#### 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_{i} of the

ith alternative to the ideal and anti-ideal solutions is calculated as

where

${d}_{i}^{+}$ and

${d}_{i}^{-}$ denote the distance of the alternative

i from the ideal and anti-ideal solutions, respectively, and

${C}_{i}\in [0,\text{}1]$.

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

and

In Equations (10) and (11), ${w}_{j}$ 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 ${r}_{ij}$ 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:

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

where

${r}_{ij}$ is the normalized rating of the alternative

i to the criterion

j, and

${x}_{ij}$ 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:

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

where

${r}_{j}^{+}$ denotes the coordinate

j of the ideal solution;

${r}_{j}^{-}$ denotes the coordinate

j of the anti-ideal solution; and

${\Theta}_{\mathrm{max}}$ and

${\Theta}_{\mathrm{min}}$ denote the sets of beneficial and non-beneficial criteria, respectively.

#### 3.2. An Extension of the TOPSIS Method Adapted for the Use of SVNNs

The typical MCDM problem that includes

m alternatives and

n criteria can concisely be presented in the following matrix form:

The entry x_{ij} in the evaluation matrix D means the rating of the alternative i with respect to the criterion j and entries w_{j} in W of the weight vector denote the weights of the criterion j, for each i = 1, … m and j = 1, ..., n.

However, many practical DM problems require the participation of more decision-makers or experts in the evaluation process. Therefore, in multiple-criteria group decision-making (MCGDM), there is more than one decision-making matrix

where

${D}^{k}$ denotes an evaluation matrix formed by the decision-maker and/or expert

k;

${x}_{ij}^{k}$ is the rating of the alternative

i with respect to the criterion

j obtained from the decision-maker and/or expert

k; and

K denotes the number of decision-makers and/or experts.

In the MCGDM process, decision-makers and/or experts often have different experiences and/or specific knowledge of the problem that has to be solved, which is why another weighting vector can be used to express the impact of the decision-makers and/or experts on the final evaluation, namely as follows:

The value ${\omega}_{k}$ is the significance or impact of the decision-maker and/or expert k on the overall evaluation.

Using the weighting vector that expresses the impact of decision-makers on the overall evaluation, the individual evaluation matrix obtained from the decision-makers and/or experts, and a sort of aggregation operator, an overall group decision-making matrix can be constructed.

Taking into consideration the foregoing facts pertaining to the MCGDM, the specifics of SVNNs and operations over them as well as the previously proposed extension of the TOPSIS method [

54,

63,

64], a thorough step-by-step procedure of the adapted TOPSIS method, as shown in

Figure 1, can be accurately presented through the following basic steps:

Step 1. Forming a team of decision-makers and assigning them relative importance to the overall evaluation. In the first step, a team of decision-makers and/or experts is formed and relative importance is assigned to each of them, if necessary. In many cases, all decision-makers and/or experts have equal importance to the final evaluation.

Step 2. Identification of acceptable alternatives and selection of criteria for their evaluation. In the second step, the team of decision-makers identified the feasible alternatives and determined a set of evaluation criteria.

Step 3. Determining the significance of evaluation criteria. In this step, the team of decision-makers and/or experts determined the weights of the evaluation criteria. A number of methods that can be used to determine criteria weights have been considered in many papers published in scientific and professional journals [

16,

65,

66,

67,

68].

Step 4. Evaluation of alternatives in relation to the selected criteria. In the fourth step, each decision-maker performs an evaluation and forms their own evaluation matrix, in which the ratings are expressed by using SVNNs. As a result of performing this step, a

K evaluation matrix is formed as follows:

where

$<{t}_{ij}^{k},{i}_{ij}^{k},{f}_{ij}^{k}>$ denotes the rating of the alternative

i with respect to the criterion

j, obtained from the decision-maker expert

k.

Step 5. Construction of an overall group evaluation matrix. In this step, the individual attitudes of the decision-makers involved in the evaluation are transformed into one overall group evaluation matrix by using a SVNWA operator (i.e., by applying Equation (8)). As a result of performing this step, a matrix of the following form is formed:

where

$<{t}_{ij},{i}_{ij},{f}_{ij}$ denotes the rating of the alternative

i in relation to the criterion

j.

Step 6. Construction of a normalized evaluation matrix. The normalization of the overall group evaluation matrix can be performed by applying Equation (3) and the following

λ:

This step is not necessary if all ratings belong to the interval [0, 1].

Step 7. Determining the ideal and negative-ideal solutions. In the case when all evaluation criteria are beneficial, the ideal and negative ideal solutions are calculated as follows:

Step 8. Obtaining the distance between each alternative and the positive ideal solution. The distances between the alternatives and the positive ideal solution can be determined by applying Equations (4) or (5).

Step 9. Obtaining the distance between each alternative and the negative-ideal solution. The distances between the alternatives and the negative ideal solution can be determined in a similar manner as the distances to the ideal solution.

Step 10. Obtaining the closeness coefficients of each alternative to the ideal solution. Applying Equations (4) and (5), SVNNs are transformed into the resulting crisp values, thus allowing the application of Equation (9) to determine the closeness coefficients to the ideal solution, as in the ordinary TOPSIS method.

Step 11. Ranking the alternatives and selection of the best one. The final ranking of the considered alternatives remains the same as in the ordinary TOPSIS method, which means that an alternative with a higher value of the closeness coefficient is more preferable.