Fuzzy Relations Research Articles

Bayesian network based probabilistic weighted high-order fuzzy time series forecasting

The present article proposes a probabilistic weighted high-order fuzzy time series (FTS) forecasting model employing Bayesian network (BN) to address complex relationships and uncertainty hidden in time series. As considerable FTS forecasting models considers the fuzzy relationships between precedent moments and the consequent moment as the complex relationships in time series, BN structure learning is utilized to discover and model dependence relationships between each moment in time series. The combination of the fuzzy relationships modeled by fuzzy logical relationships and the dependence relationships modeled by the BN provides a comprehensive establishment and representation of the complex relationships inherent in time series data. The proposed FTS forecasting method calculates the fuzzy-probabilistic weights of each fuzzy logical relationship group using the improved fuzzy empirical probabilities to model both aleatoric and epistemic uncertainty in time series. To this end, the improved fuzzy empirical probabilities are formulated by integrating fuzzy empirical probabilities with the BN to incorporate dependence relationships from the original time series into the FTS forecasting procedure. The efficiency of the proposed forecasting model is validated on fourteen publicly available time series. Experimental results confirm the better performance of the proposed method comparing with nine existing FTS models and six numeric models. Hypothesis tests also validate the robustness and reliability of the proposed method.

Exploring Roughness in Left Almost Semigroups and Its Connections to Fuzzy Lie Algebras

This paper explores the concept of Generalized Roughness in LA-Semigroups and its applications in various mathematical disciplines. We highlight the fundamental properties and structures of Generalized Roughness, examining its relationships with Fuzzy Lie Algebras, Order Theory, Lattice Structures, Algebraic Structures, and Categorical Perspectives. Moreover, we investigate the potential of mathematical modeling, optimization techniques, data analysis, and machine learning in the context of Generalized Roughness. Our findings reveal important results in Generalized Roughness, such as the preservation of roughness under the fuzzy equivalence relation and the composition of roughness sets. We demonstrate the significance of Generalized Roughness in the context of order theory and lattice structures, presenting key propositions and a theorem that elucidate its properties and relationships. Furthermore, we explore the applications of Generalized Roughness in mathematical modeling and optimization, highlighting the optimization of roughness measures, parameter estimation, and decision-making processes related to LA-Semigroup operations. We showcase how mathematical techniques can enhance understanding and utilization of LA-Semigroups in practical scenarios. Lastly, we delve into the role of data analysis and machine learning in uncovering patterns, relationships, and predictive models in Generalized Roughness. By leveraging these techniques, we provide examples and insights into how data analysis and machine learning can contribute to enhancing our understanding of LA-Semigroup behavior and supporting decision-making processes.

An application of generalized symmetric type-2 intuitionistic fuzzy variables to a transportation problem with the effect of a new ranking function

The primary objective of this study is to establish and introduce generalized symmetric type-2 intuitionistic fuzzy set, a type-2 fuzzy set extension. The major goal of such an extension is to enable the concept of fuzzy outranking relation which is known as foot print of uncertainty (FOU) to be employed in generalized symmetric type-2 intuitionistic fuzzy number (GST2IFN) for the first time. This is interesting to deal this GST2IFNs and it is complicated to find the equivalent crisp quantities due to this special structure. Hence, we have introduced a new ranking function using the mean level set of membership and non-membership functions. Besides, the arithmetic operations of the GST2IFNs are discussed. To express the effectiveness and application of the proposed GST2IF variables, a transportation problem (TP) is formulated with all cost parameters are GST2IFNs. In addition, few different algorithmic approaches are applied to find the initial basic feasible solution (IBFS) and optimal solution of the TP. Finally, the feasibility and richness of the obtained optimal solution is discussed.

A Local Adjustment Method to Improve Multiplicative Consistency of Fuzzy Reciprocal Preference Relations

Preferences that verify the transitivity property are usually referred to as rational or consistent preferences. Existent methods to improve the consistency of inconsistent fuzzy reciprocal preference relations (FPRs) fail to retain the original preference values because they always derive a new FPR. This article presents a new inconsistency identification and modification (IIM) method to detect and rectify only the most inconsistent elements of an inconsistent FPR. As such, the proposed IIM can be considered a local adjustment method to improve multiplicative consistency (MC) of FPRs. The case of inconsistent FPRs with missing values, i.e., incomplete FPRs, is addressed with the estimation of the missing preferences with a constrained nonlinear optimization model by the application of the IIM method. The implementation process of the proposed algorithms is illustrated with numerical examples. Simulation experiments and comparisons with existent methods are also included to show that the new method requires fewer iterations than existent methods to improve the MC of FPRs and achieves better MC level, while preserving the original preference information as much as possible than the existent methods. Thus, the results presented in this article demonstrate the correctness, effectiveness, and robustness of the proposed method.

Resolution of Fuzzy Relation Equations with Constraints

Resolution of Fuzzy Relation Equations with Constraints

Hesitant fuzzy time series forecasting model of higher order based on one and two-factor aggregate logical relationship

Fuzzy time series forecasting is a methodology that incorporates fuzzy logic to handle uncertainty in time series data. It allows for more flexible and accurate predictions by considering the linguistic terms and membership functions associated with each data point, enabling a better representation of complex patterns and trends within the time series. A time series forecasting model examines the relationships between present and past observations for future measurement. This article proposes a hesitant fuzzy time series forecasting (FTSF) technique for higher order based on one and two-factor aggregate fuzzy relationships. The proposed method presents a new approach to partitioning intervals into an unequal and equal length. The hesitant fuzzy sets are defined for each observation, and fuzzification is based on their maximum score function. Further, aggregated triangular fuzzy numbers are defined using an aggregation operator corresponding to the fuzzified datum. In order to enhance forecasting performance, this work also introduces a novel defuzzification rule based on aggregated fuzzy logical relationship groups (AFLRGs) for higher order one and two-factor fuzzy time series. The implementation of the proposed approach is verified on two data sets to the forecasting of enrollment data and TAIFEX and TAIEX stock index data sets, and the forecasting error is tested in term of different statistical parameters. Thus, the model validates the effectiveness of the proposed method with higher forecasting accuracy rate in the environment hesitant fuzzy information.

Clustering sustainable suppliers in the plastics industry: A fuzzy equivalence relation approach

Nowadays, pure economic supply chain management is not commonly contemplated among companies (especially buyers), as recently novel dimensions of supply chains, e.g., environmental, sustainability, and risk, play significant roles. In addition, since companies prefer buying their needs from a group of suppliers, the problem of supplier selection is not solely choosing or qualifying a supplier from among others. Buyers, hence, commonly assemble a portfolio of suppliers by looking at the multi-dimensional pre-determined selection criteria. Since sustainable supplier selection criteria are often assessed by linguistic terms, an appropriate clustering approach is required. This paper presents an innovative way to implement fuzzy equivalence relation to clustering sustainable suppliers through developing a comprehensive taxonomy of sustainable supplier selection criteria, including supply chain risk. Fifteen experts participated in this study to evaluate 20 suppliers and cluster them in the plastics industry. Findings reveal that the best partitioning occurs when the suppliers are divided into two clusters, with 4 (20%) and 16 (80%) suppliers, respectively. The four suppliers in cluster one are performing better in terms of the capability of supplier/delivery, service, risk, and sustainability criteria such as environment protection/management, and green innovation. These factors are critical in clustering and selecting sustainable suppliers. The originality of this study lies in developing an all-inclusive set of criteria for clustering sustainable suppliers and adding risk factors to the conventional supplier selection criteria. In addition to partitioning the suppliers and determining the best-performing ones, this study also highlights the most influential factors by analysing the suppliers in the best cluster.

Matrix representations of the inverse problem in the graph model for conflict resolution with fuzzy preference

The inverse analysis of the graph model for conflict resolution (GMCR) determines all the preference relationships required by each decision-maker (DM) according to a given stability definition to ensure that the given state is stable. In the paper, a matrix representation approach for the inverse graph model with fuzzy preference relations is proposed to obtain all the preference relations for each DM making the stability result based on the fuzzy stability definition, including fuzzy Nash, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability definition. For the two-DM graph model, the fuzzy preference relation over states, unilateral movements, and fuzzy unilateral improvements are refreshed by using the matrix forms. Furthermore, for the multiple DMs model, the joint movements and joint improvement for a collection including at least two DM are represented by using matrix representation. Finally, the matrix relationship for specifying the required preference relation in the framework of inverse graph model with fuzzy preferences is derived. To illustrate the usefulness of the matrix representation of inverse graph model with fuzzy preference relations, this paper demonstrates it using a real-life conflict of the Elmira groundwater contamination conflict.

Group decision making based on relative projection between fuzzy preference relations

Group decision making (GDM) has been widely focused on for coping with the complexity of a practical decision making problem. This paper proposes a relative projection-driven GDM model with fuzzy preference relations (FPRs). First, a novel additive consistency index of FPRs is constructed using the variance of relative projections between column/row vectors in two FPRs, which is called the relative projection variance consistency index (RPVCI). The thresholds of RPVCI for FPRs with acceptable additive consistency (AAC) are calculated using the simulation method. For improving consistency of FPRs, a convergent iteration algorithm is constructed by equipping subjective and objective adjustment parameters. Second, the group consensus level is captured using relative projections of individual FPRs on the collective one. An iteration model is reported to optimize the consensus degree by combining the weights of experts with a comprehensive adjustment coefficient that depends on both subjective and objective factors. Third, a novel GDM algorithm is proposed by integrating the two proposed iteration models. Comparisons with some existing models are offered by carrying out a case study to show the novelty and advantage of the proposed GDM model.

Comparison of the Chen and Sinsgh’s Fuzzy Time Series Methods in Forecasting Farmer Exchange Rates in Indonesia

Chen and Singh's Fuzzy Time Series Model is a forecasting method that uses the basi fuzzy logic in the process. The differences in the models are in the fuzzy logic relations. Chen's model uses Fuzzy Logical Relationship Groups. Meanwhile, the Singh model uses only Fuzzy Logical Relationships in the forecasting process. To find out the best model between the two models, forecasting the Farmer's Exchange Rate is carried out. Farmers' exchange rates are the option for observers of agricultural development in assessing the level of welfare of farmers in Indonesia. With changes in farmer exchange rates every month, it is necessary to forecast data in order to obtain an overview for the following month. Research used is applied research where the initial step is to study and analyze the theories related to our research, then colect the necessary data. The data used is data secondary data obtained online from the official website of the Badan Pusat Statistika (BPS). the forecasting results of the two models were compared using MAPE. The results of the comparison of the accuracy of the prediction accuracy of Chen and Singh's fuzzy time series models on farmers' exchange rates obtained Chen's MAPE fuzzy time series values ​​of 0.679% and Singh's fuzzy time series models of 0.354%. This means that the best forecasting model for farmer exchange rates in Indonesia is the Singh model.

Reduction in a fuzzy probability information system based on incomplete set-valued data

Attribute reduction for incomplete data is a hot topic in rough set theory (RST). A fuzzy probabilistic information system (FPIS) combines of fuzzy relations that satisfy the probability distribution about objects, which can be regarded as an information system (IS) with fuzzy relations. This paper studies attribute reduction in an FPIS. Based on the available information of objects on an ISVIS, the probability distribution formula of objects is first defined. Then, an FPIS can be induced by an ISVIS. Next, attribute reduction in a FPIS is proposed similar to an IS. Moreover, information granulation and information entropy in an FPIS is defined, and the corresponding algorithms are constructed. Finally, the effectiveness of the constructed algorithms is verified by k-means clustering, Friedman test and Nemenyi test.

An Extended Type-2 Fuzzy Relational Database Model for Aggregate and Grouping Operations

This paper introduces an extended type-2 fuzzy relational database model (ET-2FRDB) for aggregate and grouping operations that can represent and query uncertain and imprecise information in the real world applications. In ET-2FRDB, each fuzzy relation is represented by a type-2 fuzzy set whose membership degree of each tuple is a fuzzy number on [0, 1], the fuzzy aggregate functions, the fuzzy relational algebraic, aggregate and grouping operations are defined as extensions of those in the classical relational database model thereby the membership degree of tuples associated by using the minimum and maximum of fuzzy numbers. Also, some properties of the fuzzy relational algebraic, aggregate and grouping operations in ET-2FRDB are formulated and proven.

Q-rung orthopair hesitant fuzzy preference relations and its group decision-making application

To express the opinions of decision-makers, q-rung orthopair hesitant fuzzy sets (q-ROHFSs) have been employed extensively. Therefore, it is necessary to construct q-rung orthopair hesitant fuzzy preference relations (q-ROHFPRs) as a crucial decision-making tool for decision-makers. The goal of this paper aims to define a new consistency and consensus approach for solving q-ROHFPR group decision-making (GDM) problems. To do this, we first state the definitions of q-ROHFPRs and additive consistent q-ROHFPRs based on q-ROHFSs, an additive consistency index and acceptable additive consistent q-ROHFPRs. Second, based on minimizing the deviation, we establish an acceptable goal programming model for unacceptable additive consistent q-ROHFPRs. Third, an iterative algorithm is created for achieving acceptable consistency and reaching a rational consensus. The degree of rational consensus among individual q-ROHFPRs is quantified by a distance-based consensus index. Afterward, a non-linear programming model is formulated to derive the priority vector of alternatives, which are q-rung orthopair hesitant fuzzy numbers (q-ROHFNs). Based on this model, a GDM model for q-ROHFPRs is then developed. To demonstrate the validity and utility of the proposed GDM model, a case study on the risk assessment of hypertension is provided. The finding of sensitivity and comparison analyses supports the feasibility and efficacy of the suggested approach.

Probing the rainfall time series over northeast India through composite and binary fuzzy relation

Probing the rainfall time series over northeast India through composite and binary fuzzy relation

An indirect multicriteria group decision-making method with heterogeneous preference relations and reliabilities of decision-makers

This study proposes a new indirect method to solve multicriteria group decision-making (MCGDM) problems with heterogeneous preference relations and decision-makers’ reliabilities. To unify the heterogeneous preference relations, the transformation of multiplicative preference relations (MPRs), fuzzy preference relations (FPRs), intuitionistic fuzzy preference relations (IFPRs), and linguistic preference relations (LPRs) into distributed preference relations (DPRs) is analyzed. As consistency among DPRs is constructed under the assumption that decision-makers are bounded rationality, the abstract transformation of MPR, FPR, IFPR, and LPR into DPR is defined, and its reality depends on the preferences of decision-makers. Two important properties of abstract transformation, namely, internal consistency and restricted max-max transitivity, were theoretically verified. The reliabilities of decision-makers are involved in MCGDM by estimating the reliability of each criterion to improve solution quality. Based on the four types of abstract transformations, a process for analyzing MCGDM problems with heterogeneous preference relations and decision-makers’ reliabilities is developed. The proposed method is used to analyze a new business selection problem for an enterprise located in Changzhou, Jiangsu, China to demonstrate its applicability and validity.

Fuzzy Metrics in Terms of Fuzzy Relations

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression d(x,y)=t. Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d(x,y) and the other is a number within the range [0,+∞).

Optimization of partially monotonic functions subject to bipolar fuzzy relation equations

A method to solve a latticed optimization problem constrained by a bipolar fuzzy relation equation is presented in this paper, under the hypothesis of a partially monotonic objective function. The solving strategy consists of transforming the problem into optimizing an order-preserving function in all arguments subject to another bipolar fuzzy relation equation. As a result, all the solutions of the original optimization problem can be deduced from the extremal elements of the feasible domain of the transformed problem. The presented approach embraces the particular case of linear optimization constrained by bipolar fuzzy relation equations.

A hesitant fuzzy multiple-attribute decision-making method considering rank relation

A decision matrix is typically used to express hesitant information when solving multi-attribute decision problems in an uncertain environment. To further investigate the decision problem, this paper takes the property of matrix rank as the starting point, introduces it into the hesitant fuzzy theory, presents the concept of the rank of the hesitant fuzzy decision matrix and discusses the related properties, and then studies the hesitant fuzzy linear relation, obtaining the attribute reduction method based on the hesitant fuzzy linear relation and applying it to the multi-attribute decision making. It adds to the theoretical understanding of the hesitant fuzzy decision matrix. The aggregation operator first transforms the hesitant fuzzy information into a comprehensive decision matrix, and the row echelon transformation determines the rank of the matrix. Second, the hesitant fuzzy linear relationship is obtained using the rank property. A new hesitant fuzzy matrix is obtained after attribute reduction based on the hesitant fuzzy linear relationship, and the alternatives are sorted using the TOPSIS method. Finally, the effectiveness and superiority of the proposed method are demonstrated through a comparison of actual case analysis and existing methods, and the expected research purpose is met.

Analysis of cybercrimes and security in FinTech industries using the novel concepts of interval-valued complex q-rung orthopair fuzzy relations

Analysis of cybercrimes and security in FinTech industries using the novel concepts of interval-valued complex q-rung orthopair fuzzy relations

Semi-supervised attribute reduction based on label distribution and label irrelevance

Attribute reduction in partially labeled data, also called semi-supervised attribute reduction, is an important issue. In recent years, the research on semi-supervised attribute reduction has attracted the attention of many scholars. Unfortunately, most existing semi-supervised attribute reduction methods do not handle the information loss caused by missing labels well. Meanwhile, these methods in general only consider the relevance between attributes and labels to measure attribute correlations, which ignores the irrelevant information contained in the attributes with respect to the labels. In view of this, this paper proposes a novel semi-supervised attribute reduction algorithm considering attribute relevance, redundancy and label irrelevance from the perspective of label distribution. Firstly, the membership degree of unlabeled objects relative to labels is defined by fuzzy similarity relation, which implements information restoration and converts partially labeled data into label distribution data. Secondly, some fuzzy uncertainty measures for label distribution are defined and related properties are investigated accordingly. Additionally, considering that irrelevant information brought by attributes may lead to over-fitting, label irrelevance criterion based on fuzzy uncertainty measures is constructed. Thirdly, a novel semi-supervised attribute reduction algorithm via the maximum relevance, minimum redundancy, and minimum irrelevance is proposed. Finally, compared with the representative semi-supervised attribute reduction algorithms and supervised attribute reduction algorithm, the effectiveness of the proposed algorithm is verified by various experiments.