# Probabilistic Linguistic PROMETHEE I and II Methods for Evaluation of the Reform Scheme of Postgraduate Innovation and Entrepreneurship Education Talent Training Mode under the Big Data Environment

Authors: Wenshuai Wu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Talent training quality is an important field within higher education research. Innovating the talent training mode and deepening educational reform programs are both of great significance for enhancing the quality of postgraduate innovation and entrepreneurship education in universities. In this study, Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) I and II methods are extended with the probability linguistic term set (PLTS) to accurately express and quantitatively evaluate the reform scheme of postgraduate innovation and entrepreneurship education talent training mode under the big data environment. First, probabilistic linguistic PROMETHEE I and II methods are presented for quantitatively evaluating the reform scheme of postgraduate innovation and entrepreneurship education talent training, which have the advantages of good effectiveness and feasibility. Second, the PLTS is imported into the evaluation methods and applied to accurately depict qualitative information about the index data of the reform scheme effect by the degree of probability. Third, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) with PLTS is proposed to perform a comparative study and conduct visual analysis to verify the effectiveness of the extended probabilistic linguistic PROMETHEE I and II methods. Fourth, an empirical example illustrates the specific evaluation process, verifies the feasibility of the extended methods, and explains the effectiveness of the results. The research findings indicate that the proposed method to reform scheme evaluation can lead to better decision quality, especially in a complex fuzzy and uncertain decision-making environment.


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## Body

## 1. Introduction

Higher education is key to the success of three world-renowned bay areas, the New York Bay area, the San Francisco Bay area, and the Tokyo Bay area [1]. The development of a higher education cluster is not only one of the core contents of the study of the Guangdong-Hong Kong-Macao Greater Bay Area [2], but it is also an important source of support for the construction of a first-class bay area, which will become a new growth pole for China’s high-quality development. A survey from the innovation and entrepreneurship education alliance of China shows that Chinese graduate students are eager for innovation and entrepreneurship and hope that their universities will provide more opportunities to cultivate innovation and entrepreneurship.In recent years, postgraduate innovation and entrepreneurship education have become a hot issue in the field of higher education. Many universities have put a lot of effort into improving their organizational systems, advancing their infrastructure, carrying out extracurricular activities, and increasing financial support for the talent training of postgraduate innovation and entrepreneurship education [3–6]. However, generally speaking, insufficient attention has been paid to the talent training mode of postgraduate innovation and entrepreneurship education, and the current understanding of the talent training effect is insufficient. Some studies think mechanical replication of the traditional market with low technology as the achievements of postgraduate innovation and entrepreneurship education. Some simply understand talent training innovation as “science and technology driven innovation,” while ignoring ideology and consciousness innovation, which makes the talent training mode separate from professional education and knowledge education [7–9]. Therefore, research on the reform scheme of postgraduate innovation and entrepreneurship education talent-training mode is of great significance for universities to transform educational ideas, enhance educational modes, deepen educational reform, and improve the quality of talent training.The evaluation of the reform scheme of the talent-training mode usually involves multiple criteria, such as innovative knowledge cultivation, innovative consciousness cultivation, and innovative ability cultivation, which can be modeled as a multiple criteria decision-making (MCDM) problem. MCDM, a very popular discipline of management science and operations research [10–13], can address the selection problem of optimal alternatives according to the priority of all feasible schemes when multiple or a finite number of decision criteria exist [14–16]. The Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) method, one of the most important MCDM methods, has a wide range of applications in many different areas [17–21]. Albadvi [22] proposed a preference ranking model based on the PROMETHEE method for developing national information strategies. Cavalcante et al. [23] proposed a multicriteria model integrating PROMETHEE and the Bayesian method to address the replacement problem in service production systems. Karande and Chakraborty [24] presented an integrated PROMETHEE and GAIA method to solve four nontraditional machining process selection problems. Pawe [25] presented a NEAT F-PROMETHEE to improve the process of mapping fuzzy numbers by the correction mechanism. Corrente et al. [26] developed and applied a hierarchical SMAA-PROMETHEE model to evaluate the sustainability of European cities. Bausys et al. [27] proposed an m-Generalized q-Neutrosophic PROMETHEE method to address path selection problems for an inspection robot. PROMETHEE includes some family methods, such as the PROMETHEE I and PROMETHEE II method. Although the PROMETHEE method can be used to process and evaluate numerical data, it is unable to address qualitative data or fuzzy data. Thus, Akram and Shumaiza [20] proposed a q-rung orthopair fuzzy PROMETHEE approach to address the problems of MCDM. Akram et al. [28] proposed a bipolar fuzzy PROMETHEE method for multicriteria group decision-making to select the green suppliers. In this paper, PROMETHEE I and PROMETHEE II are extended with the probability linguistic term set (PLTS) to accurately depict qualitative information or fuzzy information for evaluation of the reform scheme.PLTS, provided by Pang et al. [29]; is a new type of linguistic variable used to accurately express qualitative data or fuzzy data. PLTS can express linguistic preference with multiple linguistic terms by making decision-makers (DMs) induce the weight of each language term in the form of a probability, which can reﬂect preference degrees of all possible linguistic information. For example, when DMs are evaluating the reform scheme of the talent training mode, based on the self-cognition and knowledge system of research problems, the DMs may consider that they are 70% sure the reform scheme effect is “very good,” 20% sure it is “good,” and 10% sure it is “bad.” Because of the advantages of accurate expression of PTLS, some MCDMs are extended with probabilistic linguistic information to accurately express qualitative data or fuzzy data [30–36]. Liao et al. [37] proposed a linear programming method with probabilistic linguistic information for solving MCDM problems. Wang et al. [38] investigated multicriteria group decision problems with PLTSs. Chang et al. [39]; based on cumulative probability-based Hellinger distance, proposed a probabilistic linguistic TODIM method for waste mobile phone recycling. Darko and Liang [40] proposed a probabilistic linguistic WASPAS method by designing and reconciling prioritized Maclaurin symmetric mean aggregation operators for patients’ prioritization. In this study, PROMETHEE I and II methods are extended with PLTS to accurately express and quantitatively evaluating the reform scheme of postgraduate innovation and entrepreneurship education talent training mode.At present, research on the reform scheme of postgraduate innovation and entrepreneurship education talent training mode is still in the stage of theoretical discussion, meaning empirical research is lacking. Plus, there are few evaluation studies on the reform effect. Moreover, since the world has entered the era of big data, big data have become a major focus of academia, industry, and government agencies [41–43]. Big data technology is gradually promoting the reform and innovation of talent training mode in universities. This study is of great significance since it explores and evaluates the reform scheme of talent training mode of postgraduate innovation and entrepreneurship education in the big data environment.The major of contributions in this paper are as follows: First, the principal contribution is that the PROMETHEE I and II methods are extended with the probabilistic linguistic environment for quantitatively evaluating the reform scheme of postgraduate innovation and entrepreneurship education talent training mode in the era of big data, since the research on the reform scheme is current in the stage of theoretical discussion, lacking empirical research. Second, the PLTS is shown to accurately depict qualitative information about the index data of the reform scheme evaluation by the degree of probability. Third, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) with PLTS is presented to conduct comparative analysis to verify the effectiveness of the extended probabilistic linguistic PROMETHEE I and II methods. The results demonstrate the advantages of good effectiveness and feasibility of the extended methods for evaluation of the reform scheme effect. Fourth, an empirical example demonstrates the specific evaluation process, proves the feasibility of the methods, and reveals the effectiveness of the results. The research findings on the reform scheme evaluation indicate that the extended methods can improve decision-making guidance and technical support for education departments, universities, and relevant teachers to guide the reform of postgraduate innovation and entrepreneurship education talent training mode.The remaining parts of this paper are organized as follows: Section2 describes some preliminaries, including PLTS, the PROMETHEE I and II methods. Section 3 extends the PROMETHEE I and II methods with the probabilistic linguistic environment. Section 4 provides details of the empirical analysis and discusses the results. Section 5 summarizes the paper.


## 2. Preliminaries

In this section, some basic concepts of the PLTS, normalization of PLTS, comparison between PLTSs, and PROMETHEE I and II methods are introduced.


### 2.1. Probabilistic Linguistic Term Set

Based on the additive linguistic term setS=Sα|α=−τ,…,−1,0,1,…,τ [44, 45], the definition of the PLTS is given by Pang et al. [29] as follows:(1)Lp=Lkpk|Lk∈S,pk≥0,k=1,2,…,#Lp,∑k=1#Lppk≤1,where Lkpk represents the linguistic term Lk associated with probability pk, and #Lp is the number of all of the different linguistic terms in Lp.Note that if∑k=1#Lppk=1, then the PLTS has the complete probabilistic information of all possible linguistic terms; if ∑k=1#Lppk<1, then the PLTS has partial probabilistic information; if ∑k=1#Lppk=0, then the PLTS has completely unknown probabilistic information.In addition, the detailed process regarding the normalization of PLTS and the comparison between PLTSs can be obtained from the work of Pang et al. [29].


### 2.2. PROMETHEE I and II Methods

The PROMETHEE method, proposed by Brans [46], is a ranking decision analysis method that constructs “values outranking relations” to distinguish the best scheme. Based on pairwise comparisons of schemes, PROMETHEE uses the preference function, attribute value, and attribute weight given by the DMs to determine the rank of each scheme by the priority relationship. This method then uses the priority relationship to define the positive outranking flow and negative outranking flow of each scheme. The positive outranking flow shows that the chosen alternative outranks other alternatives, and the negative outranking flow shows that other alternatives outrank the chosen alternative [47]. According to the negative outranking flow and positive outranking flow, the best alternative can be determined [17]. The PROMETHEE I method can obtain a partial priority relationship, and the PROMETHEE II method can get the complete priority relationship. The specific steps are given as follows:(1)
Get the standardized decision matrixA, based on the original decision matrix R=rijm×n.(2)Aij=rij−minrijmaxrij−minrijfor benefit criteria,Aij=maxrij−rijmaxrij−minrijfor cost criteria,where1≤i≤m;1≤j≤n.(2)
Compute the preference index:(3)πAa,Ab=∑j=1kpjAa,Abwj,πAb,Aa=∑j=1kpjAb,Aawj.LetAa,Ab∈A, where A is a finite number of alternatives A1,A2…Am. Furthermore, k1≤k≤n is the number of criteria; wj is the weight of criterion j, and ∑j=1kwj=11≤k≤n. pjAa,Ab and pjAb,Aa are the preference functions of the alternatives Aa and Ab.Generally speaking, in the PROMETHEE method, there are six kinds of preference functions, namely the usual criterion, quasicriterion, criterion with linear preference, level criterion, criterion with linear preference and indifference area, and Gaussian criterion. DMs usually select one type of preference function [46]. In addition, DMs can also construct new preference functions based on research problems.(3)
ObtainπAa,Ab and πAb,Aa for each pair of alternatives, where πAa,Ab represents how Aa is preferred to Ab over all the criteria, and πAb,Aa represents how Ab is preferred to Aa over all the criteria.(4)
Calculate the positive outranking flow:(4)ϕ+Aa=1n−1∑Ab∈A,a≠bπAa,Ab.(5)
Calculate the negative outranking flow:(5)ϕ−Aa=1n−1∑Ab∈A,a≠bπAb,Aa.(6)
Determine the net outranking flow:(6)ϕAa=ϕ+Aa−ϕ−Aa.The PROMETHEE I method can obtain a partial priority relationship according to the negative outranking flow and positive outranking flow. The larger the positive outranking flow and the smaller the negative outranking flow of a scheme, the better the scheme. Three conditions for judging the priority of the scheme are as follows:(1)
Aa outranks Ab, denoted as AaPAb, if(7)ϕ+Aa>ϕ+Ab,ϕ−Aa<ϕ−Ab;or,ϕ+Aa>ϕ+Ab,ϕ−Aa=ϕ−Ab;or,ϕ+Aa=ϕ+Ab,ϕ−Aa<ϕ−Ab.(2)
Aa is indifferent to Ab, denoted as AaIAb, if(8)ϕ+Aa=ϕ+Ab,ϕ−Aa=ϕ−Ab.(3)
Aa and Ab cannot be compared, which is called the incomparable situation and is denoted as AaRAb, if(9)ϕ+Aa>ϕ+Ab,ϕ−Aa>ϕ−Ab;or,ϕ+Aa<ϕ+Ab,ϕ−Aa<ϕ−Ab.The PROMETHEE II method can get the complete priority relationship according to the net outranking flow. The higher the net outranking ﬂowϕa, the better the alternative. Two conditions for judging the priority of the scheme are as follows:(10)1AaoutranksAb,if and only ifϕAa>ϕAb,2Aais indifferent toAb,if and only ifϕAa=ϕAb.


## 2.1. Probabilistic Linguistic Term Set

Based on the additive linguistic term setS=Sα|α=−τ,…,−1,0,1,…,τ [44, 45], the definition of the PLTS is given by Pang et al. [29] as follows:(1)Lp=Lkpk|Lk∈S,pk≥0,k=1,2,…,#Lp,∑k=1#Lppk≤1,where Lkpk represents the linguistic term Lk associated with probability pk, and #Lp is the number of all of the different linguistic terms in Lp.Note that if∑k=1#Lppk=1, then the PLTS has the complete probabilistic information of all possible linguistic terms; if ∑k=1#Lppk<1, then the PLTS has partial probabilistic information; if ∑k=1#Lppk=0, then the PLTS has completely unknown probabilistic information.In addition, the detailed process regarding the normalization of PLTS and the comparison between PLTSs can be obtained from the work of Pang et al. [29].


## 2.2. PROMETHEE I and II Methods

The PROMETHEE method, proposed by Brans [46], is a ranking decision analysis method that constructs “values outranking relations” to distinguish the best scheme. Based on pairwise comparisons of schemes, PROMETHEE uses the preference function, attribute value, and attribute weight given by the DMs to determine the rank of each scheme by the priority relationship. This method then uses the priority relationship to define the positive outranking flow and negative outranking flow of each scheme. The positive outranking flow shows that the chosen alternative outranks other alternatives, and the negative outranking flow shows that other alternatives outrank the chosen alternative [47]. According to the negative outranking flow and positive outranking flow, the best alternative can be determined [17]. The PROMETHEE I method can obtain a partial priority relationship, and the PROMETHEE II method can get the complete priority relationship. The specific steps are given as follows:(1)
Get the standardized decision matrixA, based on the original decision matrix R=rijm×n.(2)Aij=rij−minrijmaxrij−minrijfor benefit criteria,Aij=maxrij−rijmaxrij−minrijfor cost criteria,where1≤i≤m;1≤j≤n.(2)
Compute the preference index:(3)πAa,Ab=∑j=1kpjAa,Abwj,πAb,Aa=∑j=1kpjAb,Aawj.LetAa,Ab∈A, where A is a finite number of alternatives A1,A2…Am. Furthermore, k1≤k≤n is the number of criteria; wj is the weight of criterion j, and ∑j=1kwj=11≤k≤n. pjAa,Ab and pjAb,Aa are the preference functions of the alternatives Aa and Ab.Generally speaking, in the PROMETHEE method, there are six kinds of preference functions, namely the usual criterion, quasicriterion, criterion with linear preference, level criterion, criterion with linear preference and indifference area, and Gaussian criterion. DMs usually select one type of preference function [46]. In addition, DMs can also construct new preference functions based on research problems.(3)
ObtainπAa,Ab and πAb,Aa for each pair of alternatives, where πAa,Ab represents how Aa is preferred to Ab over all the criteria, and πAb,Aa represents how Ab is preferred to Aa over all the criteria.(4)
Calculate the positive outranking flow:(4)ϕ+Aa=1n−1∑Ab∈A,a≠bπAa,Ab.(5)
Calculate the negative outranking flow:(5)ϕ−Aa=1n−1∑Ab∈A,a≠bπAb,Aa.(6)
Determine the net outranking flow:(6)ϕAa=ϕ+Aa−ϕ−Aa.The PROMETHEE I method can obtain a partial priority relationship according to the negative outranking flow and positive outranking flow. The larger the positive outranking flow and the smaller the negative outranking flow of a scheme, the better the scheme. Three conditions for judging the priority of the scheme are as follows:(1)
Aa outranks Ab, denoted as AaPAb, if(7)ϕ+Aa>ϕ+Ab,ϕ−Aa<ϕ−Ab;or,ϕ+Aa>ϕ+Ab,ϕ−Aa=ϕ−Ab;or,ϕ+Aa=ϕ+Ab,ϕ−Aa<ϕ−Ab.(2)
Aa is indifferent to Ab, denoted as AaIAb, if(8)ϕ+Aa=ϕ+Ab,ϕ−Aa=ϕ−Ab.(3)
Aa and Ab cannot be compared, which is called the incomparable situation and is denoted as AaRAb, if(9)ϕ+Aa>ϕ+Ab,ϕ−Aa>ϕ−Ab;or,ϕ+Aa<ϕ+Ab,ϕ−Aa<ϕ−Ab.The PROMETHEE II method can get the complete priority relationship according to the net outranking flow. The higher the net outranking ﬂowϕa, the better the alternative. Two conditions for judging the priority of the scheme are as follows:(10)1AaoutranksAb,if and only ifϕAa>ϕAb,2Aais indifferent toAb,if and only ifϕAa=ϕAb.


## 3. Probabilistic Linguistic PROMETHEE I and II Methods

In this section, the probabilistic linguistic PROMETHEE I and II methods are extended and presented to quantitatively evaluate the reform scheme of postgraduate innovation and entrepreneurship education talent training mode in the era of big data.For the problem of talent training mode reform scheme evaluation with PLTS, suppose there areAii=1,2,…m alternatives and Cjj=1,2,…n criteria. Based on the additive linguistic term set S=Sα|α=−τ,…,−1,0,1,…,τ [44, 45], the DMs can evaluate the alternatives Ai for criterion Cj by PLTSs to construct decision matrix R=rijm×n. The specific steps are detailed below:Step 1: construct the original decision matrixR=rijm×n.The original decision matrix can be constructed according torij=Lijp=Lijkpijk|i∈1,m,j=1,n,k∈1,#Lp,∑k=1#Lppijk≤1, where rij is the jth criteria value with respect to the ith alternative by DMs. Lkpk represents the linguistic term Lk associated with probability pk, and #Lp is the number of all of the different linguistic terms in Lp.Then, the original decision matrixR=rijm×n can be constructed:(11)R=L11pL12p⋯L1npL21pL22p⋯L2np⋯⋯⋯⋯Lm1pLm2p⋯Lmnp.Step 2: transform the original decision matrixR into decision matrix X.(12)X=x11x12⋯x1nx21x22⋯x2n⋯⋯⋯⋯xm1xm2⋯xmn,wherexij=∑k=1#Lijprkpk/∑k=1#Lijppk according to Pang et al. [29]; rk is the subscript of linguistic term Lk, and 1≤i≤m,1≤j≤n.Step 3: get the standardized decision matrixA, based on the decision matrix X:(13)Aij=xij−minxijmaxxij−minxijfor benefit criteria,Aij=maxxij−xijmaxxij−minxijfor cost criteria,where1≤i≤m;1≤j≤n.Step 4: compute the preference index:(14)πAa,Ab=∑j=1kpjAa,Abwj,πAb,Aa=∑j=1kpjAb,Aawj,whereA is a finite number of alternatives A1,A2…Am, j,k1≤j,k≤n is the number of the criteria, wj is the weight of criterion j, and ∑j=1kwj=11≤j≤n. pjAa,Ab and pjAb,Aa are the preference functions of the alternative Aa and Ab. In this paper, the weight of each criterion can be obtained by analytic hierarchy process [48], proposed by Saaty [49, 50]. Furthermore, this study employs the linear priority relation function, presented by Hu and Jiang [51], as the preference function to induce the preference index.Step 5: calculate the positive outranking flowϕ+Aa and the negative outranking flow ϕ−Aa:(15)ϕ+Aa=1n−1∑Ab∈A,a≠bπAa,Ab,(16)ϕ−Aa=1n−1∑Ab∈A,a≠bπAb,Aa.Step 6: determine the net outranking flowϕAa:(17)ϕAa=ϕ+Aa−ϕ−Aa.The probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship according to the negative outranking flow and positive outranking flow. Three conditions for judging the priority of the scheme in the probabilistic linguistic environment are as follows:(1)
Aa outranks Ab, denoted as AaPAb, if(18)ϕ+Aa>ϕ+Ab,ϕ−Aa<ϕ−Ab;or,ϕ+Aa>ϕ+Ab,ϕ−Aa=ϕ−Ab;or,ϕ+Aa=ϕ+Ab,ϕ−Aa<ϕ−Ab.(2)
Aa is indifferent to Ab, denoted as AaIAb, if(19)ϕ+Aa=ϕ+Ab,ϕ−Aa=ϕ−Ab.(3)
Aa and Ab cannot be compared, which is called the incomparable situation and is denoted as AaRAb, if(20)ϕ+Aa>ϕ+Ab,ϕ−Aa>ϕ−Ab;or,ϕ+Aa<ϕ+Ab,ϕ−Aa<ϕ−Ab.The probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow. The higher the net outranking ﬂIIϕAa, the better the alternative. Two conditions for judging the priority of the scheme in the probabilistic linguistic environment are as follows:(21)1AaoutranksAb,if and only ifϕAa>ϕAb,(22)2Aais indifferent toAb,if and only ifϕAa=ϕAb.


## 4. Empirical Analysis

### 4.1. Datasets

Mass entrepreneurship and innovation have become a national development strategy for China’s economy. Simultaneously, China’s higher education has gradually entered the stage of “popular education” from “elite education.” In the big data environment, big data gives graduate students new opportunities and challenges for innovation and entrepreneurship. Moreover, big data have become increasingly used in the field of education, which not only brings greater development space but also poses unprecedented challenges to educational researchers. Talent training quality is a key index of education quality in universities, and it is an important field of higher education research. Transforming the educational concept, innovating the talent-training mode, and deepening the educational reform are of great significance for improving the talent training quality of postgraduate innovation and entrepreneurship education in universities. Furthermore, due to the uncertainty and fuzziness of the information environment, the decision-making process becomes more and more complex, which brings great difficulties and challenges to scientific decision-making. In this study, based on PLTS, the probabilistic linguistic PROMETHEE I and II methods are extended to accurately depict the uncertainty and fuzziness of the information involved in the reform scheme evaluation under the big data environment.An example is given to verify the effectiveness of the extended probabilistic linguistic PROMETHEE I and II methods for reform scheme evaluation. Eight reform schemesAii=1,2,3,4,5,6,7,8 are selected and combined with literature research and questionnaire survey in the era of big data, the evaluation problems of the reform scheme of postgraduate innovation and entrepreneurship education talent training mode are assessed based on PLTS according to the following five criteria: (1) C1: innovative knowledge cultivation; (2) C2: innovative consciousness cultivation; (3) C3: innovative ability cultivation; (4) C4: innovative quality cultivation; and (5) C5: innovative talent cultivation. Obviously, C1, C2, C3, C4, and C5 are all benefit criteria for the reform scheme evaluation. The detailed processes are given.


### 4.2. Empirical Evaluation

Step 1.
Construct the original decision matrixR with PLTSs.
In this subsection, the first step is to obtain evaluation information for the eight selected reform schemesAii=1,2,3,4,5,6,7,8 from the DMs, which can be expressed by the following additive linguistic term set: S=S−3=extremely bad,S−2=very bad,S−1=bad,S0=general,S1=good,S2=very good,S3=extremely good. Then, the original decision matrix can be constructed, as given in Table 1, provided by the DMs.Table 1
The original decision matrix.
C1C2C3C4C5A1S1 (0.67)S2 (0.33)S0 (0.33)S1 (0.33)S2 (0.33)S2 (0.67)S3 (0.33)S−1 (0.67)S0 (0.33)S2 (1)A2S2 (0.67)S3 (0.33)S1 (0.67)S2 (0.33)S1 (0.33)S2 (0.67)S1 (0.33)S2 (0.67)S2 (0.33)S3 (0.67)A3S0 (0.33)S1 (0.33)S2 (0.33)S1 (0.33)S2 (0.67)S1 (0.67)S2 (0.33)S−1 (0.33)S0(0.33)S1 (0.33)S1 (0.33)S2 (0.67)A4S2 (0.67)S3 (0.33)S−2(0.67)S−1 (0.33)S−1 (0.33)S0 (0.67)S−2 (0.33)S0(0.33)S1 (0.33)S−2 (0.67)S−1 (0.33)A5S2 (0.33)S3 (0.67)S2 (0.67)S3 (0.33)S−2 (0.67)S−1 (0.33)S−1 (0.33) S0(0.33)S1 (0.33)S0 (0.33)S1 (0.33)S2 (0.33)A6S−3 (0.33)S−2 (0.67)S−2 (1)S1 (0.33)S2 (0.67)S1 (0.33)S2 (0.67)S1 (0.67)S2 (0.33)A7S−2 (0.33)S−1 (0.67)S0 (0.67)S1 (0.33)S−2 (0.33)S−1 (0.33)S0 (0.33)S1 (0.67)S2 (0.33)S0 (0.33)S2 (0.67)A8S1 (0.33)S2 (0.67)S−1 (0.33)S0 (0.33)S1 (0.33)S1 (0.33)S2 (0.67)S1 (0.33)S2 (0.67)S0 (0.33)S1 (0.33)S2 (0.33)Step 2.
Transform the original decision matrixR into decision matrix X. The transformed decision matrix X can be obtained according to (12):(23)X=1.330.992.33−0.672.002.331.331.671.672.670.991.671.330.001.672.33−1.67−0.33−0.33−1.672.672.33−1.670.000.99−2.33−2.001.671.671.33−1.330.33−0.991.331.341.670.001.671.670.99.Step 3.
Get the standardized decision matrixA, based on the decision matrix X. Because C1, C2, C3, C4, and C5 are all benefit criteria for the reform scheme evaluation, the standardized decision matrix A can be easily obtained according to (13):(24)A=0.73200.69051.00000.00000.84560.93200.76910.83501.00001.00000.66400.84760.75000.28630.76960.93200.07620.33500.14530.00001.00001.00000.00000.28630.61290.00000.00000.83501.00000.69120.20000.53810.17000.85470.69350.80000.46190.83501.00000.6129.Step 4.
Compute the weight of each criterion, which can be obtained by the analytic hierarchy process [48].(25)W=w1,w2,w3,w4,w5=0.1251,0.1092,0.2293,0.2625,0.2739.Step 5.
Compute the preference index. The preference index can be computed according to (14), given in Table 2.Table 2
The preference index.
AaAbA1A2A3A4A5A6A7A8A1—0.03780.08670.45120.29300.24710.31520.1265A20.3384—0.30350.68860.48480.28510.39140.1561A30.09230.0086—0.42720.21490.19710.24570.0850A40.06320.00000.0335—0.07680.12490.12940.0165A50.14250.03370.05870.3143—0.23430.15050.0838A60.26250.00000.20680.52830.4003—0.19060.0215A70.22440.00000.14920.42660.21030.0844—0.0304A80.27100.00000.22380.54900.37880.15050.2657—Step 6.
Calculate the positive outranking flowϕ+Aa and the negative outranking flow ϕ−Aa. The positive outranking flow ϕ+Aa and the negative outranking flow ϕ−Aa can be obtained according to (15) and (16). The results are given in Table 3.Table 3
The positive outranking flow, the negative outranking flow, and the net outranking flow.
A1A2A3A4A5A6A7A8ϕ+Aa1.55752.64781.27070.44431.01781.61001.12531.8389ϕ−Aa1.39420.08011.06223.38522.05891.32351.68840.5198ϕAa0.16332.56770.2085−2.9409−1.04120.2866−0.56321.3190


### 4.3. Empirical Results

#### 4.3.1. Probabilistic Linguistic PROMETHEE I Method

According to the three conditions for judging the priority of the scheme in the probabilistic linguistic environment, given in (18)–(20), the probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship according to the negative outranking flow and positive outranking flow. The priority rank of all the schemes, produced by the probabilistic linguistic PROMETHEE I method, is given in Table 4.Table 4
The priority rank produced by probabilistic linguistic PROMETHEE I.
A1A2A3A4A5A6A7A8Priority rankA1PA4A2PA1A3PA4—A5PA4A6PA1A7PA4A8PA1A1PA5A2PA3A3PA5A6PA4A7PA5A8PA3A1PA7A2PA4A3PA7A6PA5A8PA4A2PA5A6PA7A8PA5A2PA6A8PA6A2PA7A8PA7A2PA8In order to facilitate intuitive analysis, the results for visual analysis, produced by the probabilistic linguistic PROMETHEE I method, are shown in Figure1.Figure 1
The ranking results produced by the probabilistic linguistic PROMETHEE I method.From Figure1, it is obvious that the ranking results of the eight schemes are A2≻A8≻A6≻A1≻A7≻A5≻A4 and A2≻A8≻A3≻A7≻A5≻A4. However, A6 and A1 cannot be compared with A3. These are incomparable situations, denoted as A6RA3 and A1RA3, respectively, which further illustrates that the probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship.


#### 4.3.2. Probabilistic Linguistic PROMETHEE II Method

According to the two conditions for judging the priority of the scheme in the probabilistic linguistic environment, given in (21) and (22), the probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow. The results of ϕAa, computed by (17), are given in Table 3. The higher the net outranking ﬂow ϕAa, the better the alternative. Finally, the priority rank of all the schemes, produced by the probabilistic linguistic PROMETHEE II method, is given in Table 5.Table 5
The priority rank produced by probabilistic linguistic PROMETHEE II.
SchemeϕAaRankA10.16335A22.56771A30.20854A4−2.94098A5−1.04127A60.28663A7−0.56326A81.31902In order to facilitate intuitive analysis, the results for visual analysis, produced by the probabilistic linguistic PROMETHEE II method, are shown in Figure2.Figure 2
The ranking results produced by the probabilistic linguistic PROMETHEE II method.From Figure2, it is obvious that the ranking results of the eight schemes are A2≻A8≻A6≻A3≻A1≻A7≻A5≻A4, which further illustrates that the probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow.From Figures1 and 2, we can see that the best ranking scheme is A2, the worst ranking scheme is A4, and the overall ranking trend is not much different, between the PROMETHEE I and II methods, which verifies the effectiveness and feasibility of probabilistic linguistic PROMETHEE I and II methods for the reform scheme evaluation of postgraduate innovation and entrepreneurship education talent training mode under the big data environment.


### 4.4. Further Discussion and Comparative Analysis

For the purpose of comparative analysis to further illustrate the effectiveness of the extended methods, the TOPSIS method [52], one of the classic MCDM methods, with the PLTSs is utilized for a comparative study. The specific processes are as follows:Step 1: construct the original decision matrixR with PLTSsStep 2: transform the original decision matrixR into decision matrix XStep 3: get the standardized decision matrixA, based on the decision matrix XStep 4: compute the weight of each criterion by AHPStep 5: calculate the weighted standardized decision matrixStep 6: calculate the distance from each scheme to the positive ideal solution and the negative ideal solutionStep 7: obtain the relative closeness of each schemeStep 8: sort the schemesAccording to the above steps, through simple calculation, the relative closeness of each scheme can be obtained easily. The results are given in Table6.Table 6
The relative closeness of each scheme by TOPSIS with PLTSs.
SchemeRelative closenessRankA10.56225A20.90651A30.59414A40.26678A50.44077A60.66423A70.55906A80.74722From Table6, it is obvious that the ranking results of all the schemes are 5, 1, 4, 8, 7, 3, 6, 2. That is to say, the priority rank of the eight schemes is A2≻A8≻A6≻A3≻A1≻A7≻A5≻A4, which is the same as the results by the probabilistic linguistic PROMETHEE II method. These research results further reveal and verify the good effectiveness and feasibility of the probabilistic linguistic PROMETHEE II method for reform scheme evaluation. The research results also indicate that the PLTS has good effectiveness and feasibility, as it can accurately depict qualitative information about the index data of the reform scheme effect. In addition, the results of the probabilistic linguistic PROMETHEE II method are not much different from the results of the probabilistic linguistic PROMETHEE I method. The best ranking scheme is A2, and the worst ranking scheme is A4. The research results further indicate the good effectiveness of the extended probabilistic linguistic PROMETHEE I method for reform scheme evaluation.Besides, the research on the reform scheme of the talent-training mode of postgraduate innovation and entrepreneurship education is still in the stage of theoretical discussion, lacking empirical research, and evaluation research on the reform effect is relatively limited. Therefore, this study is of great significance for exploring and evaluating the reform scheme of talent training mode of postgraduate innovation and entrepreneurship education. Moreover, the research findings for reform scheme evaluation by comparative analysis obtained in this paper can improve decision quality, especially in a complex fuzzy and uncertain decision-making environment.


## 4.1. Datasets

Mass entrepreneurship and innovation have become a national development strategy for China’s economy. Simultaneously, China’s higher education has gradually entered the stage of “popular education” from “elite education.” In the big data environment, big data gives graduate students new opportunities and challenges for innovation and entrepreneurship. Moreover, big data have become increasingly used in the field of education, which not only brings greater development space but also poses unprecedented challenges to educational researchers. Talent training quality is a key index of education quality in universities, and it is an important field of higher education research. Transforming the educational concept, innovating the talent-training mode, and deepening the educational reform are of great significance for improving the talent training quality of postgraduate innovation and entrepreneurship education in universities. Furthermore, due to the uncertainty and fuzziness of the information environment, the decision-making process becomes more and more complex, which brings great difficulties and challenges to scientific decision-making. In this study, based on PLTS, the probabilistic linguistic PROMETHEE I and II methods are extended to accurately depict the uncertainty and fuzziness of the information involved in the reform scheme evaluation under the big data environment.An example is given to verify the effectiveness of the extended probabilistic linguistic PROMETHEE I and II methods for reform scheme evaluation. Eight reform schemesAii=1,2,3,4,5,6,7,8 are selected and combined with literature research and questionnaire survey in the era of big data, the evaluation problems of the reform scheme of postgraduate innovation and entrepreneurship education talent training mode are assessed based on PLTS according to the following five criteria: (1) C1: innovative knowledge cultivation; (2) C2: innovative consciousness cultivation; (3) C3: innovative ability cultivation; (4) C4: innovative quality cultivation; and (5) C5: innovative talent cultivation. Obviously, C1, C2, C3, C4, and C5 are all benefit criteria for the reform scheme evaluation. The detailed processes are given.


## 4.2. Empirical Evaluation

Step 1.
Construct the original decision matrixR with PLTSs.
In this subsection, the first step is to obtain evaluation information for the eight selected reform schemesAii=1,2,3,4,5,6,7,8 from the DMs, which can be expressed by the following additive linguistic term set: S=S−3=extremely bad,S−2=very bad,S−1=bad,S0=general,S1=good,S2=very good,S3=extremely good. Then, the original decision matrix can be constructed, as given in Table 1, provided by the DMs.Table 1
The original decision matrix.
C1C2C3C4C5A1S1 (0.67)S2 (0.33)S0 (0.33)S1 (0.33)S2 (0.33)S2 (0.67)S3 (0.33)S−1 (0.67)S0 (0.33)S2 (1)A2S2 (0.67)S3 (0.33)S1 (0.67)S2 (0.33)S1 (0.33)S2 (0.67)S1 (0.33)S2 (0.67)S2 (0.33)S3 (0.67)A3S0 (0.33)S1 (0.33)S2 (0.33)S1 (0.33)S2 (0.67)S1 (0.67)S2 (0.33)S−1 (0.33)S0(0.33)S1 (0.33)S1 (0.33)S2 (0.67)A4S2 (0.67)S3 (0.33)S−2(0.67)S−1 (0.33)S−1 (0.33)S0 (0.67)S−2 (0.33)S0(0.33)S1 (0.33)S−2 (0.67)S−1 (0.33)A5S2 (0.33)S3 (0.67)S2 (0.67)S3 (0.33)S−2 (0.67)S−1 (0.33)S−1 (0.33) S0(0.33)S1 (0.33)S0 (0.33)S1 (0.33)S2 (0.33)A6S−3 (0.33)S−2 (0.67)S−2 (1)S1 (0.33)S2 (0.67)S1 (0.33)S2 (0.67)S1 (0.67)S2 (0.33)A7S−2 (0.33)S−1 (0.67)S0 (0.67)S1 (0.33)S−2 (0.33)S−1 (0.33)S0 (0.33)S1 (0.67)S2 (0.33)S0 (0.33)S2 (0.67)A8S1 (0.33)S2 (0.67)S−1 (0.33)S0 (0.33)S1 (0.33)S1 (0.33)S2 (0.67)S1 (0.33)S2 (0.67)S0 (0.33)S1 (0.33)S2 (0.33)Step 2.
Transform the original decision matrixR into decision matrix X. The transformed decision matrix X can be obtained according to (12):(23)X=1.330.992.33−0.672.002.331.331.671.672.670.991.671.330.001.672.33−1.67−0.33−0.33−1.672.672.33−1.670.000.99−2.33−2.001.671.671.33−1.330.33−0.991.331.341.670.001.671.670.99.Step 3.
Get the standardized decision matrixA, based on the decision matrix X. Because C1, C2, C3, C4, and C5 are all benefit criteria for the reform scheme evaluation, the standardized decision matrix A can be easily obtained according to (13):(24)A=0.73200.69051.00000.00000.84560.93200.76910.83501.00001.00000.66400.84760.75000.28630.76960.93200.07620.33500.14530.00001.00001.00000.00000.28630.61290.00000.00000.83501.00000.69120.20000.53810.17000.85470.69350.80000.46190.83501.00000.6129.Step 4.
Compute the weight of each criterion, which can be obtained by the analytic hierarchy process [48].(25)W=w1,w2,w3,w4,w5=0.1251,0.1092,0.2293,0.2625,0.2739.Step 5.
Compute the preference index. The preference index can be computed according to (14), given in Table 2.Table 2
The preference index.
AaAbA1A2A3A4A5A6A7A8A1—0.03780.08670.45120.29300.24710.31520.1265A20.3384—0.30350.68860.48480.28510.39140.1561A30.09230.0086—0.42720.21490.19710.24570.0850A40.06320.00000.0335—0.07680.12490.12940.0165A50.14250.03370.05870.3143—0.23430.15050.0838A60.26250.00000.20680.52830.4003—0.19060.0215A70.22440.00000.14920.42660.21030.0844—0.0304A80.27100.00000.22380.54900.37880.15050.2657—Step 6.
Calculate the positive outranking flowϕ+Aa and the negative outranking flow ϕ−Aa. The positive outranking flow ϕ+Aa and the negative outranking flow ϕ−Aa can be obtained according to (15) and (16). The results are given in Table 3.Table 3
The positive outranking flow, the negative outranking flow, and the net outranking flow.
A1A2A3A4A5A6A7A8ϕ+Aa1.55752.64781.27070.44431.01781.61001.12531.8389ϕ−Aa1.39420.08011.06223.38522.05891.32351.68840.5198ϕAa0.16332.56770.2085−2.9409−1.04120.2866−0.56321.3190


## 4.3. Empirical Results

### 4.3.1. Probabilistic Linguistic PROMETHEE I Method

According to the three conditions for judging the priority of the scheme in the probabilistic linguistic environment, given in (18)–(20), the probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship according to the negative outranking flow and positive outranking flow. The priority rank of all the schemes, produced by the probabilistic linguistic PROMETHEE I method, is given in Table 4.Table 4
The priority rank produced by probabilistic linguistic PROMETHEE I.
A1A2A3A4A5A6A7A8Priority rankA1PA4A2PA1A3PA4—A5PA4A6PA1A7PA4A8PA1A1PA5A2PA3A3PA5A6PA4A7PA5A8PA3A1PA7A2PA4A3PA7A6PA5A8PA4A2PA5A6PA7A8PA5A2PA6A8PA6A2PA7A8PA7A2PA8In order to facilitate intuitive analysis, the results for visual analysis, produced by the probabilistic linguistic PROMETHEE I method, are shown in Figure1.Figure 1
The ranking results produced by the probabilistic linguistic PROMETHEE I method.From Figure1, it is obvious that the ranking results of the eight schemes are A2≻A8≻A6≻A1≻A7≻A5≻A4 and A2≻A8≻A3≻A7≻A5≻A4. However, A6 and A1 cannot be compared with A3. These are incomparable situations, denoted as A6RA3 and A1RA3, respectively, which further illustrates that the probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship.


### 4.3.2. Probabilistic Linguistic PROMETHEE II Method

According to the two conditions for judging the priority of the scheme in the probabilistic linguistic environment, given in (21) and (22), the probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow. The results of ϕAa, computed by (17), are given in Table 3. The higher the net outranking ﬂow ϕAa, the better the alternative. Finally, the priority rank of all the schemes, produced by the probabilistic linguistic PROMETHEE II method, is given in Table 5.Table 5
The priority rank produced by probabilistic linguistic PROMETHEE II.
SchemeϕAaRankA10.16335A22.56771A30.20854A4−2.94098A5−1.04127A60.28663A7−0.56326A81.31902In order to facilitate intuitive analysis, the results for visual analysis, produced by the probabilistic linguistic PROMETHEE II method, are shown in Figure2.Figure 2
The ranking results produced by the probabilistic linguistic PROMETHEE II method.From Figure2, it is obvious that the ranking results of the eight schemes are A2≻A8≻A6≻A3≻A1≻A7≻A5≻A4, which further illustrates that the probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow.From Figures1 and 2, we can see that the best ranking scheme is A2, the worst ranking scheme is A4, and the overall ranking trend is not much different, between the PROMETHEE I and II methods, which verifies the effectiveness and feasibility of probabilistic linguistic PROMETHEE I and II methods for the reform scheme evaluation of postgraduate innovation and entrepreneurship education talent training mode under the big data environment.


## 4.3.1. Probabilistic Linguistic PROMETHEE I Method

According to the three conditions for judging the priority of the scheme in the probabilistic linguistic environment, given in (18)–(20), the probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship according to the negative outranking flow and positive outranking flow. The priority rank of all the schemes, produced by the probabilistic linguistic PROMETHEE I method, is given in Table 4.Table 4
The priority rank produced by probabilistic linguistic PROMETHEE I.
A1A2A3A4A5A6A7A8Priority rankA1PA4A2PA1A3PA4—A5PA4A6PA1A7PA4A8PA1A1PA5A2PA3A3PA5A6PA4A7PA5A8PA3A1PA7A2PA4A3PA7A6PA5A8PA4A2PA5A6PA7A8PA5A2PA6A8PA6A2PA7A8PA7A2PA8In order to facilitate intuitive analysis, the results for visual analysis, produced by the probabilistic linguistic PROMETHEE I method, are shown in Figure1.Figure 1
The ranking results produced by the probabilistic linguistic PROMETHEE I method.From Figure1, it is obvious that the ranking results of the eight schemes are A2≻A8≻A6≻A1≻A7≻A5≻A4 and A2≻A8≻A3≻A7≻A5≻A4. However, A6 and A1 cannot be compared with A3. These are incomparable situations, denoted as A6RA3 and A1RA3, respectively, which further illustrates that the probabilistic linguistic PROMETHEE I method can obtain a partial priority relationship.


## 4.3.2. Probabilistic Linguistic PROMETHEE II Method

According to the two conditions for judging the priority of the scheme in the probabilistic linguistic environment, given in (21) and (22), the probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow. The results of ϕAa, computed by (17), are given in Table 3. The higher the net outranking ﬂow ϕAa, the better the alternative. Finally, the priority rank of all the schemes, produced by the probabilistic linguistic PROMETHEE II method, is given in Table 5.Table 5
The priority rank produced by probabilistic linguistic PROMETHEE II.
SchemeϕAaRankA10.16335A22.56771A30.20854A4−2.94098A5−1.04127A60.28663A7−0.56326A81.31902In order to facilitate intuitive analysis, the results for visual analysis, produced by the probabilistic linguistic PROMETHEE II method, are shown in Figure2.Figure 2
The ranking results produced by the probabilistic linguistic PROMETHEE II method.From Figure2, it is obvious that the ranking results of the eight schemes are A2≻A8≻A6≻A3≻A1≻A7≻A5≻A4, which further illustrates that the probabilistic linguistic PROMETHEE II method can get the complete priority relationship according to the net outranking flow.From Figures1 and 2, we can see that the best ranking scheme is A2, the worst ranking scheme is A4, and the overall ranking trend is not much different, between the PROMETHEE I and II methods, which verifies the effectiveness and feasibility of probabilistic linguistic PROMETHEE I and II methods for the reform scheme evaluation of postgraduate innovation and entrepreneurship education talent training mode under the big data environment.


## 4.4. Further Discussion and Comparative Analysis

For the purpose of comparative analysis to further illustrate the effectiveness of the extended methods, the TOPSIS method [52], one of the classic MCDM methods, with the PLTSs is utilized for a comparative study. The specific processes are as follows:Step 1: construct the original decision matrixR with PLTSsStep 2: transform the original decision matrixR into decision matrix XStep 3: get the standardized decision matrixA, based on the decision matrix XStep 4: compute the weight of each criterion by AHPStep 5: calculate the weighted standardized decision matrixStep 6: calculate the distance from each scheme to the positive ideal solution and the negative ideal solutionStep 7: obtain the relative closeness of each schemeStep 8: sort the schemesAccording to the above steps, through simple calculation, the relative closeness of each scheme can be obtained easily. The results are given in Table6.Table 6
The relative closeness of each scheme by TOPSIS with PLTSs.
SchemeRelative closenessRankA10.56225A20.90651A30.59414A40.26678A50.44077A60.66423A70.55906A80.74722From Table6, it is obvious that the ranking results of all the schemes are 5, 1, 4, 8, 7, 3, 6, 2. That is to say, the priority rank of the eight schemes is A2≻A8≻A6≻A3≻A1≻A7≻A5≻A4, which is the same as the results by the probabilistic linguistic PROMETHEE II method. These research results further reveal and verify the good effectiveness and feasibility of the probabilistic linguistic PROMETHEE II method for reform scheme evaluation. The research results also indicate that the PLTS has good effectiveness and feasibility, as it can accurately depict qualitative information about the index data of the reform scheme effect. In addition, the results of the probabilistic linguistic PROMETHEE II method are not much different from the results of the probabilistic linguistic PROMETHEE I method. The best ranking scheme is A2, and the worst ranking scheme is A4. The research results further indicate the good effectiveness of the extended probabilistic linguistic PROMETHEE I method for reform scheme evaluation.Besides, the research on the reform scheme of the talent-training mode of postgraduate innovation and entrepreneurship education is still in the stage of theoretical discussion, lacking empirical research, and evaluation research on the reform effect is relatively limited. Therefore, this study is of great significance for exploring and evaluating the reform scheme of talent training mode of postgraduate innovation and entrepreneurship education. Moreover, the research findings for reform scheme evaluation by comparative analysis obtained in this paper can improve decision quality, especially in a complex fuzzy and uncertain decision-making environment.


## 5. Conclusion

Strengthening postgraduate innovation and entrepreneurship education and talent cultivation are key practical requirements for universities to meet in order to serve the country by helping to change the mode of economic development and building an innovative country. Additionally, in the field of education, big data will inevitably become a cutting-edge research hotspot involving educational researchers all over the world.Studies on the reform scheme of talent training mode of postgraduate innovation and entrepreneurship education are largely theoretical and lacking empirical research. Evaluation research on the effect of the talent training mode reform scheme is relatively limited. Therefore, this paper aims to propose effective methods for the reform scheme evaluation of postgraduate innovation and entrepreneurship education talent training mode under the big data environment. The main work of the paper is as follows:(1)
Two effective evaluation methods, the probabilistic linguistic PROMETHEE I and II methods, are presented to assess the reform scheme of postgraduate innovation and entrepreneurship education talent training mode under the big data environment.(2)
The PROMETHEE I and II methods are extended with PLTS for reform scheme evaluation and are shown to the advantages of good effectiveness and feasibility(3)
PLTS is imported into the evaluation methods, which can accurately express and quantitatively evaluating the reform scheme effect(4)
A case study is carried out and comparative analysis is conducted to verify the extended methods(5)
According to the comparative study and visual analysis, the extended methods to reform scheme evaluation can improve decision quality for education departments, universities, and relevant teachers to guide the reform of postgraduate innovation and entrepreneurship education talent training model, especially in the complex fuzzy and uncertain decision-making environment.In future work, the reform strategy and countermeasure implementation of postgraduate innovation and entrepreneurship education talent training mode under the big data environment will be further developed by using technology of hesitant fuzzy sets [53], probabilistic hesitant fuzzy sets [54], MCDM, machine learning, data mining, artificial intelligence, and big data to reduce the constraints of small samples for large-scale real data analysis in the complex fuzzy and uncertain decision-making environment.



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*Source: 8341052-2022-08-17.xml*