Figure - available from: Granular Computing
This content is subject to copyright. Terms and conditions apply.
![Decision rules with different values of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}](profile/Prasenjit-Mandal-3/publication/324113909/figure/fig3/AS:963461311893532@1606718485828/Decision-rules-with-different-values-of-thdocumentclass12ptminimal.gif)
Decision rules with different values of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}
Source publication
![Transform outcomes of P(A∣[xi]R^)\documentclass[12pt]{minimal}...](profile/Prasenjit-Mandal-3/publication/324113909/figure/fig2/AS:963461311905793@1606718485061/Transform-outcomes-of-PAxiRdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Decision rules with different values of θ\documentclass[12pt]{minimal}...](profile/Prasenjit-Mandal-3/publication/324113909/figure/fig3/AS:963461311893532@1606718485828/Decision-rules-with-different-values-of-thdocumentclass12ptminimal_Q320.jpg)
In this paper, we study interval-valued fuzzy probabilistic rough sets (IVF-PRSs) based on multiple interval-valued fuzzy preference relations (IVFPRs) and consistency matrices, i.e., the multi-granulation interval-valued fuzzy preference relation probabilistic rough sets (MG-IVFPR-PRSs). First, in the proposed study, we convert IVFPRs into fuzzy p...
Similar publications
We propose a staged framework for question answering over a large-scale structured knowledge base. Following existing methods based on semantic parsing, our method relies on various components for solving different sub-tasks of the problem. In the first stage, we directly use the result of entity linking to obtain the topic entity in a question, an...
There exist two formulations of the theory of rough sets, consisting of the conceptual formulations and the computational formulations. Class-specific and classification-based attribute reducts are two crucial notions in three-way probabilistic rough set models. In terms of conceptual formulations, the two types of attribute reducts can be defined...
Development of new Artificial Intelligence related data analysis methodologies with revolutionary information technology has made a radical change in prediction, forecasting, and decision making forreal-world data. The challenge arises w hen the real world dataset consisting of voluminous data is uncertain. The rough set is a mathematical formalism...
Citations
... Step 1. The consistency testing algorithm is used to eliminate the intervals that do not meet the conditions of PIPOE, and the probabilities of each interval are recalculated according to Equation (23) to obtain the revised set of probability intervals, as shown in Table 2. Step 2. According to the M1 model in the literature [35], the importance of attributes is sorted as follows: ...
Probabilistic interval ordering, as a helpful tool for expressing positive and negative information, can effectively address multi-attribute decision-making (MADM) problems in reality. However, when dealing with a significant number of decision-makers and decision attributes, the priority relationships between different attributes and their relative importance are often neglected, resulting in deviations in decision outcomes. Therefore, this paper combines probability interval ordering, the prioritized aggregation (PA) operator, and the Gauss–Legendre algorithm to address the MADM problem with prioritized attributes. First, considering the significance of interval priority ordering and the distribution characteristics of attribute priority, the paper introduces probability interval ordering elements that incorporate attribute priority, and it proposes the probabilistic interval ordering prioritized averaging (PIOPA) operator. Then, the probabilistic interval ordering Gauss–Legendre prioritized averaging operator (PIOGPA) is defined based on the Gauss–Legendre algorithm, and various excellent properties of this operator are explored. This operator considers the priority relationships between attributes and their importance level, making it more capable of handling uncertainty. Finally, a new MADM method is constructed based on the PIOGPA operator using probability intervals and employs the arithmetic–geometric mean (AGM) algorithm to compute the weight of each attribute. The feasibility and soundness of the proposed method are confirmed through a numerical example and comparative analysis. The MADM method introduced in this paper assigns higher weights to higher-priority attributes to establish fixed attribute weights, and it reduces the impact of other attributes on decision-making results. It also utilizes the Gauss AGM algorithm to streamline the computational complexity and enhance the decision-making effectiveness.
... Sun et al. [28] considered a novel MGRS with diversified binary relations for solving fuzzy MAGDM problems. For another, noticing that the structure of rough approximations is rather strict in original rough sets, PRSs-based MAGDM methods own the ability of fault tolerance by easing the strict limitations and setting suitable thresholds [12,16,17,25,27,29]. For instance, Sun et al. [25,27] proposed novel 3WD-based MAGDM approaches in fuzzy and linguistic information systems. ...
... Liang et al. [12] studied a method for 3WD by using traditional pythagorean fuzzy decision making tools. Mandal and Ranadive [16,17] discussed MGPRSs in bipolar-valued fuzzy and interval-valued fuzzy information systems along with their 3WD-based MAGDM approaches. Wang and Liang [29] presented the preference measure of multi-granularity probabilistic linguistic term sets for addressing large-scale MAGDM problems. ...
In real world, multi-attribute group decision making (MAGDM) is a complicated cognitive process that involves expression, fusion and analysis of multi-source uncertain information. Among diverse soft computing tools for addressing MAGDM, the ones from granular computing (GrC) frameworks perform excellently via efficient strategies for multi-source uncertain information. However, they usually lack convincing semantic interpretations for MAGDM due to extreme information fusion rules and instabilities of information analysis mechanisms. This work adopts a typical GrC framework named multigranulation probabilistic models to enrich semantic interpretations for GrC-based MAGDM approaches, and constructs MAGDM-oriented multigranulation probabilistic models with dual hesitant fuzzy (DHF) information in light of the MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus the full MULTIplicative form) method. After reviewing several basic knowledge, we first put forward four types of DHF multigranulation probabilistic models. Then, according to the MULTIMOORA method, a DHF MAGDM algorithm is designed via the proposed theoretical models in the context of person-job (P-J) fit. Finally, an illustrative case study for P-J fit is investigated, and corresponding validity tests and comparative analysis are conducted as well to demonstrate the rationality of the presented models.
... Such a variety of interpretations of threes makes three-way decision a useful and universally applicable theory and a practically feasible computational paradigm. In fact, there is a fast-growing interest in theory and applications of three-way decision, as evidenced from, for example, a number of recent papers published in this Granular Computing journal (Agbodah 2019;Cai et al. 2017;Li and Huang 2019;Ma et al. 2019;Mandal and Ranadive 2019;Zhang et al. 2019a, b). ...
The theory of three-way decision is about a philosophy of thinking in threes, a methodology of working with threes, and a mechanism of processing in threes. We approach a whole through three parts, in terms of three units, or from three perspectives. A trisecting–acting–outcome (TAO) model of three-way decision involves trisecting a whole into three parts and acting on the three parts, in order to produce an optimal outcome. In this paper, we further explore the TAO model in a set-theoretic setting and make three new contributions. The first contribution is an examination of three-way decision with nonstandard sets for representing concepts under the two kinds of objective/ontic and subjective/epistemic uncertainty. The second contribution is an introduction of an evaluation-based framework of three-way decision. We present a classification of trisections and investigate the notion of an evaluation space. The third contribution is, within the proposed framework, a systematical study of three-way decision with rough sets, interval sets, fuzzy sets, shadowed sets, rough fuzzy sets, interval fuzzy sets (or equivalently, vague sets, interval-valued fuzzy sets, intuitionistic fuzzy sets), and soft sets.
... Instead of utilizing crisp values to express preferences, the DM may prefer interval values [34,46] or fuzzy numbers [7,42,43,51]. In recent years, various definitions of consistency and prioritization methods for interval-valued RPRs [20,31,33,47,54] have been proposed. Wan et al. [44] introduced an index by combining the max-consistency index with the minconsistency index to measure the consistency level of interval-valued fuzzy preference relations (IVFPRs) and proposed a fuzzy logarithmic programming model to derive the interval-valued weights (IVWs). ...
... Goal-programming method Bryson [6] proposed the following goal-programming method, which also uses multiplicative RDI. However, the GP method limits the relative multiplicative deviations, such that they cannot both be greater than 1 for the tractable Model (33). To illustrate the distinction from the definitions of ξ i ij and ξ j ij , we redefine them as δ + ij , δ − ij ≥ 1, such that ...
The problem of deriving the priority vector from a pairwise comparison matrix is at the heart of multiple-criteria decision-making problems. Existing prioritization methods mostly model the inconsistency in relative preference–the ratio of two preference weights–by allowing for a small deviation, either additively or multiplicatively. In this study, we alternatively allow for a deviation in each of the two preference weights, which we refer to as the relative deviation interconnection. Under this framework, we consider both relative additive and multiplicative deviation cases and define two types of norms capturing the magnitudes of the deviations, which gives rise to four conic programming models for minimizing the norms of the deviations. Through the model structures, we analyze the signs of the deviations. This further allows us to establish the expressiveness of our framework, which covers the logarithmic least-squares method and the goal-programming method. Using numerical examples, we show that our models perform comparably against existing prioritization methods, efficiently identify unusual and false observations, and provide further suggestions for reducing inconsistency.
... Zhang (2019) defined the importance of each argument, the weighted DHFMSM, and the weighted DHFGMSM. Mandal and Ranadive (2019) defined the IVFPRs into fuzzy preference relations (FPRs), and then construct the consistency matrix, the collective consistency matrix, the weighted collective preference relations, and the group collective preference relation (GCPR). ...
In this paper, we define the new concepts of a triangular neutrosophic cubic linguistic hesitant fuzzy number (TNCLHFN), the basic operational relations of TNCHLFNs, and the score function of TNCLHFNs. Then, we develop a triangular neutrosophic cubic linguistic hesitant fuzzy weighted arithmetic averaging (TNCLHFWAA) operator and a triangular neutrosophic cubic linguistic hesitant fuzzy weighted geometric averaging (TNCLHFWGA) operator to aggregate TNCLHFN information and investigate their properties. Furthermore, a multiple attribute decision-making method based on the TNCLHFWAA and TNCLHFWGA operators, and the score function of TNCLHFN is established under a TNCLHFN environment. Finally, an illustrative example of investment alternatives is given to demonstrate the application and effectiveness of the developed approach. Triangular neutrosophic cubic linguistic hesitant fuzzy number with a wider scope of applications, and hence, the motivation for investigating into its applicability in different diseases in medical patients.
... Liang et al. (2018) proposed an optimal granulation selection method for multi-label data based on Multi-granulation Rough Sets. Mandal and Ranadive (2019) investigated interval-valued Fuzzy Probabilistic Rough Sets, based on multi-granulation approach. Skowron et al. (2016) emphasized the role of Rough Set-based methods in Interactive Granular Computing and discussed some of its important applications. ...
Feature selection is viewed as the problem of finding the minimal number of features from an original set with the minimum information loss. Due to its high importance in the fields of pattern recognition and data mining, it is necessary to investigate fast and effective search algorithms. In this paper, we introduce a novel fast feature selection algorithm for neighborhood rough set model based on Bucket and Trie structures. This proposed algorithm can guarantee to find the optimal minimal reduct by adopting a global search strategy. In addition, the dependence degree is subsequently used to evaluate the relevance of the attribute subset. Consequently, the proposed algorithm is tested on several standard data sets from UCI repository and compared with the most recent related approaches. The obtained theoretical and experimental results reveal that the present algorithm is very effective and convenient for the problem of feature selection, indicating that it could be useful for many pattern recognition and data mining systems.
... Conditional probability has been studied by many scholars. For example, Yao et al. [50] presented a DTRS model with naive Bayesian decision theory, which can be used to assess conditional probability in 3WD; Mandal et al. [29] discussed 3WDs with multi-granulation interval-valued fuzzy probabilistic rough sets; Greco et al. [5] investigated the cost of misclassification and put forward three-way probability models; Liu et al. [18] used logistic regression to estimate the conditional probability of DTRS and combined logistic regression to propose a new discriminant analysis method. ...
Three-way decision (3WD) provides a new perspective for solving practical decision-making problems, which is in line with human’s cognitive pattern. A covering information system (CIS) is an information system (IS) that consists of multiple coverings in the universe. A CIS with decision attributes which is seen as a covering decision information system (CDIS). This paper proposes three-way group decisions in a CDIS, as well as gives its application on the problem of position competition. First of all, the neighbourhood of every point in a CDIS is defined, and corresponding similarity class of this point is also obtained. Then, because of the uncertainty of risks, loss functions are acquired through group decision-making by means of interval numbers. Next, a method of three-way group decisions in a CDIS is presented. Eventually, the position competition is presented as an example to support our proposed decision-making method.
... Huang et al. (2018) also presented the inclusion measure-based optimistic and pessimistic MGDTRSs in multi-scale IF information tables, examined their properties and analyzed the three-way decision method based on the presented models. Mandal and Ranadive (2018) initiated the multi-granulation interval-valued fuzzy probabilistic rough sets and applied them three-way decisions based on intervalvalued fuzzy preference relations. In Mandal and Ranadive (2017), they also presented multi-granulation bipolar-valued fuzzy probabilistic rough sets and considered their corresponding three-way decisions over two universes. ...
This paper develops a single-granulation hesitant fuzzy rough set (SGHFRS) model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough sets model called the optimistic multi-granulation hesitant fuzzy rough sets (OMGHFRSs) and pessimistic multi-granulation hesitant fuzzy rough sets (PMGHFRSs). In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the SGHFRSs, OMGHFRSs and PMGHFRSs are also established. In order to further measure the uncertainty of multi-granulation hesitant fuzzy rough sets (MGHFRSs), the concepts of rough measure and rough measure about the parameters \(\alpha \) and \(\beta \) are presented and some of their interesting properties are examined. Finally, we give a decision-making method based on the MGHFRSs, and the validity of this approach is illustrated by two practical applications. Compared with the existing results, we also expound its advantages.
... There are a lot of scholars working on conditional probability. Yao et al. [50] proposed a DTRS model with naive Bayesian decision theory, which provides a practicable way for assessing conditional probability in 3WD; Furthermore, Yao et al. [46], [51] analyzed the related attributes of 3WD and proved the superiority of 3WD; Greco et al. [7] took into account the cost of misclassification, and presented three-way probability models under the advantage relation; Mandal et al. [31] discussed three-way decisions with multigranulation interval-valued fuzzy probabilistic rough sets. We can seen that scholars have searched for new methods to estimate conditional probability, and also studied three-way probability models under different environments. ...
Three-way decision (3WD) is a decision that conforms to human’s cognitive pattern and offer a new visual angle for solving practical decision-making problems. An information system (IS) represents relationships between objects and attributes. And a multi-source information system (MSIS) is an IS which composes of multi-source data sets. This article presents 3WD in a MSIS as well as gives its application in petroleum project investment. Considering that risks are uncertain, loss functions are first given by means of interval numbers. Then, multi-granulation decision-theoretic rough set (MGDTRS) models in a given MSIS are proposed from the point of view of multi-granulation. Next, based on the idea of decision-theoretic rough set (DTRS), the optimistic and pessimistic 3WDs in a MSIS are proposed. Finally, an example of petroleum project investment is used to support the feasibility of the proposed decision method.
... The discretization of data overlooks the differences of attribute values and causes information loss. To solve the problem, the classical rough set theory has been generalized to various aspects including fuzzy rough sets (Chen et al. 2013;Dubois and Prade 1990;Dai and Xu 2013;Dai et al. 2017;An et al. 2014An et al. , 2016Huang et al. 2014;Huang and Li 2018;Sun et al. 2015Sun et al. , 2017Mandal and Ranadive 2018;Yao et al. 2011;Yang et al. 2017a, b;Zhao et al. 2010Zhao et al. , 2015Zhang and Yang 2016;Zhang et al. 2012), covering rough sets (Zhu and Wang 2007), dominance rough sets (Greco et al. 2002;Yang et al. 2015;Zhang and Yang 2017), and so on (Min and zhu 2012;Xu and Yu 2017a;Xu and Li 2016a;Ziarko 1993). Neighborhood rough set model was introduced to deal with the data in which the values of attributes are numerical. ...
Attribute reduction is an important preprocessing step in machine learning and pattern recognition. This paper introduces condition-attribute Boolean matrix and decision-attribute Boolean matrix and proposes a new attribute reduction model based on Boolean operations according to the concept of neighborhood rough set. Some operation rules of Boolean vectors are defined and an uncertainty measure, named attribute support function, is proposed to evaluate the importance degree of candidate attributes with respect to decision attributes. An attribute reduction algorithm based on the proposed measure is designed. Ten data sets selected from public data sources are used to compare the proposed algorithm with some existing algorithms. The experimental results show that the proposed reduction algorithm is feasible and effective.
