Intelligence Approach-Driven Bidirectional Analysis Framework for Efficiency Measurement and Resource Optimization of Forest Carbon Sink in China

Abstract

A critical natural solution to combat global warming and reduce carbon emission is the forest carbon sink (FCS). Owing to variations in geographic location, policy formulation, and economic development, Chinese provinces exhibit significant disparities in forest carbon sink efficiency (FCSE). Therefore, evaluating and enhancing FCSE and optimizing resource allocation have emerged as pressing issues. This study develops a pioneering analytical framework for the systematic estimation and optimization of FCS resources. It measures FCSE, considering both dynamic and static aspects and adopting a spatial–temporal perspective, utilizing the Malmquist Index and Super Efficiency Slacks-Based Measure to analyze the primary factors influencing FCSE. The Autoregressive Integrated Moving Average method forecasts carbon sink goals for typical regions for the years 2030, 2045, and 2060. To effectively enhance FCSE and rationally optimize FCS resource allocation, this study constructs the Inverse Data Envelopment Analysis. The study’s findings indicate significant disparities in the extremes of the average FCSE across Chinese regions, with a mean value difference of 2.2188. Technological change is the primary driver of advancements in FCSE. To achieve the 2060 carbon sink goal, each input indicator requires a substantial increase. Drawing on insights into the FCS landscape, the study delineates regional disparities and offers a scientific foundation for policymakers to devise strategies and address sustainability concerns regarding FCS.

1. Introduction

1.1. Background and Motivation

Global warming has numerous detrimental effects and is a matter of significant concern, prompting nations to adopt proactive measures to reduce carbon emissions [1]. China’s greenhouse gas emissions constitute a relatively large share of the global total. Concurrently, as a signatory to the Paris Agreement, China is committed to the timely achievement of carbon targets [2]. Forests, often referred to as the lungs of the earth, play a multifaceted role in mitigating the effects of global warming, fostering regional economic development, and improving residents’ well-being [3]. In addition, forests represent a vital resource for human development, not only in terms of their capacity to absorb carbon but also in their contribution to the economy and culture [4]. In recent years, scholars have increasingly highlighted the significant role of forests as carbon sink [5]. Forests sequester atmospheric CO₂ through photosynthesis, storing carbon in plant biomass. As critical terrestrial carbon sinks, they mitigate anthropogenic emissions and regulate climate by maintaining carbon balance. Beyond carbon storage in vegetation and soil organic matter, forests stabilize ecosystems, conserve biodiversity, and protect watersheds. Their conservation and sustainable management are thus essential for climate mitigation and ecological resilience. Globally, forests represent one of the largest carbon reservoirs, accumulating carbon through tree growth and organic matter decomposition in soils and woody debris. As a responsible global actor, China is poised to assess the forest carbon sink efficiency (FCSE) and implement targeted measures to achieve the proposed policy objectives [6]. Figure 1 shows the forest carbon sink (FCS) and total carbon emission at different stages since 2004 for 30 provinces in China. The data is sourced from the China Energy Statistical Yearbook and the China Forest Resources Report.

Carbon sink is broadly categorized into two types: engineering carbon sink and ecological carbon sink. Currently, a predominant technology for engineering carbon sink is Carbon Capture, Utilization, and Storage (CCUS). Recognized widely as an essential climate change mitigation strategy, CCUS technology effectively reduces greenhouse gas emissions from industrial and energy production processes. However, advancements in engineered carbon sink have been limited in recent years [7]. Ecological carbon sink serves as a natural carbon reservoir that accumulates and stores carbon. These repositories can be found in various ecosystems, including forests, marine environments, wetlands, grasslands, and prairies. Among them, terrestrial and marine ecosystems are the most significant carbon repositories. The ocean system represents the largest carbon reservoir on Earth. There are existing studies with methods for calculating the ocean carbon sink, but the ocean carbon sink is difficult to calculate [8]. Compared with other carbon pools in terrestrial ecosystems, forests are relatively straightforward to measure and offer promising prospects for development. These development opportunities primarily involve enhancing measurement precision and management of FCSE through standardized assessment protocols. Given its rich forest resources, the potential of FCS is enormous in China. In the fight against global warming, forests play an essential role as carbon sink. However, the FCSE varies considerably from province to province in China due to a variety of factors. Therefore, assessing and improving the FCSE and optimizing resource allocation have become pressing issues requiring attention.

1.2. Literature Review

This section critically reviews the development and application of Data Envelopment Analysis (DEA) in efficiency evaluation, with a particular focus on FCSE. Two principal methodologies exist for gauging efficiency: DEA [9,10,11,12] and Stochastic Frontier Analysis (SFA) [13,14]. Among them, the DEA is widely used to assess various types of efficiencies due to its non-parametric and data-driven characteristics. DEA is regarded as one of the most popular non-parametric techniques for evaluating the performance of decision-making units (DMUs) [15]. DEA has been popularly used in industry analysis [16,17], energy [18,19,20], transit performance measurement [21], industry [22,23,24], and agriculture [25]. DEA is promising to become an important assessment tool in the energy efficiency field [26].

In terms of methodological development, DEA models have undergone several significant evolutionary advancements. Ref. [27] applied the Charnes–Cooper–Rhodes (CCR) and Banker–Charnes–Cooper (BCC) models to determine the crucial environmental factors that should be considered. Ref. [28] used the BCC model introduced to measure the efficiency of technological innovation. Conventional DEA models, known for their radial efficiency measures, fail to account for complexities like redundant inputs or suboptimal outputs in the manufacturing processes. Ref. [29] employed the Slacks-Based Measure (SBM) model to assess carbon emission efficiency. Ref. [30] used the SBM model to explore the FCSE in China. Ref. [31] evaluated the forestry sector across 30 regions in China by employing a blend of static and dynamic SBM. Ref. [32] constructed the SBM model to measure FCSE in Russia. The simple DEA model has a maximum efficiency of 1. In the event that there are multiple valid DMUs and all DMUs exhibit an efficiency of 1, further analysis of the multiple valid DMUs is not possible. To solve this problem, scholars began to use the Super SBM model [33,34,35,36]. The underlying efficiency measure can only be limited to cross-sectional data, whereas adding the Malmquist Index (MI) to the original base removes bias from the calculations. The Malmquist method of DEA is employed to examine the temporal alterations in carbon sink and the variables that influence this process across Chinese provinces, and the findings ultimately revealed significant disparities in forest carbon sink efficiency among different provinces, which aligns with the conclusions drawn in the present study [37]. Ref. [38] analyzes the dynamics of China’s FCS and the factors affecting it using the MI model, and the TC has always been a key driver behind the increasing FCSE of China. Ref. [39] explored the dynamics of agroecological effectiveness and agroecological efficiency as well as the drivers of agroecological efficiency growth through panel data analyses and DEA–Malmquist–Luenberger. Ref. [40] analyzed the impact of technology gaps and pollution emissions on China’s industrial sector using the Metafrontier Malmquist–Luenberger method from 2000 to 2016. Recent studies have shown that Inverse DEA is highly likely to be an important way to address the feedback of climate change [41]. Ref. [42] calculated the allocation scheme of the indicator on the basis of Inverse DEA and put forward some suggestions on the FCS incremental pathway. Ref. [43] applied the Inverse DEA to explore specific pathways for future CO₂ reductions. Ref. [44] constructed a new Inverse DEA for measuring CO₂ emission allocations under given assumptions. In this study, Inverse DEA is introduced into FCS to solve the infeasibility problem of resource allocation optimization.

The principal gaps identified in existing studies, as revealed by a thorough literature review and research, are as follows:

(1)

A deficiency in existing studies is the absence of a systematic assessment and analysis of FCSE using an integrated static and dynamic approach. Many studies measure FCSE solely from a static perspective or examine the main factors influencing it from a dynamic perspective.

(2)

An examination of current Inverse DEA research indicates that most studies are confined to theoretical exploration. There is scant research on the practical application of Inverse DEA and few studies investigating the optimization of FCS resource allocations.

(3)

There remains an absence of comprehensive research on the sustainable development of FCS. The development of an innovative framework for optimizing the allocation of FCS resources has yet to be developed. Long-term research is deficient in feedback. There is a gap in the research regarding how FCSE feedback can be used to inform improvements and how to achieve sustainable development.

This study proposes the following improvements to address these shortcomings:

(1)

By integrating the Super SBM and MI methods for a systematic spatial–temporal analysis of FCSE, encompassing both dynamic and static elements, and examining the main factors influencing FCSE progress.

(2)

By introducing the Inverse DEA model to the study of FCSE, the FCS resource allocation optimization is studied through Inverse DEA to solve the policy feasibility of increasing FCS.

(3)

In this study, the analysis framework consisting of four subsystems is constructed innovatively to further study the FCSE problem and the FCS resource allocation optimization problem. The optimal values of future input factors are measured, which will be a practical reference for formulating future forestry policies.

1.3. Objectives and Contributions

Aiming at the problems of measuring the FCSE and optimizing the allocation of FCS resources, this study innovatively constructs a novel analysis framework containing four sub-models. This study’s innovation and contributions primarily manifest in methodological innovation, in-depth exploration of regional disparities, guidance for optimized resource allocation, scientific foundations for policy formulation, academic contributions, and international exchanges. These novel insights and methodologies provide a novel way to research FCSE, which can promote the sustainable development of FCS.

The main objectives and work are summarized below:

(1) Measuring FCSE in 30 Chinese provinces relying on the Super SBM model. (2) To gain further insight into the factors that affect efficiency, this study decomposes the concept of efficiency using the MI. This allows us to identify the main factors influencing efficiency and the dynamics of efficiency. (3) Based on the projected FCS target value, the future allocation of input factors is measured using inverse DEA with constant control efficiency.

The primary contributions of this research may be summarized as follows:

(1)

This study innovatively constructs an analytical framework incorporating four sub-models. This study adopts an innovative methodology for assessing FCSE by integrating the Super-SBM with the MI model. Methodologically, this innovation provides a novel perspective and a comprehensive toolkit for analyzing FCSE. Utilizing this approach enables the study to precisely identify variations and trends in FCSE, thereby providing scientific insights for policy development.

(2)

The study utilizes a comprehensive approach to reveal regional disparities. It extends its analysis to the provincial level, offering an exhaustive assessment of FCSE across 30 Chinese provinces. Comparative analysis of FCSE across provinces highlights significant regional disparities, offering valuable insights for policymaking and resource optimization.

(3)

This study guides resource optimization and has a wide range of applications. This study proposes the optimal values of future input indicators for provinces with low FCSE using the Inverse DEA model. This innovative approach not only helps to optimize resource allocation and enhance the FCSE, but also provides specific directions for improvement and implementation strategies for relevant provinces.

(4)

The study contributes evidence to inform policy development. The findings provide a scientific basis for governmental entities to formulate and implement policies related to FCS. Through assessments and analyses of FCSE across various provinces, this study identifies the main factors impacting FCSE, and potential areas for enhancement, and offers targeted policy recommendations to policymakers, addressing the sustainability of FCSE.

2. Materials and Methods

Concerning the indicators of previous relevant studies, the indicators used in this study to measure the FCSE ultimately selected labor, land, and capital, with an additional energy consumption incorporated. The labor indicator selects the year-end employees in forestry, the land indicator selects the forest area, and the selection of capital investment indicators based on completed investment in forestry fixed assets as data.

2.1. Selection of Indicators

This study organizes the selection of indicators used by previous scholars in studying the FCSE problem based on the DEA method, as shown in

Table 1. Forest carbon sink efficiency input indicator reference.

Forest Carbon Sink Efficiency Indicators	1	2	3	4	5	6
Practitioners of the forestry system at the end of the year	√	√	√	√	√	√
Completion of fixed investment in forestry	√			√		√
Forest area	√	√		√	√	√
Energy consumption		√
Forestry investment completion		√	√		√
Afforestation area			√

1 = [37]; 2 = [31]; 3 = [42]; 4 = [30]; 5 = [45]; 6 = [38].

.

Labor inputs are estimated based on the number of people working in the forestry system at the end of the year in each province. The year-end forestry system staff does not include resigned staff so that it can more effectively reflect the actual labor input of the provincial forestry system in a year. Staffing in the forestry sector is closely related to FCS. Employees in the forestry sector play an essential role in protecting forest resources and promoting sustainable forest development by utilizing their professional knowledge and skills.

In terms of investment, the critical impact of economic support on improving forest efficiency is highlighted by the choice of investment in forestry fixed assets as an indicator. The infrastructure and productivity of forestry carbon sink investments have improved. This has a profound impact on the sustainable utilization and management of forest resources. The development of forestry infrastructure has benefited from increased investment in fixed assets.

Land inputs are measured through the forest area indicator. The forest area is directly related to FCS. Therefore, it is chosen as an indicator of land inputs.

Three commonly used indicators are selected for this study. Energy consumption cannot be avoided in the management and operation of the forest system, and the composition structure of regional energy and energy consumption will directly affect the region’s FCSE. The final choice of indicators for this study is shown in

Table 2. Forest carbon sink efficiency input indicator selection.

	System	Subsystem	Ind icators	Unit
Input	Labor input	Year-end forestry system practitioners	Number of state-owned economic units	People
			Collective economic units	People
			Other economic units	People
	Capital investment	The amount of investment in forestry fixed assets	Investment in fixed assets in forestry	10,000 a
			Forest industry fixed asset investment	10,000 a
	Land input	Forest area	Forest area	10,000 ha
	Energy consumption	Energy consumption	Energy consumption	10,000 tons
Output	FCS	FCS	FCS	Megaton
	Forestry production value	Forestry production value	Forestry production value	10,000 a

Note: ^a represents Yuan.

.

Given the difficulty of obtaining direct data about energy consumption in the forest sector, previous research on this topic has been invaluable in providing insight and estimates. Multiplying total regional energy consumption by the ratio of forestry output to regional GDP is one way of expressing energy consumption [46]. The resulting figure can be represented by the following formula:

$$EC={\displaystyle \frac{FPV}{TPV}}\times TEC$$

(1)

where EC denotes regional energy consumption, FPV denotes forestry production value, TPV denotes total regional production value, and TEC denotes total energy consumption.

2.2. Methodology for Calculating Forest Carbon Sink

This study does not account for differing forest types or climates. This methodological choice was made to ensure consistent application across diverse areas while maintaining computational feasibility, though we acknowledge it represents a trade-off between precision and practicality. According to the 2006 IPCC Guidelines for National Greenhouse Gas Inventories, the specific calculation formula is as follows:

$$C=A\times V\times \mathrm{D}\times \mathrm{CF}\times \mathrm{BCEF}\times (1+\mathrm{R})\times (1+\mathrm{RDW})$$

(2)

The above formula indicates the volume of timber stock in units of ${\mathrm{m}}^3$/ha. $A$ indicates the remaining land area in the same category of land use, in ha. $A\times V$ indicates total forest stock, the China Forestry Statistical Yearbook provides data on forest stock. The specific values of the parameters are shown in

Table 3. The parameter of forest carbon sink.

Parameter	Value
BCEF (The biomass expansion factor)	1.35
D (The basic density of wood)	0.441
R (The ratio of below-ground biomass to above-ground biomass)	0.404
RDW (The dead-to-live ratio of dry weight of dead wood to live biomass)	0.1727
CF (The carbon fraction of dry matter)	0.47

[47].

2.3. Data Sources and Data Processing

Owing to data limitations for Taiwan, Macao, Hong Kong, and Tibet, this study encompasses 30 Chinese provinces. The data for this study are mainly obtained from the China Forest Resources Report and the China Energy Statistics Yearbook. Some of the missing data are obtained from the China Forestry and Grassland Statistical Yearbook, the China Forestry Statistical Yearbook, and its supplementary literature.

This study selects data from 2003 to the present, segmenting the data into five-year intervals in accordance with the statistical cycle of the China Forest Resources Report for analysis. In total, it can be divided into four stages. Stage 1 is from 2004 to 2008, Stage 2 is from 2009 to 2013, Stage 3 is from 2014 to 2018, and Stage 4 is from 2019 to present.

2.4. Models for Measuring Forest Carbon Sink Efficiency and Optimizing Resource Allocation

In this study, an analysis framework consisting of four sub-models is developed to systematically assess FCSE and optimize the FCS allocation, thereby providing scientific support for the planning and scheduling of FCS resources. The first sub-model measures FCSE from a static perspective, and the second sub-model reflects changes in FCSE from a dynamic perspective and decomposes the efficiency. The third sub-model predicts the target value of the FCS and is used as an input for the next sub-model. The fourth sub-model proposes the best value of future input indicators to improve resource utilization. The primary analysis framework is described in Figure 2.

2.4.1. Forest Carbon Sink Efficiency Static Measurements (The First Sub-Model)

FCSE assesses the efficiency of the relationship between the resources invested and the amount of carbon sink generated in the carbon sink process. DEA is a commonly used efficiency measurement method. Ref. [48] pioneered the development of DEA, as published in the European Journal of Operations Research. The CCR and BCC models employ radial efficiency metrics, permitting simultaneous proportional adjustments to both input and output indicators [49]. Incorporating slack variables into the objective function facilitates the depiction of inefficiencies in the production process. The Super SBM model is adept at assessing FCSE from a static standpoint, offering insights into FCSE across time periods.

Traditional DEA models assign an efficiency score of 1 to multiple DMUs, making it impossible to further distinguish their performance. To address this limitation, the Super SBM model was adopted. As a non-radial and non-oriented approach, Super SBM evaluates efficiency by simultaneously considering input excesses and output shortfalls, thereby providing a more comprehensive and accurate measurement [50].

$$\begin{array}{l}\min \theta \\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}}\le \theta x_{ik}\\ {\displaystyle \sum _{j=1}^n{y}_{rj}{\lambda }_j}\ge y_{rk}\\ \lambda \ge 0\\ i=1,2,3,4\dots ,m;r=1,2,3,4\dots ,q;j=1,2,3,4\dots ,n\end{array}\right.\end{array}$$

(3)

Included among these, the quantity of input variables is represented by $i$. The quantity of output variables is represented by $r$. $x_{ik}$ indicates the $k$th decision-making unit. ${\displaystyle \sum _{j=1}^n{y}_{rj}{\lambda }_j}$ are a virtual effective decision-making unit. The optimal solution of the model is ${\theta }_k^{\ast }$. $y_{rj}{\lambda }_j$ is a frame of reference for the optimal solution. The efficiency value is considered valid if it is greater than or equal to 1 and invalid if it is less than 1.

2.4.2. Forest Carbon Sink Efficiency Dynamic Analysis (The Second Sub-Model)

The above only measured the static FCSE; to further obtain the dynamic changes of the efficiency and analyze the main factors affecting the efficiency, MI is introduced in this study. The MI is a measure of the total factor productivity (TFP) over time [51]. MI can measure dynamically from both time and space perspectives, which has the advantage of compensating for the shortcomings of simple DEA.

This study considers two consecutive periods at time points (t) and (t + 1). Under this model, there is a two-period frontier, and the decision unit is denoted by K. The projection of the evaluated DMUs is denoted by ′ for frontier 1 and ″ for frontier 2.

The MI productivity index for $K$ under reference frontier 1 is as follows:

$$E^1(K^1)={\displaystyle \frac{O{K}^{1^{\prime }}}{O{K}^1}},E^1(K^2)={\displaystyle \frac{O{K}^{2^{\prime }}}{O{K}^2}},{\displaystyle \frac{E^1(K^2)}{E^1(K^1)}}={\displaystyle \frac{O{K}^{2^{\prime }}/O{K}^2}{O{K}^{1^{\prime }}/O{K}^1}}$$

(4)

Concerning frontier 2, the Malmquist productivity index for K is as follows:

$$E^2(K^1)={\displaystyle \frac{O{K}^{1^{″}}}{O{K}^1}},E^2(K^2)={\displaystyle \frac{O{K}^{2^{″}}}{O{K}^2}},{\displaystyle \frac{E^2(K^2)}{E^2(K^1)}}={\displaystyle \frac{O{K}^{2^{″}}/O{K}^2}{O{K}^{1^{″}}/O{K}^1}}$$

(5)

Two Malmquist indices are obtained separately for frontiers 1 and 2. Färe proposed the MI model most widely used. The MI for evaluating DMUs is calculated as the geometric mean of the respective indices involved. The MI consists of two fundamental components: Efficiency Change (EC) and Technological Change (TC).

$$MI(x^{t+1},y^{t+1},x^t,y^t)=EC\times TC$$

(6)

The quantitative relationship between EC and TC is MI = EC × TC, and EC is as follows:

$$EC={\displaystyle \frac{E^{t+1}(x^{t+1},y^{t+1})}{E^t(x^t,y^t)}}$$

(7)

TC is defined as follows:

$$TC=\sqrt{{\displaystyle \frac{E^t(x^t,y^t)}{E^{t+1}(x^t,y^t)}}{\displaystyle \frac{E^t(x^{t+1},y^{t+1})}{E^{t+1}(x^{t+1},y^{t+1})}}}$$

(8)

MI uses historical data to measure the FCSE. An increase in the FCSE relative to the base period is indicated when the MI is greater than 1. Conversely, if the MI is less than 1, it indicates that the FCSE declined compared to the base period. The MI model further disaggregates the efficiency and measures the FCSE from a dynamic perspective.

2.4.3. Projections of Carbon Sink Targets (The Third Sub-Model)

Future carbon emissions and sink in China are full of uncertainties. It is difficult to obtain data on the carbon sink target value of the forest system in the future. ARIMA is a statistical model commonly used for time series forecasting [52,53,54,55]. ARIMA models utilize historical data and random disturbance terms to construct a fitting model. They are known for their straightforward structure, ease of computation, and high degree of accuracy in fitting [53]. The AR part contains these autoregressive terms, where p represents the autoregressive order, which indicates the number of previous observations used by the model. The MA section contains these moving average terms, where $q$ denotes the moving average order, which indicates the number of past errors used by the model. An ARIMA model is developed with the following equation:

$$X_t={\alpha }_1{X}_{t-1}+{\alpha }_2{X}_{t-2}+\dots +{\alpha }_p{X}_{t-p}+e_t-{\beta }_1{e}_{t-1}-{\beta }_2{e}_{t-2}-\dots -{\beta }_q{e}_{t-q}$$

(9)

where is the time series data, $q$ is the number of sliding mean terms, $p$ is the number of autoregressive terms, $e$ is the natural constant, $\alpha$ is the autoregressive coefficient, and is the sliding mean coefficient.

This study determines future forestry carbon sink targets based on projected carbon emission data. Since CCUS is currently undeveloped and the uptake of greenhouse gases such as carbon dioxide is largely dependent on forests, this study equates carbon neutrality with making carbon sink equal to carbon emissions. Forest absorption, as a vital mechanism of carbon sink, holds substantial importance in mitigating carbon emission in Anhui Province, contributing to 23.7% of the overall carbon sink capacity. This means that historical data on carbon emissions from 1990 to the present can be used to make projections and obtain target values for FCS.

There are many different prediction models for forest carbon sequestration targets in existing studies, and the current mainstream models are selected for the advantages and disadvantages analysis, and the results are shown in

Table 4. Analysis of advantages and disadvantages of forest carbon sink target prediction models.

Predictive Models	Advantages and Disadvantages Analysis
	Advantages	Disadvantages
ARIMA	Simple structure, only time series data needed.	Requires stationary data, limited in capturing nonlinear relationships, sensitive to parameter selection.
Long Short-Term Memory	Handles long sequences, captures long-term dependencies, suitable for complex nonlinear problems.	Complex structure, high training costs, prone to overfitting, especially with small samples.
Stochastic Impacts by Regression on Population, Affluence, and Technology	Integrates various socio-economic factors, offers flexible and comprehensive carbon emission forecasts.	Requires substantial data input, sensitive to parameter selection and model configuration.
Grey Prediction Model	Provides reasonable forecasts when data is scarce, straightforward model construction.	Predictive accuracy is constrained, may fail to provide precise forecasts for complex systems.
Bayesian Optimization Support Vector Machine	Efficiently discovers optimal solutions in high-dimensional parameter spaces, enhancing prediction accuracy.	Entails potentially onerous computational demands, necessitates robust prior knowledge.

. Combining the analysis of the above prediction models and considering the limitations of the available data, this study finally chooses the ARIMA model for prediction.

2.4.4. Measurement of Future Factor Values of Forest Inputs (The Fourth Sub-Model)

Inverse DEA is an innovative extension of the traditional DEA model [56]. It is primarily utilized for addressing resource allocation and forecasting problems with the constraint of maintaining current efficiency levels. This model is the opposite of DEA, hence it is called Inverse DEA [57]. Inverse DEA facilitates the analysis of the necessary adjustments in DMUs’ inputs and outputs to achieve the predetermined FCS targets.

Assuming that given a set of orthogonal DEA models with an efficiency value of ${\theta }_i^{\ast }(i=1,2,\dots ,n)$, for a DMU, there are two common ideas of Inverse DEA models: (1) In the case of controlling the efficiency of the unchanged, change the output or set of the target output value, solving to obtain the corresponding new inputs $x$. (2) In the case of controlling the efficiency of the ${\theta }^{\ast }$ unchanged, change the inputs or set the value of the future inputs, solving to obtain the corresponding new outputs $y$.

Due to the lag in the impact of changes in external conditions on the forest system, this study chooses to calculate the corresponding input indicator values by changing the control efficiency. Therefore, this study chooses to calculate the corresponding input indicator values by changing the FCS under the condition of constant control efficiency. This is consistent with the nature of Inverse DEA. As the output count is increased, the corresponding input count, denoted as $\Delta x$, is solved. The model of the Inverse DEA can be formulated in the following way:

$$\begin{array}{l}\min \Delta x\\ s.t.\left\{\begin{array}{l}({\theta }_k^{\ast }(x_k+\Delta x),y_k+\Delta y_k)\in T\\ \Delta x\ge 0\end{array}\right.\end{array}$$

(10)

In this study, the FCSE is held constant, and the FCS target is employed as the input. Inverse DEA is innovatively utilized to assess the prospective FCS input value, thereby making efficient use of FCS resources.

3. Results and Discussion

3.1. Static Changes in Forest Carbon Sink Efficiency (The First Sub-Model)

Considering the characteristic that the efficiency value of the Super SBM model exceeds the value of 1, FCSE values are categorized into five levels, with the following ranges: level 1 for 0 to 0.5, level 2 for 0.5 to 0.8, level 3 for 0.8 to 1, level 4 for 1 to 2, and level 5 for values from 2 to the highest observed. This study assesses FCSE in 30 Chinese provinces over the period from 2004 to the present day using the Super SBM model. According to the set levels, the overall efficiency of the FCS is shown in Figure 3, and there is an improvement in the overall FCSE.

However, some provinces are less efficient. For example, the effective FCSE of Beijing at Stage 3 changes to ineffective FCSE at Stage 4. It shows a shift from high efficiency to low efficiency. In contrast, Heilongjiang changed from an ineffective FCS (0.5439) in Stage 3 to an effective FCSE (1.0649) in Stage 4. The reasons for the decline in the FCSE may be as follows. (1) The region is undergoing a gradual process of development, with the central city’s infrastructure undergoing improvements. This leads to an influx of people into the central city, which is subsequently expanding and encroaching upon the original green space. (2) The region is in a mid to late stage of development, and satellite urban areas are being developed to relieve the pressure on the central city. Most of the newly developed satellite urban areas are suburban and occupy forested areas. The reasons for the increase in the FCSE may be the following: (1) After the rapid development of cities, people began to pay attention to the living environment, urban greening was emphasized, and there was even a return of population from the city to the countryside. (2) With the continuous development of the management methods of the forestry system and technological upgrading, the FCSE has been improved.

There are significant differences in FCSE between provinces. The majority of provinces maintain their status as either effective or ineffective over extended periods, while a few provinces experience transitions between these statuses at various stages. Fujian, Guangdong, Tianjin, Shanghai, Jilin, Zhejiang, Shandong, Jiangsu, Sichuan, Yunnan, and Hainan always remain above 1. On the contrary, the efficiency values of Liaoning, Guangxi, Gansu, Hebei, Hunan, Henan, Hubei, Xinjiang, Shaanxi, Anhui, Shanxi, and Guizhou always remain below 1.

Figure 4 visualizes regional differences in FCSE. Comprehensively analyzing the distribution areas of high FCSE has the following findings. (1) The efficient regions have been expanding, mainly in the three northeastern provinces, the Hengduan Mountain Region, the Inner Mongolia Plain, the North China Plain, the southeastern coast, and some of its neighboring provinces. (2) The FCSE remains relatively consistent throughout the four stages of change, with little fluctuation in efficiency values. This is consistent with the slow growth of forests. However, there is still a small increase in FCSE in all stages, which can be generally explained by the number of nationally and regionally relevant policies that are implemented to contribute to the increase in FCS. For example, in 2009, China promulgated the National Forest Resource Management Provisions, which provided more specific regulations on forest protection and management, and FCSE in all regions showed a clear upward trajectory. It shows that the implementation of relevant policies has a guiding role in the future impact of FCS.

It is worth noting that the efficiency of FCSE in the central region is lower than in the east and west. The eastern region has earlier economic development and a strong financial base, which is conducive to the high-quality development of the forestry system. Although the western region has a weaker economic base, it is rich in forest resources and has relatively higher efficiency than the central region.

It is found that at the same stage, the FCSE varies considerably in different provinces. For example, in Stage 2, Tianjin has an efficiency of 3.4508, while Shanxi has an efficiency of 0.2718. The reason for the large differences in forest cover in different provinces at the same stage may be due to differences in the economic development of different regions (

Table 5. Forest carbon sink efficiency at different stages in provinces.

Region	Stage 1	Stage 2	Stage 3	Stage 4	Average
Anhui	0.5333	0.5855	0.6578	0.6449	0.6054
Beijing	1.0377	1.1079	1.1746	0.2658	0.8965
Fujian	1.1283	1.2166	1.5137	1.5231	1.3454
Gansu	0.4131	0.3619	0.3704	0.3683	0.3784
Guangdong	1.3169	2.5852	2.3071	1.0177	1.8067
Guangxi	0.4598	0.4371	0.4252	0.3021	0.4061
Guizhou	0.3407	0.4072	0.6226	0.3298	0.4251
Hainan	0.8816	1.1175	2.0371	1.8042	1.4601
Hebei	0.4020	0.3439	0.5364	0.3059	0.3970
Henan	0.4954	0.4801	0.7446	0.4346	0.5387
Heilongjiang	0.6999	0.5667	0.5439	1.0649	0.7188
Hubei	0.4432	0.4710	0.5122	0.4672	0.4734
Hunan	0.4549	0.4451	0.4339	0.4215	0.4388
Jilin	1.0652	1.0753	1.1029	1.3516	1.1487
Jiangsu	1.6421	1.1754	1.1964	1.6308	1.4112
Jiangxi	0.5980	0.6143	0.5537	1.1013	0.7168
Liaoning	0.4711	0.4542	0.4402	0.4664	0.4580
Inner Mongolia	0.5586	1.0800	1.0905	1.1279	0.9643
Ningxia	0.2972	0.3371	1.4076	1.4060	0.8620
Qinghai	1.5197	0.4189	0.3690	0.3336	0.6603
Shandong	1.0283	1.4132	1.8714	1.3499	1.4157
Shanxi	0.2724	0.2718	0.3806	0.3325	0.3143
Shaanxi	0.5382	0.5509	0.4837	0.4054	0.4946
Shanghai	2.6294	2.0805	1.6029	1.8648	2.0444
Sichuan	2.1772	2.4009	1.0743	1.0764	1.6822
Tianjin	2.3033	3.4508	2.9372	1.4411	2.5331
Xinjiang	0.5530	0.4856	0.4066	0.5709	0.5040
Yunnan	1.6523	1.4940	1.2520	1.1521	1.3876
Zhejiang	1.9336	1.3565	1.4735	3.0542	1.9545
Chongqing	0.5718	0.5409	0.8246	1.0048	0.7355
Average	0.9473	0.9775	1.0116	0.9540	0.9726

).

3.2. Dynamic Changes in Forest Carbon Sink Efficiency (The Second Sub-Model)

This study applies the MI to measure the dynamics of FCS at different stages in 30 provinces in China. The TFP of Chinese FCSE fluctuates at around 1. Across all three stages, MI exceeds 1, with most provinces showing MI fluctuations around 1, except for Qinghai in Stage 2, registering at 0.22. This suggests an overall upward trend in the FCSE of China. In terms of the decomposition index, it is found that the TC mean value is greater than the EC mean value in three stages. This indicates that the improvement of FCSE within the three stages is mainly due to the improvement of the technology level. The progress of FCSE in China depends on TC (Figure 5).

The impact of TC on the FCSE can be reflected in the following areas: (1) regarding capital inputs: technological advancements typically enhance the productivity of forestry through more efficient planting, harvesting, and processing methods; (2) concerning labor inputs: the introduction of new technologies that incorporate automated equipment can diminish the demand for labor; (3) pertaining to energy consumption: technological advancements frequently offer more accurate, real-time data and enhanced monitoring capabilities, enabling more precise input planning and the elimination of unnecessary waste.

This approach enables dynamic and static measurements of FCSE across various regions, with the study methodology being universally applicable. In the latter part of the study, the research then concentrates on a specific case study in Anhui Province to scrutinize FCSE.

3.3. Forest Carbon Sink Autoregressive Integrated Moving Average Forecasts (The Third Sub-Model)

Forecasted carbon emission backward to 2060 is based on data from Anhui Province from 1990 to 2021. A projection of carbon emissions up to 2060 is produced by fitting the historical data into a suitable model.

The unit root test with intercept term and trend term is performed on the time series of the carbon emission unit root test. It is a smooth time series. According to the criterion of minimization information and the significance of t-statistics, it is considered that Autoregressive Moving Average (ARMA) (1, 3) is the best-fitting model.

Table 6. The result of Autoregressive Moving Average (1, 3).

Variable	Coefficient	Std.Error	t-Statistic	Prob.
AR (1)	1.034845	0.006899	150.0030	0.0000
MA (1)	−0.847588	0.127717	−6.636429	0.0000
MA (2)	−0.354779	0.196677	−1.803868	0.0824
MA (3)	0.743020	0.122086	6.086012	0.0000
R-squared	0.964478	Akaike info criterion		17.42131
Adjusted R-squared	0.960531	Schwarz criterion		17.60634
		Hannan–Quinn criterion		17.48162

shows the result of ARMA (1, 3).

The white noise test for the residuals of the ARMA (1, 3) model is shown in Figure 6. The Q-test p-values for all lags of the residual sequence of the ARMA (1, 3) model do not reject the null hypothesis that the sequence is a white noise series at the 1% significance level. This means that the residual sequence of ARMA (1, 3) is considered a white noise series, with no autocorrelation and partial autocorrelation, indicating that ARMA (1, 3) has adequately captured the information in the time series. Therefore, ARMA (1, 3) is an effective fitting model.

Carbon emissions and forecasts are shown in Figure 7. The actual values of carbon emission and the fitted values have a small difference, indicating a good fit, and the overall effectiveness is satisfactory.

3.4. Resource Allocation Optimization of Forest Carbon Sink (The Fourth Sub-Model)

The long investment cycle and gradual changes in FCSE suggest relatively stable efficiency values over time. This characteristic aligns well with the fundamental assumption of Inverse DEA methodology, which enables output projection while maintaining constant efficiency levels. The target of FCS predicted by ARIMA is an input to Inverse DEA. This study chooses the Inverse DEA results of FCSE in Anhui Province in 2030, 2045, and 2060, as shown in

Table 7. Target values for future input factors in Anhui.

Year	Forest Area	The Amount of Fixed Investment in Forestry	The Year-End Forestry System Practitioners	Energy Consumption
2030	613.850868	306,959.5456	21,923.83597	7,127,768.39
2045	952.981633	476,543.7735	34,035.97534	11,065,606.83
2060	1592.99633	796,586.6872	56,894.25893	18,497,178.17

.

Analyzing the forest area data in Figure 8, It can be seen that, based on Stage 4, by 2030 the inputs will have to be increased by 55%. By 2060, it will have increased to four times its original level. Forest areas directly influence carbon sink capabilities, so expanding forest areas is of heightened significance in the strategy to augment FCS. It is possible to rely on measures such as returning farmland to forests to increase forest areas, but geographical constraints limit the significant expansion of forest area. In addition to forest area, the composition of labor, capital inputs, and energy consumption needs to be improved simultaneously based on the results.

4. Conclusions

This study proposes a novel analysis framework consisting of the Super SBM-MI- Inverse DEA for a systematic study of both dynamic and static aspects of FCSE, as well as optimization of FCS resources. By employing the Super SBM model, this study analyzes the static FCSE of 30 provinces in China and employs the Malmquist model to assess dynamic FCSE at various stages, thereby identifying key factors influencing FCS enhancement through efficiency decomposition. In studying the sustainable development of FCS, this study chooses Anhui Province with relatively low efficiency as an example. This study innovatively constructs an Inverse DEA model, which provides an in-depth study of resource allocation optimization and FCS sustainable development, and also provides a scientific basis for policymakers.

The results show that the FCSE of different provinces is significantly different, and the efficiency value of some provinces is still more than 50% different from the frontier level. China’s FCSE still has much room for improvement. The high FCSE regions in China are mainly located in the southwest region, the southeast coast, the eastern part of the North China Plain, and the three eastern provinces. Technological progress is the main influencing factor in changing FCSE. Innovation plays a pivotal role in improving efficiency. In addition, it is noted in the study that different stages of urbanization, with different development priorities of the region, have different impacts on the FCSE of the region. Taking Anhui Province as an example, to reach the 2030 carbon sink target, the increased input factor for FCS is about 55%, and by 2060 it will need to triple the factor inputs.

Based on the results of static measurement and dynamic change of FCS, formulate and refine targeted policies and strategic measures for different regions and stages, with ongoing adjustments to FCS policies. Innovation plays a pivotal role in improving FCSE. Technological innovation and technological upgrading of the forestry system should be actively promoted. Regions with higher FCSE should strategically redirect a portion of their financial resources towards other carbon sink pathways to complement the stability of their original efficiency. Leveraging geographical advantages as well as economic strengths, regions should actively undertake afforestation in a rational manner to protect the ecological environment. An effective carbon sink monitoring and assessment system is essential for the systematic monitoring of carbon stocks in forests, land, and other ecosystems, facilitating prompt regulatory adjustments. Concurrently, forest restoration projects aimed at augmenting forest coverage through reforestation and natural regeneration have been initiated.

The research framework proposed in this paper is designed to be universally applicable, transcending the geographical boundaries of China. It is intended to be adaptable for use in various nations, demonstrating its flexibility and broad applicability. Furthermore, this framework can be effectively applied to assess FCSE on a smaller scale, such as within a single province or region, which could provide a more detailed and high-resolution analysis. This versatility indicates that the framework has a wide range of potential applications.

Admittedly, this study still has its shortcomings. The selection of indicators for measuring FCSE is not yet comprehensive, and the predictive models for FCS also need to be refined. In subsequent research, it is possible to further optimize and improve the indicators and incorporate more effective predictive models, updating and iterating the models.