Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision-Making Based on Power Geometric Weighted Average Operator and Prediction Model

: With respect to dynamic multi-attribute group decision-making (DMAGDM) problems, where attribute values take the form of intuitionistic fuzzy values (IFVs) and the weights (including expert, attribute and time weights) are unknown, the dynamic intuitionistic fuzzy power geometric weighted average (DIFPGWA) operator and the improved IFVs’ GM(1,1) prediction model (IFVs-GM(1,1)-PM) are proposed. First, the concept of IFVs, the operational rules, the distance between IFVs, and the comparing method of IFVs are deﬁned. Second, the DIFPGWA operator and the improved IFVs-GM(1,1)-PM are deﬁned in detail. Third, corresponding decision-making (D-M) steps are proposed. Three kinds of weights are given by the proposed determination method. Finally, an example is given to prove the effectiveness and superiority of the proposed decision-making method.


Introduction
Decision-making is an activity frequently occurring in management. This is the basic idea of modern decision theory. One of the important components in modern decision theory is multi-attribute group decision-making (MAGDM). This refers to the decision-making problem of selecting the optimal alternative or scheduling the scenario when multiple decision makers consider multiple attributes. However, many things in real life have fuzziness and uncertainty, and there is no clear statement. In the description of fuzziness and uncertainty, intuitionistic fuzzy sets have unparalleled advantages. If intuitionistic fuzzy sets are applied to MAGDM and generalized to multi-period decision-making, this decision-making process is called dynamic intuitionistic fuzzy multi-attribute group decision-making (DIF-MAGDM).
Since the Bulgarian scholar Atanassov [1] proposed the concept of intuitionistic fuzzy sets (IFSs), the research on the theory of IFSs has come a long way. Intuitionistic fuzzy sets have been widely used in many real-world cases [2][3][4][5]. Many scholars have proposed many improvements to MAGDM and multiple criteria decision making (MCDM) on IFS's methods and models. For example, Liu et al. [6] introduced intuitionistic trapezoidal fuzzy number and proposed an intuitionistic trapezoidal fuzzy prioritized ordered weighted aggregation (ITFPOWA) operator; Hashemi et al. [7] combined ELECTRE (originated in Europe in the mid-1960s. The acronym ELECTRE stands for: ELimination Et Choix Traduisant la REalité (ELimination and Choice Expressing REality)), VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje, that means: Multicriteria Optimization and Compromise idea of maximum deviation [26,27]. That is to say, the closer the evaluation value of the attribute is, the smaller the weight should be given. Otherwise, the greater weight should be given. Moreover, due to different evaluations by experts in different periods, attribute weights should also be different. For time weights, Guo et al. [28] proposed a combination of subjective and objective methods to determine the time weights. Subjective can comprehensively consider the expert's knowledge and experience, treat things with their own feelings, and make conclusions. Therefore, the combination of subjectivity and objectiveness can give more reasonable time weights. In DIF-MAGDM, the proposed DIFPGWA operator not only aggregates the information of each stage, but also captures the fine nuances of the comprehensive evaluation value that the decision makers want to reflect. It is inaccurate to make a judgment based only on the evaluation from past to today. Based on this, this paper forecasts the comprehensive evaluation value of each scheme in the next period through the improved IFVs-GM(1,1)-PM. This method can make up for the lack of DIFPGWA operator. The proposed future adjustment coefficient can combine the two results to make the final result more accurate. Moreover, the multi-period weight determination method proposed in this paper also helps to obtain more accurate results. In the second chapter, this paper introduces the basic knowledge of intuitionistic fuzzy. In the third chapter, this paper constructs the operator and prediction model. In the fourth chapter, the paper gives the corresponding decision steps and predicting steps. In the fifth chapter, the decision method proposed in this paper is applied to the selection of the company's intern employees.

Concepts and Operational Rules
For any x ∈ X, µ A (x) and ν A (x) was expressed membership degree and non-membership degree of A. Moreover, µ A (x) and ν A (x) satisfy the following properties: was expressed as a hesitancy degree of A [1].

Definition 2.
Let α = (µ α , ν α ), β = µ β , ν β be IFVs, and the operational rules are as follows [19,29]: Definition 3. Support is a similarity indicator, the closer the two data values, the greater the support. Let α 1 , α 2 , · · · , α n be IFVs. Therefore, according to the character of support sets and the operation rules of IFVs, the support is calculated as follows [13]: Sup α i , α j express the support between a i and α j , and has the following three properties:
It is easy to prove that the DIFPGWA λ(t) operator has the three properties: Commutativity, Idempotency, and Boundedness. Here no longer proof.
According to S(α k ) + a z (1) (k) = b, the time response function of S(α) = (S(α 1 ), S(α 2 ), · · · , S(α n )) can be calculated as follows: where z (1) . Thus, the predictive value of IFVs scores degree can be calculated as follows: In addition, the predictive value of hesitancy degree can be calculated as follows: According to the definition of IFVs, the IFVs-GM(1,1)-PM can be drawn as [24] However, this IFVs-GM(1,1)-PM cannot ensure that the degree of membership and non-membership are both in the range of [0, 1]. Therefore, the above IFVs-GM(1,1)-PM needs to be improved as follows: The best using condition of IFVs-GM(1,1)-PM is: a ≥ −0.5 [24].

Future Adjustment Coefficient
Definition 10. Let ψ be future adjustment coefficient in the range of [0, 1]. Assuming that α = (µ α , ν α ) is the aggregated decision value in period of [0, t], β = µ β , ν β is the predicted decision value in the period t + 1. The formula of the comprehensive evaluation value is as follows: Example 4. Suppose there are two sets of time series, as shown in Table 1. The time weight vector is (0.2, 0.3, 0.5). By observing the score degree of the Time series A and B, it is obvious that A is in the rising period and B is in the recession period. The results shown in Table 2, which is calculated by DIFPGWA λ(t) operator IFVs-GM(1,1)-PM and Future Adjustment Coefficient. Let ψ = 0.5. The result of using the DIFPGWA λ(t) operator is A ≺ B. The result obtained using the IFVs-GM(1,1)-PM is A B. The comprehensive evaluation is A B. Obviously, the results obtained using the IFVs-GM(1,1)-PM are more accurate. However, the advantage of the DIFPGWA λ(t) operator could not be negated. There are also errors in the IFVs-GM(1,1)-PM. Obviously, through the adjustment coefficient in the future, the results of the two can be integrated to get more accurate comprehensive evaluation value.

Decision-Making Steps
Description 1. In a MAGDM process, E = {e k |k = 1, 2, · · · , p} is a set of experts, A = {A i |i = 1, 2, · · · , m} is an alternative set, and C = C j |j = 1, 2, · · · , n is a set of attributes. Expert's evaluation values are expressed by IFVs. The weight vector of time under different period t l is θ(t), and θ(t) = θ(t 1 ), θ(t 2 ), · · · , θ t g T , Because each expert has his own field and expertise, experts may make irrational comments. If the experts have the same weight, the result will be unreasonable. On the other hand, if expert weights are set subjectively, especially in this multi-period situation, not only the workload is large, but also the deviation is large. Therefore, the following group mean evaluation matrix A ij (t l ) is constructed. Assessments close to the average will have a greater weight, while assessments away from the average will have smaller weight [25]: The degree of similarity between α ijk (t l ), k = 1, 2, · · · , p and α ij (t l ) is defined as , k = 1, 2, · · · , p, i = 1, 2, · · · , m, j = 1, 2, · · · , n, where , k = 1, 2, · · · , p, i = 1, 2, · · · , m, j = 1, 2, · · · , n.
In period t l , the group decision matrix B ij (t l ) = β ij (t l ) m×n is obtained by aggregating the information of each expert with IFPGWA λ ijk (t l ) operator, where where .

• Step 2. Determine attribute weights
In period t l , for attribute C j , specifies that D ij w j (t l ) = m ∑ l=1 d β ij (t l ), β sj (t l ) w j (t l ) represents the total deviation. Then, for all attributes D w j (t l ) = represents the total deviation. Then, the following model is constructed: According to Label (22), the results are shown below: Then, the results of standardization are as follows: Therefore, the attribute weight vector in period t l is w(t l ) = (w 1 (t l ), w 2 (t l ), · · · , w n (t l )) T ,

• Step 3. Determine all alternative's comprehensive evaluation value by operator in each period
All the alternative's comprehensive evaluation value by operator X(t l ) = (χ 1 (t l ), χ 2 (t l ), · · · , χ m (t l )) T is obtained by aggregating the information of each attribute with IFPGWA w j (t l ) operator in period t l , l = 1, 2, · · · , g, where where T β ij (t l ) = n ∑ j=1,j =s (1+T(βij(t l )))w j (t l ) .

• Step 4. Determine time weights
In order to prevent the error caused by the weight setting of the supervisor and objective, the two methods should be used together. The time weight is obtained as follows [28]: θ(t) = θ(t 1 ), θ(t 2 ), · · · , θ t g T are the weight of each period and θ(t l ) ≥ 0, g ∑ l=1 θ(t l ) = 1.
δ represents subjective emphasis on recent data. δ ∈ [0, 1], if δ is closer to 1, there is more emphasis on recent data. If δ is closer to 0, there is more emphasis placed on forward data.

• Step 5. Determine all alternative's comprehensive evaluation value by operator
All alternative's comprehensive evaluation value by operator Y = (Y 1 , Y 2 , · · · , Y m ) T is obtained by aggregating the information of each attribute with DIFPGWA θ(t) operator, where where .
According to S(χ i (t k )) + a z (1) . Thus, the predictive value of IFVs scores degree can be calculated as follows: In addition, the predictive value of hesitancy degree is π χ i t g+1 = max π(χ i (t 1 )), π(χ i (t 2 )), · · · , π χ i t g . According to the definition of IFVs, the IFVs-GM(1,1)-PM can be calculated as follows: Then, all alternative's predictive comprehensive evaluation value are obtained as follows:

• Step 7. Comprehensive evaluation
Let ψ be future adjustment coefficient in the range of [0, 1]. The formula for calculating the comprehensive evaluation values of all alternatives is as follows: All alternatives are sorted according to the ranking method of IFVs. If max Y 1 , Y 2 , · · · , Y m = Y i , the ith alternative is best in period g + 1. If max Y 1 , · · · , Y i−1 , Y i+1 , · · · , Y m = Y j , then the jth alternative is suboptimal in now.

Example
A company plans to recruit two employees. A total of five interviewees (A 1 , A 2 , A 3 , A 4 , A 5 ) received internship opportunities. The internship lasted for three months (t 1 , t 2 , t 3 ). At the end of each month, three experts (e 1 , e 2 , e 3 ) evaluated their performance during the month. These candidates (A 1 , A 2 , A 3 , A 4 , A 5 ) are evaluated in terms of practical ability C 1 , learning ability C 2 and innovation ability C 3 . Each expert uses the IFVs to evaluate five candidates in each period, as shown in Tables 3-5. Table 3. The three experts' evaluation value in the first month t 1 . Experts

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Step 1. Based on the above evaluation information given by the experts in each period, the group mean evaluation matrix could be constructed. Then, using Equations (18)- (20), to get the expert weights. The group decision matrices for each period are calculated by Equation (21), which is shown in Table 6. Table 6. Group decision matrix in each month. Period

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Step 2. The closer the evaluation value of the attribute is, the smaller the weight should be given. Otherwise, the greater weight should be given. The attribute weights for each period are calculated by Equations (22)- (24), which are shown in Table 7.

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Step 3. The comprehensive evaluation values are calculated by Equation (25) in each period, which are shown in Table 8.

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Step 4. In determining the time weight, experts think the recent data is more important. The time weight vectors that are calculated by Equation (26) are θ(t) = (0.154, 0.292, 0.554) T , where δ = 0.3.
. This shows that, as time goes on, experts are better at evaluating A 1 than A 3 . This means that A 1 is a growth candidate and A 3 is a negative candidate. Thus, companies should choose A 1 instead of A 3 . This also shows that A 2 , A 4 and A 5 are stable candidates. This means that performance of candidate A 2 is not only the best but also the most stable. Performances of A 4 and A 5 have been poor and do not recommend admission. The above analysis results are consistent with the comprehensive evaluation results obtained by future adjustment coefficients. The final result is

Comparative Analysis
(1) The proposed decision method in this paper was applied to the Case of [34], where δ = 0.3 and ψ = 0.3. The result is as shown in Table 12: If the weight information is unchanged, the result of using the proposed DIFPGWA operator is also The above results show that A 2 is not only the most stable but also the best. The performance of A 1 has been getting better, but the performance of A 3 has deteriorated. However, the comprehensive evaluation results are consistent with the original text, which shows that although the evaluation value of A 1 shows an upward trend, but it can not completely exceed A 3 . If the weights are unknown, the result of using the proposed DIFPGWA operator is This result is slightly different from before because the weights are different. However, the same place is that A 2 is the best and most stable.
(2) The proposed decision method in this paper was applied to the case of [35], where δ = 0.3 and ψ = 0.3. The result is as shown in Table 13: Table 13. The final sort results of the case in [35].

Decision Method Weight Result
Decision method in [35] Weights used in [ If the weight information is unchanged, the result of using the proposed DIFPGWA operator is also The comprehensive evaluation result is A 1 A 3 A 4 A 2 . A 1 and A 3 rose the fastest, and A 1 exceeded A 3 . A 4 changed from first to fourth. If the weights are unknown, the result of using the proposed DIFPGWA operator is A 3 This result is slightly different from before because the weights are different. When weights are unknown, both A 3 and A 1 perform better than A 4 . To sum up, the performance of A 3 and A 1 is not only the best but has been in a rising stage.
(3) The proposed decision method in this paper was applied to the case of [36], where δ = 0.3 and ψ = 0.3. The result is as shown in Table 14. The original result is A 1 A 2 A 3 . If the weight information is unchanged, the result of using the proposed DIFPGWA operator is This shows that the rise of A 2 exceeds A 1 , and the problem is not considered in the original text. If the weights are unknown, the result of using the proposed DIFPGWA operator is A 1 A 2 A 3 . The prediction result is A 1 A 2 A 3 . The above results are basically the same-that A 1 is the best, A 2 is the second, and A 3 is the worst.
After the above comparative analysis, it is obvious that the decision method proposed in this paper is not only effective but also more advantageous.

Conclusions
DIF-MAGDM is an important branch of modern decision theory. Because of this, the DIFPGWA operator is proposed. The operator makes up for the shortcomings of the previous decision-making method and better reflects the support relationship between attributes, between substitutions, and between periods. The DIFPGWA operator can capture the subtle details of the comprehensive evaluation value that the decision maker needs to reflect. However, judging only by aggregating evaluation information from past to present is inaccurate. Therefore, the TFVs-GM(1,1)-PM is improved in this paper. The previous IFVs-GM-(1,1)-PM predicts that the membership and membership of IFVs may be out of range. Our improved IFVs-GM(1,1)-PM limits membership and non-affiliation within [0,1]. The DIFPGWA operator and the IFVs-GM(1,1)-PM complement each other. The future adjustment coefficient is proposed, which combines the two results. This method can help decision-makers to make more accurate judgments. It is very important to determine the weights in DIF-MAGDM. In this article, attribute weights, expert weights, and time weights are all unknown. This paper adopts the idea of deriving expert weight from the evaluation value and improves the previous method. At each period, therefore, it is generally agreed that the more deviated from the common cognition, the smaller the expert weight should be, and, the closer to the common cognition, the greater the expert weight should be. However, if the experts evaluate the same value, the sum of the deviations will be zero. Therefore, it will lead to no solution to the weight of experts. Our improved approach avoids this error. For attribute weights, the decision maker can determine the weight of each period's attributes based on the idea of maximum deviation. That is, the closer the evaluation value of the attribute is, the smaller the weight that should be given. Otherwise, it should be given greater weight. Moreover, due to different evaluations of experts in different periods, attribute weights should also be different. For time weights, the decision maker can use subjective and objective methods to determine the time weights. Subjective can fully consider the knowledge and experience of experts, treat things with their own feelings, and draw conclusions. Objective can make full use of all the time information. Therefore, the combination of subjectivity and objectivity can give more reasonable time weights. The decision method proposed in this paper is applied to the selection of the company's intern employees. After three months of internship, two excellent and up-scaling employees were selected. Not only that, this article also applies the proposed decision method to other cases. The final comparative analysis shows that the decision-making method is more effective and superior. In the future, we will first apply the decision method proposed in this paper to practical cases, and second apply this method to more fields.