Multiple Fuzzy Sets Research Articles

Dual-hesitant fermatean fuzzy Hamacher aggregation operators and TOPSIS with their application to multi-criteria decision-making.

The concept of the Dual-hesitant fermatean fuzzy set (DHFFS) represents a significant advancement in practical implementation, combining Fermatean fuzzy sets and Dual-hesitant sets. This new structure uses membership and non-membership hesitancy and is more adaptable for arriving at values in a domain. Since it has the capability to treat multiple fuzzy sets over the degrees of membership and non-membership, the DHFFS greatly improves the flexibility of approaches to tackle multiple-criteria decision-making (MCDM) problems. By applying generalized T-norm (T) and T-conorm (T*) operation, improved union and intersection formulas are derived. The proposed work adopts Hamacher operations such as Hamacher T-conorm (HT*) and Hamacher T-norm (HT) that are more efficient than conventional techniques. New aggregation operators such as Hamacher weighted arithmetic, geometric, power arithmetic, and power geometric are developed for DHFFS. These operators are most beneficial when dealing with a MCDM issue. A case study is used to demonstrate the approachs' accuracy and effectiveness in real-world decision-making. The comparative and sensitivity analysis results show that these operators are more effective than traditional methods. These results show that the proposed methods are efficient and can be applied in large-scale decision-making processes, strengthening the solutions' practical implications.

Forecasting stock prices based on multivariable fuzzy time series

<abstract><p>With the development of the stock market, the proportion of the stock assets in the asset structure of the residents increases rapidly. Therefore, the research on the prediction of stocks has great theoretical significance and application potential. A key point of researching stock prices is how to pick out the main factors. In this study, principal component analysis (PCA) is applied to find out the main factors which mainly affect the stock price. Then an improved cluster analysis algorithm is proposed to fuzzy the data, and a qualitative analysis method is given to find the most suitable prediction set from the multiple fuzzy sets corresponding to the current fuzzy set. We also extend the inverse fuzzy number formula to a more general form to get the predicted value. Finally, Xishan Coal and Electricity Power (XSCE) and Taiwan Futures Exchange (TAIFEX) time series are predicted, using the proposed multivariate fuzzy time series method. The results show that the prediction error is lower than that of the previous models. The proposed method produces better forecasting performance.</p></abstract>

S-Score Table-Based Parameter-Reduction Approach for Fuzzy Soft Sets

A fuzzy soft set is a mathematical tool used to deal with vagueness and uncertainty. Parameter reduction is an important issue when applying a fuzzy soft set to handle decision making. However, existing methods neglect newly added parameters and have higher computational complexities. In this paper, we propose a new S-Score table-based parameter-reduction approach for fuzzy soft sets. Compared with two existing methods of parameter reduction for a fuzzy soft set, our method takes newly added parameters into account, which brings about greater flexibility and is beneficial to the extension of fuzzy soft sets and a combination of multiple fuzzy soft sets. Additionally, our method accesses fewer elements from the dataset, which results in lower computation compared with the two existing approaches. The experimental results from two applications show the availability and feasibility of our approach.

The cross-entropy and improved distance measures for complex q-rung orthopair hesitant fuzzy sets and their applications in multi-criteria decision-making

The complex q-rung orthopair fuzzy set (Cq-ROFS) is the extension of complex Pythagorean fuzzy set (CPFS) in which the sum of the q-power of the real part (imaginary part) of the support for and the q-power of the real part (imaginary part) of the support against is limited by one; however, it is difficult to express the hesitant information. In this study, the conception of complex q-rung orthopair hesitant fuzzy set (Cq-ROHFS) by combining the Cq-ROFS and hesitant fuzzy set (HFS) is proposed, and its properties are discussed, obviously, Cq-ROHFS can reflect the uncertainties in structure and in detailed evaluations. Further, some distance measures (DMs) and cross-entropy measures (CEMs) are developed based on complex multiple fuzzy sets. Moreover, these proposed measures are utilized to solve a multi-criteria decision-making problem based on TOPSIS (technique for order preference by similarity to ideal solution) method. Then, the advantages and superiority of the proposed measures are explained by the experimental results and comparisons with some existing methods.

Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions

Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8].

Modeling and Analysis of Mamdani Two-Term Controllers Using Non-Uniformly Distributed Multiple Fuzzy Sets and CoA/CoG Defuzzification

While fuzzy controllers with multiple fuzzy sets and non-uniformly distributed membership functions have the potential to provide better system performance, from the literature, it appears to the authors that till now, no one has found exact mathematical models of fuzzy PI/PD controllers with non-uniformly distributed multiple fuzzy sets and Center of Area (CoA) defuzzification. To bridge this gap, in this paper, analytical structures of the general Mamdani type fuzzy PI/PD controllers are presented using unequally distributed multiple fuzzy sets, Minimum (Min) t-norm, Maximum (Max) s-norm, Larsen Product (LP) inference, and CoA defuzzification. For the purpose of comparison, analytical structures of the controllers are also obtained using Center of Sums (CoS) defuzzification. Properties of the newly obtained controllers are analyzed and compared. Since digital computers are often used for simulation, a rough estimate of the computational complexities and memory requirements are provided. To show the applicability of the newly developed controllers, a nonlinear DC series motor and a nonlinear plant with dead-time are considered. The effectiveness of the proposed controllers is finally shown through a detailed system performance comparison.

Topology Learning-Based Fuzzy Random Neural Networks for Streaming Data Regression

As a type of evolving-fuzzy system, the evolving-fuzzy-neuro (EFN) system uses the structure inspired by neural networks to determine its parameters (fuzzy sets and fuzzy rules), so EFN system can inherit the advantages of neural networks. However, for streaming data regression, EFN systems still have several drawbacks: determining fuzzy sets is not robust to data sequence; determining fuzzy rules is complex as subspaces that can approximate to a Takagi–Sugeno–Kang (TSK) rule need to be obtained, and many parameters need to be optimized; and it is difficult to detect and adapt to changes in the data distribution, i.e., concept drift, if the output is a continuous variable. Hence, in this article, a novel evolving-fuzzy-neuro system, called the topology learning-based fuzzy random neural network (TLFRNN), is proposed. In TLFRNN, an online topology learning algorithm is designed to self-organize each layer of TLFRNN. Different from current EFN systems, TLFRNN learns multiple fuzzy sets to reduce the impact of noises on each fuzzy set, and a randomness layer is designed, which assigning the probability of each fuzzy set. Also, TLFRNN does not utilize TSK rules; instead uses a simple inference that considers fuzzy and random information of data simultaneously. More importantly, in TLFRNN, concept drift can be detected and adapted easily and rapidly. The experiments demonstrate that TLFRNN achieves superior performance compared to other EFSs.

Takagi-Sugeno fuzzy PID controllers: mathematical models and stability analysis with multiple fuzzy sets

This paper deals with nonlinear Takagi-Sugeno (TS) fuzzy PID controllers with multiple fuzzy sets. Two models of fuzzy PID controllers are proposed using algebraic product (AP) triangular norm, bounded sum (BS)/maximum (Max) triangular co-norm and centre of gravity (CoG) defuzzifier. The inputs are fuzzified by three or more fuzzy sets with trapezoidal/triangular type membership functions. A new rule base is proposed consisting of four rules which reduce the number of tunable parameters. The models of the fuzzy PID controllers reveal that they are (nonlinear) variable gain/structure controllers, i.e., the gains are a function of input variables and the structure of the controller changes in the input space. The variations of gain and the properties of the controllers are investigated. The bounded-input bounded-output (BIBO) stability of the closed loop system with one of the proposed models in the loop is studied. The applicability of the controllers is demonstrated with the help of two examples.

Takagi-Sugeno fuzzy PID controllers: mathematical models and stability analysis with multiple fuzzy sets

Takagi-Sugeno fuzzy PID controllers: mathematical models and stability analysis with multiple fuzzy sets

General structure of Interval Type-2 fuzzy PI/PD controller of Takagi–Sugeno type

In this paper, a new configuration of Interval Type-2 (IT2) fuzzy Proportional–Integral (PI) or fuzzy Proportional–Derivative (PD) controller of Takagi–Sugeno (TS) type is presented. An attempt is made to generalize the IT2 fuzzy PI/PD controller structure using multiple fuzzy sets. Fuzzification of the inputs is done with three or more fuzzy sets having triangular/trapezoidal membership functions. The rule base consists of only three rules to reduce the number of tuneable parameters of the controller. Minimum (Min) triangular norm and Bounded Sum (BS) triangular co-norm are used as conjunction and disjunction operators to reduce the number of rules. Karnik–Mendel (KM) type reducer and Weighted Average (WA) defuzzifier are considered to derive the analytical structure of the fuzzy controller. Properties and gain variations of the fuzzy controller are investigated. Simulation study is carried out on nonlinear dynamical systems to verify the applicability of the fuzzy controllers.

Analytical Structures of Takagi-Sugeno Fuzzy Two-Input Two-Output Proportional-Integral/Proportional-Derivative Controllers With Multiple Fuzzy Sets

In this paper, an attempt is made to generalize the analytical structures of Takagi-Sugeno (TS) fuzzy two-input two-output (TITO) proportional-integral (PI)/proportional-derivative (PD) controllers using multiple input fuzzy sets. Two models of fuzzy TITO PI/PD controllers are proposed based on two distinct control strategies. The inputs are fuzzified by multiple fuzzy sets with trapezoidal/triangular membership functions. The generalized rule base consists of nine control rules imbibing the complete control strategy and is closer in spirit to the original TS rule base. Algebraic product (AP) triangular norm, bounded sum (BS) triangular conorm, and center of gravity (CoG) defuzzifier are applied to derive the models. The models of the fuzzy TITO PI/PD controllers with multiple input fuzzy sets are (nonlinear) variable gain/structure controllers. Also, each output of the fuzzy controller is the sum of two nonlinear PI/PD controllers with variable gains. The gain variation and properties of the proposed controllers are studied. Two examples of nonlinear dynamic processes are considered to demonstrate the applicability of the proposed controllers.

Development of an integrated optimization method for analyzing effect of energy conversion efficiency under uncertainty – A case study of Bayingolin Mongol Autonomous Prefecture, China

In this study, a superiority–inferiority full-infinite mixed-integer programming (SFMP) method is developed for analyzing the effect of energy conversion efficiency under uncertainty. SFMP can effectively tackle uncertainties expressed as fuzzy sets, crisp intervals and functional intervals, it also can directly reflect relationships among multiple fuzzy sets through varying superiority and inferiority degrees with a high computational efficiency. Then the developed SFMP is applied to a real case of planning energy system for Bayingolin Mongol Autonomous Prefecture, where multiple scenarios related to different energy-conversion efficiency are concerned. Results for energy processing, energy conversion, capacity expansion, pollutant emission and system cost have been generated. It is proved that SFMP is an effective approach to deal with the uncertainties in energy systems with interactive and uncertain characteristics. A variety of uncertainties existed in energy conversion processes and impact factors could affect the modeling result. Results show that improvement of energy-conversion efficiency can effectively facilitate reducing energy resources consumption, optimizing energy generation pattern, decreasing capacity expansion, as well as mitigating pollutant emissions. Results also reveal that, for the study area, electric power has a highest energy saving potential among heating, oil processing, coal washing and refining. Results can help decision makers to generate desired alternatives that can facilitate policy enactment of conversion efficiency improvement and adjustment of regional energy structure under uncertainty.

Segmentation of gray scale image based on intuitionistic fuzzy sets constructed from several membership functions

Segmentation is the process of extraction of objects from an image. This paper proposes a new algorithm to construct intuitionistic fuzzy set (IFS) from multiple fuzzy sets as an application to image segmentation. Hesitation degree in IFS is formulated as the degree of ignorance (due to the lack of knowledge) to determine whether the chosen membership function is best for image segmentation. By minimizing entropy of IFS generated from various fuzzy sets, an image is thresholded. Experimental results are provided to show the effectiveness of the proposed method.

Learning Fuzzy Network Using Sequence Bound Global Particle Swarm Optimizer

This paper proposes an algorithm for classification by learning fuzzy network with a sequence bound global particle swarm optimizer. The aim of this work can be achieved in two folded. Fold one provides an explicit mapping of an input features from original domain to fuzzy domain with a multiple fuzzy sets and the second fold discusses the novel sequence bound global particle swarm optimizer for evolution of optimal set of connection weights between hidden layer and output layer of the fuzzy network. The novel sequence bound global particle swarm optimizer can solve the problem of premature convergence when learning the fuzzy network plagued with many local optimal solutions. Unlike multi-layer perceptron with many hidden layers it has only single hidden layer. The output layer of this network contains one neuron. This network advocates a simple and understandable architecture for classification. The experimental studies show that the classification accuracy of the proposed algorithm is promising and superior to other alternatives such as multi-layer perceptron and radial basis function network.

A fuzzy-Markov-chain-based analysis method for reservoir operation

In this study, a fuzzy-Markov-chain-based stochastic dynamic programming (FM-SDP) method is developed for tackling uncertainties expressed as fuzzy sets and distributions with fuzzy probability (DFPs) in reservoir operation. The concept of DFPs used in Markov chain is presented as an extended form for expressing uncertainties including both stochastic and fuzzy characteristics. A fuzzy dominance index analysis approach is proposed for solving multiple fuzzy sets and DPFs in the proposed FM-SDP model. Solutions under a set of α-cut levels and fuzzy dominance indices can be generated by solving a series of deterministic submodels. The developed method is applied to a case study of a reservoir operation system. Solutions from FM-SDP provide a range of desired water-release policies under various system conditions for reservoir operation decision makers, reflecting dynamic and dual uncertain features of water availability simultaneously. The results indicate that the FM-SDP method could be applicable to practical problems for decision makers to obtain insight regarding the tradeoffs between economic and system reliability criteria. Willingness to obtain a lower benefit may guarantee meeting system-constraint demands; conversely, a desire to acquire a higher benefit could run into a higher risk of violating system constraints.

A multistage fuzzy-stochastic programming model for supporting sustainable water-resources allocation and management

In this study, a multistage fuzzy-stochastic programming (MFSP) model is developed for tackling uncertainties presented as fuzzy sets and probability distributions. A vertex analysis approach is proposed for solving multiple fuzzy sets in the MFSP model. Solutions under a set of α-cut levels can be generated by solving a series of deterministic submodels. The developed method is applied to the planning of a case study for water-resources management. Dynamics and uncertainties of water availability (and thus water allocation and shortage) could be taken into account through generation of a set of representative scenarios within a multistage context. Moreover, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. The modeling results can help to generate a range of alternatives under various system conditions, and thus help decision makers to identify desired water-resources management policies under uncertainty.

Perspective rectification of document images using fuzzy set and morphological operations

In this paper, we deal with the problem of document image rectification from image captured by digital cameras. The improvement on the resolution of digital camera sensors has brought more and more applications for non-contact text capture. Unfortunately, perspective distortion in the resulting image makes it hard to properly identify the contents of the captured text using traditional optical character recognition (OCR) systems. We propose in this work a new technique, which is capable of removing perspective distortion and recovering the fronto-parallel view of text with a single image. Different from reported approaches in the literature, the image rectification is carried out using character stroke boundaries and tip points (SBTP), which are extracted from character strokes based on multiple fuzzy sets and morphological operators. The algorithm needs neither high-contrast document boundary (HDB) nor paragraph formatting (PF) information. Experimental results show that our rectification process is fast and robust.

Integrating membership functions and fuzzy rule sets from multiple knowledge sources

In this paper, we propose a GA-based fuzzy knowledge-integration framework that can simultaneously integrate multiple fuzzy rule sets and their membership function sets. The proposed two-phase approach includes fuzzy knowledge encoding and fuzzy knowledge integration. In the encoding phase, each fuzzy rule set with its associated membership functions is first transformed into an intermediary representation, and further encoded as a string. The combined strings form an initial knowledge population, which is then ready for integration. In the knowledge-integration phase, a genetic algorithm is used to generate an optimal or nearly optimal set of fuzzy rules and membership functions from the initial knowledge population. The hepatitis diagnostic problem was used to show the performance of the proposed knowledge-integration approach. Results show that the fuzzy knowledge-base resulting from using our approach performs better than every individual knowledge base.

Integrating fuzzy knowledge by genetic algorithms

We propose a genetic algorithm-based fuzzy knowledge integration framework that can simultaneously integrate multiple fuzzy rule sets and their membership function sets. The proposed approach consists of two phases: fuzzy knowledge encoding and fuzzy knowledge integration. In the encoding phase, each fuzzy rule set with its associated membership functions is first transformed into an intermediary representation and then further encoded as a string. The combined strings form an initial knowledge population, which is then ready for integration. In the knowledge-integration phase, a genetic algorithm is used to generate an optimal or nearly optimal set of fuzzy rules and membership functions from the initial knowledge population. Two application domains, the hepatitis diagnosis and the sugarcane breeding prediction, were used to show the performance of the proposed knowledge-integration approach. Results show that the fuzzy knowledge base derived using our approach performs better than every individual knowledge base.