# Statistical Measurement and Influencing Factors of Green Total Factor Productivity of China’s Tourism Industry Based on DEA-EBM Model

Authors: Zikun Xiang; Shirui Che; Sijia Zhou
Journal: Mobile Information Systems
 (2022)

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## Abstract

In order to explore the statistical measurement and influencing factors of green total factor productivity in tourism, this paper proposes the use of gray absolute correlation to measure the similarity between input variable sequences. Moreover, this paper obtains the key parameters of the combination of radial and nonradial in the model based on the proximity index of gray correlation and calculates the economic efficiency of the decision-making unit according to the steps. In addition, this paper combines the DEA-EBM model to carry out the statistical measurement of China’s tourism industry green total factor productivity and the analysis of influencing factors and verify it through data. Through research, it can be seen that the DEA-EBM model proposed in this paper can play an important role in the statistical measurement and analysis of influencing factors of green total factor production in the tourism industry.


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## Body

## 1. Introduction

Tourism is a comprehensive and linked industry. In a society with diverse cultures and ethnic diversity, tourism builds bridges for human communication and promotes mutual understanding between different races and different cultures. At the same time, there are a series of social and economic activities in the process of tourism consumption, which promote the allocation and circulation of people, materials, and funds [1]. According to relevant data from the United Nations World Tourism Organization (UNWTO), currently, tourism accounts for 10% of global GDP, 7% of world exports, and 30% of labor exports. Moreover, modern tourism is a new economy, a comprehensive industry that integrates the primary, secondary, and tertiary industries, which not only promotes the appreciation of agricultural products but also drives the development of industrial products that meet consumer needs [2]. UNWTO’s research shows that the contribution of the entire tourism sector to global warming caused by human factors has reached about 5%–14% [3]. This shows that tourism is closely related to global climate change.From the macrolevel of the entire economic field, efficiency refers to the rational allocation of resources invested so as to maximize the total surplus value that all members of society can obtain. It is a specific indicator that reflects the research object’s ability to use resources and the results obtained. For any economic entity, a high level of efficiency is an important prerequisite for its sustainable development. The issue of efficiency has become the core issue in the current economic development, and it is also a hot spot of domestic research.This paper combines the EA-EBM model to carry out the statistical measurement and analysis of the influencing factors of the green total factor productivity of China’s tourism industry, which provides a theoretical reference for the subsequent development of green tourism.


## 2. Related Work

Literature [4] studied the main source markets and compared single and multivariable models. From the model research results, it can be seen that the prediction models showed different research results in the main source markets. Literature [5] pointed out that there are many factors that influence tourism demand, and they are a series of factors and not a single factor. Literature [6] pointed out that different countries and regions have differences in prices or income, and their impact on international tourism demand is also different. Literature [7] believes that the economic development of a region or a city is too dependent on the international economic market, which may have a reactionary force on the economic development of the city and affect the development of urban tourism. Literature [8] studied the main reasons that affect rural tourism demand and found that the economic impact was limited and most rural consumers spent their expenditures on food and outdoor activities. Literature [9] conducted an in-depth investigation and exploration of the structural factors of tourism demand after readjusting the gravity model. Literature [10] used hotel overnight tourists as the research object, combined with the tourism demand model and artificial neural network model, carried out a specific analysis of the case, and proposed a prediction model to guide interested tourism suppliers to provide better product supply to meet the needs of tourists. Literature [11] uses a novel evolutionary negative correlation combined with the LSPME model to study tourism demand estimation, compares the LSPME model with other integrated models, and believes that the estimation accuracy of the LSPME model is significantly better than other integrated models. Literature [12] proposes to select a tourism demand model for measurement based on actual conditions. If the independent variable is known, it is best to use a neural network prediction model. When the independent variable is only a fuzzy value, the regression model analysis is more accurate. If the independent variable value is not known, the time series prediction model should be used. Literature [13] believes that there are many influencing factors in the generation of tourism demand, among which the most important factors are the price of tourism products, transportation, and personal disposable income. It also believes that economic growth can stimulate more tourism demand. Literature [14] proposed the concept of tourism compound cost, which included time, income, and other aspects into the model for consideration, and took the travel rate and tourism compound cost as the main measurement indicators so as to establish the tourism demand function for tourism demand and conduct calculation research. At present, domestic scholars have also begun to try to use single or combined forecasting methods to achieve forecasting goals. Literature [15] used the average absolute error and average absolute percentage error to construct multiple mathematical calculation models.Literature [16] studied the supply of tourism through qualitative and quantitative methods and concluded that the management of water resources should be strengthened in tourist destinations where water is scarce. From the perspective of supply, literature [17] believes that the tourism supply chain includes not only direct suppliers but also indirect suppliers that provide tourism products and services to meet the needs of tourists; literature [18] puts forward an in-depth analysis of the process of tourism activities. Tourism supply chain is a tourism business network formed by multiple stakeholders, which involves various private enterprises, public departments, large enterprises, and so on. Literature [19] believes that the government should undertake the construction of basic public facilities in tourist destinations. However, the construction of tourism public service system has the characteristics of public welfare, so the profit is too small and may cause some enterprises to provide poor or even unqualified tourism products.


## 3. DEA-EBM Cross-Efficiency Model for Green Tourism considering Singularity and Competition and Cooperation

In DEA-EBM, due to the quantitative relationship between the input and output of the decision-making unit and the internal correlation in the evaluation to a certain extent, the rationality of the specific efficiency evaluation method has attracted more attention.We assume that there aren DMUs, and each DMU has m inputs and s outputs. xiji=1,2,...,m represents the i-th input of unit DMUjj=1,2,...,s, yrjr=1,2,...,s represents the r-th output. For the j-th decision-making unit, its efficiency can be defined as [20](1)θj=∑r=1suryrj∑i=1mvrxijj=1,2,...,n.Among them,vii=1,...,m and urr=1,...,s represent the weight of input and output.Considering that the decision-making unit isDMUd,d∈1,2,...,n, the efficiency of this unit relative to other DMUs is called the CCR model, as shown in(2)maxEdd=∑r=1surdyrd∑i=1mvidxid,s.t.Edd=∑r=1surdyrd∑i=1mvidxid≤1j=1,2,...,n,urd≥0r=1,2,...,s,vrd≥0i=1,2,...,m.The goal of the plan is to find a set of input and output weights that are most beneficial toDMUd.By using the Charnes-Cooper transformation, model (2) can be converted into an equivalent linear program to solve, as shown in [21](3)max∑r=1suryrd=Edd,s.t.∑i=1mωixij−∑r=1suryrj≥0j=1,2,...,n,∑i=1mωixij=1,ωi≥0i=1,2,...,m,ur≥0r=1,2,...,s.Among them,Edd is the CCR efficiency of the decision-making unit Ud; that is, the optimal efficiency that the unit can achieve, reflecting the self-evaluation of the unit. The optimal weight coefficient of each decision unit DMUd,d=1,2,...,n is ωd∗=ω1d∗,...,ωmd∗,μ1d∗,...,μsd∗. Since each unit can get its own set of ωd∗, the efficiency of other units can be calculated according to co, which is called cross efficiency. Sexton et al. defined the crossover efficiency obtained by DMUp using Ud’s weight as(4)Edp=∑r=1surd∗yrp∑r=1sωrd∗xip≤1,d,p=1,2,...,n,d≠p.This model reflects the evaluation results ofDMUd on DMUp. Model (3) needs to be run n times to solve the efficiency of each DMU. Therefore, for n DMUs, n sets of input-output weights will be obtained, forming n-1 sets of cross efficiency and a CCR efficiency, thereby forming a cross-efficiency matrix (CEM), as shown in Table 1 [22].Table 1
Cross-efficiency matrix forn units.
Target DMUDMUCross-eff1 (a1)2 (a2)1 (a1)2 (a2)1 (a1)1E11E121E11E121E112E21E222E21E222E21……………………PE1pE2pPE1pE2pPE1p……………………nEn1En2nEn1En2nEn1Model (3) may have multiple sets of optimal solutions; that is, the nonuniqueness of input and output weights will destroy the use of cross-efficiency evaluation. In order to solve this problem, Sexton proposed a set of quadratic goals to optimize the input and output weights while ensuring the efficiency of model (3). It uses an aggressive strategy to calculate the cross efficiency. The aggressive cross efficiency of the decision-making unitp based on d is shown in model (5), and the efficiency value is E¯dp. The benevolent cross efficiency of decision-making unit p based on d is shown in model (6), and the efficiency value is E¯dp.(5)Minimize∑r=1surd∑p=1,j≠dnyrp,s.t.∑i=1mvid∑p=1,p≠dnxrp=1,∑r=1surdyrp−Edd∑i=1mvidxrp=0,∑r=1surdyrp−∑i=1mvidxrp≤0p=1,2,...,n,vid≥0i=1,2,...,m,urd≥0r=1,2,...,s.Maximize∑r=1surd∑p=1,p≠dnyrp,s.t.∑i=1mvid∑p=1,p≠dnxrp=1,∑r=1surdyrp−Edd∑i=1mvidxrp=0,∑r=1surdyrp−∑i=1mvidxrp≤0p=1,2,...,n,vid≥0i=1,2,...,m,urd≥0r=1,2,...,s.The above two models ensure the worst and the best overall efficiency of other units while maintaining theDMUd efficiency score.In addition, DEA-EBM also includes some other common secondary target models:(a)
Minimize∑r=1surd∑j=1,j≠dnyrk−∑i=1mvid∑j=1,j≠dnxij.(b)
It replacesMinimize1/n∑αj′−α¯′ with ∑r=1surdyrj−αd∑i=1mvidxrj≥0 in model (5). Among them, αd∈min1≤j≤nEjj,1 is a parameter that controls the range of DMUd efficiency scores.(c)
MinMaximizeαj′ or Minimize1/n∑j=1nαj′−α¯′(6)s.t.∑i=1mvidxid=1,Edd=∑r=1surdyrd,∑r=1surdyrd−∑i=1mvidxid+αjr=0j=1,2,...,n.Among them,α¯′=1/n∑j=1nαj′.Models (4) to (6) give the general DEA cross-efficiency calculation method, and model (5) treats all decision-making units as hostile situations. However, model (6) regards all decision-making units as Allies. They simply regard all decision-making units as alliance or hostile relations.The mutual information of discrete and continuous variables can be expressed as follows:Continuous type:IX,Y=∬px,ylog2px,y/pxpydxdy.Discrete type:IX,Y=∑∑px,ylog2px,y/pxpy.Among them,p(x, y) represents the joint probability distribution of random variable X and random variable Y. It can be seen from the above two forms that when the random variable X and the random variable Y are independent of each other, the mutual information I(X,Y) = 0.We set a sample setU=x1,x2,...,xn,xi∈RN. Δ is the distance function defined on U, satisfying Δxi,xj≥0. In applications, 2-norm distance (also known as Euclidean distance) is often used: ∑k=1Nxik−xjk21/2. We assume that δ≥0, and the neighborhood of labeled sample xi is δx=xi|Δxi,xj≤δ. At the same time, we are given two feature spaces R and S, and δRx and δSx are, respectively, the neighborhood of X calculated based on the distance in these feature spaces. The neighborhood has the following attributes: δR∩Sx=δRx∩δSx. In addition to the distance function given above, there are many ways to measure the distance between heterogeneous features and missing data. Definition 1 and Definition 2 refer to Huetal’s description.Definition 1.
We are given a sample setU=x1,x2,...,xn, which is described by a numerical or discrete R,S⊆F attribute subset. The neighborhood of sample xi on attribute S can be denoted as δSxi. The neighborhood uncertainty of the sample can be defined as(7)NHδxiS=−logδSxin.
Moreover, the average uncertainty of the sample set can be defined as(8)NHδS=−1n∑i=1nlogδSxin.Definition 2.
R,S⊆F is two attribute subsets, and the neighborhood mutual information defined based on R and S is(9)NMIδR;S=−1n∑i=1nlogδRxi⋅δSxinδS∪Rxi.
If we give two attribute subsetsR and S and NMIδR;S is the mutual information between these two attribute subsets, then the following equation holds:(1)
NMIδR;S=NMIδS;R(2)
NMIδR;S=NHδR+NHδS−NHδR,S(3)
NMIδR;S=NHδR+NHδS−NHδR,S
We assume that there aren decision-making units to be evaluated, and each decision-making unit has m inputs and s outputs. Moreover, we use xij and yrj to represent the i (i = 1,2, …, m) input and r (r = 1,2, …, s) output of the j-th decision-making unit, respectively.
eJ represents the average other-evaluation efficiency score of decision-making unit j, and hjj represents the self-evaluation optimal efficiency score of the model. hkj represents the evaluation value of the decision unit j when the decision unit k is optimized; that is, the decision unit j is evaluated with the optimal weight of the decision unit k. MJ represents the singularity index. Then, the formula for calculating the singular index is(10)Mj=hjj−ejej,ej=∑k≠jhkjn−1.
WhatMJ expresses is the difference and changes in the efficiency of the decision-making unit between other- and self-evaluation. The larger hjj, the smaller eJ. The larger MJ is, the easier the decision-making unit j is to be regarded as a singular person; that is, the decision-making unit j is pseudoeffective. The smaller MJ is, the closer the self-evaluation value is to his evaluation value, and the easier the final evaluation value of decision-making unit j is to be accepted.
Under the optimistic (optimal) frontier, the DEA-EBM model pursues the maximization of efficiency to determine the input and output weights. The optimistic efficiency of thek-th decision unit can be obtained by solving (11)θkk=max∑r=1surkyrk,s.t.∑i=1mvikxik=1,Edd=∑r=1surdyrd,∑r=1surdyrd−∑i=1mvidxid+αjr=0j=1,2,...,n.
Among them,vik and urk represent the weights of the i-th input and r-th output of the k-th (k = 1,2, …, n) decision-making unit, and the optimistic efficiency value of the k-th decision-making unit is within θ∈0,1. If θkk=1, the k-th decision-making unit is called optimistic and effective; otherwise, the decision-making unit is called optimistic and noneffective.
Under the pessimistic (worst) frontier, the DEA-EBM model pursues the minimization of efficiency to determine the input-output weight. The pessimistic efficiency of thek-th decision unit can be obtained by solving (12)ϕkk=min∑r=1surkyrk,s.t.∑i=1mvikxik=1,∑r=1surkyrj−∑i=1mvikxij≥0,j=1,2,...,n,urk≥0,vik≥0,r=1,2,...,s,i=1,2,...,m.
In the above formula, the pessimistic efficiency of thek-th decision-making unit is ϕkk≥1. If the force >1, the decision-making unit k is called pessimistic and invalid; otherwise, the decision-making unit is called pessimistic noninvalid.
Decision-making units are each other’s Allies, and each unit between classes is each other’s opponents. Then, we assume that thek-th decision-making unit is in the Tt category, that is, k∈Tt, and use the decision-making unit k (k = 1,2, …, n) to evaluate other decision-making units separately, so that we try to make the efficiency of the units belonging to the Tt category as high as possible, and vice versa, as low as possible. Finally, we use the arithmetic average method to fuse self-evaluation and other-evaluation efficiency to obtain the overall crossover efficiency of competition and cooperation. The model is shown in(13)min∑j≠k,j∈TtskjCP−∑jj∈TtskjCP,s.t.∑r=1surkyrj−∑i=1mvikxij+skjCP=0,j=1,2,...,n,j≠k,∑r=1surkyrk−θkk∗∑i=1mvikxik=0,skjCP≥0,j=1,2,...,m,j≠k,urk≥0,vik≥0,r=1,2,...,s,i=1,2,...,m.
Model (13) embodies the competitive and cooperative relationship between decision-making units, enabling the decision-making unit to establish a competitive or cooperative relationship with other units based on personal preference.
This paper first uses the bounded DEA-EBM model to provide an efficiency interval for each decision-making unit. The model is shown in(14)ψkk=maxmin∑r=1surkyrk,s.t.∑i=1mvikxik=1,∑r=1surkyrj−∑i=1mvikxij≤0,j=1,2,...,n,∑r=1surkyrj−α∑i=1mvikxij≥0,j=1,2,...,n,urk≥0,vik≥0,r=1,2,...,s,i=1,2,...,m.
Among them,α=maxθ∗/minφ∗0<α≤1 represents the adjustment coefficient of pessimistic efficiency, and θ∗ and φ∗ are obtained by models (11) and (12), respectively. Due to the inconsistency of the two efficiency scores, the values of optimistic efficiency and pessimistic efficiency can be adjusted to be within the interval α,1. We use model (14) to evaluate the k-th decision-making unit, and the maximum value of the objective function is ψkkU∗, which shows the best relative efficiency of the decision-making unit. The minimum value of the objective function is ψkkL∗, which represents the worst relative efficiency of the decision-making unit. The two together constitute the efficiency interval ψkkL∗,ψkkU∗ of the k-th decision-making unit.
Based on model (13), we construct a new cross-efficiency model (15). By ensuring that the evaluation efficiency of the evaluated decision-making unit is not lower than its worst relative efficiencyψkkL∗, we enable the decision-making unit to adopt different evaluation strategies for other units according to the degree of association.(15)minz1θdd−∑r=1surkyrd−z2θdd−∑r=1surkyrd,s.t.∑i=1mvikxid=1,∑r=1surkyrk−ψkkU∗∑i=1mvikxik=0,ψkkL∗∑i=1mvikxid−∑r=1surkyrd≤0,∑r=1surkyrj−∑i=1mvikxij≤0,j=1,2,...,n,urk≥0,vik≥0,r=1,2,...,s,i=1,2,...,m,z1=1,z2=0,k,d⊂H;z1=0,z2=1,k,d⊄H.
We first use model (14) to calculate the optimal relative efficiency of each decision unitψkkU∗ and the worst relative efficiency of ψkkL∗ and then evaluate decision unit d under the condition that the optimal relative efficiency of decision unit k keeps ψkkU∗ unchanged. At the same time, we ensure that the other-evaluation efficiency of unit d is not less than its worst relative efficiency ψkkL∗. As a singular index, θdd−∑r=1surk−yrd indicates the difference between self-evaluation and other evaluations. If the decision-making units k and d are in the same set, then z1=1,z2=0, which means that the decision-making unit d’s other-evaluation efficiency is as large as possible. On the contrary, it is necessary to make the decision-making unit d’s other-evaluation efficiency as small as possible. In addition, model (15) ensures that the other-evaluation efficiency of the decision-making unit d is not lower than its worst relative efficiency and can improve the acceptability of the evaluation results.
The same DEA-EBM model has many expressions. For example, model (3) is called a multiplier model of CCR. By solving this modeln times, the efficiency score of each unit from DMU to DMU can be obtained. Although model (3) is linear, the calculation of efficiency scores often needs to be transformed into a dual form, as shown in (16)maxθ,s.t.∑i=1mλjxij≤θxro,j=1,2,...,n,∑r=1sλjyrj≥yxro,λj≥0,j=1,2,...,n.
Model (16) is the envelope form of the input-oriented CCR model (Farrell model). It means that the input to the DMU shrinks as much as possible without reducing the current output level. Both inputs and outputs are highly disposable:(17)T=x,y:∑j=1nzjxij≤xi∑j=1nzjyrj≥yr,r=1,2,...,szj≥0j=1,2,...,n.
Among them,x=x1,x2,...,xn and y=y1,y2,...,yn are input and output vectors, respectively. T is the reference technology set containing all possible input-output combinations.
In the traditional DEA-EBM model as shown in Figure1, if x,y∈T and x′≥x (or y′≤y), then x′,y∈T or x,y′∈T.
Under the DEA-EBM framework, the weakly disposable reference technology is also called the environmental DEA-EBM technology, which can be expressed as model (18):(18)Te=x,y,u:∑j=1nzjxij≤xi,i=1,2,...,m∑j=1nzjyrj≥yr,r=1,2,...,s∑j=1nzjukj=ukk=1,2,...,tzj≥0j=1,2,...,n.
Among them,u=u1,u2,...,ut represents the undesired output vector, and the difference between T and Te is that, in Te, it is not feasible to reduce only the undesired output, but it is feasible to reduce the expected output and the undesired output proportionally.
We assume that there aren homogeneous units as DMUjj−1,2,...,n, and the input vector, expected output vector, and undesired output vector of each decision-making unit are expressed as x=x1,x2,...,xm, y=y1,y2,...,ys, and z=z1,z2,...,zt, respectively, which consumes m types of inputs and produces s types of expected output and t kinds of undesired output.
In order to properly describe the production process with expected output and undesired output, the following two assumptions need to be added to the production technology setT proposed by Fare:(1)
Weak disposability of output. That is, ifx,y,b∈Tand0≤θ≤1, then x,θy,θb∈T.(2)
Zero combination of undesired output and expected output. That is, ifx,y,b∈Tandb=0, then y=0. These two assumptions show that it is not feasible to reduce only undesired output, and the reduction of undesired output must be accompanied by the reduction of expected output.
There are many models that DEA-EBM technology uses to measure environmental performance in an environment with constant returns to scale. Among these models, the model only allows adjustments to undesired output.(19)PEI1=minλs.t.∑k=1Kzkxnk≤xn0n=1,...,N∑k=1Kzkymk≥ym0m=1,...,M.∑k=1Kzkujk=λuj0j=1,...,Jzk≥0k=1,2,...,K.
This paper proposes the following input-oriented EBM-GR-U model framework. The economic and ecological efficiency ofn units can be obtained by solving the model n times. This paper assumes weak disposability of pollutants that play an important role in environmental impact, such as (20)γ∗=minβ−ε∑i=1mwi−si0−xi0,s.t.βxi0−∑j=1nxijλj−si0−i=1,...,m,∑j=1nyrjλj≥yr0r=1,...,s,∑j=1nbpjλj=pbpop=1,...,t,λj≥0,sij−≥0i=1,...,mj=1,...,n.
θ in the above formula is the fraction of radial efficiency, which represents the degree of radial efficiency. ∑i=1mwi−si0−/xi0 represents the nonradial relaxation term. ε is a key parameter that combines the radial efficiency score and nonradial relaxation. wi− represents the weight of the i-th input and satisfies ∑i=1mwi−=1∀i,wi−≥0. In this formula, the values of ε and wi−i=1,...,m need to be obtained in advance. ρ represents the environmental efficiency level of DMUo. Therefore, the eco-economic efficiency of DMUo can be integrated through economic efficiency and environmental efficiency; that is, economic eco-efficiency eco−efficiency=γ∗⋅ρ.Figure 1
The general structure of the DEA-EBM model.Definition 3.
(EBM-GR-U input-oriented efficiency).
DMUo is called EBM-GR-U input valid ⇔γ∗=1.Definition 4.
The optimal solution of model (19) is expressed asθ∗,λ∗,s−∗.
The projection ofDMUoxo,yo can be defined as follows:(21)xo∗=Xλ∗=θ∗xo−s−∗,yo∗=Yλ∗,zo∗=Zλ∗,.
As shown in model (19), the values of two types of important parametersε and w=wi,i=1,...,m need to be obtained in advance, as shown in Table 2.
In addition, since the information expressed by the two types of input data overlaps to a certain extent, the eigenvalues and eigenvectors of the proximity matrix can be solved based on the idea of principal component analysis. When the closeness between the input indicators is higher, the characteristic roots compressed by principal components will be larger. According to the functional relationship betweenε and the characteristic root, ε is reduced at this time, and the nonradial features of the model are reduced; that is, the radial compression is more suitable for input at this time. Conversely, when the input index is closer, the data is more scattered, the characteristic root will be smaller, and ε will increase. At this time, it is more suitable for projection based on nonradial relaxation.Table 2
Input sequence used in correlation construction.
x1(X1)…xi(Xi)xj(Xj)xm(Xm)DMU1x11…xi1…xj1…xm1…………DMUkx1k…xik…xjk…xmk…………DMUnx1n…xin…xjn…xmnProposition 1.
We set the behavior column of the systemXi=xi1,xi2,...,xin and mark the broken line xi1−xt1,xi2−xt1,...,xin−xt1 as Xi1−xt1 and si=∫1nXi−xt1ds.Then,(1)
WhenXi is an increasing sequence, si≥0(2)
WhenXi is the attenuation sequence, si≤0(3)
WhenXi is an oscillating sequence, the symbol of si is uncertainProposition 2.
We set the behavior sequence of the system asXi=xi1,xi2,...,xin and Xj=xj1,xj2,...,xjn. The two sequences have the same length. By operating xi0=xik−xi1,k=1,2,...,n on the elements in the sequence, the zeroized images of the starting point of the two sequences can be obtained as(22)Xi0=xi01,xi02,...,xi0n,Xj0=xj01,xj02,...,xj0n.Moreover, we setsi−sj=∫1nXi0−Xj0ds.Then,(1)
WhenXi0 is above Xj0, si−sj≥0.(2)
WhenXi0 is below Xj0, si−sj≥0.(3)
WhenXi0 intersects at Xj0, the sign of si−sj is uncertain.Definition 5.
We assume that the sequencesXi and Xj have the same length and the definitions of si and sj are as shown in Proposition 1, and then we call(23)sij=εij=1+si+sj1+si+sj+sj−si.
It is the gray absolute correlation degree ofXi and Xj, the similarity index sij of these two sequences can be defined as sij=εij.Definition 6.
We assume that the two sequencesXi and Xj have the same length, and the definitions of si and sj are shown in Proposition 1. Then, the similarity index si,j of these two sequences can be defined as follows:(24)sij=εij=sjsi+si+sjsjsi+si+sj+sj−si,sij=εij=1+sjsi+si+sj1+sjsi+si+sj+sj−si.
Formulas (22) and (23) have the following four properties:(1)
Reflexivity,∀i,si,j=1(2)
Symmetry,∀i,j,si,j=sj,i(3)
Normative,∀i,j,0≤si,j≤1(4)
Proximity.Proof 1.
(1)
Reflexivity:sj−si=0,∀i,si,j=1.(2)
Symmetry:si−sj=sj−si and min1−1/m∑s1−/x/1/m∑s1−/xs.t.βxi0−∑j=1nxijλj−si0−i=1,...,m∑j=1nyrjλj≥yr0r=1,...,s∑j=1nbpjλj=pbpop=1,...,tλj≥0,sij−≥0i=1,...,mj=1,...,n.(3)
Normative: Obviously,sij=εij>0, and sj−si≥0, so sij=εij≤1.(4)
Proximity: Obviously, it is established.Step 1.
The algorithm projects all DMUs onto the front surface of the VRS.(25)min1−1/m∑i=1ms1−/xio1+1/s∑r=1ss1+/yio,s.t.xi0=∑j=1nxijλj+si−i=1,...,m,yr0=∑j=1nyrjλj−sr+r=1,...,s,bp0=∑j=1nbpjλjp=1,...,t,∑j=1nλj=1,λj≥0,si−≥0sr+≥0,∀i,j,r.
The algorithm can improve the accuracy of estimation by projecting all DMUs onto the frontier under the assumption of variable returns to scale. The projection model is shown in model (24).
The projection using slack input and output can be defined asx˜io=xio−si−∗i=1,...,my˜ro=yro−sr−∗r=1,...,s.
DMUS for N effective VRS units can be expressed asX¯=x¯1,...,x¯mY¯=y¯1,...,y¯s, and all DMUs with valid CRS and VRS are also included in this set.Step 2.
Because Step1 uses different projection models to get different projection results, you can also replace the projection based on the actual data observed.
That is, the algorithm usesX=x1⋯xm=x11⋯x1n⋮⋱⋮xm1⋯xmn and Y=y1⋯ym=y11⋯y1n⋮⋱⋮ys1⋯ysninstead of X¯ and Y¯. According to formula (19), we only consider the proximity of the input sequence, and the proximity matrix S=sij∈Rm×m is composed of elements sij=sx¯i,x¯j or sij=sx¯i,x¯j. As shown in Table 2, xi,xj is the sequence of the i-th and j-th input actual data of all units, where i,j = 1,2, ..., m. sij=εij=sxi,xj is the closeness index between the two input sequences obtained according to the actual input data and the definition of formula (21) or (22), and the final closeness matrix is shown in Table 3. According to the relevant properties of the formula, all elements in the matrix are obtained: 0≤sij=εij≤1.Table 3
Proximity matrix based on proximity index.
…Xi…XjX1…ε1i…ε1j(Sij)……………Xj…εji…εjjStep 3.
The algorithm solves the maximum eigenvalue of the similarity matrix and its corresponding eigenvector.
The matrixS is a nonnegative symmetric matrix, and the diagonal elements are all 1. According to the Perron–Frobenius theorem, S has the largest characteristic root ρx and its corresponding nonnegative characteristic vector wx. The nonnegative vector wx corresponds to the weight of each input element. According to the P–F theorem, m≥ρx≥1.
In Step 4, the algorithm uses the largest feature root and feature vector model in Step3 to calculate the values of ε and w−. If m > 1, then ε=m−ρx/m−1; and if m = 1, then ε=0, and w−=wx/∑i=1mwx.Step 4.
The algorithm uses the obtained values ofε and w− to use model (19) to calculate EBM-GR-U.
In summary, model (19) reasonably combines radial and nonradial models under the weak disposability of undesired output. Moreover, its proximity index based on gray correlation obtains the key parameters of the combination of radial and nonradial in the model and measures the economic efficiency of the decision-making unit according to the steps. Similarly, it can be calculated by building an output-oriented model.


## 4. Statistical Measurement and Influencing Factors of Green Total Factor Productivity of China’s Tourism Industry

The tourism industry chain is an input-output system of material, capital, technology, and information completed by different enterprises in the whole process from the development of tourism resources to the consumption of tourism products. We summarize the tourism industry chain into the core tourism industry chain (shown in Figure2) and the tourism-related industry chain (shown in Figure 3).Figure 2
The core tourism industry chain.Figure 3
The tourism-related industry chain.When evaluating the total factor productivity of the tourism industry and its subsectors, due to the availability of data or different research purposes, a unified evaluation index system has not been formed. Researchers make diversified choices of indicators and variables according to different specific research goals. Moreover, even in the face of the same research object, scholars will construct different evaluation index systems (see Table4).Table 4
Indicators in the study of tourism efficiency and tourism total factor productivity.
Investment indicatorsOutput indicatorsAverage price problemService levelNumber of hotelsNumber of roomsCustomer satisfactionNumber of employeesNumber of roomsHotel occupancy rateOperating expensesAccommodation incomeHotel income (dining and beverage)Other incomesNumber of employeesThe number of wages and wealthOperating costOther costsOperating incomeCustomer volumeNumber of overnight staysNumber of employeesCapitalOperating incomeAdded value, profitNumber of employees in the tertiary industryFixed asset investmentActual use of foreign capitalStar-rated hotel operating incomeNumber of employees in the tourism industryTourism net fixed assetsTotal tourism revenueTourism fixed assetsNumber of employees in the tourism industryTotal operating income of tourism enterprisesNumber of employees in the tertiary industryFixed capital stockComment on the number of scenic spotsNumber of star hotelsNumber of travel agenciesTotal tourism revenueTotal number of touristsNumber of employees in the tertiary industryFixed asset investmentOperating incomeNumber of tourists served by travel agenciesTourism net fixed assetsNumber of employees in the tourism industryTotal tourism revenueCO2 emissions from tourismThe total number of tourists includes the number of domestic tourists and the number of international tourists. The total number of tourists reflects the city’s tourism reception volume and directly reflects the tourism benefits of a city’s tourism. Therefore, the total number of tourists in this paper reflects the tourism scale of the tourist city. Total tourism revenue includes domestic tourism revenue and international tourism revenue, which is the most direct economic benefit of a city's tourism development. Moreover, it is also the tourism output value of the region and an important indicator to measure the level and quality of urban tourism development. The total factor productivity evaluation index system is shown in Table5.Table 5
Total factor productivity evaluation indicator system.
CategoryIndicator nameSpecific indicatorsInvestment indicatorsLabor forceNumber of employees in the tertiary industry/10,000 peopleCapitalUrban investment in fixed assets/100 million yuanOutput indicatorsOutput valueTotal tourism revenue/100 million yuanScaleTotal number of tourists/10,000 peopleThe model proposed in this paper should be used to calculate the total factor productivity of tourism in tourist cities. From Table6, it can be seen that the total factor productivity of tourism is showing a general upward trend.Table 6
Productivity and decomposition of tourism total factors.
TimeEffchTechchPechSechTfpch2012-20131.1350.9241.1041.0391.0392013-20141.0671.1381.0451.0301.2032014-20150.9951.1221.0001.0051.1062015-20160.9931.1880.9991.0031.1672016-20170.8961.3431.0480.8631.1922017-20181.0570.9901.0611.0071.0372018-20190.9691.1500.9741.0051.1032019-20200.9411.1960.9111.0431.1142020-20211.0241.1290.9881.0461.144Average value1.0061.1251.0131.0031.121Through data sorting (Table7), it is found that the total factor productivity of the tourist city research sample during the study period fluctuates greatly.Table 7
Analysis table of the total factor productivity indicator of urban tourism.
Years<1 city/countCity/a below averageVarianceMaxMinimumAverage value2011-201264760.0311.7220.3641.0392012-201314870.0221.7360.7171.2032013-201427970.0141.4680.8301.1062014-201512960.0141.7170.9321.1672015-20167800.0191.2060.5541.1922016-201758710.0221.9190.5141.0372017-201822770.0121.4360.5401.1032018-201919680.0171.6270.5811.1142019-202019960.0281.8920.4641.1442020-20218800.0041.3140.9061.121The technical efficiency change index is the product of pure technical efficiency and scale efficiency. The specific performance is output maximization and input minimization; that is, the maximum output is achieved when the input elements are fixed or the input element items are minimized under the given conditions of output, as shown in Table8.Table 8
Analysis table of urban tourism technical efficiency indicator.
Years<1 city/countCity/a below averageVarianceMaxMinimumAverage value2012-201345840.0411.9390.4581.1352013-201451810.0161.5860.6551.0672014-2015101920.0081.2580.7950.9952015-2016102960.0101.3880.7890.9932016-2017141820.0131.2640.4640.8962017-201848690.0191.3850.5041.0572018-2019125810.0081.3530.4720.9692019-2020128760.0131.3350.4670.9412020-202178860.0241.6990.3991.024Average value87830.0031.1760.8761.006In terms of technological progress in tourism, the overall variance of the various years fluctuates relatively small and relatively stable, indicating that the technological progress of each city is different but small, as shown in Table9.Table 9
Analysis table of urban tourism technology progress indicator.
Years<1 city/countCity/a below averageVarianceMaxMinimumAverage value2012-20131231000.0161.1780.7080.9242013-201491580.0081.4690.8951.1382014-201571110.0081.2870.9461.1222015-20160740.0011.2521.0941.1882016-20170810.0161.5991.0791.3432017-20181421390.0021.1340.8150.9902018-20193650.0021.2210.9621.1502019-20200830.0031.3121.0521.1962020-20210870.0031.2021.0501.129Average value1780.0011.1980.9831.125It can be seen from the above research that the DEA-EBM model proposed in this paper can play an important role in the statistical measurement and analysis of influencing factors of green total factor production in the tourism industry.


## 5. Conclusion

Tourism efficiency relates to the inclusive growth and sustainable development of the tourism industry. It is a comparison between input and output in the process of tourism development within a certain area. Moreover, it is an important indicator used to measure the ability of the entire tourism industry to achieve effective competition, the ability to use resources, the ability to achieve sustainable development, the ability of the tourism industry to compete in the market, the ability of input and output, and the ability of sustainable development. The level of tourism efficiency will directly determine the position of a region in the competitive environment. In addition, better industrial operation efficiency can effectively reduce production costs and increase economic returns, which plays an important role in the development of tourism. This paper combines the EA-EBM model to carry out the statistical measurement of China’s tourism industry green total factor productivity and the analysis of influencing factors and verify it through data. Through research, it can be seen that the DEA-EBM model proposed in this paper can play an important role in the statistical speed measurement and influencing factor analysis of green total factor production in the tourism industry.



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