On decision evaluation functions in generalized three-way decision spaces
Section snippets
A brief retrospection on the development of three-way decisions
In 2009, from Pawlak’s rough sets [30], decision-theoretic rough sets [46] on the basis of Bayesian decision theory and some problems in model selection [6], environmentally precautionary decision-making [7], statistical process control [36], suitable rough set model selection for data analysis [8], information filtering model on the Web [22] and Web-Based Support Systems [40], Yao put the concept of three-way decisions in [41] such that the existing models become the particular examples of
Generalized three-way decision spaces based on partially ordered sets with negations
In this section, the axiomatic definition of generalized decision evaluation functions and generalized three-way decision spaces are introduced based on the partially ordered sets with negations. We begin with the definition of negation. Definition 2.1 ([11]) Let (P, ≤ ) be a partially ordered set with the minimum element 0 and maximum element 1. A function is called a negation if it is decreasing and such that and . Further, it is said to be strong or involutive, if
In this paper, (
Generalized three-way decisions based on partially ordered sets with negations
In this section, firstly, we give the concept of generalized three-way decisions of generalized three-way decision spaces based on partially ordered sets with negations, which will be needed in Section 4. And then, as a theoretical completeness, we also give the concepts of optimistic generalized three-way decisions, pessimistic generalized three-way decisions and aggregation of multiple generalized three-way decision spaces based on partially ordered sets with negations. We must point out that
Some new types of decision evaluation functions in generalized three-way decision spaces
In this section, we propose some new types of decision evaluation functions, called generalized decision evaluation functions here, for generalized three-way decision spaces based on partially ordered sets with negations and show some generalized three-way decisions corresponding to those generalized decision evaluation functions, such as generalized decision evaluation functions and generalized three-way decisions based on fuzzy sets, interval-valued fuzzy sets, fuzzy relations, shadowed sets
A practical application
In this section, we illustrate some results obtained in this paper by using a reality example [12], [27], [33]. To be specific, we take into account an evaluation problem of credit card applicants.
For the convenience of readers, at first, we depict that example as follows.
Let be a set of nine applicants. In addition, we consider two condition attributes in this question, where the values of attribute “education” are {best, better, good} and the values of
Conclusions
In this paper, in order to get more decision evaluation functions of decision-making, we generalize decision evaluation functions to generalized decision evaluation functions. In addition, generalized three-way decision spaces based on partially ordered sets with negations are established, which provide more choices for decision-makers to make decisions in actual decision-making problems. Moreover, we propose generalized three-way decisions on generalized three-way decision spaces and
Acknowledgments
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their vital comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
References (49)
- et al.
Exploiting collective knowledge with three-way decision theory: cases from the questionnaire-based research
Int. J. Approx. Reason.
(2017) - et al.
Multi-granular mining for boundary regions in three-way decision theory
Knowl. Based Syst.
(2016) - et al.
Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis
Appl. Math. Model.
(2013) - et al.
Decision-theoretic three-way approximations of fuzzy sets
Inf. Sci.
(2014) Key concepts in model selection: performance and generalizability
J. Math. Psychol.
(2000)Do statistical inferences allowing three alternative decision give better feedback for environmentally precautionary decision-making
J. Environ. Manage.
(2007)- et al.
Criteria for choosing a rough set model
J. Comput. Math. Appl.
(2009) Three-way decisions space and three-way decisions
Inf. Sci.
(2014)Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets
Knowl. Based Syst.
(2016)Three-way decisions based on semi-three-way decision spaces
Inf. Sci.
(2017)