On transformations from semi-three-way decision spaces to three-way decision spaces based on triangular norms and triangular conorms
Section snippets
A brief introduction to the development of three-way decisions
In 2009, from Pawlak’s rough sets [26], [27] and decision-theoretic rough sets [34], [35], [36], [42] on the basis of Bayesian decision theory, Yao introduced the concept of three-way decisions in [37]. Three-way decisions, as an extension of the classical two-way decisions, possess three sorts of decision rules, that is, acceptance rules, rejection rules and uncertainty rules. And, for any object in a universe, it can be assigned into one of the three regions, that is, positive region
Preliminaries
In this section, for convenience of readers, we list some basic definitions and concepts.
Definition 2.1 [6]. Let (X, ≤ ) be a partially ordered set with the minimum element 0 and maximum element 1. A function is called a negation if it is non-increasing and such that and . Further, it is said to be strong or involutive, if
In this paper, (P, ≤ P) denotes a partially ordered set with an involution negation NP, the minimum element 0P and maximum element 1P, which is written as (P,
The two transformation methods from semi-decision evaluation functions to decision evaluation functions based on triangular norms and triangular conorms
In the following, let and be two partially ordered sets. Let U be a nonempty universe to make a decision on it, called decision universe and V be a nonempty universe where condition function is defined, named condition universe.
Definition 3.1 [5]. Let U be a decision universe and V be a condition universe. Then a mapping is called a semi-decision evaluation function of U, if it satisfies the following axioms:
Minimum element axiom
Three-way decisions transformed from semi-three-way decisions
In this section, we propose some new types of three-way decisions transformed from semi-three-way decisions based on t-norms and t-conorms, such as three-way decisions based on fuzzy sets, interval-valued fuzzy sets, fuzzy relations and hesitant fuzzy sets.
A practical application
In this section, we illustrate some results obtained in this paper by using a reality example [5], [22]. More precisely, we take into account an evaluation problem of credit card applicants. And, for the convenience of readers, firstly, 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 follow up Hu’s paper [5] to propose the construction methods from semi-decision evaluation functions to decision evaluation functions. The work studied in this paper gives us more choice for decision evaluation function in a practical problem and make the applications of three-way decisions theory more widely. We list the main contributions of this paper as follows.
- (1)
We obtain one construction method from
Acknowledgments
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable 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).
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