Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology

The aim of this paper is to introduce and study new classes of FP-multifunctions, called FP-lower (upper) irresolute and semi-irresolute, FP-presemi-open and FP-weak semi-open multifunctions, where the domain of these functions is a classical bitopological space with their values as arbitrary L-fuzzy sets in L-fuzzy bitopological space in view of Šostak’s sense. Also, we prove several characterizing theorems for these functions from different angles and we establish various properties of these multifunctions along with their mutual relationships.

This paper constructs ℒ-covering fuzzy variable precision rough set and mainly studies some properties of type I and type II ℒ-covering fuzzy lower and upper approximation operators of ℒ-fuzzy sets on X. Topology properties of type I and type II ℒ-covering fuzzy lower and upper approximation operators are also discussed through a fuzzy interior operator and fuzzy closure operator. Moreover, we investigate the dual properties and compare between ℒ-covering fuzzy variable precision rough approximation operators.

The theory of abstract convexity exists in many mathematical branches such as algebra, topology and order. The fuzzification is one of important directions towards the discussion of abstract convexity. The theory of (L, M)-convex spaces is one very general fuzzy convexity since it includes many other fuzzy convexities as its special case. In this paper, we shall introduce a subcategory of (L, M)-convex spaces and study its property. This subcategory is called enriched (L, M)-convex spaces and denoted by ELMCS. The main results are: (1) ELMCS is a topological category; (2) ELMCS is a coreflective subcategory of the category of (L, M)-convex spaces; (3) the category of M-fuzzifying convex spaces (denoted by MCS) can be embedded in the category ELMCS as a reflective subcategory.

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219434.

Nature is a huge source of inspiration for solving difficult and complex problems in science. Nature-based problem solvers always find an optimal solution (whether local or global) to any given problem; so they are sometimes considered as black-box problem solvers. Meta-heuristic algorithms that are inspired by nature (through imitating the nature) have opened a new approach to solving optimization problems. In the past decades, numerous research efforts have been focused in this specific area. In this paper, an optimization algorithm inspired by the nature has been introduced which is modeled from the behavior of the chicks of a type of bird called see-see partridge. We name the algorithm SSPCO. SSPCO optimization algorithm is an algorithm similar to the algorithm of particle swarm optimization (PSO) that the motion equation and the variables’ velocity are modeled from the behavior of this type of bird. Simulation of the algorithm was done with MATLAB software and the results of the 6 primary known algorithms and 8 advanced ones were compared with 14 unconstrained continuous functions. The results indicate that the proposed algorithm is an efficient algorithm comparing the others.

Intuitionistic fuzzy set (IFS) is an extension of fuzzy set. The basic element of an IFS is the ordered pair called intuitionistic fuzzy number (IFN). So far, some basic operational laws of IFNs are defined, but not including the logarithmic operation. In this paper, a logarithmic operational law about IFNs is defined, in which the base λ is a real number in (0, 1). Then some properties of the operational law are investigated, and four intuitionistic fuzzy information aggregation methods are proposed. Finally, the influences of logarithmic operation for IFNs and the selection of the logarithmic base λ in practice are discussed, and an example is illustrated to show the applications of the operational law and the aggregation methods.

In this paper, a Takagi-Sugeno (T-S) fuzzy hyperbolic model is proposed for the fuzzy control of a class of nonlinear systems. The consequence of the proposed model is a hyperbolic tangent dynamic model, and it is employed to represent the nonlinear system. By constructing a new Lyapunov function, the stability conditions of the open-loop T-S fuzzy hyperbolic system are derived via linear matrix inequalities (LMIs). Then, the parallel distributed compensation (PDC) method is used to design a fuzzy hyperbolic controller, and the asymptotic stability conditions of the closed-loop system are formulated via LMIs. The main advantage of the control based on T-S fuzzy hyperbolic model is that it can achieve small control amplitude via “soft” constraint control approach. Finally, the effectiveness and advantage of the proposed schemes are illustrated by a mathematical constructive example and the Van de Vusse example.

One of the most important methods in reliability analysis is Fault Tree Analysis that over time it has been extended into the more versatile method of Dynamic Fault Tree (DFT) Analysis. In most cases, exact evaluation of system reliability using fault tree due to component limited data especially owning to failure rates, is difficult. In this paper, the Fuzzy Time-To-Failure (FTTF) model based on Fuzzy Lower and Upper (L-U) bounds is developed to evaluate the reliability of system and solve aforementioned problems. This process completed by proposed Fuzzy Monte Carlo Simulation (FMCS) throughout the preferred operational time and uses the actual types of fuzzy failure distribution. FMCS is done based on Lower-Upper bounds for each event failure rates. Using fuzzy arithmetic, events FTTF are generated, and then, the Top Event failure curve and the reliability profile of the system are evaluated. The results show that the proposed method not only is feasible and powerful but can also accurate more than the other probabilistic and Possibilistic techniques. Finally, this model is implemented in an Emergency Detection System (EDS) which is a useful system in aerospace and space applications.

Among the large amount of genes presented in microarray gene expression data, only a small fraction of them is effective for performing a certain diagnostic test. It is for this reason that reducing the dimensionality of gene expression data is imperative. An improved Self-organizing map method based on neighborhood mutual information correlation measure is proposed, and then combines with Particle swarm optimization method to construct an efficient gene selection algorithm, denoted by ICMSOM-PSO. Experimental results show that the proposed method can reduce the dimensionality of the dataset, and confirm the most informative gene subset and improve classification accuracy.

Based on the extended generalized Hukuhara difference, we introduce and study a new Shapley type of value for cooperative games with fuzzy payoffs. We first propose and characterize a new interval Shapley value for interval-valued cooperative games. Those results are then extended to cooperative games with fuzzy payoffs, and the generalized Shapley function is introduced. We characterize the generalized Shapley function using the properties of generalized efficiency, generalized dummy player, generalized symmetry, and generalized additivity. At the same time, the necessary and sufficient condition for the existence of the generalized Shapley function is given. This study also shows that the generalized Shapley function is a generalization of the Hukuhara-Shapley function defined by Yu and Zhang [22]. Meanwhile, an arbitrary cooperative game with payoffs of center triangular fuzzy numbers has a unique generalized Shapley function.

The purpose of this study is to develop a hesitant fuzzy linguistic TOPSIS (The technique for order preference by similarity to ideal solution) method with a possibility-based comparison approach for addressing multi-criteria decision-making (MCDM) problems within the environment of hesitant fuzzy linguistic term sets (HFLTSs). This paper firstly analyses the existing comparison methods for HFLTSs and develops a new possibility degree formula which can address the issues in the previous ones. Then, based on the possibilities of the HFLTS binary relations, this paper defines the possibility-based outranking index to determine hesitant fuzzy linguistic positive ideal and negative ideal solutions. Subsequently, this paper introduces the concept of possibility-based comparison indices to establish a possibility-based closeness coefficient of each alternative relative to the ideal solutions. Based on a possibility-based comparison approach with the ideal solutions, this paper develops a hesitant fuzzy linguistic TOPSIS method for handling MCDM problems in which both the evaluative ratings of alternatives and the importance weights of criteria are expressed by HFLTSs. Finally, a numerical example is furnished to verify the feasibility and practicality of the proposed method and a comparative analysis with the existing methods is provided to illustrate the effectiveness and advantages of the proposed method.

In this paper, we introduce the concept of triangular cubic fuzzy numbers. We discuss some basic operational laws of triangular cubic fuzzy numbers and then develop triangular cubic fuzzy weighted average (TCFWA) operator. We also define crisp weighted possibility means of TCFNs and hamming distance between TCFNs. Furthermore, we extend the classical VIKOR method to solve the MCDM method based on triangular cubic fuzzy numbers. The new ranking method for TCFNs is used to rank the alternatives. Finally, an illustrative example is given to verify and demonstrate the practicality and effectiveness of the proposed method.

In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, the notion of doubleframed soft set is introduced and applications in LA-semigroups are discussed. Double-framed soft set is a generalization of the theory of union and intersectional soft sets. In this article, we introduce the concept of double-framed soft set in LA-semigroups and study double-framed soft LA-semigroups (resp., left, right or two-sided) ideals (briefly, DFS left (right), DFS two-sided, and DFS bi-) ideals of an LA-semigroup S over an initial universe set U. The respective examples of these notions are investigated and the related properties are discussed. The basic properties of these notions are studied and characterizations of these concepts are discussed in detail.

Although electroencephalography (EEG) brain-computer interface (BCI) has been quite successful, multi-command control is still one of the key issues for external applications. Multimodal BCI represents the direction of dealing with this problem. In our study, five healthy subjects performed the experiment cooperatively. EEG and electromyography (EMG) were recorded synchronously. For individual EEG, after Laplacian filtering, the C3 and C4 channels were determined. Then, the EEG was decomposed into the third layer by wavelet packet transform (WPT), and the average, sub-band energy and mean square deviation were calculate at particular nodes. Finally, these features were fed into support vector machine (SVM) either singly or in combination, and the EEG classification accuracy was obtained. For individual EMG, the mean absolute value (MAV) and root mean square (RMS) were calculated. Then, probabilistic neural network (PNN) was employed, and the EMG classification accuracy was also obtained. Different mental and gesture tasks were combined to represent multi-class and these commands were ranked depending on their performance. The results showed that the subjects were able to obtain multi-class with satisfactory performance by multimodal BCI. The proposed interface could support multi-command control for external applications.

In this paper, the notion of subuniverses in the theory of universal algebras is generalized to L-fuzzy setting, which is called L-subuniverses. Some characterizations and representations of L-subuniverses are given. What is more, the relationship between L-subuniverses and L-convexities is discussed when L is a complete Heyting algebra. It is shown that there exists a one-to-one correspondence between L-subuniverses with the smallest L-subset and induced L-convexities.

Extreme learning machine (ELM) has demonstrated great potential in machine learning and data mining. Smoothing strategy is an important technology for continuous optimizations. In this work, we apply a smoothing technique to replace the hinge loss function by an accurate smooth approximation. This will allow us to solve ELM as an unconstrained minimization problem directly. We term this reformulated problem as smooth ELM (SELM). A Newton-Armijo algorithm is used to solve the proposed SELM, and the resulting algorithm converges globally and quadratically. The proposed SELM with fast running speed has less decision variables and can better deal with nonlinear problems than the existing smooth support vector machine. Numerical experiments on various types of datasets including two-class datasets and multi-class datasets demonstrate that the speed of SELM is much faster than that of the existing ELM models. And compared with other popular algorithms of support vector machine and ELM, the proposed SELM achieves better or similar generalization. These demonstrate the effectiveness and fast speed of the algorithm.

In this paper, with respect to multiple criteria group decision making (MCGDM) problems in which criteria values are expressed by Pythagorean fuzzy uncertain linguistic variables (PFULVs), we propose an extended TODIM method. Firstly, we define the Pythagorean fuzzy uncertain linguistic set, and propose the operational laws, Hamming distance, score function and accuracy function of PFULVs. Then an extended TODIM method is presented to solve the MCGDM problems under the Pythagorean fuzzy uncertain linguistic environment, and a numerical example with the Pythagorean fuzzy uncertain linguistic information is given to show the effectiveness of the proposed method. Further, we analyze the influence of the different parameter on the MCGDM, and test the practicality of the proposed method.

In recent years, traditional machine learning algorithms have been gradually replaced by deep learning algorithms. In the field of computer vision, convolutional neural network is considered to be the most successful deep learning model. Based on convolutional neural network, the accuracy of image classification has been greatly improved. In this paper, a method for semantic image segmentation based on convolutional neural network is proposed. Firstly, the disparity map is introduced to improve the segmentation accuracy. To obtain the disparity map with more continuous disparity values, an image smoothing method is used to optimize the disparity map. Then, based on the AlexNet network, a fully convolutional network architecture is proposed for semantic image segmentation. The unpooling operation is employed to restore the extracted features to their original sizes. The experimental results demonstrate that the network can achieve high pixel-wise prediction accuracy and that using RGB-D image as the input of the network can reduce the noisy segmentation outputs.

The long payback period for medical care products prevents investors from immediately recognizing risks arising throughout the entire period. To avoid risk loss, delay decision should be introduced to investment decision. In this study, we illustrate investment decision from the view of three-way group decisions. Linguistic scale is widely used during assessment, but the randomness and fuzziness of linguistic information are ignored. To cover these defects, this study introduces cloud to three-way group decisions and further extends cloud to medical care product investment decision in which the weights of experts are unknown. In our proposed model, the loss functions and conditional probability described by linguistic values in decision theoretic rough sets are converted to clouds, which can handle the fuzziness and randomness of linguistic information. The corresponding three-way decision rules are also derived from a cloud perspective. In addition, we define a new derivation degree based on the score function of cloud to determine the weights of experts in three-way group decisions. To validate the feasibility of our model, comparisons with the existing model are presented.

The aim of this paper is to propose some novel multiple attribute group decision making (MAGDM) methods to deal with MAGDM problems in which the attributes are interactive in the form of interval-valued hesitant uncertain linguistic numbers (IVHULNs). Firstly, some new aggregation operators for IVHULNs based on Bonferroni mean (BM) are proposed, which are the interval-valued hesitant uncertain linguistic BM (IVHULBM) operator, the normalized weighted IVHULBM (NWIVHULBM) operator, the interval-valued hesitant uncertain linguistic geometric BM (IVHULGBM) operator and the normalized weighted IVHULGBM (NWIVHULGBM) operator. The advantages of the proposed operators are that it cannot only effectively aggregate IVHULNs, but it can also consider the interactive characteristics among attributes. At the same time, some special cases of these operators are discussed. Then, this paper demonstrates that these presented operators are able to meet four desirable properties, which are reducibility, idempotency, monotonicity, and boundedness. Moreover, to solve MAGDM problems, two approaches on the basis of the NWIVHULBM and NWIVHULGBM operators are put forward. Finally, the proposed methods are applied to a decision making problem regarding online service quality evaluation. It provides us with a useful way for MAGDM with IVHULNs.

This paper deal with certain algebraic systems called F-polygroups. We introduce and study the concept of fuzzy topological F-polygroups and prove some related properties. In particular, the fuzzy topological isomorphism theorems of fuzzy topological F-polygroups are proved.

Feature selection is one of the key problems in machine learning and data mining. It involves identifying a subset of the most useful features that produces compatible results as the original entire set of features. It can reduce the dimensionality of original data, speed up the learning process and build comprehensible learning models with good generalization performance. Nowadays, ensemble idea has been used to improve the performance of feature selection by integrating multiple base feature selection models into an ensemble one. In this paper, in order to improve the efficiency of feature selection in dealing with large scale, high dimension and imbalanced problems, a Min-Max Ensemble Feature Selection (M2-EFS) is proposed, which is based on balanced data partition and min-max ensemble strategy. The experimental results demonstrate that the M2-EFS can obtain higher performance than other classical ensemble methods in most cases, especially for large scale, high dimension and imbalanced data.

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219324.

In a formal context, the lower and upper approximations of an arbitrary set of objects are constructed by object-oriented concepts, attribute-oriented concepts, approximable concepts and weak approximable concepts, respectively. We first define the concept of approximations based on lattice-theoretic operators, and the properties of them are discussed. In order to overcome the two shortcomings in the former approximations, we present the concept of approximations based on set-theoretic operators.The study can help further understanding of data analysis using rough set theory and formal concept analysis.

Multiple attribute group decision making (MAGDM) is a very active research field in management sciences. Many practical MAGDM problems are often characterized by ambiguity and uncertainty. The aim of this paper is to develop an integrated MAGDM method with unknown weight information under trapezoidal interval type-2 fuzzy environment based on the grey relational projection (GRP) method. Firstly, to determine the comprehensive weights of attributes, a novel method is proposed by combining the analytic hierarchy process (AHP) technique under trapezoidal interval type-2 fuzzy environment and inter-attribute coefficient method. Secondly, the traditional GRP method is extended to solve MAGDM problems under trapezoidal interval type-2 fuzzy environment, i.e., the optimial alternative should have the largest grey relational projection on the trapezoidal interval type-2 fuzzy positive ideal solution (TIT2-FPIS) and smallest grey relational projection on the trapezoidal interval type-2 fuzzy negative ideal solution (TIT2-FNIS) simultaneously. Finally, an emergency medical department selection problem is taken as an illustrative example to demonstrate the calculation process of the proposed method.

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219434.

An important problem in fuzzy theory is how to determine if two elements are similar or not. There are different definitions to measure how similar or close are elements. One of the most important concepts in this context is that one of similarity. This paper aims at studying fuzzy similarities defined by fuzzy implications, logical equivalences expressed by fuzzy bi-implication operators and aggregation operators. We propose a definition of proximity between two fuzzy finite sets by using the previous operators and we study when this type of operators satisfy the formal definition of similarity.

In this article, we have proposed a new approach on the stability of a fuzzy dynamical system (FDS). Initially, we tried interval mathematics on FDS to check the stability. But when the interval mathematics fails, we have applied the quasi-level-wise system using the operators “e” and “g” and it appears that, this is the best technique to convert the unstable FDS to stable systems. Finally, the said technique is applied on different kinds of FDS and respective numerical results are presented in tabular and graphical form.

This paper proposes a novel evolutionary clustering algorithm and prepares eligible initial centroids for K-Means algorithm by global search approach of efficient hybrid knowledge of swarm intelligence algorithms. Clustering performs data grouping into subsets with common features, So that useful information can be retrieved from them. Swarm intelligence algorithms with evolutionary optimization approach have been very efficient performance in these matters. In this paper, a novel hybrid algorithm called IFAPSO has been proposed which uses swarm hybrid knowledge of Firefly and PSO intelligence algorithms to make an effective data clustering. Performance improvement of PSO and Firefly swarm algorithms and resolve the deficiency of each of them is applied and hybrid of them has been used to benefit from both algorithms into clustering problem effectively. Also this hybrid swarm algorithm overcomes the initial centers’ selection sensitivity and the limitation of local optima in K-Means. We used five benchmarks with several samples and features to evaluate our work. The comparison between proposed methods with the traditional algorithms and previous hybrid methods, suggest that there is more compactness of the resulting clusters and promising accuracy is achieved.

In this paper, we study semi-analytical methods entitled fuzzy fractional differential transform method to solve fuzzy impulsive fractional differential equations using Caupto fractional derivative. At the end first of all fuzzy differential transform method in fractional case is defined and its properties are considered completely. Then existence and uniquness theorem for the solution are proved and convergence of the proposed method is considered in details. Some examples indicate that this method can be easily applied to many linear and nonlinear problems.

Let S be a multiplicative subset of a commutative ring R. We considered a R × S as a universal and introduced the notion of lower[upper] approximations in a R × S . Also, we gave some properties of the lower[upper] approximations in a R × S . We obtained some properties of rough ideals

In this paper, as an extension of variable precision (θ, σ)-fuzzy rough sets, we introduce the notion of the generalized granular variable precision ( R _ β , R ¯ β ) -fuzzy rough sets determined by fuzzy granules with R ∈ L^X×Y on a residuated and co-residuated lattice with an involution N. We investigate the properties of R _ β and R ¯ β with e _{L ^Y} and d _{L ^Y} are (L, ⊙)-fuzzy preorder and (L, ⊕)-fuzzy co-preorder, respectively. Moreover, we investigate Alexander topologies and Alexander co-topologies induced by R _ ⊤ and R ¯ ⊥ , respectively. For an L-fuzzy relation R ∈ L^X×X and R _ β , R ¯ β for each β ∈ L, we obtain Alexandrov L-fuzzy pretopologies and Alexandrov L-fuzzy precotopologies on X, respectively. We give their examples.

In this paper, a multiple-input single-output (MISO) fuzzy rules system is decomposed into its equivalent collection of single-input single-output (SISO) fuzzy rules systems. First, the constructive and destructive linguistic models are reviewed. Under certain conditions, the final consequence of the constructive linguistic model working on a MISO fuzzy rules system is equivalent to the final consequence of the constructive linguistic model working on a collection of SISO fuzzy rules systems. Meanwhile, under another conditions, the final consequence of the destructive linguistic model working on a MISO fuzzy rules system is equivalent to the final consequence of the destructive linguistic model working on a collection of the SISO fuzzy rules systems. The decomposition of the MISO fuzzy rules system with fuzzy singleton consequent parts is also investigated. This paper shows the MISO fuzzy rules system requires many more rules than the SISO fuzzy rules system. Through the decomposing process, we can realize the inference result with fewer rules.

Crow search algorithm (CSA) is a recently proposed metaheuristic optimizer inspired by the intelligent behaviour of crows with attributes like simplicity and ease of implementation. CSA is claimed superior and more effective in optimizing a variety of constrained engineering design problems in comparison to other state-of-art algorithms. In the present work, CSA is applied to high dimensional optimization problems and it is found that CSA suffers from premature convergence which leads to lower precision and less accuracy in optimization or sometimes failure. Therefore an improvement in CSA (ICSA) is suggested to solve high-dimensional global optimization problems efficiently. The balance between exploitation and exploration capabilities of CSA is improved by introducing experience factor, adaptive adjustment operator and Lévy flight distribution in position updating mechanism of crows. Lévy flight distribution promotes continuous exploration of search space and prevents premature convergence by escaping from local optimum at any stage. The performance of ICSA is validated on high-dimensional nonlinear scalable benchmark test functions. The proposed improvement in CSA makes it highly competitive and less sensitive to function dimensions. ICSA is also found superior to other well established optimizers.

One of the problems in fuzzy group theory is concerned with classifying the fuzzy (normal) subgroups of a finite group. In this paper, we classify the fuzzy (normal) subgroups of D 2 p × ℤ q (p, q primes) and all finite groups of order n ≤ 20, by using the natural equivalence relation. In this case, the corresponding equivalence classes of fuzzy (normal) subgroups of a group G are closely connected to the chains of (normal) subgroups in G. In fact, for determining the number of these classes, we calculate the number of all chains of (normal) subgroups of G that terminate in G.

Firstly, this paper is intended to point out the lack of mathematical rigor in logic operation definition of hesitant fuzzy sets and to revise these operations. Then, new logical operations and partially ordered relations are introduced on a linearly ordered set, whose properties are studied accordingly. Finally, the concepts of hesitant set and hesitant relation are given together with their operations and corresponding properties. It can be found that some existing concepts are special examples of hesitant sets, such as shadowed sets, hesitant fuzzy sets and interval-valued hesitant fuzzy sets etc.

In this paper, a new class of games with fuzzy coalitions and fuzzy payoff value is proposed. This class of fuzzy games is based on the generalized integral form, which contains several kinds of other fuzzy games, such as fuzzy cooperative game with fuzzy payoff value, the multilinear extension game introduced by Owen, the game with proportional value proposed by Butnariu and the game with Choquet integral form given by Tsurumi et al. Also, the proposed fuzzy game is also further extension of the fuzzy game in Choquet integral form proposed by Yu et al., which is also is a kind of fuzzy games in the condition that coalition and fuzzy payoff value are both fuzzy information. The fuzzy Shapley value for this kind of fuzzy games is represented by Shapley value of corresponding fuzzy cooperative game with fuzzy payoff value. Based on Hukuhara-difference, we give the explicit Shapley value for the proposed fuzzy game. It has been seen that most of properties hold well in the proposed fuzzy game, which are processed by cooperation game with fuzzy payoff value, and fuzzy coalition game, respectively. Finally, a practical application of the proposed model is also provided.

In this paper, we will define intuitionistic fuzzy matroid and study their properties. First, we analyse the two approaches to fuzzification of matroids and decide to use an indirect approach. After some preliminaries, we define an intuitionistic fuzzy matroid as a prefect intuitionistic fuzzy pre-matroid. Second, we study two pair of classical operators imposed on intuitionistic fuzzy matroids. Finally, we induce a G-V fuzzy matroid from a given intuitionistic fuzzy matroid and study the relationship of H fuzzy matroids and intuitionistic fuzzy matroids. Besides, we introduce an approach to construction of G-V fuzzy matroids and point out the connection of fuzzy matroids and fuzzy rough set models.

Pythagorean fuzzy sets (PFSs), as an extension of intuitionistic fuzzy sets (IFSs) to deal with uncertainty, have attracted much attention since its introduction, in both theory and application aspects. The present work aims at investigating new distance measures in the PFSs and then employing them into multiple criteria decision-making application. To begin with, generalized Pythagorean fuzzy weighted averaging distance operator (GPFWAD) and generalized Pythagorean fuzzy ordered weighted averaging distance (GPFOWAD) measure are firstly introduced in the PFSs. Afterwards, probabilistic generalized Pythagorean fuzzy weighted averaging distance (P-GPFWAD) operator, probabilistic generalized Pythagorean fuzzy order weighted averaging distance (P-GPFOWAD) operator are proposed which are new distance measures and are able to integrate the (ordered) weighted averaging operator, probabilistic weight and individual distance of two Pythagorean fuzzy numbers (PFNs) in the same formulation. These generalized weighted averaging distance measures are very suitable to deal with the situation where the input data are represented in Pythagorean fuzzy numbers (PFNs). Then we present a kind of multiple criteria decision-making method with Pythagorean fuzzy information based on the developed distance measures. Finally, a numerical example is provided to illustrate the practicality and feasibility of the developed method.

The belief rules are stored out of order in the extended belief rule base (EBRB), which will weaken its reasoning performance in that all rules are visited when calculating each rule’s activation weight. This paper focuses on reducing the number of rules which are visited in the calculation of each rule’s activation weight. A new rule activation method based on VP-tree and MVP-tree is proposed to build index structure to store rules. The proposed rule activation method is based on rule similarity query, where only partial rules will be retrieved and visited while calculating each rule’s activation weight. Note that, the performance of EBRB systems based on tree index is affected greatly by the value of query threshold. However, sometimes it is difficult to determine the value of query threshold, so this paper also proposes an approach based on the k-means clustering algorithm to choose the appropriate query threshold. Some case studies show how the use of the proposed optimization method enhances the reasoning performance of EBRB systems. The proposed method has been validated to be advantageous to visit partial suitable rules instead of all rules. Beside the work performed in the EBRB, the proposed method alone can also be used in different application areas.

The aim of this paper is to develop a new methodology for solving bi-matrix games in which goals are regarded as intuitionistic fuzzy (IF) sets (IFSs) and payoffs are expressed with triangular IF numbers (TIFNs). In this methodology, a new ranking method of TIFNs is proposed and the concept of IF inequalities is interpreted. An IF non-linear programming model is constructed to obtain the solution for such a type of bi-matrix games. Then utilizing these IF inequalities and the ranking method of TIFNs proposed in this paper, the solution of any bi-matrix game with goals of IFSs and payoffs of TIFNs can be transformed into a crisp non-linear programming problem. It is shown that the bi-matrix game with goals of IFSs and payoffs of TIFNs is a generalization of the bi-matrix game with goals of fuzzy sets and payoffs of triangular fuzzy numbers. The method proposed in this paper is demonstrated with a numerical example of commerce retailers’ strategy choice problem.

In this paper, using a special family of extreme fuzzy filters F on a BL-algebra L, we construct a uniform structure ( L , K ) , and then the part K induce a uniform topology T F in L. Also, we prove that the pair ( L , T F ) is a topological BL-algebra, and some properties of ( L , T F ) are investigated. In particular, we show that ( L , T F ) is a first-countable, zero-dimensional, disconnected and completely regular space. At the same time, we give some characterizations of topological properties of ( L , T F ) , for example, ( L , T F ) is a discrete space if and only if g₁ = {1}, and ( L , T F ) is a Hausdorff space if and only if g₁ = {1}, where g = ∩ F . Finally, we naturally contact the relationship between isomorphism (algebraic invariant) and homeomorphism (topological invariant) in topological BL-algebras.

This paper investigates a non-associative generalization, more exactly, a weak t-associative (wta) generalization, of continuous t-norm-based logics. First, the wta-uninorm logic WA _t BL and its axiomatic extensions WA _t Ł, WA _t Π, and WA _t G are introduced as [0, e]-continuous wta-uninorm-based analogues of the most famous continuous t-norm-based logics, BL (Basic fuzzy logic), Ł (Łukasiewicz logic), Π (Product logic), and G (Gödel logic), respectively. Their corresponding algebraic structures and algebraic completeness results are considered. Next, completeness with respect to the algebras whose lattice reduct is [0, 1], known as standard completeness, is established for these systems by a construction in the style of Jenei–Montagna.

Internet is used as the main source of communication throughout the world. However due to public nature of internet data are always exposed to different types of attacks. To address this issue many researchers are working in this area and proposing data encryption techniques. Recently a new substitution box has been proposed for image encryption using many interesting properties like gingerbread-man chaotic map and S₈ permutation. But there are certain weaknesses in aforesaid technique which does not provide sufficient security. To resolve the security issue an enhanced version of existing technique is proposed in this paper. Lorenz chaotic map based confusion and diffusion processes in existing technique are employed. Lorenz map is used to remove strong correlation among the plain text image pixels. In diffusion stage a random matrix is generated through lorenz chaotic map and XORed with shuffled image. It the end, existing gingerbread-man chaotic map based S-box is applied to extract the final cipher text image. The proposed enhanced scheme is analysed by statistical analysis, key space analysis, information entropy analysis and differential analysis. In order to ensure the robustness and higher security of proposed scheme, results via Number of Pixel Rate Change (NPRC) and Unified Average Change Intensity (UACI) tests are also validated.

This paper is an extension work of similarity measure of soft sets initiated by Majumdar and Samanta [15, 16]. Firstly, we have discussed some properties of similarity of soft sets. After this, we have introduced some new notions such as exact fuzzy soft points, pointwise partial similarity and their properties. Also we have discussed equality of soft sets based on similarity function. Finally we have shown two applications, one for similarity measure of two face sized and another equality of same composition drugs.

In this paper, we investigate the following typical form of a class of cubic functional equations: f ( x + 2 y ) - 3 f ( x + y ) + 3 f ( x ) - f ( x - y ) - 6 f ( y )     + k [ f ( mx + y ) + f ( mx - y ) - m [ f ( x + y ) + f ( x - y ) ] - 2 ( m 3 - m ) f ( x ) ] = 0 for some rational number m and some real number k. Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form in fuzzy normed spaces.

This study presents a novel method to solve the multi-period generation and transmission expansion planning (GTEP) problem in a deregulated environment. This framework optimizes simultaneously multiple goals including economic and market indices. The investment cost as an economic criterion and the congestion cost and global welfare as the market-based criteria are taken into account in the proposed planning problem. The market reliability is also assessed considering the N-1 security criterion. An efficient combination of genetic algorithm and fuzzy technique is used to cope with non-linear nature of the proposed multi-objective optimization problem. This solving technique enables planner to adopt a perfect solution according to different levels of importance for planning objectives. The proposed GTEP methodology is implemented on IEEE 6-bus test system and IEEE 24-bus reliability test system considering a 6-year planning horizon. Different expansion planning problems are tested in order to show the impact of the proposed model on the future conditions of the case studies. To evaluate the effectiveness of the proposed optimization method, comparative studies are also provided. The obtained results justify the superiority of the proposed method in finding better expansion plan comparing to some previously reported methods.

In this paper, some types of falling fuzzy prefilters of EQ-algebras are introduced and studied. The notion of falling fuzzy prefilters of EQ-algebras are introduced, and the relationships between falling fuzzy prefilters and fuzzy prefilters are discussed. Moreover, the notions of falling fuzzy positive implicative (implicative, fantastic) prefilters are also proposed and some of their characterizations are displayed. The relationships among these special falling fuzzy prefilters are mainly investigated by using their characterizations. In particular, some conditions for a falling fuzzy positive implicative prefilter is equivalent to a falling fuzzy implicative prefilter are provided.

Emergency decision-making confronts a problem where emergency tasks coordinating to achieve emergency objectives are subject to urgent time and limited resources. Actions with uncontrollable durations make the problem more complicated. This paper proposes a novel resource-constrained Hierarchical Task Network planning approach under uncontrollable durations for emergency decision-making. The objective is to generate a dynamically controllable plan with constrained resources and execute the plan via scheduling actions dynamically. Two of the most general categories of resources in emergency decision-making, consumable resources and reusable resources, are considered. First, timed initial literals are extended to present timed resources and the constraints of timed literals are transformed to impose restrictions on actions in planning. Second, a mechanism is explored to hierarchically inherit the resource constraints of compound tasks by subtasks during Hierarchical Task Network planning. Third, two categories of resource constraints are checked, respectively. Potential resource conflicts are resolved via adding precedence constraints. Finally, experimental studies are conducted to evaluate the planning approach. The results demonstrate the effectiveness and efficiency of the planning approach for emergency decision-making.

In multi-attribute group decision making (MAGDM) problems, the information about attribute weights and the performance ratings of alternatives usually cannot be accurately quantified. This issue has motivated the development of various MAGDM models based on the fuzzy sets theory. However, these fuzzy MAGDM models mostly rely on using the extreme or expected values, but ignore the intermediate occurrences in determining the best alternatives. In order to provide a complete understanding of decision makers’ preference structure, this paper takes a stochastic perspective and proposes a simulation-based approach to facilitate MAGDM under uncertainty when both quantitative and qualitative attributes are involved. The approach not only accounts for the incomplete information about the attribute weights during decision making, but also allows for the use of comparative linguistic expressions to better capture the decision makers’ hesitancy about linguistic expressions. We apply the proposed approach to electric vehicle charging station site selection problem and highlight its effectiveness and advantages through an in-depth comparative analysis with some of the existing methods.

The flower pollination algorithm (FPA) is a recently developed meta-heuristic algorithm inspired by the pollination process of flowers. Similar to other meta-heuristic algorithms, it encounters two probable problems, i.e., entrapment in local optima and slow convergence speed, in solving challenging complex real world problems. Similar to the chaos in actual flower pollination process, this paper proposes new FPAs that employ chaotic maps for adjustment of parameters with the aim to improve the convergence rate and prevent the FPAs to get trapped on local optima. This is achieved by employing chaotic number generators every time, a random number is needed by the classical FPA. Two new chaotic FPAs have been proposed and various test problems are used for their performance evaluation. To check the effectiveness of the proposed algorithms, they are tested on various benchmark functions and engineering design problems with different characteristics having real world applications. The simulation results demonstrate that the chaotic maps are able to significantly boost the performance of FPAs.

In order to assure the stability of Takagi-Sugeno (T-S) fuzzy systems, the linear matrix inequality (LMI) should be applied. However, the LMI method cannot be applied online, because its computational complexity. In this paper, the T-S fuzzy control is transformed into a time-varying system. By using Riccati differential equation (RDE) and a special optimal-like controller, the T-S fuzzy control can be applied online. We prove that the T-S fuzzy control is stable and the trajectory tracking error converges to a bounded zone. Since RDE can be solved online, the novel T-S fuzzy control is adaptive and more simple and effective than LMI-based methods. We apply successfully this online fuzzy control to an autonomous underwater vehicle.

The concept of a fuzzy set provides a natural framework for generalizing many of the concepts of general topology to what might be called fuzzy topological space. Several types of fuzzy continuous functions and its weaker and stronger forms occur in the literature. In this paper we introduce and study the notion of fuzzy slightly γ-continuity. Fuzzy slightly γ-continuity generalize fuzzy γ-continuity. Moreover, basic properties and preservation theorems of fuzzy slightly γ-continuous functions are obtained.

The non-Archimedean normed space theory is an important research object in mathematical physics whose triangle inequality holds in a stronger form. In this note, we propose a generalized chordal distance and a non-Archimedean chordal distance for intuitionistic fuzzy sets. An illustrative example is given to calculate the constructed distances with the different parameters. And some experiments show that the new distances based on chordal distance and non-Archimedean distance are more efficient than the Euclidean-like distances in pattern recognition.

Molodtsov soft set theory provides a general mathematical framework for dealing with uncertainty. The aim of this paper is to lay a foundation for providing a new soft algebraic tool in considering many problems that contain uncertainties. In order to provide these new soft algebraic structures, we introduce the notions of (M, N)-intersectional soft hyperideals and (M, N)-intersectional soft interior hyperideals of ordered semihypergroups. The concepts of (M, N)-intersectional soft hyperideals and (M, N)-intersectional soft interior hyperideals coincide in a regular as well as in intra-regular ordered semihypergroups. We introduce the notion of (M, N)-intersectional soft simple ordered semihypergroups. Furthermore we characterize (M,  N)-intersectional soft simple ordered semihypergroups by means of (M, N)-intersectional soft hyperideals and (M, N)-intersectional soft interior hyperideals.

In this paper, we focus on the connections between rough sets and skew lattices. On the one hand, we study the special properties of the rough sets constructed by means of the congruences induced by ideals of skew lattices which are considered as a non-commutative generalization of classic lattices; On the other hand, the properties of the generalized rough sets with respect to ideals of skew lattices are investigated.

In this paper, Fully Fuzzy Linear Equation System (FFLS) that all parameters and variables are represented by triangular fuzzy numbers is discussed. FFLS has many important applications to branches of science, engineering and other disciplines. The objective of this paper is to find the feasible (strong) and approximate solution with a proposed method based on a mixed integer modeling of the nonsquare FFLS by removing all restrictions on the parameters and variables. The method is illustrated with numerical examples. Results of the numerical examples show that this method has the ability to generate a feasible (strong) and an approximate fuzzy solution and also to indicate no solution case of a nonsquare or square FFLS.

The concept of A-subset is introduced in lattice implication algebras. Firstly, the properties of the A-subset are discussed when A is a general set of lattice implication algebras. Next, the properties of A-subset are investigated when the A is an LI-ideal of lattice implication algebras. We obtain some properties of A-subset and prove that B(A)(the A-subset of B) is an LI-ideal in lattice implication algebras. Finally, the properties of A-subset are investigated in lattice implication product algebras. We prove that the A-subset of ∏ i = 1 n B i is the product of A-subset of B _i  (i = 1,  2, ⋯ ,  n) respectively ( ( ∏ i = 1 n B i ) ( ∏ i = 1 n A i ) = ( ∏ i = 1 n B i ( A i ) ) ) .

Online reviews play important roles in many Web Applications like e-business and government intelligence, since such user-generated-contents (UGC) contain rich user opinion. Opinion target and opinion word are a pair of core objects for user opinion expression in reviews. Extracting these two objects from reviews is crucial for the tasks of opinion mining. However, traditional extraction methods have various limitations such as ignoring the opinion relationship, the restriction of word span, the error propagation caused by iterative expansion, which would reduce the extraction performance. For the above deficiencies, we propose a supervised method based on the constrained word alignment model to extract opinion target and opinion word collectively at first. To tackle the time-consuming and error-prone problem of manual annotation encountered by the supervised method, we further devise a semi-supervised extraction method based on active learning. In this method, we design the sample uncertainty-based sampling strategy and the feature evidence-based one to choose the most informative samples for labeling manually. At last, a series of experiments on a real-world dataset show that our approaches outperform several state-of-the-art baselines significantly.

The problem of assessment, selection and improvement of key performance indicators in the New Service Development process is one of the most important tasks of process managers, and it has a critical effect on the considered process effectiveness which is further propagated on the competitive advantage of each service small and medium enterprises. The relative importance of the introduced key performance indicators and their values are assessed by decision makers in selected enterprises (total of 187 persons). The assessment of decision makers are described by pre-defined linguistic expressions which are modelled by using fuzzy sets theory. Aggregated relative importance is determined according to approach developed in this paper. The ranking and improvement of key performance indicators is stated as multi-criteria decision making problem that could be solved by the genetic algorithm. Priority of management initiatives that should lead to the improvement of selected key performance indicator is based on fuzzy if-then rules and single-objective genetic algorithm. In this way, more appropriate improvement strategy, which demands lower costs, may be defined. By applying the proposed model it is possible to identify weak points in organizations, to provide corrective measures, and to enhance the effectiveness of new service development process. The model presents a suitable solution for reengineering and improvement of the process performance. The application of this model could be introduced in other industrial branches.

Pythagorean fuzzy sets (PFSs), hesitant fuzzy sets (HFSs) and intuitionistic hesitant fuzzy sets (IHFSs) have attracted more and more scholars’ attention due to their powerfulness in expressing vagueness and uncertainty. Intuitionistic hesitant fuzzy set satisfies the condition that the sum of its membership’s degrees is less than or equal to one. However, there may be a situation where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that their sum is greater than 1. To overcome this shortcoming, in this paper we introduce the concept of Pythagorean hesitant fuzzy set (PHFS) which is the generalization of intuitionistic hesitant fuzzy set under the restriction that the square sum of its membership degrees is less than or equal to 1. We discuss some properties of PHFS. We define score and accuracy degree of the Pythagorean hesitant fuzzy numbers (PHFNs) for comparison in Pythagorean hesitant fuzzy numbers. Also in decision making with PHFSs, aggregation operators play a very important role since they can be used to synthesize multidimensional evaluation values represented as Pythagorean hesitant fuzzy valued into collective values. We develop distance measure between PHFNs. Under PHFS environments, we develop aggregation operators namely, Pythagorean hesitant fuzzy weighted averaging (PHFWA), Pythagorean hesitant fuzzy weighted geometric (PHFWG). We develop the maximizing deviation method for solving MADM problems, in which the evaluation information provided by the decision maker is expressed in Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. The main advantage of these operators is that it is to provide more accurate and precious results. Furthermore, we developed these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean hesitant fuzzy environment to show the validity, practicality and effectiveness of the new approach.

A lattice-valued information system is an important model in the field of artificial intelligence and the notion of homomorphisms between lattice-valued information systems is a kind of tools to study data compression in a lattice-valued information system. This paper investigates invariant characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression. Information structures in a lattice-valued information system is first proposed by using set vectors. Then, dependence and independence between information structures in the same lattice-valued information system is characterized by the inclusion degree. Finally, a complex massive lattice-valued information system can be compressed into a relatively small-scale lattice-valued information system by means of homomorphisms and it is proved that some characterizations of information structures in a lattice-valued information system under homomorphisms based on data compression are invariant, that is, some of the same data structures are obtained.

Intuitionistic fuzzy graph is a highly growing research area dealing with real life applications. In this paper, we introduce the concept of interval valued intuitionistic fuzzy graph and define magic labeling of interval valued intuitionistic fuzzy graph. Here we discuss the significance of magic labeling in interval-valued intuitionistic fuzzy graphs. We also analyse some of its properties and some structures and implement it into the operations of interval-valued intuitionistic fuzzy magic labeling graph. We have also investigated about some bounds over the size and shape of the interval-valued intuitionistic fuzzy graphs based on ε-neighborhood and confine the membership values of the vertices and edges of it.

The concepts of covering and matching in an intuitionistic fuzzy graph using strong arcs are introduced and established many interesting properties on it. The notion of paired domination in intuitionistic fuzzy graph using strong arcs is also studied. The strong paired domination number γ _spr of complete intuitionistic fuzzy graph and complete bipartite intuitionistic fuzzy graph is determined and investigated many interesting properties on it.

In M-fuzzifying interval spaces, the notion of M-fuzzifying base-point orders is introduced, by which some characterizations of M-fuzzifying geometric (resp. Peano, Pasch) interval spaces are obtained. Then notions of M-fuzzifying gated sets, M-fuzzifying gate maps and M-fuzzifying gated amalgamations are introduced. It is shown that M-fuzzifying gated sets are preserved by M-fuzzifying IP-surjective mappings, and that the M-fuzzifying Peano (resp. Pasch, modular, JHC) property is preserved by M-fuzzifying gated amalgamations. In particular, the M-fuzzifying sand-glass property is also preserved by M-fuzzifying gated amalgamations provided that the corresponding M-fuzzifying gate maps are M-fuzzifying II-mappings.

In this paper, the degree to which an L-subset of a vector space is an L-convex set is introduced and their characterizations are given. Besides, we obtain an L-fuzzy convexity induced by L-convex degrees, and analyze their L-fuzzy convexity preserving mappings and L-fuzzy convex-to-convex mappings.

The microarray data are important to detect diseases, however, there are a large number of genes with small sample size, and this leads to slow convergence speed and reducing the prediction accuracy. Therefore, reducing the dimension of data is needed as preprocessing step for classification of data. There are two methods can be used to perform the dimension reduction, namely, the feature extraction and feature selection. The feature extraction methods are transforming data into another space and then a subset of features are selected using some criteria. The projection of the measurements, using these methods, is different from the original data. Unlike feature extraction, the feature selection methods select relevant features without changing their values, however, these methods need a large time than feature extraction. There are some algorithms can simultaneously select and extract features from data to take the advantages of both methods. This paper proposed a new simultaneous feature extraction/selection method for high-dimensional microarray data. The proposed method combines fuzzy neighborhood rough set method with nonnegative matrix factorization based on multiobjective evolutionary. To evaluate the accuracy of our approach, a computational experiments were performed on seven gene microarray datasets with diverse characteristics. Experimental results illustrate that the proposed method is better than other algorithms in term of performance measures.

The main objective of this research is to give an overview of the Analytic Hierarchy Process (AHP) in neutrosophic environment. In some realistic situations, the decision makers might be unable to assign deterministic evaluation values to the comparison judgments due to his/her limited knowledge or the differences of individual judgments in group decision making. To overcome these challenges, we have used neutrosophic set theory to handle the AHP, where each pair-wise comparison judgment is represented as a triangular neutrosophic number (TNN). In this paper, neutrosophic theory is used to form AHP decision-making model for choosing the best candidates among the applications. A real life example is developed based on expert opinions from Zagazig University, Egypt. The problem is solved to show the effectiveness of the proposed neutrosophic-AHP decision making model.

In a paper by Wang and Elhag [Ying-Ming Wang and Taha M.S. Elhag, Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment, Expert Systems with Applications 31 (2006) 309-319], a fuzzy TOPSIS method on alpha level sets was introduced and a nonlinear programming solution procedure was presented. It is found that in the case that the fuzzy decision matrix is of the same dimension and needs no normalization, a pair of nonlinear programming models is incorrect for computing the relative closeness provided by the above paper. In this paper we present a correct pair of nonlinear programming models in the case of the same dimension and justify it from the viewpoint of monotonic function. An illustrated example for selecting the best supplier of metallic components used in a variety of transmission cables has been examined using the proposed programming models to fuzzy TOPSIS method and demonstrated its superiorities, rationalities.

Neutrosophic set, proposed by Smarandache considers a truth membership function, an indeterminacy membership function and a falsity membership function. Soft set, proposed by Molodtsov is a mathematical framework which has the ability of independency of parameterizations inadequacy, syndrome of fuzzy set, rough set, probability. Those concepts have been utilized successfully to model uncertainty in several areas of application such as control, reasoning, game theory, pattern recognition, and computer vision. Nonetheless, there are many problems in real-world applications containing indeterminate and inconsistent information that cannot be effectively handled by the neutrosophic set and soft set. In this paper, we propose the notation of bipolar neutrosophic soft sets that combines soft sets and bipolar neutrosophic sets. Some algebraic operations of the bipolar neutrosophic set such as the complement, union, intersection are examined. We then propose an aggregation bipolar neutrosophic soft operator of a bipolar neutrosophic soft set and develop a decision making algorithm based on bipolar neutrosophic soft sets. Numerical examples are given to show the feasibility and effectiveness of the developed approach.