**3. Results**

A problem of determining the ranks of the European countries based on two groups of criteria (*r* = <sup>2</sup>), including internal security and human development, was solved. In the first step, correlation analysis was applied to the data in Table 2 for establishing the priority of the criteria, such as internal security, *X* = (*<sup>x</sup>*1, *x*2, *x*3, *<sup>x</sup>*4) and human development, *Y* = (*y*1, *y*2, *y*3, *y*4), in each group. The larger the absolute value of the correlation coefficient of the respective criterion with the criteria of the other group, the higher its priority order. The values of the Pearson correlation coefficients are presented in Table 3.

Process criteria has the highest priority value in the Int\_Sec\_Group, due to higher correlation with Y group criteria (0.868), followed by Legitimacy (0.823), Capacity ( −0.537) and, finally, Outcomes criterion (0.467). In the human development group, GNI per Capita (0.868) has the highest priority value, Life Expectancy at birth (0.823) is second, the Expected Years of Schooling (0.772) is third and the Mean Years of Schooling (0.506) is last. Therefore, the priority order of the considered criteria is as follows

$$\mathbf{x\_4 \succ x\_1 \succ x\_3 \succ x\_2, \ y\_4 \succ y\_1 \succ y\_3 \succ y\_2}$$

and the respective weight priority (5) is

$$1 \ge w\_4^X \ge w\_1^X \ge w\_3^X \ge w\_2^X \ge 0,\ 1 \ge w\_4^Y \ge w\_1^Y \ge w\_3^Y \ge w\_2^Y \ge 0. \tag{6}$$

**Table 3.** Values of Pearson correlation coefficients of the criteria (first row) and *p*-values (second row).


\*\* Correlation is significant at the 0.01 level (2-tailed). \* Correlation is significant at the 0.05 level (2-tailed).

The second step in solving the MCDM problem is normalising the decision-making matrix elements *xij* and *yij*. Min–max normalisation equations were used in both cases. This method demonstrated the highest accuracy and was most stable compared to other normalisation techniques, when applied with SAW [30]. In the case of direct normalisation, the equations were as follows

$$
\widetilde{\mathbf{x}}\_{ij} = \frac{\mathbf{x}\_{ij} - \min\_{1 \le i \le m} \mathbf{x}\_{ij}}{\max\_{1 \le i \le m} \mathbf{x}\_{ij} - \min\_{1 \le i \le m} \mathbf{x}\_{ij}}, \\
\widetilde{y}\_{ij} = \frac{y\_{ij} - \min\_{1 \le i \le m} y\_{ij}}{\max\_{1 \le i \le m} y\_{ij} - \min\_{1 \le i \le m} y\_{ij}},
$$

while the inverse normalisation equation was applied only to the Capacity criterion, having an opposite direction with respect to the goal:

$$\widetilde{\mathbf{x}}\_{ij} = \frac{\max\_{1 \le i \le m} x\_{ij} - x\_{ij}}{\max\_{1 \le i \le m} x\_{ij} - \min\_{1 \le i \le m} x\_{ij}}.$$

In Table 4 the normalised values of the criteria *<sup>x</sup>*<sup>+</sup>*ij*, *y*+*ij* are given.

Then, the procedure of weight balancing was carried out. A possible set of weights, satisfying the conditions (1) and (6), is presented in Table 5. The elements of this set were reselected and the weighted sums *SXi* = <sup>∑</sup><sup>4</sup>*j*=<sup>1</sup> *wxj <sup>x</sup>*<sup>+</sup>*ij*, *SYi* = <sup>∑</sup><sup>4</sup>*j*=<sup>1</sup> *wyj y*+*ij* were calculated for each alternative. The minimum value of the target function was obtained for the optimal weight values *WX*∗ = *<sup>w</sup>x*<sup>∗</sup>1 , *<sup>w</sup>x*<sup>∗</sup>2 , *<sup>w</sup>x*<sup>∗</sup>3 , *<sup>w</sup>x*<sup>∗</sup>4 and *WY*∗ = *wy*<sup>∗</sup>1 , *<sup>w</sup>y*<sup>∗</sup>2 , *<sup>w</sup>y*<sup>∗</sup>3 , *<sup>w</sup>y*<sup>∗</sup>4 :

$$F\left(\mathcal{W}^{X\*},\mathcal{W}^{Y\*}\right) = \min\_{\mathcal{W}^X,\mathcal{W}^Y} \sum\_{i=1}^m \left(S\_i^X - S\_i^Y\right)^2 = \min\_{\mathcal{W}^X,\mathcal{W}^Y} \sum\_{i=1}^m \left(\sum\_{j=1}^4 w\_j^x \widetilde{\mathbf{x}}\_{ij} - \sum\_{j=1}^4 w\_j^y \widetilde{y}\_{ij}\right)^2. \tag{7}$$

*F*-*<sup>W</sup>X*<sup>∗</sup>, *WY*∗ is the minimum value of a disagreement measure between two alternative rankings (according to the criteria values *X* and *Y*). It could be interpreted as the function of assessing the weight balancing quality.



**Table 5.** The set of possible weight values 1 ≥ *wx*,*<sup>y</sup>* 4 ≥ *wx*,*<sup>y</sup>* 1 ≥ *wx*,*<sup>y</sup>* 3 ≥ *wx*,*<sup>y</sup>* 2 ≥ 0 for the criteria *X* and *Y*.

The number of possible weight combinations and, accordingly, the values of the target function (7) is 23 × 23 = 529. The function *F*-*<sup>W</sup>X*, *W<sup>Y</sup>* gained its minimum value 0.397 for the respective weight values:

$$w\_4^{x\*} = 0.6, \; w\_1^{x\*} = 0.2, w\_3^{x\*} = 0.2, w\_2^{x\*} = 0; \; w\_4^{y\*} = 0.3, \; w\_1^{y\*} = 0.3, w\_3^{y\*} = 0.2, w\_2^{y\*} = 0.2. \tag{8}$$

Next, the step length 0.05 (twice as small as in Table 5) was chosen and the optimisation procedure was repeated. However, the authors failed to ge<sup>t</sup> a better result. The minimum value of the target function (7) remained the same with the same weights (8).

At the last step of WEBIRA, the weighted sum values of the criteria *X* and *Y* were calculated for each alternative as follows

$$\mathcal{Q}\_i \Big( \mathcal{W}^{X\*} , \mathcal{W}^{Y\*} \Big) = \mathcal{S}\_i^{X\*} + \mathcal{S}\_i^{Y\*} , \ i = 1, 2, \dots, m$$

and the ranking of the alternatives based on these values was performed. The final results are presented in Table 6.

When assessing the criteria of ranking the countries, it is important to take into consideration the mutual distribution of HDI and WISPI components (the difference between the ranks of HDI and WISPI of the countries) (Figure 1). The difference between the ranks of HDI and WISPI reflects the development priorities of the countries (Table 6). Appraisal of changes in the difference between these indicators allows forecasting the trend of the country's development (e.g., development of economic potential and increasing the welfare of the population, development associated with strengthening the policy and recognising the security priorities, or harmonious development). The minimal difference between HDI and WISPI shows a balanced internal policy pursued by the countries, implying that the countries allocate their resources to the internal security and public welfare in a balanced way.

**Table 6.** European countries ranking results based on the alternative methods of WEBIRA, HDI, WISPI and cluster analysis results.


**Figure 2.** Cluster analysis results on HDI and WISPI axes.

In the following, the performed comparative analysis of WEBIRA method results with the results of cluster analysis is discussed. The clustering procedure was executed with standardised data. Based on the silhouette method, the European countries could be divided into four clusters. The partition into clusters was done by several hierarchical clustering methods and also techniques like k-means cluster analysis. The challenge when applying the hierarchical clustering methods is determining the proper distance measure. In our work, we have tested different distance measures, but there are alternative methods like A-BIRCH [35]. All of the used methods gave very similar results, so we chose the k-means method. Therefore, neither clustering approach can ultimately judge the actual quality of clustering; this needs human evaluation [36], which is highly subjective [37]. Because of these shortcomings, cluster analysis can only be used as rough initial test before applying more accurate methods. Table 6 shows the ranking of the European countries by using methods WEBIRA, HDI, WISPI and cluster analysis results. Table 6 and Figure 1 show that the WEBIRA ranking of countries is basically consistent with the results of cluster analysis. Spearman correlation coefficient of WEBIRA ranks and k-means cluster analysis results show very high correlation between them *ρS* = 0.912. All countries with the highest HDI and WISPI indicators have entered cluster 4. Cluster 3 consists of slightly lower HDI and WISPI countries with one exception—Hungary—which is the member of cluster 1. The countries with lowest HDI and WISPI are assigned to cluster 1. The most diverse is cluster 2, which consists of countries which at first glance do not have much in common, i.e., Lithuania, Latvia, Belarus, Bulgaria and Russian Federation. Figure 2 represents cluster analysis results on HDI and WISPI axes. It also shows that WISPI indicators in all cluster 2 countries except Lithuania are lower than in other countries adjacent to these in WEBIRA ranking.
