According to the framework, the first issue addressed in this paper is to examine whether the current inter-provincial CEA allocation scheme in China has any irrationality. To do so, taking the year 2020 as an example, the entropy method is initially employed to simulate the current CEA allocation method and formulate an allocation scheme. Subsequently, the SBM model is utilized to scrutinize this allocation scheme. Due to the current distribution methods in China, which mainly include historical emission methods, historical carbon intensity reduction methods, and industry baseline methods, they are all, in fact, based on the distribution of CO₂ emission. Therefore, this paper draws inspiration from Li’s study on the allocation of CEA in the cities of the BOHAI Economic Rim. Li et al. (2021a) used CO₂-related indicators, namely, population, GDP, historical CO₂, and historical cumulative net CO₂ emissions. The entropy method is then applied to allocate CEA, making the allocation scheme more closely aligned with the current practical CEA allocation scheme. Formulas 1–7 are provided for specific cases, where i denotes provinces (i = 1,2,30, as defined in the following text), and j denotes indicators (j = 1,2,30, as defined in the following text).

Normalize the indicators by defining Z_ij as the normalized value of P_ij, where P_ij represents the value of indicator j in province i: Z i j = P i j − ⁡ min ⁡ P i j max ⁡ P i j − ⁡ min ⁡ P i j (1) Z i j = max ⁡ P i j − P i j max ⁡ P i j − ⁡ min ⁡ P i j

To calculate the entropy value of a given indicator, denoted by n_ij, we need to first determine the share of that indicator in the total sum of all provinces’ values for that indicator. This share is represented by y_ij. y i j = Z i j ∑ i = 1 30 Z i j (2) n i j = − ⁡ ln ⁡ − 1 ∑ i = 1 30 y i j × ⁡ ln y i j (3)

Calculate the weight, r_j, for each indicator j: r j = 1 − n j ∑ j 1 − n j (4)

Build composite index h_i: h i = r 1 y i 1 + r 2 y i 2 + r 3 y i 3 + r 4 y i 4 (5)

Calculate the weight w_i of province i: w i = h i ∑ i = 1 30 h i (6)

Finally, the allocated amount of DEA for each province should be calculated, using C₂₀₂₀ as the total CEA amount in 2020. x i = w i × C 2020 (7)

The SBM model, as one of the DEA (Data Envelopment Analysis) models, not only calculates the input-output ratio of input variables but also considers the redundancy of input variables. Therefore, by taking the allocated CEA as the input variable, the SBM model can assess whether the current CEA allocation scheme has issues of wastefulness based on its redundancy. It is worth noting that this is to examine the rationality of the current allocation scheme, not to optimize it. The SBM formula is represented as (3.8), where x_io denotes the input variable, y_ro represents the expected output, b_ko represents the non-expected output, and s_i represents redundancy. Since data for the Tibet region is missing, this paper considers a total of 30 Decision Making Unit (DMU), and with the assistance of the DEARUN software, the redundancy of CEA for each province is calculated. θ = min λ   s − s + 1 − 1 n ∑ i = 1 n s i − x io 1 + 1 q + h ∑ i = 1 q s r + y ro + ∑ k = 1 h s k − b ko (8) s . t .   x io = ∑ j = 1 n λ j x ij + s i ‐ , i = 1 , 2 , ⋯ , m ;   y ro = ∑ j = 1 n λ j y rj ‐ s r + , r = 1 , 2 , ⋯ , q ;   b ro = ∑ j = 1 n λ j b kj + s k ‐ , k = 1 , 2 , ⋯ , h

As this article adopts a deep-path study of the distribution scheme of China’s CEA, the distribution principles are constructed based on utility, with a primary focus on fairness, efficiency, and a combination of both fairness and efficiency. Therefore, drawing inspiration from indicators used to measure income distribution fairness, the Gini Coefficient of individual utility is selected as the indicator to measure the fairness of CEA distribution. When the Gini Coefficient is 0, fairness is maximized. Based on the research of other scholars (MORADIAN H and KIA, 2021) using variable weight functions in resource allocation domains, the mean of individual utility is chosen as the main indicator to measure the efficiency of CEA distribution. Finally, using Matlab software and the FMINCON function for optimization, the variable weight parameters corresponding to fairness and efficiency values under different conditions are calculated. Ultimately, by computing the ratio of efficiency loss to fairness loss, the variable weight parameters under different distribution principles are determined. The specific formula is given by (Equation 9), where F represents fairness, E represents efficiency, F_F is the optimal solution for fairness, E_E is the optimal solution for efficiency, W_F is the fairness loss, W_E is the efficiency loss, and ρ is the ratio of fairness to efficiency loss. ρ = W E W F (9)

Among them: W E = 1 − E E E W F = 1 − F F F

As this paper adopts a utility-based perspective to construct allocation methods, it utilizes a variable weight function to formulate the objective function for maximizing CEA allocation utility, and thus, establishes a new allocation approach. The CEA objective function is primarily composed of the variable weight function and individual utility (Equation 10), where the variable weight function is represented as the normalized Hardarmard product of the constant weight vector and the state weight vector (Equation 11). The state weight vector is crucial for achieving a reasonable CEA distribution. In CEA allocation issues, governments typically focus more on provinces with higher carbon emissions. Traditional methods, such as allocating based on historical emissions, tend to indirectly encourage provinces with higher historical emissions. However, the state weight vector in the variable weight function considers the variation in weight status, enhancing the importance of smaller entities in decision-making. Therefore, the choice of the state weight vector is critical. Max U = ∑ i = 1 30 v i u i (10)

In Equation 10, U represents the total utility, v_i represents the variable weighting function, and u_i denotes individual utility. V i = W i S i X ∑ j = 1 30 W j S j X (11)

In Equation 11, W_i denotes the constant weight vector and S_i(x) represents the state variable weight vector.

Commonly, there are three types of state weighting vectors, and the specific choice needs to be considered from the perspective of balancing force values. While the importance of individuals with smaller utility is increased using the weighting function, it is also undesirable for the weights after weighting to be overly concentrated on individuals with the smallest state values. Therefore, the first two forms of weighting vectors are not suitable for application in the CEA distribution problem. On the contrary, exponential-type weighting vectors can reflect different degrees of balancing force by setting different weighting parameters. Leveraging these weighting parameters, flexibility in the distribution of CEA among provinces can be achieved. Hence, this paper mainly chooses to use this type of weighting vector to establish the weighting function for CEA distribution.

(1) The empirical formula: S i X = ∏ j = 1 j ≠ i n x j , i = 1 , 2 , ⋯ , n

The weighting function not only needs to determine the state weighting function but also define the constant weight as an indicator. Considering two aspects, on the one hand, due to significant differences in actual carbon emissions among provinces, their demand for CEA also varies. Only when demand is not considered, should the constant weights be equal among provinces, but this is obviously not suitable for CEA distribution. On the other hand, the size of the constant weight also reflects the importance assigned by decision-makers to individuals. As mentioned earlier, the government usually emphasizes provinces with higher carbon emissions due to considerations of the magnitude of emission reduction pressure. Therefore, this paper chooses to use the mean of the actual carbon emissions (D_i) of each province in the past 5 years as the constant weight for each province, as shown in Equation 12.   W i = D i ∑ i = 1 30 D i (12)

In addition, quantifying individual utility is necessary for constructing the CEA objective function, which reflects the impact of CEA distribution on each province, i.e., the satisfaction generated by each province with the allocated CEA. Although surplus CEA can be traded on the carbon emission trading platform to obtain some income with positive benefits (Zhao et al., 2022), the remaining CEA cannot be stored in the long term to offset future carbon emissions. Additionally, an excess of remaining CEA can lead to lower carbon emission permit prices, reducing income from selling. Therefore, CEA exhibits diminishing marginal utility. Hence, a marginal utility diminishing parameter p ∈ (0,1) is introduced. This parameter is challenging to directly obtain, but when using the SBM model, GDP and carbon emissions are considered as expected and non-expected outputs, respectively. Thus, the mean value of the ratio of GDP to carbon emissions in recent years is chosen as the value of the marginal utility p, i.e., the ratio of expected output to non-expected output. Therefore, the specific formula for individual utility is as shown in Equation 13, where u_i represents the utility function for each province, x_i is the actual CEA obtained, c_i is the CEA input-output ratio for each province, m is the total CEA in 2020, and p = 0.8. u i = c i x i p (13) ∑ i = 1 30 x i = m

So, by substituting the exponential weight vector into Equation 11, we obtain Equation 14. Substituting this into Equation 10 and combining it with Equation 13, we get the final CEA objective function based on the weight function, as shown in Equation 15. The constraint is that the sum of the CEA obtained by the 30 provinces in China, excluding Tibet, must not exceed the total CEA for the country, denoted as m. V i = W i e − β x i ∑ j = 1 30 W j e − β x j (14) Max U = ∑ i = 1 30 W i e − β x i ∑ j = 1 30 W j e − β x j × c i x i p (15) s . t .   ∑ i = 1 30 x i ≤ m

Based on the formulas related to the CEA fairness efficiency loss ratio and objective function, the value of the weight parameter β is obtained by comparing the efficiency principle, fairness principle, and the principle of fairness and efficiency. While the weighting function in the fairness distribution can also be used as a weight for predicting the CEA distribution, it is essential to consider that in the original weighting function, the β parameter is inversely proportional to the weighting function. Additionally, small changes in β can lead to a considerable change in the state weighting vector, causing a severe two-level differentiation of the weighting function, which does not align with the principle of fair distribution from the perspective of distribution differences. Therefore, in calculating the distribution under the fairness principle, the original weighting function formula is adjusted to the form of W_i × [1 - S_i(X)]. For the principles of fairness and efficiency, both their weighting functions should be used, leading to the issue of the weights of the two. If a uniform weight of 0.5 is adopted, it is meaningless. However, as this study obtains the β values for fairness and efficiency separately in the second step, an attempt is made to use the respective β values’ proportions as weights. Given that the efficiency parameter proportion is generally smaller, multiplying it with the corresponding β value helps avoid the problem of significant differences in distribution. Therefore, the final formula for predicting CEA distribution is as shown in Equation 16. X l = M × W i S i X ∑ j = 1 30 W j X   E f f i c i e n c y   P r i n c i p l e X l = M × W i 1 − S i X ∑ j = 1 30 W j 1 − S j X   F a i r n e s s   P r i n c i p l e X l = M × W i S i X ∑ j = 1 30 W j S j X × β E β E + β F + W i 1 − S i X ∑ j = 1 30 W j 1 − S j X × β F β E + β F F a i r n e s s   a n d   E f f i c i e n c y   P r i n c i p l e s (16)

In order to predict the total carbon emissions in China for the year 2030, a basic analysis revealed a linear correlation between GDP and time. Therefore, for predicting the total carbon emissions, this study opted to use the most widely-used model in time series forecasting - ARIMA, to forecast the GDP in 2030. Furthermore, taking into account the significant pressure China faces in reducing carbon emissions, this study set the target for reducing carbon intensity at 60%.The formula is presented as (3.17), where M represents the predicted CEA total, and I represents carbon intensity. M = GD P 2030 × I 2005 1 − 60 % (17)