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As a natural extension of three-way decisions with incomplete information, this paper provides a novel three-way decision model based on incomplete information system. First, we define a new relation to describe the similarity degree of incomplete information. Then, in view of the missing values presented in incomplete information system, we utilize interval number to acquire the loss function. A hybrid information table which consist both of the incomplete information and loss function, is used to deal with the new three-way decision model. The key steps and algorithm for constructing the integrated three-way decision model are also carefully investigated. An empirical study of medical diagnosis validates the reasonability and effectiveness of our proposed model.

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... In order to construct decision-making methods more scientifically, we combine regret theory to construct a G3WD method on an INMSDS. 3) In recent studies of incomplete information systems, defining similarity relations and supplementing missing values are the two main processing methods [13,44]. However, these methods have certain defects: similarity relations cannot be directly brought into the calculation, while the supplemental method has a large error. ...

... In order to evaluate the classification capability of our proposed method in this subsection, we further focus on the classification perspective on an INMSDS. In specific, we compare our method with the methods of Zhan et al. [13] and Liu et al. [44] from the classification perspective by using an incomplete dataset selected from the UCI database. Since both the above-mentioned methods and our method are extensions of the 3WD method, we mainly consider the strategy corresponding to the three classifications and the accuracy of the classifications when evaluating the effectiveness of the three divisions, the commonly used computational equation for the misclassification rate and accuracy are shown in Eqs. ...

... The relevant calculation results 3 are shown in Fig. 4 and Table VI. [13] x 560 0 − Liu et al.'s method [44] − 0.0029 0.9971 Fig. 4 shows the results of the three divisions of these methods, and it is especially obvious that in the results of Zhan et al.'s method [13], all objects are divided into the boundary domain, i.e., all objects take the strategy of the delayed decision, which is obviously unreasonable. In the calculation results of Liu et al.'s method [44], the boundary domain contains only two objects, i.e., two objects take the strategy of the delayed decision, which is almost consistent with the two-way decision. ...

The three-way decision theory provides a three-way philosophical thinking to solve problems, and the regret theory quantifies the risk preferences of decision makers under different psychological behaviors. On the one hand, the combination of these two theories makes models more practical by considering the psychological behaviors of decision makers. On the other hand, we can effectively combine the advantages of the three-way decision theory with the regret theory to highlight the interpretability of decision making processes. In this paper, we propose a novel approximate estimation method for incomplete utility values via the regret theory and establish a wide sense of three-way decision model on incomplete multi-scale decision information systems. First, the degree of consistency for each scale is measured via using the dependency degree, then the optimal subsystem is selected by evaluating the scale selection cost. Further, the incomplete multi-scale evaluation information is transformed into triangular fuzzy numbers via linguistic term sets. Second, in light of fuzzy evaluation values and trade-off factors, an estimation method for incomplete fuzzy subsystems is constructed, which can be used to calculate the utility difference and regret-rejoicing values for pairwise comparisons. Finally, from the perspective of human cognition, the tri-partition and the corresponding decision rules are built by the tolerance degree, and the ranking of objects is calculated by the relative closeness degree. Additionally, multi-aspect comparative and experimental analyses are performed by extensive experiments, and the feasibility, validity and stability of the constructed model are shown by parametric analyses.

... The above research is based on the problems of complete information systems (CISs) [29], but the problems of incomplete information systems (IISs) [13] are more common in the actual environment. Therefore, Liu et al. [19] defined a new relationship to describe the similarity of incomplete information, to calculate conditional probability. Considering the uncertainty in incomplete information, the authors used interval numbers to obtain the loss function in the model and then combined the incomplete information table and the loss function table. ...

... When there are too much missing data in the information table, using the possible information value filling method may lead to a large amount of calculation. When the data in the information table is not real-valued but digital valued, such as an example of medical diagnosis in [19], the method of filling in the mean value may not be applicable. The similarity degree function [19] is an effective way to express to measure similarity between objects on incomplete information tables, which has been successfully applied to IIS. ...

... When the data in the information table is not real-valued but digital valued, such as an example of medical diagnosis in [19], the method of filling in the mean value may not be applicable. The similarity degree function [19] is an effective way to express to measure similarity between objects on incomplete information tables, which has been successfully applied to IIS. Therefore, this paper proposes a method of filling missing information based on similarity degree functions. ...

Aiming at the problems of filling missing information and calculating conditional probability and loss function in incomplete information systems, this paper provides a novel three-way decision model based on incomplete information systems. Firstly, a new information table is obtained by filling in the missing information based on similarity, and the conditional probability calculation method is established by the idea of a TOPSIS combination with the information table. The relative loss function is calculated based on the risk avoidance coefficient under different attributes. Then, we propose the notion of interval relative loss function and give formulae to calculate the interval relative loss function values. In particular, the key steps of constructing the three-way decision model are summarized. Finally, a case study of medical diagnosis is provided to verify the validity of the proposed method. Moreover, the rationality and superiority of the presented method are verified by sensitivity analysis and comparative analysis.

... In the big data era, due to the diversity and complexity of various data types, the omission or loss of information may occur in data collection and storage processes. Thus, it comes as no surprise that multi-attribute decision-making (MADM) problems in realistic incomplete information systems (IISs) have appeared, such as medical diagnosis [20], automobile evaluations [37,44], air quality evaluations [45], etc. Although existing strategies of solutions can achieve an ordering or classification of objects, they often deem decision-makers as ''completely rational", hence their applicability may be limited to idealistic cases. ...

... In 1998, Kryszkiewicz [15] initiated a rough set approach for IISs. Liu et al. [20] put forward a new relation to describe the similarity of incomplete information. These authors used interval numbers to obtain the loss function, and then constructed a new 3WD model on IISs. ...

... By contrast, this paper constructed an approach to deal with the relation between the evaluation values of objects resort to the fuzzy set pair for preserving the integrity of the original information whereas solving the decision-making problem formulated by both IISs and complete information systems. Additionally, in existing 3WD approaches in an IIS framework, normally all objects can only be sorted or classified [37,20]. In order to overcome this challenge, this paper proposes a more comprehensive decision-making approach that enables decision-makers to sort and classify all objects. ...

The generalized three-way decision (G3WD) theory is featured by a trisecting-acting-outcome paradigm. Experiments in psychology and economics have indicated that risky decision-making problems should take into account the psychological characteristic of agents. Moreover, the loss of information acts as a serious challenge in G3WD. Thus, exploring valid G3WD approaches that consider the mental state of decision-makers in incomplete information systems (IISs) is imperative. This paper explores a new three-way decision (3WD) approach that combines the prospect theory with the regret theory for IISs with the support of fuzzy set pair dominance degrees. First, the notion of fuzzy set pair dominance degrees to process evaluation information in IISs is put forward. Then, an approach that obtains objective reference points and determines the weighting function in the prospect theory is given by combining 3WD with the behavioral decision theory. In light of the prospect theory, a prospect value function is obtained, which is introduced into the regret theory to obtain two utility perception functions. As a consequence, a G3WD approach based on fuzzy set pair dominance degrees and the behavioral decision theory is constructed for IISs. By using two medical cases in the KEEL and UCI datasets, the superiority, effectiveness and stability of the constructed approach are verified via corresponding comparative and experimental analyses.

... (1) Among existing methods for handling decision-making issues where data may be missing, Zhan et al.'s method [25] requires the presence of fuzzy decision attributes in original information tables, and Liu et al.'s method [4] requires the presence of interval loss functions in original information tables, leading to limitations in the use of these methods. In addition, Yang et al.'s method [41] can only rank all objects instead of classifying all objects. ...

... Thus, this paper will construct a decision-making method that combines 3WD with RT in FIISs. (3) Similarity relations [4,25] are commonly used in an IIS to deal with binary relations between objects. Since similarity relations are too strict in dealing with real-valued decision-making problems, this paper will define a priori probabilistic dominance W.J. Wang et al. relation with a broader application scope. ...

... Moreover, our method also classifies all objects into three regions. Thus, we compare the classification situations that are obtained for four methods in FIISs, including Zhan et al.'s method [25], 4 [4]. 6 The classification of two datasets in different methods is shown in Fig. 8. ...

In real world, decision-makers’ regret psychology often affects decision outcomes due to uncertain risks. Moreover, decision information may be missing in the process of data acquisitions or data storages. Three-way decision has been widely explored in the risky decision-making area by providing effective strategies to divide objects into three mutually disjoint regions. Existing three-way decision methods in fuzzy incomplete information systems rarely consider the influence of decision-makers’ psychological states on decision outcomes. In the current paper, we primarily study a new decision-making method that combines regret theory with three-way decision in fuzzy incomplete information systems. First, a prior probability tolerance dominance relation in a fuzzy incomplete information system is defined to handle a binary relation among evaluation values, and a method to calculate objective weights is designed as well. When an incomplete information system does not contain a fuzzy decision attribute value, we put forward a new method to calculate the decision attribute value of each object in the incomplete information system. Then, integrated utility perception values are obtained by combining with regret theory. Further, a regret theory-based three-way decision method with a priori probability tolerance dominance relation is proposed for fuzzy incomplete information systems. At last, the stability and validity of the presented method are verified via corresponding experimental and comparative analysis of realistic cases.

... However, in practical situations, it may not be always possible to gather all information in a decision making problem. Furthermore, the majority of 3WMADM models in IISs considered the vagueness or ambiguity in evaluation information as fuzzy set Liu et al., 2016;Yang et al., 2020b), or as IFN (Xin et al., 2021), and some of them ignored it . These models can deal with the practical problems upto a certain limit. ...

... This can lead to variations in original information, the result of which may not truly reflect the practical situation. Moreover, some of the previous 3WMADM models in an IIS only deduced the classification and passed by the ranking of alternatives (Liang et al., 2018;Liu et al., 2016). This cannot be very fruitful for DM in practical. ...

... Therefore, the similarity degree between two IVFFN objects according to the models of Zhan et al. (2021), Yang et al. (2020b) and Liu et al. (2016) is calculated as follows: ...

In the field of decision making, three-way decision making has been proven more fruitful for providing scopes to make delayed decision along with acceptance and rejection simultaneously. As a result, the decision risk and loss, which could occur due to take rapid decisions in traditional two-way decision making, are effectively reduced. Accordingly, this paper offers a novel three-way multi-attribute decision making model by combining three-way decision making and multi-attribute decision making under an incomplete information system. The incertitude in the information system is illustrated by introducing interval-valued Fermatean connection number based on interval-valued Fermatean fuzzy number and set pair analysis theory. Thereafter, the achievement of this study is five-fold. First, a possibility dominance relation is developed under incomplete information system in the basis of possibility degree measure of interval-valued Fermatean connection number. Second, we put forward a novel procedure to set up the fuzzy state set. Third, the conditional probability is estimated in light of the fuzzy state set and probability dominance relation. Fourth, the relative utility functions are determined in the virtue of regret theory. Lastly, a three-way multi-attribute decision making model is implemented in incomplete information system and exploited to deal with incomplete multi-attribute decision making problems. Eventually, the propriety, stability and superiority of the proposed model is established via conducting the comparative and experimental analysis.

... (1)In the decision-making approaches of IISs, Zhan et al. [48] and Liu et al. [26] exploited similarity relations to handle binary relations between alternatives. However, compared to the probabilistic dominance relation, the similarity relation is too strict when dealing with real-valued decision-making problems. ...

... (2)On the one hand, the existing TWD approaches in IISs can only classify all alternatives, but cannot give a complete sorting and assist decision makers to choose the optimal alternative [42,26]. On the other hand, in classic MADM approaches, all alternatives are usually sorted so that decision makers can directly make a decision and select the optimal alternative without considering the delayed decision [16,12,9,18]. ...

... ðPÞ [26]. Assume that I ¼ fA; O; V; Fg is an IDIS, the similarity degree between a i and a l in terms of the attribute o j can be given below: ...

The processing scheme of typical incomplete information systems has been widely explored nowadays, however most of these approaches may change the original data information and can not guarantee the integrity of original information systems. Meanwhile, psychological behaviors of decision makers usually have an influence on the decision-making results. In light of these facts, the paper explores a new three-way decision approach by using prospect theory for an incomplete information system, which handles the default information of an incomplete information system by virtue of a probability dominance relation. First, a probability dominance relation of an incomplete information system is defined to process a binary relation between evaluation values, and the way for calculating the relative value function in the behavior decision-making process is also given. Moreover, the conditional probability with a decision attribute in an incomplete decision information system is put forward, and an objective weight function via prospect theory is further obtained. On this basis, a prospect-theory-based three-way decision approach with a probability dominance relation for an incomplete information system is proposed. Finally, the stability, effectiveness and superiority of the presented approach are validated via corresponding experimental studies and comparative analysis with two realistic cases.

... How to determine the thresholds in IVMA3WD model under a multi-attribute environment with interval numbers is an urgent issue to be considered in our study. Many generalized 3WD models for determining the thresholds have been proposed in recent years [2,16,23,20,19], and we can divide the existing studies into two types. The first is linear rankingsbased 3WDs [2,16,23], and the second is nonlinear rankings-based 3WDs [20,19]. ...

... Many generalized 3WD models for determining the thresholds have been proposed in recent years [2,16,23,20,19], and we can divide the existing studies into two types. The first is linear rankingsbased 3WDs [2,16,23], and the second is nonlinear rankings-based 3WDs [20,19]. Using linear rankings-based 3WDs means that the ranking function for ranking fuzzy numbers is linear for the process of 3WD construction. ...

... There is a similar interpretation for the nonlinear rankings-based 3WDs. However, many traditional approaches [2,16,23] for the first type of 3WDs can not be extended directly to the second type to determine the thresholds. For the comparison of interval numbers, researchers developed different kinds of ranking methods [25,49], mainly including linear ranking functions and nonlinear ranking functions. ...

Multi-attribute decision-making (MADM) aims to rank alternatives based on their attributes and evaluation information, which provides decision support for decision-makers. Most existing MADM methods can only obtain ranking results, decision-makers usually need to subjectively choose priority alternatives to make a decision based on the preset evaluation level and ranking results, which does not meet the requirements of complex decision situations and uncertain information processing. A method is urgently needed, which can objectively produce classification results and automatically provide priority objects for decisions. Three-way decisions (3WDs) offer an effective research technique to solve decision-making problems under uncertainty and risk by objectively dividing a universal set into three pairwise disjoint parts: acceptance, non-commitment and rejection, and implementing the corresponding strategy for each part. Considering that it is difficult to accurately determine the exact value of attributes in some cases, interval numbers can be a useful concept to flexibly describe uncertain information and satisfy the decision-maker’s cognition. Therefore, under an interval-valued MADM environment, this paper proposes an evaluation-based interval-valued multi-attribute three-way decision (IVMA3WD) model from an optimization viewpoint. The model achieves an effective fusion of 3WDs and MADM problems, and further expands evaluation-based 3WD models on the totally ordered set. Specifically speaking, in order to depict uncertain information in decision-making, interval numbers are used to describe the evaluation information included in the multi-attribute evaluation matrix. Then, considering a general case that there are no decision attributes in the multi-attribute evaluation matrix, we determine the evaluation function by use of the idea of the interval-valued TOPSIS method. In addition, unlike the fixed-valued loss functions in the majority of 3WD models, the relative loss function in this article is not provided straightly but computed by attribute evaluation values expressed by interval numbers. Furthermore, based on an optimization viewpoint, we construct two optimization models and attempt to find the optimal solution to obtain the threshold pair. A housing market investment decision problem is utilized as an illustrative example to find out the applicability of our built IVMA3WD model. In the end, we adopt a sensitivity analysis and a comparative analysis to demonstrate our proposed model’s characteristics and advantages.

... In addition, scholars have conducted corresponding 3WD research based on multisource decision information systems [45], fuzzy information systems [46], intuitionistic fuzzy information systems [47], order information systems [48], lattice value information systems [49], and so on. Second, for incomplete information, Liu et al. [50] proposed a new similarity measurement method and utilized it to build a 3WD model under incomplete information systems. Luo et al. [51] developed a processing framework for four types of incomplete information and constructed a corresponding 3WD model to obtain the semantic interpretation of incomplete information. ...

... Moreover, we give a three-state decision cost matrix, combine the Choquet integral to calculate the influence between different decision costs, and construct its corresponding IF3WD model. According to current research, missing data are usually processed via direct deletion or statistical methods [50], which easily destroy the original information structure and could change the content that the data originally intended to express. Refs. ...

... Refs. [50,53] systematically analyzed the reasons for and types of missing data and processed them through tolerance and similarity relationships. However, the missing value of object x i under attribute a j is not "everything is possible" in an IFIIS; it must meet certain conditions, such as 0 ≤ a (x) + a (x) ≤ 1 . ...

As a new method of granular computing, the three-way decision (3WD) approach has unique advantages in handling uncertain and imprecise problems. Based on decision-theoretic rough sets (DTRSs) and Bayesian minimum risk theory, conditional probability and loss function are the key research issues in 3WD. Many approaches for handling deterministic and complete information have been developed. However, few studies have focused on the construction of an intuitionistic fuzzy three-way decision (IF3WD) model for an intuitionistic fuzzy incomplete information system (IFIIS). In this paper, an IF3WD model based on an IFIIS is proposed to improve the ability to process complex fuzzy incomplete information systems, which extends the application range of the traditional 3WD. Concretely, we first propose a calculation method to measure the degree of information retention of missing data and describe it in two dimensions: coarse-grained and fine-grained. Next, an intuitionistic fuzzy number approximation (IFNA) strategy for missing data is presented. Then, a loss function with three states is given. Furthermore, combined with the Choquet integral, the interaction and influence between acceptance, rejection, and delay decision costs are investigated, and the corresponding IF3WD rules are induced. Finally, the rationality and effectiveness of our proposed model are verified through case analysis and are compared with those of existing methods.

... Compared with MADM methods with two-way decision, TW-MADM methods [23,39] are more favored by scholars due to the following unique merits: (a) TW-MADM methods consider the losses triggered by taking various decision actions for all objects under different states, and the final decision is made based on the Bayesian theory with minimum losses. (b) All objects can be objectively divided into three domains, and the number of objects in each domain can be objectively determined. ...

... In the study of some MADM-driven TWD models, the loss functions are directly given in light of the knowledge and experience of experts [19,[21][22][23]47]. For the sake of reducing the subjectivity of loss functions, Jia et al. [14] combined the information of MADM problems to give the calculation method of relative loss functions according to the study of [18,42]. ...

... For the first time, we use fuzzy rough set models in the acquisition process of loss functions. In other methods, the loss functions are given subjectively [23,42,47] in most cases. In addition, there are other approaches [14,20,24] to obtain the loss functions. ...

The paper primarily explores the applicability of three-way decision (TWD) to multi-attribute decision-making (MADM), and establishes a new three-way multi-attribute decision-making (TW-MADM) method under an incomplete environment. For the sake of making rational decisions for MADM problems with fuzzy values, fuzzy rough set models are first utilized to investigate a new TWD model. By taking into account the hesitation degree of each evaluation value, a data-driven method to determine the relative loss functions is presented. Moreover, a new conditional probability calculation method is put forth via the information granularity of each object. In light of the above statement, a novel TWD model with three strategies is proposed. Afterwards, the arithmetic mean method is adopted for patching the lost data to effectively address incomplete MADM problems. Given the uncertainty of the patched data and real data, a new TW-MADM method as well as a corresponding MADM algorithm is designed. By several comparative analysis and experimental analysis, the feasibility, effectiveness, superiority and stability of the method are demonstrated. In addition, the results show that the presented method with optimistic strategies is more viable and stable than the method with compromise and pessimistic strategies.

... T HREE-WAY decision (TWD), as a granular computing methodology [1], provides an effective tool for solving uncertain decision-making problems [2][3][4][5]. It is an extension of two-way decision (2WD), which adds a deferred decision based on the deterministic decision, i.e., the acceptance and the rejection. ...

... In general, incomplete information systems (IISs) are utilized to collect and express the information on IMADM problems. At present, the research on IISs mainly includes the following three aspects: (1) discussions on various types of incomplete information [24,25]; (2) attribute reductions [26][27][28][29] and feature selections [30,31]; (3) rough set models [32][33][34][35] and TWD models [3,[36][37][38][39]. For instance, Luo et al. [25] recently summarized a framework based on a classification of four types of incomplete information. ...

... Thus, for simplicity, our study only considers the case where the unknown values are "do-not-care values". The similar operations can be found in [3,38,40]. ...

The existing three-way decision-making methods cannot effectively handle incomplete multi-attribute decision-making problems in real life, it is necessary to explore an effective three-way multi-attribute decision-making model in incomplete fuzzy decision systems. First, we consider the preference of decision-makers for each alternative and introduce the concept of pre-decisions, thus an incomplete fuzzy decision system is obtained. Then, the weighted conditional probabilities are calculated with the aid of the defined similarity relation. Subsequently, we introduce the notion of relative utility functions and then present an approach to determine the relative utility function values. Afterwards, we construct a three-way decision model in incomplete fuzzy decision systems and apply it to the modeling of incomplete multi-attribute decision-making problems. Our study not only enriches three-way decision and multi-attribute decision-making theories, but also provides a new perspective for realistic incomplete multi-attribute decision-making problems. At last, the results of comparative and experimental analyses demonstrate the validity, stability and superiority of our proposed model.

... There are a few types of incomplete data studied in the literature [19,24,28,34,35,37]. As an illustration, we consider a simple format of incomplete data represented in an incomplete information table defined as follows [28]. ...

... In other words, if f a (x) = * for an object x ∈ OB on an attribute a ∈ AT , then we do not further know the actual value of x on a due to incomplete or insufficient information. Since our primary goal is to demonstrate the usefulness of 3WD-(e p , e n ) 2 in analyzing incomplete data, we do not distinguish different semantics and cases of incomplete data discussed in a few existing works [19,24,28,34,35,37]. It is an interesting direction for future work to further explore the application of 3WD-(e p , e n ) 2 in analyzing specific types of incomplete data. ...

The theory of three-way decision has contributed to data science in many topics, such as three-way classification, three-way clustering, and three-way feature selection. Most three-way decision models are formulated based on evaluation functions that commonly consider two opposite aspects of positive and negative, alliance and conflict, etc. This idea coincides with the concept of bipolarity, which studies the two opposite poles of positive and negative. However, the connections between bipolarity and three-way decision models have not been well investigated, despite the fact that they share many common features. Therefore, this work explores their relationships and proposes a new bipolar three-way decision model. Firstly, we examine the connections between the polarity theory, including the concepts of unipolarity and bipolarity, and decision models, including the two-way and three-way decision models. The examination suggests a lack of a three-way decision model corresponding to certain types of bipolarity. Thus, secondly, we propose a new bipolar three-way decision model 3WD-(ep,en)2, which considers a pair of a positive evaluation function ep and a negative evaluation function en and applies a pair of thresholds on each function. Finally, we illustrate the usefulness and effectiveness of 3WD-(ep,en)2 in data science through its application in analyzing incomplete data. In particular, we present a computational formulation based on similarity classes and a conceptual formulation based on a conjunctive description language. The two formulations together provide a multi-view understanding of the approach.

... Yao [43] unified rough-set concept analysis and formal concept analysis based on a framework of three-way granular computing. Liu et al. [20] and Luo et al. [21] studied three-way decision based on incomplete information systems. Liu and Liang [19] researched a novel three-way decision model with order information. ...

... In the classical decision-theoretic rough set model, loss functions are real numbers [42]. Subsequently, Liang et al. [23] extended loss functions to triangular fuzzy numbers and Liu et al. [20] extended loss functions to interval-valued fuzzy numbers. As another generalization, we set loss functions as symmetric trapezoidal fuzzy numbers for practical needs in the following discussion. ...

Three-way decision is a decision-making model in line with people’s cognition and aims to think and deal with problems at three levels or three aspects. One of the main purposes of conflict analysis is to partition the set of agents into three coalitions called positive alliance, central alliance and negative alliance in order to determine the relationship between two agents. Recently, researchers combine these two closely related directions to form a new research topic: three-way conflict analysis. This paper consider the case that the attitude of an agent on an issue is a trapezoidal fuzzy number. Firstly, we provide a trapezoidal fuzzy information system for conflict analysis and then we transform attitudes of agents from trapezoidal fuzzy numbers to real numbers through the expectation of trapezoidal fuzzy numbers. Secondly, conflict analysis for a single issue is investigated and three alliances based on a pair of thresholds are obtained. As for multiple issues, it is necessary to integrate multiple attitudes for a collection of issues to one. Considering different importance of issues, we develop a new method to integrate attitudes based on the variance of trapezoidal fuzzy numbers, and then we come up with a conflict analysis model for multiple issues. Thirdly, a method to calculate thresholds is proposed based on decision-theoretic rough sets so as to acquire three alliances based on a single issue or multiple issues more reasonably. Finally, we devote to ranking all the issues according to the conflict degree among agents and our method may be instructive to promote the resolution of conflict situations.

... Since then, many concrete techniques have been developed to process various incomplete information systems [23], [24]. Among them, the aspect of similarity relations is a significant branch and it has achieved sound effects in processing incomplete information systems [25], [26], [27]. ...

As a cause of interfering with routine activities, freezing of gait (FOG) is a severe syndrome of Parkinson’s disease (PD) and usually performs as an abrupt and momentary inability to effective stepping forward. Advanced wearable acceleration sensors based on socially implemented Internet of medical things (IoMT) devices can remotely provide a platform for recognizing FOG. However, due to the diverse data acquisition modes that appear in classic IoMT devices, the obtained data may contain imprecise, hesitant, and incomplete ones. Meanwhile, the bounded rationality owned by neurologists usually has a big impact on using wearable acceleration sensors to predict illnesses. Therefore, the objective of this article lies in exploring a fuzzy intelligence learning approach based on bounded rationality in IoMT systems and providing a valid scheme for biomedical data analysis. Specifically, a brand-new three-way group decision-making approach by means of TODIM (an acronym in Portuguese for interactive multicriteria decision-making) with incomplete dual hesitant fuzzy (DHF) information and its applications in detecting FOG in PD using IoMT devices are systematically explored. First, taking advantage of DHF sets (DHFSs) when depicting realistic group decision information, the concept of multigranulation (MG) incomplete DHF information systems is built. Second, adjustable MG DHF probabilistic rough sets (PRSs) are further put forward via DHF similarity relations. Third, a three-way group decision-making approach is constructed by virtue of adjustable MG DHF PRSs and TODIM. Finally, the validity, effectiveness, and practicality of the constructed three-way group decision-making approach are investigated by a University of California, Irvine (UCI) dataset with several experimental analyses in the background of FOG detection in PD using IoMT devices. The experimental result indicates that the developed fuzzy intelligence learning approach achieves reasonable diagnostic conclusions for FOG detection in PD from the perspective of uncertain information processing abilities, decision risks, and bounded rationality.

... On the one hand, decision evaluation function plays a key role in three-way decisions (Cabitza et al. 2017;Liu et al. 2016;Yao and Azam 2015). On the other hand, decision maker will obtain different decision results through different decision evaluation functions (Qiao and Hu 2018;Qiao and Hu 2020). ...

In 2014, Hu introduced the concept of three-way decision spaces and axiomatic definition of decision evaluation functions. In three-way decision spaces, decision evaluation function satisfies minimum element axiom, monotonicity axiom and complement axiom. Since then, the research on construction method of decision evaluation functions from commonly used binary aggregation functions becomes a research hotspot. Meanwhile, uninorms, as one class of binary aggregation functions, have been successfully applied in various application problems, such as in decision making, image processing, data mining, etc. This paper continues to consider this research topic and mainly explores the new construction methods of decision evaluation functions based on uninorms. Firstly, we show two novel transformation methods from semi-decision evaluation functions to decision evaluation functions based on uninorms. Secondly, using known semi-decision evaluation functions, we give some new construction methods of semi-decision evaluation functions. Thirdly, we give some novel construction methods of decision evaluation functions and semi-decision evaluation functions related to fuzzy sets, interval-valued fuzzy sets, fuzzy relations and hesitant fuzzy sets. Based on them, decision maker can obtain more useful decision evaluation functions, thereby more choices can be used for realistic decision-making problems. Finally, we consider two real evaluation problems to illustrate the results obtained in this paper. The three-way decisions results of evaluation problem show that the construction method proposed in this paper is superior to some existing construction methods under some conditions.

... However, there are few studies on multi-attribute decisions in the heterogeneous incomplete environment. Most existing TWD models in an incomplete environment (Liu et al. 2016;Luo et al. 2020;Yang et al. 2020) tend to consider only the classification of alternatives and do not have the ranking of alternatives, which does not provide sufficient assistance to decision makers (DMs). Although there are some methods to rank alternatives, they are done in complete numerical information systems. ...

Three-way decision is a novel decision-making tool which can effectively reduce decision risk. But it still has some limitations in practical applications. On the one hand, most of existing three-way decision models are not suitable for heterogeneous environments with incomplete decision information. On the other hand, three-way decision is difficult to obtain effective decision information for multi-attribute decision-making with multiple decision makers. For these reasons, we put forward a new three-way decision model in this paper. To this end, a weighted condition probability is constructed by using tolerance block, which can fully consider incomplete information and heterogeneous decision information. Second, relative utility function is employed to fit the preference of decision makers. Moreover, a decision self-information method is presented, which is used to select the most authoritative decision maker from many decision makers. Through an example, it is found that our method is highly consistent with the optimal objectives obtained by states-of-the-art decision-making methods. Comparative experiments verify the effectiveness and feasibility of the proposed model.

... Therefore, in order to reduce the error caused by subjectivity, we propose a new method of objectively calculating the state. (3) Thirdly, since there are two states and three behaviors for each object, an object has six loss functions, each of which is either a subjective loss function [17][18][19] or an objective loss function [7,16,20]. If multiple attributes of an object are considered separately, there are six loss functions for each attribute, which requires a huge amount of calculation and lots of stored data. ...

In recent years, research on applications of three-way decision (e.g., TWD) has attracted the attention of many scholars. In this paper, we combine TWD with multi-attribute decision-making (MADM). First, we utilize the essential idea of TOPSIS in MADM theory to propose a pair of new ideal relation models based on TWD, namely, the three-way ideal superiority model and the three-way ideal inferiority model. Second, in order to reduce errors caused by the subjectivity of decision-makers, we develop two new methods to calculate the state sets for the two proposed ideal relation models. Third, we employ aggregate relative loss functions to calculate the thresholds of each object, divide all objects into three different territories and sort all objects. Then, we use a concrete example of building appearance selection to verify the rationality and feasibility of our proposed models. Furthermore, we apply comparative analysis, Spearman’s rank correlation analysis and experiment analysis to illustrate the consistency and superiority of our methods.

... Because of this advantage, three-way decision model has received extensive attention during the past decade. A number of research topics related to three-way decision model, such as three-way approximations (Deng and Yao 2014), three-way decision model for incomplete information system (Liu et al. 2016), three-way concept analysis (Yao 2017), three-way clustering (Wang and Yao 2018), three-way conflict analysis (Yao 2019), sequential three-way decision model (Yang et al. 2019), three-way fuzzy partitions (Zhao and Yao 2019), application of three-way decision model (Zhang et al. 2019;Shen et al. 2020), and multi-criterion three-way decision-making (Jia and Liu 2019;Ye et al. 2020;Zhan et al. 2021a, b;Zhang and Dai 2022;Wang et al. 2022b), have been proposed in this period. In addition, three-way decision model has been extended to many fuzzy environments, such as intuitionistic fuzzy environment (Liu et al. 2020a;Gao et al. 2020;Jiang and Hu 2021;Wang et al. 2022c), linguistic intuitionistic fuzzy environment (Liu et al. 2022b), hesitant fuzzy environment (Liang et al. 2020;Wang et al. 2021a, b;Feng et al. 2022;Wang et al. 2022a), linguistic hesitant fuzzy environment (Lei et al. 2020), interval-valued intuitionistic fuzzy environment (Jia and Liu 2021;Ye et al. 2021;Liu et al. 2022a), interval type-2 fuzzy environment (Liang et al. 2019), Pythagorean fuzzy environment (Liang et al. 2018;Lang et al. 2019;Du et al. 2022), q-rung orthopair fuzzy environment (Zhang et al. 2021a), and interval-valued q-rung orthopair fuzzy environment , to solve corresponding multi-criterion decision-making (MCDM) problems. ...

How to solve a multi-criterion decision-making (MCDM) problem with linguistic interval-valued intuitionistic fuzzy numbers (LIVIFNs) effectively is an important research topic. So far, a number of methods for solving this problem have been presented within the academia. Each of these methods can work well in specific situation. But they could produce undesirable decision-making results when the information for decision-making is insufficient or acquisition of the information needs a certain cost, since all of them are based on conventional two-way decision model. In this paper, three-way decision model is introduced into linguistic interval-valued intuitionistic fuzzy environment and a multi-criterion three-way decision-making method under this environment is presented. A specific relative loss function derived from an LIVIFN is established and corresponding three-way decision rules are developed. Based on the established function and developed rules, a three-way decision method for solving an MCDM problem with LIVIFNs is proposed. The application of the proposed method is illustrated via a practical example. The effectiveness and advantage of the method are demonstrated via an experimental comparison with some existing methods. The comparison results suggest that the proposed method is as effective as the existing methods and is more flexible than the existing methods in solving an MCDM problem with LIVIFNs.

... The theory has been implemented for feature selection [2][3][4], pattern familiarization [5,6], uncertainty reasoning [7], granular computing [8][9][10], data mining and information exploration [11][12][13]. Over the past years, it had a tremendous effect on uncertainty administration and uncertainty reasoning. ...

As it is well known the rough set is a beneficial method for rough data uncertainty analysis. However, this is a time- consuming task for many big data sets. So we utilized the concept of local rough sets in data analysis of children addicted to social media to handle big data efficiently and give some of the properties. With the results, we proved that local rough sets gave more concrete and clear information than rough sets in data analysis.

... Many kinds of data can be used to describe a patient's condition [31] (e.g., set-valued data, categorical data, Boolean data, real-valued data, and missing data), indicating that the method of finding an equivalence class based on an equivalence relation is unsuitable. As such, on the basis of previous research [6,41,42], this study uses the similarity relation to replace the equivalence relation. In addition, for different types of data, various equations need to be constructed to measure similarity. ...

In the sequential three-way decision model (S3WD), conditional probability and decision threshold pair are two key elements affecting the classification results. The classical model calculates the conditional probability based on the strict equivalence relationship, which limits its application in reality. In addition, little research has studied the relationship between the threshold change and its cause at different granularity levels. To deal with these deficiencies, we propose a novel sequential three-way decision model and apply it to medical diagnosis. Firstly, we propose two methods of calculating conditional probability based on similarity relation, which satisfies the property of symmetry. Then, we construct an S3WD model for a medical information system and use three different kinds of cost functions as the basis for modifying the threshold pair at each level. Subsequently, the rule of the decision threshold pair change is explored. Furthermore, two algorithms used for implementing the proposed S3WD model are introduced. Finally, extensive experiments are carried out to validate the feasibility and effectiveness of the proposed model, and the results show that the model can achieve better classification performance.

... Considering the complexity and diversity of representations of big data in practical problems, the following aspects are worth exploring in the future: (1) The proposed 3W-MADM-R method will be extended to different information environments, such as the applications of intuitionistic fuzzy numbers [62], hesitant fuzzy numbers [58] and incomplete environments [63]. (2) The analytical formula with the parameter ζ in the threshold can be solved and the sufficient and necessary conditions that satisfy the 3WD model can be studied. ...

Cardiovascular disease is a global leading cause of death, and timely monitoring can determine its extent. Clinicians use these diagnostic indicators to make scientific and reasonable decisions. However, when decision-makers (DMs) encounter risks in complex environments, their limited rationality may affect decision behaviors. Therefore, the paper explores a new three-way multi-attribute decision making method based on regret theory (3W-MADM-R), which uses heart disease data to make decisions in fuzzy environments. There are three main steps in developing 3W-MADM-R, i.e., (i) we propose the notion of relative outcome functions and corresponding aggregated regret-based utility functions of each object; (ii) we estimate the conditional probability via an outranked set defined by an outranking relation based on the Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE II); (iii) we construct three-way decision rules to solve the problems of clustering and ranking of objects in data analysis. In order to demonstrate the usefulness of 3W-MADM-R, we apply it to analyze heart disease data. By comparing with results of other methods, we show the feasibility, stability and superiority of the presented 3W-MADM-R method.

... By associating threes with different meanings, there are many concrete and practical 3WD models and methods. For instance, three-way classification Hu 2022;Cabitza et al. 2017;Li et al. 2017c;Zhang and Yao 2017), three-way clustering (Afridi et al. 2018;Yu et al. 2016), three-way recommendation Zhang and Min 2016;Azam and Yao 2014), three-way concept analysis Qi et al. 2016Qi et al. , 2022Ren and Wei 2016;Singh 2017), three-way conflict analysis (Lang et al. 2017;, and others (Li et al. 2017a, b;Huang et al. 2017;Liu et al. 2016;Liang et al. 2021). As a specific concrete model, recently discussed a framework of trilevel thinking, in which a whole is interpreted and investigated at three levels. ...

Uncertainty measure is one of the most significant concepts and fundamental issues in granular computing. Nowadays, there have been extensive studies on various uncertainty measures for quantifying diverse properties and associations of granules and granular structures. However, there is a lack of a systematic study for uncertain measures. Based on a trilevel thinking framework, this paper presents a systematic review and analysis of uncertainty measures used in partition-based models of granular computing. At an object level, a granule level and a granular structure level, we categorize uncertainty measures for describing the properties and the associations of objects, granules, and partitions respectively. Moreover, we illustrate how to construct an uncertainty measure at a higher level from a lower level. At last, we discuss several potential directions to design other new uncertainty measures for partition-based granular computing.

... Recently, theories such as three-way formal concept analysis [31] and three-way cognition computing [32,33] have focused on concept learning via multi-granularity from the viewpoint of cognition. The three-way fuzzy sets method [34], three-way decisions space [35], sequential three-way decisions [36], and generalized three-way decision models [37][38][39] have been proposed. Moreover, applications include the three-way recommender system [40], three-way active learning [41], three-way clustering [42], tri-partition neighborhood covering reduction [43], three-way spam filtering [44], three-way face recognition [45], and the tri-alphabet-based sequence pattern [46]. ...

Recently, predicting multivariate time-series (MTS) has attracted much attention to obtain richer semantics with similar or better performances. In this paper, we propose a tri-partition alphabet-based state (tri-state) prediction method for symbolic MTSs. First, for each variable, the set of all symbols, i.e., alphabets, is divided into strong, medium, and weak using two user-specified thresholds. With the tri-partitioned alphabet, the tri-state takes the form of a matrix. One order contains the whole variables. The other is a feature vector that includes the most likely occurring strong, medium, and weak symbols. Second, a tri-partition strategy based on the deviation degree is proposed. We introduce the piecewise and symbolic aggregate approximation techniques to polymerize and discretize the original MTS. This way, the symbol is stronger and has a bigger deviation. Moreover, most popular numerical or symbolic similarity or distance metrics can be combined. Third, we propose an along–across similarity model to obtain the k-nearest matrix neighbors. This model considers the associations among the time stamps and variables simultaneously. Fourth, we design two post-filling strategies to obtain a completed tri-state. The experimental results from the four-domain datasets show that (1) the tri-state has greater recall but lower precision; (2) the two post-filling strategies can slightly improve the recall; and (3) the along–across similarity model composed by the Triangle and Jaccard metrics are first recommended for new datasets.

... From this perspective, 3WD is an intermediate means and step to realize the two-way decision finally. In addition, the basic ideas of the 3WD have been successfully applied in many fields, including classification [9,23], pattern discovery [24], medical decision making [25], uncertainty management [26], recommender systems [27], clustering analysis and covering reduction analysis [28][29][30], incomplete data analysis [31,32]. ...

Continual learning has attracted much attention in recent years, and many continual learning methods based on deep neural networks have been proposed. However, several important problems about these methods may lead to high decision cost and affect the practical application of continual learning networks. First, continual learning networks treat all categories equally, although the unbalance of misclassification cost happens in real-world cases. Second, there is a trade-off between learning new knowledge and keep old knowledge, which leads to the forgetting of old knowledge (i.e., the catastrophic forgetting). Third, even if low confidence of a sample, the continual learning methods based on the neural network will still give a clear classification result. We propose a sequential three-way decision model for continual learning to address these problems, named Incremental Sequential Three-Way Decision model (ISTWD). Introducing cost-sensitive sequential three-way decision to continual learning network, ISTWD reduces the decision cost of continual learning, which may alleviate the potentially high cost caused by the accuracy loss in continual learning. Besides, ISTWD includes a checkpoint procedure to judge whether the process of continual learning should stop. Experimental results on CIFAR-100 and Tiny-ImageNet verify the effectiveness of our method.

... The next question is how to decide values of α and β in different real-life scenarios. To carefully address this issue, we use decision-theoretic fuzzy rough set (Feng and Mi 2016;Liang et al. 2013;Radzikowskaa and Kerre 2002;Varmaghani et al. 2021) to calculate values of α and β, a more of three-way decision approach (Hu 2014;Liang et al. 2015;Li and Zhou 2011;Li et al. 2017;Liu et al. 2016;Yang and Yao 2012) of decision-making based on Bayesian model. This way is very fruitful and already been applied in studying conflict analysis (Lang et al. 2017Li et al. 2021). ...

This paper aims to develop a novel conflict resolution model using decision-theoretic fuzzy rough set to handle more complex real scenarios by allowing decision-makers to express their opinions more freely on a scale from −1 to 1. Further, many algorithms are developed to handle change in information systems, and detailed experimental analysis is done to validate the proposed model’s efficiency and practicality.

... Liu et al. [11] summarizes the development track and evolution process of three-way decision. At present, three-way decision has been widely used in different fields, such as decision [12], medical diagnosis [13,14], face recognition [15,16], conflict analysis [17], cluster analysis [18,19] etc. ...

Three-way decision is a decision-making method based on human cognitive process, and its basic idea is to divide a universal set into three pair-wise disjoint regions to cognitive information processing. As the complexity of decision-making environment, cognitive information about alternatives given by decision-makers is uncertain and inconsistent. Picture fuzzy point operator (PFPO) is an effective tool to handle this information. In order to obtain more reasonable and effective decision results, this paper proposes three-way decision models and develops a multi-attribute three-way decision method. Then, we use the proposed method to solve a project investment problem. We define new operators on picture fuzzy numbers by a monotonically increasing binary function and a monotonically decreasing unary function. Then, we build three-way decision models based on PFPO and these new operators. Further, we fully consider the relationship between attributes and the classification of alternatives, and present a multi-criteria three-way decision method. In addition, we compare the proposed method with the existing methods by a project investment problem. We show that PFPO can handle inconsistent and changing cognitive information more accurately through an example. In a project investment problem, the decision results obtained by using the proposed method are the same as those obtained by the existing methods, which shows that the method is effective. By the analysis and comparison with these methods, it is proved that the proposed method is very suitable for dealing with multi-attribute decision-making problem with changing picture fuzzy information and consistent with human cognition.

... The main idea of 3WD is to divide the universe into three disjoint regions, then different decision strategies can be adopted for the different regions [4,5]. In recent years, 3WD theory had been successfully applied in many fields, such as neural networks [6], data mining [7], text processing [8,9], medical systems [10,11], recommendation algorithms [12,13], malware analysis [14], attribute reduction [15][16][17][18], risk decisions [19,20], incomplete information [21][22][23][24][25] and so on [26][27][28][29][30][31]. Meanwhile, the interpretation [32][33][34][35][36][37][38][39] and expansion [40][41][42][43][44][45][46][47][48][49][50] of the model were also research hotspots. ...

In some cases, the decision process of three-way decisions (3WD) is costly, and sequential three-way decisions (S3WD) may cause errors beyond tolerance. To solve the above problems, in this paper, democratic three-way decisions based on voting mechanism (D3WD-VM) is proposed from the perspective of all conditional attributes. By obtaining decision opinions of different attributes at the coarse granularity level, the final decision results is obtained. First, a voting mechanism is established to realize the idea of the democratic three-way, which is an ensemble decision space based on conditional attributes. Next, in order to make the decision results more reasonable, the normalized information gain ratio is utilized to optimize the voting weight of conditional attributes in the voting mechanism. Then, based on cognitive science, two different decision strategies are devised to make the final decision. Finally, the experimental results demonstrate that the accuracy rate and the comprehensive evaluation index of the D3WD-VM have also been improved to some extent compared with the S3WD, and the decision efficiency is better than 3WD.

... Sequential three-way decisions [13] is an iteration process that eventually leads to two-way decisions. Some generalized three-way decision models [53][54][55] are quite popular. ...

Recently, the advancement of cognitive computing and three-way decisions has enabled in-depth sequential pattern understanding through temporal association analysis. The main challenge is to obtain concise patterns that express richer semantics for multivariate time series (MTS) analysis. In this paper, we propose a tri-partition state alphabet-based sequential pattern (Tri-SASP) for MTSs. First, a tri-wildcard gap inserted between each pair of adjacent states enhances the flexibility of the method. Second, a given set of states is partitioned into positive (POS), negative (NEG) and boundary (BND) regions. The states in POS can only be used to construct a Tri-SASP, the states in NEG can only be matched by a tri-wildcard gap, and the states in BND can be used in both ways. Finally, horizontal and vertical algorithms are proposed to obtain frequent Tri-SASPs in a breadth-first manner. The experimental results on four real-world datasets show that (1) the discovered Tri-SASPs and temporal rules can enrich human cognition; (2) the two tri-partition strategies can bring us very meaningful and varied Tri-SASPs; and (3) the two algorithms are effective and scalable.

... Zhang et al. [28] combined 3WD with uncertain classification, defined two kinds of classification errors and two kinds of uncertain classification of the probabilistic rough set model, considered the cost parameters of two kinds of classification errors and two kinds of uncertain classification, got a pair of thresholds of probabilistic rough set model again, and proposed three decision-making models based on two kinds of uncertain classification. Liu et al. [29] proposed a three-way decision model based on the incomplete information system and defined a new relation to describe the similarity degree of incomplete information. Given the missing value in the incomplete information system, the loss function is obtained using an interval number. ...

Most law enforcement cases executed by the courts in China have behaviours of evading, evading, or even violently resisting execution or passively waiting for enforcement, which seriously affects the authority of legal judgments and the judiciary's credibility. erefore, we develop a hidden property evaluation model based on the probabilistic linguistic three-way multi-attribute decision-making (PL3W-MADM) method. Considering the advantages of probabilistic linguistic term sets (PLTSs) expressing the evaluation information and their probabilities on judgment debtor given by expert judges, we extend the three-way decision method to a probabilistic linguistic environment and develop the strict PL3W-MADM model and flexible PL3W-MADM model. en, the PL3W-MADM models are used to construct the hidden property evaluation model of judgment debtors. Finally, the developed hidden property evaluation model can quickly and effectively classify the judgment debtors into three categories: hidden behaviour, no hidden behaviour or lack of information, and temporary inability to judge. e results show that the developed model is more suitable for hidden property evaluation than the strict PL3W-MADM model and the flexible PL3W-MADM model.

... Based on the rough set theory model, Yao introduced the Bayes risk decision method to analyze and evaluate the risk cost of various decisions and then obtained the minimum risk cost evaluation decision result among them [2]. Nowadays, theories and methods related to three-way decisions have attracted widespread attention from scholars [3][4][5][6][7]. Moreover, it has been successfully applied to many disciplines and fields, such as attribute reduction [8][9][10], paper review [11], recommendation system [12,13], granular computing [3,14], multi-attribute decision-making [15], fuzzy clustering [16,17], concept learning [18,19], medical diagnosis [20] and face recognition [21]. ...

The method of determining probability thresholds of three-way decisions (3WDs) has always been the key of research, especially in the current environment with a large number of data and uncertainties. Among these problems, there will be correlation and similarity between them. In the light of these problems, the loss function with Probabilistic Linguistic Terms Sets (PLTSs) is introduced in the paper, and then we propose a PLTS evaluation-based approach to determine the thresholds and derive 3WDs. According to the definition and characters of PLTSs, the PLTSs loss function matrix is constructed firstly. Then using the equivalent model of Decision-theoretic rough sets (DTRSs), we construct the equivalent model (i.e., the αopt-model and the βopt-model, which are symmtrical) and try to find the optimal solution to determine the thresholds. Based on that, we propose a novel three-way decision approach under PLTSs evaluations. Finally, the validity of the method is verified by an example.

In this paper, we focus on the three-way decision model on incomplete single-valued neutrosophic information tables. Firstly, we define the minimum and maximum similarity measures between single-valued neutrosophic numbers (SVNNs) which may contain unknown values. On this basis, the notion of θ-weak similarity measure is given. Then, we introduce the conception of an incomplete single-valued neutrosophic information table (ISVNIT). For an incomplete single-valued neutrosophic information table, a new similarity relation is proposed based on the θ-weak similarity measure. Some properties are also studied. By using Bayesian decision theory and this similarity relation, we construct a three-way decision model on an ISVNIT. Finally, an example of choosing product service providers is explored to illustrate the rationality and feasibility of the proposed model. We also discuss the influence of parameters in the model on decision results.

The three-way decision approach is an emerging paradigm in the design of tools for data mining and machine learning. It switches from a two-way classification (“negative” and “positive” class) to three decisions: “negative”, “positive”, and non-commitment class. It means that when for some data it is not possible to elaborate a reliable answer they are assigned with a non-commitment class. In the paper we apply this paradigm for a cascade of neuro-fuzzy classifiers. If the first neuro-fuzzy system assigns a data item with a non-commitment class, the next neuro-fuzzy system is run for this data item. For easy items the first system is enough, but for harder ones two or more systems have to be run. Neuro-fuzzy systems elaborate interpretable fuzzy models. The models are composed of fuzzy rules that can be interpreted linguistically by humans. Application of neuro-fuzzy systems results in a cascade of interpretable models. The paper describes algorithms for training a cascade of neuro-fuzzy classifiers and for elaboration of answers. The paper presents results of numerical experiments that show that this technique can elaborate results with lower generalisation error than two-way classifiers. The implementation of the proposed system is available from a github repository.

Recently, some researchers have explored three-way decision models in the fuzzy multi-criteria environments from the perspective of criterion fuzzy concept. The main characteristics of these models include the subjective preference of decision-maker for criteria and the evaluation value information system provided by decision-maker. However, the criterion fuzzy concept only considers the decision-maker’s membership preference to each criterion, and ignores the decision-maker’s non-membership preference to each criterion. In view of this, this paper further proposes a new criterion fuzzy concept, namely the intuitionistic fuzzy concept, to express decision-maker’s preference for each criterion. At the same time, we build an intuitionistic fuzzy concept-oriented three-way decision model to solve the ranking and classification problem in the intuitionistic fuzzy multi-criteria environments with decision-maker’s preference. Firstly, based on the defined intuitionistic fuzzy concept, we construct a new intuitionistic fuzzy loss function model. Meanwhile, using intuitionistic fuzzy similarity, we propose a new conditional probability calculation method to express the correlation degree between the intuitionistic fuzzy concept and alternative’s evaluation values. Secondly, we introduce three decision-making perspectives from the membership degree and non-membership degree of expected loss to obtain the final three-way classification scheme. Moreover, by combining the thresholds of the positive and negative perspectives, we obtain the comprehensive thresholds of the comprehensive perspective, which can be used to rank all alternatives. Finally, a supplier selection problem is used to verify the feasibility and superiority of the model, and six data sets are used to verify the applicability of the model.

As a new data representation in the big data era, a multi-scale decision information system (MSDIS) realizes “multi-level, multi-angle and multi-view” evaluations in various problems. However, most of existing multi-scale data analysis models are built for complete information systems (CISs), whereas the research on incomplete multi-scale decision information systems (I-MSDISs) is not perfect. Additionally, the irrational behavior of one decision-maker (DM) often has an impact on decision outcomes. On this basis, this paper develops a prospect-regret theory-based three-way decision (3WD) model with intuitionistic fuzzy numbers (IFNs) under an I-MSDIS, which is abbreviated as PR-3WD-I-MSDIS. Specifically, we first select the optimal scale combination of I-MSDISs by the matrix theory to complete the information extraction of incomplete optimal sub-systems. Then, we further propose an attribute weight evaluation strategy under the incomplete optimal subsystem, which fully considers the neighborhood information of each object. Subsequently, the concept and calculation method of aggregated weighted fuzzy conditional probabilities are constructed from the perspective of intuitionistic fuzzy sets. Furthermore, a prospect-regret theory-based relative profit function is proposed, which sufficiently takes into account the loss and utility in decision-making processes. Finally, a novel three-way classification and ranking method is developed for solving incomplete multi-scale problems. The experimental results on real-world datasets demonstrate that the PR-3WD-I-MSDIS model achieves excellent performance.

In recent years, many scholars have explored a variety of methods integrating three-way decision (3WD) and multi-attribute decision making (MADM), which enables the classification and priority ranking of alternatives possible and fully reflects the effectiveness and advantages of 3WD in solving MADM problems. However, few of these methods can effectively deal with the MADM problems with incomplete mixed information that are frequently encountered in real-world situations. This study proposes a three-way MADM method for an incomplete mixed information system (IMIS), where the objective determination of conditional probabilities and utility functions in IMIS without decision label is the pivotal issue. To overcome this issue, we define a probabilistic similarity measure for incomplete mixed information. The probabilistic similarity measure is used to replace the distance measure of classical TOPSIS for estimating the conditional probabilities objectively. The probabilistic similarity class is introduced with arithmetic average method to design a conversion mechanism and obtain the objective relative utility functions of incomplete evaluation values. We then construct a novel 3WD model in IMIS and combine it with two customized ranking principles, to solve the incomplete mixed MADM problems from the perspective of classification and ranking in a more thoughtful and interpretable manner. Our study provides a new perspective for the research on MADM in incomplete mixed information environment. Several examples and experimental comparisons verify the effectiveness and stability of the proposed method. The experiments demonstrate that our method can meet more decision-making requirements and is more accurate and rational in some decision-making scenarios than several existing similar methods.

Three-way decision theory has emerged as an effective method for attribute reduction when dealing with vague, uncertain, or imprecise data. However, most existing attribute reduction measures in the three-way decision are non-monotonic and too strict, limiting the quality of attribute reduction. In this study, a monotonic measure called parameterized maximum distribution entropy (PMDE) is proposed for approximate attribute reduction. Specifically, considering that the classification ability under uncertainty is reflected by both the decision and the degree of confidence, a novel PMDE measure that attaches different levels of importance to the decision with the highest probability and other decisions is provided, and its monotonicity is theoretically proven. Furthermore, the idea of trisection in the three-way decision is introduced into the process of attribute reduction, and a heuristic algorithm based on the proposed measure is developed to generate an optimal three-way approximate reduct, which greatly improves the efficiency of attribute reduction. Several experiments conducted on UCI datasets show that the proposed method achieves a favorable performance with much fewer attributes in comparison with other representative methods.

The present paper introduces two models of three-way decision with ranking and reference tuple on hybrid information tables. One is the model with an importance ratio, and the other is the model with any importance ratio, where importance ratio describes the quantitative comparison of importance between two attribute subsets. A unique measure is proposed to assess the trisections generated by the two models and, correspondingly, the concepts of local optimal and global optimal trisections are proposed respectively. The two models have good properties which enable the algorithms provided in this paper to compute the optimal trisections in finite steps. Through comparison and experiments on real data, we show that the two models have strong expressive power and capture two different types of trisecting problems on hybrid information tables, and demonstrate the feasibility and practicality of our method in potential applications.

Three-way decision (TWD) theory provides us with a new perspective and methodology to improve the flexibility of traditional multiple criteria decision-making (MCDM) methods. However, the construction of TWD theory under uncertain environment is still a serious challenge. In light of this, this paper proposes a novel interval-valued TWD theory for solving MCDM problems under uncertain environment. First, several rules are defined to calculate the interval-valued thresholds under multiple criteria environment. Afterwards, on the basis of interval-valued Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, a new approach is developed to calculate the interval-valued conditional probability of each alternative. Then, a new information transformation and fusion mechanism in the framework of evidential reasoning algorithm is established to conduct the interval-valued thresholds under multiple criteria into overall interval-valued thresholds. Finally, according to the overall interval-valued thresholds and interval-valued conditional probabilities, several new decision rules for objects under different decision situations are provided to generate the classifications and rankings of alternatives. To illustrate the feasibility of the proposed method, an application about purchasing green vehicles is performed. Further detailed discussions and comparisons with other methods are also conducted to show the effectiveness and superiority of the current proposal.

In mobile edge computing (MEC), the mobile device cannot always stably connect with the same edge server due to the movement of the user. Considering the complexity and randomness of users’ movement, most of the existing service migration strategies based on two-way decisions (namely either migrate or not migrate) may make wrong decisions, which will increase the energy consumption and significantly impact users’ experience. To address such an issue, we employ three-way decisions for service migration. Specifically, first, based on users’ movement trajectory, we allocate users into three regions: migration region, non-migration region and delay region. Afterwards, we execute different operations for users in different regions accordingly. In the migration region, considering the energy consumption and service delay time, we propose a new service migration method which utilizes the latest migration radius and the lowest energy consumption for service migration. In the delay region, the edge server will keep collecting users’ movement information to prepare for the further decision making. In the non-migration region, no action is required by the edge server. Comprehensive simulation experiments using parameter settings consistent with real-world edge computing environments are conducted to show its superior performance compared with other strategies.

In this paper, we propose general three-way decision models on incomplete information tables. First, for an incomplete information table, we give an axiomatic definition of similarity degree functions on a single attribute. By use of extended aggregation functions, similarity degree functions on an attribute set are also proposed. Then we define a new kind of similarity class of objects and study its properties. On the basis of this similarity class, general three-way decision models based on two evaluation functions and one evaluation function on incomplete information tables are established, respectively. In addition, we study the properties of these general three-way decision models. Finally, we compare the general model based on one evaluation function and a pair of thresholds with four existing models. The results show that the four existing models can be regarded as special cases of this general model, which illustrates the rationality of the new proposed models.

The theory of three-way decisions, as a powerful methodology of granular computing, has been widely used in making decision under uncertainty environments. Decision tasks in incomplete hybrid data including heterogeneous and missing features are of abundance in realistic situations. To deal with these tasks, some work based on three-way decisions has been investigated. However, the losses used for evaluating objects are precise real numbers, which makes these decision models have some limitations in applications when there exist missing values in incomplete hybrid data. Thus, this paper constructs a generalized three-way neighborhood decision model by assigning the interval-valued loss function to each object and further adopting an average strategy to integrate the interval-valued loss functions of objects in each data-driven neighborhood class. Moreover, considering that the objects and attributes of incomplete hybrid data will simultaneously change over time, this paper also provides an efficient framework to dynamically maintain three-way regions of the proposed model. An approach based on matrix to compute the three-way regions is first presented by introducing the matrix operations and the matrix forms of related concepts. Then, with the simultaneous variation of objects and attributes, the matrix-based incremental mechanism and algorithm are proposed for updating the three-way regions, respectively. Experimental results on nine datasets indicate that the proposed incremental algorithm can effectively improve the computational performance for evolving data in comparison with the static algorithm.

The existing three-way conflict analysis models focus on complete situation tables and rarely involve incomplete situation tables. In this paper, we give a preliminary approach to three-way conflict analysis models based on incomplete situation tables. Firstly, we construct three-way conflict analysis models based on an incomplete three-valued situation table (ITVST). We use two different methods to establish the models. One is to establish straightly the model based on the original ITVST. The other is to convert the ITVST into a complete three-valued situation table (TVST) by supplementing the missing values with the maximum probability value method, which is called a complete TVST induced by the maximum probability value method. Then we construct the three-way conflict analysis model based on the induced complete TVST. Secondly, to deal with more uncertainty and complexity in the actual decision-making process, we generalize an ITVST to an incomplete many-valued situation table (IMVST) and establish the corresponding three-way conflict analysis models. Finally, we further extend an IMVST to an incomplete interval-valued situation table (IIVST) and construct three-way conflict analysis models based on an IIVST. We use examples to demonstrate the process of constructing conflict, neutrality, and alliance relations of agents on incomplete situation tables. Some algorithms and basic properties are also addressed.

Neighborhood classifier, a common classification method, is applied in pattern recognition and data mining. The neighborhood classifier mainly relies on the majority voting strategy to judge each category. This strategy only considers the number of samples in the neighborhood but ignores the distribution of samples, which leads to a decreased classification accuracy. To overcome the shortcomings and improve the classification performance, D-S evidence theory is applied to represent the evidence information support of other samples in the neighborhood, and the distance between samples in the neighborhood is taken into account. In this paper, a novel attribute reduction method of neighborhood rough set with a dynamic updating strategy is developed. Different from the traditional heuristic algorithm, the termination threshold of the proposed reduction algorithm is dynamically optimized. Therefore, when the attribute significance is not monotonic, this method can retrieve a better value, in contrast to the traditional method. Moreover, a new classification approach based on D-S evidence theory is proposed. Compared with the classical neighborhood classifier, this method considers the distribution of samples in the neighborhood, and evidence theory is applied to describe the closeness between samples. Finally, datasets from the UCI database are used to indicate that the improved reduction can achieve a lower neighborhood decision error rate than classical heuristic reduction. In addition, the improved classifier acquires higher classification performance in contrast to the traditional neighborhood classifier. This research provides a new direction for improving the accuracy of neighborhood classification.

The theory of three-way decision, introduced for the needs of explaining the three regions of rough sets, has developed into a more general theory of three regions in recent years. For different types of problems, we should have different types of intentions of trisecting, and only by considering the specific intentions of trisecting can we get the most accurate three regions. This is the starting point of this article. From a new perspective of trisecting, we propose two concepts on two universes, namely rankings of a set of attributes and reference tuples. These two concepts are combined together to express the original intention of trisecting in a new general meaning. At the same time, an evaluation of matching degree is proposed to formulate the trisecting. Based on the above two concepts and one evaluation method, we construct a two-universe model of three-way decision with concrete formulations, and show that the rough-set-based model proposed by Yan et al. is only equivalent to one of the eight cases of our model, with the eight cases corresponding to eight different types of intentions and hence to eight different types of problems. Therefore, the present paper extends classical rough-set-based models to a more general level on two universes. Two algorithms are provided to compute the three regions of our model, with the second one also computing the ordering of objects and hence the optimal ones.

Attribute significance is very important in multiple-attribute decision-making (MADM) problems. In a MADM problem, the significance of attributes is often different. In order to overcome the shortcoming that attribute significance is usually given artificially. The purpose of this paper is to give attribute significance computation formulas based on inclusion degree. We note that in the real-world application, there is a lot of incomplete information due to the error of data measurement, the limitation of data understanding and data acquisition, etc. Firstly, we give a general description and the definition of incomplete information systems. We then establish the tolerance relation for incomplete linguistic information system, with the tolerance classes and inclusion degree, significance of attribute is proposed and the corresponding computation formula is obtained. Subsequently, for incomplete fuzzy information system and incomplete interval-valued fuzzy information system, the dominance relation and interval dominance relation is established, respectively. And the dominance class and interval dominance class of an element are got as well. With the help of inclusion degree, the computation formulas of attribute significance for incomplete fuzzy information system and incomplete interval-valued fuzzy information system are also obtained. At the same time, results show that the reduction of attribute set can be obtained by computing the significance of attributes in these incomplete information systems. Finally, as the applications of attribute significance, the attribute significance is viewed as attribute weights to solve MADM problems and the corresponding TOPSIS methods for three incomplete information systems are proposed. The numerical examples are also employed to illustrate the feasibility and effectiveness of the proposed approaches.

Conflict occurs naturally in all walks of life at any time. When agents face a conflict, they usually have multiple alternatives and hesitate to make a decision. This paper studies conflict analysis in the case of hesitant fuzzy setting, i.e. the attitude of an agent on an issue has multiple choices. We first propose a conflict analysis model based on hesitant fuzzy information systems. Subsequently, we present the concepts of conflict, neutrality and alliance sets on one issue and multiple issues respectively, and introduce a method to compute the three sets via Bayesian decision theory. Moreover, the degree of conflict, neutrality and alliance among agents with respect to an issue are defined respectively. According to the conflict and alliance degrees, the set of issues is divided into three parts: the main reason, secondary reason and irrelevant reason. Finally, two methods of computing coalitions are discussed. One is based on Bayesian decision theory. The other is to find a complete subgraph of the coalition graph.

As a complicated cognitive process, multi-attribute decision-making usually focuses on the decision-making issue of seeking the optimal alternative or ranking alternatives under the framework of multiple attributes. The three-way decision approach with the delayed decision can more effectively reduce decision risks than traditional two-way counterparts for multi-attribute decision-making. In this article, we aim to put forward a novel three-way multi-attribute decision-making model in light of a probabilistic dominance relation with intuitionistic fuzzy sets. First, we investigate the three-way multi-attribute decision-making in light of a probabilistic dominance relation in an intuitionistic fuzzy information system. Second, we derive the conditional probability of the intuitionistic fuzzy set. Third, we evaluate the part supplier selection via the constructed model. At last, for the sake of showing the validity and applicability of the constructed three-way intuitionistic fuzzy multi-attribute decision-making model, we further perform extensive comparative analysis along with experimental analysis from diverse perspectives.

Decision-theoretic rough sets (DTRSs) as a classic model of three-way decisions have been widely applied in the field of risk decision-making. Considering situations where experts hesitate among several evaluation values, hesitant fuzzy sets, as a new generalization of fuzzy sets, can describe uncertain information flexibly in the decision-making process. In this paper, we propose a decision-theoretic fuzzy rough set (DTFRS) model in hesitant fuzzy information systems and discuss its application in multi-attribute decision-making (MADM). More specifically, we first define a novel fuzzy binary relation between two objects by using the hesitant fuzzy distance function. Then, we study the calculations of the fuzzy similarity class and the conditional probability. At the same time, based on the connection between the loss functions and the attribute values, we develop a data-driven calculation method of the relative loss functions. With these discussions, we construct a DTFRS model in hesitant fuzzy information systems and explore the related decision-making mechanism. Furthermore, a three-way decision method based on the proposed DTFRS model is established to handle MADM problems in the context of a hesitant fuzzy environment. The established method not only takes the decision risk into consideration, but also instructs us how to choose the action for each alternative and gives its corresponding semantic explanation. An illustrative example of the stock investment problem is presented to verify the efficacy of our method. Finally, we take a sensitivity analysis and a comparison analysis to show the established method’s performance and characteristics.

The decision-theoretic rough set model is adopted to derive a profit-based three-way approach to investment decision-making. A three-way decision is made based on a pair of thresholds on conditional probabilities. A positive rule makes a decision of investment, a negative rule makes a decision of non-investment, and a boundary rule makes a decision of deferment. Both cost functions and revenue functions are used to calculate the required two thresholds by maximizing conditional profit with the Bayesian decision procedure. A case study of oil investment demonstrates the proposed method.

A theory of three-way decisions is constructed based on the notions of acceptance, rejection and noncommitment. It is an extension of the commonly used binary-decision model with an added third option. Three-way decisions play a key role in everyday decision-making and have been widely used in many fields and disciplines. An outline of a theory of three-way decisions is presented by examining its basic ingredients, interpretations, and relationships to other theories.

Decision-theoretic rough set is a quite useful rough set by introducing the decision cost into probabilistic approximations of the target. However, Yao’s decision-theoretic rough set is based on the classical indiscernibility relation; such a relation may be too strict in many applications. To solve this problem, a
δ
-cut decision-theoretic rough set is proposed, which is based on the
δ
-cut quantitative indiscernibility relation. Furthermore, with respect to criterions of decision-monotonicity and cost decreasing, two different algorithms are designed to compute reducts, respectively. The comparisons between these two algorithms show us the following: (1) with respect to the original data set, the reducts based on decision-monotonicity criterion can generate more rules supported by the lower approximation region and less rules supported by the boundary region, and it follows that the uncertainty which comes from boundary region can be decreased; (2) with respect to the reducts based on decision-monotonicity criterion, the reducts based on cost minimum criterion can obtain the lowest decision costs and the largest approximation qualities. This study suggests potential application areas and new research trends concerning rough set theory.

Rough set theory has witnessed great success in data mining and knowledge discovery, which provides a good support for decision making on a certain data. However, a practical decision problem always shows diversity under the same circumstance according to different personality of the decision makers. A simplex decision model can not provide a full description on such diverse decisions. In this article, a review of Pawlak rough set models and probabilistic rough set models is presented, and a three-way view decision model based on decision-theoretic rough set model is proposed, in which optimistic decision, pessimistic decision, and equable decision are provided according to the cost of misclassification. The thresholds of probabilistic inclusion are calculated based on minimization of risk cost under respective decision bias. The study not only presents a new theoretic decision model considering the different personality of the decision makers, but also provides a practical explanation and an illustrative example
on diverse risk bias decision.
Keywords: decision-theoretic rough set; three-way view decision; risk decision making; Bayesian decision

We consider a problem of Bayesian risk decision based on probabilistic rough set over two universes. It is a new extension of classical probabilistic rough set on the same universe. We give four rough set models on probabilistic approximation space over two universes. Then we study the interrelationship between Bayesian risk decision and probabilistic rough set models over two universes. The results show that there must exist a kind of Bayesian minimum risk decision problem corresponding to one of the probabilistic rough set models over two universes. In fact, the conclusion also includes some generalized probabilistic rough set models on the same universe by other authors. And at the same time, the principal and validity of the Bayesian risk decision based on probabilistic rough set over two universes are tested by a numerical example of the medical diagnosis systems in detail. The probabilistic rough set approach over two universes gives an effective assistant for decision makers in the context of risk and uncertainty.

We present a novel approach to understanding the concepts of the theory of rough sets in terms of the inverse probabilities
derivable from data. It is related to the Bayes factor known from the Bayesian hypothesis testing methods. The proposed Rough
Bayesian model (RB) does not require information about the prior and posterior probabilities in case they are not provided
in a confirmable way. We discuss RB with respect to its correspondence to the original Rough Set model (RS) introduced by
Pawlak and Variable Precision Rough Set model (VPRS) introduced by Ziarko. We pay a special attention on RB’s capability to
deal with multi-decision problems. We also propose a method for distributed data storage relevant to computational needs of
our approach.
KeywordsRough Sets-Probabilities-Bayes Factor

Recommender systems attempt to guide users in decisions related to choosing items based on inferences about their personal opinions. Most existing systems implicitly assume the underlying classification is binary, that is, a candidate item is either recommended or not. Here we propose an alternate framework that integrates three-way decision and random forests to build recommender systems. First, we consider both misclassification cost and teacher cost. The former is paid for wrong recommender behaviors, while the latter is paid to actively consult the user for his or her preferences. With these costs, a three-way decision model is built, and rational settings for positive and negative threshold values and are computed. We next construct a random forest to compute the probability P that a user will like an item. Finally, , and P are used to determine the recommender’s behavior. The performance of the recommender is evaluated on the basis of an average cost. Experimental results on the well-known MovieLens data set show that the -pair determined by three-way decision is optimal not only on the training set, but also on the testing set.

Existing clustering approaches are usually restricted to crisp clustering, where objects just belong to one cluster; meanwhile there are some applications where objects could belong to more than one cluster. In addition, existing clustering approaches usually analyze static datasets in which objects are kept unchanged after being processed; however many practical datasets are dynamically modified which means some previously learned patterns have to be updated accordingly. In this paper, we propose a new tree-based incremental overlapping clustering method using the three-way decision theory. The tree is constructed from representative points introduced by this paper, which can enhance the relevance of the search result. The overlapping cluster is represented by the three-way decision with interval sets, and the three-way decision strategies are designed to updating the clustering when the data increases. Furthermore, the proposed method can determine the number of clusters during the processing. The experimental results show that it can identifies clusters of arbitrary shapes and does not sacrifice the computing time, and more results of comparison experiments show that the performance of proposed method is better than the compared algorithms in most of cases.

A fundamental notion of rough sets is the approximation of a set by a triplet of positive, boundary, and negative regions, which leads to three-way decisions or ternary classifications. Rules from the positive region are used for making a decision of acceptance, rules from the negative region for making a decision of rejection, and rules from the boundary region for making a decision of non-commitment or deferment. This new view captures an important aspect of rough set theory and may find many practical applications.

We investigate in this paper approximate operations on sets, approximate equality of sets, and approximate inclusion of sets. The presented approach may be considered as an alternative to fuzzy sets theory and tolerance theory. Some applications are outlined.

In the Pawlak rough set model, the positive region, the boundary region and the non-negative region are monotonic with respect to the set inclusion of attributes. However, the monotonicity property of the decision regions (positive region, boundary region or non-negative region) with respect to the set inclusion of attributes does not hold in the decision-theoretic rough set model. Therefore, the decision regions may be changed after attribute reduction based on quantitative preservation or qualitative preservation of decision regions. This effect is observed partly because three decision regions are defined by introducing the probabilistic threshold values. In addition, heuristic reduction algorithms based on decision regions may find super reducts because of the non-monotonicity of decision regions. To address the above issues, this paper proposes solutions to the attribute reduction problem based on decision region preservation in the decision-theoretic rough set model. First, the (alpha, beta) positive region distribution preservation reduct, the (alpha, beta) boundary region distribution preservation reduct and the (alpha, beta) negative region distribution preservation reduct are introduced into the decision-theoretic rough set model. Second, three new monotonic measures are constructed by considering variants of the conditional information entropy, from which we can obtain the heuristic reduction algorithms. The results of the experimental analysis validate the monotonicity of new measures and verify the effectiveness of decision region distribution preservation reducts.

Granular computing has attracted many researchers as a new and rapidly growing paradigm of information processing. In this paper, we apply systematic mapping study to classify the granular computing researches to discover relative derivations to specify its research strength and quality. Our search scope is limited to the Science Direct and IEEE Transactions papers published between January 2012 and August 2014. We defined four perspectives of classification schemes to map the selected studies that are focus area, contribution type, research type and framework. Results of mapping the selected studies show that almost half of the research focused area belongs to category of data analysis. In addition, most of the selected papers belong to proposing the solutions in research type scheme. Distribution of papers between tool, method and enhancement categories of contribution type are almost equal. Moreover, 39% of the relevant papers belong to the rough set framework. The results show that there is little attention paid to cluster analysis in existing frameworks to discover granules for classification. We applied five clustering algorithms on three datasets from UCI repository to compare the form of information granules, and then classify the patterns and define them to a specific class based on their geometry and belongings. The clustering algorithms are DBSCAN, c-means, k-means, GAk-means and Fuzzy-GrC and the comparison of information granules are based on the coverage, misclassification and accuracy. The survey of experimental results mostly shows Fuzzy-GrC and GAk-means algorithm superior to other clustering algorithms; while, c-means clustering algorithm shows inferior to other clustering algorithms.

Traditional methods of statistical process control are not well suited to controlling processes for the production of only small numbers of items. A Bayesian approach to this problem is considered in which both historical data and realistic cost assessment play a part. Two methods are proposed and developed with reference to a specific application.

By considering the various of studies on loss functions with three-way decisions, a function based three-way decisions is proposed to generalize the existing models. A "four-level" approach with granular perspective is built, and the existing models can be categorized to a "four-level" framework through different decision criteria. Our work pro-vides a novel "granularity" viewpoint on the current three-way decision researches.

Algebraic structures and lattice structures of rough sets are basic and important topics in rough sets theory. In this paper we pointed out that a basic problem had been overlooked, that is the closeness of union and intersection operations of rough approximation pairs, i.e. (lower approximation, upper approximation). We present that the union and intersection operations of rough approximation pairs are closed for classical rough sets and two kinds of covering based rough sets, but not for twenty kinds of covering based rough sets and the generalized rough sets based on fuzzy approximation space. Moreover, we proved that the union and intersection operations of rough fuzzy approximation pairs are closed and a bounded distributive lattice can be constructed.

In the previous decision-theoretic rough sets (DTRS), its loss function values are precise. This paper extends the precise values of loss functions to a more realistic stochastic environment. The stochastic loss functions are induced to decision-theoretic rough set theory based on the bayesian decision theory. A model of stochastic decision-theoretic rough set theory (SDTRS) is built with respect to the minimum bayesian expected risk. The corresponding propositions and criteria of SDTRS are also analyzed. Furthermore, we investigate two special SDTRS models under the uniform distribution and the normal distribution, respectively. Finally, an empirical study of Public-Private Partnerships (PPP) project investment validates the reasonability and effectiveness of the proposed models.

Quantitative attribute reduction exhibits applicability but complexity when compared to qualitative reduction. According to the two-category decision theoretic rough set model, this paper mainly investigates quantitative reducts and their hierarchies (with qualitative reducts) from a regional perspective. (1) An improved type of classification regions is proposed, and its preservation reduct (CRP-Reduct) is studied. (2) Reduction targets and preservation properties of set regions are analyzed, and the set-region preservation reduct (SRP-Reduct) is studied. (3) Separability of set regions and rule consistency is verified, and the quantitative and qualitative double-preservation reduct (DP-Reduct) is established. (4) Hierarchies of CRP-Reduct, SRP-Reduct, and DP-Reduct are explored with two qualitative reducts: the Pawlak-Reduct and knowledge-preservation reduct (KP-Reduct). (5) Finally, verification experiments are provided. CRP-Reduct, SRP-Reduct, and DP-Reduct expand layer by layer Pawlak-Reduct and exhibit quantitative applicability, and the experimental results indicate their effectiveness and hierarchies regarding Pawlak-Reduct and KP-Reduct.

A three-way, three-valued, or three-region approximation of a fuzzy set is constructed from a pair of thresholds (α,β) on the fuzzy membership function. An element whose membership grade equals to or is greater than α is put into the positive region, an element whose membership grade equals to or is less than β is put into the negative region, and an element whose membership grade is between β and α is put into the boundary region. A fundamental issue is the determination and interpretation of the required pair of thresholds. In the framework of shadowed sets (i.e., an example of three-way approximations of fuzzy sets), Pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions. To gain further insights into three-way approximations of fuzzy sets, we introduce an alternative decision-theoretic formulation in which the required thresholds are computed by minimizing decision cost.

In this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and Gonzalez-Rodriguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness, and formalisms that handle incomplete information often leads to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility-necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions.

Ideas of three-way decisions proposed by Yao come from rough sets. It is well known that there are three basic elements in three-way decisions theory, which are ordered set as to define three regions, object set contained in evaluation function and evaluation function to make three-way decisions. In this paper these three basic elements are called decision measurement, decision condition and evaluation function, respectively. In connection with the three basic elements this paper completes three aspects of work. The first one is to introduce axiomatic definitions for decision measurement, decision condition and evaluation function; the second is to establish three-way decisions space; and the third is to give a variety of three-way decisions on three-way decisions spaces. Existing three-way decisions are the special examples of three-way decisions spaces defined in this paper, such as three-way decisions based on fuzzy sets, random sets and rough sets etc. At the same time, multi-granulation three-way decisions space and its corresponding multi-granulation three-way decisions are also established. Finally this paper introduces novel dynamic two-way decisions and dynamic three-way decisions based on three-way decisions spaces and three-way decisions with a pair of evaluation functions.

In a probabilistic rough set model, the positive, negative and boundary regions are associated with classification errors or uncertainty. The uncertainty is controlled by a pair of thresholds defining the three regions. The problem of searching for optimal thresholds can be formulated as the minimization of uncertainty induced by the three regions. By using Shannon entropy as a measure of uncertainty, we present an information-theoretic approach to the interpretation and determination of thresholds.

The decision-theoretic rough set (DTRS) model considers costs associated with actions of classifying an equivalence class into a particular region. With DTRS, one may make informative decisions in the form of three-way decisions. Current research mainly focuses on single agent DTRS which is too complex for making a decision when multiple agents are involved. We propose a multi-agent DTRS model and express it in the form of three-way decisions. The new model seeks for synthesized or consensus decisions when there are multiple decision preferences and criteria adopted by different agents. Various multi-agent DTRS models can be derived according to the conservative, aggressive and majority viewpoints based on the positive, negative and boundary regions made by each agent. These multi-agent decision regions are expressed by figures in the form of three-way decisions.

Decision-theoretic rough sets (DTRS) play a crucial role in the risk decision-making problems. With respect to the minimum expected risk, DTRS deduce the rules of three-way decisions. Considering the new expression of evaluation information with hesitant fuzzy sets (HFS), we introduce HFS into DTRS and explore their decision mechanisms. More specifically, we take into account the losses of DTRS with hesitant fuzzy elements and propose a new model of hesitant fuzzy decisiontheoretic rough sets (HFDTRS). Some properties of the expected losses and their corresponding scores are carefully investigated under the hesitant fuzzy information. Three-way decisions and the associated cost of each object are further derived. With the above analysis, a novel risk decision-making method with the aid of HFDTRS is developed Besides the three-way decisions with DTRS, the method investigates the ranking and resource allocation by utilizing the associated costs of alternatives and a multi-objective 0-1 integer programming. Our study also offers a solution in the aspect of determining losses of DTRS and extends the range of applications.

Clustering provides a common means of identifying structure in complex data, and there is renewed interest in clustering as a tool for the analysis of large data sets in many fields. Determining the number of clusters in a data set is one of the most challenging and difficult problems in cluster analysis. To combat the problem, this paper proposes an efficient automatic method by extending the decision-theoretic rough set model to clustering. A new clustering validity evaluation function is designed based on the risk calculated by loss functions and possibilities. Then a hierarchical clustering algorithm, ACA-DTRS algorithm, is proposed, which is proved to stop automatically at the perfect number of clusters without manual interference. Furthermore, a novel fast algorithm, FACA-DTRS, is devised based on the conclusion obtained in the validation of the ACA-DTRS algorithm. The performance of algorithms has been studied on some synthetic and real world data sets. The algorithm analysis and the results of comparison experiments show that the new method, without manual parameter specified in advance, is more valid to determine the number of clusters and more efficient in terms of time cost.

As a natural extension to rough set approximations with two decision classes, this paper provides a new formulation of multi-class decision-theoretic rough sets. Instead of making an immediate acceptance or rejection decision, a third option of making a deferment decision is added to each class. This gives users the flexibility of further examining the suspicious objects, thereby reducing the chance of misclassification. Different types of misclassification errors are treated separately based on the notion of loss functions from Bayesian decision theory. The losses incurred for making deferment and rejection decisions to each class are also considered. The presented approach appears to be well suited for cost-sensitive classification tasks where different types of classification errors have different costs. The connections and differences with other existing multi-class rough set models are analyzed.

In the previous decision-theoretic rough sets (DTRS), its loss function values are constant. This paper extends the constant values of loss functions to a more realistic dynamic environment. Considering the dynamic change of loss functions in DTRS with the time, an extension of DTRS, dynamic decision-theoretic rough sets (DDTRS) is proposed in this paper. An empirical study of climate policy making validates the reasonability and effectiveness of the proposed model.

In this paper, we introduce the fuzzy interval number to decision-theoretic rough sets (DTRS), and propose a novel three-way decision model of fuzzy interval decision-theoretic rough sets (FIDTRS). The fuzzy interval number is used to describe the uncertainty of loss functions in DTRS. The corresponding propositions and criteria of FIDTRS are analyzed. An illustrative example of mines management is given to illuminate the proposed model in applications.

Email spam filtering is typically treated as a binary classification problem that can be solved by machine learning algorithms. We argue that a three-way decision approach provides a more meaningful way to users for precautionary handling their incoming emails. Three email folders instead of two are produced in a three-way spam filtering system, a suspected folder is added to allow users make further examinations of suspicious emails, thereby reducing the chances of misclassification. Different from existing ternary email spam filtering systems, we focus on two issues that are less studied, that is, the computation of required thresholds to define the three email categories, and the interpretation of the cost-sensitive characteristics of spam filtering. Instead of supplying the thresholds based on intuitive understandings of the levels of tolerance for errors, we systematically calculate the thresholds based on decision-theoretic rough set model. A loss function is interpreted as the costs of making classification decisions. A decision is made for which the overall cost is minimum. Experimental results show that the new approach reduces the error rate of misclassifying a legitimate email to spam and demonstrates a better performance for the cost-sensitivity aspect.

In classical rough set models, attribute reduction generally keeps the positive or non-negative regions unchanged, as these regions do not decrease with the addition of attributes. However, the monotonicity property in decision-theoretic rough set models does not hold. This is partly due to the fact that all regions are determined according to the Bayesian decision procedure. Consequently, it is difficult to evaluate and interpret region-preservation attribute reduction in decision-theoretic rough set models. This paper provides a new definition of attribute reduct for decision-theoretic rough set models. The new attribute reduction is formulated as an optimization problem. The objective is to minimize the cost of decisions. Theoretical analysis shows the meaning of the optimization problem. Both the problem definition and the objective function have good interpretation. A heuristic approach, a genetic approach and a simulated annealing approach to the new problem are proposed. Experimental results on several data sets indicate the efficiency of these approaches.

We investigate in this paper approximate operations on sets, approximate equality of sets, and approximate inclusion of sets. The presented approach may be considered as an alternative to fuzzy sets theory and tolerance theory. Some applications are outlined.