**5. Conclusions**

This article evaluates the efficiency of public poverty policies (PPPs) in EU-28 countries on the basis of their effort measured as social expenditure over GPD (SE). We perform this analysis for the quinquennium 2014–2018 from annual observations on variables provided by programs EU-SILC and ESSPROS of Eurostat. To obtain a single observation for the whole period, 2014–2018 for a given variable and country longitudinal observations are aggregated by means of triangular fuzzy numbers with the method in [3].

As far as the ranking of PPPs is concerned, we have ordered a set of fuzzy efficiency indexes by using their expected value. The results that we have obtained are similar to those in [7,9,10]. Better performances are attained by Anglo-Saxon welfare states (Ireland and Great Britain), Scandinavian welfare states (Finland, Sweden and Denmark) and some Visegrad pact countries like Hungary and Czechia. The less efficient countries are the Mediterranean welfare states (Italy, Greece, Spain) and some Mediterranean and Baltic former communist republics as Romania, Bulgaria, Latvia or Estonia. In intermediate positions, we find continental welfare states (as, e.g., France, Germany ... .) and a heterogeneous set of non-UE-15 states as, e.g., Cyprus, Malta or Slovakia.

To measure the relation between the efficiency of PPP with SER or with the effort done in a concrete type of social benefit, we have used the fuzzy correlation index in [34] instead that in [31] since this last may provide too uncertain outputs. Likewise, we interpret the correlation index qualitatively as a linguistic variable. We have observed that the relation between the volume of social expending and the poverty diminution despite positive, is different between EU-15 countries (that have greater SE) and non-EU-15 countries. Hence, in EU-15 countries the results are in accordance with [16] that showed that in several European countries increases in social expenses do not lead directly to reductions in poverty.

The relation between the rate of each item of social benefits with Debreu–Farrell measure also shows different behavior in EU-15 and non-EU-15 countries. In non-EU-15 countries, six of eight types of expenses are low correlated with the productivity measure. Only social exclusion miscellanea (moderate(+)) and disability benefits (strong (−)) show a significant correlation. On the other hand, in EU-15 the correlations of each type of social expense and DF is often more intense. As [10] we have found that sickness/healthcare, family/children and disability benefits expenses have a significant positive relation with the efficiency index. We have also checked that benefits due to old age and survivors have a negative strong significant relation with the efficiency of PPP.

In the case of unemployment, housing and social exclusion, the sign and the intensity of the correlation are essentially the same in EU-15 and non-EU-15 countries. Hence, the relation of DF with housing benefits is negligible, with unemployment weak(−) and lastly with social exclusion weak/moderate(+).

We have also discussed the application of other tools connected with fuzzy sets as NFSs, rough sets or GNs to quantify the observations in our problem. Instruments as IFSs or NFSs provide a more complete capture of uncertainty than FNs. However, their adjustment has a greater cost than in the case of FNs. On the other hand, GNs provide more parsimonious representations of uncertain quantities than FNs. To define a GN, it is enough to estimate its kernel and a grayness measure. Therefore, in some circumstances it can be considered that information is too simplified by GNs. Our paper evaluates poverty policies of EU-28 countries within the quinquennium 2014–2018 in such a way that for each variable/country we actually have available five annual real valued observations. By using [29] we aggregate annual observations into one TFN observation that is addressed to the whole quinquennium. We feel justified the use of TFNs because they let modeling vague observations as smooth as possible without any loss of information.

We are aware that our study has limitations. First, it is done in a concrete period with a limited sample of countries. Hence, the conclusions in our paper must be carefully interpreted since they do not necessarily apply automatically to countries/periods out of the sample. Likewise, evaluating poverty policies by using exclusively Eurostat database and performing its analysis by means of fuzzy arithmetic has limitations. It may be of interesting complementing information in Eurostat database with experts' opinion that may be extracted from structured questionnaires and/or interviews. The use of tools to deal with this kind of information that are beyond fuzzy numbers as, e.g., NFSs or hesitant fuzzy sets is fully justified. Hence, further research on the evaluation of PPPs by the use of experts' opinions and the application of Fuzzy Multicriteria Decision Methods can be a suitable complement to the methodology presented in this paper.

**Author Contributions:** A.B.-E. and F.V.-F. have developed the revision of the literature on poverty public policies. J.d.A.-S. has developed the exposition and application of fuzzy data analysis tools on the sample. All the authors have contributed to extracting conclusions from empirical results and relating them with revised literature. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has partially founded by the Social Inclusion Chair of University Rovira i Virgili.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used in this study are available in https://ec.europa.eu/ eurostat/data/database.

**Acknowledgments:** Authors acknowledge the suggestions of two anonymous referees that have improved the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.





**Table 2.** Expected value of fuzzy RRP (over 100) and E (over 1) for several values of the index of λ.

Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017).
