Type-2 Fuzzy Sets Research Articles

Counterfactuals in fuzzy relational models

Given the pressing need for explainability in Machine Learning systems, the studies on counterfactual explanations have gained significant interest. This research delves into this timely problem cast in a unique context of relational systems described by fuzzy relational equations. We develop a comprehensive solution to the counterfactual problems encountered in this setting, which is a novel contribution to the field. An underlying optimization problem is formulated, and its gradient-based solution is constructed. We demonstrate that the non-uniqueness of the derived solution is conveniently formalized and quantified by admitting a result coming in the form of information granules of a higher type, namely type-2 or interval-valued fuzzy set. The construction of the solution in this format is realized by invoking the principle of justifiable granularity, another innovative aspect of our research. We also discuss ways of designing fuzzy relations and elaborate on methods of carrying out counterfactual explanations in rule-based models. Illustrative examples are included to present the performance of the method and interpret the obtained results.

Practical consensus of T-S fuzzy positive multi-agent systems using linear programming

This paper presents the practical consensus of T-S fuzzy positive multi-agent systems with interval type-1 and type-2 fuzzy sets using observer-based control protocols. A fuzzy positive observer is first constructed for the systems. An observer-based fuzzy control protocol is designed, where an additional constant term is introduced. Some linear programming conditions are established to achieve the positivity of the original system and its observer. Then, the practical consensus of the original system is transformed into the stability of a dynamic system, where a set of new variables is defined by combining the constant term and the states of the agents. In the first step, the positivity of the new variables is guaranteed. In the second step, the stability of the dynamic system consisting new variables is addressed by using a fuzzy copositive Lyapunov function. The gain matrices of observer and control protocols are formulated based on a matrix decomposition approach. All positive and consensus conditions are described via linear programming. Finally, two examples are provided to verify the validity of the obtained results.

A least-squares framework for developing interval type-2 fuzzy semantics

The developing of IT2 fuzzy semantics is critical for computing with words (CWW), but exiting approaches lack flexibility and fail to adapt user’s diversified demands. The study proposes a least-squares framework for designing CWW encoders that construct interval type-2 fuzzy sets (IT2 FSs) to represent the semantic meanings of linguistic words. In the least-squares framework, an CWW encoder is characterized by two elements: an intra-uncertain semantic mapping and an inter-uncertain semantic family. The intra-uncertain semantic mapping transforms data intervals into type-1 fuzzy sets (T1 FSs), then an optimal IT2 FS is derived from the inter-uncertain semantic family using the least-squares method. Furthermore, several intra-uncertain semantic mappings are introduced, and a compatibility measure is defined to facilitate model selection. The least-squares framework benefits from the flexible selection of intra-uncertain semantic mappings and least-squares optimization-based construction of IT2 FSs. In experiments, the least-squares framework is applied to handle real-world online survey data and the large-scale online review data set of a Chinese life service review site, Dianping.com. Compared to the enhanced interval approach and the Hao-Mendel approach, the least-squares framework shows its favorable efficiency in experiments and statistical tests, and adapts to user-defined intra- and inter-uncertain semantic families.

A Bayesian Network Inference Approach for Dynamic Risk Assessment Using Multisource-Based Information Fusion in an Interval Type-2 Fuzzy Set Environment

A Bayesian Network Inference Approach for Dynamic Risk Assessment Using Multisource-Based Information Fusion in an Interval Type-2 Fuzzy Set Environment

Critical study of type-2 fuzzy logic control from theory to applications: A state-of-the-art comprehensive survey

The review systematically explores the theoretical foundations of type-2 fuzzy sets and the associated mathematical frameworks of the advancements in type-2 fuzzy logic systems (T2-FLS), as well as a comprehensive overview of the state-of-the-art techniques and applications of type-2 fuzzy logic systems. The review's application-oriented section looks at a number of different areas where type-2 fuzzy logic has made a big difference. These areas include robotics, autonomous vehicle control, electric and electronic control, image processing, and railway control. This overview mainly discusses the past, present, and future trends in type-2 fuzzy logic applications. The primary contribution summarizes the most essential type-2 fuzzy logic research with theoretical and practical implications. This article will provide a critical path and a strong foundation for individuals interested in this field to pursue future research.

Dementia Classification Approach Based on Non-Singleton General Type-2 Fuzzy Reasoning

Dementia is the most critical neurodegenerative disease that gradually destroys memory and other cognitive functions. Therefore, early detection is essential, and to build an effective detection model, it is required to understand its type, symptoms, stages and causes, and diagnosis methodologies. This paper presents a novel approach to classify dementia based on a data set with some relevant patient features. The classification methodology employs non-singleton general type-2 fuzzy sets, non-singleton interval type-2 fuzzy sets, and non-singleton type 1 fuzzy sets. These advanced fuzzy sets are compared with traditional singleton fuzzy sets to evaluate their performance. The Takagi–Sugeno–Kang TSK inference method is used to handle fuzzy reasoning. In the process, the parameters of the membership functions (MFs) and rules are obtained using ANFIS, and non-singleton MFs are optimized with PSO. The results demonstrate that non-singleton general type-2 fuzzy sets improve classification accuracy compared to singleton fuzzy sets, demonstrating their ability to model the uncertainties inherent in the diagnosis of dementia. This improvement suggests that non-singleton fuzzy systems offer a more robust framework for developing effective diagnostic tools in the medical domain. Accurate classification of dementia is of utmost importance to improve patient care and advance medical research.

Perceptual Computing Based Method of Evaluation in E-learning Systems

Introduction. Implementation of e-learning has transformed the education system, expanding access to knowledge and providing flexibility in learning for both students and teachers. Application of information technologies in the education system allows to improve the learning process by introducing novel methods and approaches not only to teaching but also to knowledge assessment. However, along with the growing popularity of e-learning, new challenges arise. Among them the problem of objective evaluation of students' achievements—one of the key components of learning— is particularly significant. In an e-learning environment, evaluation becomes more challenging due to the lack of physical contact and limited access to informal communication. Therefore, it is necessary to develop and apply effective evaluation methods that would ensure objectivity, reliability, and adaptability to special features of e-learning. Evaluation using fuzzy sets enables the teacher to account for uncertainty in numerical grades and to reduce their subjective influence. Of particular importance are methods based on type-2 fuzzy sets, including perceptual computing. They enable both verbal evaluation of works and aggregation of various verbal and numerical assessments to determine the final grade. Objective. To formalize a model of quiz in e-learning systems, to develop a perceptual computing based evaluation method, which will enhance the objectivity and flexibility of the evaluation process. Results. Existing fuzzy set based evaluation methods in e-learning systems are reviewed. A graph model of a quiz is proposed, consisting of problems, knowledge units, and chunks connected by arcs loaded verbally. A method is described for calculating an aggregate grade using linguistically weighted average. Conclusions. The proposed method for quiz evaluation in e-learning systems, based on perceptual computing, allows to reduce the impact of the human factor and subjectivity. In particular, verbal evaluation of open-ended questions and processing of such grades using perceptual computing based on type-2 fuzzy sets lead to objective evaluation and have significant potential for enhancing the efficiency of e-learning. Keywords: perceptual computing, e-learning, chunk, fuzzy sets.

Dynamic portfolio optimization with the MARCOS approach under uncertainty

This paper introduces a dynamic portfolio selection approach that integrates the robustness of interval type-2 fuzzy sets (IT2FSs) with the flexibility of the MARCOS (Measurement of Alternatives and Ranking according to COmpromise Solution) method, offering a novel framework for asset evaluation amidst uncertainty. The IT2FSs enhance the adaptability of asset criteria representation, while the innovative application of MARCOS within the IT2FS environment refines the asset selection process. The weighted semi-absolute deviation metric is embraced to capture portfolio risk characteristic, which ingeniously harnesses the utility function values derived from the IT2F-MARCOS framework to delineate the anticipated return profile. On this basis, a dynamic bi-objective portfolio allocation model with realistic constraints and dynamic risk preference is formulated to rebalance portfolio periodically. Empirical evidence demonstrates the robustness and effectiveness of this approach compared to benchmark indexes, different allocation strategies, and the TOPSIS-based selection method, offering significant advancements in portfolio optimization for investors navigating uncertain markets. This study endeavors to contribute to the field of portfolio management, providing a thoughtful approach that enhances both theoretical understanding and practical application.

Selection of Military Armored Vehicle Using Fuzzy EDAS method

Military armored vehicles are specially designed and equipped for defense and security activities. During military operations, the right vehicle selection is of vital importance for the success of the operation. This problem examined 5 different alternative military armored vehicles for operational and logistic success using the criteria Engine Power, Dimensions, Fire Power Range, Gradeability, and Armor Thickness. In military decision problems, some alternatives probably need to be evaluated regarding more than one conflicting criterion. The decision among these alternatives will usually involve other strategic-level decisions. Multi-criteria decision-making (MCDM) refers to making decisions in such a situation. Are there any methods and techniques available to solve MCDM problems? The evaluation according to distance from average solution (EDAS) method is a multi-criteria decision-making method whose effectiveness has been tested in other problem areas in the literature. In this study, we use the EDAS method to address the problem of military armored vehicle selection in a fuzzy environment. While a Type-1 fuzzy set is used in the fuzzy EDAS method, trapezoidal fuzzy numbers are preferred for military expert evaluation. The results of this study show that the fuzzy EDAS method is effective.

Solving Interval Type-2 Fuzzy Pentagonal Polynomial Equation Using Horner-Muller's Method

The recent boom in interval type-2 fuzzy sets has attracted many researchers to the fields of fuzzy numbers, algebraic expressions, and iterative methods. Interval type-2 fuzzy pentagonal polynomial (IT2 FPP) is one of the algebraic expressions that combine the knowledge of interval type-2 pentagonal fuzzy numbers and polynomial equations. Solving IT2 FPP in the absence of an iterative method is becoming a common encounter as the solution is often difficult, expensive, and inexact. This paper defines IT2 FPP and the iterative method Horner-Muller’s used to find the approximate solution of the IT2 FPP. The IT2 FPP is introduced by combining the generalization of interval type-2 pentagonal fuzzy numbers and fuzzy polynomials, whereas the Horner-Muller’s method is a combination of Horner’s method and Muller’s method. A numerical example is presented to illustrate the computational procedures for finding the approximate solution. The approximate solution is obtained after two iterations with zero error. A Comparative analysis is also presented to validate the consistency and efficiency of the proposed method compared to benchmark iterative methods.

Evaluation of Spare Parts Support Capacity of Civil Aircrafts Based on Type-2 Hesitant Pythagorean Fuzzy Sets and Improved Technique for Order Preference by Similarity to Ideal Solution

To improve the spare parts support capacity of civil aircrafts and given the actual lack of evaluation methods at present, the evaluation problem of spare parts support capacity was solved in this study by proposing a multi-attribute decision method based on Type-2 hesitant Pythagorean fuzzy sets and an improved Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). First, the basic definitions of Type-2 hesitant Pythagorean fuzzy sets were given, which were further promoted to Type-n hesitant Pythagorean fuzzy sets, and the basic order relation criterion of Type-2 hesitant Pythagorean fuzzy sets was introduced. Second, a complete evaluation system for spare parts supply support capacity was established with the spare parts of civil aircrafts as the study objects, and each evaluation indicator was introduced in detail. Then, the spare parts support solutions were preferentially sorted using the correlation coefficient formula of Type-2 hesitant Pythagorean fuzzy sets and improved TOPSIS. Finally, the reliability and reasonability of the proposed method were verified through an example calculation and comparative analysis. The experimental results indicate that the proposed method can acquire the evaluation results of spare parts support capacity more scientifically and can be referenced by relevant studies.

Uncertainty Minimization in the Human Visual Response Using an Interval Type-2 Fuzzy Set and Its Application to Rice Leaf Image Enhancement

Uncertainty Minimization in the Human Visual Response Using an Interval Type-2 Fuzzy Set and Its Application to Rice Leaf Image Enhancement

Group Consensus-Driven Energy Consumption Assessment Using Social Network Analysis and Fuzzy Information Fusion

With industrial development deepening, the share of industrial energy in overall consumption has notably risen. To assess industrial energy consumption accurately, this paper proposes a method employing interval type-2 fuzzy sets (IT2FSs) to represent assessment information effectively. Additionally, it analyzes decision-makers (DMs) as a social network to alleviate individual biases. IT2FSs are chosen to handle uncertainties in assessing industrial energy consumption. Addressing biases in DMs' opinions, a group consensus model aids the consensus reaching process (CRP). Industrial energy consumption is assessed using the MULTIMOORA method, yielding three results. These are fused via D-S evidence theory (DSET) to obtain the final assessment. Finally, the model's effectiveness is verified with a case study on energy consumption in the steel industry. In conclusion, this paper not only deepens the understanding of uncertainties in the energy consumption assessment process, but also provides a robust tool for various industries to optimize energy use and economic outcomes.

Human-Robot Collaboration in Mixed-Flow Assembly Line Balancing under Uncertainty: An Efficient Discrete Bees Algorithm

In the evolving landscape of manufacturing and remanufacturing, assembly lines play a crucial role. Within the context of Industry 5.0, human workers are seen as a valuable and irreplaceable resource. Human-robot collaboration is a promising production model that combines the strengths of human workers and robots, thereby enhancing production efficiency while reducing occupational risks related to ergonomics. Despite these advancements, inherent uncertainties within assembly processes, the integration of human-robot partnerships, and the dynamic nature of market demands pose significant challenges to traditional assembly methods. To address these challenges, this research introduces a novel modelling approach through a mixed-flow assembly line balancing problem designed for uncertain environments, fostering collaboration between humans and robots. The primary goal is to facilitate efficient collaboration within a type-I assembly line balancing problem framework, where predefined assembly beats guide the workflow. In this research, the use of interval type-2 fuzzy sets capabilities was investigated to address uncertainties in the assembly process. Furthermore, the potential of pairing human operators of different abilities with robots of different models for collaborative tasks at workstations was explored, enhancing flexibility and adaptability in the assembly line. In response to the complexity of the problem, this research proposes an efficient multiobjective discrete bees algorithm that incorporates innovative operators and search strategies. Rigorously tested across diverse case studies, this algorithm consistently outperforms other comparator algorithms. This research not only offers novel perspectives on addressing assembly line balancing challenges but also provides valuable insights for the effective implementation of human-robot collaborative assembly in uncertain environments.

Retraction Note: Medical image fusion based on type-2 fuzzy sets with teaching learning based optimization

Retraction Note: Medical image fusion based on type-2 fuzzy sets with teaching learning based optimization

Analyzing risk factors of tuberculosis using type-2 interval-valued trapezoidal fuzzy numbers with Einstein aggregation operators extended to MCDM

The principal motive of this work is to evolve and initiate an extension from interval-valued fuzzy sets to type-2 interval-valued fuzzy sets (T2IVFS) related to weighted aggregation functions containing the Einstein operator. The chief reason for this extension is that the constancy of the terms can also be taken into data during the aggregation operation. The main goal of this article is to compose the aggregation operators and their characteristics such as the Type-2 interval-valued fuzzy Einstein weighted arithmetic aggregating operator (T2IVFEWA), Type-2 interval-valued fuzzy Einstein weighted geometric aggregating operator (T2IVFEWG), and the characteristics are expressed. At last, to intimate the effectiveness of the suggested approach and explicate the purpose of these operators, a hybrid multi-criteria decision-making problem (MCDM) to select the best risk factor for Tuberculosis (TB) is considered and the result is compared with the outcome of the existing operators and methods. Additionally, a sensitivity analysis was conducted to verify the robustness of the proposed decision-making process.

A new defuzzification method and ranking method for type-2 fuzzy numbers

The type-2 fuzzy number (T2 FN) is a special type-2 fuzzy set whose all secondary membership functions are fuzzy numbers. This paper proposes a new defuzzification method and ranking method for T2 FNs. The defuzzification method can be expressed in a closed-form formula or can be divided into two computational steps. The first step computes the centroid of each secondary membership function to obtain a type-reduced set using the method derived from generalizing the existing alpha-cut defuzzification methods by introducing a parameter that controls the importance of secondary membership degrees, and the second step calculates the centroid of the type-reduced set. The defuzzified values are used to provide position information of T2 FNs. Secondly, we define a function to calculate the area enclosed by the image of the type-reduced set and the coordinate axes, and prove that this function is continuous. The values of this function are used to provide shape information of T2 FNs. Finally, to rank T2 FNs, we define a family of quasi-orders based on their position and shape information, and illustrate the validity and rationality of these quasi-orders through examples.

Contrasting bean analysis system based on YOLOv5 and a neural network model using the interval type-2 fuzzy set approach

Contrasting bean analysis system based on YOLOv5 and a neural network model using the interval type-2 fuzzy set approach

A Novel Risk Assessment Approach for Open-Cast Coal Mines Using Hybrid MCDM Models with Interval Type-2 Fuzzy Sets: A Case Study in Türkiye

A Novel Risk Assessment Approach for Open-Cast Coal Mines Using Hybrid MCDM Models with Interval Type-2 Fuzzy Sets: A Case Study in Türkiye

CI-MM-Dombi operator based on interval type-2 spherical fuzzy set and its applications on sustainable supply chain with risk criteria: using CI-TODIM-MARCOS method

CI-MM-Dombi operator based on interval type-2 spherical fuzzy set and its applications on sustainable supply chain with risk criteria: using CI-TODIM-MARCOS method