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• Research about the intuitionistic fuzzy set theory is analyzed systematically with quantitative analysis. • The evolution of hot topics and knowledge diffusion path are investigated. • The self-citations phenomenon is deeply discussed when performing main path analysis. • The application of the intuitionistic fuzzy set in the domain of decision making is becoming more and more dominant. Since the intuitionistic fuzzy set theory was proposed, it has received widespread attention. To get a deeper understand of this field, this paper analyzes a total of 2913 papers downloaded from the Web of Science (WoS) from 1984 to 2019 from different perspectives. On the one hand, this research identifies important themes and their interrelationships, showing the thematic evolution of this field vividly. On the other hand, this research digs out the knowledge diffusion paths of this domain with the help of global and key-route main paths. From the perspective of the thematic evolution, five thematic area are detected including Intuitionistic fuzzy relation , IVIFS-Similarity , MCDM/MADM-GDM , Aggregation operator and Extensions of IFS. Among them, MCDM/MADM-GDM has evolved from an emerging theme to a motor theme being the most important topic and GDM integrating themes MADM 、 MCDM and Aggregation operator shows continuous growth. From the perspective of knowledge diffusion paths, research topic has gradually translated from theorical construction to practical application, and several factors such as self-citation, indirect citation, hierarchy and references that may influence the result of the main path are discussed. In general, this research systematically provides scholars with the development process in this domain, which is conductive to them to fully grasp the state-of-the-art research. [ABSTRACT FROM AUTHOR]

This paper presents a Prolog-based reasoning module to generate counterfactual explanations given the predictions computed by a black-box classifier. Our approach comprises four well-defined stages that can be applied to any structured pattern classification problem. Firstly, we pre-process the given dataset by imputing missing values and normalizing the numerical features. Secondly, we transform numerical features into symbolic ones using fuzzy clustering such that extracted fuzzy clusters are mapped to an ordered set of predefined symbols. Thirdly, we encode instances as a Prolog rule using the nominal values, the predefined symbols, the decision classes, and the confidence values. Fourthly, we compute the overall confidence of each Prolog rule using fuzzy-rough set theory to handle the uncertainty caused by transforming numerical quantities into symbols. This step comes with an additional theoretical contribution to a new similarity function to compare the previously defined Prolog rules involving confidence values. Finally, we implement a chatbot as a proxy between humans and the Prolog-based reasoning module to resolve natural language queries and generate counterfactual explanations. During the numerical simulations using synthetic datasets, we study the performance of our system when using different fuzzy operators and similarity functions. [ABSTRACT FROM AUTHOR]

This paper presents a Prolog-based reasoning module to generate counterfactual explanations given the predictions computed by a black-box classifier. Our approach comprises four well-defined stages that can be applied to any structured pattern classification problem. Firstly, we pre-process the given dataset by imputing missing values and normalizing the numerical features. Secondly, we transform numerical features into symbolic ones using fuzzy clustering such that extracted fuzzy clusters are mapped to an ordered set of predefined symbols. Thirdly, we encode instances as a Prolog rule using the nominal values, the predefined symbols, the decision classes, and the confidence values. Fourthly, we compute the overall confidence of each Prolog rule using fuzzy-rough set theory to handle the uncertainty caused by transforming numerical quantities into symbols. This step comes with an additional theoretical contribution to a new similarity function to compare the previously defined Prolog rules involving confidence values. Finally, we implement a chatbot as a proxy between humans and the Prolog-based reasoning module to resolve natural language queries and generate counterfactual explanations. During the numerical simulations using synthetic datasets, we study the performance of our system when using different fuzzy operators and similarity functions. [ABSTRACT FROM AUTHOR]

• Group decision making with fuzzy information. • Multi-person decision making with the theory of expertons. • A new computer software for multi-person decision making. • Weighted averages in multi-expert analysis. Experton theory, a generalization of probabilistic set theory, that is of great usefulness to group decision analysis, was first proposed as a means of improving the processing and analysis of opinions issued by experts. This theory produces an information-fusion mathematical object, the experton, which can be used in predictive problems to justify decisions based on well-constructed reasoning. The aim of this paper is to present an aggregative method of several expertons, with the idea that some of the groups of experts involved in producing these expertons may have more influence than others in the decision-making process. In this article, we carry out an aggregation analysis of expertons, not experts, which culminates in the creation of a new mathematical object. This object, which is called the weighted average multiexperton, is coherent with an experton-type object created from a weighting of the initial data provided by all experts. Since the aggregation method presented has been devised to represent the decision-maker's attitude regarding the importance of different groups of experts, this approach represents a new tool for dealing with group decision-making problems. Additionally, the study presents some properties of the new object. Finally, the paper ends with an application for business decision-making. [ABSTRACT FROM AUTHOR]

• We introduce the first convex rough set model on a finite domain. • Convex geometries are a special type of coverings that cater for convexity. • Upper and lower approximation operators are defined from convex geometries. • Relationships with other models in rough set theory are disclosed. This paper addresses a foundational aspect of imprecision in information and knowledge. It makes a convincing case that convexity can take part in the progress of rough set theory in finite settings. To this purpose we resort to convex geometries, which constitute a special type of coverings that abstract many combinatorial features of convexity. We define convex geometry (cg) approximation spaces on a grand set , and we produce novel cg-upper and cg-lower approximation operators. Their basic properties are presented. Then we show that the model that arises has connections with well-established models in the rough set literature, both from relation and covering-based approaches. We identificate three types of subsets of the grand set that have different behaviors with respect to their cg-approximations, and we refine this classification in some benchmark cases. Finally, we produce a canonical convex geometry approximation space from any covering on a set. Examples illustrate our constructions and main results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper explores the properties of subsethood measures, including boundary conditions, monotonicity, duality, and additivity. These properties can be used as candidates of axioms to define specific subsethood measures. The standard set theory introduces the set-inclusion relation with a qualitative nature, that is, a set is either a subset of the other or not. As a quantitative generalization, a subsethood measure considers the degree of inclusion. It should keep essential characteristics of the qualitative set-inclusion and satisfy generalized properties with the qualitative set-inclusion relation as a special case. A systematic study of the relationships between the qualitative and quantitative frameworks results in the four types of properties of subsethood measures presented in this paper. A combination of different types of properties helps in constructing an appropriate subsethood measure in real-world applications. [ABSTRACT FROM AUTHOR]

Selecting the appropriate manufacturer is important in customized product development because a poor selection may delay the delivery schedule, increase costs, and even affect product quality. This selection task inherently involves ambiguous and imprecise information from decision-makers. The q-rung orthopair fuzzy (q-ROF) sets theory has been proven as a valuable instrument to model human uncertain expressions. However, the existing q-ROF score functions, an essential tool to rank q-ROF values, have some deficiencies, such as generating counterintuitive solutions and antilogarithm or division by zero problems. Furthermore, little research presented the modified best-worst methods (BWM) under q-ROF environments. Nevertheless, these models only generate crisp weights rather than q-ROF weights, which is against the original intention of the q-ROF BWM. To address these problems, this paper first introduces a novel q-ROF score function for measuring q-ROF values. A new q-ROF-BWM is then presented based on the q-ROF preference relations to determine the fuzzy criteria weights. Subsequently, an improved weighted aggregated sum product assessment (WASPAS) with q-ROF settings is presented. Based on them, an integrated q-ROF-BWM-WASPAS model is introduced to rank manufacturers. Additionally, a case study, sensitivity analysis, and several comparisons are conducted to illustrate and validate the usefulness of the developed model. [ABSTRACT FROM AUTHOR]

Distance education quality evaluation is extremely important in improving the quality of education under COVID-19. As traditional teaching-quality evaluation methods are no longer applicable, it is crucial to construct effective evaluation methods. In the evaluation of distance education quality, decision-makers have different linguistic expression preferences, and the evaluation information may be biased due to an improper grasp of the problem. In addition, the correlation between the criteria of distance education quality evaluation is common, and the results of existing evaluation methods are quite different. In this paper, to compensate for these deficiencies, we utilize the multi-granularity probabilistic linguistic term set (MGPLTS), which can reflect the linguistic expression preference of decision-makers and the importance of linguistic terms, and propose a multi-criteria group decision-making (MCGDM) method. First, the dispersion and concentration degrees are proposed as the theoretical basis for judging the hesitancy of decision-makers' evaluation information, and the decision-maker weight adjustment model is constructed. To reflect the importance and correlation of criteria, the SWARA method and the CRITIC method are constructed as criteria weight methods. To obtain reliable decision results, decision-makers' psychological expectations are taken into account, the MULTIMOORA method is improved upon, and a new integration theory is proposed to improve its robustness. Finally, through an example case of distance education quality evaluation and comparison with other methods, the effectiveness, practicability and superiority of this method are verified. [ABSTRACT FROM AUTHOR]

A rule-based system is a widely used artificial intelligence system that employs a set of rules to make decisions. The belief rule-based (BRB) classification system is an extension of fuzzy rule-based (FRB) system that handles uncertainty and imprecision in classification tasks by incorporating Dempster-Shafer evidence theory and fuzzy set theory. However, the BRB classification system suffers from the combinatorial explosion and high time complexity problems. To solve these issues, the fuzzy unordered rule induction algorithm (FURIA) is introduced into the BRB classification systems to design a novel BRB classification system in this paper. FURIA is computationally less complex and has superior classification performance. The proposed system outperforms BRB classification systems in terms of classification performance and significantly reduces the number of rules and conditions. Furthermore, the proposed system offers a more effective solution than FURIA. To evaluate the validity and superiority of the proposed system, we conduct two classification experiments, comparing it with five traditional classifiers and seven rule-based systems, respectively, in terms of classification performance and interpretability. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Few studies of conflict analysis have analyzed conflict situations from the perspective of the consistency degree of cliques. Quantifying the conflict degree is instructive for weakening or resolving conflicts. In this paper, we propose the improved Pawlak model of conflict analysis, a common consistency measurement is introduced to quantify the consistency degree of cliques, which explains the internal causes of conflict situation from a new perspective. In addition, the proposed consistency measurement reflects the fit between feasible strategy and clique, which inspires us to formulate the criteria for selecting feasible strategies in a three-valued situation table. Moreover, considering that many-valued ratings are more powerful and effective to describe vague information, a novel model of fuzzy conflict analysis is established with a fuzzy conflict function and a fuzzy consistency measurement in a fuzzy situation table, which could degenerate into the improved Pawlak model of conflict analysis. The criteria of selecting feasible strategies is illustrated by an NBA labor negotiation. Finally, the performance of proposed models is compared with other advanced models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

As a typical methodology to deal with uncertain issues, the three-way decision (3WD) has been developed rapidly in nearly ten years, both in theories and applications. Three-way classification is one of the important research fields of 3WD, which utilizes the idea of 3WD to solve classification problems. In this paper, we focus on investigating the effectiveness of three-way classification through two evaluation indicators: the classification quality (Precision , Recall , Accuracy and F 1 ) and the decision cost. The comparisons between two-way classification and three-way classification are concretely analyzed, some mathematical properties, judging conditions and decision criteria of these two classification methods are also discussed in detail. Finally, the experimental results on eight UCI data sets validate the mathematical analysis, which reveal the effectiveness of three-way classification. [ABSTRACT FROM AUTHOR]

This paper considers parameter identification problems for multivariable Hammerstein controlled autoregressive moving average (CARMA) systems. We transform a multivariable Hammerstein CARMA system into a new identification model which contains a set of bilinear parameter vectors, i.e., two sets of parameter vectors. To solve the difficulty of estimating the two sets of parameter vectors, this paper constructs two different forms of the identification model in each of which the output is linear about one set of parameter vectors, and presents an extended stochastic gradient algorithm to interactively estimate the two sets of parameter vectors by using the hierarchical identification principle. The proposed hierarchical identification algorithm has a higher computational efficiency than the over-parametrization model based stochastic gradient algorithm. The simulation results indicate that the proposed algorithm is effective, and an example of modeling a photovoltaic array system with the multivariable Hammerstein model is given. [ABSTRACT FROM AUTHOR]

Image fusion method can provide a high-quality image by merging the multiple features of different source images, and how to effectively evaluate the quality (informativeness) of image features is an important issue for image fusion. Because a considerable amount of imprecise and uncertain information exists in image fusion processes, this paper proposes a framework based on fuzzy set theory to handle the vague features, and a set of hybrid optimization methods is also designed to improve the performance. First, the two-scale decomposition method is utilized to decompose the source images and obtain a set of corresponding subimages. Second, fuzzy set theory and local spatial frequency are employed to generate preliminary decision maps by evaluating the pixel quality of the subimages. Third, a morphological method and consistency verification are utilized to optimize the decision maps to extract the focused and unfocused regions. Finally, three schemes are designed to generate the fused images according to the optimized decision maps. The experimental results show that the proposed method can achieve competitive performance compared with other methods. [ABSTRACT FROM AUTHOR]

In recent decades, sustainable investing has caught on with investors, and it has now become the norm. In the age of start-ups, with scant information on the sustainability aspects of an asset, it becomes harder to pursue sustainable investing. To this end, this paper proposes a sustainable financial portfolio selection approach in an intuitionistic fuzzy framework. We present a comprehensive three-stage methodology in which the assets under consideration are ethically screened in Stage-I. Stage-II is concerned with calculating the sustainability scores, based on various social, environmental, and economic (SEE) criteria and an evaluation of the return and risk of the ethical assets. Intuitionistic fuzzy set theory is used to gauge the linguistic assessment of the assets on several SEE criteria from multiple decision-makers. A novel intuitionistic fuzzy multi-criteria group decision-making technique is applied to calculate the sustainability score of each asset. Finally, in Stage-III, an intuitionistic fuzzy multi-objective financial portfolio selection model is developed with maximization of the satisfaction degrees of the sustainability score, return, and risk of the portfolio, subject to several constraints. The ϵ -constraint method is used to solve this model, which yields various efficient, sustainable financial portfolios. Subsequently, investors can choose the portfolio best suited to their preferences from this pool of efficient, sustainable financial portfolios. A detailed empirical illustration and a comparison with existing works are given to substantiate and validate the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, an innovative methodology for the generation of a Road Network Definition File (RNDF) using only an obstacle map as input is presented. This RNDF, which relies on a Multiclass Support Vector Machine(MSVM)-based trajectory generation method, will be used by an autonomous vehicle for transporting people in closed, unstructured areas for which no previous information is available, such as residential areas or industrial parks. The advantages of using this technique are the generation of a safe and smooth trajectory graph (making the trip more comfortable for riders by having trajectories pass as far away as possible from obstacles). Moreover, although there exist other previous Support Vector Machine (SVM) path planning methods, this is the first to use a MSVM. The advantages of doing so are that by obtaining a decision boundary for each object in the scene, all possible trajectories are computed and joined to form a graph. This is done through a combination of a Nearest-Neighbor Graph (NNG) and a Relative Neighborhood Graph (RNG). The method was tested with real data and in real conditions, yielding good results. At the end of the paper, results for two kinds of studies are presented. The first set of tests is intended to determine the best parameter values for the proposed methodology. In the second set of evaluations, the approach is compared with other state-of-the-art SVM-based methods, as well as with a classical approach, demonstrating that the method outperforms them in some aspects. Furthermore, the source code of the method is available for testing, as are some videos in which the output of the method is shown, including a comparison with previous methods. [ABSTRACT FROM AUTHOR]

The problems to find attribute reduction in rough sets and to obtain the minimal vertex cover for graphs are both NP-hard problems. This paper studies the relationship between the two problems. The vertex cover problem for graphs from the perspective of rough sets is first investigated. The attribute reduction of an information system is then studied in the framework of graph theory. The results in this paper show that finding the minimal vertex cover of a graph is equivalent to finding the attribute reduction of an information system induced from the graph. Conversely, the attribute reduction computation can be translated into the calculation of the minimal vertex cover of a derivative graph. Finally, a new algorithm for the vertex cover problem based on rough sets is presented. Furthermore, experiments are conducted to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A comparison of two objects may be viewed as an attempt to determine the degree to which they are similar or different in a given sense. Defining a good measure of proximity, or else similarity or dissimilarity between objects is very important in practical tasks as well as theoretical achievements. Each object is usually represented as a point in Cartesian coordinates, and therefore the distance between points reflects similarities between respective objects. In general, the space is assumed to be Euclidean, and a distance between points assigns a nonnegative number. From another point of view the concept of symmetry underlies essentially all theoretical treatments of similarity. Tversky (1977) provided empirical evidence of asymmetric similarities and argued that similarity should not be treated as a symmetric relation. According to Tversky’s consideration, an object is described by sets of features instead of geometric points in a metric space. In this paper we propose a new measure of remoteness between sets of nominal values. Instead of considering distance between two sets, we introduce the measures of perturbation type 1 of one set by another . The consideration is based on set-theoretic operations and the proposed measure describes changes of the second set after adding the first set to it, or vice versa. The measure of sets’ perturbation returns a value from [0, 1], and it must be emphasized that this measure is not symmetric in general. The difference between 1 and the sum of these two measures of perturbation of a pair of sets can be understood as Jaccard’s extended similarity measure. In this paper several mathematical properties of the measure of sets’ perturbation are studied, and interpretation of proximity is explained by the comparison of selected measures. [ABSTRACT FROM AUTHOR]

Aggregate nearest neighbor query, which returns an optimal target point that minimizes the aggregate distance for a given query point set, is one of the most important operations in spatial databases and their application domains. This paper addresses the problem of finding the aggregate nearest neighbor for a merged set that consists of the given query point set and multiple points needed to be selected from a candidate set, which we name as merged aggregate nearest neighbor(MANN) query. This paper proposes two algorithms to process MANN query on road networks when aggregate function is max. Then, we extend the algorithms to support other aggregate functions (e.g., sum). Extensive experiments are conducted to examine the behaviors of the solutions in terms of five parameters affecting the performance. The overall experiments show that our strategies to minimize the response time are effective. [ABSTRACT FROM AUTHOR]

This paper studies the problem of k-optimal-location-selection ( k OLS) retrieval in metric spaces. Given a set D A of customers, a set D B of locations, a constrained region R , and a critical distance d c , a metric k OLS (M k OLS) query retrieves k locations in D B that are outside R but have the maximal optimality scores . Here, the optimality score of a location l ∈ D B located outside R is defined as the number of the customers in D A that are inside R and meanwhile have their distances to l bounded by d c according to a certain similarity metric (e.g., L 1 -norm, L 2 -norm, etc.). The existing k OLS methods are not sufficient because they are applicable only to the Euclidean space, and are not sensitive to k . In this paper, for the first time, we present an efficient algorithm for k OLS query processing in metric spaces. Our solution employs metric index structures (i.e., M-trees) on the datasets, enables several pruning rules , utilizes the advantages of reuse technique and optimality score estimation , to support a wide range of data types and similarity metrics. In addition, we extend our techniques to tackle two interesting and useful variants, namely, M k OLS queries with multiple or no constrained regions. Extensive experimental evaluation using both real and synthetic data sets demonstrates the effectiveness of the presented pruning rules and the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

One important and valuable topic in rough set theory is the concept of approximation over various generalized information systems. Although rough set models and approaches over hybrid attribute sets have been studied by many researchers recently, the studies focus on the rough approximation of a crisp concept over a hybrid attribute information system. This paper considers the rough approximation of a fuzzy concept over a hybrid attribute information system. We define a hybrid indiscernibility relation by compounding the fuzzy indiscernibility (similarity) relation over numerical attributes with the equivalence relation over symbolic attributes. Then we present the lower and upper approximations of a fuzzy set based on the hybrid indiscernibility relation, i.e., the fuzzy rough set over a hybrid attribute information system. Also, some interesting properties of the hybrid fuzzy rough approximation operators are presented in detail and the relationship between the fuzzy rough set and the existing rough set model over a hybrid attribute information system is established. Meanwhile, based on the definition of the cut set of a fuzzy set of the universe of discourse, the representations of the lower and upper approximations of a fuzzy set are given with respect to hybrid attribute approximation space. Furthermore, the concept of the cut set of the hybrid indiscernibility relation of the hybrid attribute approximation space is defined, then the characterizations of the hybrid fuzzy rough approximation operators are established as well. Two different types of characterization theorems of the hybrid fuzzy rough approximation operators are presented based on the cut set of the hybrid indiscernibility relation over the hybrid attribute information system. At the same time, we study the uncertainty measure of the fuzzy rough set over a hybrid attribute information system by using knowledge granulation. The main contribution of this paper is twofold. One is to extend the existing rough set approach over hybrid attribute information systems to a fuzzy environment. Another is to present a new way to define the rough approximation of a fuzzy concept as well the uncertainty measure over hybrid attribute information systems. [ABSTRACT FROM AUTHOR]

In a recent seminal paper, Rudeanu derived necessary and sufficient conditions for the most general form of the subsumptive general solution of a Boolean equation. The present paper investigates four prominent classes of the aforementioned most general form. These include the conventional (eliminants) class, the don’t-care class, the Rudeanu class, and the complete-sum (CS) class. We show that each of these classes satisfies the Rudeanu conditions, and hence fits into the frame of the most general form of the subsumptive solution. The four classes are of variable but comparable computational complexities. The conventional (eliminants) class is the least likely to produce a compact solution, while the don’t-care class is the most likely to produce such a solution. Three examples are presented to demonstrate solutions produced by the four classes and compare them form the point of view of compactness of solutions, which determines the ease of generating the tree of particular solutions. A remaining open question that has yet to be settled is whether these four classes are exhaustive, or in other words, whether there are general solution classes other than them. We demonstrate that the don’t-care class is never inferior (and occasionally superior) to the other classes. In fact, this class seeks global minimality and has a better control on the pertinent variables and algebra generators. Both the CS class and the Rudeanu class have an advantage over the eliminants method. The CS class utilizes a canonical representation that explicitly shows all information in the simplest form. The Rudeanu class enhances the eliminants class by incorporating the consensus (Boolean resolution) concept in a way similar to that of the CS approach. [ABSTRACT FROM AUTHOR]

Abstract: Rough set theory provides a very useful idea of lower and upper approximations for inconsistent data. For incomplete data these approximations are not unique. In this paper we investigate properties of three well-known generalizations of approximations: singleton, subset and concept. These approximations were recently further generalized as to include an additional parameter , interpreted as a probability. In this paper we report novel properties of singleton, subset and concept probabilistic approximations. Additionally, we validated such approximations experimentally. Our main objective was to test which of the singleton, subset and concept probabilistic approximations are the most useful for data mining. Our conclusion is that, for a given incomplete data set, all three approaches should be applied and the best approach should be selected as a result of ten-fold cross validation. Finally, we conducted experiments on complexity of rule sets and the total number of singleton, subset and concept approximations. [Copyright &y& Elsevier]

Abstract: This paper focuses on credibilistic clustering approach. A data clustering method partitions unlabeled data sets into clusters and labels them for various goals such as computer vision and pattern recognition. There are different models for objective function-based fuzzy clustering such as Fuzzy C-Means (FCM), Possibilistic C-Mean (PCM) and their combinations. Credibilistic clustering is a new approach in this field. In this paper, a new credibilistic clustering model is introduced in which credibility measure is applied instead of possibility measure in possibilistic clustering. Also, in objective function, the separation of clusters is considered in addition to the compactness within clusters. The steps of clustering are designed based on this approach. Finally, the main issues about model are discussed, and the results of computational experiments are presented to show the efficiency of the proposed model. [Copyright &y& Elsevier]

Abstract: Retrieving representative information from large-scale data becomes an important research issue nowadays, especially in the context of mobile business/search where the screen size and navigability are limited. This paper focuses on certain aspects of representativeness in database queries and web search, and proposes an approach to extracting a subset of results from original search results in light of high coverage and low redundancy. In the paper, the notion of λ-represent is introduced, which enables us to describe the λ-represent relationship between the sets of data objects. Then, the λ-representative problem is formulated as an extension of the typical set covering problem, which leads to developing a heuristic approach (namely, LamRep) to coping with the problem effectively and efficiently. Notably, LamRep is incorporated with a “vote” mechanism, enhanced with an algorithmic acceleration strategy. Data experiments on benchmark data and a real-world example show that LamRep outperforms the other approaches. [Copyright &y& Elsevier]

In this paper, a new class of intuitionistic fuzzy implications (IFIs) known as (f I , ω) -implications is introduced which is a generalized form of Yager's f-implications in intuitionistic fuzzy environment (IFE). The basic properties of these implications are discussed in detail. It is shown that (f I , ω) -implications are not only the generalizations of Yager's f-implications, but also the generalizations of R -, (S , N) - and Q L -implications in IFE. The distributivity equations I I (T (u , v) , w) = S (I I (u , w) , I I (v , w)) and I I (u , T 1 (v , w)) = T 2 (I I (u , v) , I I (u , w)) over t-representable t-norms and t-conorms generated from nilpotent and strict t-norms in IF set theory are discussed. Also, we solve the open problems concerning characterize all of the correct solutions of the distributive equation I I (u , T 1 (v , w)) = T 2 (I I (u , v) , I I (u , w)) when t-norms are strict and nilpotent. [ABSTRACT FROM AUTHOR]

In the existing multi-factor one-order (MFOO) forecasting models, each fuzzy logical relation (FLR) has one consequent and more than one premise reflecting the association between the fuzzy values at two consecutive moments, and its premises are related to all the factors involved in forecasting. When using such FLRs to realize prediction, no matched FLR cases happened often and thus no logical prediction can be made. Two shortcomings are found for that: one is that more premises make FLRs matching more difficult; the other is that there are no enough FLRs. To overcome these shortcomings, this paper proposes two kinds of FLRs: short cross-association FLRs and long cross-association FLRs, which mean the influence on the consequent is from a part of the factors instead of all the factors. Specifically, the long cross-association FLRs aim at finding the association between fuzzy values at two non-consecutive moments. Such cross-associations exist in reality, the construction of them allows more FLRs to be mined from history observations. They can raise the possibility of finding available FLRs for forecasting. Based on the proposed FLRs, two MFOO forecasting models are proposed. Experiments show the advantage of the new FLRs and the good performance of the proposed models. [ABSTRACT FROM AUTHOR]

• An evolutionary algorithm based hyper-heuristic is proposed for the set packing problem (SPP) and the minimum weight dominating set (MWDS) problem. • Self-learning concept is employed in hyper-heuristic for a selection of heuristic. • Dynamic selection of parameter is adopted to make the approach, as much as possible, less dependent on parameter values. The empirical study shows the effectiveness of Dynamic selection of parameter. • The proposed approach has been compared with the respective state-of-the-art approaches for both the SPP and MWDS problem. • Computational results show the superiority of the proposed approach. In recent years, hyper-heuristics have received massive attention from the research community as an alternative of meta-heuristics. In a hyper-heuristic, generation or selection of an effective heuristic among a pool of heuristics is an important and challenging task in the search process. At each iteration, a suitable heuristic can take the search process toward the global optimal solution. Moreover, some additional factors such as quality and the number of heuristics also affect the performance. In this paper, we propose an evolutionary algorithm based hyper-heuristic framework that incorporates dynamic selection of parameters. To test its generality, effectiveness and robustness, we apply this approach on two different NP - hard problems - set packing problem (SPP) and minimum weight dominating set (MWDS) problem. The proposed approach for the SPP and the MWDS problem has been evaluated respectively on their respective set of benchmark instances. Computational results show that the proposed approach for the SPP and MWDS problem perform much better than their respective state-of-the-art approaches in terms of the solution quality and computational time. [ABSTRACT FROM AUTHOR]

The notion of uncertainty measure is one of the most important topics in rough set theory and has been studied in different kinds of information tables. However, few studies have focused on the interval set information table, which is regarded as one of the generalized models of single-valued information tables. This paper aims at studying the uncertainty measurements for interval set information tables. Firstly, an interval δ -similarity relation is induced based on the similarity degree. The similarity relation induces the granules, which form a covering in interval set information tables. Secondly, four types of granularity measures are defined to measure the granularity of a covering. Thirdly, the concepts of accuracy and roughness in rough set theory are respectively extended to δ -accuracy and δ -roughness for interval set information tables. Furthermore, four new combinations ofuncertainty measures by considering proposed granularity measures and δ -accuracy and δ -roughness are defined and analyzed. Theoretical analyses and experimental results illustrate that the proposed measures are effective and accurate for interval set information tables. [ABSTRACT FROM AUTHOR]

Nonmodularity is a more general concept than nonadditivity which provides a more inclusive reflection of the relationship between subsets of criteria in multicriteria decision making. In this paper, we construct the nonmodularity index to describe the kind and intensity of interaction within a subset of criteria associated with the nonmodularity, and introduce the total nonmodularity and nonadditivity (amount) index to reflect the interactions within a capacity as a whole. We prove some fundamental properties of the nonmodularity index, develop its expression in Möbius representation and prove that both nonmodularity and nonadditivity indices can serve as alternative representations of a capacity in terms of matrix form. We also discuss in detail the explicit reciprocal transform representations of the nonmodularity index, as well as the nonadditivity index with the capacity and the Möbius representation. In view of these explicit interaction indices and their intuitive merits, we develop specialized capacity identification methods in terms of nonmodularity and nonadditivity indices and formulate the corresponding linear programming problems to represent and aggregate the decision makers' preference information. [ABSTRACT FROM AUTHOR]

Existing multi-label learning approaches mainly utilize an identical data representation composed of all the features in the discrimination of all the labels, and assume that all the class labels are observed for each training sample. However, in multi-label learning, each class label might be determined by some specific features of its own, and only a partial label set of each example can be obtained for some real applications. This paper proposes a new method to learn L abel- S pecific features for multi-label classification with M issing L abels, named LSML. First, a new supplementary label matrix is augmented from the incomplete label matrix by learning high-order label correlations. Then, a label-specific data representation for each class label is learned, and the multi-label classifier is constructed simultaneously based on it by incorporating the learned high-order label correlations. A comparative study with the state-of-the-art approaches manifests the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we have proposed a variation of Barwise and Seligman's proposal for information flow among a distributed network of agents by bringing in a notion of belief structure and a notion of graded inference. In their proposal, belief profile of an agent has played an important role as it is explicitly mentioned in the principles of information flow. In contrary, while developing the formal counterpart of information flow they did not address any connection with the belief profile of an agent. Besides, they have accommodated a non-deterministic notion of derivation to design the local logic of an agent keeping a room open for unsound inferences. Our proposal here is to bring in the notion of belief structure of an agent in the development of the local logic of an agent, and convert the non-deterministic nature of consequence to a deterministic graded (four-valued) notion of consequence. For the time being, we have focused on a specific context of decision making by aggregating opinions of agents based on an approach, known as the theory of graded consequence. [ABSTRACT FROM AUTHOR]

Abstract: A special class of neighborhood systems, called 1-neighborhood systems, are important in rough set theory. By using a concept “core” originated in general topology, we define two types of rough sets based on 1-neighborhood systems in this paper. We discuss properties of these rough sets from the perspective of both common 1-neighborhood systems and several special classes of 1-neighborhood systems, such as reflexive, symmetric, transitive, or Euclidean 1-neighborhood systems. By using these properties, we discuss the relationship among several classes of 1-neighborhood systems with various special properties. We give a necessary and sufficient condition for a reflexive and symmetric 1-neighborhood system being a unary. We also prove that a reflexive and transitive 1-neighborhood system is representative. The proofs of these results show that the rough sets we defined in this paper not only have application background, but also have theoretic importance. [Copyright &y& Elsevier]

Abstract: For a graph G =(V, E), a subset F ⊆ V(G) is called an R k -vertex-cut of G if G − F is disconnected and each vertex u ∈ V(G)− F has at least k neighbors in G − F. The R k -vertex-connectivity of G, denoted by κ k (G), is the cardinality of the minimum R k -vertex-cut of G, which is a refined measure for the fault tolerance of network G. In this paper, we study κ 2 for Cayley graphs generated by unicyclic graphs. Let Sym(n) be the symmetric group on {1,2,…, n} and be a set of transpositions of Sym(n). Let be the graph on vertex set {1,2,…, n} and edge set . If is a unicyclic graph, then simply denote the Cayley graph by UG n . In particular, if is a cycle, then UG n is the well-known modified bubble sort graph MB n . We prove in this paper that if m is the length of the unique cycle in , then [Copyright &y& Elsevier]

Abstract: Compared with relational data, it is more difficult to normalize XML data. In the relational data model, semantically relevant attributes compose relations which can simplify the normalization issue. But limited by the structural characteristics, the semantic relevancies of XML data cannot be outlined explicitly. Therefore, in the existing XML normalization proposals, XML constraints hold in the unsuitable ranges and cannot authentically match the original information relevancies. In this paper, a kind of semantically relevant information sets- entity segments are used to limit the ranges where XML constraints hold. Based on entity segments, XML constraints are defined as XML attribute dependencies which can authentically reflect the original information relevancies. Simultaneously, entity primary keys are defined as the unique identifiers of entity segments, and the relationships among different entity segments are denoted by the concept of entity foreign key. To form a normalization system for XML schema design, the XML integrity rules and the XML normal form are proposed, the effect of the XML integrity rules and the XML normal form on normalizing XML data is shown by practical instances. And the information-theoretic measure is used to justify their roles further. It is concluded that entity segments are the suitable ranges where XML constraints can authentically match original information relevancies and the proposal presented in this paper is not only effective on avoiding XML data redundancies but also on keeping XML data consistencies. [Copyright &y& Elsevier]

Abstract: This paper proposes a rough set model based on formal concept analysis. In this model, a solution to an algebraic structure problem is first provided in an information system: a lattice structure is inferred from the information system and corresponding nodes are called rough concepts. How to deal with common problems in rough set theory based on rough concepts is then explored, such as upper and lower approximation operators, reducts and cores. Decision dependency has become a common form of knowledge representation owing to its properties of expressiveness and ease of understanding, so it has been widely used in practice. Finally, application of rough concepts to the extraction of decision dependencies from a decision table is studied; a complete and non-redundant set of decision dependencies can be obtained from a decision table. Examples demonstrate that application of the method presented in this paper is valid and practicable. The results not only provide a better understanding of rough set theory from the perspective of formal concept analysis, but also demonstrate a new way of combining rough set theory and formal concept analysis. [Copyright &y& Elsevier]

Abstract: Fuzzy implication operators play important roles in both theoretical and applied aspects of fuzzy sets theory. Many papers investigated various properties of different types of implications and the interrelationships among these properties. In this paper, we exploit the minimal conditions which must be satisfied for a binary operation A to generate a residual implication with additional properties. It includes several examples to clarify the situation. [Copyright &y& Elsevier]

Highlights • IFC policies are defined over a universe set of attributes. • The computation cost to determine whether a communication request shoul be permitted or denied is constant, instead of linear with the number of required attributes or receivers. • Weak attribute privacy is achieved; • Fine-grained access policies on encrypted data are supported. • The communication cost is linear with the number of required attributes and is independent of the number of receivers. To the best of our knowledge, it is the first IFC scheme enforced by using attributes. Abstract Information flow control (IFC) mechanisms regulate where information is allowed to travel. To enhance IFC, access control encryption (ACE) was proposed where both the no write-down rule and the no read-up rule are supported. Nevertheless, there are still two issues: (1) how to determine whether a communication request should be permitted or denied was not considered; (2) the commutation cost is linear with the number of receivers. Attribute-based system (ABS) can implement one-to-many communication and fine-grained access policies. In this paper, a new IFC scheme is proposed by combing ACE with ABS. Our scheme provides the following features: (1) IFC policies are defined over a universe set of attributes; (2) the computation cost to determine whether a communication request should be permitted or denied is constant, instead of linear with the number of required attributes or receivers; (3) weak attribute privacy is achieved; (4) fine-grained access policies on encrypted data are supported; (5) the communication cost is linear with the number of required attributes and is independent of the number of receivers. To the best of our knowledge, it is the first IFC scheme enforced by using attributes. [ABSTRACT FROM AUTHOR]

Highlights • We prove some properties of the Pari Mutuel Model as a coherent lower (and upper) probability. • We show that a Pari Mutuel Model is a particular case of a probability interval. • We give the maximal number of extreme points of the credal set associated with a Pari Mutuel Model and we give the form of those extreme points. • We give sufficient and necessary conditions for the Pari Mutuel Model to induce a belief function. • We show how to merge information given in terms of Pari Mutuel Models. Abstract The pari-mutuel model is a betting scheme that has its origins in horse racing, and that has been applied in a number of contexts, mostly economics. In this paper, we consider the set of probability measures compatible with a pari-mutuel model, characterize its extreme points, and investigate the properties of the associated lower and upper probabilities. We show that the pari-mutuel model can be embedded within the theory of probability intervals, and prove necessary and sufficient conditions for it to be a belief function or a minitive measure. In addition, we also investigate the combination of different pari-mutuel models and their definition on product spaces. [ABSTRACT FROM AUTHOR]

Highlights • Several selection NSGA-II modifications: population management, tournament selection, crowding distance and rank operators. • Clone prevention mechanism usage and its influence on effectiveness and efficiency. • New application of Non-dominated Tournament Genetic Algorithm (NTGA) to multiobjective MS-RCPSP. • Used 36 iMOPSE dataset ąV Pareto Front estimations for multiobjective MS-RCPSP: cost and duration of final schedule. • NTGA results are compared and analyzed by measures: HV, ED, PFS, Spacing, and Purity. Abstract In this paper a modified selection operator is presented in combination with classical Non-dominated Sorting Genetic Algorithm II (NSGA-II). It is shown that various modifications can lead to increased convergence, spread or uniformity of achieved Pareto fronts. A clone prevention method is used to increase the spread of resulting sets. Furthermore a crowding operator is removed from the NSGA-II as it serves a similar purpose as the clone prevention, even though it achieves it in a different manner. The former enforces diversity in a phenotype, while the latter enforces diversity in a genotype. Combinations of multiple selection modifiers are researched and the best configurations are identified. Multiple absolute measures are selected and applied to verify quality of the results. Additionally a relative measure is used to compare the fronts. They indicate that used selection improves convergence and spread of the front at the cost of its uniformity, while at the same time increasing the efficiency of search. Results are compared to the fronts obtained by multiple runs of a single–objective hybrid Differential Evolution with Greedy Algorithm. All experiments are performed on Multi–Skill Resource–Constrained Project Scheduling Problem. [ABSTRACT FROM AUTHOR]

Abstract Robust model predictive control (RMPC) has gained much attention in the past two decades. For classic finite-horizon RMPC, a terminal constraint set together with a terminal cost function are usually employed for the purpose of closed-loop stability. However, computational burden and feasibility region may need to be carefully addressed when the prediction horizon length becomes large, especially for the cases with additive disturbances. In this paper, a novel RMPC approach with pseudo terminal designs is proposed. Specifically, a bank of pseudo terminal sets (PTSs) are designed given a predefined terminal constraint set. Besides, a pseudo terminal cost (PTC) term is considered for each PTS. With the proposed approach, a long-horizon RMPC problem can be divided into several short-horizon ones. It is shown that both the PTSs and the corresponding PTCs can be computed off-line recursively. In the online implementation, selecting a PTS is realized in a self-triggered way. Meanwhile, a time-varying cost function, which is always an upper bound of the infinite-horizon cost function, is optimized online. [ABSTRACT FROM AUTHOR]

Abstract A major limitation of traditional High Utility Itemset Mining (HUIM) algorithms is that they do not consider that the utility of itemsets may vary over time. Thus, traditional HUIM algorithms cannot find itemsets that do not yield a high utility when considering the whole database, but still have a high utility during specific time periods. Discovering such itemsets is useful, as a product may sell exceptionally well during specific time periods but not during the rest of the year. This paper addresses this limitation of HUIM by defining the problem of mining local high utility itemsets (LHUI), and an extension to mine peak high utility itemsets (PHUI), which consists of finding the time periods where an itemset generates a utility that is much higher than usual (a peak). Algorithms named LHUI-Miner and PHUI-Miner are proposed to mine these patterns. Moreover, because the set of PHUIs can be large and some items in PHUIs don't contribute much to their peaks, a third algorithm named NPHUI-Miner is proposed to discover a smaller set of patterns called Non-redundant Peak High Utility Itemsets (NPHUIs). Experimental results show that the proposed algorithms are efficient and can find useful patterns. [ABSTRACT FROM AUTHOR]

Mixed-variable optimization problems (MVOPs) that involve continuous and discrete decision variables widely exist in industrial and scientific domains. However, how to solve MVOPs efficiently remains an open issue because the fact that continuous and discrete variables present different spatial distribution features posts a great challenge to algorithmic design. In this paper, a hybrid differential evolution (DE) framework is proposed for MVOPs. The proposed framework, namely DE MV , hybridizes the original DE and the set-based DE for evolving continuous and discrete variables, respectively. The two DEs are selected for hybridization because algorithmic analysis and experimental studies show that they share the same search mechanism. The compatibility and consistency of the two DEs is the key for enabling DE MV to coevolve different types of decision variables efficiently. Experiments are conducted on a set of artificial MVOPs converted from continuous benchmark functions and real-world engineering problems with mixed variables. Experimental results and comparisons with other representative algorithms show that DE MV is effective and efficient. [ABSTRACT FROM AUTHOR]

Uncertainty learning is an important research direction of rough set theory, wherein the most popular one is rough set-based prediction, whose goal is to extract decision rules from decision systems and then assign the corresponding decision labels for new samples in terms of the decision rules. To design efficient prediction algorithms, it is necessary and meaningful to measure the uncertainty of rough set-based prediction, especially the stability and generalization performance. In this paper, we analyze the generalization performance of rough set-based prediction algorithms in terms of algorithmic stability analysis and give the generalization error bounds. Firstly, we propose a general rough set-based prediction algorithm to predict the labels for new samples, and then define a scoring function and the corresponding loss function. Secondly, we define two kinds of algorithmic stability for this prediction algorithm in terms of their loss functions, by which two general generalization error bounds are obtained according to two different kinds of stability: strong stability and pointwise hypothesis stability. The bounds numerically imply the performance of the proposed rough set-based prediction algorithm is related to the number of samples and stability parameter. Thirdly, we adopt the confidence and max confidence, min support algorithms as the specific scoring functions instead of general scoring functions. The results show the prediction performance of the confidence algorithm is related to the number of samples and stability parameter, as well as that of max confidence, min support algorithm is associated with the number of samples and minimum support threshold. Based on these discussions, a general framework of stability and generalization error bounds analysis for rough set-based prediction is established. Finally, several experiments are performed to test the previous conclusions. [ABSTRACT FROM AUTHOR]

Signature verification is widely used in the fields of finance, access control and security. However, one of the biggest problems associated with signature verification is its instability. This instability may concern specific fragments of a signature or its entirety. The location of the unstable fragments in a signature depends on individual writing styles, which makes the signature verification more difficult, resulting in a negative impact on recognition effectiveness. In this paper the authors propose a new method of signature verification. The key stage of this approach is the determination of reference signature stability. The proposed signature stability measure, based on fuzzy set theory, is a biometric strategy that has never been used before. The fragments of one reference signature that differ from the corresponding fragments of the remaining reference signatures of the same person are treated as unstable fragments and will not be taken into consideration when comparing the reference sample with the signature being verified. The proposed method employs fuzzy sets to extract a signature’s stable fragments. To test the proposed method’s performance, seven different classifiers were used: PSO oriented, k -Nearest-Neighbor, Naive Bayes, Random Forest, RIDOR, Support Vector Machine and J48. Experiments conducted on the two independent datasets demonstrated that the method proposed here returns highly satisfactory results, outperforming the other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In a series of papers, 186 different intuitionistic fuzzy implications have been defined. For each of them, the four Barcan formulas are checked in the case of the standard intuitionistic fuzzy quantifiers. The list of the intuitionistic fuzzy implications that satisfy the Barcan formulas is given. Modifications of these formulas for the case of the extended intuitionistic fuzzy quantifiers are introduced and an open problem for them is formulated. [ABSTRACT FROM AUTHOR]