Dealing with Interval DEA Based on Error Propagation and Entropy: A Case Study of Energy Efficiency of Regions in China Considering Environmental Factors

2.1 Evaluation of the Midpoint Value θ_d

We assume that there are a set of n DMUs, and each DMUj (j = 1, 2, · · ·, n) produces s outputs using m inputs which are denoted as x_ij (i = 1, 2, · · ·, m) and y_rj (r = 1, 2, ···, s) respectively. For any evaluated DMUk (k = 1, 2, · · ·, n), the efficiency score θ_kk can be calculated by using the following CCR model.

Based on self-evaluation, the CCR model, however, fails to discriminate all DMUs when the number of the efficient DMUs is more than one. To improve the discriminatory power of DEA, cross efficiency based on peer-evaluation was proposed by Shannon[3]. The model is as follows.

where θ_kd denotes the dth DMU’s cross efficiency calculated by the optimal weights of the kth DMUs in CCR model.

Although it has been widely used because of its strong discriminatory power, cross efficiency exist two main disadvantages, namely, non-unique weights and average aggregation assumption.

To overcome the drawback of non-unique weights, we employ the modified cross efficiency proposed by Jahanshahloo et al.[8] which maximizes cross efficiency and CCR efficiency simultaneously. It should be noted that modified cross efficiency can be more objective and calculated easier than other methods. The model is as follows.

where θkk∗ is the optimum value of the kth DMU in model (3).

To overcome the drawback of average aggregation assumption, we employ a new concept of entropy from the perspective of rough set theory proposed by Liang et al.[14] to aggregate the modified cross efficiency. It should be noted that the new concept of entropy can reflect information content more overall than the entropy defined by Shannon[13]. So the new concept of entropy E_k and the entropy weight w_k of the kth evaluating DMU are defined as follows.

where pkd=θkd∑l=1nθkl,k=1,2,⋯,n,d=1,2,⋯,n.

Then θ_d, the midpoint value of ultimate cross efficiency score of DMUd, is calculated as follows.

2.2 Evaluation of the Maximum Error Estimation Δθ_d

According to θkd=∑r=1sμrkyrd∑i=1mνikxid,∑i=1mνikxid=1 and Equation (2), Δθ_kd is calculated as follows.

According to pkd=θkd∑l=1nθkl and Equation (2), Δp_kd is calculated as follows.

According to Δp_kd, and Equations (6) and (2), ΔE_k is calculated as follows.

According to ΔE_k, and Equations (7) and (2), Δw_k is calculated as follows.

According to Δw_k, and Equations (8) and (2), Δθ_d is calculated as follows.

Then the ultimate cross efficiency score of DMUdθd∼ is determined as θ_d ± Δθ_d.

2.3 Ranking DMUs Completely with Interval Efficiency

Directional distance index in the form of error distribution between DMU_d and DMU_b is defined as follows to rank all DMU_s completely in term of the calculated ultimate cross efficiency score.

So the entire directional distance index of DMU_d is expressed as follows.

The details of directional distance index can be found in the paper written by Song et al.[15].

3.1 Model Adjustment

Undesirable outputs are added to DEA models when we measure energy efficiency considering environmental factors. Undesirable outputs are generated when desirable outputs are produced by consuming inputs. We cannot increase desirable outputs and reduce undesirable outputs simultaneously. In order to reasonably deal with this kind of efficiency evaluation problem, undesirable outputs should be modeled based on the concept of weak disposability proposed by Färe and Grosskopf [35].

Assume that there are a set of n DMUs, and each DMU_j (j = 1,2, · · ·, n) produces s desirable outputs and nbad undesirable outputs using m inputs which are denoted as xij(i=1,2,⋯,m),yrjg(r=1,2,⋯,s)andyzjb(z=1,2,⋯,nbad), respectively. So model (3) and (4) can be adjusted to model (16) and (17) as follows respectively.

According to

and Equation (2), Equation (9) can be adjusted to Equation (18) as follows.

3.2 Data

29 regions (provinces, autonomous region and municipalities) in Mainland China, excluding Chongqing, Tibet, Hong Kong, Macao and Taiwan, are selected from the perspective of data availability. GDP is taken as the desirable output, CO₂ emissions as the undesirable output, capital stock, labor force and energy consumption as the three inputs. Labor force is calculated as the mean of the number of employee at the beginning and the end of the current year. Capital stock is estimated using perpetual inventory method. We estimate CO₂ emissions by the product of carbon energy consumption, carbon emission coefficient and carbon conversion coefficient. All the data are collected from China Statistical Yearbook and China Energy Statistical Yearbook. Relevant descriptive statistics are summarized in Table 1.

In the real applications, it is unreasonable that just regarding the statistical data at the end of the year as the indicator of an attribute because of various limitations such as statistical errors. In the paper, therefore, it should be noted that we construct data interval of each region by regarding the data at the beginning of the current year as the lower bound and the data at the end of the current year as the upper bound when we combine DEA with error propagation to measure energy interval efficiency of regions in China.

3.3 Results and Discussion

The results are shown in Table 2, and the relevant discussions are as follows.

The traditional CCR model fails to discriminate all DMUs when the number of efficient DMUs is more than 1. When we consider environmental factors (undesirable outputs) into it, CCR model will become weaker from the perspective of discrimination power. The assumption of weak disposability of undesirable outputs can decrease production possibility set and then increase the number of efficient DMUs. As shown in Table 2, when environmental factors considered, the number of efficient regions is 26, in other words, only Jiangxi, Guangxi and Xinjiang three are inefficient.

In contrast, we use the proposed method in Section 2 to measure energy efficiency of regions in China.

First we focus on ranking results. In the latter method in Table 2, the top five most efficient regions are Guangdong, Shanghai, Jiangsu, Zhejiang and Shandong. Guangdong has the highest energy efficiency score and it ranks number 1. In 2010, The GDP value of Guangdong is 4601.30 billion RMB, performs best among all regions and is approximately 3.5 times larger than the national average level(1477.96 billion RMB), which may be the reason why Guangdong ranks number 1. Shanghai, Jiangsu, Zhejiang and Shandong follow in turn, they are all coastal regions and their economies develop fast. The bottom five most inefficient regions are Qinghai, Anhui, Yunnan, Guangxi and Hainan. Qinghai has the lowest energy efficiency score in 2010, which may be due to its less development. The GDP value of Qinghai in 2010 is 135.04 billion RMB, approximately 11 times smaller than the national average level (1477.96 billion RMB). Similarly, Anhui, Yunnan, Guangxi and Hainan are all located in the central and west area and their economy level are relatively low. To our surprise Beijing ranks 21 in the proposed method. But it may be reasonable in some way. Although it is the capital of China, Beijing as a municipality cannot contend against some provinces like Guangdong from the perspective of inputs.

Then we focus on interval efficiency. The greatest advantage of the proposed method in Section 2 is that it can show the changing process of energy efficiency of regions. As shown in Table 2, we can see the lower bound and the upper bound of energy efficiency of each region in China in 2010. We select Guangdong as an example because of its first ranking in the proposed method. The lower bound and the upper bound of energy efficiency of Guangdong are 0.5234 and 2.1236 respectively, that is, the efficiency changing scope of Guangdong in 2010 is from 0.5234 to 2.1236.