## 3 Methodology
### Slack-Based Measure (SBM) DEA Model  
DEA is a kind of non-parametric analysis tool that can be used to assess the relative efficiency of entities, termed as decision-making units (DMUs). A DMU is considered to be efficient if it can produce more outputs with the same or fewer number of inputs. However, a conventional DEA model does not consider undesirable outputs such as air pollution and hazardous materials. Thus, the conventional DEA model cannot be used in this present research because undesirable outputs will be included.  
The Slacks-based measure (SBM) DEA model proposed in Tone [25] was found to be unable to deal with undesirable outputs [40]. The SBM DEA is a kind of non-radial and non-oriented model that uses input and output slacks to measure efficiency directly. To deal with undesirable outputs, the SBM DEA was modified in reference [41], and two variants of SBM DEA, i.e., Bad-Output and Non-Separable models, were proposed. The Bad-Output model deals with good (desirable) and bad (unwanted) outputs independently, whereas the Non-Separable model assumes that connection exists between good inputs and bad outputs. The Non-Separable model assumes that reducing unwanted outputs will reduce desirable output due to their connection. In this research, the Bad-Output model is used as there is no connection between desirable outputs and undesirable outputs.  
In reference [41], each DMU in the SBM is assumed to have the three factors--inputs, good (desirable) outputs, and bad (undesirable) outputs. The three factors are represented by the three vectors \(\mathsf{x}\in R^{m}\), \(Y^{S}\in\ R^{s1}\), and \(Y^{b}\in\ R^{s2}\), respectively.  
A DMU can be represented as \(\left(x_{o},y_{o}^{S},y_{o}^{b}\right)\) and evaluated by two kinds of output (desirable and undesirable), where \(y_{o}^{S}\) denotes the desirable output and \(y_{o}^{b}\) denotes the undesirable output. The production possibility set \(\mathrm{P}\) is defined as follows.  
\[\mathrm{P}\ =\ \left\{\left(x,y^{S},y^{b}\right)\right|\ x\ \geq X\lambda,\ y^{S}\ \leq\ Y^{S}\ \lambda,\ y^{b}\ \geq\ Y^{b}\lambda,\ L\ \leq e\lambda\ \leq U,\ \lambda\ \geq 0\right\}\]  
where \(\lambda\) is the intensity vector and \(L\) and \(U\) are the lower and upper bounds of the intensity vector, respectively.  
A DMU \(\left(x_{o},y_{o}^{S},y_{o}^{b}\right)\) is efficient if there is a vector \(\left(x_{o},y_{o}^{S},y_{o}^{b}\right)\in\mathrm{P}\) such that \(x_{0}\geq x,\ y_{o}^{S}\leq y^{S},\ y_{o}^{b}\geq y^{b}\) with at least one strict inequality. The SBM proposed by [PERSON] [25] is then modified in Equation (1).  
\[\left[SBM\right]\ \rho^{*}\ =\ \min\ \frac{1-\frac{1}{m}\sum_{i=1}^{m}\ \frac{s_{i _{0}}^{-}}{x_{o}}}{1+\frac{1}{s}\!\left(\sum_{r=1}^{s_{1}}\frac{S_{r}^{S}}{y_ {o}^{S}}\ +\ \sum_{r=1}^{s_{2}}\frac{S_{r}^{b}}{y_{o}^{b}}\right)} \tag{1}\]subject to  
\[\begin{array}{l}x_{o}=\ X\lambda\ +\ S^{-}\\ y_{o}^{g}\ =\ Y\lambda-S^{g}\\ y_{o}^{b}=\ Y\lambda\ +\ S^{b}\\ L\ \leq\ e\lambda\ \leq\ U\\ S^{-},\ S^{g},\ S^{b},\ \lambda\ \geq 0.\end{array}\]  
The vectors \(S^{-}\) and \(S^{b}\) refer to the excesses in inputs and undesirable outputs, and \(S^{g}\) represents the shortage in desirable outputs. The \(DMU_{o}\) is efficient with consideration of bad output if and only if \(\rho^{*}=1\) or in the other words, \(S^{-}\)= 0, \(S^{g}\)= 0 and \(S^{b}\)= 0.