Carbon Emissions Abatement (CEA) Allocation Based on Inverse Slack-Based Model (SBM)

Abstract

Carbon emissions abatement (CEA) is an important issue that draws attention from both academicians and policymakers. Data envelopment analysis (DEA) has been a popular tool to allocate the CEA, and most previous works are based on radial DEA models. However, as shown in our paper, these models may give biased results due to their ignorance of slackness. To avoid such problems, we propose an allocation model based on the slack-based model and multiple-objective nonlinear programming to find the CEA allocation plan, which can minimize the GDP loss. The property of nonconvexity makes the model difficult to solve. Thus, we construct an approximation algorithm to solve this model with guaranteed error bounds and complexity. In the empirical application, we take regions of china as an illustrative example and find there is a significant region gap in China. Hence, we group the regions into eastern, central, and western, and give the main results, as well as the superiority of our allocation models compared with radial models.

Appendices

Appendices

Inverse SBM Algorithm for Special Case $r=1$

Corollary A.1

Suppose that $V(\beta )$ is $\alpha$-strongly convex, then $\Vert {\hat{\beta }} - \beta ^*\Vert _1 \leqslant \frac{\varepsilon _0\Vert {\hat{\beta }}\Vert _1}{\alpha }$.

When $r = 1$, we do not have to move $(\alpha _o,\beta _k)$ on the surface of the feasible region and there exists another effective algorithm to solve model (2.9). Since $r = 1$, the MONLP is actually a nonlinear programming with a single object. For any fixed $\alpha _o$, there is an unique $\beta$ such that $\mathrm{eff}(\alpha _o,\beta ) = \theta _o$. Furthermore, by Theorem 2.1, for each fixed $\alpha _o$ the efficiency $\text {eff}(\alpha _o,y)$ is monotone. That shows that finding the root for $\text {eff}(\alpha _o,\beta ) = \theta _o$, which can be solved by bisection method, is not difficult. The detailed algorithm is presented in Algorithm 3.

It is clear that to find an $\varepsilon$ accurate solution, we need at most $\log \varepsilon$ steps. Therefore, we have the estimation for the complexity of Algorithm 3:

Proposition A.2

Assume that Assumption 2.2 holds. Let $N_1$ denote the complexity for solving LP, then the complexity for Algorithm 3 to find an $\varepsilon$ accurate solution is at most $\mathcal {O}(N_1\log \varepsilon )$.

Technical Proofs

1.1 Proof of Theorem 2.1

Proof

(2.1) can be transformed into the (B.1)

$$\begin{aligned} \begin{aligned} \min&\quad \theta (x_0,y_0,\lambda )=\frac{\frac{1}{m}\sum \nolimits _{i = 1}^m \frac{(X\lambda )_i}{x_{0i}}}{\frac{1}{r} \sum \nolimits _{i = 1}^r \frac{(Y\lambda )_i}{y_{0i}}}\\ \text {s.t. }&\quad x_0=X\lambda +s^-,\\&\quad y_0=Y\lambda -s^+,\\&\quad \lambda \in \Lambda ,\quad s^-\geqslant 0,\quad s^+\geqslant 0. \end{aligned} \end{aligned}$$

(B.1)

Denote the $\lambda$ which minimize$(\alpha _1,\beta )$ as $\lambda _1$. Due to $\alpha _1\leqslant \alpha _2$ and $X>0$, if $\alpha _1\ne \alpha _2$ we have

$$\begin{aligned} \theta (\alpha _1,\beta ,\lambda _1)>\theta (\alpha _2,\beta ,\lambda _1)\geqslant \min {\theta (\alpha _2,\beta ,\lambda )}. \end{aligned}$$

Thus we have

$$\begin{aligned} \mathrm{eff}(\alpha _1,\beta ) > \mathrm{eff}(\alpha _2,\beta ). \end{aligned}$$

Thus we complete the proof.

1.2 Proof of Proposition 2.4

Proof

When $(\alpha _o,\beta _o)$ is a solution to multi-objective nonlinear programming (2.8), then it is at least a Pareto solution. Thus assume that $\mathrm{eff}(\alpha _o,\beta _o)<\theta _o$, then there exists $\varepsilon >0$ and $e_1>0,e_1\in {\mathbb {R}}^m, e_2>0, e_2\in R_r$, such that either $\mathrm{eff}(\alpha _o - \varepsilon e_1,\beta _o) \leqslant \theta _o$ or $\text {eff}(\alpha _o,\beta _o+\varepsilon e_2) \leqslant \theta _o$ will be correct, which contradicts to the fact that $(\alpha _o,\beta _o)$ is a Pareto solution to multi-objective nonlinear programming.

Besides, let $(\alpha _o,\beta _o)$ be a pair of input and output with $\beta \leqslant y_o$, let w be the supporting vector of $(\alpha _o,\beta _o)$. Then $(\alpha _o,\beta _o)$ is the solution of the nonlinear programming with weight w.

1.3 Proof of Theorem 2.5

Proof

Since our algorithm stops at $(\alpha _o,{\hat{\beta }})$, for all $1\leqslant i,j\leqslant r$, and ${\hat{\beta }}_i \leqslant (y_o)_i$, $\frac{g_j^-}{g_i^+} \leqslant 1+\varepsilon _0$. This indicates that the KKT violation $\Vert \inf _{\mu _1,\mu _2} \mathcal {L}({\hat{\beta }}, \mu _1,\mu _2)\Vert _1 \leqslant \varepsilon _0\Vert {\hat{\beta }}\Vert _1$.

1.4 Proof of Corollary A.1

Proof

This can be directly derived from the definition of strongly convex function.

1.5 Proof of Proposition A.2

Proof

This result can be directly proved by the monotonousness of $\text {eff}(\alpha _o,\beta )$ for any fixed $\alpha _o$. Since we choose bisection to find a ${\hat{\beta }}$ such that $|\text {eff}(\alpha _o,{\hat{\beta }}) - \theta _o| \leqslant \varepsilon _0$, the iterations we take is $\log _2(\varepsilon _0)$. Furthermore, since $\text {eff}(\alpha _o,\beta )$ is Lipschitz continuous with constant $M_1$, we can prove that $|\beta ^* - {\hat{\beta }}| \leqslant M_1\varepsilon _0$ when we take $\log _2(\varepsilon _0)$ iterations. Finally, since we only have to compute one linear programming in each iteration, the total complexity is at most $\mathcal {O}(N_1\log \varepsilon )$.

1.6 Proof of Theorem 3.2

Proof

First we will prove that when $\varepsilon _0 \rightarrow 0$, the solution of Algorithm 2 will converge to the minimizer of model (3.1). By Assumption 2.2, the LES $R_{\theta _i}$ is convex; then $G(\alpha _1,\cdots ,\alpha _n)$ is convex. Assume that the solution converges to $\bar{\alpha _1^*},\cdots ,\bar{\alpha _n^*}, \bar{\beta _1^*},\cdots ,\bar{\beta ^*_n}$; then we can construct a direction $\Delta _1,\cdots ,\Delta _n$ which is a descent direction for $\tilde{\alpha _1^*},\cdots ,\tilde{\alpha _n^*}$. However, by the choice of $\tilde{\alpha _1^*},\cdots ,\tilde{\alpha _n^*}$, the direction is non-decreasing, which contradicts to our previous assumption.

Besides, since in each step we apply difference to estimate the gradient, the relative error is $\mathcal {O}(\varepsilon _0^2)$ in each step. Then the total relative error is $\mathcal {O}(\varepsilon _0)$.

1.7 Proof of Proposition 3.3

Proof

Let $(x_1,.y_1)$, $(x_2,y_2)$ be two point with $\text {eff}(x_1,y_1) = \text {eff}(x_2,y_2) = \theta$. Then

$$\begin{aligned} \begin{aligned} y_1&= \frac{\theta Y\lambda _1}{X\lambda _1}x_1,\\ y_2&= \frac{\theta Y\lambda _2}{X\lambda _2}x_2,\\ \end{aligned} \end{aligned}$$

(B.2)

Let $y^* = \frac{y_1+y_2}{2}$, $x^* = \frac{x_1+x_2}{2}$. Assume that $\text {eff}(x^*,y^*) > \theta$, we have

$$\begin{aligned} \begin{aligned} y^*> \frac{\theta Y \lambda _1}{X\lambda _1} x^*,\\ y^* > \frac{\theta Y \lambda _2}{X\lambda _2} x^*.\\ \end{aligned} \end{aligned}$$

(B.3)

Besides, we can prove that

$$\begin{aligned} \begin{aligned} \frac{x_1}{x_1+x_2}y^*> \frac{\theta Y \lambda _1}{X\lambda _1} \frac{x_1}{2},\\ \frac{x_2}{x_1+x_2}y^* > \frac{\theta Y \lambda _2}{X\lambda _2} \frac{x_2}{2}.\\ \end{aligned} \end{aligned}$$

(B.4)

Add these two term together, we have

$$\begin{aligned} y^* > \frac{y_1+y_2}{2} = y^*, \end{aligned}$$

(B.5)

which shows that $\text {eff}(x^*,y^*) \leqslant \theta$ and completes the proof.

Empirical Application

See Tables 3 4, 5, 6, 7, 8, 9, 10 and 11.

Table 3 Original data of 30 regions in China in 2016

Table 4 Comparison between radial model and SBM under CRS without grouping

Table 5 Comparison between radial model and SBM under VRS without grouping

Table 6 Comparison between radial model and SBM under CRS in Eastern China

Table 7 Comparison between radial model and SBM under CRS in Central China

Table 8 Comparison between radial model and SBM under CRS in Western China

Table 9 Comparison between radial model and SBM under VRS in Eastern China

Table 10 Comparison between radial model and SBM under VRS in Central China

Table 11 Comparison between radial model and SBM under VRS in Western China