*3.1. The TOPSIS Method*

The TOPSIS method, originated by Hwang and Yoon [19], is a very prominent and frequently used MCDM method. Compared to other MCDM methods, this method has a characteristic approach to determine the most acceptable alternative and is based upon the concept that an alternative is defined on the basis of the shortest distance to the ideal solution and the longest distance to the anti-ideal solution. The relative distance *C<sup>i</sup>* of the *i*th alternative to the ideal and anti-ideal solutions is calculated as

$$\mathcal{C}\_{i} = \frac{d\_{i}^{-}}{d\_{i}^{+} + d\_{i}^{-}} \, \prime \tag{9}$$

where *d* + *i* and *d* − *i* denote the distance of the alternative *i* from the ideal and anti-ideal solutions, respectively, and *C<sup>i</sup>* ∈ [0, 1].

The distance of each alternative from the ideal and anti-ideal solutions are computed as follows:

$$d\_{\vec{i}}^{+} = \left\{ \sum\_{j=1}^{n} + \left( w\_{j} (r\_{i\vec{j}} - r\_{\vec{j}}^{+}) \right)^{2} \right\}^{0.5} \tag{10}$$

and

$$d\_i^- = \left\{ \sum\_{j=1}^n \left( w\_j (r\_{i\bar{j}} - r\_{\bar{j}}^-) \right)^2 \right\}^{0.5}.\tag{11}$$

In Equations (10) and (11), *w<sup>j</sup>* denotes the weight of the criterion *j; r* + *j* and *r* − *j* denote the coordinate *j* of the ideal and anti-ideal solutions, respectively; and *rij* is the normalized rating of the alternative *i* to the criterion *j.*

The ordinary TOPSIS method utilizes the Euclidean distance to determine the separation measures. However, some authors such as Chang et al. [57], Shanian and Savadogo [58], and Hwang and Yoon [19], have also considered the application of the Hamming distance for that purpose:

$$d\_i^+ = \sum\_{j=1}^n w\_j |r\_{ij} - r\_i^+| \,\text{ and} \tag{12}$$

$$d\_i^- = \sum\_{j=1}^n w\_j |r\_{ij} - r\_i^-|. \tag{13}$$

In the numerous extensions of the TOPSIS method that were later proposed, the application of the Hamming distance has become more common such as in the research of Gautam and Singh [59], Izadikhah [60], and Chen and Tsao [61].

The ordinary TOPSIS method uses the vector normalization procedure for the calculation of normalized ratings, as

$$r\_{ij} = \frac{\mathbf{x}\_{ij}}{\left(\sum\_{i=1}^{n} \mathbf{x}\_{ij}^{2}\right)^{1/2}} \tag{14}$$

where *rij* is the normalized rating of the alternative *i* to the criterion *j*, and *xij* is the rating of the alternative *i* to the criterion *j.*

In some extensions of the TOPSIS method, however, this normalization procedure is followed with a simpler normalization procedure [31,62], as follows:

$$
\sigma\_{ij} = \frac{\mathbf{x}\_{ij}}{\mathbf{x}\_j^+}.\tag{15}
$$

In Equation (15), it is assumed that *x* + *j* denotes the largest rating of the criterion *j*. The ideal *A* ∗ and the anti-ideal *A* − solutions are defined by

$$A^{+} = \left\{ r\_1^{+}, r\_2^{+}, \dots, r\_n^{+} \right\} = \left\{ \max\_{\vec{i}} r\_{\vec{j}\vec{l}} | \vec{j} \in \Theta\_{\max} \right\} \left( \min\_{\vec{i}} r\_{\vec{j}\vec{l}} | \vec{j} \in \Theta\_{\min} \right) \tag{16}$$

$$A^{-} = \left\{ r\_1^-, r\_2^-, \dots, r\_n^- \right\} = \left\{ \min\_i r\_{i\mid i} | j \in \Theta\_{\max} \right\} \iota (\max\_i r\_{i\mid} | j \in \Theta\_{\min}) \right\} \tag{17}$$

where *r* + *j* denotes the coordinate *j* of the ideal solution; *r* − *j* denotes the coordinate *j* of the anti-ideal solution; and Θmax and Θmin denote the sets of beneficial and non-beneficial criteria, respectively.
