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In this paper we introduce a notion of a t-filter on residuated lattices which is a generalization of several special types of filters. We provide some basic properties of t-filters and show how particular results about special types of filters (e.g. Extension property, Triple of equivalent characteristics, and Quotient characteristics) are uniformly covered by this simple general framework.

We show that slight technical changes in the definition transform the probability of fuzzy events introduced by R. R. Yager [16] into a new concept of such probabilities having nice properties, both from an intuitive and from a mathematical point of view: monotonicity, additivity, and continuity.

In this paper we deal with conditional aggregation-based survival functions recently introduced by Boczek et al. (2020). The concept is worth to study because of its possible implementation in real-life situations and mathematical theory as well. The aim of this paper is the comparison of this new notion with the standard survival function. We state sufficient and necessary conditions under which the generalized and the standard survival function equal. The main result is the characterization of the family of conditional aggregation operators (on discrete space) for which these functions coincide.

This paper presents a new method for simultaneous modeling the uncertainty of measurement data (necessary to define the boundary shape and boundary conditions) in boundary problems. The interval parametric integral equation system (interval PIES) was developed for solving boundary problems with input data defined in this way. The motivation for conducting this research was that this topic (simultaneous consideration of uncertainties of all input data) has appeared sporadically in the literature (mainly with uncertainly defined boundary conditions or other parameters). In this paper, the uncertainty was defined using interval numbers and modeled using interval arithmetic. The direct application of both classical and directed interval arithmetic caused the overestimation and obtained solutions were useless in practice. Therefore, modification of the directed interval arithmetic was developed. The reliability of the interval PIES solutions obtained using such arithmetic was verified on 2D problems described by Laplace’s equation. The solutions were compared with the interval analytical solutions (differently obtained), as well as with the solutions of exactly defined (without the uncertainty) numerical methods. All performed tests indicated the high potential of the method. Obtained interval solutions occurred to be less overestimated and not as time-consuming as presented alternative methods.

This paper introduces an admissibilization condition for singular interval-valued fuzzy systems with a dynamic output-feedback controller using a linear matrix inequality approach. The derivation of the admissibility criterion (satisfying regularity, non-impulsiveness and stability) for the closed-loop system of the singular interval-valued fuzzy systems using the dynamic output-feedback controller is concerned. Here, the derived criterion is represented as the parameterized matrix inequalities depending on the membership functions of the system and the controller. To relax the derived parameterized matrix inequalities, this paper proposes a relaxation lemma based on the properties of the membership functions and their relations. By using this lemma, the parameterized matrix inequalities are converted into the matrix inequalities independent of the membership functions but not convex. Therefore, by introducing the structures of the variables and the congruent transformation matrix , a sufficient condition for the admissibility criterion is successfully given in terms of strict linear matrix inequalities. Two numerical examples are given to show the effectiveness of the proposed control.

In many applications, like database management systems , is very useful to have an expression to compute the cardinality of cartesian product of k sets without repeated elements; we designate this problem as T ( k ) . The value of | T ( k ) | is upper-bounded by the multiplication of cardinalities of the sets. As long as we have searched, it has not been reported a general expression to compute T ( k ) using cardinalities of the intersections of sets, this is the main topic of this paper. Given three sets with indices { 0 , 1 , 2 } , C i is the cardinality of one set, C i , j ( i j ) and C i , j , l ( i j l ) are respectively the cardinalities of the intersections of 2 and 3 sets, then the searched formulas for T ( k ) are: T ( 1 ) = C 0 ; T ( 2 ) = C 0 C 1 - C 0 , 1 ; T ( 3 ) = C 0 C 1 C 2 - ( C 0 , 1 C 2 + C 0 , 2 C 1 + C 1 , 2 C 0 ) + 2 C 0 , 1 , 2 . In this paper, we prove formulas for computing T ( k ) and its specialization when a set is contained in the next sets. For this purpose, we will use concepts like partitions of the integer k in v parts, Bell numbers, Stirling numbers of the first kind and Stirling numbers of the second kind. Additionally, we present a complexity analysis for the computation of T ( k ) .

Clone theory plays an important role in studying aggregation functions on bounded posets or bounded lattices. Several important classes of aggregation functions on a bounded lattice L form a clone, particularly the set of all aggregation functions on L, the so-called full aggregation clone on L. For any finite lattice L, this clone is known to be finitely generated and various generating sets and their constructions have been presented in recent papers. The aim of this paper is to extend previous results concerning generating sets of aggregation clones on finite chains. Namely, the objective is to discuss the minimality of certain generating bases, the so-called ( χ , ⊕ ) -generating sets.

To deal with multicriteria decision making (MCDM) problems with interaction criteria, the Choquet integral (CI) is one of effective tools. This paper first proposes the reverse Choquet integral (RCI), which defines the importance of the ordered elements in an opposite principle to the CI. To show the principle of the RCI, we offer its concrete expression in view of the Mobius representation by which one can clearly see the difference and the relationship between the CI and the RCI. Then, we propose the “bi-direction Choquet integral” (BDCI), which is a convex combination of the CI and the RCI. To get the interactions of ordered coalitions comprehensively, this paper further proposes the generalized Shapley bi-direction Choquet integral (GSBDCI). Furthermore, the hybrid generalized Shapley bi-direction Choquet integral (HGSBDCI) is proposed, which defines the importance of ordered positions and the criteria with interactions simultaneously. With respect to these types of CIs, their exponent forms are also discussed. Finally, we use an application case to show the utilization of the proposed new CIs for MCDM. The proposed new Choquet integrals provide us a very useful way to deal with MCDM problems.

In this paper, we study first order interval-valued differential equations with the boundary value condition x(0)=αx(T). By constructing proper operators, some sufficient conditions for the existence of solutions are provided for α∈R⧹{0,1}, corresponding to non-periodic boundary value problems. At the end of this paper, some examples are shown to illustrate the theorems proved.

A Boolean function in n variables is 2-rotation symmetric if it is invariant under even powers of ρ ( x 1 , … , x n ) = ( x 2 , … , x n , x 1 ) , but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ 2 to a single monomial. If the quartic MRS 2-function in 2 n variables has a monomial x 1 x q x r x s , then we use the notation 2 - ( 1 , q , r , s ) 2 n for the function. A detailed theory of equivalence of quartic MRS 2-functions in 2 n variables was given in a 2020 paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called mf 1 and mf 2 in the paper. Next to describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions 2 - ( 1 , q , r , s ) 2 n (with q r s , say), n = s , s + 1 , … can be shown to satisfy. This problem was solved for the mf 1 case only in the 2020 paper. In this paper the problem for the mf 2 case is solved, using new ideas about short functions, as defined in this paper. These short functions are also of independent interest, and further results about them are in the paper.

In this paper, the problem of exponential dissipation stability of T-S fuzzy system with state quantization is studied by using non-fragile sampled-data control. First, A Lyapunov-Krasovskii function containing all sampled-data and quantization information is constructed. Second, the better results can be obtained by using the integral inequality and the Newton Leibniz formula. In addition, a dissipative controller for non-fragile sampled-data is designed. Finally, The reasonable examples illustrate the significant improvement and value of this paper.

The framework of interval-valued fuzzy preference relations (IVFPRs) is adequate and effective to model human preference evaluations under indeterminacy. This paper analyzes three recently presented multiplicative transitivity models of IVFPRs and exposes their drawbacks. An and-like-uninorm-based functional transitivity equation is developed to introduce a multiplicative consistency notion for IVFPRs. Based on the transitivity logarithmic equation, a geometric-consistency index is further proposed to compute the inconsistency level of an IVFPR. The paper builds a logarithmic least squares model with row indeterminacy constraints and equivalently transforms it into a quadratic programming model for finding a closed-form solution of the normalized interval-valued fuzzy weights of IVFPRs. A novel method is subsequently presented to check the acceptability of an IVFPR by examining its acceptable consistency and acceptable indeterminacy. An approach including an and-like-uninorm-based maximization model is introduced to aggregate local interval-valued fuzzy weights and an interval fuzzy analytic hierarchy process is designed step-by-step. An illustrative example and a comparison study are utilized to demonstrate the performance and merits of the presented models. Meanwhile, an outstanding undergraduate student selection problem in international exchange is provided to show the application of the proposed decision method.

The paper by Z. Kurac, 2019 deals with a new property, the so-called transfer-stability, characterizing the arithmetic mean. With this property, it is possible to define special forms of arithmetic mean on finite chains. The idempotence property was required for this definition. In this paper, we neglect this necessity and deal only with transfer-stable aggregation functions. Thanks to this fact, it is possible to define these aggregation functions on any finite lattice (hereinafter “lattice”) and not only on finite chains. Transfer-stable aggregation functions can be defined on any finite lattice. Nevertheless, there is a subclass of finite lattices, the so-called transfer-stable lattices, where the behavior of the transfer-stable aggregation functions is simply described because the transfer-stability classes are linearly ordered. Therefore, the main goal of this paper is characterization of these transfer-stable lattices. The second half of the paper deals with some useful properties associated with the lattice of all k-ary transfer-stable aggregation functions.

In the paper, we describe the new measure of matching two intuitionistic fuzzy sets. The operations known in the intuitionistic fuzzy set theory are used, and the perturbation of one intuitionistic fuzzy set by another intuitionistic fuzzy set is understood as an impact of one intuitionistic fuzzy set on another intuitionistic fuzzy set. The opposite case can also be considered wherein the second intuitionistic fuzzy set is perturbed by the first one. The introduced measure of the perturbation of one intuitionistic fuzzy set by another intuitionistic fuzzy set is considered instead of commonly used distance between two intuitionistic fuzzy sets. In general, the new measure can be asymmetric and therefore can provide more information compare to a distance between intuitionistic fuzzy sets. The values of such measures of intuitionistic fuzzy sets’ perturbation are ranged between 0 and 1. In this paper specific mathematical properties of the measure of intuitionistic fuzzy sets’ perturbation are studied. The presented methodology is explained by several illustrative examples.

In this paper, we derive conditions for the existence of Pareto efficient strategy and Pareto solution under H∞ constraint for the linear quadratic (LQ) finite horizon cooperative differential game of stochastic systems with state-, control- and disturbance-multiplicative noise. Firstly, we present a stochastic bounded real lemma (SBRL) with any initial condition for the considered stochastic system. Next, a necessary and sufficient condition for Pareto optimal strategy under the H∞ constraint is researched by two cross-coupled generalized differential Riccati equations (GDREs), which is the key contribution of this paper. Another contribution is that the Pareto solution for every controller is given under the Pareto efficient strategy and worst-case external disturbance . Finally, a practical example is given to illustrate the effectiveness of our results.

Recently, Torra, Narukawa and Sugeno [Fuzzy Sets and Systems, 292 (2016) 364–379] obtained the Cauchy-Schwarz inequality for Choquet integral. In this paper, we first introduce a refined Holder’s inequality in Choquet calculus. Then by this inequality, we show that Cauchy-Schwarz inequality and Holder’s inequality for Choquet integral are equivalent only when the monotone measure is submodular or the integrands are comonotonic, thus closing the series of papers in this literature. Note that it is obvious that Holder’s inequality implies Cauchy-Schwarz’s inequality. But the converse is an interesting subject. This paper focuses on this subject.

In this paper we study the definition of the f-divergence and the Hellinger distance for non-additive measures in the discrete case. As these measures are based on the derivatives of the measures, we consider the problem of defining the Radon–Nikodym derivative of a non-additive measure. While Radon–Nikodym derivatives for additive measures exist for absolutely continuous measures, this is not the case in the non-additive case. In this paper we will define set-directional and upper, lower and interval derivatives. We will also define when two measures have the same sign. These definitions will be used to introduce alternative definitions of the f-divergence, all extending the classical definition to non-additive measures.

A Boolean function in n variables is 2-rotation symmetric if it is invariant under even powers of ρ ( x 1 , … … , x n ) = ( x 2 , … , x n , x 1 ) , but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ2 to a single monomial. If the quartic MRS 2-function in 2n variables has a monomial x1xqxrxs, then we use the notation 2-(1, q, r, s)2n for the function. This paper gives a detailed theory of equivalence of quartic MRS 2-functions in 2n variables. Such a theory was provided for the cubic MRS 2-functions in two 2015 papers of Cusick and Johns. As in the earlier papers, the two main topics in the theory are describing the affine equivalence classes of the functions under certain groups of permutations; and giving details of the linear recursions that the Hamming weights of any sequence of functions 2-(1, q, r, s)2n (with q n = s , s + 1 , … can be shown to satisfy. The discussion for both of these topics uses new ideas because the quartic theory naturally divides into two cases.

In this paper, we correct some errors in the extended first and second Gordan’s theorems in a recent paper—Information Sciences 504 (2019) 276–292.

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated In doing so, the feasible region is formed by the intersection of two inequality fuzzy systems and Dubois-Prade family of t-norms which are considered as fuzzy composition. The most well-known continuous t-norms are Archimedean such as Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family. An interesting family of t-norms that is not Archimedean has been introduced by Dubois and Prade. In this paper, the resolution of the feasible region of the problem is initialy investigated when it is defined with max-Duboise-Prade composition. A necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility. of the problem. Moreover, two procedures have also been presented with the aim of simplifying the current linear problems. A method is proposed to generate random feasible max-Dubois-Prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem. Finally, an example is described to illustrate this algorithm.

This paper focuses on obtaining an interval extension of Saaty's consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations (IMRPRs) as well as checking acceptability of IMRPRs. Four axiomatic properties are presented to characterize multiplicative consistency of IMRPRs. The paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties. A constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an IMRPR. Based on the constructed interval matrix, an ordinary interval multiplication based transitivity equation is developed to define consistency of IMRPRs. A logarithmic least square model is established to find normalized interval weights from IMRPRs and its analytic solution is obtained by the Lagrangian multiplier method. The paper introduces an index to measure uniformity of row uncertainties in an IMRPR and proposes a geometric consistency index to measure inconsistency of IMRPRs. A novel method is put forward to determine acceptability of IMRPRs by checking acceptable consistency and acceptable uncertainty. The proposed models are illustrated by eight numerical examples and a hierarchical multi-criteria decision making problem dealing with recommending undergraduate students to graduate admission.

The definable set is a core concept in rough set theory . It plays an important role in the characterizations of rough sets. In this paper, we study the lattice and matroid representations of definable sets in generalized rough sets based on relations. First, we propose the lower definable lattice, consisting of all lower definable sets with set inclusion order. Then we give some conditions under which the lower definable lattice is distributive (or geometric, or Boolean). Furthermore, we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations . On the one hand, we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive. On the other hand, we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation. Finally, we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice. We show that if a lower definable lattice is a lattice of closed sets of a matroid, then it must be an open-closed set lattice of a matroid. In addition, we prove that some lower definable lattices are not the lattices of closed sets of matroids. These results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations.

This paper focuses on answering the following question: How to estimate the quality of off-diagonal judgments in an interval multiplicative preference relation (IMPR) and obtain an analytical solution to the optimized interval priority weight vector of an IMPR. By generalizing IMPRs, an interval-multiplication-based transitivity equation is devised to define perfect consistency of generalized IMPRs. A basic interval multiplicative weight (IMW) vector is introduced to characterize different interval weight vectors with equivalency and used as a priority weight benchmark of a generalized IMPR. Two logarithmic least square models are established for determining basic IMWs of a generalized IMPR, and their analytical solutions are found by using the Lagrangian multiplier method. The paper demonstrates that any IMPR with uncertainty has no perfect consistency in terms of interval arithmetic , and constructs a generalized IMPR from an IMPR to introduce consistency of off-diagonal judgments in an IMPR. Two computation formulas are developed to obtain optimized basic IMWs of an IMPR, which can entirely capture off-diagonal judgments if the IMPR is consistent. Numerical examples are offered to illustrate how to apply the proposed models and a comparative study is made to show that off-diagonal judgments in an inconsistent IMPR are properly captured by the analytical solution based IMWs.

The purpose of this paper is to introduce the notion of locally finite covering approximation (LFC-, for brevity) space (U, C) which is a generalization of a (finite) covering approximation space. Furthermore, for a subset X of the universe U, we develop two kinds of neighborhood systems derived from the covering C, specifically K1(X) and K2(X) which are not comparable with each other. Next, the paper establishes two topologies on U, Alexandroff, and quasi-discrete topologies generated by systems K1(X) and K2(X) as bases, respectively. After suggesting many examples explaining the above notions and systems, we develop two new types of rough set structures, an H-rough set operator (H*, H*) and a Khalimsky (K-, for short) topological rough set operator ( D K − , D K + ) . The present paper investigates various properties of these operators and refers to certain advantages of them which can be used in applied science including digital geometry, information geometry, computer vision, pattern recognition, image processing, and so on. In order to accomplish this, we use some tools such as granulations, neighborhood systems, typical Pawlak’s tools, related topologies, rough set structures associated with the K-topology, and so on. Furthermore, through this work, we can obtain strong connections between covering rough set theory and applied geometry. In the present paper all sets U, C and X(⊆U) need not be finite.

In a recent paper [12] we have shown that aggregation functions on a bounded lattice L form a clone, i.e., the set of functions closed under projections and composition of functions. Moreover, for any finite lattice L we gave a finite set of unary and binary aggregation functions on L from which the aggregation clone is generated. In this paper, a general method for constructing generating sets of the aggregation clone on L is presented. Our approach is based on extending of L-valued capacities leading to so-called full systems of aggregation functions. Several full systems on L are presented (including singleton ones) and their arities are discussed.

In the authors paper (Strboja et al., 2016) [42] , the notion of the bipolar pseudo-integral has been introduced. This paper extends investigations related to this new type of bipolar integrals defined with respect to ⊕-decomposable bi-capacities. We consider the main properties of the discrete bipolar pseudo-integral. A characterization theorem for the discrete bipolar pseudo-integral is proven. Monotone convergence theorems and the Fatou type lemma for the discrete bipolar pseudo-integral are considered. In order to illustrate obtained results, many interesting examples are provided.

In a recent paper Botur et al. (2018) we have presented a generating set of the clone of idempotent aggregation functions on bounded lattices . As the main result we have shown that this clone is generated by certain ternary idempotent functions from which all idempotent aggregation functions of L can be obtained by usual term composition. The aim of this paper is to present an essential improvement of the result above by presenting a new generating set of this clone. A bit artificial ternary functions are substituted here by natural (binary) lattice a-medians and certain binary characteristic functions. Consequently, the clone is generated by its binary part and the result strengthens the essential role of medians within all idempotent aggregation functions. Moreover, we will show that for an n-element lattice L, the upper bound of binary generators is 2 n − 1 .

This paper focuses on how to find an analytic solution of optimal interval weights from consistent interval fuzzy preference relations (IFPRs) and obtain approximate-solution-based interval weights in analytic form from inconsistent IFPRs. The paper first analyzes the popularly used interval weight additive normalization model and illustrates its drawbacks on the existence and uniqueness for characterizing ]0, 1[-valued interval weights obtained from IFPRs. By examining equivalency of ]0, 1[-valued interval weight vectors, a novel framework of multiplicatively normalized interval fuzzy weights (MNIFWs) is then proposed and used to define multiplicatively consistent IFPRs. The paper presents significant properties for multiplicatively consistent IFPRs and their associated MNIFWs. These properties are subsequently used to establish two goal programming (GP) models for obtaining optimal MNIFWs from consistent IFPRs. By the Lagrangian multiplier method, analytic solutions of the two GP models are found for consistent IFPRs. The paper further devises a two-step procedure for deriving approximate-solution-based MNIFWs in analytic form from inconsistent IFPRs. Two visualized computation formulas are developed to determine the left and right bounds of approximate-solution-based MNIFWs of any IFPR. The paper shows that this approximate solution is an optimal solution if an IFPR is multiplicatively consistent. Three numerical examples including three IFPRs and comparative analyses are offered to demonstrate rationality and validity of the developed model.

This paper proposes quaternion polar harmonic Fourier moments (QPHFM) for color image processing and analyzes the properties of QPHFM. After extending Chebyshev–Fourier moments (CHFM) to quaternion Chebyshev-Fourier moments (QCHFM), comparison experiments, including image reconstruction and color image object recognition, on the performance of QPHFM and quaternion Zernike moments (QZM), quaternion pseudo-Zernike moments (QPZM), quaternion orthogonal Fourier-Mellin moments (QOFMM), QCHFM, and quaternion radial harmonic Fourier moments (QRHFM) are carried out. Experimental results show QPHFM can achieve an ideal performance in image reconstruction and invariant object recognition in noise-free and noisy conditions. In addition, this paper discusses the importance of phase information of quaternion orthogonal moments in image reconstruction.

In a recent paper we proposed the study of aggregation functions on lattices via clone theory approach. Observing that aggregation functions on lattices just correspond to $0,1$-monotone clones, we have shown that all aggregation functions on a finite lattice $L$ can be obtained as usual composition of lattice operations $\wedge,\vee$, and certain unary and binary aggregation functions. The aim of this paper is to present a generating set for the class of intermediate (or, equivalently, idempotent) aggregation functions. This set consists of lattice operations and certain ternary idempotent aggregation functions., Comment: 18 pages

Anomaly detection algorithms face several challenges, including processing speed, adapting to changes in dynamic environments, and dealing with noise in data. In this paper, a two-layer cluster-based anomaly detection structure is presented which is fast, noise-resilient and incremental. The proposed structure comprises three main steps. In the first step, the data are clustered. The second step is to represent each cluster in a way that enables the model to classify new instances. The Summarization based on Gaussian Mixture Model (SGMM) proposed in this paper represents each cluster as a GMM. In the third step, a two-layer structure efficiently updates clusters using GMM representation, while detecting and ignoring redundant instances. A new approach, called Collective Probabilistic Labeling (CPL) is presented to update clusters incrementally. This approach makes the updating phase noise-resistant and fast. An important step in the updating is the merging of new clusters with existing ones. To this end, a new distance measure is proposed, which is a modified Kullback–Leibler distance between two GMMs. In most real-time anomaly detection applications, incoming instances are often similar to previous ones. In these cases, there is no need to update clusters based on duplicates, since they have already been modeled in the cluster distribution. The two-layer structure is responsible for identifying redundant instances. Ignoring redundant instances, which are typically in the majority, makes the detection phase faster. The proposed method is found to lower the false alarm rate, which is one of the basic problems for the one-class SVM. Experiments show the false alarm rate is decreased from 5% to 15% among different datasets, while the detection rate is increased from 5% to 10% in different datasets with two-layer structure. The memory usage for the two-layer structure is 20 to 50 times less than that of one-class SVM. The one-class SVM uses support vectors in labeling new instances, while the labeling of the two-layer structure depends on the number of GMMs. The experiments show that the two-layer structure is 20 to 50 times faster than the one-class SVM in labeling new instances. Moreover, the updating time of the two-layer structure is two to three times less than for a one-layer structure. This reduction is the result of using two-layer structure and ignoring redundant instances.

In this paper, we propose a new autocratic multiattribute group decision making (AMAGDM) method for hotel location selection based on interval-valued intuitionistic fuzzy sets (IVIFSs), where the evaluating values of the attributes for alternatives and the weights of the attributes given by decision makers are represented by interval-valued intuitionistic fuzzy values (IVIFVs). The proposed method calculates the changing of the weights of the decision makers until the group consensus degree (GCD) of the decision makers is larger than or equal to a predefined threshold value. We also apply the proposed AMAGDM method to deal with the hotel location selection problem. The main contribution of this paper is that we propose a new AMAGDM method which is simpler than Wibowo's method (2013), where the drawback of Wibowo's method is that it is too complicated due to the fact that it adopts the concept of ideal solutions for determining the overall performance of each hotel location alternative with respect to all the selection criteria. The proposed AMAGDM method provides us with a very useful way for AMAGDM in interval-valued intuitionistic fuzzy environments.

This paper focuses on recovering multi-dimensional signal called tensor which is corrupted by random missing areas. The performance of the conventional tensor completion techniques deteriorate when the tensor multi-rank is large and/or large missing areas. Moreover, these techniques are weak in preserving the edges in the signals like images/videos. This paper proposes an efficient method to overcome these problems by simultaneously combining novel twist tensor total variation norm to exploit spatio-temporal correlation and tensor-Singular Value Decomposition (t-SVD) based reweighted nuclear norm to improve low multi-rank tensor recovery. The twist tensor total variation norm takes care of edges in the recovered data and aids the recovery of missing areas by utilising the similarities in the adjacent samples. The reweighted nuclear norm handles corrupted large rank tensors by sparsity enhancement via reweighting its singular values. The effectiveness of the proposed method is established by applying to video completion problem, and experimental results reveal that the algorithm outperforms its counterparts.

In most cases, ELECTRE methods just apply to traditional multicriteria decision making (MCDM) problems with independent criteria. However, there exist more or less interdependences among criteria in actual situations. A special case is MCDM with prioritizations among criteria, called prioritized MCDM. In recent years, how to deal with MCDM problems in the environment of prioritized criteria becomes hot topic increasingly. Lots of existing methods, including PROMETHEE, are modified for the prioritized MCDM, but the advantages of ELECTRE have not been exploited to the prioritized MCDM. In my opinion, design specific ELECTRE methods for prioritized MCDM problems will benefit the developments of both ELECTRE and prioritized MCDM. In this paper, we firstly reformulate the expressions of concordance and discordance indices for use in the MCDM problems with dependent criteria, especially prioritized criteria, based on the concepts of fuzzy measures and digraphs respectively. After replacing the fuzzy measure by a prioritized measure, we successfully validate the concordance of an assertion under prioritized criteria. Furthermore, we design an approach to validate the discordance of the assertion based on a digraph constructed by the criteria and their prioritizations. Finally, we design three procedures to construct outranking relations in the prioritized MCDM environment according to the ideas of ELECTRE-I, ELECTRE-Iv, and ELECTRE-IS respectively. It is meaningful of this paper to provide a new idea to solve the prioritized MCDM problems and widen the application scope of ELECTRE methods by means of the reformulations of concordance and discordance.

In this paper, optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t-norm is considered as fuzzy composition, and fuzzy inequality replaces ordinary inequality in the constraints. Unlike most optimization algorithms, in fuzzy relational inequalities with fuzzy constraints (FRI-FC) we find a near-feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp (ordinary) inequality constraints. Such solutions are called super-optima in this paper. Subsequently, an algorithm is proposed to find a super-optimum with pre-specified desirable infeasibility. For this purpose, we firstly study some structural properties of the FRI-FC problem and present a new formulization that is independent of t-norms used in the constraints of the problem. This new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters. However, it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super-optimum under a certain sufficient condition. Finally, a modified PSO is presented in which the ability of PSO to solve unconstrained problems with continuous domain, the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized FRI-FC problems. The modified PSO algorithm has been applied to the generalized FRI-FC problem defined with ten well-known continuous t-norms. Additionally, an idea of the FRI-FC problems as outer approximators for FRI problems with ordinary inequalities is also investigated.

In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers . Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provably good immunity against fast algebraic attacks. In previous literatures, only Carlet–Feng function was proven to have good immunity to fast algebraic attacks. In this paper, we first study a large family of highly nonlinear Boolean functions in terms of the immunity to fast algebraic attacks, which includes the functions of Tu–Deng, the functions of Tang et al. and the functions of Jin et al. Based on a sufficient and necessary condition for measuring the immunity of Boolean functions against fast algebraic attacks using bivariate polynomial representation , we propose an efficient method for estimating the immunity of the functions of such family. Then we prove that a family of 2 k-variable Boolean functions, including the function recently constructed by Tang et al., are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. Besides, we prove a lower bound on their nonlinearity based on the work of Tang et al. which is better than that of Carlet–Feng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet–Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang’s et al. function. To sum up, among the known functions with provably good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.

This paper looks at the challenge of object recognition from the perspective of achieving the final goals of practical real-world autonomous robot-vision applications; that of identifying a target of interest in the robot’s field of view and properly localizing its position in preparation for the higher-level goal of tracking and navigation. A unified framework is introduced that combines a multidimensional feature histogram approach with a multiscale pyramid approach for training-less color object recognition and localization, with direct application to autonomous robotic agents. This framework addresses the high computational cost associated with multidimensional processing by deriving the Multidimensional “Laplacian Feature Histogram Pyramid”, a novel approach to a unified multidimensional-multiscale histogram representation. Furthermore, a Taylor series formulation is employed to combine the multiscale levels of the multidimensional Laplacian feature histogram pyramid into one efficient multidimensional-multiscale “Laplacian-Taylor Feature Histogram” for rapid object recognition and localization. The paper describes the criteria for target detection and localization by autonomous robots and how this newly developed framework fits these needs. Comparative results demonstrate the robustness of this recognition framework to noise and localization of target objects in cluttered scenes.

The paper deals with two crucial steps in multi-criteria decision analysis, that are consistency of the judgments and priority vectors for alternatives/criteria. From a side, several consistency indices are proposed for measuring the consistency of a Pairwise Comparison Matrix. From another one, conditions weaker than consistency, such as transitivity and weak consistency, are proposed for representing further levels of coherence of a Decision Maker when he/she expresses his/her preferences by means of reciprocal Pairwise Comparison Matrix. Firstly, in this paper, a simulation is performed in order to establish a relation between random consistency index and coherence level. Then, since weak consistency ensures reliability to priority vectors proposed in literature, a second simulation is performed in order to measure, in case of no weak consistency, the reliability of these priority vectors.

Several recent papers have studied the problem of attribute reduction in concept lattices by using a discernibility matrix in various extensions of formal concept analysis. We recall the clarification and reduction method by Ganter & Wille and show that it is strictly superior to the methods based on discernibility matrix. Consequently, we show how to adapt the clarification and reduction method to the extensions of formal concept analysis considered in the recent papers.

This paper is concerned with the admissibility analysis and stabilization problems for singular fuzzy discrete-time systems with time-varying delay. The novelty of this paper comes from the consideration of a new summation inequality which is less conservative than the usual Jensen inequality, the Abel-Lemma based inequality and the Seuret inequality. Based on the inequality, sufficient conditions are established to ensure the systems to be admissible. Moreover, the corresponding conditions for the existence of desired static output feedback controller gains are derived to guarantee that the closed-loop system is admissible. The conditions can be solved by a modified cone complementarity linearization (CCL) algorithm. Examples are given to show the effectiveness of the proposed method.

This paper is concerned with the problem of designing decentralized quantized controller for networked control systems (NCSs) that are subject to time-varying transmission intervals, packet dropout, communication constraints and limited bandwidth. The convex overapproximation method proposed in Donkers and Heemels (2011) is used to handle with time-varying transmission intervals. Packet dropout process in this paper is modeled as both independent identically distributed process and homogeneous Markov jumping process. The uniform quantizer is adopted here to deal with limited bandwidth. In quantization strategy, which switches between zooming out and zooming in modes, a variable called proportionality coefficient of saturation value is used to handle with the complex coupling relationship between system states and quantized variables. Periodic protocol is used here to deal with communication constraints, under which only one node can access the network and send its information per transmission. Within this setup, we derive stability results in terms of linear matrix inequalities (LMIs). An illustrative example is given to show the usefulness of the proposed framework for the stability analysis of some classes of NCSs.

In the paper, we describe the new measure of matching fuzzy sets. The introduced measure of perturbation of one fuzzy set by another fuzzy set is considered instead of commonly used distance between two fuzzy sets. The operations known in the fuzzy set theory are used and the perturbation of one fuzzy set by another fuzzy set is understood as a measure describing changes of the first fuzzy set after adding the second one. Obviously, the opposite case can also be considered wherein the second fuzzy set is perturbed by the first one. In general, the new measure is asymmetric and can provide more information compare to a distance between fuzzy sets. The values of such measures of fuzzy sets perturbation are in range between 0 and 1. In this paper several mathematical properties of the measure of fuzzy sets perturbation are studied, and the measure of sets perturbation is compared to other selected measures.

This paper proposed a new algorithm based on kernel functions to predict activity of molecule.The proposed kernel functions used the paths of stars as vector of features.Reduction technique is proposed based on the relationship between features.The proposed algorithm was tested on two datasets and competitive results were obtained in accuracy and complexity. Information and computer science fields such as machine learning and graph theory are implemented in chemoinformatics to discover the properties of chemical compounds. This paper presents a new algorithm based on the two-class support vector machine (SVM) model, which has new kernel functions for paths of features, enabling the prediction of chemical compound activity. Initially, we extract all paths of features (star subgraphs) with certain lengths, and we encode them depending on their structure in the graphs. Then, we use these codes to construct two relationship matrices between those paths. These matrices contain common and different sub-paths between paths of stars. The number of sub-paths/paths for each compound is passed to the proposed kernel functions in the two-class SVM to predict the activity of chemical compounds. The relationship matrices created by the proposed algorithm help to reduce the number of features, which improves prediction accuracy. We apply the proposed algorithm with and without feature selection using two benchmark datasets, specifically, the monoamine oxidase (MAO) dataset and the AIDS antiviral screen database of active compound dataset, which have 68 and 2000 chemical compounds, respectively. We perform comparative experiments for the proposed kernel functions and many other two-class SVM prediction methods, and the results before feature selection show prediction accuracies of 94% and 99.5% for MAO and AIDS, respectively. After selection, the prediction accuracies are 96% and 99.5% for MAO and AIDS, respectively.

In this paper, we propose a new multiattribute decision making (MADM) method based on the interval-valued intuitionistic fuzzy weighted geometric average (IIFWGA) operator, the accuracy function of interval-valued intuitionistic fuzzy values (IVIFVs) and particle swarm optimization (PSO) techniques, where the weights of attributes and the evaluating values of alternatives with respect to attributes are represented by IVIFVs. First, the proposed method uses an accuracy function to transform the decision matrix given by the decision maker and represented by IVIFVs into a transformed decision matrix represented by real values in [1,1]. Then, it produces the optimal weights of the attributes based on the obtained transformed decision matrix and PSO techniques. It determines the weighted evaluating IVIFV of each alternative based on the IIFWGA operator, the obtained optimal weights of the attributes and the decision matrix given by the decision maker represented by IVIFVs. Finally, it calculates the transformed value of the weighted evaluating IVIFV of each alternative based on the accuracy function to get the preference order of the alternatives. The main contribution of this paper is that we propose a new MADM method based on the IIFWGA operator of IVIFVs, the accuracy function of IVIFVs and PSO techniques, which can overcome the drawbacks of the existing MADM methods for MADM in interval-valued intuitionistic fuzzy (IVIF) environments.

Stochastic processes with semicontious covariances are introduced in a flexible way.Useful optimal designs are computed.Groundbreaking financial applications for probabilities of ruin, Famas hypothesis on EMH and IS/LM model. We are encountering deep financial crisis. One of the basic problems we face is how to predict financial developments. Current modelling is clearly insufficient since continuity of autocorrelation oversmoothes the empirical financial data. We prove that geometric probability to chose positive definite continuous covariance by financial data is zero. We also have the empirical evidence for this fact from real stock data. In order to overcome this gap we introduce the novel abc class of semicontinuous covariance functions, which integrates both continuous and discontinuous covariances. The abc class of covariances is much more flexible, powerful, and realistic than its continuous counterpart. It gives answer to the question why financial markets face negative interest rates, cyclic downbreaks, and other difficulties nowadays. As it is clearly illustrated in this paper, the experimenter (e.g.the financial institution) can choose only up to some extent whether abc class member will be positive definite everywhere. To the best knowledge of the authors this fits well to the own Kolmogorovs opinions on axiomatic theory of probability. The Kolmogorovs axiomatic theory of probability was developed as one possible alternative in order to utilize the flexibility of Lebesgue integral in probability theory, but in many experiments e.g.in physics and finance, one should go beyond this axiomatics. That means one should accept negative probabilities (and consequently negative definite covariance functions). In this paper we clarify that the modelers choice of positive/negative definite functions should be considered within broader Information Theory setup. The introduced abc class is purely topologically defined, with soft regularity conditions on covariances, which are still applicable for increasing and infill domain asymptotics for regression problems, kriging, and finance. These conditions are related to semicontinuous maps of Ornstein Uhlenbeck (OU) processes. We provide several novel results for optimal design of random fields with abc covariances, random walks in finance, and probabilities of ruins related to shocks (e.g.by earthquakes). In particular, we construct a random walk model with semicontinuous covariance. The novel abc class provides a proper environment for unifying several contradictions on IS/ML model in the economic literature. All mentioned novel economical applications are hardly visible from currently used continuous covariance framework.

Rough set models on two universes are valuable generalizations of classic rough set model. These models can be applied into formal concept analysis and decision-making system. This paper focuses on catoptrical approach for rough set models over two universes using granule-based definition. In this paper, we propose a new catoptrical rough set model over dual universes. The properties of the proposed model are discussed. Consequently, a variable precision extension of the model is presented. Some properties of the extended model are obtained. Finally, we investigate a further extension of the proposed model, i.e. variable precision catoptrical rough set model with two parameters.

Pawlak’s rough membership functions not only give numerical characterizations of Pawlak’s rough set approximations, but also establishes the relationships between Pawlak’s rough sets and fuzzy sets or probabilistic rough sets. However, it is noteworthy that Pawlak’s rough membership functions have limitations when handling incomplete data that exist widely in the real world. As will be shown in this paper, one way to overcome this is to construct rough membership functions for covering-based rough sets. In this paper, we first use an example in evidence-based medicine to illustrate how to use Pawlak’s rough membership function on numerically characterizing decisions under the circumstances where data are complete. Then, we construct covering-based rough membership functions for four types of covering-based rough sets which were examined by Zhu and Wang (in IEEE Transactions on Knowledge and Data Engineering 19(8)(2007) 1131-1144 and Information Sciences 201(2012) 80-92), and use them to characterize these covering-based rough sets numerically. Finally, we present theoretical backgrounds for these covering-based rough membership functions, and illustrate how to apply them on numerically characterizing decisions under the circumstances where data are incomplete.

In this paper, we propose a new fuzzy forecasting method based on two-factors second-order fuzzy-trend logical relationship groups (TSFTLRGs), particle swarm optimization (PSO) techniques and similarity measures between the subscripts of fuzzy sets (FSs) for forecasting the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and the New Taiwan Dollar/US Dollar (NTD/USD) exchange rates. First, we propose a PSO-based optimal-intervals partition algorithm to get the optimal partition of the intervals in the universe of discourse (UOD) of the main factor TAIEX and to get the optimal partition of the intervals in the UOD of the secondary factor SF, where SF ź {Dow Jones, NASDAQ, M1B}. Based on the proposed PSO-based optimal-intervals partition algorithm, the constructed TSFTLRGs, and similarity measures between the subscripts of FSs, we propose a new method for forecasting the TAIEX and the NTD/USD exchange rates. The main contribution of this paper is that we propose a new fuzzy forecasting method based on TSFTLRGs, PSO techniques and similarity measures between the subscripts of FSs for forecasting the TAIEX and the NTD/USD exchange rates to get higher forecasting accuracy rates than the ones of the existing fuzzy forecasting methods.

This paper is concerned with the stochastic stability and stabilization problem for a class of continuous-time semi-Markovian jump systems (S-MJSs) with mode transition-dependent sojourn-time distributions. In contrast to other approaches, this paper offers the possibility to explore more refined probability models on the sojourn-time by considering the mode transition-dependent sojourn-time distributions instead of mode-dependent ones. Here, to construct a finite set of linear matrix inequalities (LMIs) from sojourn-time-dependent matrix inequalities, a less conservative relaxation method is proposed in such a way that all available constraints on time-varying transition rates can be exploited. Simulation examples report the effectiveness of the proposed method.

In a recent paper, De Baets et al. introduced the clone relation of a strict order relation. Two elements of a poset are said to be a pair of clones (or to be clones) if every other element that is greater (resp. smaller) than one of them is also greater (resp. smaller) than the other one. This clone relation played a key role in the characterization of the L -fuzzy tolerance relations and the L -fuzzy equivalence relations that a strict order relation is compatible with. In this paper, we extend the notion of clone relation to any binary relation. Although the definition of such extension is trivial, the corresponding properties significantly differ from those of the clone relation of a strict order relation. We analyse the most important ones among these properties, paying particular attention to a partition of the clone relation in terms of three different types of pairs of clones.