Practice and research have consistently demonstrated the interdependent relationship between economic development and ecological environmental protection.
Figure 1 illustrates the network structure of the economy and environment, highlighting relevant inputs and outputs in the economic development and environmental protection processes. National and regional economic development relies on various inputs such as human, material, and financial resources, including non-energy resources like humans and capital, as well as energy resources such as coal, natural gas, and crude oil. The output is intuitively expressed through the gross domestic product (GDP) [
16]. Furthermore, energy consumption processes often lead to pollution. To address environmental concerns, each region invests resources to control and mitigate pollution, thus forming a regional environmental protection subsystem.
The performance of economic development and environmental protection subsystems can be evaluated based on the input–output relationship. Subsequently, considering the coupling correlation structure, the coupling coordination model can further evaluate the correlation degree and coordination development among subsystems. The corresponding model construction is detailed below.
2.1. Super-SBM Model
Evaluating the performance of economic development and environmental protection subsystems involves a multi-index system evaluation problem. The weighting of each metric significantly impacts the results and is a focal point of multi-index evaluations. The DEA method effectively addresses the subjective issue of index weighting in multi-index evaluations and has been extensively utilized in systematic evaluation studies. Moreover, the DEA method can be tailored to suit specific evaluation systems by modifying it according to research objectives, index variance, and system structure. For an evaluation system with a network structure, two-stage and multi-stage DEA models have been constructed and applied [
19,
20]. The dynamic DEA model was constructed to consider the influence of time [
15,
21], and the super-DEA model was built to improve the identification of the DEA method [
22]. Additionally, the SBM model [
23] was constructed to deeply analyze the “crowding” or “relaxation” of each input or output. Therefore, to evaluate the performance of economic development and environmental protection subsystems in greater detail, we construct a super-efficient SBM model with stronger discriminative power.
Consider
independent decision-making units (DMUs), denoted as
.
Figure 1 illustrates a typical development system comprising two subsystems within each DMU. In the economic development subsystem, each DMU consumes energy
) and non-energy inputs
) to produce desired outputs
. In the environmental protection subsystem, each DMU consumes inputs
to reduce undesirable outputs
), which result from the consumption of energy inputs
).
The super-DEA method, with enhanced discernment, comprises two steps. The first step is to calculate the efficiency of all DMUs using the traditional DEA model. In the second step, the super-DEA model is used to re-evaluate the DMU with an efficiency of 1 obtained in the first step, and their super-efficiency value can be calculated [
24]. An input-oriented super-DEA model is constructed in this study, as the management of inputs generally offers greater convenience compared to that of outputs in system operation. Meanwhile, the SBM model can easily define and evaluate subsystem efficiency through the slack variables in the efficiency evaluation of complex multisystem structures. Finally, we construct an input-oriented SBM model as the first step in evaluating the efficiency of all DMUs. The model is expressed as follows:
In Model (1), ,, and are slack variables denoting energy input excess, non-energy input excess, and environment protection input excess. is the intensity vector. Note that Model (1) is an input-oriented SBM model under the variable returns to scale (VRS) assumption, which evaluates the pure technical efficiency (PTE) of the system.
If the influence of scale is not considered, the constraint can be omitted. The model then becomes an input-oriented SBM model under the constant returns to scale (CRS) assumption, which can evaluate technical efficiency (TE). TE is the product of PTE and scale efficiency (SE). Then, the TE, PTE, and SE of the system can be obtained and decomposed by adjusting the constraint . The effective and ineffective problems of technology and scale can then be analyzed.
The objective function calculates the maximum slack variables for all inputs.
,
, and
represent the slacks in the energy, non-energy, and environmental protection inputs, respectively. Constraint
indicates that an undesirable output is treated in terms of as few variables as possible. Existing studies have posited that if pollution can be controlled, then it can be regarded as somewhat controllable to a certain extent [
15]. Therefore, the efficiency of the economic development and environmental protection subsystems corresponding to Model (1) can be defined as follows:
Formulas (2) and (3) correspond to the efficiency of economic development (
) and environmental protection (
) subsystems, respectively, and are defined according to the input structure.
is composed of slacks in energy and non-energy inputs.
is composed of slacks in energy and pollution control inputs. The energy input plays a role in both subsystems and is similar to the shared input in related studies [
25,
26,
27]. However, the function of the energy input in this study differs from that of the shared input. In the traditional system, as part of the shared input is consumed in one subsystem, the remainder is consumed in the other. Then, the shared inputs should be suitably allocated to all subsystems for the efficiency evaluation. However, in this study, energy acts on both subsystems simultaneously, rather than partially on one. All the energy used to develop the economy, together with the production of pollutants, must be controlled in a timely manner for environmental protection. Thus, we do not allocate energy inputs to the two subsystems, and slacks in energy inputs are an essential element of efficiency in both subsystems.
The efficiencies of
and
are equal to 1 when the slacks
,
, and
in Model (1) are equal to zero. At this point, a further comparison is made between effective units, namely, DMUs with an efficiency of 1, and the corresponding super-SBM model is as follows:
,
, and
in Model (4) are slack variables denoting the energy input shortfall, non-energy input shortfall, and environment protection input shortfall of the DMU, at or beyond the frontier. Model (4) is the super-SBM model under the VRS assumption. However, the super-efficiency model under the VRS constraint [
28] or with undesirable production variables faces infeasibility problems [
29]. Model (4) must be adjusted and modified to solve the problem of a lack of feasibility. Therefore, the modified super-SBM model is constructed as follows:
,
,
,
, and
in Model (5) are slack variables denoting the energy input shortfall, non-energy input shortfall, environment protection input shortfall, desirable output excess, and undesirable output shortfall of the DMU, at or beyond the frontier, respectively. The main difference between the modified super-SBM Model (5) and super-SBM Model (4) lies in three parts: constraints
and
and the objective function. The slack
represents the allowable reduction in the desirable output, and
allows for an increase in the undesirable output. The coefficients of
and
in the objective function are
, which represents the maximum value (i.e., take
). Meanwhile, the objective minimization means that DMUs beyond the frontier are prioritized to reach the frontier by increasing energy, non-energy, and environmental inputs. The method of increasing desirable outputs or decreasing undesirable outputs is considered only when the frontier can be reached by increasing the inputs. Finally, only the inputs are considered in the efficiency definition based on the input orientation. The corresponding super-efficiency of the two subsystems is defined as follows:
Therefore, we first obtain the efficiency value of the invalid DMU using Model (1) and Formulas (2) and (3). The super-efficiency value of the effective DMU in Model (1) can then be obtained using Model (5) and Formulas (6) and (7). Finally, the efficiency values of all DMUs are determined.
2.2. Coupling Coordination Model
Coupling coordination involves coordinating and monitoring multiple systems or subsystems within a complex system. Interdependence and interaction occur between different subsystems. Thus, the degree of coupling coordination can be used to analyze and guarantee the normal operation and high performance of the entire system. Economic and environmental issues coexist within the complex system of social development. Analyzing the degree of dependence and coordination between economic development and environmental protection can facilitate a further understanding and optimization of regional coordinated development. Referring to existing studies [
6,
7], the coupling coordination model of the economy and environment can be constructed as follows:
where ECO and ENV represent the comprehensive indices of the economic development and environmental protection subsystems, respectively, with values ranging from 0 to 1. The degree of coupling (C) represents the level of interdependence among subsystems.
Table 1 illustrates the different levels of coupling. The larger C is, the higher the coupling level, which means that the higher the degree of intersystem correlation and synchronization, the better or worse the economy and environment perform simultaneously. The smaller C is, the lower the coupling level, indicating a lower degree of correlation and synchronization between systems and reflecting the opposing performance of the economy and environment.
C reflects the degree of correlation and synchronization between systems but cannot reflect whether subsystems maintain a healthy interaction and development state. The coupling coordination level can be used to analyze the coordination state between the economic and environmental subsystems; therefore, the coordination level is further calculated according to Formulas (9) and (10) as follows:
where
is the comprehensive weighted evaluation index for the two subsystems, and
and
represent the weights of the two subsystems, respectively. Generally, the economy and environment are considered equally important (i.e.,
). D is the coupling coordination level, which reflects whether the subsystems maintain a good interaction and healthy development state.
Table 2 lists the coordination types corresponding to different coupling coordination levels.
indicates the system is in a state of coordinated development, in which the two subsystems promote each other and develop healthily. The larger D is, the more coordinated the system development and the more conducive it is to joint optimization. In contrast,
indicates that the system is in a state of dysfunctional recession, in which the development between the two subsystems is unbalanced. The smaller D is, the more unbalanced the regional development and the more unfavorable the conditions are to the sustainable development of the region.
In calculating C and D, we use economic and environmental efficiencies (
and
) to represent their comprehensive index. However, owing to the super-efficiency being greater than 1, the efficiency value cannot be directly calculated in coupling coordination Models (8)–(10). The efficiency must be normalized to ensure a value between 0 and 1. Accordingly, simple Formulas (11) and (12) are used to treat the efficiency value as a comprehensive index of economic development and environmental protection as follows:
Finally, the coupling and coordination level of the regional economy and environment subsystems can be calculated and analyzed by introducing the comprehensive index from the efficiency value adjustment into the coupling formulas.