Meta-frontier analysis using cross-efficiency method for performance evaluation

European Journal of Operational Research 280 (2020) 219–229

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Meta-frontier analysis using cross-efficiency method for performance evaluation Lei Chen a, Yan Huang a,c, Mei-Juan Li a, Ying-Ming Wang a,b,∗ a

School of Economics & Management, Fuzhou University, Fuzhou 350116, China Key Laboratory of Spatial Data Mining & Information Sharing of Ministry of Education, Fuzhou University, Fuzhou 350116, China c College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China b

a r t i c l e

i n f o

Article history: Received 30 July 2018 Accepted 28 June 2019 Available online 2 July 2019 Keyword: Data envelopment analysis Technology gap Cross-evaluation strategy Meta-frontier Inefficiency

a b s t r a c t Efficiency overestimation and technology heterogeneity are important factors that affect the use of data envelopment analysis. This paper introduces a meta-frontier analysis framework into a cross-efficiency method to develop a new efficiency evaluation method. This method can be used to calculate, aggregate, and decompose the cross efficiencies relative to the meta-frontier and group-frontier. Then the technology gap between these frontiers can be measured and more detailed information regarding the inefficiency of decision-making units can be obtained. This enables decision makers to improve efficiency in a targeted manner. Subsequently, the non-uniqueness of the optimal solution is discussed for the new method, and the cross-evaluation strategy is introduced to ensure the stability of the optimal solution. Finally, two examples are presented to illustrate the effectiveness of this method. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Since the innovative work by Charnes, Cooper and Rhodes (1978), data envelopment analysis (DEA) has been widely accepted as a powerful performance assessment tool. It has been applied to various tasks, including energy efficiency assessments (Mardani, Zavadskas & Streimikiene, 2017) and bank efficiency assessments (Diallo, 2018). As a nonparametric approach, DEA evaluates the relative efficiency of homogeneous decision-making units (DMU) with multiple inputs and outputs. Furthermore, evaluation results can be obtained without the influence of the subjective wishes of decision makers. However, there are two obvious theoretical deficiencies in the traditional DEA method: evaluating DMUs under a unified technical environment (Kounetas & Napolitano, 2018) and allowing each DMU to measure its efficiency with its favorable weights (Liu, Song & Yang, 2019). The evaluation process of traditional DEA is typically performed in a unified technical environment, assuming all DMUs use the same production technology. However, owing to differences in physical, human, and other characteristics in the production process (e.g. type of machinery, size and quality to the labor force), different DMUs may have different levels of production technology (O’Donnell, Rao & Battese, 2008). Walheer (2018) noted that DMUs ∗ Corresponding author at: School of Economics & Management, Fuzhou University, Fuzhou 350116, China. E-mail address: [email protected] (Y.-M. Wang).

https://doi.org/10.1016/j.ejor.2019.06.053 0377-2217/© 2019 Elsevier B.V. All rights reserved.

with different technology levels could not be directly compared, because an evaluation result ignoring technical heterogeneity could be biased. Therefore, a meta-frontier analysis framework was introduced into the DEA method to address this issue. Metafrontier analysis splits the meta-frontier of all DMUs into several group-frontiers to help measure the technology gaps between different DMUs during the DEA evaluation process. This obtains more objective evaluation results (Carrillo & Jorge, 2018). Feng, Huang and Wang (2018); Li, Liu and Liu (2017), and Tian and Lin (2018) applied meta-frontier analysis to DEA with undesirable outputs, dynamic DEA, and sequential DEA, respectively. Additionally, the impacts of a technology gap on efficiency evaluation were measured. Although meta-frontier analysis can be used to solve the deficiency of the unified technical environment in DEA, the weights of inputs/outputs in this method are still partial to measuring DMUs’ efficiency by their own favors. Thus, evaluation results may be overestimated, and many DMUs may be evaluated as efficient, and cannot be further discriminated (Li, Zhu & Liang, 2018a). The cross-efficiency method is the main method used to address this problem. It combines self-evaluation with peer-evaluation to measure the efficiency of restricting the weight flexibility of traditional DEA. Then, the efficiency overestimation is solved accordingly (Ang, Chen & Yang, 2018). As one of the main research directions of DEA (Liu, Lu & Lu, 2016), the studies on cross-efficiency can be generally divided into two categories. The first category focuses primarily on the strategy formulation of cross-evaluation.

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The traditional aggressive/benevolent strategy, proposed by Doyle and Green (1994), regarded the others DMUs as competitors/collaborators to construct strategy formulation. Subsequently, many cross-evaluation strategies have been formulated to ensure the relative uniqueness of the optimal cross-efficiency solution in different decision-making situations. For example, the neutral strategy proposed by Wang and Chin (2010) paid attention to the further-optimized efficiency of each DMU, while some alterative strategies proposed by Davtalab-Olyaie (2019) improved the number of satisfied units according to the attitude of decision makers. The second category focuses on the rule making of cross-efficiency aggregation. Additive average is the most commonly used aggregation method, whereas other methods are recommended to improve the information utilization ratio of the cross-efficiency matrix, such as the ordered weighted averaging method (Wang & Chin, 2011) and the entropy-weight method (Song, Zhu & Peng, 2017). Unfortunately, just as with meta-frontier analysis, the cross-efficiency method can only be used to solve one of the two theoretical deficiencies of the traditional DEA method. Thus, it can solve the efficiency overestimation caused by the weights’ flexibility. However, the technology gap between different DMUs is not considered. In summary, the cross-efficiency method can be used to evaluate the performance of DMUs more objectively by combining self-evaluation and peer-evaluation. However, the evaluation process is performed in a unified technical environment. If different DMUs have diverse levels of production technology, then the results obtained by the cross-efficiency method could be biased. Therefore, to obtain a fair and objective result, decision makers should conduct cross-evaluations in the meta-frontier analysis framework. The efficiency overestimation and technical heterogeneity can be solved at a later time. To the best of our knowledge, there is no research in this area. This is probably the result of the different evaluation perspectives for cross-efficiency methods causing difficulties in measuring technical gaps. Therefore, the aim of this study is to develop a new meta-frontier cross-efficiency analysis framework to solve the problem of overestimation of efficiency in the traditional metafrontier DEA method caused by self-evaluation and the difficulty in measuring technology gaps in the cross-evaluation environment. The rest of this paper is organized as follows. Section 2 introduces the traditional cross-efficiency and meta-frontier methods. Section 3 describes a new cross-efficiency method with the meta-frontier analysis framework. Section 4 introduces the crossevaluation strategy into the new method. In Section 5, the classification method is discussed, and two examples are presented to illustrate the proposed method in Section 6. Finally, conclusions are provided in Section 7.

2. Preliminaries 2.1. Traditional cross-efficiency method Assume that there are n DMUs to be evaluated, and each DMU j (j = 1,…,n) uses m inputs, xi j (i = 1,…, m), to produce s outputs, yr j (r = 1,…,s). Then, according to the study of Charnes et al. (1978), efficiency is defined as a ratio of weighted outputs to weighted inputs, and the efficiency, θdd , of DMUd can be obtained using the following input-oriented model:

θdd = Max

s 

urd yrd

r=1

s.t.

m  i=1

vid xid = 1;

s 

urd yr j −

m 

vid xi j ≤ 0, j = 1, . . . , n;

i=1

r=1

∀vid , urd ≥ 0;

(1)

where vid , urd denote the decision variables. Model (1) is a linearization model of traditional DEA model, called the Charnes, Cooper, and Rhodes (CCR) model. Because model (1) allows each DMU to measure its efficiency with favorable weights, its efficiency result, θdd , is regarded as a self-evaluation efficiency (Sexton, Silkman & Hogan, 1986). Assume that (v∗id , u∗rd ) is the optimal solution of model (1). Sexton et al. (1986) constructed the following model to calculate the peer-evaluation efficiency of DMU j from DMUd .

s

θdj = r=1 m

i=1

u∗rd yr j

v∗id xi j

, ( j = 1, . . . , n; j = d ).

(2)

Self-evaluation and peer-evaluation are collectively called crossevaluation, and their results, cross-efficiency θd∗ , can be obtained by the following aggregation method.

θd∗ =

1  ∗ θ jd . n n

(3)

j=1

Additionally, model (1) may have multiple optimal solutions with different input/output optimal weights, and these would affect the stability of the cross-evaluation results. The remedy is to introduce a cross-evaluation strategy to generate a secondary optimization for model (1). The benevolent strategy and aggressive strategy proposed by Doyle and Green (1994) are the most commonly used cross-evaluation strategies. The aggressive strategy is as follows:

Min

s 

urd

m  i=1

s 

n 

vid

r=1

xi j = 1;

j=1, j=d

urd yrd − θdd

m 

vid xid = 0;

(4)

i=1

r=1 s 

yr j

j=1, j=d

r=1

s.t.

n 

urd yr j −

m 

vid xi j ≤ 0, j = 1, . . . , n;

i=1

∀urd , vid ≥ 0; r = 1, . . . , s; i = 1, . . . , m; where θdd denotes the optimal results of self-evaluation of DMUd obtained by model (1). Furthermore, if the minimized objective function is converted to a maximum, model (4) becomes the benevolent strategy. The benevolent strategy and the aggressive strategy regard other DMUs as collaborators/competitors, aiming to maximize/minimize their cross efficiencies on the premise that the self-evaluation optimal efficiency of DMUd remains unchanged. Then, the relative unique optimal solution of model (4) can be used to calculate cross-efficiency based on models (2) and (3). 2.2. Traditional meta-frontier analysis method O’Donnell et al. (2008) developed the meta-frontier analysis method to study the technology gap between different DMUs. According to their study, the technology set of n DMUs is T = {(x, y )|xcanproducty}, and the boundary of T is defined as the meta-frontier of all DMUs. Assuming that n DMUs can be partitioned into G groups based on their technical levels, then different groups have different technology sets: T g =

L. Chen, Y. Huang and M.-J. Li et al. / European Journal of Operational Research 280 (2020) 219–229

3.2. Cross-evaluation based on meta-frontier analysis (CMFA)

y Group-frontier 1

Meta-frontier

Group-frontier 2 A

Group-frontier 3 O

D

C

B

x

Fig. 1. Efficiency analysis based on SMFA.

{(xg , yg )|xg canproductyg} (g = 1,…,G). The boundary of T g is defined

as the group-frontier of Group g. Therefore, T =T 1 ∪ T 2 ∪ · · · T G . The efficiencies of DMUd relative to the meta-frontier and group-frontier are denoted as meta-frontier efficiency (ME) and group frontier efficiency (GE), respectively. Note that, when using model (1) to calculate GE, the number of DMUs is reduced from n to the number of DMUs in the evaluated group. The meta-technology ratio (MTR) is thus defined as:

MTR =

221

ME ≤ 1. GE

(5)

The closer the MTR is to 1, the smaller the technology gap between the meta-frontier and the group-frontier.

3. Cross-efficiency method with meta-frontier analysis framework 3.1. Self-evaluation based on meta-frontier analysis (SMFA) According to the study of Sexton et al. (1986), the process of self-evaluation in cross-efficiency is equivalent to the traditional DEA method. Thus, the traditional meta-frontier method can be regarded as SMFA. Then, the self-evaluation ME (SME), self-evaluation GE (SGE), and self-evaluation MTR (SMTR) are equivalent to traditional ME, GE, and MTR (i.e., SME = ME, SGE = GE, and SMTR = MTR). To obtain better decision information about efficiency, the inefficiency of DMUd can be further decomposed into self-evaluation technology gap inefficiency (STGI) and self-evaluation managerial inefficiency (SGMI) according to the study of Chiu, Liou and Wu (2012). STGI represents the inefficiency of DMUd in a groupspecific frontier caused by a technology gap in the perspective of self-evaluation. The SGMI represents an inefficiency originating from input excesses and output shortfalls. STGI and SGMI can be obtained by the following models.

STGI = SGE ∗ (1 − SMTR ),

(6)

SGMI = 1 − SGE.

(7)

Therefore, the self-evaluation meta-frontier inefficiency (SMTI) can be expressed as follows.

SMTI = STGI + SGMI = 1 − SME.

(8)

Assume that there is a meta-frontier comprising three groupfrontiers in Fig. 1, and A is a DMU in group 2. Because model (1) is an input-oriented DEA model, there are SME=OD/OB, SGE=OC/OB, SMTR=OD/OC, STGI=DC/OB, SGMI=CB/OB, and SMTI=DB/OB.

Based on the traditional DEA method, SMFA allows each DMU to measure its efficiency with its own favorable weights, resulting in an overestimation of the efficiency (Li, Zhu & Chen, 2018b). Therefore, the cross-efficiency method should be introduced to solve this problem. Assume that n DMUs can be partitioned into three groups. The cross-efficiency matrix obtained by traditional cross-efficiency method is shown as Fig. 2, which is obtained by models (1) and (2). Thus, according to traditional cross-efficiency method and meta-frontier analysis framework, the cross-evaluation ME (CME) can be directly obtained by model (3). This represents the crossefficiency of DMUs relative to the meta-frontier. However, if cross-evaluation GE (CGE), which represents the cross-efficiency of DMUs relative to the group-frontier, is calculated in the same way, then CME may be greater than CGE, which violates the basic properties of meta-frontier analysis: CME ≤ CGE (O’Donnell et al., 2008). This is caused by the relationship between the metafrontier and group-frontiers becoming chaotic from the different perspectives of evaluation. Thus, it is unreasonable to calculate CGE directly by traditional cross-efficiency methods. To scientifically address the relationship between different frontiers in the cross-evaluation environment, a new CMFA method is developed, and the efficiency analysis of Group 1, which has k DMUs (k ≤ n), is used as an example to illustrate the method. Step 1: Calculate the CME of DMUs. According to the metafrontier analysis method, the cross-efficiency obtained by the traditional cross-efficiency method is the CME of DMUs (i.e., θdCME = θd∗ , where θd∗ is obtained by models (1)–(3)), and θdjCME is the CME of DMU j from DMUd relative to the meta-frontier. Step 2: Construct a self-evaluation strategy. T g is a subset of T . The efficiency of each DMU relative to the meta-frontier should not be greater than its efficiency relative to the group-frontier, because the meta-frontier covers all group-frontiers (O’Donnell et al., 2008). Therefore, for Group 1, the strategy model of self-evaluation can be formulated as follows. CGE θdd = Max

s 

urd yrd

r=1

s.t.

m 

vid xid = 1,

i=1 s 

urd yr j −

vid xi j ≤ 0, j = 1, . . . , k;

(9)

i=1

r=1 s 

m 

CME urd yr j − θdj ∗

m 

vid xi j ≥ 0, j = 1, . . . , k;

i=1

r=1

∀vid , urd ≥ 0; CGE denotes the self-evaluation efficiency of DMU based where θdd d on CMFA. Thus, the third constraint is used to ensure that the CME is not greater than CGE. Step 3: Calculate the CGE matrix. Assume that (v∗id , u∗rd ) is the optimal solution of model (9). Then, the cross-efficiency Matrix G1 of Group 1 shown as Fig. 2 can be calculated by model (2). Note that the number of DMUs is reduced from n to k in model (2). CGE For example, both DMU j and DMUd belong to Group 1. Thus, θdj represents the peer-evaluation efficiency of DMU j from the DMUd relative to group-frontier. Step 4: Aggregate and decompose the CGE. According to models (3)–(8), the CGE of DMUd can be aggregated as follows:



θdCGE

1 = ∗ k

k  j=1



θ CGE , jd

(10)

222

L. Chen, Y. Huang and M.-J. Li et al. / European Journal of Operational Research 280 (2020) 219–229

Fig. 2. Cross-efficiency matrix.

where θdCGE and θ CGE denote the CGEs of DMUd from all DMUs jd and DMU j , respectively. Furthermore, the cross-evaluation MTR (CMTR) of DMUd is defined as

CMTRd =

1  1  ∗ CMTR jd = ∗ k k k

k

j=1

j=1

θ jdCME . θ jdCGE

(11)

Matrix G1, shown in Fig. 2, is not considered in the calculation of CMTR, because the peer-evaluation results from the others groups cannot reflect the technology gap of Group 1 between the meta-frontier and group-frontier. Thus, the CGE of DMUd can be decomposed as follows:

1  CGE ∗ θ jd ∗ (1 − CMTR jd ), k k

CTGId =

(12)

j=1

1  CGE ∗ (1−θ jd ), k

(13)

1  CME ∗ (1−θ jd ), k

(14)

k

CGMId =

j=1 k

CMTId =

j=1

where CTGId , CGMId , and CMTId denote DMUd ’s cross-evaluation technology gap inefficiency, cross-evaluation managerial inefficiency, and the cross-evaluation inefficiency relative to the meta-frontier, respectively. To further illustrate the relationship among traditional methods and the CMFA method, Fig. 3 is described as follows. As shown in Fig. 3, the traditional DEA method can be extended to the traditional cross-efficiency method by introducing cross-evaluation. For the traditional meta-frontier framework, although it measures efficiency relative to the meta-frontier and the group-frontier, the results (i.e., SME and SGE, respectively) are obtained only from the self-evaluation perspective. By introducing the value of CME, a new self-evaluation strategy is formulated based on the traditional DEA method. The strategy is used to calculate a set of weights relative to the group-frontier. Then, the CGE can be obtained by cross-evaluation. Based on CME and CGE, the technology gap, CMTR, and the inefficiency information, CTGI, CGMI, and CMTI, can be obtained. The aggregation of these processes is the CMFA method.

CME ( j = 1, . . . , n) are obtained by model (2), we know that θdj

(v∗id , u∗rd ), which is the optimal solution of model (1). Owing to n ≥ k, (v∗id , u∗rd ), this is a feasible solution to model (9). Assume that (v∗∗ , u∗∗ ) is the optimal solution to model id s rd ∗∗ m ∗∗ CGE = CME (9). Then, θdj ( j = 1 , . . . , k) r=1 urd yr j / i=1 vid xi j ≥ θdj for ∀DMUd . According to models (3) and (10), there are θdCME ≤ θdCGE (d = 1, . . . , k).  Theorem 2. Traditional cross-efficiency method is a special case of CMFA.

Proof. According to the proof process of Theorem 1, we know that the optimal solution of model (1) is a feasible solution of model (9). If n = k, then all DMUs are in the same group. Therefore, the constraints of model (1) are the constraints of model (9), and the s CME ∗ optimal solution of model (1) must satisfy r=1 urd yr j − θdj m v x = 0 ( j = 1 , . . . , n ) based on model (2) (i.e., the third coni=1 id i j straint of model (9) is invalid). Thus, the traditional cross-efficiency method is equivalent to CMFA in the case of n = k.  Theorem 3. CMTRd ≤ 1 and CMTId = CTGId + CGMId for ∀DMUd . Proof. According to Theorem 1, there is θdCME ≤ θdCGE (d = 1, . . . , k),  and based on model (11), CMTRd = 1k ∗ kj=1 CMTR jd ≤ 1 for ∀DMUd . According to model (12), CTGId = =

1 k

∗

k j=1

CGE ∗ (1 − θ jd

k

θ CME jd θ CGE jd

)=

1 k

∗

k

1 k

j=1

∗

k j=1

CGE ∗ (1 − CMTR ) θ jd jd

CGE −θ CME ). (θ jd jd

Because  ∗ ( kj=1

CGE ), (1−θ jd CTGId + CGMId = 1k k 1 k CGE CME ) = CMTI . + j=1 (1−θ jd ) ) = k ∗ j=1 (1−θ jd d Theorem 3 shows that the meta-frontier covers all groupfrontiers with CMTRd ≤ 1, and the inefficiency relative to the metafrontier comprises technology gap and managerial inefficiency. This is consistent with the traditional meta-frontier method. Based on the CMFA, the CGEs of Group 2 and Group 3 can be calculated, aggregated, and decomposed the same way as Group 1. These processes are not discussed in detail here. Note that all analytical processes of CMFA are based on the input-oriented DEA model, thus the results are also input-oriented. According to the actual decision-making needs, decision makers can follow this train to construct output-oriented method, and the construction process is also not discussed in detail here.

CGMId = 1k ∗

j=1

CGE −θ CME ) (θ jd jd

Theorem 1. Model (9) has a feasible solution, and θdCME ≤ θdCGE (d = 1 , . . . , k).

4. CMFA with cross-evaluation strategy

Proof. If ∀urd = 0, then any set of vid (i = 1, . . . , m) satisfying m i=1 vid xid = 1 is a feasible solution to model (1). According to

Just as with the traditional DEA method, the optimal solution of CMFA may be not unique, owing to two reasons. On one hand,

L. Chen, Y. Huang and M.-J. Li et al. / European Journal of Operational Research 280 (2020) 219–229

SMFA

223

CMFA

Self-evaluation

Crossevaluation

Traditional DEA method

Cross efficiency method

Metafrontier

SME

Groupfrontier

SGE

CME

Metafrontier

CGE

Groupfrontier

Selfevaluation strategy

SGMI

SMTR

CMTR

CGMI

STGI

SMTI

CMTI

STGI

Fig. 3. Relationship among different methods.

CME obtained by model (2) may be unstable, because the value of θdj model (1) may have multiple optimal solutions. On the other hand, model (9) may also have multiple optimal solutions. The non-uniqueness affects the application of CMFA method. The cross-evaluation strategy can be used here to avoid this problem (Li et al., 2018b). The aggressive strategy of cross-evaluation is taken as the example to demonstrate obtaining a relatively unique optimal solution of CMFA (i.e., the CMFA-AS method). Step 1: Introduce a cross-evaluation strategy to obtain the relaCME∗ . The multiple optimal solutions of model (1) is tively unique θdj a classic problem in the study of cross-efficiency. As a remedy to this problem, it is necessary to introduce a cross-evaluation strategy to ensure the uniqueness of the optimal solution for model (1). For example, the aggressive strategy of cross-evaluation is formulated as model (4) to make secondary optimization obtain the relatively unique weights of inputs/outputs (Doyle & Green, 1994). Then, the relatively unique CME of DMUs can be obtained via models (2)–(3). Step 2: Construct a new model of self-evaluation strategy to CGE∗ . Comparing the constraints of calculate the relatively unique θdd models (4) and (9), it is not certain that model (9) has a feasible solution after introducing the aggressive strategy. Thus, a new model of self-evaluation strategy is formulated as follows.

Max

s 

urd yrd −

m  i=1

s  r=1 s 

n 

vid

m 

(15)

i=1 CME∗ urd yr j − θdj ∗

r=1

m 

i=1

urd∗∗ yr j

vid∗∗ xi j

.

(16)

Min

s 

urd

m  i=1

s 

yr j

j=1, j=d

r=1

s.t.

k 

k 

vid

xi j = 1;

j=1, j=d

CGE∗ urd yrd − θdd ∗

r=1

vid xid = 0;

i=1

r=1 s 

m 

urd yr j −

m 

(17)

vid xi j ≤ 0, j = 1, . . . , k;

i=1 CME∗ urd yr j − θdj ∗

m 

vid xi j ≥ 0, j = 1, . . . , k;

i=1

∀urd , vid ≥ 0; r = 1, . . . , s; i = 1, . . . , m;

xi j = 1;

vid xi j ≤ 0, j = 1, . . . , k;

r=1

= m

Step 3: Construct a new model of cross-evaluation strategy to calculate the relatively unique CGE matrix. According to the study of Doyle and Green (1994) and model (15), the aggressive strategy can be further extended to a new model as follows.

r=1

j=1, j=d

urd yr j −

θ

s

CGE∗ dd

s 

vid xid

i=1

r=1

s.t.

m 

s m because it can achieve the same effect as r=1 urd yrd / i=1 vid xid  ∗∗  ∗∗ (Carrillo & Jorge, 2018). Assuming that (vid , urd ) is the optimal CGE∗ can be obtained solution to model (15), the relatively unique θdd as follows:

vid xi j ≥ 0, j = 1, . . . , k;

i=1

∀vid , urd ≥ 0; CME∗ denotes the relatively unique CME of DMU from where θdj j DMUd relative to the meta-frontier based on the aggressive crossevaluation strategy. The objective function is used to maximize the self-evaluation efficiency of DMUd relative to the group-frontier,

CGE∗ denotes the self-evaluation efficiency of DMU obwhere θdd d tained by model (16). The objective function of model (17) is used to minimize the efficiency of the other DMUs in Group 1, and the second constraint is used to ensure that the self-evaluation efficiency of DMUd remains unchanged. The other constraints are the same as model (15). Based on model (17), the relatively unique weights of inputs/outputs can be obtained to calculate the CGE with its efficiency matrix by models (2)–(3). Step 4: Based on the relatively unique CME and CGE matrices, CMTR, CTGI, CGMI, and CMTI can be obtained with models (11)–(14). In summary, the multiple optimal solutions of CMFA caused by the non-uniqueness of optimal solutions of models (1) and

224

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(9) are solved by the CMFA-AS method, and the stability of CMFA is greatly improved. Theorem 4. Both models (15) and (17) have a feasible solution. Proof. Assume that (vid∗ , urd∗ ) is the optimal solution of model s m ∗ CME∗ = ∗ (4). Then, θdj ( j = 1, . . . , k), (i.e., r=1 urd yr j / i=1 vid xi j s  m  ∗ CME∗  ∗ ∗ i=1 vid xi j = 0 ( j = 1, . . . , k)). Furthermore, r=1 urd yr j − θdj the first and second constraints of model (15) are the same as the first and third constraints of model (4). Thus, (vid∗ , urd∗ ) is the feasible solution to model (15). Similarly, assume that (vid∗∗ , urd∗∗ ) is the optimal solution s m ∗∗ CGE∗ = ∗∗ of model (15). Then, θdd (i.e., r=1 urd yr j / i=1 vid xi j s  ∗∗ y − θ CGE∗ ∗ m v∗∗ x = 0). u Additionally, the first, r=1 rd r j i=1 id i j dd third, and fourth constraints of model (17) are the same as the first-to-third constraints of model (15). Thus, (vid∗∗ , urd∗∗ ) is the feasible solution of model (17).  Different cross-evaluation strategies are applicable to different decision-making environments (Wu, Chu & Sun, 2016). Thus, DMUs can flexibly change the cross-evaluation strategy according to their decision needs. For instance, the aggressive strategy regards the other DMUs as competitors. Additionally, if a DMU regards the other DMUs as collaborators, a benevolent strategy can be chosen to ensure the uniqueness of the optimal solution, and the strategy model of cross-evaluation in the CMFA method can convert the objective functions of models (4) and (17) from minimum to maximum. 5. Discussion on classification method CMFA is used to evaluate the cross-efficiency of DMUs when considering the technology gap among them. Thus, classifying different DMUs into several groups is a necessary pre-requisite for applying CMFA. In this section, subjective experience classification method is introduced, and then all DMUs are divided into several groups based on their technical levels. Its specific classification process is that according to the characteristics of DMU production technology (e.g., policy differences, economic-level differences, and geographical differences), all DMUs can be divided directly by the subjective experience of decision makers or relevant experts. Actually, there are many classification methods suitable for different decision-making situations in the existing research, such as fuzzy cluster method (Guillon, Lesot & Marsala, 2019). Decisionmakers can choose appropriate classification methods to classify DMUs based on their actual needs, and then CMFA can be applied to analyze efficiency. 6. Illustrative applications In this section, two examples are provided to illustrate the proposed CMFA method. One is a numerical example, which shows the calculation process and method comparison. The other is an empirical example demonstrating its application in practice. All computational processes are accomplished with MATLAB R2014b. 6.1. Numerical example 6.1.1. Efficiency analysis of numerical example A numerical example is used to illustrate the proposed CMFA analysis method, in which there are six DMUs with two inputs and one output, as shown in Table 1. According to the efficiency of DMUs, we assume that DMU1DMU3 are Group 1, while DMU4-DMU6 are Group 2; and there are technology heterogeneity between the two groups. Then, based on

Table 1 Inputs/output and efficiency data of numerical example. DMU

1 2 3 4 5 6

Inputs

Output

x1

x2

y1

5 4 5 5 7 10

12 6 6 8 8 7

7 5 6 3 4 4

CC R θdd

1 1 1 0.474 0.5 0.571

CC R Note that θdd is the efficiency obtained by the traditional DEA method (i.e., model (1)).

the CMFA-AS method, the CME and CGE matrices are obtained as shown in Table 2. According to the CME and CGE matrices, the other efficiency information can be obtained as follows. As shown in Table 3, all DMUs are fully ranked by CME, and DMU3 is the best DMU. Whereas the CME of DMU3 is greater than its CGE, the CMEp of all DMUs is not greater than their CGE, because CME includes the peer-evaluation results of DMU3 from the DMUs of Group 2, which cannot reflect the technology gap between meta-frontier and group-frontier 1. Thus, CMTR should be calculated by CMEp and CGE. All CMTRs in Group 1 are equal to 1, indicating that there are no technology gaps between meta-frontier and group-frontier 1. Actually, Group 1 is made up CC R = 1, thus this phenomenon is reasonable. of all DMUs with θdd All CMTRs of Group 2 are less than 1. Thus, there are obvious technology gaps between meta-frontier and group-frontiers 2. Through further efficiency decomposition, we find that the inefficiency of the DMUs in Group 1 is managerial inefficiency. For instance, the CTGI and CGMI of DMU1 are 0 and 0.139, respectively. Additionally, the technology gap inefficiency has a greater influence than managerial inefficiency on the inefficiency of DMUs in Group 2. In particular, the inefficiency of DMU5 is entirely caused by technology gap inefficiency.

6.1.2. Comparison of methods In this section, the CMFA is compared with traditional cross efficiency method and SMFA method for further illustrating its effectiveness. First, traditional cross-efficiency method is directly used to calculate the CGE of DMUs, and its results are denoted as TCGE, then TCGE=CMEp . As shown in Table 3, the CME of DMU3 is greater than the value of TCGE. This means that the group-frontier 1 goes beyond the meta-frontier of all DMUs. However, Group 1 is part of all DMUs. Thus, the TCGE of DMU3 is not reasonable. CMFA effectively overcomes the problem of traditional cross-efficiency, and CME is not always greater than CGE. Additionally, CGE is also greater than TCGE for the DMUs of Group 2, owing to the technology gaps between meta-frontier and group-frontier 2. Secondly, as shown in Fig. 4, from the results of efficiency evaluation, we find that SME ≥ CME and SGE ≥ CGE. Actually, SMFA is equivalent to the traditional meta-frontier DEA method, allowing each DMU to measure its efficiency with its favorable weights. Thus, the efficiency is overestimated. CMFA overcomes this problem by combining self-evaluation with peer-evaluation, and the peer-evaluation is the cause of these efficiency differences. Additionally, it reflects that CMFA is effectively tailored to recognize the inefficiency of DMUs better than SMFA. For example, the means of SME and SGE of Group 1 are 1. This shows that SMFA failed to identify the inefficiency of all DMUs in Group 1, whereas CMFA found that managerial inefficiency exists for these DMUs, because the means of CGMI of Group 1 was 0.093.

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225

Table 2 CME and CGE matrices. CME

CGE DMU1

DMU2

DMU3

DMU4

DMU5

DMU6

DMU1

DMU2

DMU3

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6

1.000 1.000 0.583 1.000 0.583 0.583

0.893 1.000 0.833 1.000 0.833 0.833

0.857 1.000 1.000 1.000 1.000 1.000

0.429 0.474 0.375 0.474 0.375 0.375

0.408 0.480 0.500 0.480 0.500 0.500

0.286 0.358 0.571 0.358 0.571 0.571

1.000 1.000 0.583

0.893 1.000 0.833

0.857 1.000 1.000

Value

DMU

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

DMU4

DMU5

DMU6

1.000 1.000 0.656

0.952 1.000 0.875

0.667 0.736 1.000

Group1 Group2

Indicators Fig. 4. Efficiency analysis of SMFA and CMFA.

Table 3 Efficiency results of numerical example. DMU Group 1

Group 2

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6

CME

CMEp

CGE

CMTR

CTGI

CGMI

CMTI

0.792 0.899 0.976 0.417 0.478 0.453

0.861 0.909 0.952 0.408 0.493 0.500

0.861 0.909 0.952 0.885 1 0.801

1 1 1 0.473 0.525 0.629

0 0 0 0.478 0.449 0.300

0.139 0.091 0.048 0.115 0.058 0.199

0.139 0.091 0.048 0.592 0.507 0.500

Note that CME, CGE, CMTR, CTGI, CGMI, and CMTI are the efficiency indicators obtained by CMFA. The calculation of CMTR of DMU1–DMU3 does not consider the CME of DMU1–DMU3 from DMU4–DMU6, as shown in model (11). Additionally, CMEp represents the mean of part CMEs that is used to calculate CMTR.

opment direction in these areas. Therefore, many studies have used this classification approach to analyze the country’s regional economic problems (Tian, Shi & Sun, 2017). This paper adopts this classification to compare the differences of regional transportation industry between the meta-frontier and the group-frontier. Because y2 is an undesirable output, and the results are better when smaller, it does not satisfy the assumption of desirable output in traditional DEA method. To treat the undesirable output as an input is one of the most frequently used approaches to deal with undesirable output within DEA framework (Halkos & Petrou, 2019), and this paper thus selects this method to address y2 for evaluating efficiency. Then, the proposed CMFA method can be applied to analyze the efficiency of the transportation industry in China.

6.2. Empirical example 6.2.1. Data and variables To further illustrate the effectiveness of CMFA, the transportation industry of 30 regions in mainland China was evaluated. Owing to a lack of data, Tibet was not considered. For the efficiency evaluation, input indicators generally included labor, capital, and energy consumption (Cui & Li, 2014; Mardani et al., 2017). The number of employed people (x1 ), capital stock (x2 ), and total energy consumption (x3 ) of transportation industry are thus selected as inputs. Gross regional product of the transportation industry (y1 ) and carbon dioxide emissions (y2 ) are taken as outputs, representing the economic benefits and environmental effects, respectively. The data comes from China Statistical Yearbook 2008–2017 and China Energy Statistical Yearbook 2008–2017, and its descriptive statistics are summarized in Table 4. Different areas in China have different technical levels. According to the differences in natural resources, economic development level, transportation conditions, etc., all regions can be divided into east, central, and west areas (Tian & Lin, 2018). The relevant data from the China Statistical Yearbook, is summarized in Table 5. There are obvious differences in policy formulation and devel-

6.2.2. Empirical results analysis in 2016 Based on the CMFA-AS method, we evaluated the performance of China’s transportation industry in 2016, and the results are shown in Table 6. As shown in Table 6, there are two efficient DMUs in SME and eight efficient DMUs in SGE. These DMUs cannot be further assessed. However, all DMUs are fully ranked in CME and CGE, owing to their cross-efficiency and ability to comprehensively and objectively evaluate DMUs via different evaluation perspectives. When comparing the values among SME, SGE, CME, and CGE, it is evident that the efficiency of most DMUs relative to the group-frontier is greater than their efficiency relative to the metafrontier in both self-evaluation and cross-evaluation. For example, the values in Inner Mongolia were 1.0 0 0, 0.692, 0.960, and 0.642. Thus, the impact of different technical levels on efficiency is obvious, especially for the central and west areas of China. In another example, the CMTR of Gaosu was 0.592. Thus, the group-frontier of west area was the farthest from the meta-frontier. For most DMUs in the east area, the technology gaps between the metafrontier and the group-frontier were not significant, because some DMUs (i.e., Hebei, Liaoning, Jiangsu) in these areas performed well

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Table 4 Descriptive statistics for the transportation industry of China in 2007–2016. Indicators

x1 (10,0 0 0 persons)

x2 (100 million yuan)

x3 (10,0 0 0 tce)

y1 (100 million yuan)

y2 (10,0 0 0 tons)

Max. Min. Mean Std. dev.

85.40 2.81 24.17 14.96

2138.38 37.23 586.51 383.30

3096.58 64.39 881.90 584.10

1836.19 28.76 524.51 383.21

6479.32 139.55 1863.50 1226.39

Note: Because the data of capital stock was not available in China, we used the method proposed by Zhang and Wu (2004) to estimate x2 . The data takes 20 0 0 as the base year. “tce” is “ton of standard coal equivalent.” y2 was obtained by using the IPCC’s method (2006), which is one of the most commonly used methods to calculate the value of carbon dioxide emissions.

Table 5 Areas division of regions. Area

Regions

East

Beijing, Fujian, Guangdong, Hainan, Hebei, Jiangsu, Liaoning, Shangdong, Shanghai, Tianjin, and Zhejiang Anhui, Henan, Heilongjiang, Hubei, Hunan, Jilin, Jiangxi, and Shanxi Chongqing, Gaosu, Guangxi, Guizhou, Inner Mongolia, Ningxia, Qinghai, Shaanxi, Sichuan, Xinjiang, and Yunnan

Central West

with economic benefits and environmental effects. These DMUs narrow the technology gap between their own group-frontier and meta-frontier. According to the values of CTGI, CGMI, and CMTI, the inefficient information of DMUs were further decomposed. For most regions, managerial inefficiency was the primary reason for DMU inefficiency. For example, the values of CTGI, CGMI, and CMTI in Beijing were 0.015, 0.307, and 0.322. Thus, more than 95% of Beijing’s inefficiency, relative to the meta-frontier, was caused by management inefficiency. Technology gap inefficiency was also

important in the central and west areas. For example, more than 80% of Hunan’s inefficiency, relative to the meta-frontier, was caused by technology gap inefficiency. Generally, the mean of CME decreases from the east to west. This is directly proportional to the development level of the transportation industry. According to the values of CGE and CMTR, we know that the development of regions in the east and central areas is reasonably balanced. However, for the regions in the west area, the gap in development level is obvious. For example, the CGEs of Inner Mongolia and Yunnan were 0.960 and 0.219, respectively. This could be due to the continuance of different technology gaps between different regions in the west area. 6.2.3. Empirical results analysis during 2007–2016 For illustrating the stability of evaluation results, the efficiency of regional transportation industry in China during 2007–2016 is further analyzed by the proposed CMFA-AS method, and the evaluation results are shown in Fig. 5 As shown in Fig. 5, the gap of CME between the transportation industries of different areas widened during the period of 2007– 2016. The changing trends of CGE and CMTR show that the growth

Table 6 Evaluation results of China’s transportation industry in 2016. Region

SME

CME

SGE

CGE

CMTR

CTGI

CGMI

CMTI

East area Beijing Fujian Guangdong Hainan Hebei Jiangsu Liaoning Shangdong Shanghai Tianjin Zhejiang Central area Anhui Henan Heilongjiang Hubei Hunan Jilin Jiangxi Shanxi West area Chongqing Gaosu Guangxi Guizhou Inner Mongolia Ningxia Qinghai Shaanxi Sichuan Xinjiang Yunnan

0.812 0.949 0.874 0.795 0.371 1.000 0.904 1.000 0.771 0.738 0.844 0.681 0.607 0.446 0.874 0.514 0.449 0.684 0.422 0.675 0.789 0.560 0.457 0.842 0.533 0.997 0.692 0.678 0.268 0.479 0.441 0.544 0.230

0.666 0.580 0.691 0.635 0.346 0.987 0.787 0.736 0.703 0.527 0.726 0.605 0.521 0.419 0.719 0.422 0.397 0.594 0.406 0.572 0.634 0.419 0.423 0.299 0.441 0.662 0.642 0.555 0.198 0.393 0.388 0.453 0.153

0.812 0.949 0.874 0.795 0.371 1.000 0.904 1.000 0.771 0.738 0.844 0.681 0.835 0.711 1.000 0.670 0.695 1.000 0.713 0.890 1.000 0.766 0.740 1.000 0.706 1.000 1.000 0.929 0.422 0.796 0.746 0.849 0.243

0.718 0.693 0.663 0.715 0.360 0.995 0.852 0.867 0.742 0.623 0.774 0.612 0.763 0.649 0.973 0.597 0.624 0.927 0.628 0.823 0.884 0.644 0.663 0.631 0.649 0.921 0.960 0.845 0.340 0.648 0.643 0.560 0.219

0.952 0.976 0.956 0.939 0.948 0.967 0.943 0.944 0.948 0.945 0.964 0.944 0.724 0.682 0.771 0.780 0.667 0.674 0.678 0.749 0.791 0.699 0.640 0.592 0.725 0.824 0.664 0.713 0.695 0.612 0.662 0.755 0.807

0.031 0.015 0.027 0.042 0.018 0.032 0.048 0.038 0.038 0.027 0.028 0.033 0.211 0.213 0.220 0.131 0.216 0.311 0.211 0.205 0.182 0.202 0.250 0.276 0.180 0.150 0.334 0.246 0.108 0.266 0.227 0.146 0.040

0.282 0.307 0.337 0.285 0.640 0.005 0.148 0.133 0.258 0.377 0.226 0.388 0.237 0.351 0.027 0.403 0.376 0.073 0.372 0.177 0.116 0.356 0.337 0.369 0.351 0.079 0.040 0.155 0.660 0.352 0.357 0.440 0.781

0.314 0.322 0.364 0.327 0.659 0.036 0.196 0.171 0.297 0.404 0.254 0.421 0.448 0.564 0.247 0.534 0.591 0.384 0.583 0.382 0.298 0.559 0.586 0.645 0.531 0.229 0.374 0.402 0.768 0.617 0.584 0.586 0.821

SME, CME, SGE, CGE, CMTR, CTGI, CGMI, and CMTI are the efficiency indicators obtained by SMFA and CMFA.

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a 0.75 0.7 0.65

CME

Value

0.6 Whole East Central West

0.55 0.5 0.45 0.4 0.35 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Year

b 0.85 0.8

CGE Value

0.75 Whole

0.7

East

0.65

Central West

0.6

0.55 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Year

c 1 0.95 0.9 CMTR

Value

0.85 0.8

0.75

Whole

0.7

East Central

0.65

West

0.6 0.55 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Year Fig. 5. Efficiency analysis during 2007–2016.

of the east’s CME was primarily caused by the improvement of its CGE with an increase of its management level. Additionally, the high CMTR with low CGE in the east area means that there is an efficiency division problem in this area. For the central area, the technology gaps were the primary reasons for the change of its CME during 2007–2016. Meanwhile, technology gap inefficiency and management inefficiency were the reasons behind the decline of CME in the west area, and they have different influences on CME in different periods. Specifically, with decreasingly CGE, the

increase of management inefficiency was the main reason for the decline of CME in west area during 2007–2012; and the reason was replaced by the increase of technology gap inefficiency during 2013–2016. Based on the results of Table 6 and Fig. 5, some suggestions are proposed for the transportation industry in China. First, the east area should maintain its technological advantages while continuing to eliminate management inefficiency to improve its CGE. Second, the central area should focus on narrowing the technology

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gap between meta-frontier and its group-frontier to improve MTR. Third, both technology level and management level should be simultaneous improved in the west area.

Table 7 List of abbreviations. Abbreviation

Full name

Abbreviation

Full name

DEA

Data envelopment analysis Decision-making unit

CCR

Charnes, Cooper, and Rhodes Meta-frontier efficiency Meta-technology ratio

7. Conclusion DMU

As one of the most popular evaluation tools, the DEA method has attracted a great deal of attention around the world. However, efficiency overestimation and technological heterogeneity are important factors affecting the development of DEA. By introducing the meta-frontier analysis framework to the cross-efficiency method, a new CMFA method was proposed to grasp the relationship between the meta-frontier and group-frontier in the cross-evaluation environment, solving the efficiency overestimation problem of the traditional meta-frontier DEA method. Meanwhile, the technology gap between different frontiers was measured from a more comprehensive perspective, and more detailed reasons about the inefficiency of DMUs were obtained. Subsequently, the non-uniqueness of an optimal CMFA solution was discussed, and the aggressive cross strategy was chosen as the example to construct CMFA-AS, used to ensure the stability of the optimal solution. Then, CMFA can be applied scientifically and widely. Specifically, CMFA combines self-evaluation with peerevaluation, and its results, CME and CGE, can not only measure the efficiency of DMUs relative to the meta-frontier and group-frontier more objectively than SME and SGE, they also have the acuity to recognize the inefficiency of DMUs, enabling decision makers to more rationally set up benchmarks for efficiency improvements. By calculating CMTR, the inefficiency of DMUs can be further decomposed into CTGI and CGMI. CTGI represents the inefficiency caused by a technology gap, and improving the technology level of the whole group is the only way to eliminate it. CGMI represents the inefficiency caused at the management level, and DMU eliminates it via its own efforts. According to different reasons of inefficiency, a decision maker can formulate measures to improve efficiency in a targeted way. Finally, two examples were presented to illustrate the effectiveness and stability of the proposed CMFA method, and the results showed that the real efficiency of DMUs is always lower than the efficiency value obtained by traditional meta-frontier analysis. Meanwhile, the influences of technology gap inefficiency and management inefficiency on cross-efficiency can be clearly identified. However, the dynamic correlation of panel data has not been considered in the CMFA, and the degree of technological change is also not reflected in different periods. These are the limitations of this paper, and these are our main research directions for the future. Acknowledgments This research is supported by National Natural Science Foundation of China (#71801050, #71371053, #71872047, #71701050), and Social Science Planning Fund project of Fujian Province (#FJ2018C014, #FJ2017C033), Natural Science Foundation of Fujian Province (#2019J01637). Appendix Table 7 shows the full name of all abbreviations in the paper to help reader understand better. References Ang, S., Chen, M. H., & Yang, F. (2018). Group cross-efficiency evaluation in data envelopment analysis: An application to Taiwan hotels. Computers & Industrial Engineering, 125, 190–199. Carrillo, M., & Jorge, J. M. (2018). An alternative neutral approach for cross-efficiency evaluation. Computers & Industrial Engineering, 120, 137–145.

GE SMFA

SME

SGE SMTR

STGI

SGMI

SMTI

CMFA-AS

Group frontier efficiency Self-evaluation based on meta- frontier analysis Self-evaluation meta-frontier efficiency Self-evaluation group frontier efficiency Self-evaluation meta-technology ratio Self-evaluation technology gap inefficiency Self-evaluation managerial inefficiency Self-evaluation meta-frontier inefficiency Cross-evaluation based on meta-frontier analysis with aggressive strategy

ME MTR CMFA

CME

CGE CMTR

CTGI

CGMI

CMTI

TCGE

Cross-evaluation based on meta-frontier analysis Cross-evaluation meta-frontier efficiency Cross-evaluation group frontier efficiency Cross-evaluation meta-technology ratio Cross-evaluation technology gap inefficiency Cross-evaluation managerial inefficiency Cross-evaluation meta-frontier inefficiency Cross-evaluation group frontier efficiency obtained by traditional cross-efficiency method

Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. Chiu, C. R., Liou, J. L., Wu, P. I., & Fang, C. L. (2012). Decomposition of the environmental inefficiency of the meta-frontier with undesirable output. Energy Economics, 34(5), 1392–1399. Cui, Q., & Li, Y. (2014). The evaluation of transportation energy efficiency: An application of three-stage virtual frontier DEA. Transportation Research Part D, 29(6), 1–11. Davtalab-Olyaie, M. (2019). A secondary goal in DEA cross-efficiency evaluation: A “one home run is much better than two doubles” criterion. Journal of the Operational Research Society, 70(5), 807–816. Diallo, B. (2018). Bank efficiency and industry growth during financial crises. Economic Modelling, 68, 11–22. Doyle, J. R., & Green, R. (1994). Efficiency and cross-efficiency in data envelopment analysis: Derivatives, meanings and uses. Journal of the Operational Research Society, 45, 567–578. Feng, C., Huang, J. B., & Wang, M. (2018). Analysis of green total-factor productivity in China’s regional metal industry: A meta-frontier approach. Resources Policy, 58, 219–229. Guillon, A., Lesot, M. J., & Marsala, C. (2019). A proximal framework for fuzzy subspace clustering. Fuzzy Sets and Systems, 366, 34–45. Halkos, G., & Petrou, K. N. (2019). Treating undesirable outputs in DEA: A critical review. Economic Analysis and Policy, 62, 97–104. IPCC, (2006). IPCC guidelines for national greenhouse gas inventories. https://www. ipcc-nggip.iges.or.jp/public/2006gl/index.html. Kounetas, K., & Napolitano, O. (2018). Modeling the incidence of international trade on Italian regional productive efficiency using a meta-frontier DEA approach. Economic Modelling, 71, 45–58. Li, F., Zhu, Q. Y., & Liang, L. (2018a). Allocating a fixed cost based on a DEA-game cross efficiency approach. Expert Systems with Applications, 96, 196–207. Li, F., Zhu, Q., Chen, Z., & Xue, H. B. (2018b). A balanced data envelopment analysis cross-efficiency evaluation approach. Expert Systems with Applications, 106, 154–168. Li, L. B., Liu, B. L., Liu, W. L., & Chiu, Y. H. (2017). Efficiency evaluation of the regional high-tech industry in China: A new framework based on meta-frontier dynamic DEA analysis. Socio-Economic Planning Sciences, 60, 24–33. Liu, H. H., Song, Y. Y., & Yang, G. L. (2019). Cross-efficiency evaluation in data envelopment analysis based on prospect theory. European Journal of Operational Research, 273(1), 364–375. Liu, J. S., Lu, L. Y. Y., & Lu, W. M. (2016). Research fronts in data envelopment analysis. Omega, 58, 33–45. Mardani, A., Zavadskas, E. K., Streimikiene, D., Jusoh, A., & Khoshnoudi, M. (2017). A comprehensive review of data envelopment analysis (DEA) approach in energy efficiency. Renewable & Sustainable Energy Reviews, 70, 1298–1322.

L. Chen, Y. Huang and M.-J. Li et al. / European Journal of Operational Research 280 (2020) 219–229 O’Donnell, C. J., Rao, D. S. P., & Battese, G. E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique and extensions. New Directions for Program Evaluation, 32, 73–105. Song, M. L., Zhu, Q. Y, Peng, J., & Gonzalez, E. D. R. S. (2017). Improving the evaluation of cross efficiencies: A method based on Shannon entropy weight. Computers & Industrial Engineering, 112, 99–106. Tian, G., Shi, J., Sun, L., Long, X. L., & Guo, B. H. (2017). Dynamic changes in the energy–carbon performance of Chinese transportation sector: A meta-frontier non-radial directional distance function approach. Natural Hazards, 89(2), 585–607. Tian, P., & Lin, B. Q. (2018). Regional technology gap in energy utilization in China’s light industry sector: Non-parametric meta-frontier and sequential DEA methods. Journal of Cleaner Production, 178, 880–889.

229

Wang, Y. M., & Chin, K. S. (2010). A neutral DEA model for cross-efficiency evaluation and its extension. Expert Systems with Applications, 37(5), 3666– 3675. Wang, Y. M., & Chin, K. S. (2011). The use of OWA operator weights for cross-efficiency aggregation. Omega, 39(5), 493–503. Walheer, B. (2018). Aggregation of metafrontier technology gap ratios: The case of European sectors in 1995-2015. European Journal of Operational Research, 269, 1013–1026. Wu, J., Chu, J., Sun, J., & Zhu, Q. Y. (2016). DEA cross-efficiency evaluation based on Pareto improvement. European Journal of Operational Research, 248(2), 571– 579. Zhang, J., & Wu, G. (2004). The estimation of china’s provincial capital stock: 1952-20 0 0. Economic Research Journal, 10(1), 35–44.