Linear Ordering Methods Research Articles

The Financial Gap Arising from the Local Government’s Educational Tasks and the Sustainable Development of Communes in Poland

In Poland, responsibility for educational tasks has mainly been delegated to local governments, although the state still plays a decisive role. Municipalities finance these tasks using educational subsidies and targeted grants from the central budget. Because transfers from the state budget cover an ever-decreasing share of the costs of educational tasks, municipalities increasingly need to engage their own resources. Such a situation (due to the limited funds of these units) poses a threat to the proper implementation of other tasks, which may ultimately negatively affect their sustainable development. This study aimed to determine the impact of the financial gap from implementing educational tasks on the sustainable development of communes and cities with county rights. The research primarily employed linear ordering methods and Spearman’s rho correlation. It also employed the nonparametric Kruskal–Wallis test and the Page test for trends. This study utilized two types of data: secondary data from Statistics Poland covering 2477 municipalities and primary data collected through a survey of 482 local government representatives. The correlation analysis revealed that sustainable development indicators decreased as the financial gap widened between 2012 and 2022. The highest correlation coefficients were observed in all communes combined and in cities with county rights. The weakest correlations between the financial gap and sustainable development were found in urban municipalities. Representatives of local governments had similar observations. According to 63.1% of respondents, the imbalance between funds transferred from the state budget for educational tasks and the expenditure incurred by the municipality for this purpose negatively affects the sustainable development of local governments.

The Impact of Socio-Economic Factors on Digital Skills in the Population of the EU 27 Countries

This research investigates digital skills across the 27 EU countries, examining how incentive and disincentive factors shape these competencies, particularly under varying socio-economic conditions. Using a quantitative methodology, the study applies cluster analysis and linear ordering methods to classify countries by digital skills indicators, utilising data from the Eurostat Digital Economy and Society database. Key analytical methods—including Hellwig’s method, order counting, and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)—are employed to pinpoint crucial factors that stimulate digital competencies, such as investment in research, training programmes, and innovation. Findings indicate that countries within clusters featuring higher values in these stimulative factors tend to adopt a proactive approach to digital development. These clusters frequently correlate with substantial investments in skills training and comprehensive educational policies. Conversely, clusters characterised by high disincentive variables, such as limited funding and socio-economic disparities, show slower progress in digital skills development, highlighting barriers in educational and social inclusion systems. The results reveal marked spatial disparities across the EU. Leading countries typically have robust education systems promoting lifelong learning and inclusivity, while lagging countries face structural challenges, including centralised education systems and urban-rural divides. This disparity underscores the need for sustained investment in education, training, and policy support to enhance workforce digital skills. The study highlights that digital competencies are closely intertwined with each country's educational and socio-economic frameworks, recommending targeted interventions to foster lifelong digital literacy and reduce skills gaps. As the digital landscape continues to evolve, interdisciplinary research is essential for addressing both quantitative and qualitative aspects of digital skill development. Future studies should aim to bridge the digital divide through comprehensive strategies for upskilling populations, ensuring inclusive digital integration across all EU countries.

The Development of Islamic Banking in the UK – A Comparison of Islamic and Conventional Banking

The purpose of the article is to compare the development of several Islamic banks and conventional banks operating in the UK in 2014–2021. A statistical method of linear ordering of objects characterized by several variables, which is simple in interpretation, was used. The main difficulty was to take into account the time-varying quotation of the values of the considered variables. A novel method was proposed to take into account both the value of several variable and their variability over time. In this method, the emphasis can be placed on the final reslut, that is, consider only the values of the variables at the beginning and end of the period under study but the smoothness of the systematic development of the bank can be taken into account by placing emphasis on the analysis of the variable-stimulants from year to year. In the obtained ordering, Islamic banks grew more dynamically than conventional banks in all considered configurations of variables.

Global Conditions and Changes in the Level of Renewable Energy Sources

The progressing globalization of the contemporary economy impacts its volatility and unpredictability. The directions of changes in the socioeconomic development of the contemporary global economy are determined by a number of interrelated diverse factors. Factors clearly influencing the development of the modern international economy include innovation, digitization processes, instability of the economy caused by armed conflicts or pandemic outbreaks, the concept of sustainable development, climate policy, as well as issues related to the depletion of energy resources and the necessity of ensuring global energy security. The purpose of the article is to identify the factors of the development of the contemporary economy along with the analysis and evaluation of their impacts on changes in the level of renewable energy sources (RESs) in the EU countries. The time scope of the analysis covers the years 2013 and 2022 (a 10-year perspective). The study used the methods of literature study, literature criticism, statistical data analysis (statistical databases: EUROSTAT and IRENA), and linear ordering methods (TOPSIS and EDAS). The study results indicate that the levels of the RESs have changed in response to the factors diagnosed in the study. In the years studied, the leaders in terms of levels of RES development were France, Spain, and Denmark.

Enforcement actions against European banks in the years 2005–2022. Do financial penalties imposed on European banks follow any patterns?

AbstractThis study aims to identify, by year and over the entire period under study (2005–2022), the countries with the most restrictive bank sanctioning policies in terms of amounts and numbers of penalties to point out the most active institutions in terms of imposing penalties, and to determine the regularities governing the process of imposing these burdens. Using linear ordering methods, groups of countries with similar levels of supervisory stringency identified with one of the supervisory actions, which are financial penalties, are distinguished. A total of 53 banks (the 50 largest European banks and an additional three banks designated as G-SIIs by the EBA) are identified from a constructed database of more than 300 European banks with penalties imposed between 2005 and 2022. The study’s conclusions indicate that the frequency of penalties cannot be clearly equated with the restrictiveness of supervisory institutions, as evidenced by the example of Hungary (multiplication of small penalties). The positioning of countries regarding the number of penalties, their value and grouping using linear ordering allows the conclusion that highly developed countries are characterized by relatively high restrictiveness. The results also show the difference between the Anglo-Saxon and continental models. The analysis in this article is the first comprehensive study of penalties imposed on banks by different types of institutions in 36 jurisdictions from the pre-GFC period up to the year in which the COVID-19 pandemic expired. The multi-faceted approach also provides a basis for formulating practical regulatory implications by redirecting sanctioning policy towards banks to their management and reducing the frequency of penalties while increasing their severity.

Application of linear ordering methods to assess the level of inequality in the context of sustainable development goal 10 in the European Union

Application of linear ordering methods to assess the level of inequality in the context of sustainable development goal 10 in the European Union

Management of local finances in crisis conditions on the example of voivodeship governments in 2019-2021

The management of local finances is an intricate process consisting of decisions made by authorized bodies, which, as a result, after taking into account the established goals, both current and strategic, should result in the maximization of socio-economic effects. Indicator analysis is considered one of the important instruments of local government financial management. In the article, based on the data published by the Ministry of Finance, the National Council of Regional Chambers of Accounts and the Central Statistical Office, the financial management of the provincial cross-section of Poland in 2019-2021 was reviewed. The idea of the study was to assess the management of local finances during the crisis caused by the COVID-19 pandemic in synthetic terms, based on the TOPSIS multi-criteria linear ordering method and benchmarking, and to try to identify the factors determining this process. The research was carried out by means of financial analysis of 21 indicators, which were compiled for 16 local government provinces in Poland. The results of the research made it possible to determine the classes of financial condition of the analyzed units. The COVID-19 pandemic caused that in 2020, none of the studied local government provinces ranked in the first class characterizing a very high level of financial condition according to the TOPSIS method, although as many as 9 units were assigned to the second class characterizing a high level of financial condition. Local government provinces in 2019-2021 were characterized by very high development and investment potential, as well as the self-financing rate, they accumulated budget surpluses at the expense of an apparent slowdown in investment activity.

A FAMILY OF 82Q EXPONENTIAL HYBRID METHODS FOR SOLVING IVPS IN ODES

In a recent paper, a new class of exponential general linear methods of second stage seventh order for the numerical solution of first order initial value problems in ordinary deferential equations was derived. In this paper, we derive the new class of higher order extended exponential general linear methods for the numerical solution of first order initial value problems in ordinary deferential equations. The numerical results obtained by the new method for some problems show its superiority in efficiency and accuracy for solving problems for which the proposed general linear method is appropriate

Analysis of Fractional Linear Multi-Step Methods of Order Four from Super-Convergence

We analyze two implicit fractional linear multi-step methods of order four for solving fractional initial value problems. The methods are derived from the Grunwald-Letnikov approximation of fractional derivative at a non-integer shift point with super-convergence. The weight coefficients of the methods are computed from fundamental G unwald weights, making them computationally efficient when compared with other known methods of order four. We also show that the stability regions are larger than that of the fractional Adams-Moulton and fractional backward difference formula methods. We present numerical results and illustrations to verify that the theoretical results obtained are indeed satisfied.

Ranking Cities According to Their Fundamental Power: A Comparison Among Results of Different Methods of Linear Ordering

City managers search for new concepts of city development in order to satisfy growing demands and manage risk. The research assumes that city management requires the combination of ideas of sustainable development, a smart city and resilience. These three concepts are the components of the so-called “Fundamental Power of the City”. The aim of this paper is to present and test the operationalization of the new hybrid approach to city development through the index named the Fundamental Power of the City Index which includes 80 variables. The research applies six linear ordering methods to investigate the functionality of the Index. These are used to investigate each city’s performance and compare it with other cities by ranking them according to a chosen measure. The research considers 18 voivodship capital cities of Poland. The research draws attention to different approaches of development assessment although analysis shows that most linear ordering methods produce similar rankings. Investigation shows that human and social capital factors contribute to high positions in the ranking which is consistent with the development strategies formulated by the leading cities. The results of the analysis also identify the most urgent problems of the municipalities. The research confirms that the Fundamental Power of the City concept, by including the requirements for sustainable, smart and resilient cities, fills the gaps between them and provides a holistic approach to city management.

Disproportions in the level of innovation in European Union countries

Purpose: One of the most important areas of reflection on innovation and innovativeness is regional policy and issues related to the development of the economy. The phenomena of globalisation and pressure to create a knowledge-based economy means that not only enterprises, but also countries are forced to optimise the use of their potential on a macro, meso or micro scale. For this reason, the issue of innovativeness is connected with particular economic entities and sectors of the economy, with the activity of state, regional and local authorities and the European Union. In the conditions of deepening European integration, stimulating innovation of territorial socio-economic systems is an important instrument of economic policy at the international, national and regional levels. The aim of the study is to assess the level of innovativeness of the region in spatial and dynamic terms. Multivariate Statistical Analysis will be used to determine the aggregated indicator based on data taken from Eurostat database. Design/methodology/approach: The implementation of the objective will be achieved through the evaluation and comparison of the level of innovativeness of EU countries with the use of a synthetic measure, estimated by using one of the model methods of linear ordering of objects in the years 2015, 2018 and 2021. In the proposed method, the synthetic measure is determined using the distance of the examined objects from the model objects. The dynamic approach will allow to determine the directions of changes. The level of innovation of regions (EU countries) determined by a number of indicators that were adopted in four areas: human capital, finance, business activity and intellectual property protection. Findings: The region's innovativeness is a multidimensional phenomenon, which is directly unmeasurable, hence the need to use statistical methods when measuring it. Obtained results will allow to assess the studied phenomenon, build a ranking and identify countries with a high, medium or low level of innovation. The results of the analyses confirmed that it is advisable to consider innovation by plane, because in this case there is a greater differentiation of countries. Due to the level of innovation, the highest positions in the ranking were taken by Sweden, Germany, Finland and Austria, at the end of the ranking were countries from Central and Eastern Europe, Poland, Latvia, Bulgaria and Romania. In the area of innovative activities, Estonia ranked very high being classified in the group I of the most innovative countries. The country is characterized by high employment in innovative enterprises, significant expenditure on innovation and a large share of Small and Medium Enterprises (SMEs) introducing product innovations. The fact that the countries of Central and Eastern Europe are reducing the distance to more innovative EU countries should be assessed positively, as evidenced by lower values of the coefficients of variation in 2021 compared to 2015. Keywords: innovation of regions, dynamics, Multivariate Statistical Analysis. Category of the paper: research paper.

Analysis of banking sector stability using the taxonomic measure of development

Motivation: The stability of the financial system means that the entire system performs its key functions properly. It is a prerequisite for sustained economic growth. Maintaining stability of the banking sector is of particular importance for the firmness of the financial system, as it plays a key role in financing the economy, transmission of monetary policy impulses, and monetary settlements. Therefore, in the context of financial security, it is extremely important to analyse the banking stability. Aim: Assessment of the stability of banking sectors of the Central and Eastern European EU Member States and a comparative analysis and classification of the CEE countries in terms of selected indicators characterising the stability of their banking sectors using the model method of linear ordering, i.e. the Hellwig’s method. The analysis will verify the thesis whether banks in countries of lower national income (according to the World Bank’s classification) perform worse in terms of stability than banks in countries of higher national income. To illustrate the situation in the field of banking sector stability more clearly, the authors presented the dynamics of all variables considered from the point of view of the analysis (dynamics were presented for 2015 and 2019 in relation to the base year, which was assumed to be 2011). Results: A multivariate analysis was used in the comparative analysis of banking sectors development in Central and Eastern Europe. For this purpose, the development pattern method was used so that a synthetic indicator of the development of the banking sector with regard to its stability was calculated. Based on the proposed measure of development, a ranking of the Central and Eastern European EU Member States was prepared for 2019. It should be emphasized that the thesis outlined in the article was not empirically confirmed, namely, banks from countries of lower national income (according to the World Bank classification) did not have much worse results than banks from countries of higher national income.

Assessment of the level of economic development of Polish voivodships by means of linear ordering methods in the dynamic approach

The research on the economic development of voivodships is important in the context of the policy of levelling regional disparities and has been necessitated by the fact the disproportions in the level of the economic development of Polish voivodships have persisted for years. The main aim of this study was to determine the direction and scope of changes affecting the level of economic development of particular voivodships. To this end, the study used linear ordering methods in the dynamic approach. The additional aim, of the methodological nature, was to compare two methods of the normalisation of variables – the Strahl quotient transformation and the zero unitarisation method. The research was conducted on the basis of relevant data from Statistics Poland’s Local Data Bank database for the years 2010–2020. The results of the study include the calculation of the synthetic measure of the level of development of Polish voivodships in the studied period, the compilation of the relevant ranking of the voivodships, and the comparison of changes thereof. The performed analysis demonstrated that all the voivodships developed economically in the examined period, but the changes took place at a different pace and the disparities between them remained. The indicator of economic development grew to the largest extent in the voivodships which were already the most developed in economic terms, i.e. in the Mazowieckie, Dolnośląskie, Wielkopolskie and Małopolskie voivodships. Both of the applied methods of the variable normalisation led to similar conclusions. The differences between them concerned the value of the development level indicator, and affected to some degree the assessment of the scope and direction of changes, along with the assessment of the distance between the objects.

FINANCIAL SITUATION AND SUSTAINABLE DEVELOPMENT OF TERRITORIAL UNITS – A COMPARATIVE ANALYSIS OF POLISH VOIVODESHIPS

This paper compares and statistically evaluates two complex phenomena emerging from the example of Polish voivodeships in the years 2017-2019: the resulting financial situation and the level of sustainable development. The authors applied a model-free research method of linear ordering for objects, taking into account the dynamics of this approach. Through this method, the authors determined the financial situation and sustainable development of Polish voivodeships by using the values of synthetic measures. The results of the study confirm previous findings about differentiation of Polish voivodeships as regards to both their financial situations and their levels of sustainable development. Conclusions from this study may be the basis for comparison between the territorial units examined, helping future policy makers choose rationally. The methodological approach has great practical value, and might be used in comparative research: e.g., by territorial units in different countries to determine their own financial situations and levels of sustainable development.

New fractional linear multi-step methods of order four with improved stabilities

New fractional linear multi-step methods of order four with improved stabilities

Fiscal stability in EU countries

The article presents the evolution of the concept of fiscal stability towards the sustainability of public finances. The aim of the study is to join the discussion on the factors shaping fiscal stability and to verify their significance during disturbances caused by the financial and economic crisis and the impact that the COVID-19 pandemic had on the economy. The result of the undertaken research is a ranking of EU countries according to the sustainability of their public finances, created using one of the linear ordering methods, TOPSIS. In addition, the study allowed the indication of the factores determining the stability of public finances and its affecting its sustainability. The analysis was based on Eurostat data for 2010, 2015, 2017 and 2020. The study showed that the variation among the analysed countries’ sustainability of public finances decreased significantly in 2020 as compared to the previous years. Moreover, relatively significant changes in the ranking were observed in the analysed period in terms of the adopted criterion, which primarily resulted from changes in the countries’ level of economic growth and the size of their public debt in relation to GDP. In each of the studied years, Greece proved to have had the most unstable public finance. In the years 2010 and 2015, Ireland, Malta and Slovakia, respectively, occupied the first positions in the ranking, while in 2020 Romania, Lithuania, Poland and Bulgaria became the leaders.

Poland on the Path towards Sustainable Development—A Multidimensional Comparative Analysis of the Socio-Economic Development of Polish Regions

The aim of the paper is to conduct research and present the results of a taxonomic analysis assessing the 2019 and 2020 implementations of socio-economic sustainability goals in the Lubuskie Province as compared with other Polish regions. To this end, a multidimensional comparative analysis was performed to determine the level of socio-economic development of the Lubuskie Province and then benchmark it against other regions in the context of sustainable development. The research was based on independent indicators developed with the use of linear ordering methods and accounting for aspects of socio-economic development such as the economic potential, innovation of the economy, sustainable production patterns, demographic changes, job market, and social integration. Synthetic measures were developed with the use of the proposed partial indicators (independent variables), which, according to the authors, best illustrate the socio-economic development and sustainability at the regional level. The regions were then grouped by level of economic development, and the groupings were assessed for their performance. The research shows that in 2020, compared to 2019, the overall level of economic development deteriorated in all Polish regions, but the pandemic has so far had significantly less impact on the social area.

A variational splitting of high-order linear multistep methods for heat transfer and advection–diffusion parabolic problems

The main result of this paper is the generalization of the concept of variational splitting proposed by Behnoudfar et al. (2018) into higher-order time-stepping schemes. The key novelty of the technique proposed here is the following: when using the isogeometric finite elements, one can factorize the matrix of the linear system and obtain the linear complexity. This matrix has the form M+ηK, where M is the mass matrix, K is the stiffness matrix. Following the technique of Behnoudfar et al. (2018), and factorizing M+ηK one would get the additional error of order η2. We propose the technique to couple the possibility of factorization of M+ηK with the higher-order time-stepping schemes and preserve this higher order. Thus, we introduce a variational splitting for implicit, linear multistep methods of order s for parabolic partial differential equations. Our technique exploits the tensor-product structure of the discretization in space (mesh and basis functions) to derive an update strategy with a linear cost regarding the total number of degrees of freedom for multi-dimensional problems. As an example, we consider two popular methods: Adams–Moulton (AM) and backward differentiation formulae (BDF). We derive the corresponding coefficients for AM and BDF methods. We present a C++ implementation of the time-integration schemes, targeting shared-memory Linux clusters. We present numerical results for the heat transfer problem and advection–diffusion problem. We also discuss the limitations of the method and possible extensions overcoming this limitation by using our method as a preconditioner for an iterative solver.

Assessing the Effectiveness of Selected European Innovation Systems

The growing importance of innovation in the modern economy has revived the interest of economic sciences in studies on the mechanisms that govern innovation and its impact on economic development. This growth of interest induced the concept of the national system of innovation (NSI), which occupies an important place in the innovation policy of all developed market economies. The economic literature distinguishes various typologies of innovation systems. The aim of the article is to assess the effectiveness of the system of European integration, the socio‑democratic system, and the mutated system, measured by the level of innovation of the economies that belong to these systems, in 2014 and 2019. The article analyzes the literature on the subject of innovation systems. The method of linear ordering, which makes it possible to build a synthetic measure calculated using the Hellwig method, is used to assess the effectiveness of innovation systems. The article formulates a research hypothesis that the most effective innovation systems are the socio‑democratic system and the system of European integration. This hypothesis has been positively verified.

Is small beautiful? Techinical efficiency and environmental sustainability of small-scale family farms under the conditions of agricultural policy support

The need to guarantee the efficiency of the agricultural sector and to shape production in order to limit its negative impact on the natural environment is one of the most important priorities of the European Union's (EU) Common Agricultural Policy (CAP). In particular, the relationship between efficiency and environmental balance under the policy support for small-scale family farms has not been widely established. Answering this question is important because small-scale farms are the basis for the functioning of agricultural sectors in many regions of the world. Such a situation takes place also in the countries of Central and Eastern Europe (CEE), which, apart from some exceptions (e.g. the Czech Republic), are characterised by a fragmented agrarian structure. Hence, the article is an attempt to identify the connection between the economic and ecological performance of small-scale farming in five countries. As three countries under study belong to the European Union's agricultural support system and two are outside this system, the next target is to check whether farms in countries covered by the CAP showcase higher economic and environmental indicators. The study was based on surveys conducted in small-scale farms in Poland (710 farms), Lithuania (1000), Romania (900), Serbia and Moldova (550). To find the results, data envelopment analysis (DEA), single-factor analysis of variance (the ANOVA method) and linear ordering methods were used. Two main conclusions were drawn: First, regardless of the support system (EU vs. national), the researched farms are characterised by relatively low technical efficiency and relatively high environmental sustainability. Thus, it can be concluded that they are rather weak economically but environmentally friendly. Second, there are no statistically significant differences in the environmental balance and technical efficiency for farms in EU countries (Poland, Lithuania and Romania) and non-EU countries (Serbia and Moldova), despite the fact that the latter obtained significantly lower agricultural support.