Elsevier

Information Sciences

Volume 507, January 2020, Pages 733-754
Information Sciences

On decision evaluation functions in generalized three-way decision spaces

https://doi.org/10.1016/j.ins.2018.07.032Get rights and content

Abstract

Recently, Hu proposed the notion of three-way decision spaces based on partially ordered sets with involutive negations by the axiomatic definition. However, involutive negation is so strong that imposes restrictions on the use of the theory of three-way decision spaces from theoretical point of view. Therefore, this paper attempts to extend the concept of three-way decision spaces from partially ordered sets with involutive negations to partially ordered sets with negations. At first, we generalize three-way decision spaces and three-way decisions based on general partially ordered sets with negations such that the existing three-way decisions are the specific cases of generalized three-way decision spaces discussed in this paper. And then, we give some new decision evaluation functions in generalized three-way decision spaces (e.g., decision evaluation functions based on fuzzy sets, interval-valued fuzzy sets, fuzzy relations, shadowed sets and hesitant fuzzy sets), which enriched the class of decision evaluation functions. In particular, they provide more choices for decision-makers in realistic decision-making problems. Finally, in order to illustrate the practical applications of the results presented in this paper, we analyse a practical example of an evaluation problem of credit card applicants.

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 N:PP is called a negation if it is decreasing and such that N(1)=0 and N(0)=1. Further, it is said to be strong or involutive, if NN=idP.

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 U={x1,x2,,x9} be a set of nine applicants. In addition, we consider two condition attributes AT={education,salary} 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).

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