Decision self-information based on parameterized fuzzy β neighborhood and its application in three-way multi-attribute group decision-making

: As a special kind of entropy, decision self-information effectively considers the uncertainty information of both the lower and upper approximations. However, it is limited to rough binary relations, which limits its application to complex problems. In addition, parameterized fuzzy β covering, as an extension of the covering-based rough set model, can effectively characterize the similarity between samples. We combine decision self-information with a parameterized fuzzy β neighborhood to propose decision self-information in fuzzy environments, and we study its important properties. On this basis, a three-way multi-attribute group decision-making algorithm is established, and a practical problem is solved. The effectiveness of the proposed method is verified by experimental analysis.


Introduction
Zakowski [1] first proposed the covering-based rough set model [2], which is a natural extension of the classical rough set model and an effective tool to deal with uncertain information. However, like traditional rough sets, covering-based rough sets deal with discrete attributes that belong or do not belong in a dataset, which limits their application in complex environments. To this end, Dubois and Prade [3] introduced the concept of fuzzy rough sets and extended rough set theory to the fuzzy environment, and scholars have proposed various improved fuzzy rough set models. Ma [4] defined two pairs of fuzzy approximation operators in the covering-based fuzzy approximation space, which show the properties and topological importance of the complementary neighborhood. D'eer et al. [5] discussed the relationship between various fuzzy covering-based fuzzy rough set models. Ma [6] proposed the concept of fuzzy β covering and fuzzy β neighborhoods. Zhan et al. [7] improved the fuzzy β neighborhood, proposed a covering-based variable-precision fuzzy rough set model, and applied it to multi-attribute decision-making. Zhang et al. [8] explained the fuzzy binary relation in fuzzy β approximation space from the perspective of pessimism and optimism, which makes up for the defect that the fuzzy β neighborhood operator cannot obtain the fuzzy binary relation between objects. However, the fuzzy β covering-based model proposed by Ma cannot guarantee that the lower approximation is included in the upper approximation. Subsequently, Zhang et al. [9] and Huang et al. [10] proposed a parameterized fuzzy β covering-based model that guarantees that the lower approximation is included in the upper approximation while reducing the influence of noisy data. Dai et al. [11,12] constructed four kinds of fuzzy β neighborhood operators with reflexivity by using fuzzy logic operators and used fuzzy β covering relations to describe the similarity between samples.
Information entropy [13] is another important and effective method to characterize information uncertainty, which is widely used in the fields of artificial intelligence, multi-attribute decision-making, attribute reduction, and information security. In recent years, information entropy has been combined with rough set theory in various types of entropy models [14,15]. Liao et al. [16] considering the scale diversity between different attributes, proposed a new uncertainty measure, which provides effective support for some decision-making constrained by test cost. Li et al [17]. proposed an uncertainty measurement method for fuzzy relational information systems, and gave an axiomatic definition of granularity measurement. Wang et al. [18][19][20] constructed various types of entropy according to different binary relations, among them a special form of entropy, decision self-information [21], which takes into account uncertainty information in both the lower and upper approximations. However, decision self-information is limited to rough binary relations, which limits its application to complex problems. We combine decision self-information with parameterized fuzzy β covering to enable its application in fuzzy environments.
In an increasingly complex social environment, multi-attribute decision-making problems are part of daily life. Traditional decision-making methods [22,23] are insufficient to solve complex uncertainty problems in real life, and many methods have been proposed [24][25][26][27]. Zhang et al. [28] constructed a reflexive fuzzy α neighborhood operator, proposed a fuzzy α rough set model based on the fuzzy neighborhood operator, and applied it to multi-attribute decision-making. Wang and Miao [29] proposed exponential hesitant fuzzy entropy and gave a hesitant fuzzy multi-attribute decision-making model based on the entropy weight method. Yao [30] proposed three ideas to solve complex and uncertain multi-attribute decision-making problems. In recent years, the three-way decision model has been successfully applied in various fields [31][32][33][34]. Zhang et al. [35] proposed a classification and ranking decision method based on three-way decision theory and the TOPSIS model. Ye et al. [36] established a three-way multi-attribute decision-making model in an incomplete environment. Zhang et al. [37] proposed a three-way decision-making model based on a utility function, and Zhan et al. [38] proposed a relative utility function and established a three-way multi-attribute decision-making model based on utility theory in incomplete fuzzy information systems. Decision research using behavioral theory is a hot topic recently, applying regret theory to multi-attribute decision making can reflect the risk attitude and psychological behavior of decision makers and improve the scientificity of decision making [39][40][41]. The above models have one thing in common: they involve only one decision-maker or multiple decision-makers that agree. However, due to different backgrounds, decision-making experience, and subjective preferences, the opinions of decision-makers may diverge and cannot be compromised. We select one of multiple decision-makers who is most suitable to make a decision.
We combine parameterized fuzzy β covering and decision self-information, propose decision selfinformation based on a parameterized fuzzy β neighborhood to determine the most suitable decisionmaker, and propose a three-way multi-attribute group decision-making model based on a parameterized fuzzy β neighborhood. The classification and ranking results of all alternatives can be obtained. The effectiveness of the proposed method is experimentally verified.

Parameterized fuzzy β neighborhood
The parameterized fuzzy β covering [10], as an extension of the covering-based rough set model, can effectively characterize the similarity between samples.
Let ℂ , ,⋯, be the fuzzy covering group of , ∈ 0,1 , and let ,ℂ be a fuzzy covering information list. If ⊆ ℂ, then for all ∈ , the fuzzy neighborhood of with regard to is Given real numbers ∈ 0,1 and ∈ , the parameterized fuzzy neighborhood is defined as where is the fuzzy neighborhood radius. Let ,ℂ be a fuzzy covering information list, ∈ 0,1 , and ⊆ ℂ . Then for all ∈ ℱ , the lower and upper approximations of are respectively 1, .

Three-way decision-related theories
Based on the three-way decision model [30], Zhang [37] and Zhan et al. [38] proposed a threeway decision model using utility theory to improve classification accuracy.
Suppose the state set , indicates that an object belongs to states and . , , is an action set, where , , represent acceptance, delay, and rejection, respectively. Table 1 gives the corresponding utility and relative utility of alternatives in the two states of the three actions.
, , and denote the utility of alternative in taking actions , , and , respectively, in . Similarly, , , and denote the utility of alternative in adopting , , and , respectively, in . The relative utility function can be understood as follows. When the utility of action is used as the criterion and ∈ , , , and 0 are the relative utility functions of , , and , respectively; similarly, when the utility of action is used as the criterion and ∈ , 0, , and are the relative utility functions of , , and , respectively; where , , , and .  (2) Since ⊆ ⊆ ℂ, we obtain , ⊇ , . Furthermore, from the structure of ̅ , we obtain ̅ , ⊇ ̅ , or ̅ , ⩾ ̅ , , so we also obtain ⩾ . Proposition 3.2 shows that both the definite and possible decision indexes are monotonic. As the number of attributes increases, the decision-making index increases, as does the decision-making consistency. As the number of attributes increases, the possible decision indicators decrease, and the decision uncertainty decreases. Definition 2. Let ⊆ ℂ and be the target set obtained from the decision attribute. Then the accuracy and roughness of the decision index are determined as , 1 .
(2) The proof is similar to that of (1). Proposition 3.3 shows that the accuracy and roughness of the definite decision index are monotonic. Definition 3. Let ,ℂ, be a fuzzy covering decision information list, ⊆ ℂ, and the target set obtained from the decision attribute. Then the definite decision self-information definition of ,ℂ, is ln .
⩾ . Proof. The proof is similar to that of Proposition 3.3. Proposition 3.5 shows that the accuracy and roughness of the possible decision index are monotonic.

Definition 5. Let
,ℂ, be a fuzzy covering decision information list, ⊆ ℂ, and the target set obtained from the decision attribute. Then the possible decision self-information definition of ,ℂ, is ln .
Proof. By Proposition 3.5, we know that ⩽ and ⩾ . Therefore, ⩾ . Next, we propose another two types of decision self-information to characterize the uncertainty of fuzzy information, and we consider using both upper and lower approximation information to measure the uncertainty of the target concept. Definition 6. Let ⊆ ℂ, and let be the target set obtained from the decision attribute. Then the corresponding accuracy and roughness of the decision index are: It is clear that by Proposition 3.1, 0 ⩽ , ⩽ 1.
⩾ . Proof. The proof is similar to that of Proposition 3.3. Proposition 3.7 shows that the precision and roughness of relative decision indicators are monotonic. Definition 7. Let ,ℂ, be a fuzzy covering decision information list, ⊆ ℂ, and the target set obtained by the decision attribute. Then the relative decision self-information definition of ,ℂ, is ln .
Proof. By Proposition 3.7, we know that ⩽ and ⩾ . Therefore, , , , , , , with data as shown in Table 2. Let 0.6, 0.3. According to the fuzzy covering decision information list ,ℂ, , the parameterized fuzzy domain is obtained, as shown in Table 3.  Table 4. Furthermore, we can obtain the values of three kinds of decision self-information of , as shown in Table 5.

Parameterized fuzzy neighborhood class
Next, we construct the parameterized fuzzy neighborhood class and convert it to the classic set.

Construction of three-way decision-making model
Next, we construct conditional probabilities based on parameterized fuzzy neighborhood classes, and establish a three-way decision model.  Proof. Since is the complement of the fuzzy set , then ∀ ∈ , 1 . Therefore,  Table 3: . Furthermore, as an example, we can calculate the conditional probability of : According to the relative utility function studied by Zhan et al. [38], the standard deviation of the utility values of all alternatives given by the decision-maker is used to measure the dispersion of the decision-maker's preference: where ∑ / is the average of the utility values of all the alternatives below state . The larger the value of , the better the decision-maker's ability to distinguish all alternatives, i.e., the greater the priority difference. According to the utility value , taking into account the priority difference, we calculate the relative utility function of taking action in state , .
Similarly, we calculate the relative utility function of taking action in state : A risk coefficient, ∈ 0.5,1 , is introduced to calculate the relative utility function of adopted behavior under different states, i.e., and . According to the relative utility function [38] and our constructed conditional probability, the expected utility values of three behaviors of all objects are calculated as: According to the Bayesian decision rule, the action with the greatest utility value should be selected, so the following three decision rules can be obtained: if  Table 6. It is further possible to calculate thresholds and . Some important results can be seen in Table 7. , , , , .

Three-way multi-attribute group decision-making model based on parameterized fuzzy neighborhood
We consider that the relative decision self-information contains both upper and lower approximation information. Hence, we build a three-way multi-attribute group decision model based on relative decision self-information to solve real-life problems.

Problem statement
In the real world, the uncertainty and complexity of the social environment bring certain difficulties to decision-makers, and an important decision can require multiple decision-makers, whose evaluations can differ due to their knowledge, experience, and subjective factors. When they cannot reach an agreement, we need to choose the most suitable decision-maker. The parameterized fuzzy covering, as an extension of the covering-based rough set model, provides an effective method to deal with uncertain information. We establish a three-way multi-attribute decision-making model based on parameterized fuzzy neighborhoods to solve the uncertain multi-attribute decision-making problem in the real world when multiple decision-makers disagree.
The parameterized fuzzy neighborhoods of all alternatives are obtained based on fuzzy covering decision information list ,ℂ, , and the upper and lower approximations of all decisionmakers are further obtained. We use relative decision self-information to measure the uncertainty of all decision-makers and select the one with the smallest entropy value. We construct conditional probabilities using parameterized fuzzy neighborhoods and use this to further revise the decisionmaker's decision preference. We calculate the relative utility function values of all the alternatives. Using classification rule 2 2 and comparing the magnitude between the conditional probability and thresholds and , we determine the final decision action for each alternative. Finally, we can calculate the expected utility value of all the alternatives to take the final decision action, All alternatives can be sorted according to the expected utility values and priorities of the three domains. We sort according to the expected utility value of each domain, , ∈ and | | ; then ≻ . Then we consider the priority of each domain as ≻ ≻ .

Three-way multi-attribute group decision-making algorithm based on parameterized fuzzy neighborhood
According to the above properties and decision rules, we can obtain a three-way multi-attribute group decision-making algorithm based on parameterized fuzzy neighborhood.
Input: Fuzzy covering decision information list ,ℂ, , evaluation of all alternatives by decision-makers , ,⋯, and Output: The most suitable decision-maker, and the classification and ranking of each alternative Step 1 The decision information list ,ℂ, is covered by fuzzy , and the parameterized fuzzy neighborhoods ℂ , of all alternatives are calculated; Step 2 Calculate the lower approximation ℂ , and upper approximation ̅ ℂ , based on the neighborhood of parameterized fuzzy for all decision-makers, where 1,2,⋯, ; Step 3 Calculate the decision self-information ℂ of all decision-makers; Step 4 Find the smallest value of the decision self-information ℂ min ℂ , ℂ ,⋯, ℂ , and then the most suitable decision-maker is ; Step 5 According to Definitions 8 and 9, calculate the conditional probabilities of each alternative, | , ; Step 6 Calculate the relative utility function values and thresholds and for all alternatives from the relative utility function in Section 3.2; Step 7 According to the decision rule, 2 2 obtains the domain corresponding to the final decision behavior of all alternatives; Step 8 Calculate expected utility value of all alternatives; Step 9 Compare the priorities of ≻ ≻ and the expected utility values of the alternatives in each domain to rank all the alternatives.
The pseudo-code program is as follows:

Numerical example
We use examples from the literature [25] to verify the effectiveness of the proposed method. Example 4. An investment company intends to select some projects for investment, and decisionmakers make choices based on the benefits that each project can bring. There are eight investment projects , ,⋯, , which the company considers from five aspects ℂ , , , , , which represent expected benefits, environmental factors, market saturation, social benefits, and energy conservation. and are cost attributes, and the rest are benefit attributes. The attribute weight 0.3,0.1,0.3,0.2,0.1 is transformed to the evaluation result of the benefit standard, as shown in Table 8. Three experts are evaluating these eight projects, with results as shown in Table 9.
, , , , .  From the relative utility function and the decision preference of , two thresholds and conditional probabilities can be obtained, as shown in Table 11.
To more intuitively show the relationship between the conditional probability and the threshold, we show a comparison chart between them, as shown in Figure 1. From the decision rule 2 2 , the final decision classification result of expert can be obtained as: , , , , , , , , ∅.
The expected utility of all investment projects can then be calculated, as shown in Figure 2, from which a complete ranking can be obtained: The company can make decisions on which projects to invest in based on the final decision classification and ranking results of expert .

Experiment analysis
To illustrate the effectiveness of our method, we compare it with state-of-the-art and traditional decision-making methods, i.e., the methods of Zhan et al. [38], Ye et al. [34], and Zhang et al. [35], the TOPSIS method [23], and the WAA operator method [22]. The classifications and ranking results of different methods are shown in Tables 12 and 13.   Table 13 includes the ranking results of experts 1,2,3 . It can be found that the results of expert are most similar to those of other methods, and the optimal objects are all , while the optimal results of expert are , indicating that experts and are different. By the method in this paper, expert can be selected from the three experts 1,2,3 for decision-making, with results basically consistent with those of other methods, which shows that the proposed method is effective. To observe the difference between the ranking results of our and other methods, we compare the ranking results of different methods in Figure 3.
To further illustrate the effectiveness of the proposed method, is used to analyze the correlation between the ranking results of different methods, as shown in Table 14.
A ratio greater than 0.8 between the ranking results of two methods indicates that the correlation between them is significant. It can be seen from the table that the differences between the proposed method and the other methods are greater than 0.8, indicating the effectiveness of the method.
From the above analysis, we can find that the results obtained by using the decision information of expert is the most reasonable. Due to the lack of decision-making experience of expert on this issue, the results obtained by using the decision-making information of expert is not ideal. Therefore, the proposed model can effectively improve the scientificity of decision-making while comparing the decision-making information of many experts and avoiding incorporating the lack of experience expert information.

Conclusions
Decision self-information is a special kind of entropy and is an effective tool to characterize uncertain information. In this paper, the parameterized fuzzy neighborhood was combined with decision self-information to extend it to the fuzzy environment and apply it to multi-attribute group decision-making. We defined three kinds of decision-making self-information, studied their important properties, and defined the parameterized fuzzy neighborhood class and the corresponding conditional probability to establish a three-way decision-making model. We applied relative decision self-information, including both upper and lower approximation information, to three-way multiattribute group decision-making, solving the problem of disagreement among multiple decisionmakers in the real world. A three-way multi-attribute group decision-making algorithm based on a parameterized fuzzy neighborhood was proposed and was used to solve a practical example. An experimental analysis showed the effectiveness of the proposed method. The main contributions of this paper are listed as follows: (1) In this paper, the advantages of parametric fuzzy neighborhood satisfying reflexivity and effectively reducing the influence of noise data are used to construct decision self-information based on parametric fuzzy neighborhood. This measure can effectively describe the target concept in fuzzy environment.
(2) In order to avoid incorporating inexperienced expert information in the process of group decision-making, we construct a three-way multi-attribute group decision-making algorithm based on parametric fuzzy neighborhood to measure multiple experts and select the most suitable experts for decision-making. The advantage of doing so is that the process can both compare the decision-making information of multiple experts and avoid fusing the information of inexperienced experts.
In solving multi-attribute decision-making problems, we will consider the impact of risk aversion or benefit maximization of psychological behavior on decision-making, which is a direction worthy of further study. In addition, group consensus decision-making based on regret theory will be one of our future research directions.