Comprehensive fuzzy concept-oriented three-way decision and its application
Introduction
In 1982, Pawlak [1] proposed rough set theory, but one of its flaws is that it did not consider the fault tolerance of uncertainty and involved a narrow range of applications. To solve this problem, Yao and Wong [2] proposed decision-theoretic rough set model using Bayesian decision process. Subsequently, Yao [3], [4] proposed three-way decision concepts based on the decision-theoretic rough set theory. At present, there are many researches on decision-making theory. As two important parts of decision-making theory, three-way decision theory and the multi-criteria decision-making (MCDM) theory are playing an important role in the field of decision-making. Based on the three-way decision theory and MCDM theory, this paper attempts to combine the two theories from the perspective of fuzzy concept, and proposes a new three-way MCDM method. We hope that this method can complete the scheme classification, the scheme ranking and optimal scheme selection at the end. Before introducing our method in detail, we first describe the three-way decision theory and MCDM theory.
Three-way decision, simply speaking, is that people make three types of judgments in decision based on existing knowledge: acceptance decision, rejection decision and delay decision. After the three-way decision was put forward, many researchers expanded and developed this theory [5], [6], [7]. For examples, Yao [8] further analyzed many attributes of three-way decision, and elaborated the advantages of three-way decision. Liu et al. [9] introduced linguistic intuitionistic fuzzy numbers into the loss function and proposed a threshold determination method based on a single optimization model to deduce three-way decision. Campagner et al. [10] introduced the concept of reduction in the machine learning field, and proposed three reductions based on three-way decision, possibility theory and probability theory in order to study the impact of Ground Truth on the performance of machine learning model. Campagner et al. [11] proposed a set of learning algorithms called three-way-in and three-way-out approach based on three-way decision and evidence theory, in which three-way decision is used as a tool to enable uncertain and approximate reasoning and inference. Wang et al. [12] constructed a regret-based three-way decision model under interval type-2 fuzzy environment, with the purpose of improving the three-way decision to deal with risks and uncertainties. All in all, many researchers have made contributions to the development of three-way decision [13], [14].
When talking about the core content of three-way decision, these two aspects are always inseparable, namely conditional probability [15], [16] and loss function [17], [18]. In decision-theoretic rough set theory, conditional probability is also called evaluation function, which can connect evaluation with various measurement functions. The conditional probability represents the probability of an scheme belonging to a certain state under a certain description. During the period of prosperity and development of three-way decision, many researchers conducted researches on conditional probability. For examples, Yao and Zhou [19] proposed Bayesian decision rough set model, which can estimate conditional probability independently on the basis of Bayesian theory and naive probability. Liu et al. [20] proposed to use the grey correlation degree to calculate the conditional probability, which is used to solve the problem that there is no decision attribute in the MCDM model. For the deeper discussion of condition probability, we can study the description of the scheme and the state of the scheme in its formula. On the one hand, the description of the scheme refers to the granule of scheme, we can express it by equivalent granule [21], [22], dominance granule [23] or fuzzy granule [24]. For examples, Ye et al. [25] defined fuzzy -neighborhood, and constructed a new three-way decision model of fuzzy -granule. Zhan et al. [26] constructed the outranking granule by ELECTRE method, and proposed a three-way decision model based on outranking relation. On the other hand, the state of an scheme is either a classical set or a fuzzy set. Meanwhile, the state of the scheme is proposed subjectively in many researches, and some researchers put forward it objectively. For instance, Zhang et al. [27] put forward two objective fuzzy sets with opposite characteristics according to TOPSIS method, and obtained two three-way decision models. In order to further deepen the study of conditional probability, based on two fuzzy concepts with opposite semantics in [27], we introduce weight coefficient and propose the new objective fuzzy concepts, namely comprehensive fuzzy concepts. Moreover, we propose a new conditional probability formula combined with the four fuzzy granules obtained by the fuzzy -neighborhood operator.
The other core element of three-way decision is loss function. Determining the loss value of loss function is one of the important ways to study the nature of three-way decision. Due to the loss function involves the setting of threshold parameters in the model, many researchers also studied the loss function. Liang et al. [28] introduced the form of uncertainty evaluation into the loss function, proposed a new decision-theoretic rough set model and extended the range of loss value. Zhang et al. [29] combined three-way decision with uncertain classification, obtained a pair of thresholds of probabilistic rough set model, and proposed a three-way decision model based on two kinds of uncertain classification. Liang et al. [30] considered introducing the Pythagoras fuzzy number into the loss function, giving the loss function a new interpretation, and constructed a new fuzzy decision-theoretic rough set model based on the Bayesian decision process. Recently, in dealing with MCDM problem, in order to obtain the final model objectively, Jia and Liu [31] put forward the concept of relative loss function and establish the conversion method from the evaluation value in MCDM to the relative loss function.
MCDM, as an important part of analytical decision theory, is widely used in investment, medicine, construction and other management projects [32], [33]. There are many methods to solve MCDM tasks, including TOPSIS method [34], [35], ELECTRE method [36], EDAS method [37] and PROMETHEE method [38]. Recently, researchers have developed some rough decision methods by combining classical methods with rough set models. For examples, Zhang et al. [39] proposed a new TOPSIS method and applied it to the selection question of biological nano-materials. Jiang et al. [40] proposed a new method to solve multi-attribute decision-making by combining variable precision fuzzy rough set model based on covering and two classical decision methods. Zhang et al. [41] used fuzzy logic operators to propose two pairs of -fuzzy rough set models, and combined them with the PROMETHEE method to propose a new decision method to solve the MCDM problem on fuzzy information. Zhang et al. [42] constructed a new reflexive fuzzy -neighborhood operator, and based on this new operator, they proposed a new fuzzy rough set model to apply to the MCDM problem. There are also some researchers who have proposed new three-way decision models to solve the MCDM problem in different environments. For examples, Wang et al. [43] combined an objective membership function of the hesitation fuzzy sets, and proposed the MCDM model based on the three-way method with hesitation fuzzy information in the hesitation fuzzy environment. Wang et al. [44] defined the relative loss function in the hesitant fuzzy environment according to the properties of the relative loss function, and provided an aggregate loss function through the hesitant fuzzy weight average operator to solve the multi-attribute decision-making problem in the hesitant fuzzy environment. From these researches, we discover that the MCDM methods based on three-way decision are being widely studied.
According to the above descriptions, we find that the existing three-way decision methods are mainly for qualitative clustering, while the purpose of the existing MCDM methods is to rank all schemes. Thus, based on the idea of three-way decision and MCDM, some researchers have combined the two and put forward the concept of three-way MCDM. Next, we briefly introduce the differences between our method and three-way decision methods, MCDM methods and existing three-way MCDM methods.
- (1)
Traditional MCDM methods have their own characteristics, and the main purpose is to rank all schemes. Among them, TOPSIS method [35] ranks all schemes by constructing intimacy coefficient. Differently, in this paper, drawing on the ideas of the TOPSIS method, our method takes intimacy coefficient as fuzzy concept, and then constructs three-way MCDM models through fuzzy concept, which can rank and classify all schemes.
- (2)
In the existing researches on three-way decision, we find that most researchers are mainly concerned with promoting three-way decision in various environments (the fuzzy environment [12], the intuitionistic fuzzy environment [34], the hesitant fuzzy environment [44], etc.) and obtaining the classification results of all schemes. Differently, our method is based on the fuzzy environment, combined with the advantages and ideas of the MCDM method, to rank and classify all schemes.
- (3)
There are two components of conditional probability, namely the granule of scheme and the fuzzy concept. At present, some existing three-way MCDM methods [25], [26] mainly focus on the structure of the granule of scheme. Differently, the research foundation of the three-way MCDM method proposed in this paper mainly considers fuzzy concepts.
Based on the above description, we observe that from the perspective of fuzzy concepts, it is an interesting topic to study the combination of three-way decision and MCDM. In light of this, we introduce the research motivations of this paper as follows:
- (1)
By studying a large number of references, we discover that some studies are subjective in proposing the state of the scheme (or called concept expression) in the conditional probability formula, which may lead to the imprecision and inaccuracy of decision-making results. Therefore, in order to objectively give the definition of the concept, we combine the concepts of the optimal distance and the worst distance which mentioned in the classical TOPSIS method, and propose the new concepts called comprehensive fuzzy concepts. Using the newly proposed comprehensive fuzzy concepts, we can not only objectively calculate the loss function and threshold, but also further optimize the three-way decision model.
- (2)
In the fuzzy decision-making environments, the fuzzy -neighborhood operator has been regarded as an effective tool for expressing the fuzzy binary relationship between schemes, and it is very helpful for processing fuzzy data. However, there are few researches on the combination of fuzzy -neighborhood operators and three-way decision model. In light of this, we try to introduce four fuzzy granules related to fuzzy -neighborhood operator for research of three-way decision. Meanwhile, in order to reduce the subjectivity in the decision process, we combine fuzzy granule with the comprehensive fuzzy concepts to establish an objective formula for determining the conditional probability, thereby developing a new three-way decision method.
Further, the innovations in this paper are explained as follows:
- (1)
This paper combines the cost fuzzy concept with the benefit fuzzy concept to propose the new comprehensive fuzzy concepts for the first time.
- (2)
This paper combines the fuzzy neighborhood granules obtained by four fuzzy neighborhood operators with the comprehensive fuzzy concepts for the first time to propose a new calculation method of conditional probability.
- (3)
This paper combines the comprehensive fuzzy concepts, the three-way decision theory and the MCDM theory for the first time to propose a three-way MCDM method based on the comprehensive fuzzy concepts.
In this part, we introduce the structure of the paper. In Section 2, we review the knowledge of the fuzzy neighborhood operators, MCDM and decision-theoretic rough fuzzy set models. In Section 3, we propose the three-way decisions based on comprehensive fuzzy concepts. In Section 4, we introduce a new three-way MCDM method based on comprehensive fuzzy concepts, and give some examples to illustrate the effectiveness of the method. In Section 5, through comparative analysis, Spearman rank correlation analysis and Kendall rank correlation coefficient analysis, we demonstrate the rationality and stability of the method proposed in this paper. At the same time, we also discuss the similarities and differences between our method and existing methods and the advantages of our method. In Section 6, we conduct experiments on the parameters involved in our proposed three-way model to test the stability of our method. Finally, Section 7 summarizes this paper and plans future research directions.
Section snippets
Preliminaries
In this section, we will introduce some basic knowledge points used in this paper, such as fuzzy -coverings, MCDM and decision-theoretic rough fuzzy set models.
Three-way decisions based on comprehensive fuzzy concepts
In this section, for the benefit fuzzy concept and the cost fuzzy concept, we combine the two fuzzy concepts to obtain the definition of the comprehensive benefit fuzzy concept and the comprehensive cost fuzzy concept. Then we analyze the decision-theoretic fuzzy rough set based on these two comprehensive fuzzy concepts. Finally, combining four types of fuzzy neighborhood operators with these two comprehensive fuzzy concepts, we obtain the three-way decision model.
A new three-way MCDM method based on comprehensive fuzzy concepts
In this section, according to the three-way decisions based on comprehensive fuzzy concepts proposed in Section 3.3, we propose a new method to solve energy efficiency issues, and draw the flowchart of the method. Furthermore, we obtain the final ranking by this method, which shows that the method is feasible. In addition, we test the effectiveness of the proposed method, and the final results show that the method is feasible and effective.
The comparison analysis, Spearman rank correlation coefficient analysis and Kendall rank correlation coefficient analysis
In this section, we first use the data in Table 16 to compare our method with the other five methods, and analyze the comparison results of distinct ranking methods. Then we use Spearman rank correlation coefficient (SRCC) and Kendall rank correlation coefficient to analyze the relationship between the ranking results obtained by different methods. Finally, we discuss the proposed method and other existing methods from the three perspectives of similarities, differences and advantages, and we
Experimental analysis
In this section, we mainly conduct experiments on the parameters involved in proposed three-way model, with the purpose of testing the stability of our method. The experiments are operated under the software MATLAB R2020a on a personal computer with an Intel Core i5-4200H, 2.80 GHz CPU, 12.00 GB of memory, and 64-bit Windows 7.
In the three-way MCDM method proposed in this paper, the change of parameters and are worthwhile discussed. Therefore, in this section, we will discuss the
Conclusion
In this paper, we propose a new three-way decision model based on two comprehensive fuzzy concepts, and the main contributions of this paper are as follows:
- (1)
First of all, according to the cost fuzzy concept and benefit fuzzy concept, we put forward the definitions of comprehensive benefit fuzzy concept and comprehensive cost fuzzy concept. Furthermore, we discuss the relationship between two comprehensive fuzzy concepts and relative loss function, then we analyze the relationship between the
CRediT authorship contribution statement
Xiangbin Liu: Supervision, Validation, Writing - review & editing. Wang Mao: Conceptualization, Methodology, Investigation, Writing - original draft, Writing - review & editing. Jianhua Dai: Conceptualization, Supervision, Validation, Methodology, Writing - review & editing. Kai Zhang: Conceptualization, Methodology, Investigation, Validation, Writing - review & editing.
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
This work was supported by the National Natural Science Foundation of China (61976089, 61473259), the Natural Science Foundation of Hunan Province (2021JJ30451), the Hunan Provincial Science & Technology Project Foundation (2018TP1018, 2018RS3065) and Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210431).
References (49)
- et al.
A decision theoretic framework for approximating concepts
Int. J. Man-Machine Studies
(1992) Three-way decisions with probabilistic rough sets
Inform. Sci.
(2010)- et al.
Three-way decisions based multi-attribute decision making with probabilistic dominance relations
Inform. Sci.
(2021) - et al.
A three-way decision approach with risk strategies in hesitant fuzzy decision information systems
Inform. Sci.
(2022) - et al.
Three-way decision and conformal prediction: Isomorphisms, differences and theoretical properties of cautious learning approaches
Inform. Sci.
(2021) The superiority of three-way decisions in probabilistic rough set models
Inform. Sci.
(2011)- et al.
On three perspectives for deriving three-way decision with linguistic intuitionistic fuzzy information
Inform. Sci.
(2022) - et al.
The three-way-in and three-way-out framework to treat and exploit ambiguity in data
Int. J. Approx. Reason.
(2020) - et al.
TWD-R: A three-way decision approach based on regret theory in multi-scale decision information systems
Inform. Sci.
(2021) - et al.
A novel approach of three-way decisions with information interaction strategy for intelligent decision making under uncertainty
Inform. Sci.
(2021)
A decision-theoretic fuzzy rough set in hesitant fuzzy information systems and its application in multi-attribute decision-making
Inform. Sci.
Decision-theoretic rough fuzzy set model and application
Inform. Sci.
Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes
Inform. Sci.
A multiple attribute decision making three-way model for intuitionistic fuzzy numbers
Int. J. Approx. Reason.
A three-way decision method based on Gaussian kernel in a hybrid information system with images: An application in medical diagnosis
Appl. Soft Comput.
A decision-theoretic rough set model with q-rung orthopair fuzzy information and its application in stock investment evaluation
Appl. Soft Comput.
A novel decision-making approach based on three-way decisions in fuzzy information systems
Inform. Sci.
A new classification and ranking decision method based on three-way decision theory and TOPSIS models
Inform. Sci.
Triangular fuzzy decision-theoretic rough sets
Int. J. Approx. Reason.
Three-way decision model with two types of classification errors
Inform. Sci.
Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information
Inform. Sci.
A novel three-way decision model under multiple-criteria environment
Inform. Sci.
TOPSIS-WAA method based on a covering-based fuzzy rough set: An application to rating problem
Inform. Sci.
A ranking model of Z-mixture-numbers based on the ideal degree and its application in multi-attribute decision making
Inform. Sci.
Cited by (7)
BTWM-HF: A behavioral three-way multi-attribute decision-making method with hesitant fuzzy information
2024, Expert Systems with ApplicationsFusion decision strategies for multiple criterion preferences based on three-way decision
2024, Information FusionThree-phase multi-criteria ranking considering three-way decision framework and criterion fuzzy concept
2024, International Journal of Approximate ReasoningThree-way decision for three-stage ranking pattern with criterion fuzzy concept
2023, Information SciencesAn attribute fuzzy concept-oriented three-way utility decision model in multi-attribute environments
2023, Applied Soft Computing