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We deal with solutions of classical linear equations a · x = b and y · a = b , applying a particular lattice valued fuzzy technique. Our framework is a structure with a binary operation · (a groupoid), equipped with a fuzzy equality. We call it a fuzzy quasigroup if the above equations have unique solutions with respect to the fuzzy equality. We prove that a fuzzy quasigroup can equivalently be characterized as a structure whose quotients of cut-substructures with respect to cuts of the fuzzy equality are classical quasigroups . Analyzing two approaches to quasigroups in a fuzzy framework, we prove their equivalence. In addition, we prove that a fuzzy loop (quasigroup with a unit element) which is a fuzzy semigroup is a fuzzy group and vice versa. Finally, using properties of these fuzzy quasigroups, we give answers to existence of solutions of the mentioned linear equations with respect to a fuzzy equality, and we describe solving procedures. Quasigroups and other related structures are an algebraic tool successfully applied up to now in coding theory and cryptology. In our work we propose related applications.

Fuzzy classification rules are more interpretable and cope better with pervasive uncertainty and vagueness with respect to crisp rules. Because of this fact, fuzzy classification rules are extensively used in classification and decision support systems for disease diagnosis. But, most of the rule mining techniques failed to generate accurate and comprehensive fuzzy rules. This paper presents a hybrid decision support system based on Rough Set Theory (RST) and Bat optimization Algorithm (BA) called RST-BatMiner. It consists of two stages. In the first stage, redundant features have been removed from the data set through RST-based QUICK-REDUCT approach. In the second stage, for each class BA is invoked to generate fuzzy rules by minimizing proposed fitness function. Further, an Ada-Boosting technique is applied to the rules generated by BA to increase the accuracy rate of generated fuzzy rules. Moreover, to generate comprehensive fuzzy rules, a new ≠ (not equal) operator along with = (equal) operator is introduced into BA encoding scheme. The proposed RST-BatMiner builds consolidated fuzzy ruleset by learning the rules associated with each class separately. The proposed RST-BatMiner is experimented on six bench-mark datasets namely Pima Indians Diabetes, Wisconsin Breast Cancer, Cleveland Heart disease, iris, wine and glass, in order to validate its generalization capability. These experimental results show that except for wine dataset the proposed RST-BatMiner yields high accuracy and comprehensible ruleset when compared to other state-of-the-art bio-inspired based fuzzy rule miners and Fuzzy Rule Based Classification Systems (FRBCS) in the literature. In the case of wine dataset, the proposed RST-BatMiner yields second highest accuracy along with comprehensible ruleset.

In this study, we establish the existence of the α-core in fuzzy games. Using the existence results, we identify a class of fuzzy games comprising a dense residual subset of fuzzy games such that every point of the α-core is essential. We also show that every fuzzy game in this class has an essential component of its α-core.

In many practical applications of classifiers, not only high accuracy but also high interpretability is required. Among a wide variety of existing classifiers, Takagi–Sugeno–Kang (TSK) fuzzy classifiers may be one of the best choices for achieving a good balance between interpretability and accuracy. In order to further improve their accuracy without losing their interpretability, we propose a highly interpretable deep TSK fuzzy classifier HID-TSK-FC (deep shared-linguistic-rule-based TSK fuzzy classifier) based on the concept of shared linguistic fuzzy rules. The proposed classifier has two characteristics: One is a stacked hierarchical structure of component TSK fuzzy classifiers for high accuracy, and the other is the use of interpretable linguistic rules with the same set of linguistic labels for all inputs. High interpretability is achieved at each layer by using the same set of linguistic values for all inputs, including the outputs from the previous layers in the stacked hierarchical structure. We show that a linguistic rule with the outputs from the previous layers as its inputs is equivalent to a fuzzy rule with a nonlinear consequent or a linear consequent with a certainty factor. We also show that HID-TSK-FC is mathematically equivalent to a novel TSK fuzzy classifier with shared interpretable linguistic fuzzy rules. Promising performance of HID-TSK-FC is demonstrated through extensive computational experiments on benchmark datasets and a real-world application case.

In this paper, a novel fuzzy neural network with intuitive, interpretable, and correlated-contours fuzzy rules (IC-FNN), for function approximation, is presented. The surfaces of these fuzzy rules are similar to the surfaces of the hills in the function landscape. Contours of the hills could be correlated and nonseparable with different shapes and directions. Thus, to obtain nonseparable and correlated fuzzy rules, a proper optimization problem is introduced and solved. To form contours with different shapes, a novel shapeable membership function with an adaptive shape is introduced to define the fuzzy sets. Next, based on a hierarchical Levenberg–Marquardt learning method, the parameters of the extracted fuzzy rules are fine tuned. The performance of the proposed method is evaluated in real-world regression and time-series prediction problems, and compared with other existing methods. According to these experiments, the proposed method could construct more parsimonious structures with higher accuracy, in comparison to the existing methods. Although the performance of the proposed method for complex and correlated functions is premier, for simple and uncorrelated cases, it is appropriate but with a more complex structure.

Right invariant fuzzy quasi-orders for fuzzy automata are broadly studied in the recent literature, as they arise as solutions to particular systems of fuzzy relation equations and inequalities. Some of their applications include determinization and state reduction procedures, as well as simulations and bisimulations for fuzzy automata. In this paper we provide a procedure for computing the greatest right invariant fuzzy quasi-order for a given fuzzy automaton over a complete residuated lattice. The proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite. When the previous condition is not satisfied, we show that the greatest right invariant fuzzy quasi-order can be obtained by taking the limit value of the convergent array of fuzzy quasi-orders for fuzzy automata over BL-algebras on the real unit interval [ 0 , 1 ] . Analogous procedures for computing the greatest left invariant fuzzy quasi-order, as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented. In addition, the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented.

Based on the Xie–Beni index and an improved particle swarm optimization algorithm, a novel identification method for the Takagi–Sugeno fuzzy model is proposed in this paper. Firstly, Xie–Beni indices with a fuzzy c-means clustering algorithm are adopted to find the rule number of the Takagi–Sugeno fuzzy model. By utilizing the particle swarm optimization algorithm, the initial membership function and the consequent parameters of the fuzzy model are obtained. In addition, through an improved fuzzy c-regression model and orthogonal least-square method, the premise structure and consequent parameters can be obtained to establish the Takagi–Sugeno fuzzy model. Some well-known models are used to demonstrate that the proposed method outperforms some existing methods.

This paper further investigates axiomatic characterizations of lower fuzzy rough approximation operators determined by a semicontinuous fuzzy implication I. The I-lower fuzzy rough approximation operators on a general fuzzy relation can be characterized by only one axiom. The axiomatic characterizations of I-lower fuzzy rough approximation operators are studied in two approaches, which characterize abstract I-lower fuzzy rough approximation operators with ordinary fuzzy operations on fuzzy sets and product operation, respectively. Considering the properties of fuzzy implications, different axioms are proposed for I-lower fuzzy rough approximation operators, which are determined by an S-implication, an R-implication and a fuzzy implication satisfying left neutrality property and regular property, respectively. When the fuzzy relation is special, such as serial, reflexive, symmetric and so on, I-lower fuzzy rough approximation operators can also be characterized by a single axiom.

Rule-based fuzzy cognitive maps (RBFCMs) have been developed for modeling nonmonotonic, uncertain, cause-effect systems. However, the standard reasoning and impact accumulation mechanisms developed for RBFCMs assume that the level of variation that a fuzzy set represents is directly linked with the shape of the fuzzy set. It poses a big restriction on how the corresponding fuzzy sets have to be constructed. In this paper, we propose a new reasoning and impact accumulation mechanisms which take into consideration standard semantics of fuzzy sets, where their uncertainty is measured by fuzziness. New type of complex fuzzy relationships and reasoning on them is introduced to model a joint impact of several causal nodes on one effect node. With these new mechanisms, RBFCMs become much more flexible, provide more means to capture complexity of real-world systems, and are less computational demanding than standard mechanisms. The advantages of the new RBFCMs are demonstrated using different examples and compared with standard mechanisms.

Fuzzy time series forecasting is an emergent research topic. In fuzzy time series model design, accuracy of forecast is dependent on two major issues: (1) Efficient data partitioning (2) Establishing Fuzzy logical relationships for Prediction. In this study, a new data partitioning technique based on rough-fuzzy approach has been proposed. Then, for the prediction purpose, a novel rule selection criterion has been adopted. In addition to that a mechanism is devised to deal with the situation when there is no matching rule present in the training data. Motivation for the present work is to overcome the drawback of existing high-order fuzzy time series models by avoiding the computations of complicated fuzzy logical relationship considering all previous states at a time and then explicit matching of those rules. The proposed work produces output of improved accuracy with selective rules only. In this high order model fuzzy logical relationships for each time lag are established separately and predictions are combined at the end to produce final result. Performance of the model is evaluated using TAIEX dataset. This idea also outperforms the some of the recent fuzzy time series forecasting models using the same dataset, in terms of forecast accuracy.

In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We investigate relations between ranks and prove that row rank, column rank and Schein rank of idempotent fuzzy matrices are equal. In particular, ranks and corresponding decompositions of fuzzy matrices representing fuzzy quasi-orders are studied in detail. We show that fuzzy matrix decomposition by ranks can be used in the state reduction of fuzzy automata. Moreover, we prove that using rank decomposition of fuzzy matrices improves results of any state reduction method based on merging indistinguishable states of fuzzy automata.

This study develops the approximate fuzzy Decision Making Trial and Evaluation Laboratory (AFDEMATEL) to analyze uncertain influential factors. The approximate fuzzy arithmetic operations under the weakest t-norm (Tω) arithmetic operations to evaluate sustainable supply chain management based on AFDEMATEL. The fuzzy DEMATEL is one of important decision-making method under uncertain environment. The fuzzy DEMATEL had to be developed for clearly display expert’s options with linguistic variables. The fuzzy operations usually adopt α-cut arithmetic in fuzzy DEMATEL. In this research, the fuzzy DEMATEL technology is substituted with the AFDEMATEL technology. In the sustainable supply chain management example, the AFDEMATEL is employed to find fuzzy cause and effect relationships among criteria. Particular note should be made of the following: [1] the fuzzy DEMATEL with α-cut arithmetic model cannot exactly handle fuzzy cause and effect relationships under uncertain environment, and the fuzziness accumulation phenomenon of the α-cut arithmetic may influence final fuzzy cause and effect relationships; and [2] the approximate fuzzy arithmetic operations gives a justifiable fuzziness spread to analyze fuzzy cause and effect relationships. In the case of selection of cans suppliers, the AFDEMATEL examines the influential factors. Proposed method provides justifiable fuzzy cause and effect relationships with approximate fuzzy arithmetic and can analyze across quadrants phenomenon under uncertain environment, since decision-makers usually want to accurately estimate uncertain influential factors.

This paper is concerned with network-based fuzzy H∞ control for a class of T-S fuzzy systems under an event-triggered communication scheme. Considering the fact that the network-induced delays are inevitable, the closed-loop system is modeled with asynchronous membership functions. To deal with the asynchronous membership functions, a new condition that limits the deviation bounds of membership functions, is introduced into the event-triggered scheme (ET scheme). Besides, by introducing a fuzzy event-triggered weighting matrix, a new ET scheme is proposed. Based on the proposed ET scheme, a novel criterion for the asymptotic stability and H∞ performance analysis is established in terms of linear matrix inequalities. Then some sufficient conditions and an algorithm to co-design the controller and the parameters of the ET scheme are presented. The effectiveness of the proposed method is illustrated through a mass-spring-damper system.

The idea of network-induced delay and packet disordering processing unit (DDPU) using fuzzy rules is presented for the first time to handle the network-induced delay and packet disordering synchronously in T–S fuzzy descriptor systems communicated by network. Particularly, there are two crucial variables in DDPU, which come from the delays of two packets arriving at actuator successively, to characterize the levels of those two network problems, respectively. Under the assumptions of bounded network-induced delay and bounded survival interval of a packet, these two variables are then served as premise variables of the DDPU module. Then, it is quite reasonable that four IF–THEN fuzzy rules are employed, provided that every fuzzy premise variable is described by two fuzzy sets. Furthermore, considering that the derivative of time-varying input delay is 1, a new Lyapunov–Krasovskii functional is designed for T–S fuzzy descriptor systems with input delay. A sufficient condition for the admissibility problem of T–S fuzzy descriptor systems is obtained in terms of strict linear matrix inequalities (LMIs). Finally, example results are presented to illustrate the usefulness and effectiveness of the proposed fuzzy DDPU method.

This paper proposes the use of mixed fuzzy clustering (MFC) algorithm to derive Takagi–Sugeno (T–S) fuzzy models (FMs). Mixed fuzzy clustering handles both time invariant and multivariate time variant features, allowing the user to control the weight of each component in the clustering process. Two model designs based on MFC are investigated. In the first, the antecedent fuzzy sets of the T–S model are obtained from the clusters obtained by the MFC algorithm. In the second, FMs based on fuzzy c-means (FCM) are constructed over the input space of the partition matrix generated by MFC. The proposed fuzzy modeling approaches are used in health care classification problems, where time series of unequal lengths are very common. MFC-based T–S FMs outperform FCM-based T–S FMs in four out of five datasets and k -nearest neighbors classifiers in five out of five datasets. Dynamic time warping performs better than the Euclidean distance in one dataset and similarly in the remaining. Given the different nature of time variant and invariant data, the choice of a clustering algorithm that treats data differently should be considered for model construction.

The paper presents an adaptation of pentagonal fuzzy number. Different type of pentagonal fuzzy number is formed. The arithmetic operation of a particular type of pentagonal fuzzy number is addressed here. The difference between two pentagonal valued functions is also addressed here. Demonstration of pentagonal fuzzy solutions of fuzzy equation is carried out with the said numbers. Additionally, an illustrative example is also taken with the useful graph and table for usefulness for attained to the proposed concept.

The paper is concerned with the $H_\infty$ and $l_2$ – $l_\infty$ filtering design problems for discrete-time nonlinear switched systems with quantized measurements using the Takagi–Sugeno (T–S) fuzzy model. The systems under consideration inherently combine features of the switched hybrid systems and the T–S fuzzy systems. The sector bound approach is employed to deal with quantization effects. Based on the fuzzy-basis-dependent Lyapunov function, sufficient conditions are established such that the filtering error system is stochastically stable and a prescribed noise attenuation level in an $H_\infty$ or $l_2$ – $l_\infty$ sense is achieved. Both numerical and practical examples are provided to show the feasibility and efficiency of the design schemes.

In this paper, we consider the state-feedback and static output feedback control problems of continuous singular interval-valued Takagi–Sugeno fuzzy systems with mismatched membership functions. First, based on the Lyapunov stability theory, a state-feedback control criterion is proposed to guarantee the closed-loop system to be admissible. Second, the result is extended to the static output feedback control problem, and a sufficient condition for synthesis is derived in terms of strict linear matrix inequalities, which eliminates the disadvantages in some existing results, such as same output matrices and the bilinear matrix inequality problem. To obtain less conservative results, the information of mismatched membership functions is employed. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.

This paper proposes a modified particle swarm optimization (PSO) algorithm that can be used to solve a variety of fuzzy nonlinear equations, i.e. fuzzy polynomials and exponential equations. Fuzzy nonlinear equations are reduced to a number of interval nonlinear equations using alpha cuts. These equations are then sequentially solved using the proposed methodology. Finally, the membership functions of the fuzzy solutions are constructed using the interval results at each alpha cut. Unlike existing methods, the proposed algorithm does not impose any restriction on the fuzzy variables in the problem. It is designed to work for equations containing both positive and negative fuzzy sets and even for the cases when the support of the fuzzy sets extends across 0, which is a particularly problematic case.

Network operators must perform many tasks to ensure smooth operation of the network, such as planning, monitoring, etc. Among those tasks, regular testing of network performance, network errors and troubleshooting is very important. Meaningful test results will allow the operators to evaluate network performanceof any shortcomings and to better plan for network upgrade. Due to the diverse and mainly unquantifiable nature of network testing results, there is a needs to develop a method for systematically and rigorously analysing these results.In this paper, we present STAM (System Test-result Analysis Method) which employs a bottom-up hierarchical processing approach using Fuzzy logic. STAM is capable of combining all test results into a quantitative description of the network performance in terms of network stability, the significance of various network erros, performance of each function blocks within the network. The validity of this method has been successfully demonstrated in assisting the testing of a VoIP system at the Research Instiute of Post and Telecoms in Vietnam.The paper is organized as follows. The first section gives an overview of fuzzy logic theory the concepts of which will be used in the development of STAM. The next section describes STAM. The last section, demonstrating STAM’s capability, presents a success story in which STAM is successfully applied.

This paper presents a novel direct solution approach for fully fuzzy transportation problems in which all of the model parameters as well as decision variables are considered as fuzzy numbers. In detail, a fuzzy decoding procedure based on constrained fuzzy arithmetic operations and a fuzzy ranking technique is first introduced for solution of the problem directly without any fuzzy to crisp transformation process. Then, this decoding procedure is embedded into a metaheuristic, namely priority-based PSO algorithm for generating new solution vectors and seeking for better fuzzy acceptable solutions. By making use of the constrained fuzzy arithmetic concept, the proposed approach is also able to handle the decision maker's attitude toward risk. In order to show validity and applicability of the proposed approach, numerical examples on both balanced and unbalanced fully fuzzy transportation cases are generated and solved. The computational results have shown that relatively more precise and information efficient solutions can be obtained from the proposed approach for risk-averse and partially risk-averse decision makers. Furthermore, the proposed approach is also able to produce fuzzy solutions for risk seekers with high degree of uncertainty similar to the other methods available in the literature.

Based on the optimization design technology and fuzzy uncertainty theory, this paper proposes a novel inverse analysis method for membership function identification in steady-state heat transfer problem with fuzzy modeling parameters. The system subjective uncertainties associated with expert opinions are quantified as fuzzy parameters, which can be converted into interval variables by level-cut strategy. By means of the errors between measured and calculated temperature data, the parameter identification process is executed as a nested-loop optimization model. To avoid the considerable computational cost caused by nested-loop, an interval vertex method is presented to replace the inner-loop for predicting the temperature response bounds. The eventual membership functions of input fuzzy parameters are constructed by using the fuzzy decomposition theorem. Comparing results with traditional Monte Carlo method, a numerical example about 3D air cooling system is provided to verify the feasibility of proposed method for fuzzy parameter identification in engineering.

This paper generalizes recent results on multi-instant observer design for discrete-time Takagi–Sugeno fuzzy systems through a valid ranking-based switching approach. The approach hereby develops a concentrated subdivision of spanning space composed of normalized fuzzy weighting functions and then substantially produces a new ranking-based switching mechanism. By taking advantage of this ranking-based switching mechanism, a class of new fuzzy multi-instant observers are achieved and more relaxed design conditions with respect to the recent work can be obtained for ensuring the asymptotically stability of the developed state estimation error system. Two illustrative examples are provided to validate the effectiveness of the result given in this study.

Trafficking in persons is the most heinous and condemnable organized crime of our time. The eradication of this crime must involve the full cooperation of all. In this paper, we provide a mathematical method to model such cooperation. We use a new concept in fuzzy graph theory, namely, that of incidence. We introduce fuzzy influence graph and characterize influence cutpairs in fuzzy influence graphs since their removal increases the number of connected components of a fuzzy network and thus weakens the potential flow in the network. Some of the properties of influence pairs and influence cutnodes are also studied.

In this paper, a notion of fuzzy copula function is introduced by defining joint distribution function of two fuzzy random variables. Using some lemmas, it is proven that the extended fuzzy copula satisfies many desired properties used for non-fuzzy data. The proposed fuzzy copula is then applied to construct some common non-parametric measures of association between two fuzzy random variables. The proposed methods is then illustrated via some numerical examples.

This paper presents an efficient approach for the use of recursive least square (RLS) learning algorithm in Takagi-Sugeno-Kang neural fuzzy systems. In the use of RLS, reduced covariance matrix, of which the off-diagonal blocks defining the correlation between rules are set to zeros, may be employed to reduce computational burden. However, as reported in the literature, the performance of such an approach is slightly worse than that of using the full covariance matrix. In this paper, we proposed a so-called enhanced local learning concept in which a threshold is considered to stop learning for those less fired rules. It can be found from our experiments that the proposed approach can have better performances than that of using the full covariance matrix. Enhanced local learning method can be more active on the structure learning phase. Thus, the method not only can stop the update for insufficiently fired rules to reduce disturbances in self-constructing neural fuzzy inference network but also raises the learning speed on structure learning phase by using a large backpropagation learning constant.

In this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets. Solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise. We show that solutions to each of these systems form a complete lattice, and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations. We also demonstrate the application of solutions to these systems in data reduction.

This paper proposes new algorithms based on the fuzzy c-regressing model algorithm for Takagi-Sugeno (T-S) fuzzy modeling of the complex nonlinear systems. A fuzzy c-regression state model (FCRSM) algorithm is a T-S fuzzy model in which the functional antecedent and the state-space-model-type consequent are considered with the available input-output data. The antecedent and consequent forms of the proposed FCRSM consists mainly of two advantages: one is that the FCRSM has low computation load due to only one input variable is considered in the antecedent part; another is that the unknown system can be modeled to not only the polynomial form but also the state-space form. Moreover, the FCRSM can be extended to FCRSM-ND and FCRSM-Free algorithms. An algorithm FCRSM-ND is presented to find the T-S fuzzy state-space model of the nonlinear system when the input-output data cannot be precollected and an assumed effective controller is available. In the practical applications, the mathematical model of controller may be hard to be obtained. In this case, an online tuning algorithm, FCRSM-FREE, is designed such that the parameters of a T-S fuzzy controller and the T-S fuzzy state model of an unknown system can be online tuned simultaneously. Four numerical simulations are given to demonstrate the effectiveness of the proposed approach.

The mathematical models reported in the literature so far have been found using Center of Sums (CoS) defuzzification method only. It appears that no one has found models using Center of Area (CoA) or Center of Gravity (CoG) defuzzification method. Although there have been some works reported to deal with modeling of fuzzy controllers via Centroid method, all of them have in fact used CoS method only. In this paper, for the first time mathematical models of the simplest Mamdani type fuzzy Proportional Integral (PI)/Proportional Derivative (PD) controllers via CoG defuzzification are presented. L-type and Γ-type membership functions over different Universes of Discourse (UoDs) are considered for the input variables. L-type, Π-type and Γ-type membership functions are considered for the output variable. Three linear fuzzy control rules relating all four input fuzzy sets to three output fuzzy sets are chosen. Two triangular norms namely Algebraic Product (AP) and Minimum (Min), Maximum (Max) triangular co-norm, and two inference methods, Larsen Product (LP) and Mamdani Minimum (MM), are used. Properties of the models are studied. Stability analysis of closed-loop systems containing one of these controller models in the loop is done using the Small Gain theorem. Since digital controllers are implemented using digital processors, computational and memory requirements of these fuzzy controllers and conventional (nonfuzzy) controllers are compared. A rough estimate of the computational time taken by the digital computer while implementing any of these discrete-time fuzzy controllers is given. Two nonlinear plants are considered to show the superiority of the simplest fuzzy controller obtained using CoA or CoG defuzzification method over the simplest fuzzy controller obtained using CoS method and reported recently. Real-time implementation of one of the developed controller models is done on coupled tank experimental setup to show the feasibility of the developed model.

Probabilistic implications and probabilistic S-implications, introduced recently by Grzegorzewski, have attracted attention of researchers engaged in fuzzy implications. Since these two families are based on copulas, they form a kind of bridge between probability theory and fuzzy logic. In this paper we generalize the aforementioned two classes and propose a new approach for constructing fuzzy implications which combines a given a priori fuzzy implication and a semicopula.

Basaran [Calculating fuzzy inverse matrix using fuzzy linear equation system, Applied Soft Computing, 12 (2012), 1810–1813] proposed a method for finding the inverse of a fuzzy matrix by assuming all the elements of the fuzzy inverse matrix as non-negative fuzzy numbers, while some of the elements of fuzzy matrix inverse may also be negative fuzzy numbers. Keeping the same in mind, Mosleh and Otadi [A discussion on “Calculating fuzzy inverse matrix using fuzzy linear equation system”, Applied Soft Computing, 28 (2015), 511–513] assumed (i, j) element x ˜ ij = ( x ij , α ij , β ij ) of the fuzzy inverse matrix as a non-negative fuzzy number if the value of xij obtained by Basaran's approach, is a non-negative real number and a negative fuzzy number if the value of xij is negative real number. In this paper, it is shown that the fuzzy multiplicative inverse of a fuzzy matrix, obtained by considering this assumption, is also not an exact fuzzy multiplicative inverse. Furthermore, the required modifications, in Mosleh and Otadi's approach, to obtain the exact multiplicative inverse of a fuzzy matrix are suggested.

This paper addresses a fuzzy dynamic output feedback H∞ control design problem for continuous-time nonlinear systems via T-S fuzzy model. The stability of the fuzzy closed-loop system which is formed by a T-S fuzzy model and a fuzzy dynamic output feedback H∞ controller connected in a closed loop is investigated with Lyapunov stability theory. The proposed fuzzy controller does not share the same membership functions and number of rules with T-S fuzzy systems, which can enhance design flexibility. A line-integral fuzzy Lyapunov function is utilized to derive the stability conditions in the form of linear matrix inequalities (LMIs). The boundary information of membership functions is considered in the stability analysis to reduce the conservativeness of the imperfect premise matching design technique. Two simulation examples are provided to demonstrate the effectiveness of the proposed approach.

In this presentation, we continue and expand our previous work on fuzzy subsethood and entropy measures. After a minor alteration to the axioms of fuzzy inclusion, we are able to produce new possible fuzzy inclusion and entropy indicators. We believe that these measures could be used in applications which require or properly exploit fuzzy inclusion and entropy measurements (e.g., image processing, feature selection, fuzzy controllers, similarity measures). Possibly they could offer us more information or lead to alternative ways of solving specific problems of these areas of research. We back up this by introducing a general method of global image thresholding which effectively uses some of these measures. Unlike other common techniques of global image thresholding, this method does not depend on histogram concativity analysis nor does it rely on optimizing some statistical measure (e.g. variance minimization) of the gray-level information. It only needs specific attributes of the image which are measured by some of our fuzzy inclusion and entropy indicators. It’s more of an adaptable process rather than a “strict” procedure and we believe that it can be easily adjusted to meet the needs of different domains or fields of research.

To analyse the component of fuzzy output entropy, a decomposition method of fuzzy output entropy is first presented. After the decomposition of fuzzy output entropy, the total fuzzy output entropy ...

Model checking of linear-time properties based on possibility measures was studied in previous work (Y. Li and L. Li, Model checking of linear-time properties based on possibility measure, IEEE Transactions on Fuzzy Systems, 21(5)(2013), 842-854). However, the linear-time properties considered in the previous work was classical and qualitative, possibility information of the systems was not considered at all. We shall study quantitative model checking of fuzzy linear-time properties based on generalized possibility measures in the paper. Both the model of the system, as well as the properties the system needs to adhere to, are described using possibility information to identify the uncertainty in the model/properties. The systems are modeled by {\sl generalized possibilistic Kripke structures} (GPKS, in short), and the properties are described by fuzzy linear-time properties. Concretely, fuzzy linear-time properties about reachability, always reachability, constrain reachability, repeated reachability and persitence in GPKSs are introduced and studied. Fuzzy regular safety properties and fuzzy $\omega-$regular properties in GPKSs are introduced, the verification of fuzzy regular safety properties and fuzzy $\omega-$regular properties using fuzzy finite automata are thoroughly studied. It has been shown that the verification of fuzzy regular safety properties and fuzzy $\omega-$regular properties in a finite GPKS can be transformed into the verification of (always) reachability properties and repeated reachability (persistence) properties in the product GPKS introduced in this paper. Several examples are given to illustrate the methods presented in the paper., Comment: arXiv admin note: text overlap with arXiv:1409.6466

The aim of this work, is to evaluate the value of a fuzzy integral by applying the Newton-Cotes integration rules via a reliable scheme. In order to perform the numerical examples, the CADNA (Control of Accuracy and Debugging for Numerical Applications) library and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method are applied based on the stochastic arithmetic. By using this method, the optimal number of points in the fuzzy numerical integration rules and the optimal approximate solution are obtained. Also, the accuracy of the fuzzy quadrature rules are discussed. An algorithm is given to illustrate the implementation of the method. In this case, the termination criterion is considered as the Hausdorff distance between two sequential results to be an informatical zero. Two sample fuzzy integrals are evaluated based on the proposed algorithm to show the importance and advantage of using the stochastic arithmetic in place of the floating-point arithmetic.

The problem of admissibility analysis and control synthesis for Takagi–Sugeno fuzzy descriptor systems is investigated. First, based on Nonquadratic fuzzy Lyapunov function and fully using the information of fuzzy membership functions, a new relaxed sufficient condition ensuring a fuzzy descriptor system to be admissible (regular, impulse-free, and stable) is proposed, in which it is not necessary to require every fuzzy subsystem to be stable. Second, the other sufficient condition for the admissibility is obtained without the information of time derivatives of fuzzy membership functions. Following the analysis, both parallel and nonparallel distributed compensation controllers are designed, linear matrix inequalities conditions are given to construct the controllers. Finally, some examples are provided to illustrate the main results in this paper less conservative than some earlier related results.

Deep learning (DL) is an emerging and powerful paradigm that allows large-scale task-driven feature learning from big data. However, typical DL is a fully deterministic model that sheds no light on data uncertainty reductions. In this paper, we show how to introduce the concepts of fuzzy learning into DL to overcome the shortcomings of fixed representation. The bulk of the proposed fuzzy system is a hierarchical deep neural network that derives information from both fuzzy and neural representations. Then, the knowledge learnt from these two respective views are fused altogether forming the final data representation to be classified. The effectiveness of the model is verified on three practical tasks of image categorization, high-frequency financial data prediction and brain MRI segmentation that all contain high level of uncertainties in the raw data. The fuzzy dDL paradigm greatly outperforms other nonfuzzy and shallow learning approaches on these tasks.

In this paper, generalized fuzzy soft rough matrices and their operations which are more essential to make theoretical studies in the fuzzy soft rough sets are defined. Further, based on the analysis of generalized fuzzy soft rough matrices, several algebraic properties are established. Using the notions of generalized fuzzy soft rough matrices, a novel method for choosing an optimum choice in a multi-criteria decision-making problem is developed. Moreover, the proposed method is compared with the well-known existing methods of fuzzy soft matrices, and the effectiveness of the proposed method has been demonstrated through numerical example.

In this paper, we define the intuitionistic circular fuzzy matrix and introduce the necessary and sufficient conditions for an intuitionistic fuzzy matrix to be circular. Also, we study some properties of intuitionistic circular fuzzy matrices