**1. Introduction**

The European countries are exposed to various hybrid dangers (e.g., political differences, military aggression, financial and economic crises, natural and technogenic catastrophes, social upheavals, criminal offences, etc.). This problem is closely associated with the internal security of the states. The institutions, ensuring the internal security of the states, also guarantee their economic stability. The European Union (EU) is becoming a centre of sustainability of the European values and stability. The systematic and consistent attitudes towards the topical issues of internal security are formed by the EU member states through their joint regulations. However, the particular EU states develop their security systems depending on numerous internal and external factors. Therefore, their internal security systems have some specific features. The states not belonging to the EU also demonstrate their distinctive properties, though in the world involved in the global processes similar tendencies of changes in the internal safety systems can be observed [1]. This field of research has been barely explored. The World Internal Security and Police Index (WISPI) [2] focuses on describing both the effective rendering of security services and the outcome of the rendered services. WISPI is considered the first international index for measuring the indicators of the internal security worldwide, as well as ranking the states according to their ability to provide security services and boost security performance.

The economic and human development of a particular country relate to the state of security in this country [3]. Higher development levels, in terms of GDP per capita, are capable of providing social and individual prosperity or human development. It is not clear whether other interrelations between prosperity indicators exist on different levels of economic development. Social and human development and security status indicators improve with economic development. Public well-being increases with income rise at all levels of economic development [4]. Focusing on people instead of economic outcomes provides a wider range of options for policy-makers [5]. Countries' development policies should strive to remove any obstacles that impede people's freedoms: political freedom provides individuals to enjoy the freedom of political expression; economic facilities allow the use of economic resources for the purpose of consumption, production or exchange; social opportunities are made possible access to education and health; transparency guarantees relate to openness and the prevention of any type of corruption; and protective security allows a social safety network that protects individuals from misery [6]. Human development paves the way for economic development and security. State policy not only secures educational programs but also promotes development through innovation and expansion of new programs [7]. Meanwhile relationship exists between pro-governmen<sup>t</sup> militias and various types of human rights violations [5]. A relationship between a given governmen<sup>t</sup> regime's security repertoire and the likelihood of control and violence against civilians exists [8]. Uncontrollable human rights violations have a harmful effect on the positive country's image [9]. Since the governmen<sup>t</sup> is a source of legitimate authority, laws and regulations also provide important cues about which course is supported and protected by the government. A legal country system that protects certain interests with certain methods sends a signal to world societal participants that these interests and these methods should be determined as a dominant image of the country [10].

Human development is a process, which seeks to expand the possibilities to create an environment where people can live long, healthy and creative lives. Human Development Index (HDI) [11] is one of the most widely used composite indicators of socioeconomic development of a country. People who have achieved high or very high human development level represent 51 percent of the global population. Researchers are extremely interested in factors influencing HDI [12]. Overwhelming evidence of the direct positive effects of economic freedom on human development is provided by a large number of the cross-country studies [13]. However, the 'original sin' of HDI involves neglecting the environmental and social sustainability and personal security issues [14].

A wide variety of approaches and evaluation techniques are used in the field of security research; however, there are some gaps, particularly if researchers aim to study the internal security of the whole country. The aim of the present study is to propose a new approach to identify a method of ranking the countries for evaluating the internal security of the European countries, using indicators such as the Human Development Index [11] and the Internal Security and Police Index [2]. Thus, the combining of HDI indicators with the World Internal Security and Police Index can provide an integrated evaluation approach for filling this gap. However, the conventional security system's modelling tools and models, such as expert-based or other approaches, do not propose any integral internal security metric, covering all types of threats, to which the countries and citizens are exposed.

There is not much research in the literature dedicated to studying HDI and particularly WISPI by means of mathematical modelling. Most research is related to the separate dimensions of HDI—public health, economic development and quality of life. The most commonly used methods are various tools of mathematical statistics, i.e., correlation, regression analysis and some econometric models.

The study by Zaborskis et al. [15] introduces several methods for measuring family affluence inequality in adolescent life satisfaction (LS) and assesses its relationship with macrolevel indices (Gross National Income, Human Development Index and the mean Overall Life Satisfaction score). Poisson regression estimations and correlation analysis were used in this research. Murray et al. [16] investigated how preterm delivery rates differ in a country with a very high human development index and explored rural vs. urban environmental and socioeconomic factors that may be responsible for this variation. A multiple linear regression was used for this purpose. The study by Liu et al. [17] employs a panel smooth transition vector error correction model (PST-VECM) to explore the education-health causality. The paper by Sayed et al. [18] discusses the rank reversal issue in multicriteria decision-making (MCDM) techniques. The proposed methodology of the Goal Programming Benefit-of-the-Doubt (GP-BOD) aims to overcome this problem and obtain consistent and stable rankings for the human development index (HDI) framework. The paper by Carvalhal Monteiro et al. [19] proposes a new Human Development Index (HDI) classification method using the combination of the ELECTRE TRI method with statistical tools to define classes and class profiles for the HDI.

We could not find any quantitative investigation of WISPI in the literature. The synergy of HDI and WISPI as a research object is unprecedented in the scientific literature. However, the task of ranking the countries according to HDI and WISPI interrelation is an interesting and relevant issue.

MCDM methods usually rank countries by set of homogeneous (having the same nature) indicators. If several criteria groups having different nature exist, for example, subjective and objective, external and internal evaluations of alternatives, other methodologies should be proposed. KEMIRA [20] is the MCDM method implemented by maximising compatibility of two or more subsets of criteria, thus it is naturally appropriate for solving our task. In this research, a modification of KEMIRA called WEBIRA [21] has been applied to the case of two groups of evaluation criteria. The advantage of WEBIRA is that its efficiency does not decrease with increasing number of alternatives as other MCDM methods [22]. It also remains stable with increasing number of criteria [21].

Prioritisation of criteria is a separate issue of the WEBIRA method that needs to be addressed before solving the optimisation task. In this sense, WEBIRA is not a fully objective method for determining criteria weights. The problem of criteria prioritisation can be solved by applying wide range of objective or subjective (expert-based) methods. Examples of expert-based methods are Analytic Hierarchy Process (AHP) [23], Kemeny median method [24], Stepwise Weight Assessment Ratio Analysis (SWARA) [25], a fuzzy inference system (FIS) approach [26], etc. However, when dealing with country rating task, we need to look for alternative methods for prioritising criteria, because we do not have information about criteria assessments by experts. Objective methods for criteria weighting are based on initial data values and their structure (entropy-based methods [27], mathematical programming models [28], IDOCRIW [29], etc.).

There are three main steps of WEBIRA: (1) criteria priority setting separately in every subset; (2) criteria weight determining by solving optimisation problem; and (3) ranking of alternatives by applying one of MCDM methods. One of the novelty elements of this article is to use correlation analysis to set criteria priority. Statistical methods are traditionally used in weighting attributes. Thus, CRICTIC (Criteria Importance Through Intercriteria Correlation), developed by Diakoulaki et al. [30], aims to determine objective weights of relative importance in MCDM problems by considering correlation coefficient values between criteria and standard deviations of each criterion for alternatives. High correlation is considered as some kind of double counting, so assigned weights are inversely proportional to the correlation coefficient value. Our methodological assumption is based on the maximisation of compatibility between two different groups of indicators. A suitable way to measure compatibility is to apply intergroup correlation coefficients. Unlike correlations within groups, where attributes with strong correlation are undesirable, high correlation of the attribute with the attributes of other groups indicates that the interdependence between the two group's indicators became higher; such indicator is more preferable in the decision-making process.

This idea arose from the ultimate goal of this research—to evaluate countries by combining several dimensions: economic prosperity, comprehensive education, healthy lifestyle, safe environment and human security. Thus, two groups of criteria—X and Y—were distinguished and the optimisation task has been solved according to weight balancing procedure. This procedure ensures that the criteria for the two groups in the final order of alternatives are maximally aligned with each other.

### **2. Materials and Methods**

### *2.1. Criteria and Their Definitions*

The Human Development Index is a summary measure of average achievement in key dimensions of human development: (1) a long and healthy life; (2) knowledge; and (3) a decent standard of living. The knowledge dimension consists of two subdimensions: (1) mean of years of schooling for adults aged 25 years and more and (2) expected years of schooling for children of school entering age. The HDI is the geometric mean of normalised indices for each of the three dimensions [11]. In the present work, four components of HDI are used: the ability to lead a long and healthy life, measured by life expectancy at birth (years) (*y*1); the ability to acquire knowledge, measured by the mean number of years of schooling (*y*2); the expected years of schooling (*y*3); and the ability to achieve a decent standard of living, measured by the gross national income (GNI) per capita (PPP \$) (*y*4) [11]. HDI makes an assessment of diverse countries with very different price levels. To compare economic statistics across countries, the data must first be converted into a common currency. For this reason GNI per capita is measured in purchasing power parity (PPP) international dollars (PPP \$). One PPP dollar (or international dollar) has the same purchasing power in the domestic economy of any country as US\$1 has in the US economy.

World Internal Security and Police Index (WISPI) measures the capacity and efficiency of police and security service providers to address the internal security issues worldwide through the four domains, i.e., capacity, process, legitimacy and outcomes (Table 1) [2]. Domain content can be explained by answering these questions:

Capacity: Do security providers have the resources needed to address security violation? Process: Are the resources directed towards violence prevention used effectively? Legitimacy: Are security providers trusted by the people? Do they abuse their position? Outcomes: Do people feel safe in their neighbourhoods? Are crime rates low?

Each WISPI domain acquires values from 0 to 1. The higher the numerical value of the country's respective domain, the higher the position of that country in the corresponding rating. WISPI measures the ability of police and internal security service to protect society as well as provides broader measure of human security.


**Table 1.** World Internal Security and Police Index, Domains and Indicators [2].

The initial data matrix, maximum and minimum values of indicators are presented in Table 2.



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**Table 2.** *Cont.*

### *2.2. General Description of WEBIRA Method*

⎪⎪⎨

Let the initial data be the results of the performed measurements, expert evaluations, etc., presented in *m* × *n*-dimension matrix *X* = - *xij m*×*n* . The element *xij* of the decision-making matrix is the estimate of the alternative *i* (*i* = 1, 2, . . . , *m*) based on using the criteria *j* (*j* = 1, 2, . . . , *<sup>n</sup>*). A data normalisation procedure is required because there are different criteria measurement units. There is a variety of data normalisation formulas, in this case a min–max normalisation was used:

$$\begin{cases} \widetilde{\mathbf{x}}\_{ij} = \frac{\mathbf{x}\_{ij} - \min\_{1 \le i \le m} \mathbf{x}\_{ij}}{\max\limits\_{1 \le i \le m} \mathbf{x}\_{ij} - \min\_{1 \le i \le m} \mathbf{x}\_{ij}}, \text{ for the direct normalization,} \\ \qquad \widetilde{\mathbf{x}}\_{ij} = \frac{\max\limits\_{1 \le i \le m} \mathbf{x}\_{ij} - \mathbf{x}\_{ij}}{\max\limits\_{1 \le i \le m} \mathbf{x}\_{ij} - \min\_{1 \le i \le m} \mathbf{x}\_{ij}}, \text{ for inverse normalization.} \end{cases}$$

The choice of min–max normalisation is based on the results of previous studies [22,31], which revealed that min–max normalisation ensures the best stability of the SAW method compared to other well-known normalisation procedures such as max, sum, vector, logarithmic, etc. Stability of the min–max method was the highest for cases of both more and less separable alternatives. All the variables after the min–max normalisation gain their values between 0 and 1.

If new countries with values not in the range analysed initially would be introduced, normalisation would be performed again and all values after normalisation would also range between 0 and 1. However, after introducing new countries (cases) and recalculation of correlation coefficients the priority of criteria, data structure and, subsequently, the overall rating of the countries, could be changed. For this reason, only European countries were involved in the investigation.

The normalised decision-making matrix has the following form, *X*+ = - *x*+*ij m*×*n*, 0 ≤ *x*+*ij* ≤ 1. Let *wj*, *j* = 1, 2, . . . , *n* be criteria weights, satisfying the conditions as follows

$$\sum\_{j=1}^{n} w\_{\backslash} = 1, \ 0 \le w\_{\backslash} \le 1. \tag{1}$$

The Simple Additive Weighting (SAW) method [32] is a well-known and widely used MCDM tool. SAW with the weights *wj*, *j* = 1, 2, ... , *n* can be applied to solve MCDM problem. The aggregated value based on using the SAW criteria was calculated for each alternative as follows

$$S\_{\vec{i}} = \sum\_{j=1}^{n} w\_{\vec{j}} \widetilde{\mathbf{x}}\_{\vec{i}j}, \; i = 1, 2, \dots, m. \tag{2}$$

The values 0 ≤ *x*+*ij* ≤ 1 in Equation (2) were normalised so that the higher *x*+*ij* value would correspond to the better evaluation of the *i*-th alternative *Si*.

The weighted coefficients (1) are usually determined by using various methods that could be based on expert judgement (subjective methods) or the objective weight assessing methods [33]. WEBIRA is objective weight assessing method which is appropriate for solving our problem for two reasons. The first is the absence of highly qualified expert judgements. This prevents the use of subjective methods such as Analytic Hierarchy Process (AHP), Delphi, Stepwise Weight Assessment Ratio Analysis (SWARA), etc. The second reason is the structure of the data. The set of criteria (indicators) naturally and logically could be divided to two groups of criteria. The idea of WEBIRA method is weights determining procedure when the rankings of alternatives in the few groups of criteria maximally match each other. This goal is achieved by performing a so-called weight balancing procedure aimed at minimising a certain objective function.

Suppose that *n* criteria are being divided to *r* groups. The coefficient calculation scheme is introduced when there are *r* normalised data matrices *Xk*:

*Mathematics* **2019**, *7*, 293

$$\|X^k = \|\mathbf{x}\_{ij}^k\|\_{m \times n^k}, 0 \le \mathbf{x}\_{ij}^k \le 1, \ k = 1, 2, \dots, r,\\ \sum\_{k=1}^r n^k = n. \tag{3}$$

The aggregated values for each matrix *k* = 1, 2, ... ,*r* obtained by using SAW criteria were as follows

$$S\_i^k = \sum\_{j=1}^{n^k} w\_j^k \hat{x}\_{ij}^k \; i = 1, 2, \dots, m, \; k = 1, 2, \dots, r. \tag{4}$$

The coefficients *w<sup>k</sup> j*in Formula (4) satisfy the inequalities

$$1 \ge w\_1^k \ge w\_2^k \ge \dots \ge w\_{n^k}^k \ge 0, \ k = 1, 2, \dots, r. \tag{5}$$

The optimisation problem is formulated where the minimum value of the function has to be found:

$$F\left(\mathcal{W}^1, \mathcal{W}^2, \dots, \mathcal{W}^r\right) = \sum\_{k=1}^{r-1} \sum\_{l=k+1}^r \sum\_{i=1}^m \left| S\_i^k - S\_i^l \right|^{\mathcal{S}}$$

by checking the value of the function above with each vector *W<sup>k</sup>* = *w<sup>k</sup>* 1, *w<sup>k</sup>* 2,..., *w<sup>k</sup> n<sup>k</sup>* , *k* = 1, 2, ... ,*r* satisfying the inequalities (5) and the relationships (1). The parameter's *δ* value is *δ* = 2 throughout the paper. The inequalities (5) can be determined by using various methods of processing the expert assessments; in Krylovas et al. [24], it has been proposed to apply Kemeny median [34] for this purpose. This method for prioritising criteria and determining weights which satisfy Formulas (1) and (5) is named the KEmeny Median Indicator Ranks Accordance (KEMIRA) method. The order of preference of the weighted coefficients can be determined by using other methods. In this paper correlation analysis is applied to the solution of this problem. Therefore, a group of the methods given in Krylovas et al. [21] is referred to as WEBIRA (WEight Balancing Indicator Ranks Accordance).

Suppose that *A* = {1, 2, . . . , *m*} is a set of the available alternatives, while the subsets of the set *A* are denoted as follows

$$\begin{array}{l} A\_{\mathfrak{a}}^{+} = \left\{ i \in A : S\_{i}^{1} > \mathfrak{a}, S\_{i}^{2} > \mathfrak{a}, \dots, S\_{i}^{r} > \mathfrak{a} \right\}, \\ A\_{\mathfrak{a}}^{-} = \left\{ i \in A : S\_{i}^{1} \le \mathfrak{a}, S\_{i}^{2} \le \mathfrak{a}, \dots, S\_{i}^{r} \le \mathfrak{a} \right\}, \\ A\_{\mathfrak{a}}^{\pm} = A \backslash (A\_{\mathfrak{a}}^{+} \cup A\_{\mathfrak{a}}^{-}), \end{array}$$

*A*<sup>+</sup> *α* denotes the sets of the undoubtedly superior alternatives, *A*− *α* are the sets of undoubtedly inferior alternatives and *A*<sup>±</sup> *α* denotes the sets of alternatives whose assessment is doubtful. Note that when 0 ≤ *Sk i* ≤ 1, *A*<sup>+</sup> 0 = *A*− 1 = *A*, *A*<sup>+</sup> 1 = *A*− 0 = ∅. The functions *<sup>F</sup>*+(*α*), *<sup>F</sup>*−(*α*), *<sup>F</sup>*<sup>±</sup>(*α*) are determined as the number of elements of the respective sets *A*<sup>+</sup> *α* , *A*− *α* , *A*<sup>±</sup> *α* . It is obvious that *<sup>F</sup>*+(*α*) + *<sup>F</sup>*−(*α*) + *<sup>F</sup>*<sup>±</sup>(*α*) = *m*. *<sup>F</sup>*+(*α*), *<sup>F</sup>*−(*α*), *<sup>F</sup>*<sup>±</sup>(*α*) are stepwise functions, having the first type points of discontinuity. The values of the functions can help assess the quality of weight balancing. In the ideal case, *<sup>F</sup>*<sup>±</sup>(*α*) ≡ 0. In this research, the authors deal with *A*<sup>+</sup> 0 = *A*− 1 = *A*.
