Cross-efficiency aggregation based on interval conditional entropy: An application to forest carbon sink efficiency

Abstract

Cross-efficiency evaluation is an extension of data envelopment analysis (DEA), which can effectively distinguish between decision-making units (DMUs) through self- and peer-evaluation. The cross-efficiency of each DMU in a set of DMUs is measured in terms of intervals when the input–output data are represented by the number of intervals. Based on the interval cross-efficiency matrix, the interval entropy is defined in terms of the likelihood. Then, considering the influence of peer evaluation, the interval conditional cross-efficiency entropy is proposed and an aggregation model of the interval conditional cross-efficiency entropy is presented to create a ranking index for DMUs. Finally, a simple example is provided to illustrate the effectiveness of the proposed method, which is applied to the evaluation of forest carbon sink efficiency in China. The results indicate that the final cross-efficiencies of all 30 provinces range from 0 to 0.6. Among these provinces, those with a relatively high efficiency include Guangdong, Guizhou, Hainan, Shandong, and Qinghai.

Keywords

Data envelopment analysis interval data cross-efficiency entropy likelihood

1 Introduction

Data envelopment analysis (DEA) is a non-parametric method that can evaluate the relative efficiency of a set of decision-making units (DMUs) with multiple inputs and outputs. Its essential principle is the comprehensive analysis of the input and output data of DMUs, obtain the relative index of each DMU efficiency, and then rank all of the DMU efficiencies to determine the most effective DMUs. Charnes et al. [1] first proposed a DEA model to measure the relative efficiencies of DMUs. Their model simply divided the DMUs into effective and non-effective units; however, it could not distinguish between DMUs with equal efficiencies. The traditional DEA method is based on the concept of self-evaluation, and it evaluates its own efficiency value from the perspective that is most beneficial to ensure that its own efficiency is relatively optimal. As a result, the DMU being evaluated may overestimate its own advantages and avoid its disadvantages. To solve this problem, Sexton et al. [2] introduced the concept of peer evaluations and proposed a cross-efficiency evaluation method in which all DMUs used their input and output self-evaluation weights to evaluate each other. The cross-evaluation method could effectively avoid the disadvantages of the traditional DEA method, and thus it has since attracted the attention of many scholars.

The relevant research can be divided into the following main categories. The first type of research has mainly focused on formulating strategies for cross-efficiency. When evaluating the efficiencies of DMUs, there will be multiple groups of weights for the optimal cross-efficiency obtained using the cross-efficiency evaluation method. To solve this problem, Doyle and Geen [3] proposed aggressive and benevolent strategies. However, aggressive and benevolent strategies may lead to two different evaluation results, and the choice of strategy is a difficult problem. To avoid the above problems, Wu et al. [4] developed a bargaining game model for cross-efficiency evaluation. In the bargaining game model, each DMU is regarded as an independent participant, and the bargaining solution between the CCR efficiency and cross-efficiency is obtained using the classic Nash bargaining game model. The results show that the bargaining efficiency is a Pareto solution. Wang and Chin [5] proposed a neutral model. In this model, the DMU does not consider the efficiency evaluation of other DMUs but only determines the weight from its own perspective. The cross-efficiency calculated in this manner is more neutral and logical. Based on the target recognition model, Wu et al. [6] considered the ideal and non-ideal cross-efficiency targets of all DMUs to build several secondary target models.

The second type of research involves the aggregation of cross-efficiency matrices. The entropy weight method is the mainstream method for aggregating cross-efficiency matrices. Extensive research has been conducted on the aggregation of cross-efficiency matrices using the entropy weight method. Wu et al. [7] extended information entropy theory to solve the ranking problem of the cross-efficiency matrix. Qi and Guo [8] calculated the Shannon entropies of the input and output weights. The different optimal weight sets were aggregated into a common weight set, which was then used to determine the importance of different DMUs. Song et al. [9] introduced the Shannon entropy to obtain public objective weights for cross-efficiency matrix aggregation to avoid subjective factors. Song and Liu [10] proposed a variance coefficient method based on the Shannon entropy to weight the interval cross-efficiency of DMUs by considering the mean and variance of the information entropy. Si and Ma [11] combined the relative entropy evaluation and grey relational analysis methods to rank the efficiency of DMUs. Lee [12] determined the final ranking of DMUs by calculating the information entropy of the cross-efficiency scores for each DMU in different models. Scholars have also used other methods to aggregate the cross-efficiency matrices. Oukil and Amar [13] developed two programs based on OWA to aggregate the cross-efficiency using the positive attribute of the preference voting matrix. Wu et al. [14] used the entropy weight method to aggregate the index weights, further generate a group of public weights, and then use the measurement of alternatives and ranking according to compromise solution (MARCOS) method to rank the DMUs. Their method has been used in a wide range of fields, such as advanced manufacturing technology selection [15], economic–environmental performance measurement [16], performance measurement of nations participating in the Olympic Games [17], public resource management evaluation [18], portfolio selection [19], baseball player ranking [20], sustainable development efficiency evaluation [21], and university research evaluation [22].

The traditional DEA model assumes that the input–output data of all DMUs are specific numbers, but there are often many uncertain factors in the actual efficiency evaluation activities, such as measurement error and data noise. Therefore, it is more practical to study interval efficiency in uncertain environments. Cooper and Yu [23] first discussed the solution of DEA models with interval data and developed the IDEA method, in which IDEA models can be transformed into ordinary linear programming forms. Scholars have since conducted more in-depth studies. Liu et al. [24] combined AHP to study DEA models with interval data and used the AHP method to construct a preference cone to reflect the preference of DMUs. Wu and Huang [25] also considered the optimism and pessimism of DMUs with interval data to measure the interval efficiency. The lower limit of efficiency was obtained by calculating the worst relative efficiency of the ADMU. Interval DEA is also widely used, such as in performance evaluations of the manufacturing industry under the constraint of environmental pollution [26], performance evaluations of banks [27], post-earthquake studies of reservoir dams [28], and measurement of hospital efficiency [29].

Scholars have also conducted in-depth research on cross-efficiency in uncertain environments. These studies can be divided into two main categories: the construction of interval cross-efficiency matrices and the aggregation of interval cross-efficiency matrices. In the construction of interval cross-efficiency matrices, An et al. [30] constructed an interval multiplicative preference relations (IMPRs) matrix of DMUs based on pairwise comparisons of the dominant relations of the relative efficiency between each pair of DMUs. Liu [31] used the cross-efficiency results of the benevolence and aggressiveness as the upper and lower bounds, respectively, to construct the interval cross-efficiency. Liu et al. [32] combined interval possibility degree theory and group decision-making theory to realise the cross-evaluation based on intuitionistic fuzzy preference under the best frontier to achieve full sorting of DMUs. Wu et al. [33] modified the aggressive cross-efficiency model and used a new MARCOS method to calculate the utility score of DMUs.

In terms of the aggregation of interval cross-efficiency matrices, Yang et al. [34] constructed interval cross-efficiencies of DMUs by considering the ownership weight information of each DMU in the cross-evaluation and aggregating them using the SMAA2 method. Wu et al. [35] proposed a new TOPSIS method to rank interval cross-efficiencies. This improved TOPSIS technique could change subjective weights into objective weights through an optimisation model. Wang et al. [36] proposed a cross-efficiency entropy model to calculate the aggregation weight of the interval cross-efficiency matrix to solve the problem of inaccurate input or output data of DMUs. Lu and Liu [37] constructed a numerical index for DMUs with interval cross-efficiency by searching for the minimum information entropy of the DMUs within the upper and lower bounds of the interval cross-efficiency. Liu [31] transformed the signal-to-noise ratio originally used for optimising industrial processes into a basis for integrating the interval cross-efficiency matrix. Wang et al. [38] developed a new integrated ranking technique for cross-efficiency intervals using the plant growth simulation algorithm to generate the generalised Fermat–Torricelli point and aggregate the interval cross-efficiency matrix. In recent years, interval cross-efficiency has also been applied to the selection of the best web services [39], the evaluation of biomass materials [40], the selection of suppliers [41], and performance evaluations of tourist hotels [42].

In the context of global warming, the issue of energy conservation and emission reduction has received great attention, for example, Nikjoo et al. [43] focuses on green communications, Mohajer et al. [44] on green networking. Global carbon sinks are a core issue in global climate change re-search, and carbon sinks in forest ecosystems are an important component of global carbon sinks. In recent years, numerous studies have focused on the development of forest carbon sinks. Many scholars have focused on measuring forest carbon sink efficiency, and the most important method is DEA. Long et al. [45] calculated the carbon sink efficiency using the CCR model. Ao et al. [46] measured the production efficiency of carbon sequestration in bamboo forests using a three-stage DEA model, which eliminated the impact of environmental factors on the efficiency. He et al. [47] discussed carbon sink efficiency from the perspective of resource allocation. They determined that the province with the largest increase in forestry carbon sinks is Guangdong, which has increased by 169,362 megatons, while the province with the smallest increase is Tianjin, which has only increased by 619 megatons. Wei and Shen [48] used the DEA-SBM model to calculate the efficiency of forest carbon sinks in 30 provinces in China from 2005 to 2018. They concluded that the average annual forest carbon sink efficiency in China was only 0.29. Wang et al. [49] used the DEA–Tobit panel regression model to measure carbon sink efficiency. The mean values of comprehensive efficiency for the seventh to ninth forest inventory periods were 0.421 and 0.336, respectively.

In addition, scholars have studied the factors affecting forest carbon sinks and the measurement of carbon sinks. Shao et al. [50] measured the forest car-bon sinks of 13 provinces in northern China, of which Inner Mongolia, Xinjiang, Heilongjiang, and Shanxi had the best development of forest carbon sinks. Lin and Ge [51] investigated the relationship between institutional freedom and forest carbon sinks. They found that when economic development is high, institutional freedom has a positive impact on forest carbon sinks. Zhu et al. [52] studied the impact of cross-sectoral climate policy on forest carbon sinks. The results showed that cross-sectoral climate policies had a positive impact on forest carbon sinks in the region and its neighbouring areas.

Based on the above literature, three problems can be identified. (i) Entropy information is a good method for determining uncertainty in cross-evaluation, but reasonably describing entropy information in an interval environment is a problem. (ii) Most of the existing literature only considers the relationship between self-evaluation and peer evaluation while ignoring comparisons of the relationship between peer evaluation and peer evaluation to aggregate the cross-efficiency matrix. (iii) Evaluation of forest carbon sink efficiency is complex and diverse, and thus more rational methods to evaluate forest carbon sink efficiency are needed. Moreover, there may be cases in which the data are not real, and it is necessary to develop methods by which the efficiency of forest carbon sinks can be evaluated in such cases.

To address the above problems, this study includes the following elements. (i) To address the first problem, a likelihood degree is introduced to extend the information entropy to the interval domain. (ii) To address the second problem, the interval conditional entropy method is proposed to further modify the relationship between peer assessment and self-assessment using peer-to-peer assessment. (iii) To address the third problem, the proposed interval conditional entropy method is used to evaluate the efficiency of forest carbon sinks.

The contributions of this study are as follows.

Contribution to the interval cross-efficiency set. The likelihood is introduced to define the interval information entropy, which not only considers the uncertainty of cross-efficiency but also reduces the information loss in the process of converting interval data into exact data. Furthermore, to consider the relationship between peer evaluations, an interval conditional entropy method is proposed based on the interval information entropy method. This method further revises the relationship between self-evaluation and peer evaluation using peer-to-peer evaluation. It combines the perspectives of self-evaluation and global peer evaluation to better match the diverse characteristics of the evaluated DMUs in interval cross-efficiency evaluation. Moreover, it makes full use of all of the cross-efficiency evaluation information to accurately measure the uncertainty information of the DMU evaluation results, produce a more realistic interval cross-efficiency assembly scheme, and achieve a complete ranking of DMUs.

Contribution to forest carbon sink efficiency analysis. Compared with the CCR model or other self-evaluation models commonly used in the existing literature, the interval conditional entropy method for evaluating forest carbon sink efficiency in this study is not only able to evaluate the cross-efficiency under uncertain environments but also takes into account the relationship between self-evaluation and peer evaluation and the relationship between peer inter-evaluations. Therefore, the evaluation results are more reasonable and objective.

The remainder of this paper is organised as follows. Section 2 introduces the methods and definitions covered in this study in terms of both precise numbers and interval numbers. Section 3 aggregates the cross-efficiency evaluation results of the DMUs based on the interval conditional entropy and gives the aggregation solution based on the fuzzy consistency matrix. Section 4 applies the innovative aggregation method in this study to an arithmetic example used by others and compares the results with other aggregation methods to demonstrate the superiority of the proposed method. Section 5 presents an application of the forest carbon sink efficiency assessment, and Section 6 presents conclusions.

2 Preliminaries

2.1 Traditional work on precise numbers

Suppose there are n DMUs and each DMU _j (j = 1,2, ... , n) has m inputs and s outputs. Then, x_ij(i = 1,2, ... , m) represents the i-th input of DMU _j and y_rj(r = 1,2, ... , s) represents the r-th output of DMU _j . The CCR efficiency of DMU _a (a = 1, ... , n) can be obtained using the following CCR model [1]: $\begin{matrix} Max θ_{aa} = \sum_{r = 1}^{s} u_{ra} y_{ra} \\ s . t . \sum_{i = 1}^{m} v_{ia} x_{ij} - \sum_{r = 1}^{s} u_{ra} y_{rj} ⩾ 0, j = 1, 2, \dots, n; \\ \sum_{i = 1}^{m} v_{ia} x_{ia} = 1; \\ v_{ia}, u_{ra} ⩾ 0, i = 1, \dots, m, r = 1, \dots, s . \end{matrix}$ (1)

Let $v_{ia}^{*}$ and $u_{ra}^{*}$ be the optimal solutions of the above model, then the cross-efficiency of DMU _j is measured by DMU _a as the ratio of weighted output to weighted input obtained when the input and output levels of DMU _j and input and output weights derived for DMU _a are used. Therefore, the cross-efficiency of DMU _j evaluated by DMU _a can be obtained with the following equation [2]: $E_{aj} = \frac{\sum_{r = 1}^{s} u_{ra} y_{rj}}{\sum_{i = 1}^{m} v_{ia} x_{ij}}$ (2)

In management, entropy primarily expresses the amount of information obtained. As an important tool for information measurement, the information entropy method proposed by Shannon solves the problem of quantifying information.

An information system can be expressed as IS = (U, A = C ∪ D, V, f), where

U is a non-empty finite set of objects, U = { x₁, x₂, …, x_n };

A is a non-empty finite set of attributes, A = C ∪ D, where C is the conditional attribute set and D is the decision attribute;

$V = \underset{a \in A}{\cup} V_{a}$ , where V_a is the set of values of a for every a ∈ A;

f : U × A → V_a is an information function, and the value of x for attribute a is f (x, a) ∈ V_a for ∀x ∈ U.

Definition 1. Assuming the existence of an information system IS = (U, A = C ∪ D, V, f), let P ⊆ C, where P is a subset of the conditional attribute set C, and the scalar of U under P is U/ - P = { X₁, X₂, …, X_n }. Then, the information entropy of P is defined as $H (P) = - \sum_{i = 1}^{k} f (X_{i}) {log}_{2} f (X_{i})$ [51].

Definition 2. Given an information system IS = (U, A = C ∪ D, V, f), the scalar of U under the conditional attribute P and decision attribute D are U/ - P = { X₁, X₂, …, X_n } and U/ - D = { d₁, d₂, …, d_n }, respectively [52]. The conditional information entropy of P with respect to the decision attribute D is then defined as follows: $H (D | P) = - \sum_{i = 1}^{k} f (X_{i}) \sum_{j = 1}^{n} \frac{f (X_{i} \cap d_{j})}{f (X_{i})} log \frac{f (X_{i} \cap d_{j})}{f (X_{i})}$ (3)

2.2 Traditional work on interval numbers

Definition 3. Let $\bar{a} = [a^{L}, a^{U}]$ be an interval number where a^L, a^U ∈ R . I(R) represents a set of interval numbers. If a^L = a^U, $\bar{a}$ degenerates into a real number. The length of $\bar{a}$ is L (a) = a^U - a^L and the midpoint of $\bar{a}$ is m (a) = (a^U + a^L)/2 [55].

Let $\bar{a}, \bar{b} \in I (R)$ be two interval numbers, and let a^L ⩾ 0, b^L ⩾ 0. Arithmetic operations can be defined between them as follows [56]:

$\bar{a} + \bar{b} = [a^{L} + b^{L}, a^{U} + b^{U}];$

$\bar{a} - \bar{b} = [a^{L} - b^{U}, a^{U} - b^{L}];$

$\bar{a} \times \bar{b} = [a^{L} b^{L}, a^{U} b^{U}];$

$\bar{a} / - \bar{b} = [a^{L} / - b^{U}, a^{U} / - b^{L}] .$

Moreover, $\bar{a} = \bar{b}$ if and only if a^L = b^L.

To compare two interval numbers, we introduce the likelihood to express the priority between them. Here, $\bar{a} ⩾ \bar{b}$ denotes that $\bar{b}$ is no greater than $\bar{a}$ . The likelihood of $\bar{a} ⩾ \bar{b}$ is defined as follows [57, 58]: $p (\bar{a} ⩾ \bar{b}) = \frac{max {0, a^{U} - b^{L}} - max {0, a^{L} - b^{U}}}{a^{U} - a^{L} + b^{U} - b^{L}}$ (4)

It is clear that $0 ⩽ p (\bar{a} ⩾ \bar{b}) ⩽ 1$ and $p (\bar{a} ⩾ \bar{b}) + p (\bar{b} ⩾ \bar{a}) = 1$ .

Some useful properties of the likelihood are as follows [56, 58]:

$p (\bar{a} ⩾ \bar{b}) = 1$ if and only if a^L ⩾ b^U;

$p (\bar{a} ⩾ \bar{b}) = 0$ if and only if a^U ⩾ b^L;

$p (\bar{b} ⩾ \bar{a}) ⩾ 0.5$ if and only if $m (\bar{a}) ⩾ m (\bar{b})$ , and $p (\bar{b} ⩾ \bar{a}) = 0.5$ if and only if $m (\bar{a}) = m (\bar{b})$ ;

For three interval numbers $\bar{a}, \bar{b}, \bar{c}$ , if $p (\bar{a} ⩾ \bar{b}) ⩾ 0.5$ and $p (\bar{b} ⩾ \bar{c}) ⩾ 0.5$ , then $p (\bar{a} ⩾ \bar{c}) ⩾ 0.5$ .

Properties (a) and (b) indicate that the two interval numbers do not overlap. Property (c) describes the relationship between the likelihood and the midpoint of the interval number. Property (d) defines the transitivity of likelihood.

In an uncertain environment, traditional cross-efficiency evaluation methods are no longer applicable. Therefore, the interval cross-efficiency model proposed by Wang et al. [59] is introduced. Because of the uncertainty, the input and output data of DMU _j are represented by the interval numbers $\bar{x_{ij}} = [x_{ij}^{L}, x_{ij}^{U}]$ (i = 1, 2, …, m ; j = 1, 2, …, n) and $\bar{y_{rj}} = [y_{rj}^{L}, y_{rj}^{U}]$ (r = 1, 2, …, s ; j = 1, 2, …, n), respectively. Then, the interval CCR efficiencies of all DMUs can be obtained using the following equation [59]: $\begin{matrix} Max E_{aa}^{U^{CCR}} = \sum_{r = 1}^{s} u_{ra}^{U} y_{ra}^{U} \\ s . t . \sum_{i = 1}^{m} v_{ia}^{U} x_{ij}^{L} - \sum_{r = 1}^{s} u_{ra}^{U} y_{rj}^{U} ⩾ 0, j = 1, 2, \dots, n; \\ \sum_{i = 1}^{m} v_{ia}^{U} x_{ia}^{L} = 1; \\ v_{ia}^{U}, u_{ra}^{U} ⩾ 0 . \end{matrix}$ (5)

$\begin{matrix} Max E_{aa}^{L^{CCR}} = \sum_{r = 1}^{s} u_{ra}^{L} y_{ra}^{L} \\ s . t . \sum_{i = 1}^{m} v_{ia}^{L} x_{ij}^{L} - \sum_{r = 1}^{s} u_{ra}^{L} y_{rj}^{U} ⩾ 0, j = 1, 2, \dots, n; \\ \sum_{i = 1}^{m} v_{ia}^{L} x_{ia}^{U} = 1; \\ v_{ia}^{L}, u_{ra}^{L} ⩾ 0 . \end{matrix}$ (6)

In Equations (5) and (6), DMU _j is evaluated. $E_{aa}^{U^{CCR}}$ is the upper bound of the optimal self-evaluation efficiency of DMU _a in the best production state, i.e. that in which DMU _a has the minimum input and maximum output. Similarly, $E_{aa}^{L^{CCR}}$ represents the lower bound, where DMU _a is in the worst production state with the maximum input and minimum output.

The interval cross-efficiencies of DMUs are calculated using the benevolent interval DEA cross-efficiency evaluation model proposed by Huang and Wang [60] to perform the cross-efficiency evaluation of DMUs using interval numbers. $\begin{matrix} Min \begin{matrix} \sum_{r = 1}^{s} s_{aj}^{U_{1}} + \end{matrix} s_{aj}^{U_{2}} \\ \begin{matrix} \begin{matrix} s . t \end{matrix} \sum_{i = 1}^{m} v_{ia}^{U} x_{ia}^{L} = 1; \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{U} y_{ra}^{U} = E_{aa}^{U^{CCR}}; \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{U} y_{rj}^{U} - \sum_{i = 1}^{m} v_{ia}^{U} x_{ij}^{L} ⩽ 0 \begin{matrix} , j = 1, 2, \dots, n; \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{U} y_{rj}^{U} - E_{jj}^{U^{CCR}} \sum_{i = 1}^{m} v_{ia}^{U} x_{ij}^{L} + s_{aj}^{U_{1}} = 0 \begin{matrix} , j = 1, 2, \dots, n; \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{U} y_{rj}^{L} - E_{jj}^{U^{CCR}} \sum_{i = 1}^{m} v_{ia}^{U} x_{ij}^{U} + s_{aj}^{U_{2}} = 0 \begin{matrix} , j = 1, 2, \dots, n; \end{matrix} \\ \begin{matrix} u_{ra}^{U}, v_{ia}^{U} ⩾ 0, s_{aj}^{U_{1}}, s_{aj}^{U_{2}} ⩾ 0, r = 1, 2, \dots, s, \end{matrix} i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (7)

The core idea of Equation (7) is to maintain the condition in which the self-evaluated efficiency of DMU _a is optimal and then maximise the cross-efficiency value. The first and second constraints of Equation (7) ensure that the self-evaluated efficiency of DMU _a is equal to the optimal efficiency, $E_{aa}^{U^{CCR}}$ , calculated in Equation (5), and the third constraint ensures that DMU _a does not evaluates DMU _j with an efficiency exceeding one. In Equation (7), the weights $v_{ia}^{U}$ and $u_{ra}^{U}$ for DMU _a to evaluate DMU _j are from model (5). $s_{aj}^{U_{1}}$ and $s_{aj}^{U_{2}}$ are the deviations between the interval cross-efficiency of DMU _j and the upper CCR interval efficiency of DMU _a . Therefore, the objective function of Equation (7) aims to minimize the total deviation between the interval cross-efficiency of all DMUs and the upper CCR interval efficiency of DMU _a [60].

Using the optimal weight solution of Equation (7) to evaluate DMU _j , the interval cross-efficiency of DMU _j evaluated by DMU _a is obtained to solve for n-time linear programming and obtain the interval cross-efficiency matrix, $\bar{E^{U}} = {([E_{a^{U} j^{L}}, E_{a^{U} j^{U}}])}_{n \times n}$ , for n DMUs. Here, E_{a
^U
j
^L} and E_{a
^U
j
^U} refer to the lower and upper bounds, respectively, of the peer-evaluated efficiency of DMU _a evaluated against DMU _j under the condition of the optimal self-evaluated efficiency.

Similarly, when both DMU _a and DMU _j are in the worst state, the maximized cross-efficiency value of DMU _j is: $\begin{matrix} Min \begin{matrix} \sum_{r = 1}^{s} s_{aj}^{L_{1}} + \end{matrix} s_{aj}^{L_{2}} \\ \begin{matrix} \begin{matrix} s . t \end{matrix} \sum_{i = 1}^{m} v_{ia}^{L} x_{ia}^{U} = 1; \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{L} y_{ra}^{L} = E_{aa}^{L^{CCR}}; \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{L} y_{rj}^{U} - \sum_{i = 1}^{m} v_{ia}^{L} x_{ij}^{L} ⩽ 0 \begin{matrix} , j = 1, 2, \dots, n; \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{L} y_{rj}^{U} - E_{jj}^{L^{CCR}} \sum_{i = 1}^{m} v_{ia}^{L} x_{ij}^{L} + s_{aj}^{L_{1}} = 0 \begin{matrix} , j = 1, 2, \dots, n; \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \sum_{r = 1}^{s} u_{ra}^{L} y_{rj}^{L} - E_{jj}^{L^{CCR}} \sum_{i = 1}^{m} v_{ia}^{L} x_{ij}^{U} + s_{aj}^{L_{2}} = 0 \begin{matrix} , j = 1, 2, \dots, n; \end{matrix} \\ \begin{matrix} u_{ra}^{L}, v_{ia}^{L} ⩾ 0, s_{aj}^{L_{1}}, s_{aj}^{L_{2}} ⩾ 0, r = 1, 2, \dots, s, \end{matrix} i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (8)

Here, we evaluate DMU _j by using the optimal weight solution of Equation (8). Similarly, we solve for n-time linear programming to obtain the interval cross-efficiency matrix, $\bar{E^{L}} = {([E_{a^{L} j^{L}}, E_{a^{L} j^{U}}])}_{n \times n}$ , for n DMUs.

Finally, the corresponding elements of the interval cross-efficiency matrices $\bar{E^{U}} = {([E_{a^{U} j^{L}}, E_{a^{U} j^{U}}])}_{n \times n}$ and $\bar{E^{L}} = {([E_{a^{L} j^{L}}, E_{a^{L} j^{U}}])}_{n \times n}$ are weighted and averaged to obtain the overall combined interval cross-efficiency matrix $\bar{E} = {([E_{aj}^{L}, E_{aj}^{U}])}_{n \times n}$ , as summarised in Table 1.

Table 1

Comprehensive interval cross-efficiency matrix, $\bar{E}$

DMU_i
DMU_j	1	2	...	n
1	$[E_{11}^{L}, E_{11}^{U}]$	$[E_{21}^{L}, E_{21}^{U}]$	...	$[E_{n 1}^{L}, E_{n 1}^{U}]$
2	$[E_{12}^{L}, E_{12}^{U}]$	$[E_{22}^{L}, E_{22}^{U}]$	...	$[E_{n 2}^{L}, E_{n 2}^{U}]$
...	...	...	...	...
n	$[E_{1 n}^{L}, E_{1 n}^{U}]$	$[E_{2 n}^{L}, E_{2 n}^{U}]$	...	$[E_{nn}^{L}, E_{nn}^{U}]$

3 Methodology

3.1 Entropy based on the interval cross-efficiency

Information entropy is a measure of the degree of disorder in a system, and it can be used to determine the degree of change in a given indicator. The degree of change in an indicator reflects the amount of information it provides. The greater the entropy, the greater the change; the greater the amount of information, the greater the uncertainty, the greater the role played in the comprehensive evaluation, and the greater the corresponding weight. Therefore, this study introduces information entropy to determine the weights.

However, this study is concerned with the information entropy in an interval environment. To express the information relationships between interval numbers, the likelihood method is introduced.

Based on the definition of entropy in Section 2.1 and the definition of likelihood in Section 2.2, the information system, IS = (U, A = C ∪ D, V, f), in the interval cross-efficiency setting can be redefined as follows:

U is the set of all DMUs, where U = { DMU₁, DMU₂, …, DMU_n };

C is the set of DMUs to be evaluated, D is the set of evaluated DMUs, and A = C ∪ D;

$V = {\bar{e} | e^{L} ⩾ 0, e^{U} ⩽ 1}$ ;

$f (DM U_{a}, DM U_{j}) = P (\bar{E_{aj}} ⩾ \bar{E_{jj}}) \in V_{a}$ , and a ∈ A for ∀DMU_j ∈ U, where $P (\bar{E_{aj}} ⩾ \bar{E_{jj}}) = \frac{max {0, E_{aj}^{U} - E_{jj}^{L}} - max {0, E_{aj}^{L} - E_{jj}^{U}}}{E_{aj}^{U} - E_{aj}^{L} + E_{jj}^{U} - E_{jj}^{L}}$ .

Therefore, we define the interval entropy as follows.

Definition 6. Assuming an interval cross-efficiency information system IS = (U, A = C ∪ D, V, f), let P ⊆ C, where P is a subset of the conditional attribute set C and represents the interval cross-efficiency scores $\bar{E_{aj}} (j = 1, 2, \dots, n)$ of DMU_j(j = 1,2, ... ,n) evaluated by DMU_a; the scalar of U under P is $U / - P = {\bar{E_{a 1}}, \bar{E_{a 2}}, \dots, \bar{E_{an}}}$ . Then, the interval entropy of P, i.e. the interval entropy of DMU_a is defined as $H (DM U_{a}) = - \sum_{j = 1}^{n} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{jj}) {log}_{2} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{jj})$ (9)

It is clear that the higher the value of H(DMU_a), the greater the amount of information provided by DMU_a and the greater its role in the evaluation will be, larger evaluation weight of DMU_a.

3.2 Interval Conditional entropy based on interval cross-efficiency

However, the interval entropy method only considers the evaluation results of DMU_a on DMU_j. However, the objectivity of the evaluation result of DMU_a on DMU_j needs to be analysed from a global perspective; therefore, the evaluation scores of DMU_a on DMU_j must be compared with the evaluation scores of all DMUs on DMU_j. The following formulais uesd: $P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij}) (i = 1, 2, \dots, n)$ (10)

Clearly, the larger the value of $P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij})$ , the smaller the overlap of the interval cross-efficiency of DMU_j derived from the evaluation of DMU_a and other DMUs will be, i.e. the higher the uncertainty. In addition, the evaluation of DMU_a is more objective.

It is necessary to consider the relationship between the evaluation results of DMU_a and other DMUs on DMU_j with the amount of information carried by the results of the evaluation of DMU_a during the interval cross-efficiency matrix aggregation. The relationship between DMU_a and DMU_j can be defined using the conditional entropy theory. Therefore, the interval conditional entropy is defined using Equation (10) to improve Equation (9).

Definition 4. Given an interval cross-efficiency information system IS = (U, A = C ∪ D, V, f), the scalar of U under conditional attribute P and decision attribute D are $U / - P = {\bar{E_{a 1}}, \bar{E_{a 2}}, \dots, \bar{E_{an}}}$ and $U / - D = {\bar{E_{1 j}}, \bar{E_{2 j}}, \dots, \bar{E_{nj}}}$ (j = 1, 2, …, n), respectively. From the cross-efficiency matrix, U/P is the self-evaluation corresponding to the columns of the matrix. Similarly, U/D is the peer evaluation corresponding to the rows of the matrix. Then, the interval information entropy of P with respect to D, i.e. the interval conditional entropy of DMU_a with respect to the decision attribute D, is defined as follows: $\begin{matrix} H (D | DM U_{a}) = - \sum_{j = 1}^{n} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{jj}) \\ \sum_{i = 1}^{n} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij}) {log}_{2} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij}) \end{matrix}$ (11) where $P (\bar{E_{aj}} ⩾ \bar{E_{jj}}) = \frac{max {0, E_{aj}^{U} - E_{jj}^{L}} - max {0, E_{aj}^{L} - E_{jj}^{U}}}{E_{aj}^{U} - E_{aj}^{L} + E_{jj}^{U} - E_{jj}^{L}}$ and $P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij}) = \frac{max {0, E_{aj}^{U} - E_{ij}^{L}} - max {0, E_{aj}^{L} - E_{ij}^{U}}}{E_{aj}^{U} - E_{aj}^{L} + E_{ij}^{U} - E_{ij}^{L}}$ ; $\sum_{i = 1}^{n} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij})$ ${log}_{2} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{ij})$ represent the uncertain influence of DMU_j on the evaluation of DMU_a and $- \sum_{j = 1}^{n} P ({\bar{E}}_{aj} ⩾ {\bar{E}}_{jj})$ represents the uncertainty information from DMU_a itself. Conditional entropy explains the uncertain influence of DMU_j on the evaluation of DMU_a, which is based on the uncertain influence of DMU_j on the evaluation of DMU_i. Because the uncertain influence of DMU_j on the evaluation of DMU_i is newly added information, it will reduce the uncertainty of the system. With Equation (11), the confusion of the results of the DMU_a evaluation can be more accurately measured from a global perspective and the importance of its evaluation can be determined.

3.3 Aggregation by interval conditional entropy

Huang et al. [61] constructed a fuzzy consistency matrix based on the cross-efficiency evaluation results, so as to distribute weights to the DMUs and complete the aggregation of the cross-efficiency matrices. In this paper, according to the thoughts of Huang et al. [61], the consistency preference of experts for DMUs can be obtained from the interval conditional entropy of the DMUs. Thus, definition 5 is constructed as follows.

Definition 5. Let U = {DMU₁, DMU₂, ⋯ , DMU_n} be the set of DMUs, and let H_S be the interval conditional entropy of DMU_a. Reordering the DMUs according to H_S from large to small yields . The degree of fuzzy preference of DMU_i over DMU _j can be expressed as follows: $p_{i j}^{*} = \frac{1}{2} [1 + \frac{\sum_{t = 1}^{j} H (D M U_{t}^{*}) - \sum_{t = 1}^{i} H (D M U_{t}^{*})}{n - 1}]$ (12) where ${DMU}_{t}^{*}$ refers to the DMU in which the information interval conditional entropy of the probability degree has rank t after H_S is arranged from large to small, and the subscript t in ${DMU}_{t}^{*}$ represents the position of the DMUs. The fuzzy preference matrix generated by Equation (12) has been shown to be a fuzzy consistency preference matrix. See Theorem 1.

Aggregating the fuzzy consistency preference matrix by column gives the weights of multiple indicators, and the decision maker ranks the indicators according to these weights. Xu [62] proposed a minimum variance method for the aggregation of fuzzy consistency preference matrices as follows: $ω_{i} = \frac{\sum_{j = 1}^{n} p_{ji} + \frac{n}{2} - 1}{n (n - 1)}, i = 1, 2, \dots, n .$ (13)

Theorem 1. Let $P^{*} = (p_{ij}^{*})_{n \times n}$ be the fuzzy preference matrix generated using Equation (12); then, $P^{*} = (p_{ij}^{*})_{n \times n}$ is the fuzzy consistency preference matrix.

Consequently, the final interval cross-efficiency of DMU_j can be aggregated as follows: $\bar{E_{j}} = \sum_{a = 1}^{n} ω_{a} \bar{E_{aj}}, j = 1, 2, \dots, n .$ (14)

The weight vector $w^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})$ is then obtained by aggregating the fuzzy consistency preference matrix p using Equation (13). The weight vector w = (w₁, w₂, ⋯ , w_n) is obtained by regrouping the DMUs according to their pre-sorting position relationships, and the global cross-efficiency vector, $\bar{E} = ({\bar{E}}_{1}, {\bar{E}}_{2}, \dots, {\bar{E}}_{n})$ , of the DMUs is calculated through combination Equation (14).

The evaluation results of the cross-efficiency matrix are used to construct a fuzzy consistency matrix with a certain degree of objectivity. And the consistency of fuzzy preference and the displacement invariance and strong order-preserving nature of the ranking results brought about by the allocation of indicator weights are able to make the decision results more reasonable and stable.

As described by the theoretical analysis, the main steps for the interval cross-efficiency matrix aggregation based on the interval condition entropy method are as follows.

Step 1: Calculate the interval CCR efficiency of the DMUs. Using the input-output data to calculate Equations (5) and (6), the interval CCR efficiency upper bound, $E_{aa}^{U^{CCR}}$ , and lower bound, $E_{aa}^{L^{CCR}}$ , are obtained for each DMU.

Step 2: Calculate the interval cross-efficiency of the DMUs. Using the input-output data and interval CCR efficiencies $([E_{aa}^{L^{CCR}}, E_{aa}^{U^{CCR}}])_{n \times 1}$ obtained in Step 1, Equations (7) and (8) are applied to obtain the interval cross-efficiency matrices $\bar{E^{U}} = {([E_{a^{U} j^{L}}, E_{a^{U} j^{U}}])}_{n \times n}$ and $\bar{E^{L}} = {([E_{a^{L} j^{L}}, E_{a^{L} j^{U}}])}_{n \times n}$ for each DMU. The matrices $\bar{E^{U}}$ and $\bar{E^{L}}$ are then weighted and averaged using the averaging operation algorithm to obtain a combined interval cross-efficiency matrix $\bar{E} = {([E_{aj}^{L}, E_{aj}^{U}])}_{n \times n}$ .

Step 3: Calculate the interval conditional entropy of the DMUs. The interval conditional entropy of each DMU is obtained by first calculating the likelihood between the self-evaluated efficiency and the peer-evaluated efficiency of each DMU and the likelihood of the peer-to-peer evaluated efficiency of each DMU, and then substituting these into Equation (11).

Step 4: Calculate the aggregation weights. A fuzzy consistency preference matrix is constructed based on the interval conditional entropy using Equation (12), and the aggregation weight, ω_a, of the fuzzy consistency matrix is calculated using Equation (13).

Step 5: Calculate the final cross-efficiency. Equation (14) is used to aggregate matrix $\bar{E_{aj}}$ to obtain the final interval cross-efficiency, $\bar{E_{j}}$ , of the DMUs, and the median of the final interval cross-efficiency is used to rank all DMUs.

4 Numerical examples

4.1 Calculation results and analysis

To illustrate the application of the conditional entropy model proposed in this study to the evaluation of interval cross-efficiency, the data from Kao and Liu [63] is used to evaluate the efficiency of 24 banks (DMUs) in Taiwan. The inputs and outputs are expressed in intervals. There are three input and three output indicators, as follows:

X₁ = total deposits, X₂ = interest expenses, X₃ = non-interest expenses, Y₁ = total loans, Y₂ = interest income, and Y₃ = non-interest income.

Equations (5) and (6) are used to obtain the optimal CCR efficiency scores $E_{aa}^{U^{CCR}}$ and $E_{aa}^{L^{CCR}}$ , for each DMU in the best and worst conditions, respectively. Then, following Step 2, the CCR efficiency results are substituted into Equations (7) and (8) to calculate the comprehensive interval cross-efficiency matrix for each DMU.

For the interval cross-efficiency matrix aggregation, the following two aspects are considered: (1) the relationship between peer-evaluation and self-evaluation and (2) the relationship between peer-evaluation and the remaining DMUs peer-evaluation to obtain a global perspective. Equation (9) is used for the first aspect, and Equation (11) is used for both aspects. Table 2 lists the results of the calculations using Equations (9) and (11). A comparison of the results of this calculation is shown in Fig. 1.

Table 2
Entropy values of DMU_a

DMU Interval entropy Interval conditional entropy DMU Interval entropy Interval conditional entropy

1 11.591 87.904 13 9.840 62.565

2 11.114 87.001 14 6.207 41.308

3 11.435 87.643 15 11.119 74.986

4 12.541 88.259 16 12.298 87.392

5 8.514 51.578 17 12.548 91.839

6 12.424 84.835 18 11.598 85.758

7 11.524 73.749 19 8.229 38.583

8 11.074 74.642 20 5.635 27.394

9 8.391 44.338 21 11.595 85.545

10 11.898 86.606 22 11.971 85.764

11 11.558 86.004 23 4.047 25.726

12 12.491 92.930 24 12.495 95.788

DMU	Interval entropy	Interval conditional entropy	DMU	Interval entropy	Interval conditional entropy
1	11.591	87.904	13	9.840	62.565
2	11.114	87.001	14	6.207	41.308
3	11.435	87.643	15	11.119	74.986
4	12.541	88.259	16	12.298	87.392
5	8.514	51.578	17	12.548	91.839
6	12.424	84.835	18	11.598	85.758
7	11.524	73.749	19	8.229	38.583
8	11.074	74.642	20	5.635	27.394
9	8.391	44.338	21	11.595	85.545
10	11.898	86.606	22	11.971	85.764
11	11.558	86.004	23	4.047	25.726
12	12.491	92.930	24	12.495	95.788

Fig. 1

Entropy values obtained by different methods.

The results in Table 2 indicate that the interval conditional entropy of each DMU is greater than its interval entropy because, from a global perspective, the entropy values of these DMUs are increased. For example, the interval entropy values of DMU₄ and DMU₂₄ are relatively similar, even though the interval entropy value of DMU₄ is greater than that of DMU₂₄. However, with the interval conditional entropy method, the entropy value of DMU₄ is much smaller than that of DMU₂₄. This is because, from a global perspective, the entropy value of DMU₄ is no longer determined by the relationship between the self-evaluation interval and peer-evaluation interval, but should be considered in accordance with the evaluation of the other DMUs. It is clear that the relationship between the results of the evaluation of DMU₂₄ and the results of the evaluation of other peers plays an important role. Moreover, the distribution of the entropy values in the two different models can be explore. Figure 1 indicates that the entropy value calculated using Equation (3) fluctuates around approximately 10 with a linear distribution. Compared with the results of Equation (3), the value of the conditional entropy calculated using Equation (10) shows fluctuating waves distributed between 20 and 100, and the value of the interval entropy of each DMU varies widely. This indicates that the improved conditional entropy has a stronger distinguishing ability than the original entropy, and its values are more discrete. The conditional entropy values exhibit greater differences because the interval conditional entropy method is considered from a global perspective, comparing peer evaluations between DMUs based on both self-evaluation and peer evaluation. This is equivalent to adding global uncertainty to the information entropy, which can lead to a greater difference in entropy values and thus to a greater ability to distinguish between the conditional entropy.

It can be observed from the above analysis that the interval conditional entropy method is superior to the interval information entropy method. Therefore, the calculation results for the information entropy are not given below, and only the calculation results for the interval conditional entropy method are provided, as listed in Table 3. In Table 3, the DMU with the maximum efficiency based on the interval conditional entropy is DMU₂₄, and the DMU with the minimum efficiency is DMU₂₃. Meanwhile, the evaluation weight of DMU₂₄ is the largest among all DMUs, and the DMU with the smallest evaluation weight is DMU₂₃. This shows that the evaluation weight and efficiency are positively correlated, but this positive correlation is not absolute. Taking DMU₁ as an example, its efficiency ranking is second; however, its weight ranking is fourth rather than second. Therefore, the efficiency value of DMUs with high weights is often large. Because the self-evaluation efficiency of DMUs is the highest, DMUs with a larger evaluation weight can experience a certain advantage in the efficiency aggregation process. This reflects an important advantage of the interval conditional entropy method, i.e. the self-evaluation efficiency is much more important than the peer evaluation efficiency.

Table 3

Efficiency results based on interval conditional entropy

DMU	Weights	Lower limit	Upper limit	midpoint	Ranking
1	0.0472	0.761	0.956	0.859	2
2	0.0447	0.694	0.941	0.818	14
3	0.0464	0.713	0.960	0.836	6
4	0.0480	0.736	0.978	0.857	3
5	0.0365	0.621	0.895	0.758	18
6	0.0398	0.701	0.961	0.831	10
7	0.0377	0.611	0.847	0.729	19
8	0.0383	0.464	0.669	0.566	23
9	0.0362	0.697	0.690	0.693	21
10	0.0439	0.710	0.967	0.839	5
11	0.0430	0.674	0.926	0.800	15
12	0.0498	0.718	0.978	0.848	4
13	0.0370	0.718	0.949	0.834	7
14	0.0359	0.701	0.942	0.822	12
15	0.0391	0.511	0.793	0.652	22
16	0.0455	0.686	0.955	0.821	13
17	0.0489	0.605	0.974	0.789	16
18	0.0414	0.699	0.963	0.831	8
19	0.0357	0.710	0.939	0.825	11
20	0.0356	0.680	0.885	0.783	17
21	0.0406	0.611	0.805	0.708	20
22	0.0422	0.679	0.982	0.831	9
23	0.0355	0.366	0.615	0.491	24
24	0.0508	0.775	0.981	0.878	1

4.2 Analysis and comparison with other methods

To better illustrate the effectiveness of the above method, the aggregation method proposed by Guo and Mu [64] is selected to aggregate the interval cross-efficiency values in this case. This method aggregates the cross-efficiency matrix using only the likelihood. In the following analysis, an aggregation method that utilises only the degree of likelihood is compared with one that combines entropy and the likelihood. A comparison between the values of the interval conditional entropy and the likelihood is shown in Fig. 2.

As shown in Fig. 2, the values calculated using only the degree of likelihood are similar to the values calculated using only the information entropy in Fig. 1. The values are all distributed between 0 and 10 with little differentiation, but the degree of fluctuation in the degree of likelihood values is slightly greater than that of the information entropy values. Again, the dispersion of the interval conditional entropy values is much greater than that of the degree of likelihood values, which allows a clearer distinction between DMUs and simplifies the subsequent efficiency ranking problem. The interval conditional entropy values for DMU₅, DMU₉, DMU₁₄, DMU₁₉, DMU₂₀ and DMU₂₃ are much lower than those of the other DMUs, which indicates that these DMUs are much less efficient. However, this conclusion cannot be drawn from the method proposed by Guo and Mu because the likelihood of the efficiency of these DMUs does not show a tendency to be biased toward lower values between the likelihood of the efficiency of all decision units. In addition, the interval conditional entropy method also introduces conditional entropy to modify the information entropy, i.e. the comparison between peer evaluations is taken as a conditional reference value to correct the relationship between self-evaluation and peer evaluations.

Fig. 2

Comparison between interval conditional entropy and likelihood.

The evaluation weights and final cross-efficiency scores of the DMUs obtained using the two methods are shown in Figs. 2 and 3, respectively.

The ranking results of the two methods are also compared, as summarised in Table 4. It can be seen from Table 4 that the final cross-efficiency rankings calculated by these two methods are quite different; for example, DMU₁₁ ranks 15th and 9th respectively, and DMU₁₆ ranks 13th and 4th, respectively. The rankings of these two DMUs obtained by the two methods thus differ by nine places. The method proposed by Guo and Mu converts uncertain data into deterministic data, and the obtained data may be too rigid and lead to the loss of decision information, thereby reducing the flexibility of the fuzzy decision and the reliability of the decision results. Moreover, the likelihood matrix generated by the method proposed by Guo and Mu is a fuzzy complementary judgment matrix that does not meet the consistency, which leads to inaccurate weight allocation. The interval conditional entropy method directly generates a fuzzy consistency matrix through ranking of the likelihood, which reduces the information loss of the weight and makes the system more stable. Therefore, the weight assignment of the interval conditional entropy method is more reliable than that of the method proposed by Guo and Mu.

Table 4

Cross-efficiency comparison of two methods

DMU	Cross-efficiency based			Cross-efficiency based
	on interval conditional			on the method proposed
	entropy method			by Guo and Mu
	Lower	Upper	Midpoint	Lower	Upper	Midpoint
	limit	limit	(Ranking)	limit	limit	(Ranking)
1	0.761	0.956	0.859 (2)	0.897	0.980	0.939 (2)
2	0.694	0.941	0.818 (14)	0.823	0.962	0.893 (8)
3	0.713	0.960	0.836 (6)	0.825	0.982	0.904 (6)
4	0.736	0.978	0.857 (3)	0.868	0.962	0.915 (5)
5	0.621	0.895	0.758 (18)	0.689	0.905	0.797 (17)
6	0.701	0.961	0.831 (10)	0.698	0.950	0.824 (15)
7	0.611	0.847	0.729 (19)	0.703	0.859	0.781 (19)
8	0.464	0.669	0.566 (23)	0.493	0.707	0.600 (23)
9	0.697	0.690	0.693 (21)	0.751	0.586	0.668 (22)
10	0.710	0.967	0.839 (5)	0.754	0.952	0.853 (14)
11	0.674	0.926	0.800 (15)	0.805	0.971	0.888 (9)
12	0.718	0.978	0.848 (4)	0.849	0.997	0.923 (3)
13	0.718	0.949	0.834 (7)	0.805	0.918	0.862 (13)
14	0.701	0.942	0.822 (12)	0.814	0.946	0.880 (10)
15	0.511	0.793	0.652 (22)	0.605	0.864	0.735 (21)
16	0.686	0.955	0.821 (13)	0.839	1.000	0.920 (4)
17	0.605	0.974	0.789 (16)	0.739	1.000	0.869 (11)
18	0.699	0.963	0.831 (8)	0.820	0.980	0.900 (7)
19	0.710	0.939	0.825 (11)	0.762	0.885	0.823 (16)
20	0.680	0.885	0.783 (17)	0.661	0.824	0.742 (20)
21	0.611	0.805	0.708 (20)	0.717	0.848	0.783 (18)
22	0.679	0.982	0.831 (9)	0.732	1.000	0.866 (12)
23	0.366	0.615	0.491 (24)	0.465	0.686	0.576 (24)
24	0.775	0.981	0.878 (1)	0.907	1.000	0.954 (1)

In summary, compared with the method proposed by Guo and Mu, the interval conditional entropy method has the following advantages.

The interval conditional entropy method defines the entropy based on the likelihood of the interval number, which improves the description of the interval number uncertainty.

From a global perspective, the interval conditional entropy method takes the effect of peer-to-peer evaluation as a precondition for considering the relationship between self-evaluation and peer evaluation. In other words, for the effect of DMU_a’s evaluation of DMU_j, instead of considering only the uncertainty between DMU_j’s self-evaluation efficiency and DMU_a’s peer evaluation efficiency of DMU_j, the uncertainty between DMU_a’s peer evaluation efficiency of DMU_j and the evaluation efficiency of all decision units of DMU_j is first calculated, and then incorporated into the uncertainty between DMU_j’s self-evaluation efficiency and DMU_a’s peer evaluation efficiency of DMU_j.

The fuzzy preference matrix generated by the interval conditional entropy method in the process of efficiency aggregation meets the consistency, which makes the weight distribution results more reliable.

5 Application

In this section, the interval efficiency of forest carbon sinks is evaluated for 30 provinces in China. According to three production factors (manpower, capital, and land), the forest land area, forestry personnel, and total forestry fixed investment are selected as input indices. The data are obtained from the China Statistical Yearbook (2017). The output index is the forest carbon sink. These data are obtained from Chen et al. [65]. Owing to certain errors in the input indicators, such as a lack of accurate accounting for private capital in the total investment, the lack of urban forest area in the land area, and the fact that forestry practitioners mainly refers to the staff on the payroll, interval treatment is performed on the forestry input indicators. The forest area and forest land area are used as the upper and lower limits of the interval, respectively. The data collected by forestry practitioners are the lower limit of the interval, and the upper limit of the interval is based on the lower limit plus the rural population multiplied by a certain ratio. The forestry investment uses the original data as the lower limit, and the upper limit of the interval is increased by 10% based on the lower limit. At the same time, there are measurement errors in the data collection of the forest carbon sinks. Chen et al. [65] noted that such errors are in the range of at least 10%. To unify the standard, all of the data are multiplied to obtain the interval data of the input and output, as indicated in Table 5.

Table 5
Input and output interval data of forest carbon sink efficiency

forestry personnel (x₁) total forestry fixed investment (x₂) forest land area (x₃) forest carbon sink(y₁)

Province L U L U L U L U

Beijing 8216 8509 1052474 1157721 58.81 101.35 21.79422 26.63738

Tianjin 763 1029 29878 32865.8 11.16 15.62 11.87387 14.51251

Hebei 20725 24108 4671 5138.1 439.33 718.08 215.1761 262.993

Shanxi 22790 24369 17257 18982.7 282.41 765.55 195.0671 238.4153

Inner Mongolia 89669 90630 51254 56379.4 2487.9 4398.89 973.4518 1189.774

Liaoning 26690 28110 54350 59785 557.31 699.89 229.9656 281.069

Jilin 93074 94252 42330 46563 763.87 856.19 333.1773 407.2167

Heilongjiang 252449 253987 1068658 1175524 1962.13 2207.4 776.52 949.08

Shanghai 1259 1556 64443 70887.3 6.81 7.73 13.99687 17.10729

Jiangsu 14031 16539 151012 166113.2 162.1 178.7 166.5909 203.6111

Zhejiang 9659 11469 7718 8489.8 601.36 660.74 259.4338 317.0857

Anhui 21268 24177 273659 301024.9 380.42 443.18 217.7509 266.14

Fujian 19869 21246 30703 33773.3 801.27 926.82 338.3622 413.5538

Jiangxi 35015 37113 24679 27146.9 1001.81 1069.66 335.0089 409.4553

Shandong 19757 23701 54286 59714.6 254.6 331.26 219.9975 268.8858

Henan 28067 32831 18796 20675.6 359.07 504.98 222.8019 272.3134

Hubei 26247 28649 188436 207279.6 713.86 849.85 329.2432 402.4083

Hunan 43899 47012 382471 420718.1 1011.94 1252.78 419.252 512.4191

Guangdong 25526 28893 7413 8154.3 906.13 1076.44 520.8062 636.541

Guangxi 38251 40732 8850982 9736080 1342.7 1527.17 630.0898 770.1097

Hainan 4865 5254 775 852.5 187.77 214.49 84.79276 103.6356

Chongqing 6513 7618 40509 44559.9 316.44 406.28 140.5105 171.735

Sichuan 40917 45002 182437 200680.7 1703.74 2328.26 778.0821 950.9892

Guizhou 14923 16855 8676 9543.6 653.35 861.22 377.786 461.7385

Yunnan 30474 33033 140193 154212.3 1914.19 2501.04 1015.939 1241.703

Shaanxi 33835 35492 284472 312919.2 853.24 1228.47 310.0599 378.9621

Gansu 36763 38171 199480 219428 507.45 1042.65 369.1038 451.1268

Qinghai 4086 4367 89769 98745.9 406.39 808.04 274.1154 335.0299

Ningxia 8248 8535 7747 8521.7 61.8 180.1 36.35969 44.43963

Sinkiang 18613 19851 561486 617634.6 698.25 1099.71 744.965 910.5128

	forestry personnel (x₁)	total forestry fixed investment (x₂)	forest land area (x₃)	forest carbon sink(y₁)
Beijing	8216	8509	1052474	1157721	58.81	101.35	21.79422	26.63738
Tianjin	763	1029	29878	32865.8	11.16	15.62	11.87387	14.51251
Hebei	20725	24108	4671	5138.1	439.33	718.08	215.1761	262.993
Shanxi	22790	24369	17257	18982.7	282.41	765.55	195.0671	238.4153
Inner Mongolia	89669	90630	51254	56379.4	2487.9	4398.89	973.4518	1189.774
Liaoning	26690	28110	54350	59785	557.31	699.89	229.9656	281.069
Jilin	93074	94252	42330	46563	763.87	856.19	333.1773	407.2167
Heilongjiang	252449	253987	1068658	1175524	1962.13	2207.4	776.52	949.08
Shanghai	1259	1556	64443	70887.3	6.81	7.73	13.99687	17.10729
Jiangsu	14031	16539	151012	166113.2	162.1	178.7	166.5909	203.6111
Zhejiang	9659	11469	7718	8489.8	601.36	660.74	259.4338	317.0857
Anhui	21268	24177	273659	301024.9	380.42	443.18	217.7509	266.14
Fujian	19869	21246	30703	33773.3	801.27	926.82	338.3622	413.5538
Jiangxi	35015	37113	24679	27146.9	1001.81	1069.66	335.0089	409.4553
Shandong	19757	23701	54286	59714.6	254.6	331.26	219.9975	268.8858
Henan	28067	32831	18796	20675.6	359.07	504.98	222.8019	272.3134
Hubei	26247	28649	188436	207279.6	713.86	849.85	329.2432	402.4083
Hunan	43899	47012	382471	420718.1	1011.94	1252.78	419.252	512.4191
Guangdong	25526	28893	7413	8154.3	906.13	1076.44	520.8062	636.541
Guangxi	38251	40732	8850982	9736080	1342.7	1527.17	630.0898	770.1097
Hainan	4865	5254	775	852.5	187.77	214.49	84.79276	103.6356
Chongqing	6513	7618	40509	44559.9	316.44	406.28	140.5105	171.735
Sichuan	40917	45002	182437	200680.7	1703.74	2328.26	778.0821	950.9892
Guizhou	14923	16855	8676	9543.6	653.35	861.22	377.786	461.7385
Yunnan	30474	33033	140193	154212.3	1914.19	2501.04	1015.939	1241.703
Shaanxi	33835	35492	284472	312919.2	853.24	1228.47	310.0599	378.9621
Gansu	36763	38171	199480	219428	507.45	1042.65	369.1038	451.1268
Qinghai	4086	4367	89769	98745.9	406.39	808.04	274.1154	335.0299
Ningxia	8248	8535	7747	8521.7	61.8	180.1	36.35969	44.43963
Sinkiang	18613	19851	561486	617634.6	698.25	1099.71	744.965	910.5128

Table 6 presents the final efficiency values calculated using the interval conditional entropy method. The results show that the final cross-efficiencies of all 30 provinces ranged from 0 to 0.6, which indicates that there is still much room for the development of forest carbon sinks in China.

Table 6

Comparison of the efficiency results

DMU	Weights	Lower limit	Upper limit	Midpoint (Ranking)
Beijing	0.0275	0.026	0.058	0.042 (30)
Tianjin	0.0345	0.209	0.461	0.335 (19)
Hebei	0.0283	0.145	0.705	0.425 (12)
Shanxi	0.0378	0.095	0.841	0.468 (10)
Inner Mongolia	0.0323	0.089	0.566	0.328 (22)
Liaoning	0.0355	0.138	0.515	0.326 (23)
Jilin	0.0298	0.187	0.505	0.346 (17)
Heilongjiang	0.0389	0.100	0.307	0.204 (28)
Shanghai	0.0308	0.197	0.405	0.301 (24)
Jiangsu	0.0339	0.266	0.697	0.482 (9)
Zhejiang	0.0271	0.309	0.721	0.515 (8)
Anhui	0.0383	0.146	0.444	0.295 (25)
Fujian	0.0413	0.215	0.618	0.416 (13)
Jiangxi	0.0367	0.187	0.482	0.334 (20)
Shandong	0.0401	0.181	0.894	0.538 (4)
Henan	0.0303	0.154	0.769	0.462 (11)
Hubei	0.0407	0.165	0.501	0.333 (21)
Hunan	0.0395	0.112	0.404	0.258 (26)
Guangdong	0.0329	0.263	0.883	0.573 (1)
Guangxi	0.0279	0.077	0.151	0.114 (29)
Hainan	0.0264	0.414	0.724	0.569 (3)
Chongqing	0.0334	0.194	0.584	0.389 (14)
Sichuan	0.0318	0.150	0.608	0.379 (15)
Guizhou	0.0350	0.243	0.898	0.570 (2)
Yunnan	0.0267	0.268	0.774	0.521 (7)
Shannxi	0.0313	0.088	0.370	0.229 (27)
Gansu	0.0288	0.093	0.661	0.377 (16)
Qinghai	0.0361	0.233	0.835	0.534 (5)
Ningxia	0.0293	0.084	0.608	0.346 (18)
Xinjiang	0.0372	0.234	0.832	0.533 (6)

Among these provinces, the more well-developed include Guangdong, Guizhou, Hainan, Shandong and Qinghai. Guangdong is controlled by a humid marine monsoon climate and has excellent natural conditions that are ideal for plant growth and conducive to the development of forest carbon sinks. Guizhou, with a forest cover of 62.12%, has the advantage of abundant forest resources, thus laying a good foundation for the development of forest carbon sinks. Hainan is located in the tropics and is rich in tropical rainforest resources, with good water and heat resources, providing unique conditions for the development of forest carbon sinks. Shandong is actively engaged in afforestation and forestry industry development. Its forestry resources are poor, but its forestry industry is developing rapidly, with less than 1% of the country’s forest resources, creating more than 10% of the country’s forestry output value, which is known as the “Shandong phenomenon” in the industry.

The most poorly developed provinces include Shaanxi, Heilongjiang, Guangxi, and Beijing. Shaanxi has serious soil erosion problems, which can lead to drought phenomena; moreover, water resources are more dispersed, and the sandiness of the land has skyrocketed, leading to serious desertification problems. Because of the shortage of water resources, plant growth is difficult, making it more difficult to develop forest carbon sinks. The forestry industry in Heilongjiang is relatively underdeveloped and needs increased investment and a gradual improvement in the forestry market conditions. Guangxi is one of the most serious forest biohazards in China, which is detrimental to the development of forest carbon sinks. The forest area of Beijing is dominated by planted forests, with a small area of natural forest and a lack of forest resources. Although it has a strong economy, its forestry industry lags behind and should be driven by other developed industries.

The development of forest carbon sinks in the remaining provinces is at an intermediate level. Among them, Fujian, Jiangxi, and Jilin have abundant forest resources, but they are not very efficient. These provinces have high forest stocks, but because of tree species and forest management, their carbon sink capacity has not yet been fully utilised, and more investment is needed to improve the efficiency of carbon sinks.

6 Conclusion

In the aggregation of interval cross-efficiency matrices, efficiency aggregation from only the perspective of information entropy often leads to information loss. This study eliminates this defect and considers the application of conditional entropy in the interval cross-efficiency matrix aggregation. The following aspects are summarised through analysis of this method.

In this study, the information entropy is first defined based on the likelihood, which not only considers the uncertainty of the cross-efficiency but also reduces the loss of information in the process of converting interval data into accurate data.

However, information entropy only considers the relationship between self-evaluation and peer evaluation, thus ignoring the impact of peer-to-peer evaluation. Therefore, this study proposes an interval conditional entropy method based on the interval information entropy method. This method further corrects the relationship between self-evaluation and peer evaluation by using mutual evaluations among peers. It combines the perspective of self-evaluation and the perspective of global peer evaluation, which is more consistent with the diversified characteristics of the evaluated DMUs in interval cross-efficiency evaluation. This method can make full use of all of the cross-efficiency evaluation information to measure the uncertainty information of the DMU evaluation results accurately, produce more realistic interval cross-efficiency aggregation schemes, and implement the full ranking of the DMUs.

In addition, the cross-efficiency aggregation in this study is conducted in an interval environment. Future research can be carried out in the following two directions: (1) this study starts from the two endpoints of interval numbers to conduct cross-efficiency aggregation, and future studies can further consider the variability within the interval; (2) further consideration should be given to how cross-efficiency aggregation can be carried out in a fuzzy number setting.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos. 72001042 and 72001043), the China Postdoctoral General Science Fund (No. 2021M700781) and Fujian Agricultural and Forestry University Science and Technology Innovation Special Fund Project (Social Sciences) (NO. CXZX2021029).

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