1 Introduction

Data envelopment analysis (DEA), a non-parametric efficiency measure for evaluating the efficiency of decision-making units (DMUs) with multiple inputs and multiple outputs [1, 2], has been widely conducted for efficiency evaluation in banks, cities, hospitals, schools and the military.

As a non-parametric production analysis, DEA does not need to assume production functions, input–output weights or decision makers’ (DMs’) preference relationships for indicators before evaluation. Halme et al. pointed out in 1999 that traditional DEA models implicitly assume that there is no objective difference in importance between indicators [3]. Therefore, the evaluated DMU selects the indicator weights that are most favorable to itself and obtains the optimal self-evaluation efficiency. The flexibility in the selection of indicator weights becomes a major advantage of the DEA model. However, overly flexible weight selection often leads to efficiency evaluation results that do not correspond to the actual situation. Firstly, when evaluating DMUs, the DEA method often places weights on only a few groups of inputs and outputs and ignores the remaining indicators by assigning them zero weights, resulting in these indicators not participating in the evaluation and reducing the comprehensiveness, comparability and reasonableness of the evaluation results. Roll and Golany even suggested that some control should be applied to changes in the weights owing to the flexibility of DEA [4]. Furthermore, the classical DEA (CCR) model is a self-evaluation DEA of efficiency based on the DMU’s optimal weight vector, and the optimal weights selected by individual DMUs are often different, which amounts to the DMUs not being evaluated under the same criteria. It is thus difficult to evaluate and rank DMUs directly using self-assessed efficiency, and the results are not convincing.

In response to the shortcomings of the classical DEA models, Sexton et al. introduced the peer-evaluation mechanism and proposed the DEA cross-efficiency evaluation method [5]. The method considers the effect of the optimal weight of each DMU on the efficiency of all the other DMUs. The method is based on a set of common weight combinations of the peer-evaluation of DEA efficiency and can achieve the full ranking of DMUs. It has thus quickly become one of the main research directions in DEA theory. The research on the DEA cross-efficiency evaluation method has mainly involved cross-efficiency aggregation methods and cross-efficiency evaluation strategies. The cross-efficiency aggregation method was designed to address the irrationality of the average aggregation method, whereas the study of cross-evaluation strategies has focused on the uniqueness of the weights [6]. So-called weight uniqueness refers to the fact that when DMUs use classical DEA (CCR, BCC) models to select optimal weights, there may be multiple sets of weights that make the efficiency optimal, and selecting different weights yields different cross-efficiency evaluation results [7]. Sexton et al. pointed out the possibility of this problem when proposing the cross-efficiency evaluation method and advised the use of the secondary-goal approach to solve the problem [5]. Adopting this idea, scholars have proposed various secondary-goal models to select the unique optimal weights for DMUs. Doyle and Green proposed the most commonly used secondary-goal models, namely benevolent and aggressive cross-efficiency evaluation models [7], where the main idea is to maximize or minimize the cross-efficiency of other DMUs while keeping a DMU’s own CCR efficiency constant. Both models require DMs to make extreme choices in their decision making. To this end, Wang and Chin developed a neutral cross-efficiency model in which each DMU determines weights only from its own perspective and does not affect other DMUs [8]. Wu et al. innovatively considered the DMUs’ willingness to accept and satisfaction with the results of the DEA cross-efficiency evaluation and proposed a new secondary-goal model based on the concept of satisfaction with the optimal weights [9]. Liu et al. proposed using the optimal ranking of DMUs as a secondary goal [10].Zhu et al. proposed the concept of a fairness utility to construct a new secondary goal model. [11] Davtalab-Olyaie et al. proposed a general and more discriminating secondary goal model using multi-objective programming, including several benevolent, aggressive and neutral secondary goal, and a weighted average cross-efficiency evaluation model. [12] Although these cross-efficiency-focused secondary-goal models effectively address weight uniqueness, the aggregation of cross-efficiency still does not fundamentally address the problem of zero weights and excessive differences.

Anderson et al. argued that cross-efficiency eliminates impractical weighting schemes because the effects of inconsistent evaluation metrics and zero weighting are offset in the aggregation of cross-efficiency [13]. The reality is that when a particular indicator is given a weight of zero in each DMU’s efficiency evaluation, the indicator still amounts to being ignored and the evaluation remains incomplete. Scholars have thus proposed secondary-goal models that enhance the balance of the selected weights and fundamentally avoid unreasonable weights, rather than expecting to cancel out unreasonable weights when aggregating cross-efficiencies. Ramón et al. introduced the concept of weight similarity between input–output weights in DEA cross-efficiency evaluation and developed a two-step procedural method to find the minimum dissimilar weights [14], effectively avoiding zero weights. Similar studies were conducted by Wu et al. and Theodoridis et al., who proposed some different weight-restricted models to reduce the large variation in weights and effectively reduce the number of zero weights [15, 16]. Wang et al. adopted a generic set of normalized weights and the idea of a bargaining game to minimize the bias between the self-evaluation efficiency of each DMU and the peer-evaluation efficiency [17].

Traditional DEA models based on classical additive measure theory assume that the input and output indicators are largely independent of each other. Yue et al. [18] pointed out that traditional DEA methods do not consider the correlation and interaction of indicators, resulting in the loss of a large amount of decision-making information; hence, the efficiency values obtained in real-life applications are less scientific and accurate. The use of the Choquet integral is a widely accepted fuzzy integration method that deals with the interactions of attributes (criteria). By combining the fuzzy integral with the theoretical framework of DEA, the Choquet integral is introduced to replace the weighted average operator and fully exploit the implied interaction information such that the aggregation results are more comprehensive and objective. Ji et al. and Xia et al. improved on the traditional DEA models using the Choquet integral [19, 20]. Yue et al. extended a serial structured network DEA model using the Choquet integral [18]. Pereira et al. used the Choquet integral to incorporate the DMs’ preference information into an ADD model [21].

In summary, we find that when calculating the cross-efficiency scores of DMUs, little consideration has been given to the DMUs’ willingness to accept and satisfaction with the DEA cross-efficiency evaluation results. Wu et al. creatively proposed a cross-efficiency evaluation model based on the DMUs’ satisfaction but did not consider the rationality of indicator weights. The present paper thus extends the satisfaction model proposed by Wu et al. by including the idea of the minimum dissimilarity of weights proposed by Ramón et al. and proposes an improved satisfaction cross-efficiency method, aiming to select more reasonable optimal weights for DMUs. In this process, to enhance the differentiation of the evaluation results of the DEA method, the 2-additive Choquet integral is used in a more feasible attempt to reflect the interactions between inputs or outputs. The main contributions of this paper are as follows.

  • Restrictions on indicator weights are introduced to the method proposed in this paper involving DMUs’ satisfaction to avoid unreasonable weight selection and address the problem of zero weights or excessive differences.

  • The 2-additive Choquet integral is combined with an extended satisfaction secondary goal evaluation model for situations that there are interactions between indicators.

  • The selection (preferences) and ranking of DEA cross-efficiency evaluation methods are discussed from a satisfaction perspective in relation to several aspects of ethical principles in social choice.

  • The method proposed in this paper is applied to the study of W–E–F nexus input–output efficiency in various provinces of China.

The rest of the paper unfolds as follows. In Sect. 2, we review the underlying theory related to the Choquet integral and DEA. In Sect. 3, we propose an improved cross-efficiency assessment method that is more in line with actual situations by extending the idea of weight similarity and combining it with a satisfaction secondary-goal model in a context that indicator interactions are considered, and we demonstrate the validity and innovation of the method through an example analysis. In Sect. 4, we apply the method to an empirical study on the assessment of the W–E–F nexus efficiency in China. A summary is presented in Sect. 5.

2 Preliminaries

2.1 Choquet Integral

The Choquet integral refers to a nonlinear aggregation operator based on fuzzy measures and is widely used in the study of attribute interaction (association) problems [22]. Its measurement results reflect not only the strength but also the type of interaction between indicators. Assume that the finite set of indexes is \(X=\{x_1,x_2,\ldots ,x_n\}\), P(X) is the power set of X, and the index evaluation value is expressed by function \(f:X \rightarrow R\).

Definition 1

[23] Let X be a finite set and P(X) be the set of all subsets of X. The set function \(\mu : P(X)\rightarrow [0,+\infty )\) satisfies (i) \(\mu (\emptyset )=0\) and (ii) \(\forall A,B \in P(X)\) and \(A \subset B\), then \(\mu (A) \le \mu (B)\), \(\mu\) is called fuzzy measure defined on P(X). This fuzzy measure is a special generalized fuzzy measure.

Definition 2

[24] The Möbius transform of any set function \(\mu : P(X)\rightarrow R\) on X is

$$\begin{aligned} a(T)=\sum \limits _{K\subseteq T}(-1)^{|T-K |}\mu (K). \end{aligned}$$
(1)

Therefore, the Möbius transform of the fuzzy measure \(\mu\) in Definition 1 satisfies (i) \(a(\emptyset )=0\) and (ii)\(\sum \nolimits _{S:i\in S\subseteq T}a(S)\ge 0, \ \ \forall T\subseteq X, \ \ \forall i\in T\).

Correspondingly, its (discrete) Choquet integral can be expressed as [25]

$$\begin{aligned} C_\mu (X)=\sum \limits _{T \subseteq X} a(T)\mathop {\wedge }\limits _{i\in T}x_i, \end{aligned}$$
(2)

where \({\text{``}}\wedge {\text{''}}\) represents the minimum operator, i.e. \(\min \lbrace x_i,i\in T \rbrace\).The coefficient a(T) is the Möbius representation of the fuzzy measure.

In practical applications, when the finite set \(|X |=n\), the determination of a general fuzzy measure requires the determination of \(2^n\) parameter values. When the value of n is large, the determination of the fuzzy measure becomes difficult [26]. Grabisch proposed that the adoption of k-additive fuzzy measures effectively reduces the number of parameters to be determined [27]. The 2-additive fuzzy measure is a commonly used special form of the k-additive fuzzy measure and assumes that there are pairwise interactions between attributes (indicators) [28]. When \(k=2\), the calculation of the Choquet integral is further simplified by including the Möbius transform, which only requires the determination of a total of \(n(n+1)/2\) single-point sets and two-point sets. 2-additive Choquet integrals are expressed as

$$\begin{aligned} C_\mu (x)=\sum \limits _{i\in X}a(\lbrace i \rbrace )x_i+\sum \limits _{\lbrace i,j \rbrace \subseteq X}a(\lbrace i,j \rbrace )(x_i \mathop {\wedge } x_j), \ \ x\in {\mathbb {R}}_+^n, \end{aligned}$$
(3)

where \({\text{``}}\wedge {\text{''}}\) is the minimum operator and \(x_i \mathop {\wedge } x_j\) stands for the smaller of \(x_i\) and \(x_j\), i.e. \(\min \lbrace x_i,x_j \rbrace\).

The Möbius representation of 2-additive fuzzy measure needs to satisfy \(a(\emptyset )=0\), \(a(\{i\})\ge 0\), \(\forall i\in N\), \(a(\{i\})+\sum \nolimits _{j\in T}a(\{i,j\})\ge 0\), \(\forall i\in X\), \(\forall T\subseteq X{\setminus }\{i\}\).

To simplify expressions in the following, we denote \(a(\emptyset )\), a(S), \(a(\{i\})\), \(a(\{i,j\})\) and \(a(S {\setminus } \{i\})\) by \(a_{\emptyset }\), \(a_S\), \(a_i\), \(a_{ij}\) and \(a_{S{\setminus } i}\) respectively.

In the case of considering attribute (criterion) interactions, the global importance degree of a single attribute is not fully determined by the importance of the single point set alone but also by the importance of the subsets containing the attribute [29]. Thus, according to the individual attribute importance index or Shapley value proposed by Shapley in cooperative game theory, in the 2-additive situation, the global importance degree and interaction index of an attribute are defined as follows.

Definition 3

[26] The global importance degree of the \(i-th\) criterion over criterion interactions is \(I_i=a_i+\frac{1}{2}\sum \nolimits _{j\in N {\setminus } i}a_{ij}\), \(i\in N\). Where \(a_{ij}\), \(i,j\in N\) is the interaction index, reflecting the direction and intensity of interaction between the two criteria.

2.2 Data Envelopment Analysis (DEA)

DEA was proposed by Charnes, Cooper and Rhodes in 1978 as a non-parametric efficiency measure for evaluating the efficiency of DMUs with multiple inputs and outputs [30]. DEA is based on the concept of relative efficiency and adopts convex analysis and linear programming to calculate and compare the relative efficiency between DMUs of the same type (homogeneous DMUs) and to evaluate the object to be evaluated accordingly.

2.2.1 DEA Efficiency Evaluation Models

Assume that there are n homogeneous DMUs, denoted DMU\(_j\) \((j=1,2,\ldots ,n)\), which make up the DEA evaluation system. Each DMU has m input indicators and s output indicators. Let \(x_{tj}\) \((t=1,\ldots ,m)\) and \(y_{rj}\) \((r=1,\ldots ,s)\) respectively denote the values of all input and output indicators of DMU \(_j\). For each DMU \(_j\), there is a corresponding efficiency evaluation index

$$\begin{aligned} E_j=\frac{\sum \nolimits _{r=1}^{s}\omega _ry_{rj}}{\sum \nolimits _{t=1}^{m}\upsilon _tx_{tj}},\quad j=1,\ldots ,n, \end{aligned}$$
(4)

where \(\upsilon _t\) \((t=1,\ldots ,m)\) and \(\omega _r\) \((r=1,\ldots ,s)\) are the input and output weights, respectively. It follows that changes in the efficiency of each DMU\(_j\) depend on changes in indicator weights.

Charnes et al. proposed the well-known CCR model in 1978 [30] to assess the efficiency of a particular DMU\(_d\) relative to other DMU\(_j\). For computational convenience, the original fractional programming model can be transformed into an equivalent linear programming model by means of the Charnes-Cooper transformation [31]:

$$\begin{aligned}&{\text {Max}} \ E_d = \sum \limits _{r=1}^{s}u_ry_{rd} \nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \sum \limits _{r=1}^{s}u_ry_{rj}-\sum \limits _{t=1}^{m}v_tx_{tj} \le 0,j=1,\ldots ,n, \\ \sum \limits _{t=1}^{m}v_tx_{td}=1, \\ v_t \ge 0, u_r \ge 0,t=1,\ldots ,m;\quad r=1,\ldots ,s.\\ \end{array}\right. } \end{aligned}$$
(5)

2.2.2 DEA Cross-Efficiency

The classical DEA (CCR) model is a typical (extreme) self-evaluation efficiency analysis model having its own optimal weight vector, which makes the evaluation results less comparable. To address the shortcomings of the classical DEA model, Sexton et al. introduced the peer-evaluation mechanism and proposed the DEA cross-efficiency evaluation method [5], expressed by

$$\begin{aligned} E_{dj}=\frac{\sum \nolimits _{r=1}^{s}u_{rj}^{*}y_{rd}}{\sum \nolimits _{t=1}^{m}v_{tj}^{*}x_{td}},j=1,\ldots ,n. \end{aligned}$$
(6)

By peer-evaluation, we refer to the use of the optimal weight vector of another DMU\(_j\) to evaluate the efficiency of DMU\(_d\), as shown in Eq. (6). Here, \(v_{tj}^{*}\) \((t=1,\ldots ,m)\) and \(u_{rj}^{*}\) \((r=1,\ldots ,s)\) denote the optimal solution of DMU\(_j\) in model (5). The \(n-1\) peer-efficiencies of DMU\(_d\) are obtained by varying \(j (j=1,\ldots ,n)\). Including the optimal self-evaluation efficiency of DMU\(_d\), the final cross-efficiency value of DMU\(_d\) is obtained through weighted average aggregation; i.e., \({\overline{E}}_d=\frac{1}{n}\sum \nolimits _{j=1}^n E_{dj}\).

The study of cross-evaluation strategies focuses on solving the problem of the non-uniqueness of weights. The above cross-efficiency strategy is considered an arbitrary (stochastic) strategy. To address this problem, Sexton et al. introduced a secondary-goal model [5]. Under the premise that the primary goal (i.e., the optimal self-evaluation efficiency of CCR) is guaranteed to remain unchanged, the weight selection is made based on the secondary goal. Adopting this idea, Wu et al. creatively proposed the concept of a DMUs’ satisfaction with the optimal weights of other DMUs and took the DMUs’ willingness to accept and satisfaction with the DEA cross-efficiency evaluation results as a secondary goal [9].

Definition 4

[9] The satisfaction of DMU\(_j\) with the weights chosen by DMU\(_d\) is expressed as

$$\begin{aligned} \begin{aligned} {\text {SD}}_{dj}&=\frac{\sum \nolimits _{r=1}^{s}u_{rd}y_{rj}/\sum \nolimits _{t=1}^{m}v_{td}x_{tj}-E_{dj}^{\min }}{E_{dj}^{\max }-E_{dj}^{\min }}\\&=\frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}},E_{dj}^{\max }\ne E_{dj}^{\min },\forall j. \end{aligned} \end{aligned}$$
(7)

Here, \(E_{dj}^{\max }\) and \(E_{dj}^{\min }\) denote the ideal and non-ideal cross-efficiency targets attainable by DMU\(_j\). The calculation can be found in the literature [9]. \(s_{dj}\) and \(\varphi _{dj}\) denote the distance from the peer-efficiency of DMU\(_j\) to its ideal cross-efficiency target and non-ideal cross-efficiency target, respectively. So, there are formulas \(\sum \nolimits _{r=1}^{s}u_{rd}y_{rj}-E_{dj}^{\max }\sum \nolimits _{t=1}^{m}v_{td}x_{tj}+s_{dj} = 0,\forall j,j\ne d\) and \(\sum \nolimits _{r=1}^{s}u_{rd}y_{rj}-E_{dj}^{\min }\sum \nolimits _{t=1}^{m}v_{td}x_{tj}-\varphi _{dj} = 0,\forall j,j\ne d\). By transforming these two equations, we can obtain \(E_{dj}^{\max }=\frac{\sum \nolimits _{r=1}^{s}u_{rd}y_{rj}+s_{dj}}{\sum \nolimits _{t=1}^{m}v_{td}x_{tj}} \ \ (a)\) and \(E_{dj}^{\min }=\frac{\sum \nolimits _{r=1}^{s}u_{rd}y_{rj}-\varphi _{dj}}{\sum \nolimits _{t=1}^{m}v_{td}x_{tj}} \ \ (b)\). Substitute (a) and (b) into \({\text {SD}}_{dj}=\frac{\sum \nolimits _{r=1}^{s}u_{rd}y_{rj}/\sum \nolimits _{t=1}^{m}v_{td}x_{tj}-E_{dj}^{\min }}{E_{dj}^{\max }-E_{dj}^{\min }}\), then \({\text {SD}}_{dj}=\frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}}\) can be obtained. Thus, the satisfaction of DMU\(_j\) with the weights chosen by DMU\(_d\) can be expressed in a simplified way.

It is noted that there may be cases that \(E_{dj}^{\max } = E_{dj}^{\min }\) for some DMU\(_j\) and DMU\(_d\). This means that the cross-efficiency of DMU\(_j\) with respect to DMU\(_d\) does not change regardless of which set of optimal weights is chosen by DMU\(_d\). For any such DMU\(_j\), DMU\(_d\) does not need to consider the satisfaction of that DMU\(_j\) when choosing the optimal weights.

In some practical situations, where a DMU is sometimes a region, sometimes a product, sometimes an event, sometimes a person (consensus, group decision-making, etc.). The satisfaction of the DMU is actually a reflection of the acceptance of the evaluated results (the peer-evaluation efficiency score) by the stakeholders behind the DMU. The closer the efficiency value is to the maximum cross-efficiency target for that DMU, the higher the recognition and acceptance of the assessed score by that DMU, namely, the higher the satisfaction level. In some cases, a high level of satisfaction with the outcome of the decision can avoid some conflicts and disputes and is more conducive to the project (process) moving forward.

On the above basis, Wu et al. developed a satisfaction cross-efficiency evaluation model shown in Model 8.

$$\begin{aligned}&\max \ {\text {SD}}_d \nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \sum \limits _{r=1}^{s}u_{rd}y_{rj}-E_{dj}^{\max }\sum \limits _{t=1}^{m}v_{td}x_{tj}+s_{dj} = 0,\forall j,j\ne d,E_{dj}^{\max }\ne E_{dj}^{\min }, & \quad (8{\text {-}}1) \\ \sum \limits _{r=1}^{s}u_{rd}y_{rj}-E_{dj}^{\min }\sum \limits _{t=1}^{m}v_{td}x_{tj}-\varphi _{dj} = 0,\forall j,j\ne d,E_{dj}^{\max }\ne E_{dj}^{\min }, & \quad (8{\text {-}}2) \\ \sum \limits _{r=1}^{s}u_{rd}y_{rd}-E_d^{*}\sum \limits _{t=1}^{m}v_{td}x_{td}=0, & \quad (8{\text {-}}3) \\ \sum \limits _{t=1}^{m}v_{td}x_{td}=1, & \quad (8{\text {-}}4) \\ \sum \limits _{r=1}^{s}u_{rd}y_{rj}-\sum \limits _{t=1}^{m}v_{td}x_{tj}\le 0, \forall j,j=1,\ldots ,n, & \quad (8{\text {-}}5) \\ \frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}}\ge {\text {SD}}_d,\forall j,j\ne d, E_{dj}^{\max }\ne E_{dj}^{\min }, & \quad (8{\text {-}}6) \\ v_{td},u_{rd} \ge 0, & \quad (8{\text {-}}7) \\ s_{dj},\varphi _{dj}\ge 0,\forall j,j\ne d, E_{dj}^{\max }\ne E_{dj}^{\min }, & \quad (8{\text {-}}8)\\ \end{array}\right. } \end{aligned}$$
(8)

This satisfaction secondary-goal model aims to maximize the satisfaction of all other DMU\(_j\) with respect to the optimal weights chosen by DMU\(_d\), while ensuring that the self-evaluation CCR efficiency of DMU\(_d\) remains unchanged. In model (8), Eqs. (8-1) and (8-2) ensure that the cross-efficiency of each DMU\(_j\) relative to DMU\(_d\) is between its ideal and non-ideal cross-efficiency target, Eqs. (8-3) and (8-4) ensure that the efficiency of DMU\(_d\) is the CCR efficient, and Eqs. (5)–(6) combined with the objective function refers to the maximization of the satisfaction of DMU\(_j\) that is the least satisfied.

The secondary-goal model solves the problem of the non-uniqueness of cross-efficiency assessment weights but still does not avoid the problem of excessive differences in indicator weights in DEA when choosing weights, especially for the scheme that some indicators have zero weights. In response, Ramón et al. devised a two-step procedural method to control the weight boundaries of extremely efficient and non-extremely efficient DMUs in goal programming, seeking the weight with the least difference and excluding unreasonable weighting schemes in the cross-efficiency evaluation [14]. Here, only the first step of the two-step procedure is used to introduce the basic idea of the method, as shown in model (9).

$$\begin{aligned}&\max \ \theta _d \nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \sum \limits _{t=1}^{m}v_{td}x_{td}=1, & \quad (9{\text {-}}1), \\ \sum \limits _{r=1}^{s}u_{rd}y_{rd}=E_d^{*}, & \quad(9{\text {-}}2), \\ \sum \limits _{r=1}^{s}u_{rd}y_{rj}-\sum \limits _{t=1}^{m}v_{td}x_{tj}\le 0, j=1,\ldots ,n, &\quad (9{\text {-}}3) \\ z_I \le v_t \le h_I, t=1,\ldots ,m, &\quad (9{\text {-}}4) \\ z_O \le u_r \le h_O, r=1,\ldots ,s, &\quad (9{\text {-}}5) \\ \frac{z_I}{h_I}\ge \theta _d, &\quad (9{\text {-}}6) \\ \frac{z_O}{h_O}\ge \theta _d, &\quad (9{\text {-}}7) \\ z_I,z_O \ge 0. & \quad (9{\text {-}}8)\\ \end{array}\right. } \end{aligned}$$
(9)

The basic idea of the above model is to find the least different (least dissimilar) optimal input and output weights in the secondary-goal model while ensuring that the optimal self-evaluation efficiency of the CCR model for DMU\(_d\) remains unchanged. Equations (9-1) to (9-3) show that the secondary-goal model allows for all the optimal weights in the CCR model for DMU\(_d\). Equations (9-4) and (9-5) force all the input weights and all the output weights to vary between the bounds \(z_I\) and \(h_I\) and \(z_O\) and \(h_O\), respectively. Obviously, \(\theta _d \in (0,1]\), and the closer \(\theta _d\) is to 1, the smaller the range of weights that can be changed and the smaller the difference in weights. Therefore, Eqs. (9-6) and (9-7) combined with the objective function indicate maximizing the smallest of the two ratios. This process can also be referred to as “weight balance”. The significance of weight balance is to lessen large differences in weighted data (weighted inputs and weighted outputs) and to effectively reduce the number of zero weights for inputs and outputs, allowing all input and output indicator data to participate fully in the evaluation, effectively avoiding unreasonable weighting schemes in DEA evaluation and making the results more scientifically comprehensive.

2.2.3 DEA Models Based on the 2-Additive Choquet Integral

In all classical DEA models, the values of input and output indicators are aggregated using a weighted average operator, with the underlying assumption that the variables are independent of each other. However, in the actual evaluation process, there are often interactions (correlations) between input (output) variables that cannot be ignored. Xia et al. proposed DEA models based on the Choquet integral in the case of the correlation of indicators [20].

To avoid the situation that the application of DEA methods to evaluate the relative efficiency of DMUs may lead to poor differentiation of the evaluation results, that is, the efficiency of the DMUs are all 1, Cooper et al. suggested that the number of DMUs to be evaluated satisfies \(n \ge \max \{3*(m+s),m*s\}\) if the DMUs need to be well distinguished [32], where m and s are respectively the numbers of inputs and outputs of the DMUs. It is seen that the number of indicators is limited by the number of DMUs and the number of indicators should not be excessive. However, when considering the interaction between indicators, it is equivalent to adding some (interactive) indicators on the basis of the original number of indicators, what’s more, the number of interactive indicators increases exponentially. Therefore it is not a bad compromise to consider only the two-by-two interaction of indicators, and a similar treatment can be found in the paper of Gong et al. [33] At the same time, the computational effort is less in the 2-additive case. We therefore consider DEA models based on the 2-additive Choquet integral.

As an example, consider the problem of evaluating the efficiency of DMU\(_d\) relative to other DMUs. First, let the input and output indicator weights respectively satisfy

$$\begin{aligned} Q_{v-2}&=\bigg \{v=\{v_t,t\in M\}\ne 0|v_{\emptyset }=0, v_t \ge 0, \forall t \in M, \nonumber \\&\quad \quad v_t+\sum \limits _{p\in T}v_{tp}\ge 0,\forall t\in M, \forall T \subset M{\setminus } t \bigg \}, \nonumber \\ Q_{u-2}&=\bigg \{u=\{u_r,r\in S\}\ne 0|u_{\emptyset }=0, u_r \ge 0, \forall r \in S, \nonumber \\&\quad \quad u_r+\sum \limits _{q\in R}u_{rq}\ge 0,\forall r\in S, \forall R \subset S{\setminus } r \bigg \}, \end{aligned}$$

where v and u are the Möbius transform for the 2-additive fuzzy measures of the input and output variables, equivalent to a in Eq. (1). Thus, the linear expression based on the 2-additive Choquet integral CCR (2-CHCCR) model is

$$\begin{aligned}&{\text {Max}} \ E_d = \sum \limits _{r\in S}u_r^dy_r^d + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^d \wedge y_q^d)\nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \sum \limits _{t\in M}v_t^dx_t^d + \sum \limits _{\{t,p\} \subset M}v_{tp}^d\left( x_t^d \wedge x_p^d\right) =1, \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \\ \ \ \ \ \ \ \ \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg )\le 0,j=1,2,\ldots ,n, \\ (v,u)\in (Q_{v-2},Q_{u-2}). \\ \end{array}\right. } \end{aligned}$$
(10)

3 Improved Method of Evaluating the Satisfaction Cross-Efficiency

In this section, in the context of indicator interactions, we propose an improved method for assessing satisfaction cross-efficiency by extending the ideas of Ramón et al. [14] and including the satisfaction secondary goal proposed by Wu et al. [9].

The models constructed in this paper satisfy the basic assumptions there are n homogeneous DMUs forming the DEA evaluation system, with each DMU having m input indicators and s output indicators; there are pairwise interactions between input (output) indicators; and the DMUs are independent of each other. Table 1 lists the mathematical symbols used and their meanings.

Table 1 Symbols and meanings
Full size table

3.1 2-Additive Choquet Integral DEA Models Involving the DMUs’ Satisfaction and Weight Restrictions

The improved satisfaction cross-efficiency evaluation method proposed in this paper has three main steps, involving the optimal self-evaluation efficiency model, ideal and non-ideal cross efficiency target models and the optimal satisfaction model.

STEP 1: We solve for the optimal self-evaluation efficiency of each DMU. Under the premise of indicator interactions and based on the idea of the minimum dissimilarity of weights [14], we simplify the two-step weight constraint approach by setting bounds directly on the global importance degrees of the input (output) indicators by introducing a weight ratio parameter \(\theta\) \((\theta \ge 1)\) that restricts the indicators to shift within the range of differences allowed by the weights, avoiding large differences between the weights. Here, \(\theta\) can be transformed on its own according to the specific situation and requirements, or the boundary ratios can be obtained using a method in the literature [14] for reference. Thus, the linear CCR model based on the 2-additive Choquet integral (2-CHCCR) is

$$\begin{aligned}&{\text {Max}} \ E_{d\_{\text {CCR}}} = \sum \limits _{r\in S}u_r^dy_r^d + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^d \wedge y_q^d)\nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \sum \limits _{t\in M}v_t^dx_t^d + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^d \wedge x_p^d)=1, \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \\ \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg )\le 0,j=1,2,\ldots ,n, \\ I_t^d=v_t^d+\frac{1}{2}\sum \limits _{p\in M {\setminus } t}v_{tp}^d, \forall t, t\in M, \\ O_r^d=u_r^d+\frac{1}{2}\sum \limits _{q\in S {\setminus } r}u_{rq}^d, \forall r, r\in S, \\ z_I \le I_t^d \le \theta z_I, \forall t, t\in M, \\ z_O \le O_r^d \le \theta z_O, \forall r, r\in S, \\ z_I,z_O \ge 0, \\ (v,u)\in (Q_{v-2},Q_{u-2}). \\ \end{array}\right. } \end{aligned}$$
(11)

STEP 2: We continue to solve for the ideal and non-ideal cross-efficiency targets of DMU\(_k\) relative to DMU\(_d\) under interaction scenarios and weight restrictions [9]:

$$\begin{aligned}&E_{dk} = {\text {Max}}/{\text {Min}} \ \sum \limits _{r\in S}u_r^dy_r^k + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^k \wedge y_q^k)\nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \sum \limits _{t\in M}v_t^dx_t^k + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^k \wedge x_p^k)=1, \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^d + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^d \wedge y_q^d)\bigg )-E_d^{*}\bigg (\sum \limits _{t\in M}v_t^dx_t^d \\ +\sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^d \wedge x_p^d)\bigg ) = 0,\\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \\ \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg )\le 0,j=1,2,\ldots ,n, \\ I_t^d=v_t^d+\frac{1}{2}\sum \limits _{p\in M {\setminus } t}v_{tp}^d, \forall t, t\in M, \\ O_r^d=u_r^d+\frac{1}{2}\sum \limits _{q\in S {\setminus } r}u_{rq}^d, \forall r, r\in S, \\ z_I \le I_t^d \le \theta z_I, \forall t, t\in M, \\ z_O \le O_r^d \le \theta z_O, \forall r, r\in S, \\ z_I,z_O \ge 0, \\ (v,u)\in (Q_{v-2},Q_{u-2}). \\ \end{array}\right. } \end{aligned}$$
(12)

This model shows the best and worst cross-efficiencies that DMU\(_k\) can achieve with the optimal weights chosen by DMU\(_d\), keeping the optimal self-evaluation efficiency (CCR efficiency) of DMU\(_d\) unchanged. We get \(E_{dk}^{\max }\) when the target is maximum and \(E_{dk}^{\min }\) when the target is minimum. In the satisfaction model, these are referred to as the ideal cross-efficiency target and non-ideal cross-efficiency target. The cross-efficiency targets for all DMU\(_j\) can then be found, as shown in Table 2. We then filter all evaluated DMUs that meet \(E_{dj}^{\max } \ne E_{dj}^{\min }\).

Table 2 Ideal and non-ideal cross-efficiency targets
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STEP 3: As shown in model (13), an improved satisfaction cross-efficiency model involving the DMUs’ satisfaction and weight restrictions in the indicator interaction scenarios is proposed.

$$\begin{aligned}&{\text {Max}} \ {\text {SD}}_d \nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-E_{dj}^{\max }\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg ) +s_{dj} = 0,\forall j,j\ne d,E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (13{\text {-}}1) \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-E_{dj}^{\min }\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg ) -\varphi _{dj} = 0,\forall j,j\ne d,E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (13{\text {-}}2) \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^d + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^d \wedge y_q^d)\bigg )-E_d^{*}\bigg (\sum \limits _{t\in M}v_t^dx_t^d + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^d \wedge x_p^d)\bigg ) = 0, &\quad (13{\text {-}}3) \\ \sum \limits _{t\in M}v_t^dx_t^d + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^d \wedge x_p^d)=1, &\quad (13{\text {-}}4) \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg )\le 0,\forall j,&\quad (13{\text {-}}5) \\ \frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}}\ge {\text {SD}}_d,\forall j,j\ne d, E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (13{\text {-}}6) \\ I_t^d=v_t^d+\frac{1}{2}\sum \limits _{p\in M {\setminus } t}v_{tp}^d, \forall t, t\in M, &\quad (13{\text {-}}7) \\ O_r^d=u_r^d+\frac{1}{2}\sum \limits _{q\in S {\setminus } r}u_{rq}^d, \forall r, r\in S, &\quad (13{\text {-}}8) \\ z_I \le I_t^d \le \theta z_I, \forall t, t\in M, &\quad (13{\text {-}}9) \\ z_O \le O_r^d \le \theta z_O, \forall r, r\in S, &\quad (13{\text {-}}10) \\ s_{dj},\varphi _{dj}\ge 0,\forall j,j\ne d, E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (13{\text {-}}11)\\ z_I,z_O \ge 0, &\quad (13{\text {-}}12) \\ (v,u)\in (Q_{v-2},Q_{u-2}). &\quad (13{\text {-}}13) \\ \end{array}\right. } \end{aligned}$$
(13)

In model (13), the objective function is to maximize the satisfaction of DMU\(_d\), reduce the weight differences while keeping the CCR optimal self-evaluation efficiency constant, and improve the satisfaction of other DMUs with the weights chosen by DMU\(_d\). Equations (13-1) and (13-2) ensure that the peer-efficiency of each DMU\(_j\) relative to DMU\(_d\) falls between its ideal and non-ideal cross-efficiency targets. Equations (13-3) to (13-5) indicate that the model allows for all the optimal weights in the CCR model for DMU\(_d\). Equation (13-6) indicates the maximization of the minimum satisfaction among all DMUs. Equations (13-7) to (13-10) control the global importance degree of all input and output indicators to vary within the specified boundaries to narrow the differences in the importance of indicators. All the DMUs select the optimal weight under these constraints and objectives. The cross-efficiency scores are then obtained for evaluation and ranking.

3.2 Satisfaction Evaluation Methods that Take Ethical Principles into Account-Fairness, Utilitarianism and Equity

As an important development in modern economics, social choice theory is mainly concerned with analyzing the relationship between individual preferences and collective choices. The fundamental question in its study is whether various social decisions respect individual preferences and whether different social states can be fairly ranked or evaluated in some other way [34]. Social choice theory is applicable to not only aspects of the public sphere such as politics but also the selection and ranking of social activities, such as those of culture, sports and the economy, and it is of great value in terms of improving the efficiency of social decision-making and enhancing the level of social welfare [34].

This paper argues that social choice theory is equally applicable to the selection and ranking of DEA cross-efficiency evaluation methods. We may consider the choice of weights for a particular DMU\(_d\) to be evaluated as a collective social choice. Whether the optimal weight chosen by this DMU\(_d\) respects or matches the individual preferences of the remaining DMUs (i.e., the extent to which individual preferences are satisfied) is the key issue of concern. We adopt the concept of satisfaction proposed by Wu et al. to express the acceptability of the remaining DMUs with respect to the optimal weights chosen by the DMU\(_d\) to be evaluated. We indirectly reflect the degree of respect for the DMU by measuring the degree of deviation from individual preferences, where not only the ideal preferences (targets) but also the non-ideal preferences (targets) are considered.

González–Pachón et al. pointed out that when dealing with social choice issues, we should not ignore or obscure the ethical principles involved in decision-making issues [35], which serve to constrain people’s behavior and regulate the relationships between individuals and society, as well as between people. Therefore, ethical principles should not be ignored in DEA cross-efficiency evaluations either. In the context of social choice, the three ethical principles are the Rawlsian or minimax principle (fairness and fraternity), the Benthamite or utilitarian principle (freedom) and equity from Marx’s political perspective.

Rawls’s theory of fairness (or fraternity), which advocates respect for all members of society and emphasizes maximizing the welfare of the individuals who benefit least, is linked to the minimax principle. This corresponds to the idea behind the approach in Sect. 3.1, maximizing the satisfaction degree of the DMU with the least satisfaction.

Bentham’s utilitarianism principle emphasizes the determination of the best overall social welfare taking into account the interests of each individual while preserving the maximum individual freedom. According to the utilitarian principle, the constraint \(\frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}}\ge {\text {SD}}_d,\forall j, E_{dj}^{\max }\ne E_{dj}^{\min }\) in model (13) can be replaced with \({\text {SD}}_d = \sum\limits _{\forall j, E_{dj}^{\max }\ne E_{dj}^{\min }}\frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}}\), whereas the remaining constraints and the objective remain unchanged. This implies the maximization of the overall satisfaction by maximizing the sum of the satisfaction of all DMUs.

The theory of equity from Marx’s political view refers to the equal distribution of total welfare among all members of society, thus providing maximum equity. However, in the DEA cross-efficiency evaluation method, the choice of weights is subject to multiple conditions and it is almost impossible to choose weights such that all DMUs have the same satisfaction degree. We therefore introduce Adams’s equity theory for illustration. According to Adams’s equity theory, the dissatisfaction and satisfaction of DMUs are almost always derived from the equity perception after a DMU compares its own satisfaction with that of other DMUs, with an emphasis on an equal distribution. The Italian economist Gini proposed the Gini coefficient based on the Lorenz curve to judge the income equity of residents [36]. The value range of the Gini coefficient is [0,1], and a distribution of income is more equal when the Gini coefficient is closer zero. Here, we construct the Gini coefficient from a satisfaction equity perspective, translate it into one of the constraints and seek to select the most equitable weighting choices possible. The parameter \(\alpha\) in the objective can be selected according to the requirements of the decision-making objective for the degree of equity. The model (14) is expressed as follows.

$$\begin{aligned}&{\text {Max}} \ {\text {SD}}_d - \alpha {\text {Gini}}_d \nonumber \\&{\text {s.t.}}\nonumber \\&{\left\{ \begin{array}{ll} \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-E_{dj}^{\max }\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg ) +s_{dj} = 0,\forall j,j\ne d,E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (14{\text {-}}1) \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-E_{dj}^{\min }\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg ) -\varphi _{dj} = 0,\forall j,j\ne d,E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (14{\text {-}}2) \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^d + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^d \wedge y_q^d)\bigg )-E_d^{*}\bigg (\sum \limits _{t\in M}v_t^dx_t^d + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^d \wedge x_p^d)\bigg ) = 0, &\quad (14{\text {-}}3) \\ \sum \limits _{t\in M}v_t^dx_t^d + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^d \wedge x_p^d)=1, &\quad (14{\text {-}}4) \\ \bigg ( \sum \limits _{r\in S}u_r^dy_r^j + \sum \limits _{\{r,q\} \subset S}u_{rq}^d(y_r^j \wedge y_q^j)\bigg )-\bigg (\sum \limits _{t\in M}v_t^dx_t^j + \sum \limits _{\{t,p\} \subset M}v_{tp}^d(x_t^j \wedge x_p^j)\bigg )\le 0,\forall j,&\quad (14{\text {-}}5) \\ {\text {SD}}_d = \sum \limits _{\forall j, E_{dj}^{\max }\ne E_{dj}^{\min }}\frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}},&\quad (14{\text {-}}6) \\ {\text {SD}}_{dj}=\frac{\varphi _{dj}}{s_{dj}+\varphi _{dj}},\forall j, E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (14{\text {-}}7) \\ {\text {Gini}}_d=\frac{1}{2n\sum \limits _j {\text {SD}}_{dj}}\sum \limits _i\sum \limits _j |{\text {SD}}_{di}-{\text {SD}}_{dj} |,\forall i,j, &\quad (14{\text {-}}8) \\ I_t^d=v_t^d+\frac{1}{2}\sum \limits _{p\in M {\setminus } t}v_{tp}^d, \forall t, t\in M, &\quad (14{\text {-}}9) \\ O_r^d=u_r^d+\frac{1}{2}\sum \limits _{q\in S {\setminus } r}u_{rq}^d, \forall r, r\in S, &\quad (14{\text {-}}10) \\ z_I \le I_t^d \le \theta z_I, \forall t, t\in M, &\quad (14{\text {-}}11) \\ z_O \le O_r^d \le \theta z_O, \forall r, r\in S, &\quad (14{\text {-}}12) \\ s_{dj},\varphi _{dj}\ge 0,\forall j,j\ne d, E_{dj}^{\max }\ne E_{dj}^{\min }, &\quad (14{\text {-}}13)\\ z_I,z_O \ge 0, &\quad (14{\text {-}}14) \\ (v,u)\in (Q_{v-2},Q_{u-2}). &\quad (14{\text {-}}15) \\ \end{array}\right. } \end{aligned}$$
(14)

3.3 Numerical Example

In this section, we illustrate the performance of the proposed approach and compare it with the performances of classical DEA cross-efficiency procedures using a small data set [9, 20] that has frequently been used in the related literature. The dataset comprises five DMUs with three inputs and two outputs as shown in Table 3.

Table 3 Example data
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3.3.1 Comparative Analysis of Indicator Weights

As mentioned earlier, we are particularly concerned with the issue of zero weights in that the use of zero weights means that some of the indicators considered are excluded from the evaluation. As a simple example, Tables 4 and 5 show the weight allocations and efficiency scores of the DMUs for the classical CCR model without and with a weight restriction. We see that there are many zero weights when there is no weight restriction; the weights of the third input indicator \(v_3\) for A, B, C and D are all zero. This means that the cross-efficiency evaluation is also not guaranteed to offset the effect of zero weights. In Table 5, however, the constraint of the indicator weight ratio not only effectively avoids the occurrence of zero weights but also limits the differences between the weights better, and the distribution of weights is more balanced. Although the weight constraint seems to reduce the efficiency score, such evaluation results are more scientific in terms of the comprehensiveness and rationality of the evaluation process.

Table 4 Weight choice and efficiency scores without a weight constraint
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Table 5 Weight choice and efficiency scores with a weight constraint \((\theta = 10)\)
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Another focus of this paper is an efficiency evaluation in the case of indicator interactions. At this point, the weight of an indicator is no longer just about the importance of the indicator individually but the degree of global importance of each indicator. We thus set limits for the allowable differences in the global importance degree of the indicators. Here, we compare the efficiency evaluation scores of the classical CCR method with those of the 2-CHCCR method that considers indicator interactions under the same indicator weight constraints. Table 6 and Fig. 1 show that, as concluded by Xia et al. [20], the decision-making information space is enlarged and the optional weight combinations becomes more available after considering indicator interactions, which is conducive to obtaining better optimized values and improving the efficiency score of the DMU.

Table 6 Comparison of efficiency scores between CCR and 2-CHCCR models \((\theta = 10)\)
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Fig. 1
figure 1

Efficiency scores for CCR and 2-CHCCR methods

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3.3.2 Sensitivity Analysis of Parameter \(\theta\)

The parameter \(\theta\) for limiting the indicator weight ratios introduced in this paper effectively avoids zero weights. Its value indicates the degree of restriction in allowing for differences in indicator importance in the DMU assessment. A sensitivity analysis of parameter \(\theta\) is conducted to further explore the effect of the selection of \(\theta\) on the assessment results.

Table 7 and Fig. 2 clearly show that the efficiency scores of DMUs gradually decrease as \(\theta\) decreases; i.e., a smaller value of \(\theta\) corresponds to a lower efficiency score of the DMU. This result is due to a tighter constraint on indicator importance differences corresponding to a smaller range of weights for individual DMUs and fewer available combinations of weights, with the potential for smaller efficiency scores. In practical application scenarios, the DMs are perfectly placed to select the appropriate parameter values according to the decision-making requirements.

Table 7 Efficiency scores of the 2-CHCCR method with different \(\theta\) constraints
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Fig. 2
figure 2

Trends of the 2-CHCCR efficiency score versus the \(\theta\) constraint

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3.3.3 Comparative Analysis of Cross-Efficiency Strategies

We evaluate and rank the efficiencies of DMUs using the 2-CHCCR model (self-evaluation efficiency), an arbitrary strategy, the well-known aggressive and benevolent strategies proposed by Doyle and Green [7], and the satisfaction-based strategy proposed in this paper.

Table 8 shows that (1) the 2-CHCCR model evaluates B and C as efficient with an efficiency of 1 but cannot further distinguish the ranking and (2) several different cross strategies yield different cross-efficiency scores and different ranking results. Specifically, all the cross-efficiency scores of DMUs are lower than the CCR self-evaluated efficiency; the aggressive cross-efficiency scores of DMUs are clearly lower than the benevolent cross-efficiency and have the same ranking as the arbitrary cross-efficiency; and the results of the satisfaction strategy proposed in this paper are closer to those of the benevolent strategy in terms of both efficiency scores and efficiency ranking, the efficiency scores of satisfaction strategy are relatively higher among these different strategies. In fact, there is no absolute advantage or disadvantage of these cross-efficiency evaluation methods and they just provide DMs with more options for decision making.

Table 8 CCR efficiency, cross-efficiency and ranking results of 2-CHCCR \((\theta =10)\)
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3.3.4 Satisfaction Cross-Efficiency Analysis Based on the Principles of Fairness, Utilitarianism and Equity

Another interesting topic is the preferences and ranking of DMUs from a satisfaction perspective in relation to several aspects of ethical principles in social choice theory. We calculated the satisfaction and cross-efficiency of each DMU from the perspectives of fairness, utilitarianism and equity, respectively. The results are given in Table 9.

Table 9 and Fig. 3 show that Rawls’s fairness theory emphasizes maximizing the welfare of the individuals who benefit the least, and the minimal satisfaction of DMUs guided by the theory of fairness is the greatest among the three principles. Taking DMU C as an example, the minimum satisfaction under fairness theory is 0.560, whereas the minimum satisfaction is 0.382 and 0.544 under utilitarianism and equity theory, respectively. No longer bound by the DMU of minimal satisfaction, Bentham’s utilitarianism principle truly maximizes overall satisfaction and preserves maximum individual freedom. On the one hand, the average values of satisfaction under utilitarianism are the largest, and the satisfaction cross-efficiencies are largest for all DMUs except DMU E, which has a slightly smaller cross-efficiency. On the other hand, under utilitarianism, the standard deviations (SDs) are also the largest among the three principles, with relatively large differences in satisfaction. The SDs of satisfaction guided by the equity theory are the smallest, and the DMUs have the most balanced distribution of satisfaction. However, we note that this is achieved at the expense of overall satisfaction, and the average values of satisfaction under the equity theory are the smallest. In fact, the fairness theory of increasing minimum satisfaction follows the idea of equity to some extent.

Table 9 Satisfaction evaluation results under different ethical principles
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Fig. 3
figure 3

Trends in mean and SD of satisfaction under different ethical principles

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4 Assessment of the W–E–F Nexus Input–Output Efficiency in China

In this section, we apply the improved satisfaction DEA cross-efficiency method to the empirical study of the W–E–F nexus input–output efficiency evaluation in China to further demonstrate the availability of the method.

4.1 Case Background

The rapid growth of the global population and increasing economic prosperity are putting enormous pressure on the sustainability of resources. Water, energy and food are essential for human production and life, and are important to economic development, environmental protection and social security. However, resource scarcity and environmental constraints are challenging the sustainable development of many aspects of society. In January 2011, the Global Risks Report (2011) of the World Economic Forum noted that resource scarcity can lead to social and political instability, geopolitical conflict and irreparable environmental damage and for the first time listed the W–E–F nexus as one of three key risk clusters to focus on that year, highlighting that any single optimization strategy that does not consider the W–E–F nexus will have unintended and serious consequences [37]. In the same year, the German federal government convened a security conference on the W–E–F nexus, which also pointed to the complex nexus of water, energy and food [38]. Since then, the W–E–F nexus has been the subject of much scholarly inquiry around the world. Olsson argued that the securities of water, energy and food are closely linked and that the concept of a nexus illustrates the scarcity of natural resources and the urgent need for integrated planning and management [39]. Machell et al. describe the interdependence of the production and consumption of water, energy and food as the W–E–F nexus [40]. In the context of global resource scarcity, water, energy and food constrain or contribute to each other, and all three are involved throughout an overall system [41], which is the W–E–F nexus. We know that the production of energy resources often requires large amounts of fresh water, and in the energy sector, almost all production and conversion processes require water resources, which in turn must be powered by energy for extraction, treatment and redistribution [42]. Food production is even more dependent on water and energy. In addition to water for agriculture, the modern production situation in agriculture is characterized by mechanical power, agricultural diesel, fertilizers and pesticides, all of which are indirect energy products. In turn, food products can be converted into many forms of energy through biomass. However, this is only the tip of the iceberg of the W–E–F nexus. In fact, the complex interactions are difficult to break down and quantify accurately and completely. It is therefore important to consider the complex system of the W–E–F nexus as a whole and to bring it into a unified framework for research and analysis. At the same time, the complex interactions between water, energy and food cannot be ignored.

This section thus considers water, energy and food as a system, incorporates demographic, economic and environmental factors into the indicator framework, and applies our improved satisfaction DEA cross-efficiency method to measure and analyze cross-section data for 30 provincial areas across China from 2011 to 2020. Through spatial and temporal comparisons, we analyze the efficiency and changes of W–E–F nexus inputs and outputs in each provincial area, indirectly reflecting the efficiency of W–E–F nexus resource allocation and its effect on the economy and environment and providing a reference for decision-making and theoretical support for improving the security of water, energy and food as well as nexus efficiency.

4.2 Indicators and Data

Aiming to ensure the relevance, importance and data accessibility of indicators, we refer to the DEA W–E–F input–output indicator system constructed by Li et al. [43] and Sun et al. [44] and finalize the indicator system shown in Table 10 under the premise of ensuring the number and linearity of the indicators.

Table 10 W–E–F nexus input–output indicator system
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In this paper, input and output data of water, energy and food for 30 provinces, municipalities and autonomous regions in China from 2011 to 2020 are selected as the production possibility set, and the Tibet Autonomous Region is excluded in view of the availability of energy data. The raw data for this study include the total water consumption, total energy consumption, food consumption expenditure per capita, resident population, gross domestic product per capita, chemical oxygen demand emission, (general) industrial solid waste generation, SO2 emission and smoke (dust) emission (particulate matter emission after 2016). These data are obtained from the 2012–2021 provincial and regional statistical yearbooks as well as the China Statistical Yearbook, China Energy Statistical Yearbook and China Environmental Statistical Yearbook. Owing to a change in statistical caliber, the per-capita food consumption expenditure in 2011 and 2012 is calculated from urban and rural per-capita food consumption expenditure data combined with urban and rural resident population data. The entropy method is used to construct the environmental pollution index, and the inverse is taken as one of the output indicators.

4.3 Empirical Analysis

Adopting the improved satisfaction DEA cross-efficiency method, applying the indicator system in Table 10 and using MATLAB R2018b software, we calculate the cross-sectional data of 30 provinces, municipalities and autonomous regions for 2011–2020 to obtain the annual W–E–F input–output cross-efficiency for each region and rank the efficiency according to the average value of cross-efficiency in all years. The results are given in Table 11.

Table 11 and Fig. 4 show that the input–output efficiency of the national W–E–F nexus is low from 2011 to 2020, with a 10-year national average value of 0.517, and that the national efficiency continually declines from 2013 to 2020, from 0.570 to 0.447. Among the seven geographic regions, East China (0.594), North China (0.591) and Central China (0.564) have high average values of W–E–F nexus input–output efficiency over the 10-year period whereas the remaining regions have efficiency lower than the national average, with the efficiency being lowest in Northwest China (0.398), and there is obvious regional heterogeneity in the W–E–F nexus input–output efficiency. Specifically, among the 30 provinces, municipalities and autonomous regions, Beijing (1.000), Shanghai (0.713), Jiangsu (0.696), Shaanxi (0.670) and Tianjin (0.624) have the highest average efficiency levels. Among them, Beijing is the only effective DMU with an efficient allocation of water, energy and food resources and a strong capacity for sustainable development. The majority of regions of China still have room to improve their W–E–F nexus inputs-outputs efficiency. Inner Mongolia (0.382), Heilongjiang (0.357), Qinghai (0.351), Ningxia (0.327) and Xinjiang (0.255) have low levels of efficiency and urgently need to adjust their allocations of water, energy and food resources.

Table 11 Satisfaction cross-efficiency for the W–E–F nexus input–output by region from 2011 to 2020
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Fig. 4
figure 4

Trends in satisfaction cross-efficiency averages of the W–E–F nexus input–output by region from 2011 to 2020

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The above results are based on satisfaction cross-efficiency guided by the fairness theory of the maximization of minimum satisfaction. Table 12 gives the satisfaction by province for each year from 2011 to 2020. The choice of weights for regions that do not appear in the table or have no entry for a particular year does not affect the optimal efficiency of the other regions; i.e., \(E_{dj}^{\max }=E_{dj}^{\min }\). There is thus no need to consider the satisfaction of other regions with the choice of weights for such regions. Finally, the results show that satisfaction with the vast majority of regions is high, except for low satisfaction with individual provinces in individual years. The average satisfaction level is above 0.8 in every year except 2014. A higher satisfaction level corresponds to the peer-evaluation efficiency of the DMU in the cross-efficiency evaluation process being closer to its ideal cross-efficiency target, implying a higher level of recognition and acceptance of the cross-efficiency evaluation results by region.

Table 12 Satisfaction of cross-efficiency by region from 2011 to 2020
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5 Conclusions

To address the problem of the optimal weighting of evaluation indicators and the interactions in DEA cross-efficiency evaluation, this paper proposed a new 2-additive Choquet integral DEA cross-efficiency evaluation method that considers the DMUs’ satisfaction and weight balancing. The implementation of the method is divided into three main steps, involving the optimal self-evaluation efficiency model under the 2-additive weight constraint, the ideal and non-ideal cross-efficiency target model and the optimal satisfaction model. We illustrated the performance of the proposed method by applying it to a mathematical arithmetic example that is frequently found in the relevant literature and to the real problem of evaluating the input–output efficiency of the W–E–F nexus. The results showed that the proposed method not only avoids the problem of zero weighting but also effectively limits differences in the importance of indicators, safeguarding the comprehensiveness and rationality of the evaluation process and making the evaluation results more scientific. Additionally, considering indicator interactions expands the decision-making information space, giving DMUs more weight choices, which is conducive to obtaining better optimized values and improving the efficiency scores of DMUs. Furthermore, we discussed the preferences and ranking of DEA cross-efficiency evaluation methods from the satisfaction perspective in relation to several aspects of ethical principles in social choice theory, providing DMs with more decision-making positions. Overall, the method proposed in this paper overcomes some of the problems of traditional DEA cross-efficiency evaluation methods, enriches the research perspectives of DEA methods and provides reasonable and comprehensive evaluation results. In future research, we hope to explore more scientific and feasible DEA cross-efficiency strategies. Additionally, more attention should be paid to issues about DMUs in the evaluation process, such as how to avoid large differences between the weights provided by different DMUs and how to deal with the possible relationships between DMUs. In terms of research perspective, inspired by articles by Zhang et al. and Xiao et al. [45, 46], we will continue to try to integrate consensus models with DEA.