A Book in Honour of Professor Vijay Verma

Gianni Betti and Achille Lemmi

This is the fourth book edited by Gianni Betti and Achille Lemmi; it contains a balance between theoretical/methodological and empirical chapters prepared by a group of contributors who are a mix of luminaries in socioeconomic fields (Jacques Silber, Vijay Verma, Francesca Bettio, Stephan Mussard, Monica Pratesi, Maria Noel Pi-Alperin, Tomasz Panek, among others) and a group of young and very promising researchers.

In the first of the three previous books they со-edited, Betti and Lemmi assembled advanced thinking about the multi-dimensional measurement of poverty, including the theoretical background, applications to cross-sections and longitudinal aspects of multi-dimensional fuzzy poverty analysis, with particular attention to the transitory, or temporary, conditions that often occur during transitions to a market economy (Lemmi and Betti, 2006a).

Next, Betti and Lemmi (2008a) collected an impressive set of contributions that emerged from the conference in Certosa di Pontignano in May 2005 in honour of two social scientists at the turn of the twentieth century: C. Gini and M. O. Lorenz. In particular, the main themes that Betti and Lemmi (2008b) presented in this second book were related to: innovation in the theory and methods on income inequality and concentration analysis, decomposition of inequality and empirical applications in applied economic analysis worldwide. These research themes are a consistent and relevant part of the recent economic, econometric and statistical methods and applications in the international academic literature.

Finally, in the third book, Betti and Lemmi (2013a) proposed an up-to- date and innovative survey of new methods for estimating poverty at the local level, as well as the most recent multi-dimensional methods used to examine the dynamics of poverty (Betti and Lemmi, 2013b). They argued that the most useful measures of poverty and inequality for policymakers and researchers are finely disaggregated into small geographic units. This volume was the first attempt to compile the most recent research results about local estimates of multi-dimensional deprivation (Betti and Lemmi, 2013c).

This collection is dedicated to the memory of Professor Vijay Verma, who passed away in September 2018, and also contains his new research in Chapter 7 on the jack-knife repeated replication technique for variance estimates of fuzzy poverty measures, with a foreword from his wife, Gillian Verma.

Verma greatly contributed to the development of the fuzzy, multi-dimensional and longitudinal approach to poverty measurement; his original ideas and suggestions have inspired many researchers over the past two decades. Directly or indirectly, his ideas are found in most of the chapters in this book.

The relationship between Vijay Verma and this volume’s editors evolved from purely academic links to a strong and long-lasting friendship, with common respect and trust. These common interests, in terms of research and a lust for life, drove Vijay to move from London to Siena in 2002; a few years later, Gillian joined Vijay, and they became fully ‘Sienese’.

In the early 1990s, Lemmi presented the then-brand-new fuzzy approach to poverty measurement at a Eurostat meeting on official poverty in Stockholm, which had previously been discussed at the Italian National Institute of Statistics (Istat), accompanied by the Istat researchers Giuliana Coccia and Mauro Masselli. After the presentation, Achille desperately looked for a place to smoke a cigarette, finding an open space where a nice-looking bearded gentleman was puffing on his cigarette. Achille joined him, and they started to discuss the presentation and the new and interesting idea of fuzzy versus Boolean states. ‘The bearded gentleman’ offered Achille congratulations on both the presentation and the new ideas in it. Afterwards, Achille returned to Mauro, who exclaimed enthusiastically in Roman dialect: ‘Anvedi! Avemo fatto un figurone! [Wow, we made a good impression!]. Do you know who he is? He is Vijay Verma, the father of ECHP [European Community Household Panel survey]’.

On that day, a fruitful collaboration began, under the direction of David Rose based at the Institute for Social and Economic Research (ISER) at the University of Essex, where several researchers from the University of Siena - Giulio Ghellini, Bruno Cheli, Nicoletta Pannuzi and Gianni Betti - later spent some months working with Vijay, under the supervision of Achille. This first collaboration ended with the first two seminal papers on the fuzzy- set approach to poverty measurement: Cheli et al. (1994) and Cheli and Lemmi (1995).

Gianni Betti first met Vijay Verma during the lectures that he delivered as part of the PhD course in applied statistics at the University of Florence during the academic year 1996-1997. Later, in November 1997, Gianni joined the ISER at the University of Essex, where Vijay was reproducing the British Household Panel Survey as the British ECHP. Vijay supervised Gianni in the calculation of special weights to be used in a pseudo-panel for estimating longitudinal equivalence scales (Betti, 1998), which became the basis for a collaboration to estimate the aggregated weights in the multi-dimensional and fuzzy approach. In the spring of 1998, Vijay and Gianni spent many mornings in Wivenhoe Park, one of the ‘squares’ at Essex University, defining the prevalence-correlation weights, first published in Betti and Verma (1998) and later presented at the sixth Islamic Countries Conference on Statistical Sciences in Lahore, Pakistan (Betti and Verma, 1999). Nearly a decade later, the paper was finally published, with some improvements, as Betti and Verma (2008).

The present collection gathers chapters of very high academic quality on multi-dimensional indices in the social sciences. Over the past three decades, several methodological studies in these fields have been developed, especially on poverty measurement, and multi-dimensional indices have recently become a key research topic in other fields, such as marital disruption, sustainability, violence against women, educational mismatch and quality of life. Most noteworthy is the use of fuzzy logic in these works. Fuzzy-set theory has been shown to be a powerful tool for describing the multi-dimensionality and complexity of social phenomena, replacing the classical crisp approach, which generally tends to overestimate or underestimate social dynamics.

Fuzzy-set theory, introduced by Zadeh in 1965, emerged in response to evidence that real situations are often characterised by imprecision, uncertainty and vagueness and cannot be described properly with the classical set theory, which represents reality with a simple true-false binary logic. Indeed, in the classical crisp approach, sets are characterised by sharp and clearly defined boundaries; thus, an item might fully belong or not belong at all to a set according to a bivalent condition. By contrast, in fuzzy-set theory, an item can belong to a set with partial degrees of membership between 0 and 1, not only with the extreme values.

For all these reasons, the objective of this collection is to explore the most up-to-date fuzzy-set methods for the measurement of socio-economic phenomena from a multi-dimensional and/or dynamic perspective. The chapters were selected based on three criteria: scientific quality, their ability to represent the leading paths in current scientific research, and the existence of a sort of scientific 'file rouge’ among the contributions in order to present a homogeneous, useful and updated group of discussions.

We invited the most authoritative researchers worldwide to submit high-quality original research and surveys as chapters on the fuzzy measurement of socio-economic phenomena. The main themes are related in particular to descriptions of the evolution and recent research findings on multi-dimensional fuzzy poverty and social exclusion (Part 1, consisting of Chapters 2-7), the extension of the fuzzy multi-dimensional method to measuring the quality of life (QoL; Part 2, comprising Chapters 8-13), and the extension of the methods to specific social science fields, such as the labour market, educational mismatch, sustainability, industrial processes and violence against women (Part 3, made up of Chapters 14-20).

After this introduction, in Chapter 2, Bruno Cheli, Achille Lemmi, Nicoletta Pannuzi and Andrea Regoli describe the ‘Evolution of the Fuzzy-Set Approach to Multi-Dimensional Poverty Measurement’. The relevant key points in the chapter, which are preliminary to any discussion of the methods used for a multi-dimensional analysis of poverty, are the selection of the relevant dimensions and the indicators used to measure people’s achievements in these dimensions. They also mention the related issue of the choice of deprivation thresholds and the weights assigned to each dimension, but they do not discuss them in a full way as done in Betti et al. (2008). Rather, they simply state that the binary distinction between a ‘bad state’ and a ‘good state’ is too sharp because deprivation is likely to occur in degrees. Beginning with this consideration, they retrace and update the fuzzy-set approach (Zadeh, 1965) for measuring multi-dimensional poverty, which leads to the integrated fuzzy and relative method. This chapter is not only the basic reference point for the remaining chapters in the first section but also a starting point in the development of a fuzzy approach to the measurement of any phenomenon in the social sciences.

Chapter 3, by Jose Espinoza-Delgado and Jacques Silber on ‘Using Rippin’s Approach to Estimate Multi-Dimensional Poverty in Central America’, describes the mainstream approach to the measurement of multi-dimensional poverty in developing countries, which is insensitive to inequality among the poor and also overlooks intra-household inequality. Consequently, the authors of this chapter propose a departure from the mainstream approach and take an individual-based and inequality-sensitive view of multi-dimensional poverty when only dichotomised variables are available. They estimate multi-dimensional poverty among individuals between the ages of 18 and 59 living in Guatemala, El Salvador, Honduras, Nicaragua and Costa Rica. Overall, the most interesting finding is that people in Guatemala have the highest potential for multi-dimensional poverty, followed by those in Nicaragua; people in Costa Rica, by contrast, have by far the lowest probability of being poor. Honduras and El Salvador are in the middle, with Hondurans having a larger probability of multi-dimensional poverty than Salvadorans.

Chapter 4 analyses the measurement of poverty using the framework of the 2030 UN Agenda for Sustainable Development, in which one of the most important goals is to ‘eradicate poverty, in all its forms and dimensions’. This has been particularly necessary since 2008, when the global financial crisis started, and 2015, with the failure to achieve the millennium development goal of halving extreme poverty in the world. The chapter authors, Gianni Betti, Federico Crescenzi and Francesca Gagliardi, respond to the question ‘Can a neighbouring region influence poverty?’ using a fuzzy and longitudinal approach. They also discuss the adoption of a longitudinal measure proposed by Verma et al. (2017), which is based on the fuzzy-set approach to multi-dimensional poverty: the ‘fuzzy at-persistent-risk-of-pov- erty rate’. Then they estimate this measure at the regional level with small area estimation techniques by introducing a spatial correlation model. In this way, the approach takes into account whether a neighbouring region can influence poverty in all its forms and dimensions: the multi-dimensional dimension, the regional dimension and the longitudinal dimension.

In Chapter 5, Hossein Khoshbakht, Francesca Gagliardi and Ali Asadi use a multi-dimensional and fuzzy-set approach to measure poverty at the province level in Iran, offering the first attempt to use this approach for estimating poverty at the local level in that country. They go beyond conventional studies of poverty based simply on a dichotomy between those who are and are not poor defined in relation to a poverty line drawn only on the basis of income or total expenditure. They use the methodology in the Household Budget Survey in Iran, with a large set of indicators to examine the latent dimensions of non-monetary poverty for the period 2016-2017.

Another interesting application on an emerging country is reported in Chapter 6, ‘China’s Multi-dimensional Poverty and Trade’, by Elisabetta Croci Angelini and Yang Liu. They explain that, over the past 40 years of reform and opening-up, China’s economy has maintained rapid growth. Its achievements in poverty reduction have been remarkable: more than 700 million Chinese have been lifted out of poverty. Meanwhile, the volume of China’s import and export trade has risen several fold, and, in 2017, China became the world’s largest importer and exporter. By conducting an analysis at the provincial level, this chapter explores the links between poverty and trade and how poverty restricts participation in international trade. Assessments of poverty usually rely on a monetary variable, but in this chapter the authors adopt a multi-dimensional measurement of poverty calculated with a fuzzy-set methodology at the provincial level and compare it to the volume of trade, revealing an interesting relationship between multi-dimensional poverty and global trade.

Chapter 7, the final chapter in Part 1, presents the new and final research of Vijay Verma, in collaboration with Gianni Betti and Francesca Gagliardi. In this chapter, they further extend variance estimation to longitudinal multi-dimensional fuzzy poverty measures. The measures considered are based on fuzzy representations of individuals’ propensity for deprivation in monetary and different non-monetary dimensions and are derived from sample surveys with complex designs and fairly large samples. In particular, the chapter adopts a new longitudinal measure based on the fuzzy-set approach to multi-dimensional poverty proposed by Verma et al. (2017): the ‘fuzzy at-persistent-risk-of-poverty’ rate. The authors present a practical methodology for variance estimation - in particular, the jack-knife repeated replication method - for multi-dimensional measures of poverty and deprivation of households and individuals in a longitudinal context. They quantitatively illustrate the calculation procedures and difficulties in producing reliable and robust estimates of sampling errors for such measures. Some of the problems encountered are identified, and solutions are provided in the context of actual conditions.

Part 2 is devoted to phenomena related to the quality of life. Over the past few decades, QoL has become a key issue in modern society and one of the most important goals for individuals. QoL implies, first, that the minimum conditions required for humans to thrive are met and, second, that opportunities and skills adequately match (Veenhoven, 2000; Betti, 2017).

On this basis, in Chapter 8, Antonella D’Agostino, Giulio Ghellini, Maria Navarro and Angeles Sanchez present an ‘Overview of the Quality of Life in Europe’. They focus on the measurement of QoL in Europe, considering it a latent concept that can be studied as a foundational measurement model. That is, QoL is assumed to be defined by a set of indicators: objective or social indicators that reflect people’s objective conditions within a given cultural or geographic area and indicators of subjective well-being that reflect individual judgements of well-being. In this light, the main goal is to show the utility of applying a fuzzy-set approach in this framework, using microdata collected through sample surveys.

The second aspect of QoL, financial literacy measurement, is addressed by Albert Hizgilov and Jacques Silber in Chapter 9. This chapter takes a multi-dimensional approach with a special emphasis on the fuzzy approach. Various weighting schemes (the weights given to the different aspects or questions related to financial literacy) are used to check whether selecting a specific weighting scheme has an impact on the overall degree of financial literacy. This study also places special emphasis on the fuzzy approach advocated by Rippin (2013) in the context of multi-dimensional poverty measurement.

An empirical illustration, based on data collected in a survey conducted by Israel’s Central Bureau of Statistics in 2012, shows that, regardless of the weighting scheme adopted, financial literacy is higher among males, the relationship between age and financial literacy takes a U shape, and married people have a higher financial literacy score. Other things held constant, financial literacy is generally lower among Arab Muslims and among the unemployed, and it rises with education. Finally, it appears that the conclusions to be drawn from focusing on the Rippin approach do not depend on whether the attributes of financial literacy are substitutes or complements.

In Chapter 10, Hanna Dudek and Wieslaw Szczesny focus their attention on multi-dimensional material deprivation (MD) in the four Visegrad Group (V4) countries, using zero-inflated beta regression modelling. This approach has many benefits. First, it enables the direct measurement of living standards by looking at the ‘enforced lack’ of ‘necessities’. The ‘enforced lack’ approach means that an item is counted as lacking if people cannot afford it - distinguishing between those who choose not to obtain a specific item and those forced to forgo it because they lack the necessary economic resources. Thus, the lack of items is not due to lifestyle preferences and choice but, rather is the result of the enforced lack of items, that is, things that people would like to possess (have access to) but cannot afford. Analysing MD not only provides a broad picture of living standards but also captures elements of persistent poverty because the lack of goods is usually associated with a prolonged shortage of resources. Moreover, an analysis of common items in the European Union (EU) that captures the same aspect of deprivation enables comparability across countries. The chapter provides the first econometric evidence of the factors that affect household MD in the V4 countries using a fuzzy multi-dimensional approach. Moreover, it investigates the differences between the risk and the intensity of the extent to which both indicators are subject to the same determinants.

In Chapter 11, Gaia Bertarelli, Antonella D’Agostino, Caterina Giusti and Monica Pratesi present a multi-dimensional and fuzzy approach for measuring educational poverty (EP) in Italy. They extend previous work by providing a fuzzy multi-dimensional measure of EP based on digital as well as literacy and quantitative skills in the dimension ‘friends and skills’. The first step consists of exploring the fuzzy nature of the phenomenon, as it results from four dimensions. Indeed, as the EP Index is multi-dimensional, the integrated fuzzy and relative methodology can be used under the assumption that EP is a vague predicate manifested in different shades and degrees (fuzzy concept), rather than an attribute that is simply present or absent for individuals. The chapter measures the degree of EP in the Italian regions and in local areas at the intersection of the regions by defining an overall EP fuzzy measure and four EP dimension- specific measures.

In Chapter 12, Tomasz Panek and Jan Zwierzchowski present fuzzy and multi-dimensional measures of the degree of social exclusion risk, providing evidence of social exclusion in the population over age 50 in Poland. Because of the progressive ageing of European society, the fight against the social exclusion of older people has become an increasing challenge in the EU member countries. Monitoring the threat of social exclusion among the elderly and exploring the factors which determine it is needed in order to develop an environment in which people can live well into old age. The chapter constructs a tool to help in achieving these goals. The authors propose a new methodology to measure the degree of social exclusion risk using a fuzzy multi-dimensional approach.

Chapter 13, the last chapter in Part 2, by Stephane Mussard and Maria Noel Pi-Alperin, is on ‘Socio-economic Health Inequality Indices: A Fuzzy Approach Applied to European Countries’. In this chapter, the authors propose a family of socio-economic health inequality indexes based on fuzzy sets. These indexes depend on two parameters: the first embodies the behaviour of the decision-maker with respect to income redistribution, and the second represents the sensibility of the decision-maker with respect to one exogenous risk factor. The indexes enable robust ranking between health deprivation matrices with respect to the risk factors studied. They are empirically applied to 13 European countries. It is shown that in many countries the level of education of parents is the main driver of risk that increases socio-economic health inequality.

Chapter 14, by Francesca Bettio, Gianni Betti and Elisa Ticci, begins Part 3 and is on ‘The Fuzzy Perspective on Violence against Women:

Challenges and Advancements’. In this chapter, the authors discuss how the fuzzy-set approach advances the measurement of gender-based violence by accounting for the frequency and severity of these different acts in addition to their prevalence. As argued elsewhere (Bettio, Tied and Betti, 2020), severity scales that enable going beyond prevalence measurement have long been proposed in the literature but suffer from limitations that can be overcome using fuzzy measurement, e.g. the costly gathering of information and various selectivity biases. One novel dimension of violence measurement that this chapter explores is aggregation measures. Separate fuzzy indexes for, respectively, physical, sexual and psychological violence can be aggregated into a single index by means of a simple average for the same individual and then for selected population sub-groups. Possible alternatives to this basic aggregation are standardising the index (with respect to the reference population group) or considering indexes of fuzzy intersections and fuzzy unions.

Chapter 15, by Tony Venditti, Nguyen Duy Phuong Tran and Anh Dung Ngo, offers an innovative approach for studying a system safety analysis of industrial processes. Industrial processes and machines (industrial systems) pose risks from equipment failure and worker accidents. Occupational laws and regulations, as well as safety codes and standards, call for risk assessment to effectively identify measures that prevent these risks from materialising. Risk assessment involves identifying the risk factors and estimating their importance in terms of their associated probability of occurrence and the severity of the consequences they can present. Probabilities are commonly expressed in terms of linguistic expressions (e.g. ‘very low’, ‘low’, ‘moderate’ and ‘high’). Therefore these assessments are subjective and vague. Other methods, such as fault trees, are also often used to identify the combinations of events which can lead to accident-prone events. If data are available, such as failure and human error probabilities, the probability of occurrence of accidents could be calculated. But in real life, these values are not usually known. Furthermore, these analyses are often performed without full knowledge of the processes being examined. So, a degree of uncertainty is associated with the data and the knowledge used in the analyses. In these situations, fuzzy numbers are an attractive method for dealing and taking into account this uncertainty. A numerical fuzzy probability assessment or linguistic expression, instead of a single data point, can instead belong to several sets in various degrees of memberships. Using the rules of fuzzy logic, a fuzzy number can then be calculated as a single number encapsulating the underlying associated uncertainty. In this chapter, fuzzy concepts are explained and applied to industrial risk estimation methods and fault tree analysis.

Chapter 16, by Bruno Cheli, Alessandra Coli and Andrea Regoli, discusses ‘A Fuzzy Approach to the Measurement of Employment and Unemployment’. This takes a new perspective using fuzzy sets, unlike other analyses of the workforce, which are traditionally based on a clear-cut distinction between those who are employed and those who are unemployed, which form complementary sets. However, this distinction appears too rigid because it obscures all the nuances between people who are fully employed and those who work only occasionally but need to, or wish to, work more. Furthermore, this way of proceeding involves a significant loss of statistical information, which could be used for portraying and measuring the phenomenon more accurately. In the chapter, employment is considered not as a simple condition that is either present or absent but, rather, as a matter of degree. In logical terms, this implies moving from a Boolean conception to a ‘fuzzy’ one. The final goal is to define fuzzy measures of employment and unemployment using the available information on the weekly number of hours worked and on the need or desire of workers to work more hours.

In Chapter 17, Besma Belhadj and Firas Kaabi explore ‘The Relationship between Employment and Poverty Using Fuzzy Regression’, with models linking poverty to employment. Sometimes, ordinary linear regression models, such as the model linking poverty to employment, do not satisfy the conditions under which a least-squares estimator would be consistent. In these cases, a consistent estimator needs to be found. For these reasons, Belhadj and Kaabi construct estimators for a linear fuzzy regression model and establish consistency in these estimators.

In Chapter 18, Lorenzo Mori and Duccio Stefano Gazzei propose a new approach using fuzzy clustering for geo-marketing. Marketing techniques encompass many subjects: neuroscience yielded neuro-marketing, which focuses on channels for emotional communication; out of digitisation grew digital marketing, which played a decisive role in the development of e-com- merce; finally, others include relationship marketing, buzz marketing and guerrilla marketing. In this chapter, the authors are mainly interested in geo-marketing, which uses geo-location to plan and implement marketing activities. Interest has grown in techniques to determine the particular geographic zone where potential consumers live, work, or do something else. This chapter uses consumer data from a big Italian bank with a simple but efficient technique to divide customers into different clusters. Eurisko segmentation logic is a method that enables clustering of the population using socio-demographic information.

In Chapter 19, Francesca Gagliardi, Laura Neri, Edmira Shahu and Aurora Hoxha investigate ‘Satisfaction in Higher Education: A Multi- Dimensional and Fuzzy Approach’. In this chapter, they measure satisfaction in higher education using a multi-dimensional and fuzzy methodology. As a complex phenomenon, satisfaction is difficult to measure in general because many factors are involved in its definition, and a univariate indicator offers a limited view of the concept. In their analysis, the authors identify three dimensions to describe student satisfaction. To overcome the oversimplification of a binary variable, they calculate a degree of membership in the set of those who are satisfied, based on these three dimensions. The findings in this chapter suggest that some dimensions are more informative than others, confirming that this approach yields useful results that can also help in designing policies.

Finally, in Chapter 20, Luigi Palumbo, Tiziana Laureti and Ilaria Benedetti discuss ‘A Multi-dimensional Clustering on Fuzzy Metrics to Classify CPG Pricing and Price Promotion Strategies: The Case of Pasta in Italy’. Although retailers compete along many dimensions, pricing strategy is clearly one of the most important drivers for increasing sales, especially in consumer packaged goods (CPG). Previous studies reveal that the classification of pricing and price promotion strategies involves complexity that cannot be reduced to a binary choice between high reliance on price promotion (HiLo) and an everyday-low-price (EDLP) approach. This chapter offers a fuzzy and multi-dimensional framework for analysing price promotion strategies in the CPG market over time and geographic regions using a dataset of weekly prices and price promotions for pasta brands across different retailers and regions in Italy. Retailers’ promotional strategies are divided into five dimensions: price level, price variability, price promotion frequency, price promotion intensity and price promotion variability. Membership functions are designed to transform those metrics into fuzzy measures. The authors obtain nine groups which identify different pricing and price promotion strategies and describe their distribution across brands, retailers and geographic regions.

References

Betti G. (1998), Intertemporal equivalence scales and cost of children using BHPS. ERSC Research Centre on Micro-Social Change Working Papers. Paper 11/98. University of Essex, Colchester.

Betti G. (2017), What impact has the economic crisis had on quality of life in Europe? A multi-dimensional and fuzzy approach. Quality & Quantity, 51(1), pp. 351-364.

Betti G., Cheli B., Lemmi A., Verma V. (2008), The fuzzy set approach to multidimensional poverty: the case of Italy in the 1990s, in Nanak Kakwani and Jacques Silber (eds.), Quantitative approaches to multi-dimensional poverty measurement. New York: Palgrave MacMillan, pp. 30-48.

Betti G., Lemmi A. (2008a), Advances on income inequality and concentration measures. London: Routledge.

Betti G., Lemmi A. (2008b), Editors’ introduction, in G. Betti, A. Lemmi (eds.), Advances on income inequality and concentration measures. London: Routledge, pp. 3-11.

Betti G., Lemmi A. (2013a), Poverty and social exclusion: new methods of analysis.

London: Routledge/Taylor 6c Francis Group.

Betti G., Lemmi A. (2013b), Introduction, in G. Betti, A. Lemmi (eds.), Poverty and social exclusion: new methods of analysis. London: Routledge/Taylor 6c Francis Group, pp. 1-6.

Betti G., Lemmi A. (2013c), Poverty and social exclusion in 3D. Multidimensional, longitudinal and small area estimation, in G. Betti and A. Lemmi (eds.), Poverty and social exclusion: new methods of analysis. New York: Routledge/Taylor &C Francis Group, pp. 301-315.

Betti G., Verma V. (1998), Measuring the degree of poverty in a dynamic and comparative context: a multi-dimensional approach using fuzzy set theory. Working Paper no. 22. Dipartimento di Metodi Quantitativi, Universita di Siena.

Betti G., Verma V. (1999), Measuring the degree of poverty in a dynamic and comparative context: a multi-dimensional approach using fuzzy set theory. Proceedings of the ICCS-V1, 11, pp. 289-301, Lahore, Pakistan, August 27-31.

Betti G., Verma V. (2008), Fuzzy measures of the incidence of relative poverty and deprivation: a multi-dimensional perspective. Statistical Methods and Applications, 12(2), pp. 225-250.

Bettio F., Ticci E., Betti G. (2020), A fuzzy index and severity scale to measure violence against women. Social Indicators Research, 148(1), pp. 225-249.

Cheli B., Ghellini G., Lemmi A., Pannuzi N. (1994), Measuring poverty in the countries in transition via TFR method: the case of Poland in 1990-1991. Statistics in Transition, 1, pp. 585-636.

Cheli B., Lemmi A. (1995), A ‘Totally’ fuzzy and relative approach to the multidimensional analysis of poverty. Economic Notes, 24, pp. 115-134.

Lemmi A., Betti G. (2006a), Fuzzy set approach to multi-dimensional poverty measurement. New York: Springer.

Lemmi A., Betti G. (2006b), Introduction, in A. Lemmi, G. Betti (eds.), Fuzzy set approach to multi-dimensional poverty measurement. New York: Springer, pp. 1-7.

Rippin N. (2013), Considerations of efficiency and distributive justice in multidimensional poverty measurement. PhD dissertation, George-August-Universitat Gottingen, Germany. https://www.die-gdi.de/uploads/media/Rippin_Efficiency_ and_Distributive_Justice_Online_Publication.pdf.

Veenhoven R. (2000), The four qualities of life. Journal of Happiness Studies, 1, pp. 1-39.

Verma V., Betti G., Gagliardi F. (2017), Fuzzy measures of longitudinal poverty in a comparative perspective, Social Indicators Research, 130(2), pp. 435-454.

Zadeh L.A. (1965), Fuzzy sets. Information and Control, 8, pp. 338-353.

2 Evolution of the Fuzzy-Set

 
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