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In this paper, we propose a method for solving the system of fuzzy relation equations (SFRE) with extended max–min composition for inverse inference problems. The properties of interval and constrained solutions with granular and relational structure of the solution set are investigated. The extended max–min SFRE can be represented in the form of the max–min subsystems aggregated using the min operator or dual min–max subsystems aggregated using the max operator. When decomposing the SFRE, the set of solutions can be decomposed into the lower and upper subsets bounded by the same aggregating solutions. Each lower (upper) subset is defined by the unique greatest (least) or aggregating solution and the set of minimal (maximal) solutions. Following Bartl et al. (Fuzzy Sets Syst 187:103–112, 2012), to avoid excessive granularity and ensure interpretability of the interval solutions when restoring causes through observed effects, the constraints in the form of linguistic modifiers are imposed on the measures of causes significances. The interval solutions are modeled by the complete crisp solutions, that is, the maximum solutions for the vectors of binary weights of the linguistic modifiers. The search for approximate solutions of the SFRE amounts to solving the optimization problem using the genetic algorithm. Due to the properties of the solution set, the genetic search for the lower and upper subsets is parallelized for each aggregating solution. The developed method makes it possible to simplify the search for the solution set based on the constraints on accuracy (interpretability) of the applied problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new optimization problem with respect to a generalized form of fuzzy relational equations (FRE) in which fuzzy equality replaces ordinary equality in the constraints (FRE-FC). Fuzzy constraints enable us to attain optimal points (called super-optima in this paper) that are better solutions than those resulted from the resolution of the similar problems with ordinary equality constraints. Some structural properties of the FRE-FC problems are studied and a new formulation is presented in which the fuzzy constraints (equations) are precisely modeled. Subsequently, a new PSO-based algorithm is proposed to solve the FRE-FC problems defined by arbitrary continuous t-norms. The proposed algorithm is tested with different test problems generated by ten well-known continuous t-norms used in the literature. Moreover, the generated solutions for these problems, are also compared with some well-known meta-heuristic methods which have been applied to many practical optimization problems. It is shown that the optimal intensity of electromagnetic radiation problem can be formed as a special case of FRE-FC problems in which fuzzy composition is defied by max-product composition. [ABSTRACT FROM AUTHOR]

Systems of fuzzy relational equations belong to key mathematical problems related to the correct behavior of fuzzy systems, especially of fuzzy inference systems. Indeed, if we find a fuzzy relation that solves a given system of fuzzy relational equations, we found a model of a fuzzy rule base related to the given system that will behave correctly in the sense that it will preserve the fundamental modus ponens property. This problem with numerous results becomes absolutely unexplored as soon as we allow partiality in the system. Partial fuzzy sets have only partially defined membership degrees, i.e., for some elements, the membership degree is undefined, i.e., partial propositions have only partially defined truth-values. There are many origins from which we may obtain propositions with undefined truth-values, e.g., non-denoting or irrelevant terms, inconsistent (both true and false) propositions, or truth-values are simply unknown. Partial logics belong to classical topics in logic so, the extensions to partial fuzzy logic in recent years were not surprising. However, we should not dare to deal with partial fuzzy sets in fuzzy systems without any guarantee of the preservation of fundamental properties such as modus ponens. This article is focusing on filling this gap. [ABSTRACT FROM AUTHOR]

Max-t-norm compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations, especially, max-min and max-product compositions. However, the associativity is not forcefully needed in many cases. In this paper, the max-overlap function composition is considered for the same optimization model. Then some properties of the solution set are obtained. According to these properties, the characterization of the optimal solution for the optimization model is proposed. Furthermore, a simple value matrix with rules is proposed to reduce problem size. Thus, a solution procedure is presented for determining optimal solutions without translating such an optimization problem into two sub-problems. A numerical example is provided to illustrate the procedure. [ABSTRACT FROM AUTHOR]

Managerial decision-making is a complex process that has several problems. The more heterogeneous the system, the more immeasurable, non-numerical information it contains. To understand the cognitive processes involved, it is important to describe in detail their components, define the dependencies between components, and apply relevant algorithms for scenario modelling. Fuzzy cognitive maps (FCMs) is the popular approach for modeling a system's behavior over time and defining its main properties. This work develops a new algorithm for scenario analysis in complex systems represented by FCMs to provide support for decision-making. The algorithm allows researchers to analyze system-development scenarios to obtain the required change to the system's components that leads to the target state. The problem of determining a system's initial state is most conspicuous when constructing a compound or unbalanced fuzzy maps. Currently, a brute force algorithm is used to calculate the steps needed to approach a target, but that takes exponential time. The paper describes a new algorithm to obtain the initial values of the controlled concepts in fuzzy cognitive maps using the theory of neutrosophic fuzzy equations. This approach reduces the time needed to find the optimal solution to a problem, and it allows inverse problems to be solved in the fuzzy cognitive maps as a part of the scenario-modeling framework. [ABSTRACT FROM AUTHOR]

Bipolar max-⁎ equations, with ⁎ a triangular norm, have recently become a popular research topic embedded in the broad field of fuzzy relational equations. In this paper, we lift the work from the restrictive setting of the real unit interval — obfuscating the underlying lattice-theoretical essence — to the general setting of complete distributive symmetric residuated lattices, allowing to build upon the vast body of knowledge on unipolar sup-⁎ equations on complete distributive residuated lattices. We determine the full solution set, with particular emphasis on the extremal solutions, of a bipolar sup-⁎ equation in case the right-hand side is a join-irreducible element. The results are illustrated by means of ample examples. [ABSTRACT FROM AUTHOR]

When a system of max-min equations is inconsistent, which is frequently encountered in modelling with fuzzy relations, the approximate solutions may be considered instead within an admissible error bound. This paper tackles the linear optimization problem over the approximate solutions with respect to the L ∞ and the L 1 residual error, respectively. It demonstrates that such an optimization problem can be reformulated as either a generalized set covering problem for the L ∞ scenario or a mixed integer linear programming problem for the L 1 scenario. As a result, it may be solved to optimality with the aid of a state-of-the-art optimization solver and hence does not demand a tailored solving method except for very large instances. [ABSTRACT FROM AUTHOR]

This paper focuses on a classical problem of computing max-min inverse fuzzy relation. The resolution for the problem is useful for solving the well known problems of fuzzy abductive/backward reasoning. We propose a simple analytical method for finding its exact solutions or approximate solutions. Some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Fuzzy inference systems have been widely investigated from different perspectives, including their logical correctness. It is not surprising that the logical correctness led mostly to the questions on the preservation of modus ponens. Indeed, whenever such a system processes an input equivalent to one of the rule antecedents, it is natural to expect the modus ponens to be preserved and the inferred output to be identical to the respective rule consequent. This leads to the related systems of fuzzy relational equations where the antecedent and consequent fuzzy sets are known values, the inference is represented either by the direct product related to the compositional rule of inference or by the Bandler-Kohout subproduct, and the fuzzy relation that represents the fuzzy rule base is the unknown element in the equations. The most important question is whether such systems are solvable, i.e., whether there even exists a fuzzy relation that models the given fuzzy rule base in such a way that the modus ponens is preserved. Partiality allows dealing with partially defined truth values which allows to deal with situations, where we cannot define the truth value for a given predicate. The partial logics have been recently extended to partial fuzzy logics and the partial fuzzy set theory has been developed. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe. This background leads naturally to the problem of solvability of partial fuzzy relational equations, which are equations with partial fuzzy sets in the role of antecedents and consequents and with partial fuzzy relation as the model of the given fuzzy rule base. Such a setting led to a recent publication that uncovers the solvability and even the shape of the solutions. In this contribution, we revisit the problem and consider a specific case that allows partiality only in the input. In other words, antecedents, as well as consequents expressing the knowledge, are fully defined. Thus, the model of the fuzzy rules is one of the standard and fully defined relations. We investigate what happens if the input differs from one of the antecedents only by a few undefined values. This mimics the situation when a description of an observed object misses a few values in its feature vector. We show that under specific conditions, we still may preserve the modified modus ponens, i.e., that the inferred output is identical with the fully defined consequent of the respective rule. [ABSTRACT FROM AUTHOR]

In this article, we study the inconsistency of systems of max-min fuzzy relational equations. We first compute by an explicit analytical formula the Chebyshev distance (defined by the L-infinity norm) between the second member of an inconsistent system and the set of second members of consistent systems defined with the same matrix: that of the inconsistent system. Then we study the set of Chebyshev approximations of the second member of the inconsistent system, which are the second members of the consistent systems defined with the same matrix as the inconsistent system, whose distance to the second member of the inconsistent system is equal to the computed Chebyshev distance. This set is non-empty and associated to the approximate solutions set of the inconsistent system in the following sense: an approximate solution is a solution vector of a consistent system defined with the same matrix as the inconsistent system, whose second member is a Chebyshev approximation of the second member of the inconsistent system. As main results, we describe both the structure of the approximate solutions set and that of the set of the Chebyshev approximations of the second member of the inconsistent system. As an application of our results, we show that our work can help to find approximate values for the rule parameters of a possibilistic rule-based system. [ABSTRACT FROM AUTHOR]

A minimal solution to a fuzzy relational equation is, loosely speaking, a solution such that the values of all its elements cannot be decreased. To find some minimal solution is simple – one can just identify the greatest solution to the equation and then lower the values of all its elements as much as possible. However, finding all minimal solutions is significantly more difficult since when decreasing the values one needs to go through all combinations of the elements. The paper shows that when considering an n-element scale of truth values equipped with Łukasiewicz or Goguen operations, the set of all minimal solutions can be represented by constrained solutions which allows to skip a considerable amount of these combinations. [ABSTRACT FROM AUTHOR]

This paper focuses on the general decomposition problem of fuzzy relations, which can be regarded as an extension of the common one, modeled by fuzzy relational equation (FRE) X ∘ X = R. The considered decomposition problem is described by an FRE X ∘ Y = R. For the purpose of deriving all solutions to the decomposition problem, the structure of solutions to the corresponding FRE is investigated first, and a necessary and sufficient condition for the existence of solutions is obtained. Then, based on semi-tensor product (STP) of matrices, an algebraic approach is proposed. Furthermore, all solutions to the general decomposition problem can be computed via the proposed method. In order to demonstrate the effectiveness of the presented approach, two numerical examples are given. [ABSTRACT FROM AUTHOR]

A gradient aggregate asymptotical smoothing algorithm is proposed for training fuzzy neural networks (FNNs), which develops the smoothing algorithm (SA) described in Li et al. (2017). By introducing the framework of asymptotical approximation to the algorithm, any degree of approximation for the error function of max–min FNNs can be obtained by an aggregate smoothing function with a variable precision parameter. The algorithm minimizes a sequence of asymptotically approximate functions using the steepest descent algorithm for solving a nondifferentiable max–min optimization problem of max–min FNNs. The proposed update rule on the precision parameter reconciles the conflict between the high-accuracy approximation and the numerical ill-conditioning. The algorithm is globally convergent under Armijo line search. As shown in the simulation results for three artificial examples and a real-world problem in fault diagnosis, compared with SA, the proposed algorithm can efficiently deal with the numerical oscillations and has better performance. [ABSTRACT FROM AUTHOR]

Abstract In a recent paper, De Baets et al. have studied the compatibility of a(n) (strict) order relation with a fuzzy relation, and have characterized the fuzzy tolerance (and, in particular, fuzzy equivalence) relations that a given strict order relation is compatible with. We extend this study by considering the left- and right-compatibility of a(n) (strict) order relation with a fuzzy tolerance relation and vice versa. We characterize the fuzzy tolerance relations that are compatible with a given (strict) order relation. Conversely, we provide a representation of the fuzzy tolerance relations that a given strict order relation is left- or right-compatible with. Specific attention is paid to the case of fuzzy equivalence relations. We conclude by pointing out that the representation theorems in the above-mentioned paper need some minor rectification. [ABSTRACT FROM AUTHOR]

Abstract We present new results regarding the problem of factorization of matrices with grades, or, equivalently, decomposition of fuzzy relations. In particular, we examine geometry of factorizations and the role of fuzzy concept lattices in factorizations of matrices with grades. The results make it possible to reduce input data and enable a more focused search for factors in the search space, and are intended to guide the design of greedy and other approximation algorithms for the decomposition problem, which itself is NP-hard. To demonstrate usefulness of these results, we propose a new factorization algorithm based on these results. Our experiments demonstrate improvements in the quality of factorizations due to the new approach. We conclude by presenting further research topics implied by our findings. [ABSTRACT FROM AUTHOR]

Abstract Relation is one of the important concepts in rough set models. Some uncertainty measures for relations were proposed in recent years by generalizing information entropy. However, the existing theories considered only special relations and do not address mixed-type data well. They always transformed each attribute into a binary relation with the same structure regardless of attribute types. As a result, the structure information can be lost for other types of attributes. In this paper, a new theory about uncertainty measures for general fuzzy relations is proposed. The proposed theory can overcome the weakness of the existing theories. We first introduce a novel entropy to compute the uncertainty information of fuzzy binary relations and then present the concepts of joint fuzzy entropy, conditional fuzzy entropy and fuzzy mutual information. The basic properties of these uncertainty measures are studied. Finally, we apply the proposed theory to perform attribute reduction of heterogeneous data sets. The experimental results show that our proposed method is feasible and effective in some cases. The proposed theory can provide a fundamental framework that integrates all kinds of methods for the uncertainty information of special relations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the degree of the symmetry of fuzzy relations on a set X. Based on the fuzzy e -equality derived from uninorms with unit element e , the degree of the symmetry of fuzzy relations are defined by two different approaches, the type-I degree of symmetry and the type-II degree of symmetry. The main work of this paper include: first, we discuss some basic properties of those two degrees, especially for the relationship between them. In particular, for continuous t-norms, we give a necessary and sufficient condition such that the type-I degree of symmetry and the type-II degree of symmetry are equal. And then, we obtain that the type-II degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type-I degree of symmetry with respect to whether they preserve the fuzzy e -equalities or not. In the meantime, for a special case with continuous t-norms, we provide a necessary and sufficient condition such that the type-I degree of symmetry preserves the fuzzy equalities. Finally, for conjunctive left-continuous idempotent uninorms with neutral element e ∈ (0 , 1 ] , we find out a symmetric fuzzy relation S which is close enough to the given fuzzy relation R with respect to fuzzy e -equality such that the degree of fuzzy e -equality of S and R is the type-II degree of symmetry of R. In particular, we obtain analogous results for continuous t-norms and one class of left-continuous t-norms. [ABSTRACT FROM AUTHOR]

Abstract In this paper new notions of fuzzy equivalences interpreted as fuzzy connectives are introduced. One of the axioms of such fuzzy equivalences is transitivity defined by the use of a fuzzy conjunction or both a fuzzy conjunction and a parameter α from the unit interval, that is why they are called fuzzy α – C -equivalences. In particular, fuzzy C -equivalences, fuzzy semi- C -equivalences as well as fuzzy weak C -equivalences are considered. Dependencies between the new types of fuzzy equivalences are presented. Moreover, the problem of preservation of axioms of such defined fuzzy equivalences by aggregation functions is examined. As a conclusion, ways of generating new fuzzy equivalences of the considered type from given ones are indicated. [ABSTRACT FROM AUTHOR]

Abstract T -generable indistinguishability operators are operators E that can be expressed in the form E = T (E μ 1 , E μ 2 ,... , E μ m ) , where T is a t -norm and E μ is the fuzzy relation generated by the fuzzy subset μ. In this paper we analyse their relation with powers with respect to the t -norm T and with quasi-arithmetic means. For non-strict continuous Archimedean t-norms they are completely characterised as generable by crisp equivalence relations. These fuzzy relations are used to define a method, called JADE , useful for feature selection and classification tasks. JADE is based on minimising the distance between two indistinguishability measures: the one given by weighting the attribute-values describing the domain objects and the other one given by the correct classification taken as an equivalence relation. The preliminary experiments we carried out with JADE are promising concerning the accuracy in solving classification tasks. We also report some issues of the method that could be improved in the future. [ABSTRACT FROM AUTHOR]

L∘U-factorization was recently used to solve the max-product fuzzy relation equation by Molai (Inf Sci 234:86-96, 2013). Considering the forward and backward substitutions play an important role in this method, this paper firstly amend the forward and backward substitutions for solving max-product fuzzy relation equation with L∘U-factorization. And then, the computational complexities of improved forward and backward substitutions are analyzed in detail. Finally, we find that the L∘U-factorization acts as splitting an irredundant covering of max-product fuzzy relation equation into two parts. It therefore cannot change the fact that finding the solutions of max-product fuzzy relation equation with L∘U-factorization is an NP-hard problem. On the contrary, the computational expense will linearly increase with the number of minimal solutions of L∘y=b. [ABSTRACT FROM AUTHOR]

The paper introduces a new approach to preference structure, where from a weak preference relation derive the following relations: strict preference, indifference and incomparability, which by aggregations and negations are created and examined. We decomposing a preference relation into a strict preference, an indifference, and an incomparability relation. This approach allows one to quantify different types of uncertainty in selecting alternatives. In presented preference structure we use interval-valued fuzzy relations, which can be interpreted as a tool that may help to model in a better way imperfect information, especially under imperfectly defined facts and imprecise knowledge. Preference structures are of great interest nowadays because of their applications, so we propose at the end the algorithm of decision making by use new preference structure. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate the aggregation of diverse families of binary fuzzy relations, fuzzy filters, and fuzzy lattices. Some links between these families and their images via an aggregation are explored. [ABSTRACT FROM AUTHOR]

Numerous fuzzy time series (FTS) forecasting methods have been proposed in scientific literature and has achieved growing attention in practice. Most of them are based on modeling fuzzy relationships(FRs) of the past data. In this paper, a new hybrid forecasting model based on the concept of time-variant FR group (TV-FRG), particle swarm optimization technique (PSO) and refinement technique in the defuzzization stage is presented to forecast the enrolments of the University and the stock market. PSO technique is utilized to adjust for obtaining the effective length of each interval in the universe of discourse for the forecasting model. Most of the existing forecasting models simply ignore the repeated FRs without any proper justification or accept the number of recurrence of the FRs without considering the appearance history of these FRs in the grouping FRs process. Therefore, the appearance history of the fuzzy sets on the right-hand side of the FRs is considered to establish the FR groups, called the TV-FRGs. Furthermore, the high-order TV-FRGs are used in order to obtain more accurate forecasting results in the defuzzication stage. To calculate these high-order TV-FRGs values, a refined defuzzication technique is developed, and incorporated in the proposed model. To verify the effectiveness of the proposed model, two numerical simulations are examined with the case of University enrollments and Taiwan futures exchange (TAIFEX). The experimental results show that the proposed model achieves good forecasting results compared to other existing forecasting models based on the high-order FTS. These promising results bring a significant meaning for the future work on the development of FTS and PSO algorithm in real-world forecasting applications. [ABSTRACT FROM AUTHOR]

This paper describes a set of solution-set-invariant coefficient matrices of the fuzzy relation equation with max-min composition A ⊙ x = b or ⋁ j ∈ { 1 , 2 , ⋯ , n } ( a ij ∧ x j ) = b i for all i ∈ { 1 , 2 , ⋯ , m } with 1 ≥ b 1 ≥ b 2 ≥ ⋯ ≥ b m > 0. An algorithm is given for determining the set scripfontA = {B ∣ {x ∣ B ⊙ x = b} = {x ∣ A ⊙ x = b}} completely. [ABSTRACT FROM AUTHOR]

Some fuzzy set and fuzzy relational inequations and equations are considered in the framework of a meet-continuous codomain lattice. The existence of maximal solutions to some fuzzy set inequations is proven. An algorithm is given for calculating the closure of a fuzzy set (relation) under the composition with a given fuzzy relation. Another algorithm for finding the transitive closure is proven to work in this framework. [ABSTRACT FROM AUTHOR]

In many small-data-learning problems, owing to the incomplete data structure, explicit information for decision makers is limited. Although machine learning algorithms are extensively applied to extract knowledge, most of them are developed without considering whether the training sets can fully represent the population properties. Focusing on small data which contains nominal inputs and continuous outputs, this paper develops an effective sample generating procedure based on fuzzy theories to tackle the learning issue by data preprocessing. According to the derived fuzzy relations between categories and continuous outputs, the possibilities of the combinations of categories (virtual samples) can be aggregated when continuous outputs are given. Proper virtual samples are further selected by using fuzzy alpha-cut on the possibility distributions, and these are added to the training sets to form new ones. In the experiment, sixteen datasets taken from the UC Irvine Machine Learning Repository are examined with back-propagation neural networks and support vector regressions. The results reveal that the forecasting accuracies of the two models are significantly improved when they are built with the proposed new training sets. Moreover, the results also indicate the proposed method outperforms bootstrap aggregating and the synthetic minority over-sampling technique-Nominal-Continuous with the greatest amount of statistical support. [ABSTRACT FROM AUTHOR]

This paper considers application of the fault tree method to analyze types and reasons for failure of an annular blowout preventer (BOP). In combination with fuzzy relation analysis theory, we have calculated the fuzzy importance of basic events which can lead to failure plus the correlation coefficient for the minimum cut sets. Consequently, we can obtain the degree of correlation between the top event and the minimum cut sets. We have determined the key factors influencing stable operation of the annular BOP. Our results can provide a theoretical basis for use and improvement of an annular BOP. [ABSTRACT FROM AUTHOR]

This paper is a short note contribution to the topic of fuzzy relational inference systems and the preservation of their desirable properties. It addresses the two main fuzzy relational inferences – compositional rule of inference (CRI) and the Bandler–Kohout subproduct (BK-subproduct) – and their combination with two fundamental fuzzy relational models of fuzzy rule bases, namely, the Mamdani–Assilian and the implicative models. The goal of this short note article is twofold. Firstly, we show that the robustness related to the combination of BK-subproduct and implicative fuzzy rule base model was not proven correctly in [24] . However, we will show that the result itself is still valid and a valid proof will be provided. Secondly, we shortly discuss the preservation of desirable properties of fuzzy inference systems and conclude that neither the above mentioned robustness nor any other computational advantages should automatically lead to a preference of the combinations of CRI with Mamdani–Assilian models or of the BK-subproduct with the implicative models. [ABSTRACT FROM AUTHOR]

Satisfiability of three naturally proposed axioms, which are desirable for fuzzy inference system, was discussed by Moser and Navara in [36] . The fuzzy rule base models considered in this study were mainly the Mamdani–Assilian ones and partly also logically motivated implicative fuzzy systems. The investigation leads to the fact that often none of these fuzzy systems do satisfy the given axioms simultaneously. Therefore, Moser and Navara proposed so-called conditionally firing rules and proved that the simultaneous satisfaction of the three axioms is feasible under very mild and practically non-restrictive conditions. Note that the inference mechanism, tailored to the fuzzy rules in the investigation, was the compositional rule of inference (abb. CRI) and similarly, the concept of conditionally firing rules were stemming from the CRI and the Mamdani–Assilian model of a fuzzy rule base. For the implicative interpretation of fuzzy rules, the original axioms proposed by Moser and Navara were later on re-defined by Štěpnička and Mandal with the aim to express the meaning of the original axioms initially proposed for the Mamdani–Assilian rules. Furthermore, conditionally firing implicative rules were proposed based on the so-called Bandler–Kohout subproduct (abb. BKS) inference mechanism and a study similar to the one by Moser and Navara was undertaken for them. The study encompasses the satisfaction of the modified axioms for the use of implicative rules and the BKS. Furthermore, some additional results confirming the applicability of the conditionally firing rules satisfying the Moser–Navara axioms are also provided, including the universal approximation ability. However, the solvability criterions for fuzzy relational equations give a strong preference to the usage of the other combinations, namely, to the combination of CRI with the implicative interpretation of fuzzy rules and to the combination of BKS with the Mamdani–Assilian interpretation of fuzzy rules. Therefore, also these combinations of fuzzy inference mechanisms and interpretations of fuzzy rules should be investigated with respect to the satisfaction of Moser–Navara axioms. The main purpose of this article, which extends the short conference contribution [47] by Štěpnička, is to provide such an investigation jointly with the review on conditionally firing rules and supplementary results. [ABSTRACT FROM AUTHOR]

This paper focuses on fuzzy relational structures. In order to collect the knowledge that has been acquired with regard to fuzzy relational structures, the concept of collective relations is proposed and characterizations of collective relations are investigated. A method on computing the largest collective relation is further given. Moreover, based on substructures of fuzzy relational structures, a decomposition theorem of collective relations is obtained and weak collective relations and strong collective relations are put forward. Finally, the fuzzy rough set analysis of fuzzy relational structures is discussed. [ABSTRACT FROM AUTHOR]

This paper presents result on the lattice structure resulting from T-transitive fuzzy orderings. Some representation theorems are also proved on the new structure. [ABSTRACT FROM AUTHOR]

The product of gene expression works together in the cell for each living organism in order to achieve different biological processes. Many proteins are involved in different roles depending on the environment of the organism for the functioning of the cell. In this paper, we propose gene ontology (GO) annotations based semi-supervised clustering algorithm called GO fuzzy relational clustering (GO-FRC) where one gene is allowed to be assigned to multiple clusters which are the most biologically relevant behavior of genes. In the clustering process, GO-FRC utilizes useful biological knowledge which is available in the form of a gene ontology, as a prior knowledge along with the gene expression data. The prior knowledge helps to improve the coherence of the groups concerning the knowledge field. The proposed GO-FRC has been tested on the two yeast ( Saccharomyces cerevisiae ) expression profiles datasets (Eisen and Dream5 yeast datasets) and compared with other state-of-the-art clustering algorithms. Experimental results imply that GO-FRC is able to produce more biologically relevant clusters with the use of the small amount of GO annotations. [ABSTRACT FROM AUTHOR]

The hesitant fuzzy linguistic preference relations (HFLPRs) facilitate decision makers to express hesitant assessments in fuzzy decision making problems. It is important for the use of consistent HFLPRs since inconsistency leads to unreliable decision making results. This paper aims at proposing the additive and multiplicative consistency improvement processes for HFLPRs such that they reach an acceptable level. Some properties of the additive and multiplicative consistency of HFLPRs are investigated, based on which two algorithms for improving the additive consistency and multiplicative consistency are introduced. Two numerical examples are analyzed by using the algorithms and the comparative analysis between the proposed model and other models is also given. A multi-criteria decision making model based on HFLPRs is proposed, and a practical example on performance evaluation of commercial banks is presented to show the feasibility of the proposed model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, as an extension of variable precision (θ, ω)-fuzzy rough sets, we introduce the notion of the generalized granular variable precision (R̲β,R̅β)-fuzzy rough sets determined by fuzzy granules with R ∊ LXxY on a residuated and coresiduated lattice with an involution N. We investigate the properties of R̲β and R̅β with eLY and dLY are (L, ☉)-fuzzy preorder and (L, ⊗)-fuzzy co-preorder, respectively. Moreover, we investigate Alexander topologies and Alexander cotopologies induced by R̲⊤ and R̅⊥, respectively. For an L-fuzzy relation R ∊ LXxX and R̲β, R̅β for each β ∊ L, we obtain Alexandrov L-fuzzy pretopologies and Alexandrov L-fuzzy precotopologies on X, respectively. We give their examples. [ABSTRACT FROM AUTHOR]

Relational Medoid based fuzzy relational clustering (FRC) algorithms perform better than center based FRC. However, in medoid based FRC the selection of medoid is solely random and sometimes lead to inconsistent results. In this paper, a subtractive medoids selection based fuzzy relational clustering (SMS-FRC) method is proposed. In SMS-FRC algorithm inherent geometry and density of pairwise dissimilarity values are preferred over random initial values of medoids. The SMS-FRC is applied to identify clusters of user sessions from server log data, based on their browsing behavior. The concept of augmented sessions is used to derive the page relevance based intuitive augmented dissimilarity matrix. The experiments are performed on a publicly available log data from NASA web server. The generated clusters are evaluated using various fuzzy cluster validity measures, and results are compared with relational fuzzy c-medoids (RFCMdd) clustering algorithm. The results suggest the quality of fuzzy clusters discovered using SMS-FRC clustering is better than that of those obtained with the relational fuzzy c-medoids algorithm. [ABSTRACT FROM AUTHOR]

The aim of this paper is, first, to recall fuzzy relational compositions (products) and, to introduce an idea, how excluding features could be incorporated into the theoretical background. Apart from definitions, we provide readers with a theoretical investigation. This investigation addresses two natural questions. Firstly, under which conditions (in which underlying algebraic structures) the given three natural approaches to the incorporation of excluding symptoms coincide. And secondly, under which conditions, the proposed incorporation of excluding features preserves the same natural and desirable properties similar to those preserved by fuzzy relational compositions. The positive impact of the incorporation on reducing the suspicions provided by the basic “circlet” composition without losing the possibly correct suspicion is demonstrated on a real taxonomic identification (classification) of Odonata. Here, we demonstrate how the proposed concept may eliminate the weaknesses provided by the classical fuzzy relational compositions and, at the same time, compete with powerful machine learning methods. The aim of the demonstration is not to show that proposed concept outperforms classical approaches, but to show, that its potential is strong enough in order to complete them or in order to be combined with them and to use its different nature. [ABSTRACT FROM AUTHOR]

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

A fuzzy inference framework based on fuzzy relations is developed, adapted and applied to temperature and humidity measurements from a specific coffee crop site in Brazil. This framework consists of fuzzy relations over possibility distributions, resulting in a model analogous to a Bayesian inference process. The application of this fuzzy model to a data set of experimental measurements and its correspondent forecasts of temperature and humidity resulted in a set of revised forecasts, that incorporate information from a historical record of the problem. Each set of revised forecasts was compared with the correspondent set of experimental data using two different statistical measures, MAPE (Mean Absolute Percentage Error) and Willmott's D. This comparison showed that the sets of forecasts revised by the fuzzy model exhibited better results than the original forecasts on both statistical measures for more than two thirds of the evaluated cases. [ABSTRACT FROM AUTHOR]