Staged Resource Allocation Optimization under Heterogeneous Grouping Based on Interval Data: The Case of China’s Forest Carbon Sequestration

Abstract

In interval data envelopment analysis (DEA), the production possibility set is variable, which causes traditional resource allocation optimization methods to yield results with limited reachability. This study aims to improve existing resource allocation optimization models so that they can produce meaningful results when handling interval data. Addressing this topic can enhance the applicability of existing models and improve decision-making accuracy. We grouped decision-making units (DMUs) based on heterogeneity to form production possibility sets. We then considered the characteristics of the worst and best production possibility sets in the interval DEA to establish multiple benchmark fronts. A staged optimization procedure is proposed; the procedure provides a continuous optimization solution, offering a basis for decision-makers to formulate strategies. To illustrate this, we provide a numerical example analysis and a case study on forest carbon sequestration. Finally, by applying our method to China’s forest carbon sink data, we show that it better meets the practical needs in China. The practical implication of this procedure is that it provides a basis for decision makers to formulate strategies based on interval data. The theoretical implication is that it extends the application of DEA models to interval data.

1. Introduction

Data envelopment analysis (DEA) is a non-parametric method used for performance evaluation, first introduced by Charnes et al. (1978) [1]. In DEA, the objects being evaluated for efficiency are considered DMUs, such as companies, projects, or provinces. Resources used in the production process are regarded as inputs, such as funds, labor, and materials. The results produced by the process are considered outputs, such as products and services. Subsequently, based on the concept of efficiency in DEA, Golany et al. (1993) [2] applied the DEA model to resource allocation optimization. As of now, numerous scholars have conducted research on resource allocation optimization based on DEA [3,4,5]. Resource allocation optimization is divided into benchmark setting and resource allocation. Therefore, research mainly focuses on two dimensions: (1) the optimality of benchmark setting; and (2) the reachability of resource allocation optimization.

For the optimality of benchmark setting in DEA, efficient DMUs can form a Pareto-efficient frontier. Therefore, the initial idea among scholars for benchmarking was to use the efficient frontier to determine benchmarks [2]. Subsequent scholars have introduced expert preferences and management goals to ensure the optimality of benchmarks during benchmarking processes [3,6]. Some scholars have relaxed the convexity assumption of DEA to make the benchmarks more aligned with practical needs, ensuring the optimality of the benchmarks [7,8].

Regarding the reachability of resource allocation optimization, despite the usefulness of DEA in benchmarking, it often sets unrealistic targets. These targets can differ significantly from the current conditions of DMUs, making resource allocation optimization challenging. Many scholars have proposed suitable methods to help DMUs achieve their targets with less effort, thereby improving the reachability of resource allocation optimization [9,10,11,12,13,14]. Some scholars have also divided the process of resource allocation optimization into multiple stages. They set more reachable short-term goals, allowing DMUs to progressively achieve long-term objectives through these stages [4,15,16,17]. In recent years, scholars have expanded the range of alternatives by relaxing the convexity assumption in DEA, making the benchmarks more aligned with the evaluated DMUs [18,19].

Meta-frontier is a method used to analyze heterogeneous groups. Scholars often use DEA to calculate the efficiency of various groups and conduct economic analyses based on these calculations [20,21,22,23]. In addition, some scholars have also introduced the concept of meta-frontier into research on resource allocation optimization [24,25].

When input–output data are in the form of interval data, traditional DEA methods cannot address the uncertainty in efficiency evaluation. Cooper et al. (1999) [26] first proposed the interval DEA model. Subsequently, scholars initiated a series of theoretical studies on interval DEA. In interval DEA, the optimal production point for DMUs is the combination of minimum inputs and maximum outputs, forming the best production possibility set for all DMUs. Conversely, the worst production point for DMUs is the combination of maximum inputs and minimum outputs, forming the worst production possibility set for all DMUs. Scholars have proposed efficiency calculation models for the best and worst production possibility sets, and the efficient production points derived from each production possibility set constitute the corresponding efficient frontier [27,28,29]. Currently, there are two main viewpoints in research: first, interval efficiency measurement based on bilateral production possibility sets; scholars have proposed interval efficiency evaluation models to obtain the maximum interval efficiency evaluation value, using both the best and worst production possibility sets as evaluation reference sets, and discussed the interval sensitivity of various models [30,31]. The second approach involves interval efficiency measurement based on unified production possibility sets; adjusting the width of interval efficiency through optimizing overall efficiency and considering the likelihood ranking method of decision-maker preference information can solve the full-ranking problem of interval efficiency [28,29,32]. Interval DEA has also been widely applied in the field of forest carbon sequestration. Huang et al. (2022) [33] used an interval data-based cross-efficiency evaluation model to assess the development of forest carbon sequestration areas in China. The following year, Huang et al. (2023) [34] constructed a benchmark based on interval data to evaluate the efficiency of forest carbon sequestration in China, providing guidance for the development path of forest performance management in China. Chen et al. (2024) [35] improved the aggregation method of the interval cross-efficiency model and evaluated the cross efficiency of forest carbon sequestration across various provinces in China.

The Chinese government is working to balance economic development with ecological protection in order to achieve the dual carbon goals. The industrial sector needs to use carbon-offset mechanisms to balance the carbon dioxide emissions they produce. Carbon-offset measures primarily involve compensating for emissions through forest carbon sinks. Currently, the evaluation of forest carbon sink performance needs to consider two factors. First, forest carbon sinks are affected by climate, which is an uncertain factor. This requires the use of interval data in our analysis. Second, there is uneven development across different regions in China, which means that heterogeneity issues need to be considered during the analysis. Most existing resource optimization models use the production possibility frontier as the benchmark to set standards [8,18,19]. In real numbers, only one production possibility set needs to be considered. However, with interval data, considering just the best and worst cases results in two production possibility sets, leading to four types of efficiency evaluation models [34]. Considering heterogeneity while using interval data further increases the number of production possibility sets that need to be considered. Thus, resource allocation optimization with interval numbers involves numerous production possibility sets, complicating the development of optimization plans. In summary, optimizing forest carbon sequestration resources requires facing two main issues: (1) how to select the benchmark, and (2) how to reduce the complexity of resource optimization and improve the feasibility of solutions.

In the present paper, we propose a staged resource allocation optimization procedure under heterogeneous grouping in interval DEA. This paper categorizes DMUs into “global improvement type” and “intra-group improvement type” under heterogeneous grouping. Achieving global optimality is pursued by aiming for the global best frontier. A staged benchmarking approach is adopted to enhance the procedure’s reachability. Additionally, the technological gap proposed by Walheer (2023) [36] is introduced to mitigate the difficulty of achieving the final goal for intra-group improvement-type DMUs. We then improve the benchmark point calculation model proposed by Ramón et al. (2018) [4] based on the needs of forestry carbon sink project decision-makers. Finally, we contrast our approach with traditional methods using a numerical example and apply our method to carbon sequestration in China’s forest carbon sink dataset.

Our method constructs optimization solutions for each DMU using real-number benchmark points, whereas Huang et al. (2023) [34] focused more on the construction of benchmark surfaces. Additionally, our method considers heterogeneity issues, making the final optimization solutions more reasonable. The advantage of the method applied in this paper is that the staged solution is highly reachable, and the real-numbered benchmark points help decision makers understand and apply the results more clearly. A limitation of our method is that it does not provide a dynamic adjustment plan. This paper’s main contributions to the field include: (1) proposing a staged resource allocation optimization procedure that can handle interval data and produces solutions composed of real-number benchmark points; (2) proposal of a DMU classification method, enabling the determination of the number of stages required for optimization based on the production status of DMUs; (3) using technical gap adjustments to benchmark surfaces to enhance the reachability of the optimization solutions.

The rest of the paper is structured as follows. In Section 2, we define some necessary concepts. In Section 3, we present some DEA models that will be utilized in this paper. In Section 4, we elaborate on our staged resource allocation optimization procedure under heterogeneous grouping in interval DEA. Next, we illustrate the difference between our method and traditional approaches through a simple example in Section 5. In Section 6, we apply our method to carbon sequestration in China’s forest carbon sink dataset. Section 7 concludes.

2. Problem Description and Formulation

The main issues that our method needs to address include: (1) selecting appropriate benchmark surfaces for the resource allocation optimization procedure; and (2) enhancing the accessibility of the solutions. To address these issues, the following definitions and assumptions are introduced in this paper.

Throughout the paper we suppose that we have $n$ DMUs which use $m$ inputs to produce s outputs. These are denoted by $(X_j,Y_j),j=1,2,\dots ,n.$ It is assumed that $X_j=(x_{1j},\dots ,x_{mj})\ge 0$, $X_j\ne 0$, $j=1,2,\dots ,n$, and $Y_j=(y_{1j},\dots ,y_{sj})\ge 0$, $Y_j\ne 0$, $j=1,2,\dots ,n$. It is also assumed that $T=\left\{(x,y)|y \mathrm{can} \mathrm{be} \mathrm{produced} \mathrm{by} x\right\}$; the corresponding production possibility set is:

$$PPS=\left\{(x,y)|{\displaystyle \sum _{j=1}^n{\lambda }_j{X}_j\le x,}{\lambda }_j{Y}_j\ge y,{\lambda }_j\ge 0\right\}$$

(1)

Definition 1.

Let $\overline{a}=\left[a^L,a^U\right]$ be the number of intervals, $a^L,a^U\in R$. If $a^L=a^U$, then $\overline{a}$ will degenerate into real numbers. Inputs can be represented as $\overline{x_{ij}}=\left[x_{ij}^L,x_{ij}^U\right],(i=1,2,\dots ,m)$, outputs can be represented as $\overline{y_{rj}}=\left[y_{rj}^L,y_{rj}^U\right],(r=1,2,\dots ,s)$, $x_{ij}^L,y_{rj}^L>0$. The combination of minimum inputs and maximum outputs for DMUs forms the best production point, and the best production points of all DMUs constitute the best production possibility set $PP{S}^U$. Conversely, the combination of maximum inputs and minimum outputs for DMUs forms the worst production point, and the worst production points of all DMUs constitute the worst production possibility set $PP{S}^L$. As follows:

$$PP{S}^U=\left\{(x^L,y^U)|{\displaystyle \sum _{j=1}^n{\lambda }_j{X}_j^L\le x^L,}{\lambda }_j{Y}_j^U\ge y^U,{\lambda }_j\ge 0\right\}$$

(2)

$$PP{S}^L=\left\{(x^U,y^L)|{\displaystyle \sum _{j=1}^n{\lambda }_j{X}_j^U\le x^U,}{\lambda }_j{Y}_j^L\ge y^L,{\lambda }_j\ge 0\right\}$$

(3)

3. Solution Approach

Wang et al. (2005) [28] proposed to use the best production possibility set $PP{S}^U$ as a unified evaluation reference set to measure interval efficiency. The model is as follows:

$$\begin{array}{l}Min {\theta }_d^{UU},\\ {\displaystyle \sum _{j=1}^n{x}_j^L{\lambda }_j}\le {\theta }_d^{UU}{x}_d^L\\ {\displaystyle \sum _{j=1}^n{y}_j^U}{\lambda }_j\ge y_d^U\\ {\lambda }_j\ge 0,j=1,\dots ,n.\end{array}$$

(4)

$$\begin{array}{l}Min {\theta }_d^{UL},\\ {\displaystyle \sum _{j=1}^n{x}_j^L{\lambda }_j}\le {\theta }_d^{UL}{x}_d^U\\ {\displaystyle \sum _{j=1}^n{y}_j^U}{\lambda }_j\ge y_d^L\\ {\lambda }_j\ge 0,j=1,\dots ,n.\end{array}$$

(5)

In models (4) and (5), variable $(x_j^L, y_j^U), j=1,\dots ,n$ represents a point within the $PP{S}^U$, indicating that both Model (4) and Model (5) are based on the $PP{S}^U$ for efficiency calculation. In Model (4), variable $(x_d^L, y_d^U)$ represents the input–output values of the best production point for DMU_d. In Model (5), variable $(x_d^U, y_d^L)$ represents the input–output values of the worst production point for DMU_d. ${\theta }_d^{UU*}$ and ${\theta }_d^{UL*}$ are the optimal solutions of Models (4) and (5), respectively. The first superscript denotes the production possibility set serving as the reference for evaluation, while the second superscript represents the production point being evaluated. The * indicates the optimal solution, meaning the efficiency value of the DMU being evaluated. For example, ${\theta }_d^{UU*}$ represents the efficiency value for the best production point of DMU_d evaluated under the reference set $PP{S}^U$. Similarly, ${\theta }_d^{UL*}$ represents the efficiency value of the worst production point of DMU_d. Subsequent content uses similar expressions.

As a comparison, Huang and Wang (2017) [29] proposed an interval efficiency evaluation model based on the worst production possibility set as a unified reference set $PP{S}^L$. That model is as follows:

$$\begin{array}{l}Min {\theta }_d^{LU},\\ {\displaystyle \sum _{j=1}^n{x}_j^U{\lambda }_j}\le {\theta }_d^{LU}{x}_d^L\\ {\displaystyle \sum _{j=1}^n{y}_j^L}{\lambda }_j\ge y_d^U\\ {\lambda }_j\ge 0,j=1,\dots ,n.\end{array}$$

(6)

$$\begin{array}{l}Min {\theta }_d^{LL},\\ {\displaystyle \sum _{j=1}^n{x}_j^U{\lambda }_j}\le {\theta }_d^{LL}{x}_d^U\\ {\displaystyle \sum _{j=1}^n{y}_j^L}{\lambda }_j\ge y_d^L\\ {\lambda }_j\ge 0,j=1,\dots ,n.\end{array}$$

(7)

In Models (6) and (7), variable $(x_j^U,y_j^L), j=1,\dots ,n$ represents a point within the $PP{S}^L$, indicating that both Model (4) and Model (5) are based on the $PP{S}^L$ for efficiency calculation. The variables meanings related to DMU_d are the same as in both Model (4) and Model (5). ${\theta }_d^{LU*}$ and ${\theta }_d^{LL*}$, respectively, represent the efficiency of the DMU_d best and worst production points under $PP{S}^L$ as the evaluation reference set.

4. Staged Resource Allocation Optimization Procedure under Heterogeneous Grouping in Interval DEA

We propose a resource allocation optimization method for handling interval data. This method is used to design a staged improvement plan for DMUs, optimizing from the worst production point to the frontier of the best possible production set. Our methodology first groups DMUs based on their heterogeneity and classifies them according to their efficiency. Then, we construct multiple benchmark surfaces by considering the worst and best production status of the DMUs. Finally, we select the appropriate benchmark surfaces based on the group and type of each DMU and use the model to calculate the benchmark points for each stage.

4.1. Definition of Efficiency and Frontier Construction under Heterogeneous Grouping

This paper divides the $n$ DMUs into $k$ groups, where $PP{S}_c^L$ and $PP{S}_c^U$ (c = 1, 2, …, k) respectively denote the worst production possibility set and the best production possibility set of the c-th group of DMUs. Similarly, ${PPS}^L$ and ${PPS}^U$, respectively, represent the worst production possibility set and the best production possibility set of all DMUs. We define the efficiency of the worst and best production points of DMUs in ${PPS}_c^L$, ${PPS}_c^U$, ${PPS}^L$, and ${PPS}^U$ as follows.

Definition 2.

Definition of efficiency types of DMUs:

We define ${\theta }_{cd}^{LL}$ as the efficiency value of DMU_d worst production point, calculated using Model (7) on $PP{S}_c^L$. The first superscript $L$ denotes the worst production possibility set. When this subscript is $U$, it signifies the best production possibility set. The second superscript $L$ represents the worst production point. When this subscript is $U$, it signifies the best production point. The first subscript c indicates that the evaluated DMU belongs to group c. When this subscript is g, it signifies reference to the global production possibility set. The superscripts and subscripts of the following letters are named in this way. Using a similar expression, we use Model (7) to define ${\theta }_{gd}^{LL}$, Model (5) to define ${\theta }_{cd}^{UL}$ and ${\theta }_{gd}^{UL}$, Model (6) to define ${\theta }_{cd}^{LU}$ and ${\theta }_{gd}^{LU}$, and Model (4) to define ${\theta }_{cd}^{UU}$ and ${\theta }_{gd}^{UU}$.

(1) 

L-cL efficient. When ${\theta }_{cd}^{LL}=1$, it means that DMU_d’s worst production point is efficient at $PP{S}_c^L$, which is L-cL efficient.

(2) 

U-cL efficient. When ${\theta }_{cd}^{UL}=1$, it means that DMU_d’s worst production point is efficient at $PP{S}_c^U$, which is U-cL efficient.

(3) 

L-cU efficient. When ${\theta }_{cd}^{LU}=1$, it means that DMU_d’s best production point is efficient at $PP{S}_c^L$, which is L-cU efficient.

(4) 

U-cU efficient. When ${\theta }_{cd}^{UU}=1$, it means that DMU_d’s best production point is efficient at $PP{S}_c^U$, which is U-cU efficient.

Similarly, under the global production possibility set, we can use ${\theta }_{gd}^{LL}$, ${\theta }_{gd}^{UL}$, ${\theta }_{gd}^{LU}$, and ${\theta }_{gd}^{UU}$ to define L-gL, U-gL, L-gU, and U-gU efficiency.

Remark 1.

In interval DEA, the worst production point of any DMU is likely to be efficient only within $PP{S}_c^L$ and ${PPS}^L$ [29].

According to the definition above, we proceed to provide definitions for each frontier.

Definition 3.

Definition of efficient frontier:

(1) 

Lower group efficient frontier. The worst production point of the L-cL efficient DMU forms the lower group efficient frontier $\partial (T_{\mathrm{c}}^L)$.

(2) 

Upper group efficient frontier. The best production point of the U-cU efficient DMU forms the upper group efficient frontier $\partial (T_{\mathrm{c}}^U)$.

(3) 

Lower global efficient frontier. The worst production point of the L-gL efficient DMU forms the lower group efficient frontier $\partial (T_g^L)$.

(4) 

Upper global efficient frontier. The best production point of the U-gU efficient DMU forms the upper group efficient frontier $\partial (T_g^U)$.

4.2. DMU Improvement Types

In traditional resource allocation optimization procedures, benchmarks with limited reachability are often set for some DMUs [4]. This paper identifies these DMUs based on the following two conditions and defines them as “intra-group improvement type”. The remaining DMUs are defined as “global improvement type”. Global improvement types aim for the global frontier as their improvement target, while the intra-group improvement type initially targets the group frontier for improvement before aiming for the global frontier. The differentiation method is as follows.

Condition 1.

The DMU is L-cL efficient.

In traditional resource allocation optimization procedures, DMUs with higher efficiency are more likely to achieve objectives. We define DMUs that satisfy Condition 1 as “global improvement type”, while those that do not satisfy Condition 1 are assessed in the next condition to determine whether they are defined as “intra-group improvement type”.

Condition 2.

Only outputs are considered; the weights of the composed points on the frontier $\partial (T_{\mathrm{c}}^L)$, summing to 1, cannot be linearly combined to form the worst production point of the evaluated DMU.

Remark 2.

For the assessment of Condition 2, we propose the following method.

Let DMU_j belong to group $c$; $Y_{rj}^L(r=1,2,\dots ,s)$ represents the r-th output value of the worst production point of DMU_j. Let $y_r^{ML}=\underset{i\in \partial (T_{\mathrm{c}}^L)}{max}{y}_{ri}^L(r=1,2,\dots ,s)$ represent the maximum value of the r-th output among all production points on the lower bound of the lower of efficient frontier $\partial (T_{\mathrm{c}}^L)$. When any output of DMU_j satisfies $Y_{rj}^L>y_r^{ML}(r=1,2,\dots ,s)$, it indicates that the worst production point of DMU_j has an output exceeding the lower group efficient frontier. That means DMU_j satisfies Condition 2.

Under the assumption of convexity of the frontier, when the output of the DMU to be improved exceeds the output values of all points on $\partial (T_{\mathrm{c}}^L)$, it needs to reduce output to improve to the efficient frontier. However, in certain specific applications, increasing output indicators might be required. Therefore, it is not appropriate to target the group efficient frontier for improvement. DMUs that do not meet Condition 1 and also fail to meet Condition 2 are defined as intra-group improvement type. DMUs that meet Condition 2 are defined as global improvement type.

Improvement type determination: A DMU is classified as intra-group improvement type when it fails to satisfy both conditions mentioned above, and as global improvement type when it satisfies either one of them.

4.3. Staged Resource Procedures Optimization

In Interval DEA, the upper global efficient frontier represents the best production state that DMUs can achieve. Therefore, we take the upper global efficient frontier as the final goal for resource allocation optimization, allowing the production potential of DMUs to be fully realized. To ensure the reachability of the resource allocation optimization procedure, multiple intermediate goals can be established [4,15,37]. In this paper, different staged resource allocation optimization procedures are proposed for intra-group improvement type and global improvement type, respectively, while ensuring the improvement space and reachability at each stage.

4.3.1. Global Improvement Type DMUs Resource Allocation Optimization Procedure

For DMUs of global improvement type, $\partial (T_g^L)$ serves as their first-stage improvement target, while $\partial (T_g^U)$ serves as their second-stage improvement target. Below, we discuss each stage in detail.

Global improvement type DMUs are divided into two scenarios in the first stage. We discuss these two scenarios using DMU_d as an example:

(1)

When ${\theta }_{gd}^{LL}<1$, targeting $\partial (T_g^L)$ can improve the efficiency of DMU_d.

(2)

When ${\theta }_{gd}^{LL}=1$, targeting $\partial (T_g^L)$ cannot improve the efficiency of DMU_d. The worst production point of DMU_d is on $\partial (T_g^L)$, so there is no need for improvement in the first stage.

We propose the following theorem to discuss the second stage of the resource allocation optimization procedure.

Theorem 1.

Let DMU_d be any DMU on $\partial (T_g^U)$, and let DMU_j be any DMU. Then, based on $PP{S}^U$ for efficiency calculation, it can be concluded that ${\theta }_{gd}^{UU}>{\theta }_{gj}^{UL}$.

Proof. 

According to the definition of the worst and best production points of DMUs in interval DEA, it can be inferred that ${\theta }_{gj}^{UU}>{\theta }_{gj}^{UL}$. Since DMU_d is located on $\partial (T_g^U)$ and satisfies ${\theta }_{gd}^{UU}\ge {\theta }_{gj}^{UU}$, if ${\theta }_{gd}^{UU}\le {\theta }_{gj}^{UL}$, then it can be inferred that ${\theta }_{gj}^{UU}\le {\theta }_{gd}^{UU}\le {\theta }_{gj}^{UL}$, which contradicts with ${\theta }_{gj}^{UU}>{\theta }_{gj}^{UL}$. This completes the proof. □

The resource allocation optimization procedure in the first stage can optimize the DMUs to $\partial (T_g^L)$. The improvement target of the resource allocation optimization procedure in the second stage is $\partial (T_g^U)$. $\partial (T_g^L)$ is composed of the worst production points. According to Theorem 1, when efficiency is calculated based on ${PPS}^U$, the efficiency value of any point on $\partial (T_g^U)$ will be greater than the efficiency value of the worst production point of any DMU. Therefore, the second stage can further optimize the efficiency of global improvement type DMUs.

This paper improves the two-stage DEA benchmarking model proposed by Ramón et al. (2018) [4]. In the two-stage benchmarking model proposed by Ramón et al. (2018) [4], the benchmark points require that outputs must exceed those of DMUs, while inputs must be less than those of the evaluated DMUs. This condition means that when using the model to derive benchmark points for resource allocation optimization, it does not allow for improving the efficiency of DMUs by increasing outputs while simultaneously increasing inputs. However, in certain industries, decision makers prioritize increasing outputs and may be less concerned about increasing inputs to achieve these goals. For instance, in China’s forestry carbon sequestration projects, under the backdrop of China’s dual carbon goals, the government places greater emphasis on increasing carbon sequestration, even if it requires further increases in inputs. Therefore, relative to the original model, in the first stage, we introduce $s_{id}^{aE-}-s_{id}^{bE-} (s_{id}^{aE-},s_{id}^{bE-}\ge 0 )$ as a relaxation variable for the first constraint, allowing the benchmark point’s inputs to exceed those of the evaluated DMUs. When calculating the benchmark point for the first stage, we designed the first constraint ${\displaystyle \sum _{j\in E}{\lambda }_j^E{x}_{ij}}=x_{id}-(s_{id}^{aE-}-s_{id}^{bE-})$. We use the second constraint, $x_{id}-(s_{id}^{aE-}-s_{id}^{bE-})\ge 0$, to ensure that the inputs of the benchmark point is non-negative. For outputs, we use $s_{rd}^{E+} (s_{rd}^{E+}\ge 0)$ as a relaxation variable for the third constraint, ensuring that the benchmark point’s output is greater than or equal to that of the evaluated DMU. The above three constraints are the first three constraints for the first stage in Model (10). Similarly, for the second stage, we designed corresponding constraints based on this concept. Observing the first three constraints of the second stage in model (10), the structural difference compared with the first stage lies in the different starting points. The starting point for the first stage is $\left(x_{id},y_{rd}\right)$, while the starting point for the second stage is $\left(x_{id}-(s_{id}^{aE-}-s_{id}^{bE-}),y_{rd}+s_{rd}^{E+}\right)$.

We minimize the distance from the evaluated DMU $(X_d,Y_d)$ to the benchmark frontier of the first stage, to determine the benchmark point $({\widehat{X}}_d^1,{\widehat{Y}}_d^1)$ of the first stage. Thus, we adopt Formula (8) as the objective function of the first stage, as shown below.

$${\Vert (X_d,Y_d)-({\widehat{X}}_d^1,{\widehat{Y}}_d^1)\Vert }_1^{\omega }={\displaystyle \sum _i^m\frac{(s_{id}^{aE-}+s_{id}^{bE-})}{x_{id}}+}{\displaystyle \sum _r^s\frac{s_{rd}^{E+}}{y_{rd}}}$$

(8)

$${\Vert (X_d,Y_d)-({\widehat{X}}_d^2,{\widehat{Y}}_d^2)\Vert }_1^{\omega }={\displaystyle \sum _i^m\frac{(s_{id}^{ae-}+s_{id}^{be-})}{x_{id}}+}{\displaystyle \sum _r^s\frac{s_{rd}^{e+}}{y_{rd}}}$$

(9)

In Formula (8), $s_{id}^{aE-}+s_{id}^{bE-}$ represents the absolute value relaxation variable of $s_{id}^{aE-}-s_{id}^{bE-}$. Similarly, in the second stage, we adopt a similar improvement approach as in the first stage and use Formula (9) as the objective function for the second stage. Subsequently, we ensure through Constraints 2 and 11 of Model (10) that the inputs of benchmark points in the first and second stages do not become negative. The improved model is shown as follows:

$$\begin{array}{ll}Min {\displaystyle \sum _i^m\frac{(s_{id}^{aE-}+s_{id}^{bE-})}{x_{id}}+}{\displaystyle \sum _r^s\frac{s_{rd}^{E+}}{y_{rd}}}+{\displaystyle \sum _i^m\frac{(s_{id}^{ae-}+s_{id}^{be-})}{x_{id}}+}{\displaystyle \sum _r^s\frac{s_{rd}^{e+}}{y_{rd}}}& \\ \mathrm{The} \mathrm{first} \mathrm{stage}& \\ {\displaystyle \sum _{j\in E}{\lambda }_j^E{x}_{ij}}=x_{id}-(s_{id}^{aE-}-s_{id}^{bE-})& i=1,\dots ,m\\ x_{id}-(s_{id}^{aE-}-s_{id}^{bE-})\ge 0& i=1,\dots ,m\\ {\displaystyle \sum _{j\in E}{\lambda }_j^E{y}_{rj}}=y_{rd}+s_{rd}^{E+}& \mathrm{r}=1,\dots ,s\\ {\displaystyle \sum _{j\in E}{\lambda }_j^E}=1\\ -{\displaystyle \sum _{i=1}^m{v}_i^E{x}_{ij}}+{\displaystyle \sum _{r=1}^s{u}_r^E{y}_{rj}}+u_d^E+d_j^E=0& j\in E\\ v_i^E{x}_{id}\ge 1& i=1,\dots ,m\\ u_r^E{y}_{rj}\ge 1& \mathrm{r}=1,\dots ,s\\ d_j^E\le M^E{b}_j^E& j\in E\\ {\lambda }_j^E\le M^E(1-b_j^E)& j\in E\\ \mathrm{The} \mathrm{second} \mathrm{stage}& \\ {\displaystyle \sum _{j\in e}{\lambda }_j^e{x}_{ij}}=(x_{id}-(s_{id}^{aE-}-s_{id}^{bE-}))-(s_{id}^{ae-}-s_{id}^{be-})& i=1,\dots ,m\\ (x_{id}-(s_{id}^{aE-}-s_{id}^{bE-}))-(s_{id}^{ae-}-s_{id}^{be-})\ge 0& i=1,\dots ,m\\ {\displaystyle \sum _{j\in e}{\lambda }_j^e{y}_{rj}}=(y_{rd}+s_{rd}^{E+})+s_{rd}^{e+}& \mathrm{r}=1,\dots ,s\\ {\displaystyle \sum _{j\in e}{\lambda }_j^e}=1& \\ -{\displaystyle \sum _{i=1}^m{v}_i^e{x}_{ij}}+{\displaystyle \sum _{r=1}^s{u}_r^e{y}_{rj}}+u_d^e+d_j^e=0& j\in e\\ v_i^e{x}_{id}\ge 1& i=1,\dots ,m\\ u_r^e{y}_{rj}\ge 1& \mathrm{r}=1,\dots ,s\\ d_j^e\le M^e{b}_j^e& j\in e\\ {\lambda }_j^e\le M^e(1-b_j^e)& j\in e\\ s_{id}^{aE-},s_{id}^{bE-},s_{rd}^{E+}\ge 0& i=1,\dots ,m\\ s_{id}^{ae-},s_{id}^{be-},s_{rd}^{e+}\ge 0& \mathrm{r}=1,\dots ,s\\ {\lambda }_j^E,d_j^E\ge 0& j\in E\\ {\lambda }_j^e,d_j^e\ge 0& j\in e\\ b_j^E\in \left\{0,1\right\}& j\in E\\ b_j^e\in \left\{0,1\right\}& j\in e\\ u_d^E, u_d^e \mathrm{free}& \end{array}$$

(10)

In Model (10), variable $x_{id},i=1,\dots ,m$ represents the input of the worst production point of the evaluated DMU, while variable $y_{rd}, \mathrm{r}=1,\dots ,s$ represents the output. Variables $E$ and $e$ represent the improvement objectives of the first and second stages, respectively. For global improvement type DMUs, E and e are $\partial (T_g^L)$ and $\partial (T_g^U)$, respectively. $M^E$ and $M^e$ are large positive quantities. $s_{id}^{aE-}-s_{id}^{bE-}$ is the slack variable for adjusting the i-th input of DMU_d to the first-stage target, while $s_{rd}^{E+}$ is the slack variable for adjusting the r-th output of DMU_d to the first-stage target. $s_{id}^{ae-}-s_{id}^{be-}$ is the slack variable for adjusting the i-th input of DMU_d to the second-stage target, and $s_{rd}^{e+}$ is the slack variable for adjusting the r-th output of DMU_d to the second-stage target. The inputs of DMUd at the first-stage improvement target point are $x_{id}-(s_{id}^{*aE-}-s_{id}^{*bE-})$$(i=1,2,\dots ,s)$, and the outputs are $y_{rd}+s_{rd}^{*E+}$$(i=1,2,\dots ,s)$; the inputs at the second-stage improvement target point are $(x_{id}-(s_{id}^{*aE-}-s_{id}^{*bE-}))-(s_{id}^{*ae-}-s_{id}^{*be-})$$(i=1,2,\dots ,s)$, and the outputs are $(y_{rd}+s_{rd}^{*E+})+s_{rd}^{*e+}$$(i=1,2,\dots ,s)$. When global improvement type DMUs belong to the second scenario, the model calculation results of the first stage will be the same as their input–output, indicating that no improvement is needed.

4.3.2. Intra-Group Improvement Type DMUs Resource Allocation Optimization Procedure

In this section, we extend the computational model for technical efficiency (TE) and technological gap (TG) proposed by Walheer (2023) [36] to interval DEA. The model is as follows:

$$\begin{array}{l}T{E}^U=Max {\theta }^U\\ {\displaystyle \sum _{j\in \partial (T^U)}^{ }{\lambda }_j{y}_j^U}\ge {\theta }^U{y}_D\\ {\displaystyle \sum _{j\in \partial (T^U)}^{ }{\lambda }_j{x}_j^L} \le x_D\\ {\lambda }_j\ge 0,{\theta }^U\ge 0\end{array}$$

(11)

$$\begin{array}{l}T{E}^L=Max {\theta }^L\\ {\displaystyle \sum _{j\in \partial (T^L)}^{ }{\lambda }_j{y}_j^L}\ge {\theta }^L{y}_D\\ {\displaystyle \sum _{j\in \partial (T^L)}^{ }{\lambda }_j{x}_j^U} \le x_D\\ {\lambda }_j\ge 0,{\theta }^L\ge 0\end{array}$$

(12)

In Models (11) and (12), the reference point ${DMU}_D(x_D,y_D)$ is composed of the maximum output and minimum input values of all DMUs. In Model (11), variable $\left(x_j^L,y_j^U\right),j\in \partial (T^U)$ represents a point on the frontier $\partial (T^U)$. In Model (12), variable $\left(x_j^U,y_j^L\right),j\in \partial (T^L)$ represents a point on the frontier $\partial (T^L)$. $\partial (T^U)$ and $\partial (T^L)$ represent the upper and lower frontier, respectively. Model (11) is used to evaluate the TE on the upper frontier; the optimal solution ${\theta }^{U*}$ is the TE score. Model (12) is used to evaluate the TE on the lower frontier; the optimal solution ${\theta }^{L*}$ is the TE score. Applying Model (11), we can calculate the efficiency scores $T{E}_g^U$ and $T{E}_c^U$ for the frontier $\partial (T_g^U)$ and $\partial (T_c^U)$. Through Model (12), we can calculate the efficiency scores $T{E}_g^L$ and $T{E}_c^L$ for the frontier $\partial (T_g^L)$ and $\partial (T_c^L)$. Based on the technical efficiency scores on the global frontier, we calculate the technological gap $T{G}_c^U=T{E}_c^U/T{E}_g^U$ for $T{E}_c^U$ and $T{G}_c^L=T{E}_c^L/T{E}_g^L$ for $T{E}_c^L$. Multiplying the outputs of all points on $\partial (T_g^U)$ by $T{G}_c^U$ yields the virtual frontier $\partial *(T_c^U)$, and multiplying the outputs of all points on $\partial (T_g^L)$ by $T{G}_c^L$ yields the virtual frontier $\partial *(T_c^L)$.

The resource allocation optimization procedure for intra-group improvement type DMUs is considered in two scenarios. Assuming DMU_d is an intra-group improvement type in group c, it will encounter the following two scenarios.

(1)

When $T{G}_c^U=1$, DMU_d does not need to undergo the third stage. When $T{G}_c^L=1$, DMU_d operates using model (10) in a manner consistent with the global improvement type. When $T{G}_c^L<1$, let $E=\partial *(T_c^L)$ $e=\partial (T_g^U)$, computation is performed using Model (10).

(2)

When $T{G}_c^U<1$, DMU_d needs to undergo the third stage of improvement. Let $E=\partial *(T_c^L)$ $e=\partial *(T_c^U)$; computation is performed using Model (10) to obtain the benchmarks for the first two stages of DMU_d. Finally, computation is carried out using Model (13). The model is as follows:

$$\begin{array}{ll}Min {\displaystyle \sum _i^m\frac{(s_{id}^{a-}+s_{id}^{b-})}{x_{id}}+}{\displaystyle \sum _r^s\frac{s_{rd}^{+}}{y_{rd}}}& \\ {\displaystyle \sum _{j\in \partial (T_g^U)}{\lambda }_j{x}_{ij}}=x_{id}-(s_{id}^{a-}-s_{id}^{b-})& i=1,\dots ,m\\ x_{id}-(s_{id}^{a-}-s_{id}^{b-})\ge 0& i=1,\dots ,m\\ {\displaystyle \sum _{j\in \partial (T_g^U)}{\lambda }_j{y}_{rj}}=y_{rd}+s_{rd}^{+}& \mathrm{r}=1,\dots ,s\\ {\displaystyle \sum _{j\in \partial (T_g^U)}{\lambda }_j}=1& \\ -{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}}+{\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}}+u_d+d_j=0& j\in \partial (T_g^U)\\ v_i{x}_{id}\ge 1& i=1,\dots ,m\\ u_r{y}_{rj}\ge 1& \mathrm{r}=1,\dots ,s\\ d_j\le M{b}_j& j\in \partial (T_g^U)\\ {\lambda }_j\le M(1-b_j)& j\in \partial (T_g^U)\\ s_{id}^{a-},s_{id}^{b-}\ge 0& i=1,\dots ,m\\ s_{rd}^{+}\ge 0& \mathrm{r}=1,\dots ,s\\ {\lambda }_j,d_j\ge 0& j\in \partial (T_g^U)\\ b_j\in \left\{0,1\right\}& j\in \partial (T_g^U)\\ u_d^E \mathrm{free}& \end{array}$$

(13)

The construction approach of Model (13) is similar to Model (10), transforming the two-stage model into a one-stage model. Model (13) takes as input the inputs and outputs of the benchmarks for the second stage of intra-group improvement type DMUs. Model (13) yields the inputs for the third stage benchmark as $x_{id}-(s_{id}^{a-}-s_{id}^{b-}) (i=1,\dots ,m)$ and the outputs as $y_{rd}+s_{rd}^{+} (\mathrm{r}=1,\dots ,s)$.

4.4. Staged Resource Allocation Optimization Procedural Steps under Heterogeneous Grouping in Interval DEA

4.4.1. Procedural Steps

Step 1: Divide n DMUs into k groups based on heterogeneity.

Step 2: Calculate the efficiency of the worst production point for each DMU. Use Model (7) to calculate the efficiency of the worst production point for each DMU in ${PPS}^L$.

Step 3: Determine the type of DMUs. Based on the calculation results of Step 2, determine whether the DMUs simultaneously satisfy both conditions 1 and 2 mentioned earlier. If the a DMU does not satisfy both conditions 1 and 2 simultaneously, then it belongs to the intra-group improvement type; otherwise, it belongs to the global improvement type.

Step 4: Construct the efficient frontiers $\partial (T_g^L)$, $\partial (T_g^U)$, $\partial (T_c^L)$, and $\partial (T_c^U)$ according to Definition 3.

Step 5: Calculate TE and TG. Calculate the TE values of frontiers $T{E}_g^L$, $T{E}_g^U$, $T{E}_c^L$, and $T{E}_c^U$ using Models (11) and (12). Based on the TE of each frontier, derive TG values $T{G}_c^L$ and $T{G}_c^U$, deriving virtual frontiers $\partial (T_c^{L*})$ and $\partial (T_c^{U*})$ through $T{G}_c^L$ and $T{G}_c^U$.

Step 6: Run the resource allocation optimization model. Calculate benchmarks for stages of global improvement type DMUs using Model (10). Compute benchmarks for stages of intra-group improvement type DMUs using Models (10) and (13).

Step 7: Complete staged resource allocation optimization.

4.4.2. Technical Process Diagram

After determining the improvement objectives for each stage of the two types of DMUs and combining the calculation models for each stage, the process for resource allocation optimization is shown in Figure 1.

Input: Worst input–output values of DMUs, efficiency values ${\theta }_{cd}^{LL}$$(c=1,2,\dots ,k)$, and the technological gap $T{G}_c^U$$(c=1,2,\dots ,k)$.

Output: Benchmarks for each stage of the resource allocation optimization procedure.

Figure 1 illustrates the steps that DMUs need to follow using the method proposed in this paper, making it easier for readers to understand the procedure described in Section 4.4.1.

5. Numerical Example

Assume there is a set of DMUs represented by interval data: $T$ = {1, 2, 3, 4, A, B, C}, each with two inputs and one output. We categorize these DMUs into two groups based on their characteristics: {1, 2, 3, 4} and {A, B, C}.

Table 1. Worst production points of DMUs.

Group	DMU	${\mathit{x}}^{\mathit{U}}$	${\mathit{x}}_{\mathbf{2}}^{\mathit{U}}$	${\mathit{y}}^{\mathit{L}}$	${\mathit{\theta }}_{\mathit{g}\mathit{d}}^{\mathit{L}\mathit{L}}$	${\mathit{\theta }}_{\mathit{c}\mathit{d}}^{\mathit{L}\mathit{L}}$
Group1	1	5	4	10	1	1
	2	8	3	6	0.8	0.8
	3	7	5	9	0.72	0.72
	4	6	9	10	0.83	0.83
Group2	A	4	8	9	1	1
	B	5	4	8	0.8	1
	C	5	10	5	0.5	0.5

shows the inputs and output under their worst production conditions, while

Table 2. Best production points of DMUs.

Group	DMU	${\mathit{x}}^{\mathit{L}}$	${\mathit{x}}_{\mathbf{2}}^{\mathit{L}}$	${\mathit{y}}^{\mathit{U}}$	${\mathit{\theta }}_{\mathit{g}\mathit{d}}^{\mathit{U}\mathit{U}}$	${\mathit{\theta }}_{\mathit{c}\mathit{d}}^{\mathit{U}\mathit{U}}$
Group1	1	4	3	14	1	1
	2	7	2	8	0.86	0.86
	3	5	3	10	0.71	0.71
	4	3	6	13	1	1
Group2	A	3	7	10	0.77	1
	B	4	3	11	0.79	1
	C	4	9	9	0.52	0.68

presents the inputs and output under their best production conditions.

Table 1 and Table 2 present the input–output data for DMUs under their worst and best production conditions, respectively. Table 1 displays the global efficiency ${\theta }_{gd}^{LL}$ and intra-group efficiency ${\theta }_{cd}^{LL}$ of DMUs at their worst production points. Table 2 displays the global efficiency ${\theta }_{gd}^{UU}$ and intra-group efficiency ${\theta }_{cd}^{UU}$ of DMUs at their best production points.

5.1. Sensitivity Analysis

Assuming the worst optimal input difference is $\Delta X=X^U-X^L=\left\{x^U-x^L,x_2^U-x_2^L\right\}$ and the input difference is $\Delta Y=Y^U-Y^L=\left\{y^U-y^L,y_2^U-y_2^L\right\}$, we can derive that the fuzzy data for DMUs are $\left(X,Y\right)=\left(X^U-\alpha \Delta X,Y^L+\alpha \Delta Y\right)$, $\alpha \in \left[0,1\right]$. The fuzzy inputs and outputs of all DMUs can be used to construct a fuzzy point set and $T^{*}$. When $\alpha =0$, it can form the worst production possibility set, while when $\alpha =1$, it can form the best production possibility set. We use Model (14) to calculate the efficiency of the worst production points for DMUs under the fuzzy production possibility set. The model is as follows:

$$\begin{array}{l}\min {\theta }_d\\ {\displaystyle \sum _{j=1}^n{X}_j{\lambda }_j\le \theta X_d^U},\\ {\displaystyle \sum _{j=1}^n{Y}_j{\lambda }_j\ge Y_d^L},\\ {\lambda }_j\ge 0,j=1,2,\dots ,n.\end{array}$$

(14)

In Model (14), $X_d^U$ and $Y_d^L$ represent the inputs and outputs of the worst production point of the evaluated DMU. $X^j$ and $Y^j$ represent the inputs and outputs of the j-th DMU in $T^{*}$, while ${\lambda }_j$ denotes the weights associated with this DMU. In Model (14), ${\lambda }_j$ is an unknown variable. The optimal solution ${\theta }_d^{*}$ of Model (14) represents the efficiency of the evaluated DMU_d. We calculated the efficiency of the DMUs for five different values of $\alpha$ using Model (14). The results are shown in

Table 3. Efficiency of DMUs under the fuzzy production possibility set.

DMU	$\mathit{\alpha }\mathbf{=}\mathbf{0}$	$\mathit{\alpha }\mathbf{=}\mathbf{0.25}$	$\mathit{\alpha }\mathbf{=}\mathbf{0.5}$	$\mathit{\alpha }\mathbf{=}\mathbf{0.75}$	$\mathit{\alpha }\mathbf{=}\mathbf{1}$
1	1.00	0.86	0.86	0.75	0.57
2	0.80	0.68	0.68	0.58	0.43
3	0.72	0.61	0.61	0.53	0.39
4	0.83	0.72	0.72	0.63	0.42
A	1.00	0.86	0.86	0.75	0.48
B	0.80	0.69	0.69	0.60	0.45
C	0.50	0.43	0.43	0.38	0.23

.

Table 3 presents the efficiency scores of DMUs based on the worst production points under the fuzzy production possibility set. It details the efficiency scores for DMUs for five different values of $\alpha$. When $\alpha =0$, the fuzzy production possibility set is equivalent to the worst production possibility set. As shown in Table 3, the efficiency scores in the column for $\alpha =0$ match those in column ${\theta }_{gd}^{LL}$ of Table 1. As $\alpha$ increases, the fuzzy production possibility set increasingly approximates the best production possibility set. During this process, the efficiency scores of the DMUs gradually decrease. This result indicates that as $\alpha$ increases, the difficulty of optimizing DMUs to the best frontier of the fuzzy production possibility set becomes progressively greater. This phenomenon proves that our proposed approach, which involves optimizing in stages from the worst production points to the boundary of the frontier of the best production possibility set, is feasible.

5.2. Step Analysis

In this section, we apply the procedural steps outlined in Section 4.4.1 to the data presented in Table 1 and Table 2.

From ${\theta }_{cd}^{LL}$ and Y values in Table 1, it can be observed that DMU₂, DMU₃, DMU₄, and DMU_C simultaneously satisfy the aforementioned conditions 1 and 2. Thus, they are classified as intra-group improvement type, while DMU₁, DMU_A, and DMU_B are classified as global improvement type. With this, steps 1–3 are completed.

According to Table 1 and Table 2, the constituent points of $\partial (T_g^L)$ are the worst production points of DMU₁ and DMU_A, and these two DMUs do not need to undergo first-stage improvement. The constituent points of $\partial (T_g^U)$ are the best production points of DMU₁ and DMU₄; the constituent point of $\partial (T_1^L)$ is the worst production point of DMU1; the constituent points of $\partial (T_1^U)$ are the best production points of DMU₁ and DMU₄; the constituent points of $\partial (T_2^L)$ are the worst production points of DMU_A and DMU_B; the constituent points of $\partial (T_2^U)$ are the best production points of DMU_A and DMU_B. With this, step 4 is completed.

According to Table 1, the reference point input values for Model (11) are 8 and 10, respectively, with an output value of 5. Through Model (11), the TE of each frontier obtained in step 4 and the TG with respect to the global frontier are calculated. TE and TG values are both shown in

Table 4. Frontier adjustment table.

Group	Frontier	TE	TG	DMU	Y	Y*
Global	$\partial (T^L)$	2	1	1	10	10
				A	8	8
	$\partial (T^U)$	2.8	1	1	14	14
				4	13	13
Group1	$\partial (T_1^L),\partial (T_1^{L*})$	2	1	1	10	10
				A	8	8
	$\partial (T_1^U),\partial (T_1^{U*})$	2.8	1	1	14	14
				4	13	13
Group2	$\partial (T_2^L),\partial (T_2^{L*})$	1.8	0.9	1	10	9
				A	8	7.2
	$\partial (T_2^U),\partial (T_2^{U*})$	2.2	0.79	1	14	11
				4	13	10.2

. In Table 4, DMU represents the points forming the global efficient frontier, and Y represents their output values, while Y* represents the output values adjusted by TG. With this, step 5 is completed.

Table 4 shows the TE and TG for each frontier, along with the output values before and after the adjustment of the points constituting each frontier. Y represents the output value before adjustment, while Y* represents the output value after adjustment. From Table 4, it is evident that for Group 1, both the upper and lower frontiers of TG = 1, indicating no adjustment is required for the frontiers. Moreover, for the intra-group improvement type DMUs within Group 1, only two stages of improvement are needed. In Table 3, $\partial (T_1^{L*}),\partial (T_1^{U*}),\partial (T_2^{L*})$, and $\partial (T_2^{U*})$ represent the adjusted frontier. Global improvement type DMUs use $\partial (T^L)$ and $\partial (T^U)$ as their improvement objectives for two stages. For intra-group improvement type DMUs in Group 1, $\partial (T_1^{L*})$ and $\partial (T_1^{U*})$ are used as the improvement objectives for two stages. For intra-group improvement type DMUs in Group 2, $\partial (T_2^{L*})$, $\partial (T_2^{U*})$, and $\partial (T_g^U)$ are used as the improvement objectives for three stages. Based on the above results, resource allocation optimization for all DMUs are calculated using Models (11)–(13). With this, steps 7–8 are completed, and the results are shown in

Table 5. Results of numerical example.

Type	DMU	Stage	X	X2	Y
Intra-group improvement type	2	1	5	4	10
		2	3.67	4	13.67
	3	1	4.75	5	9.75
		2	3.33	5	13.33
	4	1	5	4	10
		2	3.67	4	13.67
	C	1	4	7	6.4
		2	3	6	10.21
		3	3	6	13
Global improvement type	1	1	Skip
		2	3.67	4	13.67
	A	1		Skip
		2	3	6	13
	B	1	5	4	10
		2	3.67	4	13.67

.

Table 5 presents the results of the resource allocation optimization. In Table 5, X and X₂ represent the input values for each stage’s objective, while Y represents the output value.

5.3. Comparative Analysis between Different Methods

The traditional resource allocation optimization model under heterogeneous grouping targets the improvement of the group frontier as the objective for DMUs (Cook et al., 2017 [24]). In the interval DEA method we adopt, improvement targets for DMUs are set as the upper group frontier. Furthermore, to contrast with our method, an additional stage is introduced based on the original method, with improvement targets set as the upper global frontier for DMUs. The results are shown in

Table 6. Traditional results of numerical example.

Type	DMU	Stage	X	X2	Y
Group1	1	1	3.67	4	13.67
		2	3.67	4	13.67
	2	1	4	3	14
		2	4	3	14
	3	1	3.33	5	13.33
		2	3.33	5	13.33
	4	1	3	6	13
		2	3	6	13
Group2	A	1	3	7	10
		2	3	6	13
	B	1	3.75	4	10.75
		2	3.67	4	13.67
	C	1	3	7	10
		2	3	6	13

.

The traditional method does not classify DMUs and considers only intra-group optimality without considering global optimality. In this section, for a more intuitive comparison between the traditional method and our method, we introduce a step into the traditional method whereby DMUs are improved to achieve global optimality. Based on the data in Table 5 and Table 6, we compare the two methods from two perspectives: reachability and improvement space.

Considering reachability, we analyze reachability from the perspectives of improvement magnitude and improvement difficulty. Considering improvement magnitude, let us take the process of improving DMU₂ to intra-group optimality as an example. In the traditional method, in the first stage, the input X needs to be improved from 8 to 3.67, input X₂ from 3 to 4, and output Y from 6 to 13.67. This requires a 54% reduction in input X and a 128% increase in output Y, indicating a significant magnitude of adjustment and a lack of reachability. With the use of this paper’s method, DMU₂ is identified as an intra-group improvement type, and it is adjusted over two stages to $\partial (T_2^{U*})$. Throughout each stage of the adjustment process, the maximum adjustment magnitude for DMU₂ is observed when output Y is adjusted from 6 to 10 in the first stage, requiring a 67% adjustment. This indicates that relative to the traditional method, our method exhibits sufficient reachability. Considering improvement difficulty, let us take the process of DMU_C improving to global optimality as an example. The traditional method requires a decrease of 2 in input X₂ while increasing output Y by 3. In our method, the input structure of $\partial (T_2^{U*})$ is similar to that of $\partial (T_g^U)$. After improving DMU_C to $\partial (T_2^{U*})$ and then to $\partial (T_g^U)$, only an increase of 2.79 in output value is needed. Compared with the traditional method, our method has lower adjustment difficulty.

Considering improvement space, in the traditional method, some DMUs may reach a point where there is no space for improvement in the final stage. For example, for DMUs in Group 1, there is no improvement space between the first and second stages in the traditional method. In this paper’s method, during the process, it is recognized that the technological level of $\partial (T_1^U)$ is equivalent to that of $\partial (T_g^U)$. Therefore, the third-stage improvement is abandoned, and the resource allocation optimization procedure is completed in two stages.

In summary, the method proposed in this paper has the following advantages: (1) After categorizing the DMUs, targeted improvements are made in stages for the DMUs with poor production performance. This approach reduces the adjustment magnitude for each stage compared with traditional methods, thus increasing the reachability of the resource allocation optimization procedure. (2) Using the adjusted frontier as the improvement objective for the first two stages for intra-group improvement type DMUs facilitates easier completion of the final stage of improvement. (3) In the process of resource allocation optimization, we can assess whether there is space for improvement in the final stage. Based on this assessment, we can decide whether to proceed with a three-stage improvement or a two-stage improvement.

6. Example Analysis of Forest Carbon Sequestration

We conducted computational analysis using data from 30 provinces (municipalities, autonomous regions), each characterized by three input variables and one output variable. Specific data can be found in

Table 7. Chinese Forestry Carbon Sink Input–Output Indicator Data.

Group	DMU	Investment	Area	Practitioners	Carbon Sink	Group Efficiency	Global Efficiency
Eastern	Beijing	2,388,438	34,084	9639	2823.74	0.19	0.11
	Tianjin	448,385	16,538	704	533.23	0.20	0.16
	Hebei	1,431,233	520,644	20,426	15,915.79	0.20	0.16
	Liaoning	347,031	157,599	26,668	34,465.17	1.00	0.45
	Shanghai	240,345	5003	1397	520.86	0.24	0.14
	Jiangsu	731,374	44,366	18,758	8161.21	0.43	0.22
	Zhejiang	1,423,907	75,527	9993	32,571.55	1.00	0.67
	Fujian	1,142,097	214,092	22,038	84,500.07	1.00	0.79
	Shandong	1,245,878	168,297	19,164	10,613.82	0.16	0.11
Central	Shanxi	1,074,677	347,355	23,681	14,972.05	0.31	0.13
	Jilin	875,511	102,929	93,018	117,353.7	1.00	1.00
	Heilongjiang	1,288,251	118,264	238,224	213,984.3	1.00	1.00
	Anhui	1,056,938	138,746	17,514	25,703.67	0.73	0.30
	Jiangxi	2,288,186	269,354	40,886	58,697.63	0.73	0.33
	Henan	1,977,923	196,493	25,367	24,003.62	0.47	0.19
	Hubei	3,321,823	473,105	25,502	42,295.33	0.80	0.34
	Hunan	907,967	574,543	45,678	47,170.19	0.50	0.24
	Guangdong	128,620	244,726	26,512	54,166.94	1.00	1.00
	Guangxi	1,173,403	220,068	40,967	78,492.91	1.00	0.52
	Hainan	7,378,570	15,704	8202	17,771.95	1.00	1.00
Western	Inner Mongolia	1,733,035	720,285	93,147	176,911.5	0.50	0.47
	Chongqing	765,752	275,195	5586	23,956.19	0.88	0.88
	Sichuan	2,168,729	400,370	53,089	215,600.3	0.84	0.84
	Guizhou	2,990,078	346,974	21,423	45,394.37	0.44	0.44
	Yunnan	1,127,236	353,812	47,011	228,537.4	1.00	1.00
	Shaanxi	1,141,571	333,451	31,679	55,454.77	0.36	0.36
	Gansu	1,364,271	369,085	39,343	29,181.96	0.15	0.15
	Qinghai	576,211	220,576	20,796	5635.23	0.06	0.06
	Ningxia	295,003	92,254	10,239	967.5769	0.02	0.02
	Xinjiang	914,730	229,331	17,452	45,439.09	0.54	0.54

.

6.1. Data Source

In addition to the data on forest carbon sequestration volume, the data used in this paper were sourced from the China Forestry and Grassland Statistics Yearbook and the China Statistical Yearbook. As the recently published China Forestry and Grassland Statistics Yearbook included forestry data for the year 2019, data from that year were selected for analysis. Similarly, data on forest carbon sequestration volume were obtained from the 2019 dataset on aboveground and belowground vegetation carbon storage in Chinese forests, provided by the National Tibetan Plateau Science Data Center. This dataset was generated by integrating high-resolution active microwave remote sensing, long time-series passive microwave, and optical remote sensing information using regression and machine learning algorithms.

We used investment, area, and practitioners as input variables, and carbon sink as the output variable, specifically as follows:

Investment: Annual completed forestry investment, in CNY 10,000s;

Area: Afforested area, in hectares;

Practitioners: Number of forestry practitioners;

Carbon sink: Output of forest carbon sequestration, in ten thousand tons.

Table 7 presents a set of input–output data for Chinese forest carbon sequestration, including both group efficiency and global efficiency for each province. This study lowered the data in Table 7 by 10% as the lower bound of interval numbers and increased them by 10% as the upper bound of interval numbers, thereby obtaining a set of interval data. Among them, if the number of forestry practitioners is adjusted by simply ±10%, decimals may occur, which is unreasonable as the number of personnel should be an integer, so rounding has been adopted to make it an integer. In Table 7, group efficiency and global efficiency, respectively, represent the efficiency values of DMUs within the group and globally. Due to the adjustment of data to interval numbers, the input and output variables of each DMU are multiplied by the same coefficient, which does not change their efficiency values in the DEA model. Therefore, the worst and best production points of DMUs in both intra-group and global efficiency are consistent with the efficiency values in Table 7. This article refers to the seventh Five-Year Plan passed by the National People’s Congress of China when grouping China into three groups: East, Central, and West. The eastern region is the most economically developed area in China. The central region has abundant energy and mineral resources with a relatively good industrial foundation. The western region possesses abundant natural resources and huge development potential. Based on the two conditions given above and the data in the table, it is concluded that Liaoning, Zhejiang, Fujian, Jilin, Heilongjiang, Guangdong, Guangxi, Hainan, and Yunnan are global improvement type DMUs, while the rest are intra-group improvement type DMUs.

6.2. The Final Result of Resource Allocation Optimization Procedure

After calculating the technological level of the frontier, the technological levels of the upper western frontier and global frontier were found to be the same. Therefore, DMUs in the western group do not need to undergo the third stage of resource optimization and allocation. The final resource optimization and allocation scheme obtained was as follows.

Table 8. Results for intra-group improvement type.

DMU	Stage	Investment	Area	Practitioners	Carbon Sink
Beijing	1	7,476,403	37,492.4	10,638.6	6885.91
	2	6,640,713	14,133.6	7381.8	7228.15
	3	6,640,713	14,133.6	7381.8	19,549.14
Tianjin	1	8,116,427	17,274.4	9022.2	5913.95
	2	6,597,280	14,865.01	9022.2	7860.50
	3	6,617,559.52	14,865.01	7440.28	19,691.20
Hebei	1	1,574,356.3	223,934.9	25,544.43	15,849.03
	2	1,574,356.3	171,930.8	25,544.43	20,593.46
	3	1,574,356.3	174,177.07	20,177.06	50,634.26
Shanghai	1	8,116,427	17,274.4	9022.2	5913.95
	2	6,640,713	14,133.6	7381.8	7228.16
	3	6,640,713	14,133.6	7381.8	19,549.14
Jiangsu	1	804,511.4	248,253.9	27,488.7	17,018.15
	2	804,511.4	198,496.1	22,121.33	20,468.12
	3	804,511.4	198,496.1	22,121.33	55,357.72
Shandong	1	1,370,466	230,375.7	26,059.36	16,158.66
	2	1,370,466	178,371.5	26,059.36	21,056.01
	3	1,370,466	180,617.9	20,691.99	51,885.25
Shanxi	1	1,182,145.7	236,324.6	26,534.97	38,839.01
	2	1,182,145.7	186,566.8	21,167.6	49,663.12
	3	1,182,144.7	186,566.8	21,167.6	53,040.72
Anhui	1	1,162,632	236,941	26,584.25	38,909
	2	1,162,632	184,936.9	26,584.25	54,515.3
	3	1,162,632	184,936.9	26,584.25	58,222.9
Jiangxi	1	2,517,004.6	193,813.8	44,974.6	52,827.87
	2	2,517,004.6	133,024.9	44,974.6	65,996.42
	3	2,517,004.6	133,024.88	44,974.6	70,484.84
Henan	1	2,175,715.3	203,315.3	27,903.7	37,413.56
	2	2,175,715.3	155,180.5	18,658.31	43,955.17
	3	2,175,715.3	155,180.5	18,658.31	46,944.57
Hubei	1	3,654,005.3	159,955.5	28,052.2	38,065.79
	2	3,654,005.3	102,988.1	28,052.2	47,055.73
	3	3,654,005.3	102,988.1	28,052.2	50,256
Hunan	1	998,763.7	239,873.5	32,360.07	42,453.17
	2	998,763.7	192,359.7	21,630.74	50,716.62
	3	998,763.7	192,359.7	21,630.74	54,165.87
Inner Mongolia	1	1,906,339	258,889.2	102,461.70	159,220.39
	2	1,906,339	153,413.60	102,461.70	159,220.39
Chongqing	1	842,327.2	247,059.3	27,393.19	45,871.66
	2	842,327.2	195,055.2	27,393.19	60,188.16
Sichuan	1	1,649,057	356,179.4	58,397.9	194,040.3
	2	1,649,057	237,623.3	58,397.9	194,040.3
Guizhou	1	3,289,086	172,815.4	23,565.3	40,854.93
	2	3,289,086	116,779.3	23,565.3	47,393.73
Shaanxi	1	1,255,728	230,864.4	34,846.9	49,909.29
	2	1,255,728	178,439.6	34,846.9	65,666.72
Gansu	1	1,500,698	226,261.7	25,730.45	43,167.54
	2	1,500,698	176,503.9	20,363.09	51,086.2
Qinghai	1	633,832.1	253,645.5	27,919.75	46,728.02
	2	633,832.1	203,887.7	22,552.39	56,404.94
Ningxia	1	6,569,636	66,136.68	12,928.67	22,347.89
	2	6,569,636	15,330.52	10,066.25	22,347.89
Xinjiang	1	1,006,203	241,882.5	26,979.32	45,198.58
	2	1,006,203	189,878.4	26,979.32	59,182.68

and

Table 9. Results for global improvement type.

DMU	Stage	Investment	Area	Practitioners	Carbon Sink
Liaoning	1	381,734.1	260,263.9	29,334.8	48,386.74
	2	381,734.1	209,279.3	29,334.8	63,748.32
Zhejiang	1	5,984,238	89,000.45	15,122.7	29,314.39
	2	5,984,238	32,325.5	15,122.7	29,314.39
Fujian	1	1,256,307	264,527.7	31,347.94	76,050.06
	2	1,256,307	190,296	31,347.94	76,050.06
Jilin	1	Skip
	2	963,062.1	113,221.9	102,319.8	139,393.9
Heilongjiang	1	Skip
	2	1,417,076	130,090.4	158,165.7	192,585.9
Guangdong	1	Skip
	2	141,482	217,448.1	28,557.19	63,916.68
Guangxi	1	1,290,743	242,074.8	45,063.7	70,905.85
	2	1,290,743	173,020.6	45,063.7	75,171.69
Hainan	1	Skip
	2	6,523,477	17,274.4	9022.2	21,536.4
Yunnan	1	Skip
	2	1,239,959.6	266,263.2	51,712.1	205,683.67

, respectively, present the benchmarks for intra-group improvement type DMUs and global improvement type DMUs. For example, in Table 8, Beijing’s optimization benchmarks for the first stage are CNY 74,764,030,000, 37,492.4 hectares, and 10,638.6 people in inputs. The goal is to improve production and operational technology to achieve an output of 6885.91 million tons of forest carbon sequestration. In the second stage, the benchmarks for inputs are CNY 66,407,130,000, 14,133.6 hectares, and 7381.8 people, with an output target of 7228.15 million tons of forest carbon sequestration. Table 9 follows a similar format, with the key difference being the presence of “Skip.” For instance, for Jilin, “Skip” indicates that Jilin does not require optimization in the first stage and proceeds directly to the second stage of optimization.

In Table 8, Beijing and Shanghai are first-tier cities in China, with high levels of urbanization and highly developed economies. However, they have very little forest area. The construction of urbanization occupies a large amount of land, which is not conducive to the development of forest carbon sinks. They are classified as intra-group improvement type in the results. The process of the final resource allocation optimization procedure shows that Beijing and Shanghai need to make significant improvements to achieve their improvement goals. For example, in Beijing, all three inputs need to be improved, and a large increase in forest carbon sink output is needed after adjusting the inputs, especially in the first stage of adjusting forestry investment, which requires more than doubling the amount of forestry investment.

In Table 9, Liaoning, Zhejiang, Fujian, Jilin, Heilongjiang, Guangdong, Guangxi, Hainan, and Yunnan are classified as global improvement type. These provinces all rank in the top 20 nationwide in terms of forest coverage, possessing favorable natural conditions conducive to the development of forest carbon sinks. Particularly in Heilongjiang and Yunnan, not only is the forest coverage high, but the forest area also ranks in the top three nationwide. In the final results, their worst production points lie at the lower bound of the global frontier surface. Therefore, these two DMUs do not require improvement in the first stage, only improvement in the second stage.

Comparing the two tables, for DMUs in the western group such as Ningxia, Qinghai, and Gansu, the forest coverage and forest area are both very low. Among the western regions, Yunnan has very favorable natural conditions, as described above, and may even only require one stage of improvement to achieve the final goal. Yunnan has raised the technological level of the entire western group, bringing the upper western frontier and global frontier to the same technological level. Intra-group improvement type DMUs in the western group only need to undergo two stages. This results in DMUs with poor natural conditions such as Ningxia, Qinghai, and Gansu needing significant improvements to achieve the final goal. For example, in Ningxia, in the first stage of improvement, the forest carbon sink output needs to be improved from 870.82 to 22,347.89, which is already beyond reach. This is also a flaw in the methodology of this article.

7. Conclusions

The DEA model has been widely applied in resource allocation optimization in the realm of real numbers. Scholars use the frontier of the production possibility set as the benchmark for the optimization process to ensure the optimality of the solution. However, when analyzing interval data, the production possibility set is variable. In real-number DEA models, selecting the benchmark becomes challenging when dealing with interval data, making it difficult to provide a meaningful solution. The first issue in constructing a resource allocation optimization model based on interval data is how to choose appropriate benchmarks. The second issue is enhancing the reachability of the optimization solution. The model in this study was improved based on the characteristics of forest carbon sink production processes to meet decision makers’ needs.

To address the aforementioned issues, we propose the staged resource allocation optimization under a heterogeneous grouping program. In this program, we consider multiple production possibility set benchmarks based on the worst–best production scenarios. From a long-term perspective on resource optimization, we introduce a staged improvement approach. This method allows decision makers to develop continuous optimization strategies based on multiple objectives, which is more attainable compared with directly targeting best practice performance. Additionally, we use the meta-frontier approach to address heterogeneity issues and enhance the benchmark point calculation model, ensuring that our program meets the needs of forest carbon sink project decision makers.

This method has been proven effective in evaluating China’s forest carbon sinks. Specifically, it identifies the allocation of efforts needed to achieve best practice performance. The allocation is based on the various frontiers generated by interval data under the worst–best scenario. Classifying these provinces enables some initially more efficient provinces to shorten the period required to achieve the final goal. For example, certain intra-group improvement type DMUs can develop strategies to achieve best practice performance through three improvement stages, while others within the same group may require only two stages. In fact, some global improvement type DMUs may even achieve best practice performance in just one stage.

In terms of theoretical implication, this study extends real-number resource optimization models to interval data. Practically, the method is suitable for industries that need to consider interval data and heterogeneity analysis. For example, in the electronics manufacturing industry, fluctuations in raw material costs and changes in market demand result in interval data, and regional market differences also need to be taken into account. However, this study still has the following limitations: (1) our method does not provide a dynamic adjustment plan; and (2) this study considers only the best and worst production possibility sets in the interval DEA but does not discuss the production possibility sets in between. Future research directions could consider constructing production possibility sets that lie between the worst and best cases, to address these two issues.