Extended GRA Method Based on Cumulative Prospect Theory with Probabilistic Hesitant Fuzzy Information for MAGDM and its Application to Green Supplier Selection

: An extended grey relational analysis (GRA) method is introduced in this article to reduce the limitations of the classical GRA method using the cumulative prospect theory (CPT) which takes into account psychological factors such as the risk appetite of decision makers. Moreover, the circumstance of probabilistic hesitant fuzzy (PHF) which assigns probabilistic values to DMs’ different levels of hesitation shows its superiority when making decisions in a complex environment. Meanwhile the weighting vector of each attribute is calculated according to the entropy which is calculated by the different prospect decision elements. Thus, in this paper, we proposed an extended GRA method based on cumulative prospect theory in the probabilistic hesitant fuzzy circumstance and applying the model in the selection of the green supplier. At last, the comparative analysis and the simulation analysis are made to show the practicability of this newly proposed method.


Introduction
Multi-attribute decision making (MADM), as a branch of the multi-criteria decision making (MCDM), has discrete decision variables and a limited number of options which are evaluated [1][2][3][4][5]. To solve many MADM issues, a sea of decision-making methods has been developed over the past few decades. The more important ones, for example, like VIKOR (a multi-criteria compromise solution ordering method)method [6][7][8], MABAC (the multi-attribute border approximation area comparison) method [9][10][11][12], CODAS(combinative distance based assessment) method [13][14][15][16] EDAS (the evaluation based on distance from average solution) method [17][18][19][20][21][22], MOORA (the multiobjective optimization ratio analysis)method [23][24][25][26], TODIM method [27][28][29][30] and Taxonomy method [31,32] and so on. These methods each have different advantages and disadvantages due to their calculation principles and among them, the GRA (grey relational analysis) method which is the part of the grey system theory introduced by Deng [33] and now it is widely applied in many fields. The grey system theory includes the GRA method is introduced to solve the issues under a complex environment with the unknown information. Specially, the GRA method is appropriate for solving the issues which is with complicated interrelationships between multiple factors and variables( Kuo, Yang and Huang [34]), and the specific application cases are as follows. Zhu, Yuan and Ye [35] examined the multi-timescales of European carbon market with GRA method and the empirical mode decomposition. Tang, Zhu, Liu, Jia and Zheng [36] extended the TOPSIS with GRA to evaluate the urban sustainability. Tan, Chen and Wu [37] combined the AHP (analytical hierarchy process) with GRA to evaluate the environmentally conscious product design. Liu, Liu and Hu [38] evaluated the risks of mergers and acquisitions by GRA method under interval-valued intuitionistic fuzzy circumstance. However, the distance from the negative ideal point ignores the subjective factors such as the risk reference which influence the behavior of the decision makers in practice. Therefore, we extend the GRA method with cumulative prospect theory (CPT)Tversky and Kahneman [39] which improves the random dominance of prospect theory and shows the decision maker's performance under different environment.
With the conception of the fuzzy set (FS) by Zadeh [40], the extension of it such as the intuitionistic fuzzy set(IFS) [41] and hesitant fuzzy set (HFS) [42]. The HFS depict the complex decision-making environments in the real world using the memberships to describe each of the attribute. As the environment continues to be studied, the probabilistic hesitant fuzzy set (PHFS) was proposed by Xu and Zhou [43] to extend the original environment into a more general form.
PHFS which solves the problem of DMs' choice of preference uses the probability to express different membership degrees and the HFS is the special form whose probabilities of the possible numbers are equal. Then the basic calculation formulas and related concepts has been proposed one after another such as the probabilistic hesitant fuzzy geometric operators [43] and distance measurement [44]. To avoid the different distance between probabilistic hesitant fuzzy elements (PHFEs), a new process of normalization was proposed by Li, Chen, Niu and Wang [45]. The PHFS is more consistent with behavior of DMs when making decisions, therefore, it is utilized in many fields. The TOPSIS method has been extended in Interval-Valued Pythagorean Fuzzy environment [46], Gaussian Interval Type-2 Fuzzy [47], Hesitant Fuzzy Linguistic environment [48] and so on.
Thus, this paper is about to extended GRA method with CPT in probabilistic hesitant fuzzy circumstance.
The contributions to this article are as follows: (i) the CPT-GRA method expands the application of the classical GRA method for a better depict of the DMs' activity affected by psychological factors. (ii) the unknown weighting vectors of the attributes is also obtained by entropy weighted methods under probabilistic hesitant fuzzy environment. (iii) in the extended GRA which is based on CPT will be used in more practical situations.
The following is the main thread of the article. The basic knowledges are introduced in section 2. In section 3, an extended GRA method model of MAGDM is set up. In section 4, the newly proposed method is applied in the selection of green supplier. In section 5, the comparative analysis and the simulation analysis are made to show the practicability and validly. In section 6, we draw a conclusion and prospect the future application.

Preliminary Knowledge
In this section, we excerpt some basics of probabilistic hesitant fuzzy set and novel GRA method based on cumulative prospect theory.

Probabilistic Hesitant Fuzzy Sets
Definition 1 [43]. Set y as a fixed set, a probabilistic hesitant fuzzy set (PHFS) on y is represented as equation 1: where the function Step 2. Fix the second element of normalized PHFEs. If t t  and 1 1 2 With the novel normalization method proposed, some calculation algorithms are introduced as follows.
With the calculation of score and deviation, we can compare two PHFEs 1 1 by the following laws: l r  is the l-th largest elements, l t  is the l-th largest probability of the membership.
Definition 6 [45]. Let 1 1 ( ) k t and 2 2 ( ) k t be two PHFEs. And their normalized forms are (1)  The multiple attribute decision-making matrix is shown as follows: Step 1. Obtain the normalized decision matrices In a real case, different metrics or different directions my appear in attributes which may cause the wrong result. Therefore, it is of necessity to normalize every element into the comparative one. The normalize steps are as follows.
Firstly, standardize for the directionality of attributes by using Eq. (9) the attribute is the positive attribut where kg w  represents the value of normalized decision matrix.
Then normalize the decision matrices using the steps in Definition 2.
Step 2. Integrate the normalized group decision matrices into a decision matrix by using the PHFWA which calculated by Eq. (10) Step 3. Step 4. Compute out the weights among attributes The weighting vector of each attribute is completely unknown in this condition, so it's either given directly or computed. Owing to the fact that in probabilistic hesitant fuzzy environment every prospect decision element should have their corresponding different attributes weight. We use the entropy weighting method and its calculating law is as follows.
where n h denotes the entropy of the n-th attributes and calculate it by Eq. (13) Step5. Definite the positive ideal solution (PIS) g  and negative ideal solution (NIS) g  separately.
Step 6. Calculate the distances of between the positive and negative ideal point using Eq.
where #r denotes the value of the possible membership of every element.
Step 8. Figure (19) where  and  are the parameters and  is the parameter deriving from the experiment of Tversky and Kahneman [39] .88) which express the sensitivity of the risk aversion.
Step 9. Obtain the transformed probability weights for the gains and losses ( ) mn    which can be calculated by the following equations: where where  and  represent the attitude of decision maker towards gains and losses and these parameters are from the experiment of Tversky and Kahneman [39] (  =0.61 and  =0.69).
Step 10. Obtain the prospect matrix m where m V denotes the optimal comprehensive prospect value of alternative m W .
Step 11. Rank the optimal comprehensive prospect value of each scheme in a descending order. Step 1: normalize and integrate matrix Step 2:determine the PIS and NIS

Prospect matrix
Step 3: obtain the optimal alternative Cumulative Prospect Theory

the numerical example
The concept of green supplier selection, also known as the environmentally conscious supply chain, is proposed to develop the product production processes such as the material acquisition, processing, packaging, warehousing, transportation and use to scrap, so that they have the least impact on the environment (negative effect) and the highest resource efficiency. In recent years, with the concept of environmental protection deeply rooted in people's hearts, the green supplier selection is taken seriously by more and more enterprises. Normally, six evaluation principles to evaluate the green supplier: 1 P cost factor; 2 P product factor; 3 P green supply chain management; 4 P green imagination of enterprise; 5 P service capability; 6 P enterprise qualification. Considering an enterprise need to select the most optimal supplier from five different green supplier selection  Step 1. Standardize the decision matrices and its results are shown in Table 4 to 6. Step 2. Integrate the standardized decision matrices into group decision matrix using the Eq.
(10), and the overall decision matrix is shown in Table 7.   Step 4. Compute out the weights among attributes using the entropy and its result is as follows.
Step 5. Determine the positive ideal solution (PIS) g  and negative ideal solution (NIS) g  in table 9 and 10.  Step 6. Calculate the distance between the positive and negative ideal point separately, and the results are shown in Table 11 and 12. Step 7. Figure out the grey relational coefficient of each alternative and the grey relational coefficient between PIS and NIS are shown in Table 13 and 14.  Step 8.  Table   15 and 16.

Comparative Analysis
In this section, we will first apply the same data into different methods to testify the effectiveness of the newly proposed method. Then in order to further prove the applicability and feasibility of the method, sensitive and simulation analysis are also applied in this article.

Comparative Analysis with other methods
we will use kinds of methods whose classical TODIM (PHF-TODIM) method [50], PHFWA operator [43], PHFWG operator [43], probabilistic hesitant fuzzy TOPSIS (PHF-TOPSIS) method [51] and the novel TODIM method proposed by Tian, Niu, Ma and Xu [52] to compare with GRA based on CPT method. To make a better comparison, we use the same data from the Table 1 to Table 3 and the attributes weight vector which is obtained from Step 4 is as follows: {0.160, 0.156, 0.175, 0.143, 0.220, 0.145} n o 

Comparison with PHF-TODIM method
The classical TODIM method under probabilistic hesitant fuzzy condition proposed by Zhang, Du and Tian [50] is one of the most commonly used evaluation methods to solve the selection issues.
Thus, we compare it with the developed GRA method. The following is the detailed process of this method.
Firstly, use the data of Table 7 (25) Finally, the overall dominance degree of the alternative can be obtained by Eq. (26).

Comparison with Probabilistic Hesitant Fuzzy Weighting Aggregation Operators
Aggregation operator is used as the most intuitive measurement data to solve the MAGDM problems. The result of probabilistic hesitant fuzzy aggregation operators is shown in Table 18 and 19.  The result  W W W W W    

Comparison with PHF-TOPSIS method
The TOPSIS method which is similar to the GRA method in theory and principles under probabilistic hesitant fuzzy circumstance proposed by Wu, Liu, Wang and Zhang [51] can be applied to testify the effectiveness of the developed method.
Firstly, used the value of Table 7

Comprehension Comparison
According to Table 16

Sensitive and Simulative Analysis
In order to better prove the stability and effectiveness of the new method, we apply the sensitive analysis and the simulation analysis. According to the results of these two analyses, the superiority of the developed method has been proved.

Sensitive Analysis of the Extended GRA and the GRA method with Probabilistic
Hesitant Fuzzy Information Based on the Distinguishing Parameter  .
To further illustrate the superiority of the extended GRA method based on CPT under probabilistic hesitant fuzzy circumstance, this section we calculate the extend GRA method and classical GRA with probabilistic hesitant fuzzy information by changing the distinguishing parameter  .
The distinguishing parameter  is the only changeable parameter in both extended GRA and classical GRA method. By changing the value of  ( 0 1    ), and the numerical results of the extended GRA method and classical GRA method can be shown in Table 22 and Table 23 and the normalized data of the Table 22 and 23 to draw the relevancy degree changes curve with MATLAB software can be shown in Figure 2.  To better represent trends in the graph, we further standardized the results of Table 22 ad Table   23. According to the Figure 2, when  varies, the curve of the extended GRA is more smooth than the one obtained by the classical GRA. In the comparison of alternative 1 and 2, the fluctuation range of the extended GRA method is bigger than the one obtained by the GRA method, while the fluctuation range of extended GRA method in alternative 3 is smaller than the one gained by classical GRA method. Moreover, no matter what the parameters are, the final ranking is all equal.

Sensitive Analysis of the Extended GRA with Probabilistic Hesitant Fuzzy
Information and Hesitant Fuzzy Information Based on the Distinguishing Parameter  .
In this section, we use the extended GRA method with the probabilistic hesitant fuzzy information and fuzzy hesitant fuzzy information separately to find the sensitivity of the extended method to parameters in different environment. The numerical result of the probabilistic hesitant fuzzy environment and fuzzy hesitant fuzzy environment are shown in Table 22 and Table24. Besides, the standardized relevancy degree changes curve with MATLAB software can be shown in Figure 3.  In summary, the results in Figure 2 and 3 show that the extended GRA based on CPT in probabilistic hesitant fuzzy circumstance has a fluctuant stability under different schemes by comparing with the classical GRA method in the same environment. The Figure 3 shows us that compared with hesitant fuzzy environment, the probabilistic hesitant fuzzy environment contains more information and is more stable.

Simulation Analysis
The sensitive analysis performed above is based on a sample analysis, and there we use MTLAB software as a tool,1,000 samples were randomly generated for analysis. The ranking results are shown in Table 25 and Figure 4 to 8.
The data of Table 25 shows that in the probabilistic hesitant fuzzy circumstance compared with TODIM method, there are 688 out of 1,000 random samples have the same optimal results.  With the calculation of a mass of random samples, the results which has a lot of same ranking results show us the applicability of extended GRA method base on cumulative prospect theory.
While the different results by comparing with the other four methods show us that these methods have some differences because of their different priorities when making decisions.

Conclusions
The GRA, a method to measure the degree of correlation between factors according to the degree of similarity or difference between the development trends of factors is applied in complex environment with unknown information, while it ignores the psychological factors which may affect the results of the decision-making. Meanwhile, As the development of the decision-making, the complexity of the environment is increasing and fitting. Thus, in this paper, we developed classical GRA method with the cumulative prospect theory in probabilistic hesitant fuzzy environment (PHF-CPT-GRA). What's more, we use the entropy method to obtain the weighting vector of each attributes. To testify the feasibility and stability of newly proposed method, we use four different methods to compare with it by using the sensitivity analysis and simulation analysis with the MATLAB software. The advantages of PHF-CPT-GRA method are as follows: (1) It can help DMs make as accurate decisions as possible in the complex probabilistic hesitant fuzzy circumstance.
(2) It takes into account the psychology of DMs and combines classical GRA method and cumulative prospect theory considering the different propensity of DMs to risk.
In view of the above analysis, in the future, we firmly believe that PHF-CPT-GRA method will be applied in the more fields. Meanwhile, the integration of other classical method and CPT with more complex and uncertain information also needs more attention.
Authors' contribution Ningna Liao, Guiwu Wei and Xudong Chen discussed and constructed the mathematical models, found their applications and wrote the paper together.

Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.

Conflict of interest
The authors declare that they have no conflict of interest.

Data Availability
The data used to support the findings of this study are included in the article.