*2.2. DEA Slack-Based Measure Model*

With the presence of undesirable outputs, there is one popular DEA model that can deal with this problem called Slack-based measure model (SBM), which was proposed by Tone in 2003 [29] and extended the SBM model proposed in 2001 by Tone [30]. In this study, we followed the equations proposed by Tone (2003) and are detailed as follows.

Suppose that n represents the number of decision-making units (DMUs) and each DMU has inputs, desirable outputs, and undesirable outputs. 

Let us decompose the output matrix Y into -Yg, Yb , where Yg, Yb denote good (desirable) and bad (undesirable) output matrices, respectively. For a DMU (*x*o, *y*o), the decomposition is denoted as - *x*o, *y* g o, *y*<sup>b</sup> 0 .

The production possibility set is defined by:

$$\mathbf{P} = \left\{ (\mathbf{x}, \mathbf{y}^{\mathbf{g}}, \mathbf{y}^{\mathbf{b}}) \Big| \mathbf{x} \ge \mathbf{X}\boldsymbol{\lambda}, \mathbf{y}^{\mathbf{g}} \le \mathbf{Y}^{\mathbf{g}}\boldsymbol{\lambda}, \mathbf{y}^{\mathbf{b}} \ge \mathbf{Y}^{\mathbf{b}}\boldsymbol{\lambda}, \mathbf{L} \le \mathbf{e}\boldsymbol{\lambda} \le \mathbf{U}, \boldsymbol{\lambda} \ge \mathbf{0} \right\} \tag{8}$$

where λ is the intensity vector while L and U are the lower and upper bounds of the intensity vector, respectively. Then, a DMU - *x*o, *y* g o, *y*<sup>b</sup> o is efficient in the presence of bad outputs, if there is no vector - *x*, *y*g, *y*<sup>b</sup> ∈ P such that *x*<sup>o</sup> ≥ *x*, *y* g <sup>o</sup> <sup>≤</sup> *<sup>y</sup>*g, *<sup>y</sup>*<sup>b</sup> <sup>o</sup> <sup>≥</sup> *<sup>y</sup>*<sup>b</sup> with at least one strict inequality.

In accordance with this definition, we modified the SBM in Tone (2001) as follows.

$$\lceil SBM \rceil \rho^\* = \min \frac{1 - \frac{1}{m} \sum\_{i=m}^m S\_i^-}{1 + \frac{1}{S\_1 S\_2} \left( \sum\_{r=1}^{S1} \frac{S\_r^3}{y\_{rv}^3} + \sum\_{r=1}^{S2} \frac{y\_r^b}{y\_{rv}^b} \right)} \tag{9}$$

Subject to

*x*<sup>o</sup> = *X*λ + *S*<sup>−</sup> *y g* <sup>o</sup> = *Yg*λ − *s<sup>g</sup> yb* <sup>o</sup> = *Yb*λ + *S<sup>b</sup>* L ≤ eλ ≤ U *S*<sup>−</sup> ≥ 0, *S<sup>g</sup>* ≥ 0, *S<sup>b</sup>* ≥ 0, λ ≥ 0

where vector *<sup>S</sup>*<sup>−</sup> <sup>∈</sup> *<sup>R</sup>m*: excesses in inputs and *Sb* <sup>∈</sup> *RS*2: excesses in bad outputs and *<sup>S</sup><sup>g</sup>* <sup>∈</sup> *RS*<sup>1</sup> is the shortage in the good outputs. With the presence of bad outputs, the DMU - *x*o, *y* g o, *y*<sup>b</sup> o is efficient if and only if ρ<sup>∗</sup> = 1, *i*.*e*., *S*−∗ = 0, *Sg*<sup>∗</sup> = 0, and *Sb*<sup>∗</sup> = 0.

## **3. Empirical Results**
