Research on Hybrid Multi-Attribute Three-Way Group Decision Making Based on Improved VIKOR Model

: In the era of internet connection and IOT, data-driven decision-making has become a new trend of decision-making and shows the characteristics of multi-granularity. Because three-way decision-making considers the uncertainty of decision-making for complex problems and the cost sensitivity of classiﬁcation, it is becoming an important branch of modern decision-making. In practice, decision-making problems usually have the characteristics of hybrid multi-attributes, which can be expressed in the forms of real numbers, interval numbers, fuzzy numbers, intuitionistic fuzzy numbers and interval-valued intuitionistic fuzzy numbers (IVIFNs). Since other forms can be regarded as special forms of IVIFNs, transforming all forms into IVIFNs can minimize information distortion and effectively set expert weights and attribute weights. We propose a hybrid multiattribute three-way group decision-making method and give detailed steps. Firstly, we transform all attribute values of each expert into IVIFNs. Secondly, we determine expert weights based on interval-valued intuitionistic fuzzy entropy and cross-entropy and use interval-valued intuitionistic fuzzy weighted average operator to obtain a group comprehensive evaluation matrix. Thirdly, we determine the weights of each attribute based on interval-valued intuitionistic fuzzy entropy and use the VIKOR method improved by grey correlation analysis to determine the conditional probability. Fourthly, based on the risk loss matrix expressed by IVIFNs, we use the optimization method to determine the decision threshold and give the classiﬁcation rules of the three-way decisions. Finally, an example veriﬁes the feasibility of the hybrid multi-attribute three-way group decision-making method, which provides a systematic and standard solution for this kind of decision-making problem.


Introduction
With the rapid popularization of the internet and the internet of things, the generation and collection speed of various decision-making data in economic production and life is rapidly increasing. Due to the limitations of data collection technology and expert judgment [1,2], the decision-making data show the characteristics of incompleteness, uncertainty, incongruity, fuzziness and hesitation [3,4]. For this kind of decision-making problem with complex decision data and uncertain evaluation information, the traditional optimization mechanism model based on function relationship becomes more difficult in decision analysis and problem-solving. In fact, there is a large amount of effective decision information hidden in the decision data. Based on the existing decision data, we use scientific data processing technology to objectively analyze and evaluate them and transform them into effective decision indicators and knowledge, which can provide reliable and reasonable suggestions and decision support for decision-makers. This data-driven decision-making has become a new trend in modern decision-making [5][6][7]. significance to discuss the three-way decisions under a hybrid multi-attribute environment, especially in the case of attributes represented by intuitionistic fuzzy numbers or IVIFNs with more fuzzy information.
The representative studies on the three-way decisions under intuitionistic fuzzy or interval-valued intuitionistic fuzzy multi-attribute environments are shown in Table 1. Table 1. The representative three-way decision methods under intuitionistic fuzzy or interval-valued intuitionistic fuzzy multi-attribute environment.

Method Basic Principle Characteristics
Jia and Liu [32] • The conditional probability is calculated by TOPSIS.
• It is easy to understand the geometric position proximity to the ideal points, but it does not take into account the inherent characteristics of the data, such as the similarity with the ideal points.
Liu et al. [33] • The conditional probability is calculated based on the grey correlation degree between each scheme and the ideal scheme.

•
The losses are determined based on the preference coefficient and the distance from the ideal point, and then the threshold is determined by the Bayesian deduction formula.
• It reflects the similarity with the ideal scheme represented by a positive ideal point and reflects the inherent characteristics of data.

•
The loss function has certain objectivity, but the risk-taking level needs to be determined according to the personal preference coefficient.
Gao et al. [34] • The conditional probability is calculated by VIKOR.

•
The attribute weights are calculated according to the method of maximizing the deviation.
• From the whole perspective of all attributes, the group utility and individual regret relative to the ideal point can be considered, and the factors are more comprehensive.
Xue et al. [35] • The comprehensive evaluation value of each scheme is obtained by intuitionistic fuzzy additive operation between each attribute value and the attribute weight.
Combining the hesitation degree and threshold pair, the threshold of each scheme is obtained, and then the classification of each scheme is given.
• The calculation is simple, but the attribute weight is not fully used when calculating the conditional probability value with an intuitive fuzzy logic operation.
Xue et al. [36,37] • Based on the intuitionistic fuzzy possibility measure, the threshold pair and three decision classifications are determined, and then the selected schemes are further ranked based on the decision risk.

•
The classification based on probabilistic positive region, negative region and boundary region has clear meaning, but the attribute weight is not considered, and the schemes in negative and boundary regions cannot be further sorted.
Liu et al. [38] • Three-way decision rules are formed based on the intuitive fuzzy similarity, risk costs and closeness degree between schemes, combined with the ordering method of an intuitive fuzzy number.

•
The classification based on similarity is easy to understand, but attribute weights are not considered.
Ye et al. [39] • The interval-valued intuitionistic fuzzy weighted averaging operation is used to aggregate the group opinions on the losses, and the score and accuracy of the expected loss are used to determine the classification of each scheme.

•
The weights of experts are determined by grey correlation analysis.

•
The classification of each scheme is determined based on the expected loss after the aggregation of the loss of each expert, which fails to reflect the attribute value of the scheme.
Liu et al. [40,41] • Based on the optimization model and Karush-Kuhn-Tucker condition, a new method to determine the threshold is proposed.
• It provides an idea for determining the threshold pair of risk losses expressed by intuitionistic fuzzy numbers and IVIFNs.
The main methods for determining conditional probability in three-way decisions include TOPSIS [32], grey correlation analysis [33] and VIKOR [34]. Two methods are used to determine the decision thresholds: one is to use the optimization method to determine the thresholds based on the subjective risk loss matrix [40,41]; the second is to determine the losses based on the preference coefficient and the distance from the ideal points and then use the formula derived from Bayesian decision to determine the thresholds [33]. In addition, some scholars put forward the method of weight determination based on deviation [34], and some scholars put forward the method of grey correlation analysis to determine the weights of experts in group decision-making [39].
The above literature provides a good foundation for this study. However, the existing studies still have the following contents that may be deepened. Firstly, there is a lack of discussion on the hybrid multi-attribute three-way decision, even the study on the intervalvalued intuitionistic fuzzy three-way decision is relatively lacking. Secondly, there are few discussions about expert weight and attribute weight in the interval-valued intuitionistic fuzzy three-way group decisions. In fact, the interval-valued intuitionistic fuzzy group decision matrix contains a lot of information. It is of great significance to make effective use of the information and give the scientific weights of experts and attributes for decision results. Thirdly, the determination method of conditional probability in the three-way decision can be further improved. For example, the advantages of VIKOR, TOPSIS and grey correlation analysis can be fully integrated to form a grey correlation improved VIKOR model, which can give the conditional probability more objectively. In order to make up for the above deficiencies, we will discuss the hybrid multi-attribute three-way group decisionmaking method. The attribute values of different forms are unified into IVIFNs with the least information distortion. Based on the IVIFNs group decision matrix, the expert weight and attribute weight are determined. Then the conditional probability is determined by using the improved VIKOR model based on grey correlation, and the three-way decision rules can be formed by comparing with the threshold pair based on optimization.
The rest of this paper is organized as follows. Section 2 proposes research preliminaries, including interval-valued intuitionistic fuzzy sets and three-way decisions. Section 3 proposes a hybrid multi-attribute three-way group decision method based on an improved VIKOR model. Section 4 provides an illustrative example to verify the validity of the method. Section 5 summarizes the conclusions of this study.

Interval-Valued Intuitionistic Fuzzy Sets
Definition 1 [15]. Let X be a non-empty set and an IVIFS A in X is expressed as follows: where, µ L A (x) and µ R A (x), respectively, represent the upper and lower boundaries of the membership degree µ A (x) of the element x in X belonging to A; v L A (x) and v R A (x), respectively, represent the upper and lower boundaries of the non-membership degree v A (x) of the element x that belong to A. For each x ∈ X, it satisfies the conditions: Definition 2 [15]. For an IVIFS A, the hesitation degree of element x in A is: Definition 3 [42]. For an IVIFS A, the fuzzy degree ∆ A (x) of element x belonging to A is given as follows: where: Mathematics 2022, 10, 2783

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Definition 4 [15]. The complement of an IVIFS A is given as follows: Definition 5 [15]. Given three IVIFNs , their basic operations are summarized as follow: Definition 6 [42]. Let IVIFS(X) be the set of all IVIFSs in X, a real-valued function E: IVIFS(X) [0, 1] is called an entropy measure for IVIFSs if it satisfies the following axiomatic requirements: (1) E A = 0, if and only if Ais an exact set, namely.
(4) For a constant a in (0, 1), let ∆ L a , ∆ R a , π L a and π R a be the sets of all IvIFSs a respectively and is strictly monotone increasing with respect to π L A (x) on π L a and π R A (x) on π R a , respectively.

Definition 7.
In [43], for an IVIFS A in X, X = {x 1 , x 2 , . . . , x n }, the authors define the following entropy function: It is not difficult to prove that the above entropy function satisfies the axiomatic condition of interval-valued intuitionistic fuzzy entropy in Definition 6. Definition 8 [44]. Given two IVIFSs A and B in X, X = {x 1 , x 2 , . . . , x n }, the cross entropy of them is defined as follows: where: Obviously, 0 ≤ D A , B ≤ n ln 2, and D A , The cross entropy can also be called the relative entropy or divergence measure, Mathematics 2022, 10, 2783 6 of 21 which indicates the discrimination degree of IVIFS A from B. Since the cross entropy formula does not satisfy the symmetry, we rewrite it as follows: It is not difficult to prove that the following relationships hold: Definition 9 [15]. Let α j = a j , b j , c j , d j ( j = 1, 2, · · · , n) be a set of IVIFNs, intervalvalued intuitionistic fuzzy weighted averaging operator is as follows: where ω = (ω 1 , ω 2 , · · · , ω n ) T is the weighting vector of the IVIFNs α j (j = 1, 2, · · · , n).

Three-Way Decision
Assuming U is a finite nonempty set, R is an equivalence relation defined on U, and apr (α,β) = (U, R) is a probabilistic rough approximation space, then for X ⊆ U, let 0 ≤ β ≤ α ≤1, the upper and lower (α, β)-approximation sets of apr (α,β) can be expressed as [25]: where [x] is the equivalence class of X with respect to R.
In the above formula, | represents the conditional probability of classification, and |·| represents the cardinality of elements in the set. (α, β)-upper and lower approximation sets divide the domain into three parts, i.e., positive domain POS (α,β) (X), negative domain NEG (α,β) (X) and boundary domain BND (α,β) (X) [27]: The thresholds α and β are often given artificially in advance, and so are too subjective and difficult to obtain. Decision rough set introduces Bayesian theory into probability rough set and uses loss function to construct the division strategy of three-way decision with the minimum overall risk, which promotes the development of rough set theory. The decision rough set describes three-way decision processes through the state set Ω = {X, ¬X} and the action set A = {a P , a B , a N }. The state set Ω = {X, ¬X} represents two states of events, that is, belonging to concept X and not belonging to concept X. The action set A = {a P , a B , a N } indicates that three action strategies of acceptance, delay and rejection can be adopted for different states. Considering that different actions will cause different losses, we record that λ PP , λ BP and λ NP , respectively, represent the losses of actions a P , a B and a N when x ∈ X, and λ PN , λ BN and λ NN , respectively, represent the losses of actions a P , a B and a N when x / ∈ X. Therefore, the expected losses of actions a P , a B and a N can be expressed as: According to Bayesian decision criteria, we select the action set with the minimum expected loss as the best action scheme, and obtain the following three decision criteria [27]: Because Pr(X|[x] ) + Pr(¬X|[x] ) = 1, the above rules (P)~(N) are only related to the conditional probability Pr(X| [x] ) and the loss function λ •• (• = P, B, N). Generally, the loss of accepting the right thing is not greater than that of delaying to accept it, and both of them are less than the loss of rejecting the right thing. The loss of rejecting the wrong thing is not greater than that of delaying rejecting it, and both of them are less than the loss of accepting the wrong thing. Therefore, these loss parameters satisfy the following relationships: 0 ≤ λ PP ≤ λ BP < λ NP , 0 ≤ λ NN ≤ λ BN < λ PN , and the decision rules (P)~(N) can be rewritten as [27]:

Hybrid Multi-Attribute Three-Way Group Decision Based on Improved VIKOR Model
Several experts evaluate multiple programs based on multiple indicators. Quantitative indicators may be expressed as exact real numbers, or as interval numbers with minimum and maximum boundaries. Qualitative indicators may be expressed by proper linguistic expressions (values of some linguistic variables), fuzzy numbers, intuitionistic fuzzy numbers or IVIFNs. In accordance with the actual situation, all experts adopt the same expression for the same indicator of each scheme. For this hybrid multi-attribute group decision-making problem, scholars have proposed two different methods. One is to directly construct a hybrid multi-attribute decision matrix and apply TOPSIS, prospect theory, or other methods to make decisions [46,47]. Another is to transform different forms of attributes into the same form and construct a decision model based on a single form of attributes [48][49][50][51]. IVIFNs are more flexible and practical in dealing with fuzziness and uncertainty, and other forms of expression can be regarded as special forms of IVIFNs. Therefore, transforming hybrid multi-attribute values into IVIFNs can minimize information distortion. Moreover, after being transformed to the same form, we can effectively calculate the expert weight and attribute weight. Therefore, we choose the latter method for the hybrid multi-attribute group decision-making. The overall decision-making steps are shown in Figure 1.
single form of attributes [48][49][50][51]. IVIFNs are more flexible and practical in dealing with fuzziness and uncertainty, and other forms of expression can be regarded as special forms of IVIFNs. Therefore, transforming hybrid multi-attribute values into IVIFNs can minimize information distortion. Moreover, after being transformed to the same form, we can effectively calculate the expert weight and attribute weight. Therefore, we choose the latter method for the hybrid multi-attribute group decision-making. The overall decision-making steps are shown in Figure 1.  For the intuitionistic fuzzy number (uij (k) , vij (k) ), we can transform it to an IVIFN as follows: by an IVIFN.
For the intuitionistic fuzzy number (u ij (k) , v ij (k) ), we can transform it to an IVIFN as follows: r For a real number x ij (k) , we first use the linear proportion, vector normalization, extreme value transformation, or other methods to make dimensionless processing. For example, the calculation formula of the linear proportion method is as follows: where J 1 is an indicator of benefit type that the larger the better, and J 2 is an indicator of cost type that the smaller the better. Then we transform y ij (k) into an intuition- istic fuzzy number r ij (k) = (y ij (k) , 1 − y ij (k) ), and transform r ij (k) into an IVIFN r For an interval number [x ij L(k) , x ij R(k) ], we first carry out dimensionless processing. For example, the calculation formula of the linear proportion method is as follows: , here q is an odd positive number, the IVIFN corresponding to the q linguistic evaluation granularity can be expressed as [52]: where µ q 0 = v q 0 = 0.5 − 1 2q , 0.5 . Then, for a linguistic variable value s ij (k) , we determine the linguistic evaluation value of the corresponding level in the q granularity, and then express it with the corresponding IVIFN.
In this way, we can transform the hybrid multi-attribute decision-making matrix R (k) into an interval-valued intuitionistic fuzzy decision matrix R k = r

Determination of Expert Weight Based on Entropy and Cross Entropy
In multi-attribute group decision-making, the smaller the difference between the evaluation value of a decision-maker and other decision-makers, the greater weight should be given to this decision-maker. At the same time, the higher the effectiveness of information in a decision-maker's evaluation matrix, that is, the smaller the redundancy, the greater the weight of this decision-maker. In evaluating the redundancy and difference of information, we introduce entropy and cross-entropy to measure them, respectively, and then build a model to determine the weights of experts.
For the evaluation matrix of a single decision maker, we use entropy E (k) to express the redundancy of evaluation information, and the formula is as follows: where E (k) j represents the entropy of the jth indicator obtained from the decision matrix of the kth expert. According to Definition 7, its expression is as follows: Based on the entropy of each expert, we can calculate the expert weight as follows: To reflect the difference between a single decision-making matrix and the other decision-making matrices, we define the cross entropy as follows: According to Definition 8, the formula of D * r (k) , r (t) is as follows: Because 0 ≤ D * r (k) , r (t) ≤ 2mn ln 2, 0 ≤ D (k) ≤ 2 ln 2. Then, based on the cross-entropy, we can calculate the expert weight as follows: By aggregating w (k) 1 and w (k) 2 with weight coefficients γ and (1-γ), respectively, we can calculate the final expert weight as follows:

Determination of Group Comprehensive Evaluation Matrix
Combined with all the experts' weights, we apply the interval-valued intuitionistic fuzzy weighted averaging operator to calculate the group comprehensive evaluation matrix X = x ij n×m , where:

Determination of Attribute Weight Based on Entropy
Based on the group comprehensive evaluation matrix, we apply the entropy value method to determine the weight of each attribute: where:

Determination of Conditional Probability
The determination of conditional probability is the key to a three-way decision. The VIKOR method originates from TOPSIS and can take group utility and individual regret into account. Grey correlation analysis can make full use of sample information to reflect the internal law of sample data. We use the VIKOR method improved by grey correlation analysis to determine the conditional probability, and the concrete steps are as follows: Step 1: According to the evaluation matrix X, the positive and negative ideal solutions are as follows: x where: Step 2: Calculate the group utility value S i and the individual regret value R i of the ith scheme: where d(x, y) represents the distance between two IVIFNs x and y, which can be calculated according to Definition 10. The smaller the value of S i , the higher the group utility. The smaller the value of R i , the smaller the individual regret.
Step 3: Determine the best and the worst group utility values as follows: The best and the worst individual regret values are: Step 4: Calculate the grey correlation degree between the ith scheme and the positive and negative ideal solutions as follows: where: In the above formula, ρ ∈ [0, 1] is the distinguishing coefficient. The smaller the value of ρ, the greater the distinguishing ability. Generally, ρ is taken as 0.5.
Step 5: Calculate the group utility value and individual regret value of the ith scheme based on grey correlation analysis as follows: Both the group utility value and the individual regret value are indicators that the smaller the better. Then the best and the worst group utility values are, respectively: The best and the worst individual regret values are: Step 6: Determine the benefit ratio of the ith scheme based on the VIKOR-grey correlation analysis method as follows: where σ represents the compromise coefficient between group utility and individual regret, 0 ≤ σ ≤ 1. If σ > 0.5, it represents the principle of conformity.
Step 7: The smaller the benefit ratio of the ith scheme, the greater the probability that it belongs to the acceptable state Z. The conditional probability can be calculated as follows:

Determination of Decision Thresholds
The threshold pair (α, β) is another key parameter of a three-way decision, which is determined by the loss function. In practice, it is difficult for decision-makers to give the exact value of risk loss of each action under different states. They prefer to use uncertain expressions, such as interval number, fuzzy number, linguistic variable value, intuitionistic fuzzy number and IVIFN. According to the linear or nonlinear ordering rules of various uncertain forms, scholars proposed the corresponding determination methods of the threshold pair [40,41,53,54]. Considering the deficiency of large information distortion in linear ordering, Liu et al. proposed a generalized scalable and nonlinear sorting method to determine the threshold pair for the risk loss matrix represented by IVIFNs from the perspective of optimization [41].
The expert group expresses the risk loss values of three actions a P (acceptance), a B (delay) and a N (rejection) under two states Z (acceptable) and Z C (unacceptable) as IVIFNs, as shown in Table 2.
Then the optimization model for solving α and β is as follows [41]:

Classification and Sorting of Schemes
According to the value of the threshold (α, β), we can classify schemes: (1) If the conditional probability of the ith scheme Pr(Z|[G i ] ) ≥ α, the scheme G i can be accepted; (2) If Pr(Z|[G i ] ) ≤ β, the scheme G i shall be rejected; (3) If β < Pr(Z|[G i ] ) < α, the scheme G i can be used as a candidate scheme and needs further evaluation.
In addition, the larger the value of Pr(Z|[G i ] ), the greater the possibility of selecting the scheme G i . If α = β, the three-way decision model degenerates into a two-way decisionmaking model. If Pr(Z|[G i ] ) ≥ α, we accept the scheme G i ; Otherwise, we reject the scheme G i .

An Illustrative Example
We use the latent dirichlet allocation topic model to mine customers' demand factors for mobile phone performance, and extract six features, namely appearance (A 1 ), fast response (A 2 ), endurance (A 3 ), screen definition (A 4 ), running fluency (A 5 ) and battery heating (A 6 ). We organize four experts D 1 , D 2 , D 3 and D 4 from China Mobile Communications, China United Network Communications and China Telecommunications in the field of mobile communication performance evaluation to evaluate the above characteristics of the five mobile phone brands G 1~G5 . In order to verify the feasibility of the method proposed in this paper, after discussion with experts, the forms of different indicators are set as follows: (1) A1 is evaluated in the form of percentage real number. The prettier the mobile phone, the larger the value of A1. (2) A2 is evaluated in the form of percentage interval number. For example, if an expert thinks that a mobile phone responds well to various functional requirements, according to the percentage system, it can be regarded as more than 80, but less than 85, then he can give an evaluation value of [80, 85]. The evaluation matrices of the four experts are shown in Tables 3-6, respectively. We will select the brands that can be agented, rejected and pending from the five mobile brands.   The four experts jointly give the risk loss matrix represented by IVIFNs, as shown in Table 13. We substitute the data in the above table into the nonlinear programming models (45) and (46) and obtain that α = 0.608646 and β = 0.122339. It can be seen that the conditional probabilities of G 3 and G 4 are greater than α, indicating that the two mobile phone brands can be chosen as an agent. If the conditional probability of G 1 is less than β, this mobile phone brand shall be excluded. The conditional probabilities of G 2 and G 5 are between α and β, so they need to be further investigated In order to reflect the difference between the improved VIKOR model and other conditional probability models, we calculated the conditional probability results and the three classification results under TOPSIS, grey correlation analysis and traditional VIKOR models. The results are shown in Table 14. It can be seen that the conditional probability results of grey correlation analysis are too close to effectively distinguish the differences between brands. TOPSIS results of different brands are different to some extent, but brands G 1 , G 2 and G 3 are all pending, which indicates that the distinction is not obvious enough. Of course, this is related to the risk loss matrix given by decision-makers. However, considering only the proximity to positive and negative ideal points, it is difficult to reflect the intrinsic characteristics of data. Nor does it capture decision-makers' attitudes to utility and regret. The results of the VIKOR method are similar to those of improved VIKOR, but there are differences in brand G 1 , which is greatly related to the addition of grey correlation analysis results reflecting the inherent characteristics of data. In general, the improved VIKOR model can not only reflect the proximity to the ideal points, but also reflect the inherent characteristics of data and decision-makers' trade-offs on utility and regret, and the results of it are relatively objective.

Conclusions
For the hybrid multi-attribute decision-making problem, we propose a three-way group decision-making method based on the improved VIKOR model. Based on the transformed interval-valued intuitionistic fuzzy decision matrix, we apply entropy and cross-entropy to determine the expert weights and obtain the group comprehensive evaluation matrix. Then, we use entropy to obtain attribute weights. By using the improved VIKOR method by grey correlation analysis, we determine conditional probability. By comparing the conditional probability with the decision threshold pair based on optimization, we obtain the classification rules of the three-way decision. The example analysis shows that the method has good three-way classification and can provide support for actual management decision-making. This study has the following features and benefits: First, it considers the hybrid multi-attribute environment, especially the interval-valued intuitionistic fuzzy environment containing more fuzzy information, which is closer to the actual decision-making and has better universality. Second, considering the group decision-making environment, the hybrid multi-attribute evaluation matrix is given by each expert, which is more consistent with reality. Moreover, the proposed expert weight determination method can not only reflect the differences among experts' opinions, but also reflect the uncertainty degree of each expert's evaluation opinion, and the obtained weights are more reasonable and objective. Different from scholars' studies, this paper mainly has three aspects of innovation. First, from the perspective of research, it expands the research of hybrid multi-attribute decision-making and three-way group decision-making. Second, it deepens the research on expert weights and attribute weights in interval-valued intuitionistic fuzzy group decision making and improves the objectivity of weights. Thirdly, an improved VIKOR model based on grey correlation analysis is proposed to determine the conditional probability, which improves the scientificity of the conditional probability.
There are some shortcomings in this study. First, in the determination of expert weights and attribute weights, only one form of interval-valued intuitionistic fuzzy entropy is considered. In fact, there are many forms of interval-valued intuitionistic fuzzy entropy that meet the axiom conditions. How do they affect the weight results and final results, and whether there will be contradictory conclusions? These are not tested. Second, for the risk loss matrix represented by IVIFNs, we use the threshold determination method based on the optimization model, but there is another interactive threshold determination method, that is, to determine the losses based on the preference coefficient and the distance from the ideal points, and then calculate the thresholds. How much is the difference between the results of these two methods? In addition, is it more advantageous to combine the two, that is, to first determine the threshold loss matrix in an interactive way and then determine the thresholds by an optimization method? These aspects are also not explored. Third, we adopt the method of conditional probability determination of the improved VIKOR model. In fact, the prospect theory based on an ordinary utility curve is being gradually introduced to determine conditional probability. Limited by the fact that the prospect theory based on the IVIFN decision matrix is not perfect, we have not conducted research on this aspect. Based on the shortcomings of the method, further research can be conducted in the following aspects. First, we can analyze the influence of other forms of interval-valued intuitionistic fuzzy entropy on expert weights, attribute weights, and the final results. Second, based on the risk loss matrix expressed by IVIFNs, we can discuss the impact of other threshold determination methods on the decision results. Third, we can further improve the prospect theory based on the IVIFN decision matrix and introduce it into the determination of the conditional probability of a three-way decision.