1. Introduction
When decision making units (DMUs) have multiple inputs and outputs, data envelopment analysis (DEA) is a well-known non-parametric programming technique for assessing the efficiency of these DMUs. The maximum of the ratio of a DMU’s weighted sum of outputs to its weighted sum of inputs is defined as the efficiency score of this DMU. If the efficiency score of a DMU is equal to 1, it is considered as efficient. Otherwise, it is inefficient. Usually, inefficient DMUs are considered as performing worse than efficient ones. Since DEA was proposed by Charnes et al. [
1], it has been widely applied to various cases of performance evaluation [
2,
3,
4,
5,
6,
7]. DEA models (both CCR (Charnes, Cooper and Rhodes) and BCC (Banker, Charnes, and Cooper) models) classify units into two groups: efficient and inefficient in the Pareto sense (see Sinuay-Shten et al. [
8]). In addition, DEA is not able to rank the efficient DMUs that all have an efficiency score of 1. In order to solve this problem, the cross-efficiency method was developed by Sexton et al. [
9]. The cross-efficiency method, as a DEA extension, could obtain the efficiency of each DMU by linking the weights of all DMUs. Its primary advantage is that all DMUs can be completely ranked [
10]. In addition, the cross-efficiency method could eliminate unrealistic weight results [
11].
With these advantages, the cross-efficiency evaluation has been extensively applied in various performance evaluation problems [
12,
13,
14,
15]. In spite of its wide applications, cross-efficiency evaluation still has some defects, such as non-uniqueness of the DEA optimal weights [
9]. Usually, the optimal weights obtained by traditional DEA models are non-unique. If a set of weights are selected arbitrarily, then cross-efficiency scores will be arbitrarily generated [
16]. To solve the problem of weight non-uniqueness, Sexton et al. [
9] improved the cross-efficiency evaluation method by incorporating a secondary goal model. Following the idea of Sexton et al. [
9], a number of scholars have proposed secondary goal models. For example, Liang et al. [
17] proposed three secondary goal models, and each secondary goal corresponds to a practical case scenario. Based on the models of Liang et al. [
17], Wang and Chin [
18] proposed the improved models by replacing the target efficiency. Jahanshahloo et al. [
19] improved traditional cross-efficiency evaluation by considering symmetric weights. Wu et al. [
20] and Contreras [
21] proposed improving the ranking position of the evaluated DMU when choosing weights. In the study of Lim [
22], the secondary goal was proposed to minimize (or maximize) the cross efficiencies of evaluated DMUs. Maddahi et al. [
23] proposed a proportional weight assignment secondary goal, making weights be assigned proportionally to input or output evaluated DMUs. In these secondary models, most models are benevolent or aggressive. In the benevolent (aggressive) model, the selected weights for evaluated DMUs are to make the cross-efficiencies of other DMUs as large (small) as possible. Different from the above ideas, scholars also have proposed new cross-efficiency models from different perspectives. For example, Cook and Zhu [
24] proposed a units-invariant multiplicative DEA model, directly generating the unique cross-efficiency scores. Based on the Pareto optimality, Wu et al. [
25] proposed the Pareto improvement cross-efficiency evaluation, which could obtain Pareto-optimal cross efficiencies for all DMUs.
Besides secondary goals in the cross-efficiency evaluation, scholars also have examined aggregating cross-efficiencies for obtaining cross-efficiency matrix. For example, Wu et al. [
26] introduced Shapley cooperative game theory into cross-efficiency evaluation, considering each DMU as a player, and proposed a Shapley DEA model to obtain all the weights of cross-efficiencies. Wang et al. [
27] considered that all cross-efficiencies of DMUs should have different preference weights. From the preference deviation degree, they proposed three different models for aggregating cross-efficiencies. Angiz et al. [
28] argued that the DMU may be more concerned about whether the assigned weights improve their ranking when weights are selected for their cross-efficiencies. Based on this idea, they proposed a ranking preference model. Yang et al. [
29] regarded the cross-efficiency as the independent evidence, and thus the evidence reasoning method was used to aggregate cross-efficiencies.
The traditional DEA or cross-efficiency models assume that the data of DMUs are known exactly. However, because of the existence of uncertainty, the data may be given in a fuzzy form. Therefore, a number of studies examined how to evaluate the efficiencies of DMUs with fuzzy data. For example, Cooper et al. [
30] proposed an imprecise DEA (IDEA) model, which can be transformed into a linear programming model based on a series of variable alternations and scale transformations. However, Lee et al. [
31] argued that IDEA model was complicated, and may lead to a rapid increase in computation burden. To solve this problem, Despotis and Smirlis [
32] proposed two improved models. Through these two improved models, the lower and upper efficiency of each DMU could be obtained. Wang et al. [
33] pointed out that Despotis and Smirlis’s model [
32] used two different production frontiers to measure the efficiencies of DMUs, and this may lead to the efficiencies of DMUs’ lack of comparability. To deal with such an issue, Wang et al. [
33] proposed the new DEA models based on a common frontier to obtain the interval efficiency of each DMU and a minimax regret-based approach was then used for ranking the interval efficiencies of all DMUs. To determine the range of interval efficiency of each DMU, Azizi and Jahed [
34] introduced a virtual ideal DMU into the DEA model. The efficiency of ideal DMU is definitely the largest among all the DMUs, so the worst and the best relative efficiencies of each DMU can be obtained. Then, the worst and the best relative efficiencies constitute an interval for the overall performance evaluation of each DMU. Wang and Chin [
35] proposed the fuzzy DEA models based on two pairs of expected value models to measure the optimistic and the pessimistic efficiencies of DMUs. They integrated two extreme efficiencies through a geometric average for obtaining the overall performances of the DMUs. Dotoli et al. [
36] proposed a novel approach by integrating the DEA cross-efficiency technique with the fuzzy logic framework. This approach not only maintains the cross-efficiency DEA discriminative power but also deals with uncertainty.
Although the existing cross-efficiency methods have well examined how to be aggressive or benevolent to DMUs when evaluating the efficiencies, the maximum discrimination of DMUs has been largely ignored. In addition, there is a paucity of research on aggregating interval cross-efficiency matrices. To fill these gaps, the present study proposes a new cross-efficiency method based on the entropy theory. In this new approach, the model of Wang et al. [
33] is first extended into cross-efficiency evaluation to obtain the intervals of all cross-efficiency values. Then, the DEA entropy model is used to calculate the weights of all interval cross-efficiencies. Finally, all DMUs are evaluated and ranked according to the distance to ideal positive cross-efficiency. This approach is illustrated and verified by a demonstrative case using data from China’s primary schools. We conclude that the proposed approach is effective to evaluate DMUs with interval data and can provide complete and fair results for all DMUs.
The rest of the paper is organized as follows.
Section 2 introduces the interval DEA models and
Section 3 presents the cross-efficiency evaluation method with interval data. The cross-efficiency model based on Shannon entropy is discussed in
Section 4, followed by a numerical demonstration using data from Chinese primary schools in
Section 5. Conclusions are presented in
Section 6.
2. Interval DEA Models
There are
n DMUs to be evaluated, and each DMU has
s different outputs and
m different inputs. Input
i and output
r for
are denoted as
and
, respectively. The input and output data may be imprecise because of uncertainty and thus only their bounded intervals
and
, with
and
, are provided. For measuring the efficiencies of the DMUs with interval data, Despotis and Smirlis [
32] proposed a linear problem model to generate the lower and upper bounds of the efficiency for each DMU, as shown in Model (1):
However, Wang et al. [
33] pointed out that Despotis and Smirlis [
32] used two different production frontiers to obtain interval efficiency, and thus all DMUs cannot be compared on the basis of a common evaluation criterion. In order to calculate the lower and upper bound of the efficiency of
, Wang et al. [
33] proposed two linear formulations to generate the bounded interval
, as follows:
and
In Models (2) and (3), is to be evaluated. and are the weights of the input i and output r, respectively. (or ) is the lower (or upper) efficiency for . is the non-Archimedean infinitesimal. From Models (2) and (3), it is clear that .
3. Cross-Efficiency Evaluation Method with Interval Data
Models (2) and (3) are the self-assessment models. The self-evaluated DEA model enables each DMU to choose the most favorable weights for evaluating its efficiency. This may lead to more than one DMU is assessed as efficient, and such DEA-efficient DMUs cannot be further distinguished (Wang and Chin, [
37]). To solve this problem, Sexton et al. [
9] proposed a cross-efficiency DEA method by introducing the concept of peer evaluation. However, the method of Sexton et al. [
9] has a problem of multiple optimum weight solutions. A weight scheme obtained by Sexton et al. [
9] may be favorable to one DMU, but not to another. To address this ambiguity of weight selection, Doyle and Green [
10] proposed the aggressive and benevolent formulations by introducing a secondary goal into the cross-efficiency method. In the case of aggressive (or benevolent) formulation, the secondary goal could choose the weights to make the efficiency of the target DMU the best that it can be, and all other DMUs worst (or best). However, Oukil and Amin [
38] pointed out that the aggressive and benevolent models of Doyle and Green [
10] used a common set of weights for all DMUs, which would not guarantee maximum discrimination among DMUs. To improve discrimination, Oukil and Amin [
38] proposed using different weights for cross-efficiency scores. The purpose of our present study is to effectively discriminate all DMUs with interval data. Therefore, our study adopts the viewpoint of Oukil and Amin [
38]. In our study, the model [
33] is extended to obtain the lower and upper cross-efficiencies for each DMU. Model (4) can calculate the low cross-efficiency values for interval data:
Similarly, the upper cross-efficiency values of interval data can be computed with Model (5):
After all cross-efficiency values are computed, an interval efficiency matrix can be obtained as shown in
Table 1. In each column, [
,
] represents the lower and upper bounds of the cross-efficiency scores of
by using the weights of
.
Models (4) and (5) are built upon the classical cross-efficiency DEA framework, and we consider only the CRS (constant returns to scale) condition in the present study. Models (4) and (5) are inappropriate to be extended to the case of VRS (variable returns to scale). The reason is that integrating the concept of the VRS into the cross-efficiency DEA framework may yield negative cross-efficiency scores (Cook and Zhu [
39] and Lim and Zhu [
40]).
The DEA method mainly has two orientation modes—inputs and outputs. Under the form of the multiplier DEA model, the input orientation mode is expressed as maximizing the ratio of the DMU’s sum of weighted outputs to its sum of weighted inputs. Output orientation mode is formulated as maximizing the ratio of the DMU’s sum of weighted inputs to its sum of weighted outputs (Cooper et al. [
41] and Cook and Bala [
42]). Therefore, per these definitions, Models (4) and (5) are input orientation modes.
4. The Cross-Efficiency Model Based on Shannon Entropy
Shannon entropy is a useful and effective mathematical concept for measuring uncertainty [
43]. Incorporating Shannon entropy into DEA has attracted the interests of a number of scholars [
44,
45,
46]. In this section, Shannon entropy is utilized to calculate the weights of interval cross-efficiency. The weights are obtained by making the distance between the self-evaluation entropy score and peer evaluation entropy score as small as possible. The cross-efficiency entropy model is proposed as in the following steps:
• Step 1: Determining the entropy value of interval cross-efficiency.
As defined in
Table 1,
is the interval cross-efficiency matrix, and the elements
and
represent the interval efficiency that
accords to
. After normalizing the matrix
E, the entropy value of interval cross-efficiency can be defined as follows:
Definition 1. For , the lower (or upper) entropy value of lower (or upper) cross-efficiency score is defined as:where and . Theorem 1. Entropy values can be added.
For each (e.g., ), it has n lower cross-efficiency scores (, ) and n upper cross-efficiency scores (, ). After Step 1, the lower (or upper) entropy of is equal to the sum of its n entropy cross-efficiency scores. That is, • Step 2: Calculating the weights based on the proposed cross-efficiency entropy model.
Definition 2. For , the distance entropy function between cross-efficiency score and its self-evaluation efficiency score is defined as:where (or ) and (or ) are the entropy values of cross-efficiency score (or ) and efficiency score (or ) of Models (2) or (3), respectively. Entropy is a measurement of uncertainty, and we assume that the reasonable weights should make distance entropy of all cross-efficiencies be the smallest and thus uncertainty of interval cross-efficiency would be the smallest. Therefore, cross-efficiency entropy model are as follows: Model (10) essentially is a multi-attribute decision making method. According to characteristics of multi-attribute decision making model (Wang and Parkan [47], Wang and Luo [48], and Tzeng and Huang [49]), the sum of weights is equal to 1. Therefore, the weights in Model (10) also need to satisfy this constraint. Then, according to Lagrangian sufficiency theorem, the weight can be determined as follows: • Step 3: Determining the weighted normalization decision matrix.
The weighted normalization value is calculated by
where
is the weight of attribute
, for all
.
• Step 4: Determining the positive ideal solutions.
After Step 3, there are two weighted normalization matrix (
and
). In this step, the maximum value of each row in each matrix is found as the positive ideal solution:
• Step 5: Calculating the Euclidean distance from the positive solutions:
Therefore, the final distance from the positive solutions is .
• Step 6: Determining the rank of all alternatives on the basis of their relative Euclidean distance from the positive solutions.
The smaller the is, the better the alternative will be. The best alternative is the one with the smallest relative Euclidean distance to the ideal solutions.
Theorem 2. The constraints of Model (10) are non-empty convex set.
Proof. Let C be the constraints of Model (7). It is evident that C is non-empty. Now, assume both and . For any , we have . Therefore, C is convex. □
Theorem 3. The objective function is a convex function in the definition domain.
Proof. For
and
C is a non-empty convex set, the objective function
has the continuous second partial derivatives. The Hessian matrix of the objective function is:
Hessian matrix is always positive definite. Therefore, the objective function is a strongly convex function. □
Theorem 4. is the global optimal solution of Model (10).
Proof. C is a non-empty convex set, the objective function is a convex function, thus Model (10) is a convex programming model, and the generated is the global optimal solution of Model (10). □
6. Conclusions
Traditional cross-efficiency models assume that the data of all DMUs are precise. However, this assumption is not always correct in the real world. In many real circumstances, the outputs and inputs of DMUs are not perfectly precise, which may only have a range in an interval form. In these cases, traditional cross-efficiency models cannot evaluate the efficiencies of DMUs. To address this problem, the present study proposes a new approach. In this approach, we firstly extend traditional cross-efficiency models for obtaining the interval efficiency of each DMU. Then, the distance entropy model is utilized to calculate the weights of interval cross-efficiency scores. Finally, all DMUs are assessed and ranked by the distance to the positive ideal cross-efficiency. A demonstrative case using data from China’s primary schools is used to illustrate the newly proposed model. Through this real case, we can conclude that the proposed method is convenient to solve problems with interval data of DMUs, and can provide complete and fair results for all DMUs.
The method proposed in this paper can be further expanded in the future studies. The DEA Shannon entropy model is proposed based on the cross-efficiency method. The proposed model can also be extended to other DEA models with intervals in the future studies. In addition, our study collected the imprecise data in an interval form. However, in some real cases, a proportion of data are missing. Under this circumstance, it is difficult to extend our method to evaluate the DMUs with missing data, but this direction is worth further investigation.