\{1\}$-inverse of $A$ is introduced. This method is illustrated by interesting examples.

Fuzzy rule interpolation (FRI) predicts an accountable outcome of a possible course of action in sparse fuzzy rule base system (FRBS). The geometry based linear fuzzy rule interpolation (GLFRI) is extended for multi-dimensional fuzzy rule base interpolation. Expansion/contraction (EC) of triangular, trapezoidal and complex polygonal fuzzy sets has been also proposed which enables the proposed FRI method to incorporate with fuzzy rules which include triangular, trapezoidal, hexagonal or complex fuzzy sets. The study further extends to introduce the process of backward rule base interpolation.It has been shown that the scale and move transformation-based FRI method can yield a non-convex fuzzy consequent which can be avoided by using the proposed method. The proposed method performs better without any risk of obtaining non-convex fuzzy consequent. The efficiency of proposed forward and backward FRI methods is projected with several numerical examples. A detailed comparison of EC transformation with scale and move transformation is also presented here.

In this paper, based on the weighted metric, the problem of approximating a general continuous fuzzy number $u$ with derivable $\underline{u}(r)$ and $\overline{u}(r)$ (left and right cut functions of level value $r$) for $r \in (0,1]$ by using the $m-n-$step type fuzzy number is studied. Two kinds of approximations ($I-$nearest approximation and $II-$nearest approximation which satisfy different conditions) of using $m-n-$step type fuzzy numbers to approximate general continuous fuzzy numbers are defined, and the relationship between the $I-$nearest approximation and $II-$nearest approximation is obtained. Then the methods of the two kinds of approximations are respectively presented by the theorems more specifically. At last, an example is given to show the effectiveness and usability of the methods set up by us.

One of the useful distributions in modeling mortality (or failure) data is the univariate Gompertz--Makeham distribution. To examine the relationship between the two variables, the extended bivariate Gompertz--Makeham distribution is introduced, and its properties are provided. Also, some reliability indices, including aging intensity and stress-strength reliability, are calculated for the proposed model. Here, a new copula function is constructed based on the extended bivariate Gompertz--Makeham  distribution. Some of its features including dependency properties, such as dependence structure, some  measures of dependence, and tail dependence,  are studied.The estimation of the  parameters of new copula is presented, and at the end, a simulation study and a performance analysis based on the real data are presented.  So, by analyzing the mortality data due to COVID-19, the appropriateness of the proposed model is examined.

Randomness and fuzziness of argumentation have attracted the interest of many researchers.However, though each of these two properties is discussed in the past, seldom literature considers both of them.The purpose of this paper aims to explore semantics of the argumentation frameworks with these two attributes at the same time.Firstly, we introduce probabilistic-fuzzy matrices to describe the arguments with randomness and fuzziness,and define the mathematical form of the probabilistic-fuzzy argumentation frameworks.In these frameworks, an argument has finite fuzzy states and each fuzzy state has a probability.This provides a mathematical foundation for the follow-up work.Then, we introduce a method of modifying the probabilities of the fuzzy states,which proposes a feasible way to revise the probabilities.Formally, it is the revision of the probabilistic-fuzzy matrices of arguments.Finally, based on this process, we set up an extension semantics system for probabilistic-fuzzy frameworks.The semantics enriches the theory of argumentation, and propose a way to check the probabilities.

The aim of this paper is to introduce a concept of type-2 $L$-preorders for $L$ being a complete residuated lattice. It can be considered as a common framework of $L$-preorders and hemimetrics, and also contains various kinds of fuzzy metrics, including Morsi fuzzy metrics, KM-fuzzy metrics and modular metrics, as natural examples. It is shown that the category of $L$-preordered sets can be reflectively and coreflectively embedded in that of type-2 $L$-preordered sets. A type-2 $L$-preorder can be supplied as different models for further study.