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Aiming of this paper is to introduce large-scale group decision-making (LSGDM) into three-way decisions (3WDs) with decision-theoretic rough sets (DTRSs) and propose LSGDM based 3WDs under linguistic assessments. There are two parameters involved of the 3WDs with DTRSs such as conditional probability and loss functions. Here we mainly focus on the calculation of loss function using LSGDM approach. LSGDM problem is characterized by a large number of experts, multiple clusters and a mass of evaluation data given by the experts. In some cases, experts are unwilling to revise their opinions to reach a consensus. So, the proper management of experts opinions and their non-cooperative behaviors (NCBs) is necessary to establish a consensus model. An appropriate adjustment of the credibility information is also essential. Using the clustering method, the proposed model divides the experts with similar evaluations into a subgroup. In each cluster, the experts' opinions are then aggregate. In order to measure the level of consensus among clusters, the cluster consensus index (CCI) and group consensus index (GCI) have been developed. Then, using a tool for managing the NCBs of clusters includes two components: (1) for identifying the NCBs of clusters, NCB degree has defined using CCI and GCI; (2) for consensus improvement, implemented the weight punishment mechanism to NCBs clusters. Finally, we have designed a rule for classifying the objects into three regions, and the associated cost of each object is derived for ranking the objects in each region. An example is offered for selection of the Energy project to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This article introduces a general framework of multi-granulation fuzzy probabilistic rough sets (MG-FPRSs) models in multi-granulation fuzzy probabilistic approximation space over two universes. Four types of MG-FPRSs are established, by the four different conditional probabilities of fuzzy event. For different constraints on parameters, we obtain four kinds of each type MG-FPRSs over two universes. To find a suitable way of explaining and determining these parameters in each kind of each type MG-FPRS, three-way decisions (3WDs) are studied based on Bayesian minimum-risk procedure, i.e., the decision-theoretic rough set (DTRS) approach. The main contribution of this paper is twofold. One is to extend the fuzzy probabilistic rough set (FPRS) to MG-FPRS model over two universes. Another is to present an approach to select parameters in MG-FPRS modeling by using the process of decision-making under conditions of risk. [ABSTRACT FROM AUTHOR]

Multi-granulation rough sets (MGRSs) and decision-theoretic rough sets (DTRSs) are two important and popular generalizations of classical rough sets. The combination of two generalized rough sets have been investigated by numerous researchers in different extensions of fuzzy settings such as interval-valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs), bipolar-valued fuzzy sets (BVFSs), etc. Pythagorean fuzzy (PF) set is another extension of fuzzy set, which is more capable in comparison to IFS handle vagueness in real world. However, few studies have focused on the combination of the two rough sets in PF settings. In this study, we combine the two generalized rough sets in PF settings. First, we introduce a type of PF subset (of subset of the given universe) of the PF Set (of the given universe). Then we establish two basic models of multi-granulation PF DTRS (MG-PF-DTRS) of PF subset of the PF set based on PF inclusion measure within the framework of multi-granulation PF approximation space. One model is based on a combination of PF relations (PFRs) and the construction of approximations with respect to the combined PFR. By combining PFRs through intersection and union, respectively, we construct two models. The other model is based on the construction of approximations from PFRs and a combination of the approximations. By using intersection and union to combine the approximations, respectively, we again get two models. As a result, we have total four models. Further for different constraints on parameters, we obtain three kinds of each model of the MG-PF-DTRSs. Then, their principal structure, basic properties and uncertainty measure methods are investigated as well. Second, we give a way to compute PF similarity degrees between two objects and also give a way to compute PF decision-making objects from incomplete multi-source information systems (IMSISs). Then we design an algorithm for decision-making to IMSISs using MG-PFDTRSs and their uncertainty measure methods. Finally, an example about the mutual funds investment is included to show the feasibility and potential of the theoretic results obtained. [ABSTRACT FROM AUTHOR]

Preference analysis is a class of important issues in multi-criteria decision making. The rough set theory is a powerful approach to handle preference analysis. In order to solve the multi-criteria preference analysis, this work improves the fuzzy multi-granulation decision-theoretic rough set model with additive consistent fuzzy preference relation, and it is used to analyze data from different sources, i.e., multi-source (fuzzy) information system. More specifically, we introduce the models of optimistic and pessimistic fuzzy preference relation multi-granulation decision-theoretic rough sets. Then, their principal structure, basic properties and several kinds of uncertainty measure methods are investigated as well. An example is employed to illustrate the effectiveness of the proposed models, and comparisons are also offered according to different measures of our models and existing models. [ABSTRACT FROM AUTHOR]

This article introduces general framework of multi-granulation bipolar-valued fuzzy (BVF) probabilistic rough sets (MG-BVF-PRSs) models in multi-granulation BVF probabilistic approximation space over two universes. Four types of MG-BVF-PRSs are established, by the four different conditional probabilities of BVF event. For different constraints on parameters, we obtain four kinds of each type MG-BVF-PRSs over two universes. To find a suitable way of explaining and determining these parameters in each kind of each type MG-BVF-PRS, three-way decisions (3WDs) are studied based on Bayesian minimum-risk procedure, i.e., the multi-granulation BVF decision-theoretic rough set (MG-BVF-DTRS) approach. The main contribution of this paper is twofold. One is to extend the fuzzy probabilistic rough set (FPRS) to MG-BVF-PRS model over two universes. Another is to present an approach to select parameters in MG-BVF-PRS modeling by using the process of decision making under conditions of risk. [ABSTRACT FROM AUTHOR]

This paper is devoted to propose generalized L-fuzzy variable precision rough sets, based on generalized residuated lattices with conjunctions not necessarily commutative over two universes, as a generalization of the notion of L-fuzzy variable precision rough sets based on residuated lattices with commutative conjunctions over one universe only. Then we define and investigate several classes of generalized L-fuzzy variable precision rough sets. The topological properties and granular representation of generalized L-fuzzy variable precision rough sets are also given. Our main concern is to propose a real number valued function for each approximation operator in generalized L-fuzzy variable precision rough sets and to measure its approximating ability. We observe that the functions of lower and upper approximation operators are natural generalizations of belief and plausibility functions respectively as given in the evidence theory. Using these functions, accuracy measure and degree of roughness is defined for generalized L-fuzzy variable precision rough sets. As an application of generalized L-fuzzy variable precision rough sets, I-fuzzy variable precision rough sets are proposed on the unit interval I = [0, 1], based on generalized residuated lattices induced by left-continuous pseudo-t-norms. [ABSTRACT FROM AUTHOR]