*3.3. Undesirable Outputs Model*

The undesirable outputs model is utilized to calculate the performance of DMUs when its outputs obtain undesirable outputs. In this study, the undesirable outputs model is applied to deal with good (desirable) and (bad) (undesirable) outputs. We utilize an undesirable outputs model to compute the efficiency of the electrical energy consumption in 42 countries. The DMUs are the 42 countries, these countries are set up n DMU (*<sup>a</sup>*0, *b*0) (*n* = 1, 2, ... , *s*). Let the input factor be A, desirable factor (*B<sup>d</sup>*), and undesirable factor (*Bu*). Then, the production possibility is given by:

$$P = \left\{ \left( a, b^d, b^u \right) , a \ge X\lambda; b^d \le B^d\lambda; b^u \ge B^u\lambda; L \le e\lambda \le U, \lambda \ge 0 \right\} \tag{1}$$

The intensity vector is *λ*, it means that the above definition corresponds to the constant return to scale technology [41], and the lower and upper bounds of the intensity vector are *L* and *U*, respectively (*e* = (1, ... 1) ∈ *R*+, *L* ≤ 1, *U* ≥ 1). There is at least one strict inequality when formulating the efficiency of one DMU (*<sup>a</sup>*0,*b<sup>d</sup>* 0, *bu* 0 ) without vector (*<sup>a</sup>*0,*b<sup>d</sup>* 0, *bu* 0 ) ∈ *P* and *a*0 ≥ *a*, *bd* 0 ≤ *<sup>b</sup>d*, *bu* 0 ≥ *bu*. According to the SBM of Tone [42], the objective function of the undesirable model is formulated as follows:

$$\rho^{\*} = \min \frac{1 - \frac{1}{k} \sum\_{i=1}^{k} \frac{s\_i^{-}}{a\_{i0}}}{1 + \frac{1}{s} \left(\sum\_{r=1}^{s\_1} \frac{s\_r^{d}}{b\_{ro}^{d}} + \sum\_{r=2}^{s\_1} \frac{s\_r^{u}}{b\_{ro}^{u}}\right)} \tag{2}$$

Subject to:

> *a*0 = *A λ* + *s*<sup>−</sup> *bd* 0 = *Bλ* − *sd bu* 0 = *Bλ* + *s<sup>u</sup> <sup>s</sup>*<sup>−</sup>,*sd*,*su*, *λ* ≥ 0.

The excess in inputs, bad outputs and shortages in good outputs are *<sup>s</sup>*<sup>−</sup>, *<sup>s</sup>u*, *sd*, respectively. The number of factors in *s<sup>u</sup>* and *sd* are *s*1 and *s*2, respectively, and *s* = *s*1 + *s*2. Using an optimal solution as *ρ*<sup>∗</sup>, *s*<sup>−</sup>\* , *sd*\* and *su*\* for determining the efficiency of country by undesirable outputs when *ρ*∗ = 1, *s*<sup>−</sup>\* = 0, *sd*\* = 0, and *su*\* = 0. When the DMU is inefficient, *ρ*∗ can be improved in order to become

efficient by moving the excesses in inputs and bad outputs, simultaneously increasing the shortfalls in good outputs [42] as follows:

$$\begin{aligned} a\_0 - s^{-\*} &\Rightarrow \; a\_0\\ b\_0^d + s^{d\*} &\Rightarrow \; b\_0^d\\ b\_0^u - s^{u\*} &\Rightarrow \; b\_0^u \end{aligned} \tag{3}$$

The above program was transformed into an equivalent linear program by Charnes and Cooper [43]. Let the dual variable vectors be *x*, *yd*, *y<sup>u</sup>*. Based on the dual side of the linear program, the dual program in the variable *x*, *yd*, *yu* for constant return to scale [30] is defined as below:

$$
\mathbf{x} \mathbf{a} \mathbf{x} \mathbf{y}^d b\_0^d - \mathbf{x} a\_0 - \mathbf{y}^u b\_0^u. \tag{4}
$$

Subject to:

$$\begin{aligned} &y^d B^d - \mathbf{x}A - y^\mathbf{u} B^\mathbf{u} \le 0\\ &\mathbf{x} \ge \frac{1}{k} \begin{bmatrix} \frac{1}{a\_0} \end{bmatrix} \\ &y^d \ge \frac{1 + y^d b\_0^d - \mathbf{x}a\_0 - y^\mathbf{u} b\_0^\mathbf{u}}{s} \begin{bmatrix} 1\\ \frac{1}{b\_0^d} \end{bmatrix} \\ &y^\mathbf{u} \ge \frac{1 + y^d b\_0^d - \mathbf{x}a\_0 - y^\mathbf{u} b\_0^\mathbf{u}}{s} \begin{bmatrix} \frac{1}{R\_0^d} \end{bmatrix} \end{aligned}$$

The virtual prices of inputs, desirable and undesirable outputs are replaced by the dual variables *x*, *yd*, *yu* respectively. The profit *ydbd* − *xa* − *yubu* [30] does not exceed zero for every DMU, and the profit *<sup>y</sup>dbd*0 − *xa*0 − *yubu*0 for the DMU concerned when the dual program aims at obtaining the optimal virtual costs and prices for each DMU.

In addition, we set *w*1 ∈ *R*+, *w*2 ∈ *R*<sup>+</sup> as the weights of desirable and undesirable outputs, respectively. The weights of bad and good outputs are converted to relative weights with their mathematical expression [30] as follows:

$$\rho^\* = \min \frac{1 - \frac{1}{k} \sum\_{i=1}^k \frac{s\_{i\mu}^-}{a\_{io}}}{1 + \frac{1}{k} \left(\mathcal{W}\_1 \sum\_{r=1}^{s\_1} \frac{s\_r^d}{b\_{ro}^d} + \mathcal{W}\_2 \sum\_{r=1}^{s\_2} \frac{s\_r^u}{b\_{ro}^d}\right)}}. \tag{5}$$

Subject to:

$$W\_1 = \frac{sw\_1}{w\_1 + w\_2}.$$

$$W\_2 = \frac{sw\_2}{w\_2 + w\_1}.$$

$$(w\_1 \ge 0, w\_2 \ge 0).$$

Consequently, if *ρ*∗ < 1, the country is inefficient so the excesses in inputs and undesirable outputs must be removed, and the shortfalls in desirable outputs must be increased. A country reaches efficiency when *ρ*∗ = 1.
