Carbon emission allocation in China based on gradually efficiency improvement and emission reduction planning principle

Abstract

Carbon emission allocation in China is one of the most important research fields in recent years. The existing carbon allocation approaches are primary based on efficiency maximized principle or efficiency invariance principle. Such two principles, however, are both extreme cases which are particularly difficult to realize in reality. In this study, we proposed an alternative approach based on gradually efficiency improvement planning and emission reduction planning principles. Based on them, the efficiency is planning to be steadily improved after allocating carbon emission year by year rather than achieve the maximized target (efficient) in a single step or maintain the original efficiency without any improvement. Meanwhile, the total carbon emission is also planning to be reduced by a certain percent according to the promise by China in Copenhagen climate conference. Finally, the proposed method is applied to allocate the carbon emission quotas among provinces in China, and some realistic conclusions are obtained to guide in reality.

Appendices

Appendix A: The proof of Theorem 1

Theorem 1

To each DMU$_{d}$, the efficiency based on the gradually efficiency improvement principle ($\rho _d -\rho _d e_d^*+e_d^*$) is a lower bound of CCR efficiency after carbon emission allocated ($e_{d}^{allo\_CCR}$), that is, $\rho _d -\rho _d e_d^*+e_d^*\le e_{d}^{allo\_CCR}$.

Proof

According to the Theorem 1, the CCR efficiency $e_{d}^{allo\_CCR}$ with the allocated carbon emission is calculated by

$$\begin{aligned} \begin{array}{l} e_d^{allo\_CCR} =M\hbox {a}x\;\frac{\sum _{r=1}^s {\mu _r } y_{rd} }{\sum _{i=1}^m {\upsilon _i } x_{id} } \\ s.t.\;\left\{ {\begin{array}{l} \sum \limits _{r=1}^s {\mu _r } y_{rj} -\left( \sum \limits _{i=1}^m {\upsilon _i } x_{ij} +R_j^*\right) \le 0,\quad \forall j \\ \mu _r, \upsilon _i \ \ge 0,\quad \forall r,i \\ \end{array}} \right. \quad \\ \end{array}\quad \quad \quad \quad \quad \qquad (\hbox {A.1}) \end{aligned}$$

where $R_{j}^{*}$ is the optimal allocated carbon emission in the model (6).

Assume $(u_{r}^*,v_{i}^*,R_{r}^*)(\forall r,i,j)$ is the optimal solution of the model (6), so it satisfies the first constraint of the model (4) as follows:

$$\begin{aligned} \sum _{r=1}^s {u_r^{*} } y_{rj} =(\rho _j -\rho _j e_j^*+e_j^*)\left( \sum _{i=1}^m {v_i^{*} } x_{ij} +R_j^*\right) ,\quad \forall j \end{aligned}$$

From section 3.1, we know that $\rho _d -\rho _d e_d^*+e_d^*\le 1$. Therefore, we have

$$\begin{aligned} \sum _{r=1}^s {u_r^{*} } y_{rj} \le \left( \sum _{i=1}^m {v_i^{*} } x_{ij} +R_j^*\right) \quad \forall j, \end{aligned}$$

which indicates that $(u_r^*, v_i^*, R_j^*),(\forall r,i,j)$ is also the feasible solution of the model (A.1). As the objective function of the model (A.1) is the maximized function, we can obtain that

$$\begin{aligned} \rho _d -\rho _d e_d^*+e_d^*=\frac{\sum _{r=1}^s {u_r^*y_{rd} } }{\sum _{i=1}^m {v_i^*x_{id} +R_d^*} }\le Max\frac{\sum _{r=1}^s {\mu _r y_{rd} } }{\sum _{i=1}^m {\upsilon _i x_{id} +R_{d}^*} }=e_{d}^{allo\_CCR}. \end{aligned}$$

which completes the proof of the Theorem 1. $\square$

Appendix B: An algorithm to solve model (6)

Model (6) is a multi-objective programming, so we present an algorithm to solve model (6) to obtain a unique carbon emission allocation inspired by Li et al. (2013).

Let $\mathop {\max }\limits _{1\le d\le n}R_d=\varphi$, then model (6) can be transformed into the following model:

$$\begin{aligned} \begin{array}{l} \mathop {Min}\limits _{u,v} \varphi \\ \hbox {s.t.} R_j \le \varphi ,\quad \forall j \sum \limits _{j=1}^n {u_r y_{rj} } =(\rho _j -\rho _j e_j^*+e_j^*)\left( \sum \limits _{i=1}^m {v_i x_{ij} } +R_j \right) \quad \forall j\\ \sum \limits _{j=1}^n {R_j } =R_0 (1-\alpha )^{t}\\ u_r, v_i, R_j \ge 0,\forall r,i,j\\ \end{array} \end{aligned}$$

(7)

Step 1 Let l=1, the optimal solution to model (7) is represented by ($R_{1j}^*$, $u_{1r}^*$, $v_{1i}^*$, $\forall j,r,i)$. If $R_j =\varphi _1^*$, then denote the DMU set with a same allocated carbon emission $\varphi _1^*$ as $J_1 =\{j|R_j =\varphi _1^*,\;\forall j\in J_0 \},$ then $R_j^*=R_{1j}^*,\;\forall j\in J_1$ and the number of DMUs with the same allocated carbon emission $\varphi _1^*$ is denoted by $n_{1}$. Then the other DMUs form a set denoted as $J_2 =\{j|R_j <\varphi _1^*,\;\forall j\in J_0 \}$.Apparently, $J_0 =J_1 \cup J_2 ,{}\;J_0 =\{1,2,\ldots ,n\}$ holds. If $n_1 =m+s$, then the procedure stops and the optimal solution ($R_{1j}^*$, $u_{1r}^*$, $v_{1i}^*$, $\forall j,r,i)$ is unique. The reason can be seen in Li et al. (2013). If $n_1 <m+s$, then go to step 2.

Step 2 Let $l=l+1$, calculate the following general model:

$$\begin{aligned} \begin{array}{l} \varphi _l^*=Min\varphi \\ R_j =\varphi _1^*,\;j\in J_1 \\ R_j =\varphi _2^*,\;j\in J_3 \\ \vdots \\ R_j =\varphi _{l-1}^*,\;j\in J_{2l-3} \\ R_j \le \varphi ,\;j\in J_{2l-2} \\ \sum \limits _{j=1}^n {u_r y_{rj} } =\left( \rho _j -\rho _j e_j^*+e_j^*\right) \left( \sum \limits _{i=1}^m {v_i x_{ij} } +R_j \right) \quad \forall j \\ \sum \limits _{j=1}^n {R_j } =R_0 (1-\alpha )^{t} \\ u_r, v_i, R_k \ge 0,\forall r,i,k \\ \end{array} \end{aligned}$$

(8)

From model (8), the optimal ($R_{lj}^*,u_{lr}^*,v_{li}^*,\forall j,r,i$) are calculated. Similarly, $J_{2l-2}$ can also be grouped into two subsets: $J_{2l-1} =\{j|R_j =\varphi _l^*,\;\forall j\in J_{2l-2} \}$, then $R_j^*=R_{lj}^*,\;\forall j\in J_{2l-1}, \quad J_{2l} =\{j|R_j <\varphi _l^*,\;\forall j\in J_{2l-2} \}=J_{2l-2} -J_{2l-1}$. In addition, the numbers of DMUs with the same allocated carbon emission $\varphi _1^*$ is denoted as $n_{l}$. If $n_1 +n_2 +\cdots +n_l <m+s$, then run step 2 again. Until $n_1 +n_2 +\cdots +n_l =m+s$, the procedure stops and the optimal solution ($R_{lj}^*$, $u_{lr}^*$, $v_{li}^*$, $\forall j,r,i$) is obtained uniquely. The reason can also be seen in Li et al. (2013). As a consequence, the carbon emission allocation can be uniquely determined as ($R_j^*=R_{lj}^*,\;\forall j\in J_0$).