# A Regret-Enhanced DEA Approach to Mapping Renewable Energy Efficiency in Asia's Growth Economies
## 2 Methodology
### Self-Efficiency Data Envelopment Analysis  
The basic idea of DEA is to find a way to compare the efficiency of different decision-making units (DMUs). This is carried out by constructing a production frontier, which is the set of all DMUs that are technically efficient. A DMU is said to be technically efficient if it cannot produce more output without using more inputs or produces the same output with fewer inputs. According to CRR (Charnes, Cooper, and Rhodes) model, based on \(n\) inputs (\(i=1\ldots I\)) and \(m\) outputs (\(j=1\ldots J\)), the self-technical efficiency \((E_{k})\) of DMU \(kth\) (\(k=1\ldots K\)) is estimated by solving the model in Equation (6) [56]. In this model, \(y_{jk}\) and \(x_{ik}\) express the \(jth\) output and \(ith\) input of kth DMU, respectively. In addition, \(v_{jk}\) and \(u_{ik}\) express the weights with respect to the \(jth\) output and the \(ith\) input of the \(k\)th DMU.  
\[\max\ E_{k}=\frac{\sum_{j=1}^{J}v_{jk}v_{jk}}{\sum_{i=1}^{J}u_{ik}x_{ik}}\] (6) _subject to_ \[\frac{\sum_{j=1}^{J}v_{jk}v_{jk}}{\sum_{i=1}^{J}u_{ik}x_{ik}}\leq 1 \qquad\qquad l=1,\ldots,K\] \[u_{ik}\geq 0 \qquad\qquad i=1,\ldots,I;\ k=1,\ldots,K\] \[v_{jk}\geq 0 \qquad\qquad j=1,\ldots,J;\ k=1,\ldots,K\]  
The CCR model is a simple and intuitive way to measure the efficiency of DMUs. However, it has some limitations. One limitation is that it assumes constant returns to scale (CRS). This may not be realistic in all cases, as some DMUs may be able to produce more output by increasing inputs at a decreasing rate. Therefore, another DEA model is proposed to estimate the efficiency of DMU based on slack measurement by [PERSON] [57]. The slack-based measure (SBM) model, as formulated in Equation (7), assumes CRS, similar to the CCR model. However, a variable returns to scale (VRS) version of the SBM model can be obtained by incorporating a convexity constraint on the intensity variables, as introduced in [PERSON] (2002) [58]. This makes it a better choice for situations where the assumption of CRS is not realistic. By solving the model in Equation (7), the \(kth\) is considered SBM-efficient if \(\beta=1\).  
\[\min\ \beta=\frac{\frac{1-\frac{1}{2}\sum_{i=1}^{l}\frac{s_{i}^{-}}{s_{k}^{-}}}{1 +\frac{1}{2}\sum_{j=1}^{l}\frac{s_{j}^{-}}{j_{jk}}}}{\sum_{i=1}^{l}\frac{s_{j} ^{-}}{j_{jk}}} \tag{7}\]  
_subject to_  
\[\begin{array}{c}x_{ik}=\sum\limits_{i=1}^{M}\sigma_{k}x_{ik}+s_{i}^{-}\\ y_{jk}=\sum\limits_{i=1}^{M}\sigma_{k}y_{jk}+s_{j}^{+}\\ \sigma_{k},s_{i}^{-},s_{j}^{+}\geq 0\end{array}\qquad\begin{array}{c}i=1, \ldots,I\\ j=1,\ldots,J\end{array}\] \[\begin{array}{c}\sigma_{k},s_{i}^{-},s_{j}^{+}\geq 0\end{array}\qquad \begin{array}{c}\forall i\in I,j\in J,k\in K\end{array}\]  
Here, \(s_{i}^{-}\), and \(s_{j}^{+}\) express the negative slack (\(s_{i}^{-}\)) and positive slack (\(s_{j}^{+}\)) variables, respectively. The \(\sigma\) is a non-negative vector in \(\mathbb{R}^{k}\) to modify the production possibility set. However, these traditional DEA models sometimes provide an efficiency value of 1 for multiple DMUs at the same time.  
Because the model in Equation (6) is a nonlinear programming model, it was converted into a linear programming model by [PERSON] and [PERSON] as follows [59]:  
\[\max\ E_{kk}=\sum_{j=1}^{l}v_{jk}y_{jk}\]  
_subject to_  
\[\begin{array}{c}\sum_{i=1}^{l}u_{ik}x_{ik}=1\\ \sum_{j=1}^{J}v_{jk}y_{jl}-\sum_{i=1}^{l}u_{ik}x_{il}\leq 0\\ u_{ik}\geq 0\\ v_{jk}\geq 0\end{array}\qquad\begin{array}{c}i=1,\ldots,I;\ k=1,\ldots,K\\ j=1,\ldots,J;\ k=1,\ldots,K\end{array} \tag{8}\]  
By solving the model in Equation (8), the optimized value of the objective function \(E_{kk}^{*}\) is the self-efficiency value of \(kth\) DMU with the optimal solution \(v_{jk}^{*}\) and \(u_{ik}^{*}\).