A decision-theoretic fuzzy rough set in hesitant fuzzy information systems and its application in multi-attribute decision-making
Introduction
Nowadays, due to the increasing complexity and uncertainty of decision-making problems, research on the theories and methodologies of decision-making under uncertainty has caught extensive attention from diverse domains, and various methods for solving decision-making issues in uncertain environment have been proposed [28], [36]. This paper details our proposed DTFRS model in hesitant fuzzy information systems and its application to MADM problems. The proposal comes from our work combining three-way decision theory [43], [44], [45] with hesitant fuzzy sets (HFSs) [37], [38]. In what follows, we briefly review some related studies of DTRSs and HFSs.
In light of Bayesian decision procedure, Yao [43] proposed DTRSs, which bridge the rough set theory and the risk decision theory. DTRSs can generate three kinds of decision rules through the minimum expected risk, namely, acceptance decisions, uncertainty decisions and rejection decisions [35]. With the introduction of DTRSs, they have attracted the attention of many researchers and have been used in many domains, such as feature selection [11], knowledge granulation [10], [31], [39], conflict analysis [17], [13], pattern recognition [32], formal concept analysis [16], and so on. The studies of DTRSs mainly include two aspects: one is the accurate and reasonable expression of the loss functions, and the other is the effective computation and estimation of the conditional probability [22].
Expression of the loss functions using crisp values is difficult for decision-makers because of the complexity and uncertainty found in realistic decision-making situations. To handle this problem, Zhao and Hu [50] considered fuzzy sets and interval-valued fuzzy sets as the evaluation forms of the loss functions, and introduced fuzzy and interval-valued fuzzy DTRS models. Liang et al. [21] proposed a new DTRS model by measuring the loss functions with intuitionistic fuzzy sets. Liang et al. [22] also took into account Pythagorean fuzzy sets as a new evaluation format of the loss functions and explored a Pythagorean fuzzy DTRS model. Sun et al. [34] utilized linguistic terms to represent the loss functions and established a decision-theoretic rough fuzzy set model. Abdel-Basset et al. [1] used neutrosophic sets to denote the loss functions and constructed a corresponding DTRS model. Tang et al. [36] used q-rung orthopair fuzzy sets to depict the loss functions and developed a q-rung orthopair fuzzy DTRS model. Recently, Liang and Liu [19] discussed a new DTRS model by using HFSs as an expression form of the loss functions. However, these studies do not consider the differences of objects and do not give a computational method of the loss functions. Fortunately, under a multi-attribute environment, Jia and Liu [12] proposed a novel three-way decision model by utilizing attribute values to derive the relative loss functions. Liu et al. [26] also used the attribute values to calculate the relative loss functions under an intuitionistic fuzzy multi-attribute environment. Similarly, Liang et al. [20] and Lei et al. [14] studied the relative loss and benefit functions induced from attribute values with interval type-2 fuzzy sets and hesitant fuzzy linguistic term sets, respectively.
The conditional probability in most DTRS models is calculated using the information granule in decision information systems, and one of the preconditions for determining the conditional probability is the decision attribute [22], [26]. For example, Liu and Liang [24] used a dominating (or dominated) class of objects to compute the conditional probability in an ordered information system. Liu et al. [25] utilized an L-level similarity class of objects to arrive at the conditional probabilities in an incomplete information system. However, we may confront a universal circumstance in the actual decision environment that the information system does not have the class label or the decision attribute. For instance, in many MADM problems [5], there are no decision attributes and only conditional attributes in information tables. To address this issue, Liang et al. [22] utilized the TOPSIS method to calculate the conditional probability under a Pythagorean fuzzy MADM environment. Recently, Liu et al. [26] effectively used grey relational analysis to estimate the conditional probability in an intuitionistic fuzzy MADM problem.
The notion of HFSs was initially introduced by Torra [37], which can effectively describe the uncertainty of complicated problems and the vagueness of people’s cognition. Ever since the introduction, HFSs have developed quickly both in theory and application. In theory, Xia and Xu [38] proposed some hesitant fuzzy aggregation operators. Xu and Xia [40] investigated diverse distance and similarity measures of HFSs. Liao et al. [23] discussed the multiplicative consistency of hesitant fuzzy preference relations. Recently, Hu [9] pointed out that there is a lack of mathematical rigor in logic operation definition of HFSs and revised these operations. In application, Ebrahimpour and Eftekhari [3] introduced an innovative method based on HFSs to deal with feature selection on the high dimensional data sets. Sun et al. [33] developed a new approach to pattern recognition issues with HFSs using grey relational analysis. Xu and Zhang [41] explored a new MADM method with the aid of TOPSIS in the hesitant fuzzy context and discussed its application to energy policy selections. Xia and Xu [40] utilized hesitant fuzzy aggregation operators to fuse uncertain information in MADM problems. Feng et al. [4] proposed an MADM method using possibility theory under hesitant fuzzy linguistic environment and studied its application in investments.
In recent years, research on hybrid models by combining HFSs [37] with rough set theory [29], [30] has become an increasingly important branch of HFS theory. In 2014, Yang et al. [42] introduced the concept of hesitant fuzzy rough sets based on the constructive approach and the axiomatic approach. In light of the fact that the performance of two different but highly related universes is better than that of a single universe for describing the actual decision-making problems, Zhang et al. [49] proposed a hesitant fuzzy rough set over two universes and studied its applications in decision-making. In 2015, Liang and Liu [19] considered the loss functions of DTRS models with hesitant fuzzy values (HFVs) to construct a hesitant fuzzy DTRS model and studied a risk decision-making method. In the same way, Li and Huang [15] utilized HFVs to characterize the cost and revenue functions in investment decision problems and further developed a hesitant fuzzy three-way investment decision model. In 2020, Zhang et al. [47] studied a multi-granularity DTRS over two universes in the hesitant fuzzy linguistic context and applied it to person-job matching problems. Similarly, Zhang et al. [48] proposed an interval-valued hesitant fuzzy multi-granularity DTRS over two universes by integrating interval-valued hesitant fuzzy sets, multi-granularity rough sets over two universes and DTRSs. Recently, considering decision-makers’ bounded rationality and criteria interaction, Lei et al. [14] established a behavioral hesitant fuzzy linguistic multi-granularity DTRS over two universes using prospect theory and Choquet integral, then they established a three-way group decision method to solve green supplier selection problems.
In this paper, we develop a DTFRS model in the hesitant fuzzy information system by combining DTRSs with HFSs and discuss how it can be applied to MADM problems under hesitant fuzzy environment. The motivations of this paper are summarized as follows:
- (1)
Uncertainty and complexity can cause an increasingly risk impact on our decision-making processes. Meanwhile, experts may hesitate among several evaluation values for lack of information and the uncertainty that persists in people’s cognition. In this situation, we encounter the following two challenges: (a) How to represent the evaluation values of objects reasonably and accurately by considering experts’ knowledge and understandings. (b) How to establish a scientific and effective decision model to reduce decision risks and explain the decision results objectively. As a solution to these challenges, we utilize HFVs [40] to represent evaluation values in information systems. Then, we propose a DTFRS model in the hesitant fuzzy information system, which can consider decision risks and direct us regarding the selection of each object’s decision action.
- (2)
The loss functions in traditional DTRS models arise straightly and are constant values for all objects. Based on the results in [12], [14], [20], [26], we develop a data-driven approach to compute the relative loss functions of each object using attribute evaluation values. In addition, in most DTRS models, the conditional probability is calculated by equivalence relations in information tables, which seems to be a stringent condition that may limit the applicability of DTRS models. Therefore, to overcome this limitation, we define a new fuzzy similarity relation by utilizing the hesitant fuzzy distance function, which can depict the relationships among objects flexibly.
- (3)
The traditional MADM methods are all bulit based on two-way decision theory, which illustrates that the decision result is an either-or matter. Unfortunately, this result can be oversimplified because it can often ignore the necessity of further testing to obtain a final decision result beyond the either-or dichotomy in practical decision-making processes. Therefore, we need to design a new model for MADM. The intuitive approach is to add uncertainty to the final decision results for further investigation. This idea coincides with the thought of three-way decisions [43]. Therefore, we utilize the proposed DTFRS model to establish a three-way decision method to solve MADM problems under hesitant fuzzy environment, dividing all alternatives into three decision regions and obtaining corresponding decision actions objectively.
The remainder of this paper is organized as follows. Section 2 briefly reviews the theory of DTRSs as well as the basic concepts of HFSs. In Section 3, we develop a DTFRS model in hesitant fuzzy information systems based on DTRSs and HFSs. In Section 4, we use the proposed DTFRS model to establish a three-way decision method to MADM under hesitant fuzzy environment. Section 5 presents a stock investment example to show the application of our proposed three-way decision method and verifies its effectiveness by a sensitivity analysis and a comparison analysis. In Section 6, we conclude our work and sketch a plan for future research.
Section snippets
Preliminaries
In this section, for the convenience of readers, we give some basic definitions and concepts that are used throughout the paper.
A decision-theoretic fuzzy rough set in hesitant fuzzy information systems
Considering the various advantages of HFSs, we propose a DTFRS model in hesitant fuzzy information systems. First, we present the definition of hesitant fuzzy information systems. Then, according to the connection between the loss functions and the attribute values, we discuss a calculation method to obtain the relative loss functions by HFVs. Finally, in light of the Bayesian decision process, we establish a DTFRS model and discuss its decision rules.
A three-way decision method to MADM based on the DTFRS model
As an important branch in the field of decision-making research, MADM helps decision-makers select or rank appropriate alternatives according to multiple attributes. With the increasing complexity and uncertainty of decision-making problems, it is necessary to develop an effective and feasible method for avoiding decision risks and acquiring high utilities. This section aims to propose a three-way decision method to MADM under hesitant fuzzy environment by using the constructed DTFRS model. We
An illustrative example
In this section, we take the proposed three-way decision method to solve a stock investment problem under hesitant fuzzy environment to illustrate its effectiveness. Moreover, a sensitivity analysis and a comparison analysis are conducted to illustrate the characteristics and advantages of our method.
Conclusions
In this article, we propose a DTFRS model in the hesitant fuzzy information system and establish a novel three-way decision method to deal with MADM problems. The main contributions of this article are summarized as follows. (1) A new fuzzy similarity relation between two objects is defined by utilizing the hesitant fuzzy distance function. (2) Based on the attribute values expressed by HFVs, we develop a method to calculate the relative loss functions and establish a DTFRS model in the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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. 11971365 and 11571010).
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