Agricultural eco-efficiency is the ratio of the economic value added by agricultural production to the consequences of environmental impact[27]. DEA method has become the most commonly used method for AEE evaluation[28]. The main reason is that the DEA method ignores the influence of random errors and can overcome the impact of non-technical factors on the frontier production function [18]. It has advantages, including simultaneous processing of multiple input-output elements and nonparametric processing of effective boundary. Obtaining the current output level at a lower input level is more conducive to achieving the goal of sustainable agricultural development[29]. By controlling the state of economies of scale, orientation, disposability of elements, and production frontier functions, the work uses the SUPER-ML index to measure the AEE of various regions in China and further analysis of the temporal and spatial differences in decomposition efficiency, the slackness of input-output indicators, and inefficiency.

ML index

Malmquist index usually to analyzes the panel data of observed values at multiple time points. Färe first used the DEA method to calculate the Malmquist index(MI) and further decomposed it into technical efficiency change (EC) and technical change (TC), which commonly used to analyze productivity changes, and the effect of technological efficiency and technological progress on productivity change. Chung introduced the directional distance function into the Malmquist index to deal with the problem of unexpected output, which is called the Malmquist–Luenberger (ML)index[30]. The core is to solve the problem of unexpected output. The Fixed Malmquist index takes the single-phase front of a fixed period as the reference front for calculating MI (t-1, t) in each period [31]. The MI and its decomposition efficiency model are as follows:

$$MI\left(t-1,t\right)=\frac{Score\_f(x\_t,y\_t)}{Score\_f(x\_t-1,y\_t-1)}$$

$$EC\left(t-1,t\right)=\frac{Score\_f(x\_t,y\_t)}{Score\_f(x\_t-1,y\_t-1)}$$

\(TC\left(t-1,t\right)=\frac{MI\left(t-1,t\right)}{EC\left(t-1,t\right)}\) \(=\frac{\left(\frac{Score\_f(x\_t,y\_t)}{Score\_f(x\_t-1,y\_t-1)}\right)}{\left(\frac{Score\_f(x\_t,y\_t)}{Score\_t-1(x\_t-1,y\_t-1)}\right)}\)

$$=\frac{\left(\frac{Score\_f(x\_t,y\_t)}{Score\_t(x\_t,y\_t)}\right)}{\left(\frac{Score\_f(x\_t-1,y\_t-1)}{Scor{e}_{t}-1(x\_t-1,y\_t-1)}\right)}$$

Where: Score_f (x_t, y_t) represents the DEA efficiency value obtained from the reference fixed front. Among them, \(\frac{Score\_f(x\_t,y\_t)}{Score\_t(x\_t,y\_t)}\) reflects the distance between the front of phase t of the fixed front; \(\frac{Score\_f(x\_t-1,y\_t-1)}{Score\_t-1(x\_t-1,y\_t-1)}\) reflects the distance between the front of phase t-1 of the fixed front. The ratio of the two reflects the change of the t-period front compared with the t-1 period front.

Strong Disposability-Undesirable-EBM-Super Efficiency model

Tone and Tsutsui proposed the EBM (Epsilon-Based Measure) model. It is a hybrid model that includes two types of distance functions: radial and SBM. The non-point source pollution and carbon emission in agricultural production as unexpected output. The projection direction of the evaluated DMU in the model is to increase good output and reduce bad output (unexpected output). Referring to the setting of the model by Wang, select the Super Efficiency-EBM model with four inputs, one expected output, and four unexpected outputs. In the planning formula, Max represents the strong, effective frontier. Record 30 provinces as decision-making units (DMU) as xij, in period t(t=1,...,T), there are k(k=1,...,30) DMU, each decision making unit has m input X=(\({\text{x}}_{1},{\text{x}}_{2},\cdots ,{\text{x}}_{\text{m}}\))\(\in {\text{R}}_{+}^{\text{m}}\), n expected output Yr=(r=1,2,…,n)\(\in {\text{R}}_{+}^{\text{n}}\) and J non-expected output Yj=(\(j=1,...,\text{j})\in {\text{R}}_{+}^{\text{j}}\), X={xij}∈RM×N, Y={yij}∈RM*N, and X>0, Y>0 respectively are input and output matrices (Yin, Zhu, 2020). The model is as follows:

$${\rho }^{*}=max\frac{\theta -{\epsilon }_{x}{\sum }_{i=1}^{4}\frac{{W}_{i}^{-}{S}_{i}^{-}}{{x}_{ik}}}{\phi +{\epsilon }_{y}\sum _{r=1}^{}\frac{{W}_{r}^{+}{S}_{r}^{+}}{{y}_{rk}}+{\epsilon }_{b}\sum _{p=1}^{j}\frac{{W}_{p}^{{b}_{-}}{S}_{p}^{{b}_{-}}}{{b}_{pk}}}$$

s.t.\(\sum _{j=1,j\ne jo}^{k}{x}_{ij}{\lambda }_{j}+{S}_{i}^{-}=\theta {X}_{ik},i=\text{1,2},\text{3,4}\)\(\sum _{j=1,j\ne jo}^{k}{y}_{rj}{\lambda }_{j}-{S}_{r}^{+}=\phi {y}_{rk},r=1\)

\(\sum _{p=1}^{2}{b}_{pj}{\lambda }_{j}+{S}_{p}^{{b}_{-}}=\phi {b}_{pk},p=\text{1,2},\text{3,4}\) \({\lambda }_{j}\ge 0,{S}_{i}^{-},{S}_{r}^{+},{S}_{p}^{{b}_{-}}\ge 0\)

Where: ρ* is the best efficiency under the condition of constant return to scale. \({\theta }\) is the planning parameter of the radial part. \({\epsilon }_{x}, {\epsilon }_{y}\) is key parameter. Satisfy 0\(\le {\epsilon }_{x},{\epsilon }_{y}\le 1.{ W}_{i}^{-}\) is the importance of input indicators, it meets \(\sum _{i=1}^{4}{W}_{i}^{-}\)=1; \({x}_{ik}\) and \({y}_{rk}\) are the i inputs and the r outputs of decision-making of DMUk. \({S}_{i}^{-}\) is the relaxation of input element i. \({\phi }\) is the output expansion ratio. \({S}_{r}^{+}\) is the relaxation variable of the expected output of class r. \({S}_{p}^{{b}_{-}}\) is the relaxation variable of p-type unexpected output. \({W}_{r}^{+},{W}_{p}^{{b}_{-}}\)is the weight of both index. \({b}_{pk}\) is the p unexpected output of DMUk, q is the number of unexpected outputs, j is the DMU, \({\lambda }_{j}\)is the linear of combination coefficient. \({j}_{0}\) represents the super efficiency value of DM\({U}_{j}\) on the new effective frontier excluding DM\({U}_{jo}\) when the commented decision unit is DM\({U}_{jo}\).

Efficiency decomposition

According to the idea of ecological efficiency measurement, we carried out further research and decomposed the calculation formula of the ineffective rate of each input-output element. The projection value is the projection of the evaluated DMU on the leading edge. The relative gap between the current value and the projection value represents the inefficiency of each input and output of the DMU. The improvement value of inefficient DMU includes two parts: the one is proportional movement, and the other is slack movement. Combined with Cooper's decomposition idea, the efficiency decomposition formula and inefficiency decomposition formula of input-output elements are as follows:

$${e}_{i}=\frac{{x}_{ik}^{T}-{s}_{i}^{-}}{{x}_{ik}^{T}},{e}_{r}=\frac{{y}_{rk}^{g,T}}{{y}_{rk}^{g,T}+{s}_{r}^{+}} , {e}_{q}=\frac{{y}_{qk}^{b,T}-{s}_{q}^{-}}{{y}_{qk}^{b,T}}$$

$$i{e}_{i}=\frac{1}{2m} \frac{{s}_{i}^{-}}{{x}_{ik}^{T}} , i{e}_{r}=\frac{1}{2({s}_{1}+{s}_{2})} \frac{{s}_{r}^{+}}{{y}_{ik}^{g,T}} , i{e}_{q}=\frac{1}{2({s}_{1}+{s}_{2})} \frac{{s}_{q}^{-}}{{y}_{qk}^{b,T}}$$

Where: \({s}_{i}^{-}\),\({s}_{r}^{+}\),\({s}_{q}^{-}\)expression relaxation variables of input, expected output, and unexpected output. ei, er, eq represent the efficiency decomposition formula of input factors, expected output, and unexpected output. iei, ier, ieq represent the inefficiency decomposition formula. The efficiency decomposition formula obtains the efficiency level of each element, and the inefficiency decomposition formula obtains the decomposition of the total inefficiency, reflecting the promotion potential of each element to agricultural ecological efficiency.