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.


Introduction
Every city is a complex, multidimensional unit, dependent on many internal and external factors.Cities are areas with a high and growing concentration of population.Almost 60% of the earth's inhabitants live in cities and the ratio is increasing.The forecasts show that over 80% of the population will live in cities by the end of this century (United Nations, 2019).Although cities occupy less than 5% of the earth's land area, they consume more than 75% of the natural resources and emit 60-80% of the global greenhouse gases.This is a consequence of their crucial role in the world's development process, as economic activity and social, technical, administrative and political affairs are concentrated in cities (Henderson, 2010;Zhang, 2011;Beal & Adam, 2017).Therefore, managing the city is such an important and at the same time difficult task.Moreover, considering that cities' managers have been often pursuing the objective of growth at the expense of sustainability, a new approach to city management is needed.
The sustainable development is the basic and fundamental concept which city management should follow.The origins of the 2030 Agenda for Sustainable Development can be found in the Millennium Agenda, which was the first global sustainability-related plan focused on fighting poverty and its consequences (United Nations, 2015).Based on the "5 Ps": people, planet, prosperity, peace and partnerships, the Agenda has established 17 Sustainable Development Goals (SDGs) and 169 specific targets to be pursued in a 15-year period (Akenroye et al., 2018).SDGs contribute to a better understanding of the complexity of the city when it is managed through the interconnected social, economic and environmental dimensions (Tanguay et al., 2010;Ahvenniemi et al., 2017;Giddings et al., 2002) and respective 169 targets.But the 2030 Agenda does not promote only sustainable development.It also promotes resilience (Sianes et al., 2022).Finally, the technological revolution offers tools to create smart cities.These issuesrespectively, sustainable development, the smart city and resilience-create conceptual research grounds and determine city management.
Management is a process that is supposed to lead to the accomplishment of identified aims in an efficient and effective way, which can be measured and monitored by performance indicators.Various indices and indicators can be used for strategic analysis (diagnosis) to identify a city's potential, help to set up development aims and enable monitoring the performance and comparison with other entities.Measuring socioeconomic development is a challenge and new measures are needed (Stiglitz et al., 2018) as, on one hand, some frameworks lack dedicated indicators and, therefore, they are not holistic; on the other, the number and variety of ratios drive the need to reduce a large amount of information to support decision-making (Wendling et al., 2018;Bibri, 2019;Ahvenniemi et al., 2017;Anderson, 2020).Many methods can be applied to capture the position of a city in relation to other cities via the construction of rankings.In particular, linear ordering methods can be used.These methods include different approaches to standardizing variables and then combining multiple variables into a single (aggregated) measure that can serve as a result according to which cities are ranked.
Given the above, the research central thesis is that city management requires a new hybrid paradigm combining the ideas of sustainable development, the smart city and resilience, called the "Fundamental Power of the City" which are implemented via its Index named the Fundamental Power of the City Index (hereinafter: FPCI).This raises conceptual and methodological questions about the concept of the Fundamental Power of the City and functionality of its Index.The following research questions (RQs) can be posed: RQ1: What is the background of the concept and related assessment metric?RQ2: Is the choice of method relevant?Do different methods of linear ordering provide similar results?RQ3: What are the implications of the rankings for the city stakeholders?
In order to answer the stated questions, the study applies a critical literature review and survey conducted for major cities in Poland as the concept should be locally driven.The local focus is particularly strong in the literature on sustainable development (Chaminade, 2020).The research applies different linear ordering methods to investigate the functionality of the Index.Composite indicators' advantages and disadvantages, requirements relating to data selection and guiding principles for the construction of a composite indicator (Cash et al., 2002;OECD, 2008) were taken under consideration.
The remainder of this paper is organized as follows.The following Sect. 2 provides background for the development of the new hybrid paradigm based on the sustainable development, smart city and resilience -the phenomenon to be measured.Section 3 introduces data and methods.This section includes the information about cities selected for the analysis, sources of data (variables) and procedural information about the selected linear ordering methods that are applied.In Sect. 4 results are presented and discussed.The final Sect. 5 offers the study's discussion and conclusion.It also presents the study limitations and possibilities for further research.
This research responds to the demands relating to redefinition of the city development paradigm and to filling the gaps between sustainable, smart and resilient city concepts.It supports the development of strong and holistic models for the city transformation which was claimed among others by Giddings et al. (2002) and Chaminade (2020).This paper also develops further the research published by authors in 2021 (Wojewnik-Filipkowska et al., 2021) by adding and comparing different methods of linear ordering.The literature and data for the survey have also been updated compared to the first study.Finally, it responds to a demand for empirical testing of different conceptual frameworks (Zhao et al., 2021), procedures (Szymańska et al., 2011;Bąk, 2016), and their correlation (Ahvenniemi et al., 2017).The novelty of the study refers to the concept, as yet unpublished in international journals, of the Fundamental Power of the City and its operationalization via the Index, including its robustness verification using six different linear ordering methods.

Sustainable, Smart and Resilient City: Need for a Hybrid Development Model
In 2018 the European Commission presented a long-term vision aimed at reducing the effects of climate change.This strategy is a tool for implementing the European Green Deal and for achieving climate neutrality.The activities should be local -as confirmed by the New Leipzig Charter (van Lierop, 2020).The New Leipzig Charter, accepted by the unofficial council of EU ministers on urban matters, concentrates on the transformative power of cities.It identifies three dimensions of European cities: a fair city (with equal access to services, education, leisure and health care -for all residents), green (with clean air, water and soil; access to green and recreational areas) and productive (i.e., competitive, attractive for investment, innovative).These concepts reflect three spheres of the city: fair city refers to the social city dimension, green city -to the natural environment and productive city -to the economic dimension.
The New Leipzig Charter is then grounded in the concept of sustainable development.While weak sustainability, presenting the environmental, social and economic themes with equal weighting was popularized based on the Brundtland Report (Morandín-Ahuerma et al., 2019), the concept of strong sustainability should be implemented.
Strong sustainability assumes that natural capital is not replaceable-the three dimensions of sustainable development are nested and eco-centric.The economy is dependent on society and both are dependent on the environment (Giddings et al., 2002).Life beyond the biosphere is just not possible and so is an unlimited growth strategy.The consequences of this include many contemporary problems such as excessive local and global resource consumption, air pollution, urban sprawl and climate problems.These challenges threaten not only each individual citizen but also the entire city -its society and infrastructure, which in turn may cause a social, economic and environmental crisis.Residents are already experiencing heat waves, above-average rainfall, flooding, drought and strong winds (Shi et al., 2010).In large cities the intensity of these phenomena will increase due to the specific conditions occurring in urban areas, such as intensive development, the density of the technical infrastructure and a high degree of surface sealing.As a result, social, economic and environmental shocks can happen.They are interrelated and may occur simultaneously.For example, an economic recession may be the result of or may be linked to environmental or social risks (Simmie & Martin, 2010).
In the situation of shock, the city finds itself in three situations.It can simply try to avoid shock.The second possibility is that the city absorbs the shock and its effects naturally, which is possible thanks to the resources and its characteristics (competencies, potential) activated to reduce social, economic and environmental losses.Thirdly, the city can not only absorb the shock effects but also use them to take a new, better path of development where quality of life is higher than before (Wieteska-Rosiak, 2018).This is a general idea of the concept of resilience.Urban resilience means that the city maintains or returns to the path of development after different shocks.More specifically, resilience is the ability of a vulnerable system, community or society to quickly and effectively capture, absorb, adapt and rebuild itself after disruptions of any nature (Jha et al., 2013).Therefore, the concept of resilience can be perceived as a developed and more advanced concept of sustainability.
To meet the aforementioned challenges cities can implement modern technology.The use of modern technology relates to the concept of the smart city.The term "smart city" has been widely used in urban research, but it is not presented in a holistic way.The prime association is innovative (smart) technology; however, a smart city is not based only on innovative technologies (Girard, 2013;Hollands, 2008).
According to a study by Giffinger et al. (2007) the concept of smart cities embraces the following six smart aspects: the economy (competitiveness), people (social and human capital), governance (participation), mobility (transport and ICT), the environment (natural resources) and living (quality of life)."A Smart City is a city well performing in a forwardlooking way in these six characteristics, built on the 'smart' combination of endowments and activities of self-decisive, independent and aware citizens" (Giffinger et al., 2007).A smart city promotes the use of modern technology in everyday urban life, for instance transport technologies which improve the urban traffic and the inhabitants' mobility.In the relationship between the city government, administration and citizens smart means the employment of new channels of communication (e.g.e-government).In the economy smart is related to the business of information and communication technologies (ICT) and the creation of business parks.It means that cities in their sustainable development must act smarter, which includes employing IT technologies, but are not limited just to them.Ahvenniemi et al. summarize the relation between sustainable and smart cities stating that the general goal of smart cities is to improve sustainability with the help of technology.Thus, they recommend the use of a more accurate term "smart sustainable cities" instead of smart cities (Ahvenniemi et al., 2017).Ahvenniemi et al. also provide a literature analysis of selected smart city and urban sustainability assessment frameworks.They conclude that a large gap between smart city and sustainable city frameworks exists.Their conclusion connects with the matter of weak sustainability: the sustainability is weak if technology and technology-related innovation is seen only as key instruments to replace natural capital for other forms of capital.Technological progress reduces the dependence on natural resources and the ecological impact of growth, but the action should be subordinated to social and, to a certain extent, economic welfare within planetary boundaries as the substitutability of natural capital is not always possible (Chaminade, 2020).This approach led to the synergies of different approaches (Schravena et al., 2021;Wojewnik-Filipkowska et al., 2021).
The literature on the subject describes numerous concepts of city development besides the just, green and productive city enumerated in the Leipzig Charter.Competitive, creative, attractive, vulnerable, wired, knowledge, entrepreneurial, vital and viable, digital cities are discussed in the literature (de Jong et al., 2015).Moreover, these concepts are not exclusive as hybrid concepts develop.
Besides the above-mentioned concept of the smart sustainable city (Wojewnik-Filipkowska, 2017; Ahvenniemi et al., 2017) also the smart slow city or innovative eco-city have been proposed.The idea of the smart slow city concept is to use modern technology to prioritize the quality of life for its citizens.It can be considered as the next level of slow cities development (Farelnik & Stanowicka, 2016).An innovative eco-city can also be perceived as a new level of development of the eco city as contemporary urban development is based on innovation and cities are accelerators of progress and innovation.So the innovative eco-city addresses both ecological sustainability and the new requirements of economic prosperity which is innovation (Łobejko, 2015).
An important issue which might be problematic with the above mentioned concepts is that they evolve and overlap one another, not to mention shortcomings of the individual concepts.Lack of clear definition and interpretation makes it difficult to implement the development model and in result, actions taken in the name of a certain concept are insufficient or ineffective.Lack of concrete policies and actions, a focus on the short-term and shallow solutions, insufficient consideration of social aspects while focusing on economic and environmental issues, and too much faith in technology and innovation are justified critiques.Still, criticism of single concepts is important because it helps to identify and solve problems related to this concept and improve its implementation also in combined strategies of development.Some criticism may refer to the arbitrary selection of the models, but the most important advantage is a holistic approach helping to address the interactions between different aspects of such development.
City development strategy can be therefore built based on selected models separately or jointly.According to Drobniak, the hybridization of development models is a process of joining elements of various models rooted in sustainable development to achieve a higher quality of life in the city (Drobniak, 2019).Sustainable development is therefore a basic concept for city development.Taking into consideration current opportunities and challenges which are the technology revolution and various global crises, we propose to select sustainable, smart and resilient concepts to form the Fundamental Power of the City as presented in Table 1.
Having so many options, city management creates a complex and multi-level system, which ultimately poses a first challenge to the identification of the appropriate strategy of development and, the second challenge -its implementation.The implementation can be supported by managerial methods, which include ratio analysis and benchmarking.This however constitutes the further problematic issue of appropriate indicators as sustainability assessment itself is one of the most complex types of appraisal methodologies (Sala et al., 2015), not to mention the smartness and resilience assessment.The questions (problems) refer to the selection of proxies and the managerial approach -whether decision making should be based on a set of single proxies or (and) a composite index.If the composite index applies -what are the proper statistical methods that need to be applied?
Aggregation and cross-country analysis are not a new concept even though cross-country comparison may be difficult to perform due to problems in the availability and comparability of the data (Chelli et al., 2013).The review over aggregated indicators performed by Ahvenniemi et al. (2017) showed different frameworks of the sustainable and smart city and authors left out weighing and normalization in their comparative analysis.
The aggregation procedure is applied to the Fundamental Power of the City Index.The Index derives from combined measures of sustainable, smart and resilient development.It is a proposal created using a selection of methods of linear ordering.

Data and Methods
The purpose of this effort is to show the methodology for the design, formulation and robustness of the Fundamental Power of the City Index, as well as the possibility of its regional application.
The first step is to identify and organize the main proxies according to the framework of a sustainable, smart and resilient city.
The variables selection is based on the relevant literature in reference to the sustainability, smartness and resilience assessment.Three sets of variables are formed, one for each of the three aforesaid dimensions.The choice of variables is also determined by the formal requirements for the quality of synthetic measures to be obtained.The requirements include silence, credibility and legitimacy.The indicated criteria are generally coherent with the proposals of Agenda 21 and OECD methodology referring to the availability and reliability of the information data source, currency of the statistical data, and data completeness (Barrera-Roldán et al., 2002).
Salience is the term used to describe the relevance of information.High salience information is timely and informs about priority issues.Whether the information is considered to meet the requirements of scientific plausibility and technical adequacy is referred to as credibility.The sources of information, as well as the facts, hypotheses and causal explanations they invoke, must be regarded as reliable and/or believable.A procedure is legitimate if it is fair and objective.Legitimacy is the conviction that a procedure is reasonable and takes different interests, concerns and unique circumstances into account (Cash et al., 2002).
Selecting a set of frameworks was challenging because of the great number of existing performance measurement systems, not only in reference to sustainability assessment (Ahvenniemi et al., 2017;Sala et al., 2015).The selected city indices are presented in Appendix 1 -the list includes rankings based on sustainability, smart and resilient criteria.It also includes rankings which cannot be classified within any kind of city development model and it is closed by two Polish rankings remarkable because of their importance for the SDG promotion (Arcadis, 2021) and scope of the research (Szymańska et al., 2011).Finally, the set of proxy variables measuring the sustainability of the city is based on the 17 Sustainable Development Goals formulated in Agenda 2030 (United Nations, 2015).The set of variables for the smartness of a city is based on the research conducted by Giffinger et al. (2007Giffinger et al. ( , 2010)).The set of variables for the resilience of a city is inspired by several studies (Jha et al., 2013;Drobniak, 2015;Hill et al., 2008;Cutter et al., 2010).For the purpose of the research to check the completeness of data, also the following indexes were analyzed: IESE Cities in Motion (IESE Business School, 2021), Global Livability Index (Economist Intelligence Unit, 2021b), Green City Index (Economist Intelligence Unit, 2021a), Smart City Index (IMD World Competitiveness Center, 2021), Global Power City Index (Institute for Urban Strategies, 2021), City Resilience Index (Arup International Development, 2020).Furthermore, the standard of International Organization for Standardization such as ISO 37 120, ISO 37 122, ISO 37 123 (Marsal-Llacuna, 2016) and recommendations of the International Telecommunication Union for smart sustainable cities (Huovila et al., 2019; ITU-T, 2016) were considered.Finally, for the purpose of variable selection, research by Tanguay et al. (2010), Mori & Christodoulou (2012), Deakin & Reid (2014), Mori & Yamashita (2015) and Dall'O' et al. (2017) were considered.
The second step considers the choice of the object of analysis.For this research, Polish cities with county rights (66 in total) were initially selected due to their urbanization level, and the choice of variables was made based on these cities.However, the final choice of cities is limited to 18 voivodeship capital cities due to their influence on regional development.In addition, this choice is determined by the transparent presentation of results of the analysis.In general, the proposed methodology can be applied to both small and big cities.
The third step refers to data collection.Data is collected from the Central Statistical Office database (GUS, accessed 01.07.2022) and National Electoral Commission (PKW, accessed 19.02.2021).Statistics by GUS relate to the year 2020, except for the attendance in the latest elections to voivodeship regional councils which refers to the latest elections held in 2018; data by PKW on attendance in the latest European Parliament elections held relates to 2019.The materials also include published official development strategies of the cities, available on cities' official websites (the list in Appendix 2), useful to compare the results of their ranking and their current strategy.
After the initial collection of variables is obtained, variability and correlation between variables was measured.Variables characterized by low variability (variability coefficient below 5%) or strong correlation with other variables (Pearson's correlation coefficient above 0.8 or below − 0.8) were removed.These threshold values were chosen based on previous studies present in the literature, which usually assume values between 0.5 and 0.9 for the correlation coefficient threshold and 5-10% for the variation coefficient threshold (Tarka, 2012;Oleńczuk-Paszel & Sompolska-Rzechuła, 2013;Kowalczyk-Rólczyńska, 2016;Zalewska, 2017;Horbaczewska, 2020).100 variables were obtained as a result, among them 32 for constructing the sustainable city's indicator, 48 are for that of the smart city and 20 for that of the resilient city (see Fig. 1).See also the detailed proxies in Appendix 3.
The Fundamental Power of the City Index uses all the variables included in the sustainable, smart and resilient city indicators but the repeating variables are removed so that in the combined Index each variable occurs only once.In this way, an Index including 80 variables is obtained.The selected cities are classified according to this combined Index.
Each variable is characterized as a stimulant or dis-stimulant.For the stimulant, higher values are preferrable and for the dis-stimulant lower values are preferable (Hellwig, 1972).Next, six different linear ordering methods were used to rank the cities according to the Fundamental Power of the City Index, specifically those listed in Table 2, and their results were successively compared to each other.The calculations were performed in Microsoft 365 Excel.
The first method used is the Hellwig's Measure of Economic Development Method (Hellwig, 1972).This method was originally used by the authors in the book (Wojewnik-Filipkowska et al., 2021) developing the Index.As a first step, all variables are standardized according to the formula: where z ij and x ij are respectively the standardized and the observed value of variable j for city i, whereas x j and s j are the mean and the standard deviation of variable j calculated over cities, for i = 1,2,…,n and j = 1,2,…,k, being here n = 18 and k = 80.
(1)  Next, two vectors of pattern and anti-pattern objects (cities) are created.A pattern is assigned considering the best possible value for each variable, that is the maximum in the dataset for stimulants and the minimum for dis-stimulants.For the anti-pattern the worst possible values are assigned, so the minimum value in the dataset for stimulants and the maximum value in the dataset for dis-stimulants.The Euclidean distance d 0 between the pattern and anti-pattern and Euclidean distance d i0 between pattern and each object is then calculated according to the formulas: where z 0j and z −0j are the value of variable j into the pattern and into the anti-pattern vec- tors respectively.
Taxonomic measure m 1i is then obtained for each city i using the formula1 : The second linear ordering method used is TOPSIS, created by Hwang and Yoon (1981) also described by Jahanshahloo et al. (2006).In this method variables are normalized as follows: Next, the vectors of the pattern and anti-pattern objects are prepared as explained above.The Euclidean distance d i0+ of each city from the pattern and Euclidean distance d i0− of each city from the anti-pattern are calculated as follows: where y 0j and y −0j represent respectively the value of variable j into the pattern and into the anti-pattern vectors.
The final taxonomic measure m 2i for the city i in the TOPSIS method is: (2) The next four linear ordering methods apply a different approach as they do not use the pattern and anti-pattern in the calculation of the taxonomic measure.The first is the method of standardized sums (Balicki, 2013).Variables are standardized as follows: Taxonomic measure m 3i for the city i is then calculated by averaging the standardized values by varying variables: The fourth method considered in this paper is a variation of the method of standardized sums, where standardization is done according to the formula: where Me j and R j are the median and the range (distance between the maximum and the minimum value) of variable j respectively.
Taxonomic measure m 4i for the city i is then calculated by averaging the standardized values over variables: The first approach (used in the calculation of m 3 ) uses classical measures -the mean and standard deviation.The second ( m 4 ) employs the median and range, the posi- tion measures.Both approaches are used to make the variables uniform.The first one achieves this in relation to the standard deviation (all standardized variables have a standard deviation equal to 1), the second in relation to the range of values (range of transformed variables is now equal to 1).
Next, the method of distances is used (Balicki, 2013).Standardization of variables is performed according to the formula: where max i x ij and min i x ij are the maximum and the minimum value of variable j respectively. (8) for stimulants x ij for dis − stimulants Taxonomic measure m 5i for the city i is then calculated by averaging the standardized values: Since in the case of two variables (length of unsurfaced roads and number of road accident casualties) two cities recorded a value of zero, it was not possible to include them in the calculation using this method.Standardization for these two cities would require division by zero, thus these two variables were removed.
The final method is that of ranks (Balicki, 2013).For each variable, the values are sorted from highest to lowest (for stimulants) and lowest to highest (for dis-stimulants).To each city a rank r ij is assigned and the ranks are averaged to create the last taxonomic measure:

Results
The cities in the ranking are listed according to decreasing values of m 1 .Warszawa ranked highest in all the rankings, which proves its well-established leadership position (see Table 3).
Hellwig's measure ( m 1 ) can take values between 0 and 1, with 0 signifying that the city is an anti-pattern and 1 that it is a pattern (ideal object).In the ranking shown in Table 3 no city achieves a value close to 1. Warszawa, the capital, is the only city with a value over 0.5.The second and third ranked cities (Opole and Kraków) score values close to 0.48.Values for the worst ranked cities (Łódź and Bydgoszcz) are lower than 0.3.Similarly, in the TOPSIS method ( m 2 ) the values that can be achieved fall in the range from 0 to 1, as it is also based on measuring the distance from the pattern and the anti-pattern.Most cities in the ranking achieve a value close to 0.5.Warszawa again has the maximum value of 0.62.These two first rankings are similar, with a couple of differences, namely Opole which ranks 2nd according to Hellwig's method but 16th according to the TOPSIS method, and Zielona Góra which ranks 15th according to Hellwig's method and 6th according to the TOPSIS method.These differences can be attributed to different methods of standardizing variables and different formulas for calculating the final measure.
In both methods based on standardized sums ( m 3 and m 4 ) the values are not restricted to any range.Positive values indicate that the city obtained an average result above the "average object", with "average object" defined by the mean in the method with classical measures and the median in the method with position measures.Both rankings are in fact similar, with Katowice and Wrocław switching ranks being the only difference.Both also resemble the ranking prepared using Hellwig's method ( m 1 ).
In the method of distances ( m 5 ), the measure typically takes values between 0 and 1.Most considered cities fall in the middle of this range, achieving a value close to 0.5.In the method of ranks ( m 6 ), the actual values achieved by cities for each variable are not considered, instead only the rank according to each variable is taken into account, which is then averaged over the whole set of 80 variables.As seen in Table 3, the average rank for most cities falls in the middle of the possible ranks: 1-18.( 14) Ranking according to the method of ranks ( m 6 ) is very similar to some other ones (spe- cifically Hellwig's method ( m 1 ), method of standardized sums ( m 3 and m 4 )).On the other hand, method of distances ( m 5 ) produces a slightly different ranking from those obtained applying the four methods just mentioned.The main differences are the position of Opole (ranked 11th, as opposed to rank 2 in Hellwig's method ( m 1 ) and 3 in the method of stand- ardized sums ( m 3 and m 4 )) and Lublin (ranked 3rd, as opposed to rank 8th in Hellwig's method ( m 1 ) and 9th in the method of standardized sums ( m 3 and m 4 )).Table 4 presents descriptive statistics calculated for each of the six methods used.
For the two indices built using the pattern and anti-pattern the mean value is around the middle of possible values -0.396 for m 1 and 0.515 for m 2 .The values according to the second method are much less variable, as illustrated by the variation coefficient (9%, compared with 18% in the case of the first method) and minimum and maximum values.Indices based on methods three and four have a similar mean value (zero for m 3 and 0.015 for m 4 ).This means it is not possible to calculate a variation coefficient for m 3 .The minimum and maximum values show however that m 3 has a greater spread of values than m 4 .The variation coefficient for m 4 is very high, due to the very low (almost zero) mean value.The variability of the last two indices is similar to that of m 1 .
To investigate further similarities and differences between the measures obtained by applying the six indices Pearson's correlation coefficient was calculated.Results are shown in Table 5.
As shown in Table 5, most methods produce strongly correlated values.Correlations between values for the method of ranks ( m 6 ) and other methods are negative because in m 6 lower values are preferable, while in other methods higher values are preferable.Hell- wig's method ( m 1 ), both versions of the method of standardized sums ( m 3 and m 4 ) and the method of ranks ( m 6 ) show a very high correlation between their values (over |0.975|).Two other methods -TOPSIS ( m 2 ) and the method of distances ( m 5 ) produce values that are less strongly correlated with the other indices.The values of correlation coefficient between the results of TOPSIS and other methods are around 0.62-0.69 in absolute value, signifying a medium correlation.The values of the correlation coefficient between the results of the method of distances and other methods are slightly higher, around 0.66-0.72 in absolute value.In all the methods Warszawa ranks number 1.The 2nd and 3rd places differ depending on the method used and are taken by Opole (methods m 1 and m 6 ), Kraków, Wrocław, Rzeszów and Lublin.Similarly, the cities ranked last also differ depending on the method applied.Last place is occupied by Bydgoszcz, Łódź and Gorzów Wielkopolski.Second and third to last are also Opole, Szczecin and Kielce.The first three and the last three cities in rankings according to each method and their location on the map of Poland are shown in Fig. 2. It has been prepared by PowerBI.

Discussion and Conclusions
The first two methods applied, Hellwig's Measure of Economic Development ( m 1 ) and TOPSIS ( m 2 ) use pattern and anti-pattern objects to rank cities.The next two methods dif- fer from each other in the way the variables are standardized: the first is a method of standardized sums ( m 3 ) which uses classical measures (mean and standard deviation), the second a variation of this method ( m 4 ) which uses position measures (median and range).The last two methods are that of distances ( m 5 ) and that of ranks ( m 6 ).
In our opinion, of great importance is the fact relevant to the question of rankings' implications that the choice of the linear ordering method can have an influence on the results and therefore both city stakeholders and managers should be aware of the risks underlying the results' control.There is a threat that the ranking is manipulated due to the deliberate choice of the method.In addition, besides the availability of the data which seems a subjective obstacle, the choice of variables used for constructing sustainable, smart and resilient city indicators and as result the Fundamental Power of the City Index is a subjective one which can affect the results and rankings obtained.Although this choice in the research is made based on the appropriate literature review and data quality considerations, the subjectivity involved in the decision should be considered a disadvantage (Mori & Christodoulou, 2012;Tanguay et al., 2010).Still, the results of the conducted analysis show that most methods of linear ordering produce similar rankings.The correlation between the results of Hellwig's method ( m 1 ), methods of standardized sums ( m 3 and m 4 ) and method of ranks ( m 6 ) is very high (above 0.9) and the rankings obtained by these methods differ only in minor details.TOPSIS and the method of distances produce slightly different rankings, but still moderately correlated with the results of other methods.
In rankings prepared using methods that apply the pattern and anti-pattern none of the cities is ranked close to the perfect object.Most cities achieve average results.In ranking by Hellwig's method most cities obtain results around or below 0.5, signifying that their performances are closer to those of the anti-pattern rather than pattern.In TOPSIS and the method of distances ranking most cities achieve values around 0.5, again showing the average results.The standardized sums method ranking confirms this, by showing cities achieving average values (between − 0.5 and 0.7 in the method using classical measures for standardization and between − 0.2 and 0.2 in the method using position measures).In both of these methods the value "0" means the city achieves an average result being in line with the average over all cities.In the last considered method, the method of ranks, cities obtain Fig. 2 The first three and the last three cities in the ranking according to different methods of linear ordering-year 2020.Source: own work based on data from (GUS, accessed 01.07.2022;PKW, accessed 19.02.2021) average ranks between 7.3 and 11.8, hence middle to high values in the possible range of values from 1 to 18, where high values mean a low position of a city in the ranking.
In all the methods Warszawa, the capital city with approximately 1.8 million inhabitants, ranks 1st.The City's strategic management is aimed at the development of innovation through smart, inclusive and sustainable growth; the managers focus on the efficient adaptation of the most productive technologies, but also the implementation of their own innovations, increase in the effectiveness of research and development, financing, strengthening the local potential and human resources development.The city's fundamental power is confirmed by the high results of variables of a sustainable, smart and resilient city while still the overall results were not close to the pattern.
In most rankings it is the same set of cities which reach the top.Among them are Kraków (0.802 mln inhabitants) and Wrocław (0.674 mln inhabitants), the most populous cities in Poland after Warszawa.Kraków builds the fundamental power of the city mainly thanks to variables related to a sustainable society (expenditure on protection of cultural heritage), smart people (readers of public libraries, net international migration) and smart life (low number of crimes against public safety and safety in transport).Strategic management in Wrocław allows to get good results in the smart life category (higher average floor area of dwelling per person, lower number of families awarded aid), smart governance (higher percentage of children cared for in nurseries and lower percentage of children for whom parents claim family allowance) and sustainable environment (expenditure on environmental protection).
Also Opole (0.126 mln inhabitants), Rzeszów (0.196 mln inhabitants) and Lublin (0.332 mln inhabitants) are highly ranked in terms of the Fundamental Power Index.These are representatives of medium sized cities according to their population.Opole builds its power due to its strategy of a city that is engaged and socially responsible; Rzeszów derives its power mainly due to its society, to a lesser extent its economy; Lublin in the resilient and sustainable society categories.These conclusions based on values of indices are consistent with the development strategies formulated by the cities.
Cities closing the rankings also require comments in relation to the strategies formulated.In 2013, the City Council of Bydgoszcz adopted a new strategy document.It focused on improving the quality of life of residents and increasing the competitiveness of the city, assuming that the most urgent problems to be solved were infrastructural deficiencies.The strategy points to the problem of the capital intensity of these investments, which confirms that the position of Bydgoszcz is justified by the negative indicators in this area and the time of catching up with infrastructure is long.
In the case of Łódź, a city with an industrial character, a special focus is devoted to attractiveness factors for the development of modern industry and services.Łódź indicates in its strategy a comprehensive offer combining different areas of life, which allows for increasing the attractiveness of the city for young people.Despite many road investments, there is still a problem with mobility efficiency.The strategy so far has also not paid enough attention to the challenges posed by the climate crisis, including the implementation of blue-green infrastructure.Therefore, the low assessment of Łódź is also justified, despite taking action to improve the situation.
Negative demographic phenomena and the place of the second capital of the voivodship seem to be the main reasons for the generally low position of Gorzów Wielkopolski in the ranking.The strategy in this area is, in a sense, too universal, outlining various actions that create the conditions for a balance between professional, family and personal life.The smart city as the dominant development direction of Kielce was indicated in the strategy adopted in 2022.Kielce aims to be a city for everyone, efficient and active, developing.
Szczecin formulates actions in its strategy along the vision of being an open and tolerant city, an attractive place to live and work, respecting cultural heritage and the natural environment.
The core of strategic activities in Opole is technology, the region, greenery and the environment, citizens and science.Therefore, the city focuses on sustainable and intelligent development.Opole is the only city which takes the leading place or one of the last places in rankings depending on the method applied.As already indicated, these differences might be attributable to various standardization techniques and various final measure calculation procedures.In relation to previous studies based on data from 2019 (Wojewnik-Filipkowska et al., 2021), Kielce and Opole worsened their place in the ranking.The other cities at the end of the ranking have similar results.Given the long-term nature of the strategy, it is difficult to expect a significant improvement in such a short period of time taking into consideration also that all the cities continue their development while the ranking provides comparison within a set of the cities.Still, the biggest problems of the above-mentioned cities are illustrated by the value of variables related to technical infrastructure and economic activity -the latter may be the result of the general economic slowdown.
In terms of methodological discussion, it is worth in our opinion referring to one research particularly.The most comprehensive scientific study in terms of the number of indicators and spatial scope was presented by Szymańska et al. (2011).The analysis included 38 large cities (with 100,000 and more than 100,000 inhabitants) and 59 variables in a total of 10 groups.This study also provides information on how the choice of different computational methods and data normalization techniques affects the city rankings and classifications.We can confirm the conclusions of Szymańska et al. (2011).Comparing the positions of cities obtained by different methods of normalization while maintaining the same input variables, the positions of individual cities have changed.On the other hand, generally, high positions in the rankings of the same cities maintain their well-established strong position.Regardless of the calculation technique, the same cities were in the top position.Both in the study of Szymańska et al. (2011), as well as in the present study, Warszawa and Krakow ranked high.Similarly, Szczecin and Bydgoszcz had the final places in the rankings in both studies.This also indicates that the development and development strategy is not a matter of just a moment of time to provide rapid results.It is rather a timelasting process of planning, organization, motivation and control which are basically functions of management, here city management, where multicriteria analysis can be used to support the decision-making.Nevertheless, both studies are primarily methodical.They are part of the trend of searching for appropriate methods of ranking.
From the concept to local practice, we can indicate that the analyzed cities in Poland are characterized by factors which build their fundamental power in the social area (human capital), culture and strengthening relations between residents (social capital).On the other hand, technical infrastructure and economic activity (economic dimension, productive and material resources, institutional, infrastructural and economic resilience) represent the spheres that need improvement.Analyzing the research results in the light of the content of strategic documents it can be also concluded that sustainable development is a commonly accepted standard, most cities refer to the concept of smartness, but none of them acknowledges resilience.It seems that resilience is a relatively new concept.
Finally, to answer the question about the concept background and related assessment metric (RQ 1), the Fundamental Power of the City concept is built on the grounds of sustainable development, the smart city and city resilience, and therefore fills the gaps between them.This concept fulfills the aims and assumptions of the Brundtland Report and Leipzig Charter.It meets the requirement of a holistic paradigm indicated in the literature review and deliver synergies of different approaches.Considering strategic development in a holistic way, the concept of the Fundamental Power of the City can help to transform and improve cities.The transformation may take place thanks to the implementation of the Fundamental Power of the City Index.It can support the strategic diagnosis and monitoring of the city.
According to the recommendation formulated by Tanguay et al. (2010) in relation to sustainable development, the components and constituent categories should be covered as broadly as possible while the number of variables should be minimized.A similar conclusion was drawn by Mori and Christodoulu (2012).They also claim that it is necessary to create a new city sustainability index in order to measure the cities impact on the environment, human life and the economy to provide local authorities with guidance toward sustainable paths of development and our research responses to that demand.Their research claims that there is no index/indicator that satisfies the formulated requirements among which is the need to assess the performance of cities in both developed and developing countries in an equal manner.
In this respect we draw a slightly different conclusion, mainly because the local focus, not national, is particularly strong in the literature on sustainable development (Chaminade, 2020).This point of view is also expressed by Dall'O et al. (2017) claiming that existing indicators for smart cities are tailored for large cities and do not fit the small ones.We also believe that keeping an equal manner of approach is not possible and not proper if the local values and individual city advantages (but also problems) are to be recognized and appreciated (or mitigated).On the other hand, we confirm that the research limitation may refer to using variables calculated per capita, as the total value of the variable may still matter even if the relevance per capita may be small.
We agree with Mori and Christodoulou (2012) that a synthesized composite index encourages the trade off among the environmental, economic and social dimensions which imply weak sustainability.The Fundamental Power of the City Index concerns not only sustainable but also smart and resilient variables.It enables analysis within sub concepts and subdimensions comprised in the Index, which allows considering the Fundamental Power of the City as a strong, not a weak concept.The composite Index is a clear and transparent tool for benchmarking and decision making on the condition of conscientious variables and aggregation method selection.It is possible to decompose the Index into sustainable, smart and resilient cities rankings to find out the contribution of each performance assessment to the fundamental power of the city.
We confirm the expressed limitation of the composite indexes in general.The selection of variables is a subject of arbitrary decision determined, among others, by the city's uniqueness.The consequence might be that the comparison of cities of different countries is limited as the universal index may leave the specificity of a city or country behind.The solution might be to create a unique set of variables which respects this specificity but still cover the requirements of sustainability, smartness and resilience.But not only the development aim must be clear (whether city strategy development is towards sustainable, smart or/and resilient city) but also the aim of the assessment itself.If the aim of the analysis is to help to identify the most urgent problems of the unique municipality, a unique set of variables can be used.This, however, simultaneously limits opportunities for benchmarking.On the other hand, if the aim of assessment refers to benchmarking, a universal set of proxies should be used so that Index can be comparable.Apart from the assessment aim, the limitation referring to variable selection can also result from the lack of the availability of data, particularly in relation to small administrative units like cities.In this sense, the problem of subjectivity of variables selection can be solved indirectly by applying the mentioned rules of silence, credibility, and legitimacy and also rules of availability and reliability of the information data sources.We think that using sophisticated ratios based on data not provided on free and transparent basis could require additional costs.We believe that the Index should be accessible also for less prosperous municipalities.
In reference to the second research question about relevance of the method choice (RQ 2a), several methods could be applied in order to select predictor variables.For modelling dependent variable stepwise selection with a statistical test, Akaike's Information Criterion AIC or Schwarz's Bayesian Information Criterion BIC, regression trees (Murtaugh, 2009) or Hellwig's approach (Kajewska-Szkudlarek & Łyczko, 2021) could be used.When no response variable is present in the study (as is the case in this paper), the choice regarding variables can be made using Hellwig's parametric method (Młodak, 2006).The drawback of this method is its focus on the direct relationship between variables, not taking into account indirect relationships that might (and often do) exist between economic and social variables.In this research the decision as to which variable(s) among the correlated ones were removed was made based on merits, taking into account the interrelations between different variables.Although most methods produce strongly correlated values, when it comes to details, we cannot confirm obtaining similar results using different methods of linear ordering.Therefore, the answer to the research question of similar results of linear ordering methods is negative (RQ 2b).These differences can be attributed to different methods of standardizing variables and different formulas for calculating the final measure.
We believe that our approach is beneficial for local initiatives and may form grounds for future research.Deakin and Reid (2014) conclude that there is a demand for more complex and advanced methods to build the systemic and multi-model capacity of the city.There are final concluding issues which connects with the research questions of assessment criteria (RQ 1: What is the background of the development concept and related assessment metric?) and methods (RQ 2: Is choice of the method relevant?Do different methods of linear ordering provide similar results?) in a general way.But in particular, these issues refer to policy implementation and the implication of rankings on city stakeholders and managers and answers the remaining question (RQ3).They include: • Methodologies for city development assessment must be trans-disciplinary and holistic; however, there is no single best measure and certain measures may be more suitable in certain situations, especially when the assessment aim refers to problem identification and not benchmarking.
• Use of different metrics, ratios, or indices must relate to an understanding of what is measured in terms of conceptual foundations, their potential biases and limitations, but also why it is measured.• City development is facilitated both top-down (e.g.Agenda 2030) and down-top (e. g. Liepzig Charter) so the development assessment applies both to the local and national scales enabling stand-alone and comparative analysis.• Development of suitable procedures and methods requires awareness of their advantages and disadvantages, including being sensitive to methodological issues, political influence, and should be adjusted to the aim of development.• The Index usefulness but also usefulness of any managerial tools to enhance positive impacts depends on the formulation of clear and well-founded aims decomposed into actions, as development is achieved by means of action taken and, as already mentioned, the aim of the assessment itself.• When applied in a systematic way, the Index can provide information relating to the path of the city towards development, its stability, progress and the trend, therefore it can be useful for public managers and provide support for decision-making.However one must acknowledge the methodological requirements and limitations as it has been already discussed above.
Our future research considers verification of the Fundamental Power of the City Index's implementation with the use of dedicated dashboards within individual municipalities.Additionally, the search towards approach and methods consistent with the concept of strong sustainability, smartness and resilience will be continued.We are also interested in extending our preliminary analysis of Polish cities to cities of different countries, including comparative research.We are aware that might be not an easy task as the collection of the data may be difficult, no assessment approach can fit all and data may not be comparable.Still, we believe that the concept can be informative to policy makers, managers and citizens.

Fig. 1
Fig. 1 Summary of the sets of variables in the study.Source: own elaboration pollutants (gases) from entities harmful to the environment per 1000 inhabitants (t/year) pollutants (gases) from entities harmful to the environment per 1000 inhabitants (t/year) change in own revenue derived from a share in revenue from corporate income taxes per inhabitants general education high schools per 1000 inhabitants Stimulant

Table 1
Components of the new paradigm of the Fundamental Power of the City -a sustainable, smart and resilient city.Source:

Table 2
List of linear ordering methods applied in the research.
Source: own elaboration Acronyms Method m 1 Hellwig's Measure of Economic Development Method

Table 3
Indices' values and rankings of Polish cities according to the six methods -year 2020.Source:

Table 4
Descriptive statistics for each of the six methods based on data for 2020.Source: own work based on data from(GUS, accessed 01.07.2022;PKW, accessed 19.02.2021)

Table 8
Variables of sustainable city indicator.