Attributes inequality in multidimensional poverty measures fuzzy modeling

Poverty is a multidimensional one, which means that the poor can suffer multiple disadvantages at the same time. This paper aims to further develop and refine the multidimensional poverty measure using Fuzzy Sets Theory (FST). The application of FST starts with properly conveying the realities of attributes inequality and poverty proposing appropriate and justified membership functions for both variables. And then, applying fuzzy rules to integrate attributes inequality in multidimensional poverty measures. We obtain a class of fuzzy multidimensional poverty integrated indices. An application based on individual well-being data from Tunisian households in 2015 is presented to illustrate the use of proposed concepts.


Introduction
The poor may have poor health or malnutrition, a lack of clean water or electricity, poor quality of work or little schooling. However, he can suffer multiple disadvantages at the same time. And to capture the reality of poverty it must focus on many factors and move to a multidimensional measure of poverty. Multidimensional poverty measure can be used to create a more comprehensive picture. It reveals the range of different disadvantages that the poor experiences. As well as providing a headline measure of poverty, multidimensional measure can be broken down to reveal the poverty level in different areas of a country and among different sub-groups of people.
Inequality is about the distances that separate individuals and groups living in a society. Inequality, however, is not a simple concept. Distances can be measured in several different dimensions and as a consequence inequality can refer to rights, capabilities, income, and well-being and so on. This suggests that fighting current inequality may help to make social mobility a more concrete phenomenon and reduce a particularly disturbing feature of inequality: its persistence. Too much inequality produces segregation and polarization which put in jeopardy the 'cement of society.' This implies the much debated disappearance of the middle class-which the data in many cases are not able to confirm-raising preoccupations precisely on the ground of its social sustainability. For all these reasons inequality reduction (and more generally, inequality control) should rate high on the political agenda next to poverty. In fact, three I's of poverty (Jenkins and Lambert 1997) are present: It is not only important to reduce the Incidence and Intensity, but also Inequality. However, inequality must be considered in poverty measurement.
Consideration of Inequality in poverty measurement has been the norm since Sen (1976). Consideration of inequality is natural for measures in cardinal approach-Approaches for Cardinal data (Chakravarty et al.1998 Alkire andFoster 2011, Belhadj 2016), but not straightforward for measures in counting approach. However, in this last case, inequality can be captured across deprivation counts.
To fine-tune a poverty measure to capture inequality, Bossert et al. (2009) uses symmetric or generalized mean across deprivation counts. Also, Jayaraj and Subramanian (2009) and Rippin (2012) use weights deprivation counts by themselves (like FGT). These approaches are merely used for ranking but are not suitable for understanding inequality within groups and between groups.
To capture inequality in measure poverty, first, we must create a poverty index that is sensitive to inequality. Then, use a separate inequality measure to analyze inequality among the poor and provide more information. An advantage of a separate inequality measure is that if decomposable, it can be used to analyze inequality within groups and between groups.
In this paper, we apply Fuzzy Sets Theory (FST) to propose a (1) separate attributes inequality measure and (2) multidimensional poverty indices sensitive to attributes inequality like the combination of multidimensional poverties and the concept of inequality of attributes. In fact, there are in the literature several multidimensional poverty indices that provide numbers with a compelling, commonsense interpretation. Why should we want to go beyond these approaches? A good answer to this question is that it may be appropriate to consider whether poverty measurement should take into account more information about the income distribution. If the distribution amongst the poor is of concern then we may need a more sophisticated poverty measure (Watts 1968;Sen 1976). Why could the FST be useful? A good answer to this question is that FST allows, using the so-called membership function, to fully formalize the complicated reality. This approach is able to consider specific aspects-principles-of the comparison of income distributions in terms of poverty and inequality to illuminate central issues without resorting to formal axioms to express these principles. Some of the axioms that have been suggested may be less persuasive than their counterparts in inequality and poverty analyses and some may actually obscure the insights available from the simpler approaches.
This paper is structured as follows. Section 2 presents a brief description of FST and its application to poverty. Section 3 uses FST to derive alternative attributes inequality measure. Section 4 uses FST to derive families of multidimensional poverty indices sensitive to attributes inequality. Section 5 illustrates the use of proposed measures using Tunisian household data of 2015, and Sect. 6 concludes.

Fuzzy logic, poverty
Fuzzy logic (Zadeh 1965) is a form of multi-valued logic derived from FST to deal with approximate reasoning rather than the precise one. Let Y be an income distribution, y i the income of the ith observation unit i ¼ 1; :::; n ð Þ such as 0 y 1 ::: y n ð Þ . In fuzzy set theory with fuzzy logic, how a given observation unit can be considered poor or not? We assign, following this approach, to every observation unit a degree of poverty ranging from 0 to 1 according to the corresponding income. This can be done by introducing the so-called membership function, l A y i ð Þ, of an observation i to the subset 'poor,' A. The set membership values can range (inclusively) between 0 and 1. Also, the degree of poverty can range between 0 and 1 and is not constrained to the two true values (true (1), false (0)) as in classic predicate logic (Novák et al. 1999). l A y i ð Þ allows us to express to which extent the ith observation is poor. It quantifies the degree according to which y i belongs to A.
Membership functions were first introduced Zadeh (1965). They characterize fuzziness (i.e., all the information in fuzzy set), whether the elements in fuzzy sets are discrete or continuous. Membership functions can be defined as a technique to solve practical problems by experience rather than knowledge. They are represented by graphical forms. We propose, in Sects. 3.2 and 4.2, membership functions of variables Inequality of attributes (Eqs. 2 and 3) and Poverty (Eq. 15). They are defined as the process of transforming a crisp sets to fuzzy sets or fuzzy sets to fuzzier sets. Basically, this operation translates accurate crisp input values of inequality of attributes and poverty into linguistic variables.
In fuzzy logic there are three basic operations on fuzzy sets: union, intersection and complement. For example, let l B y i ð Þ be membership function of the subset 'very poor,' B and l A y i ð Þ be membership function of the subset 'moderately poor,' A.
The union of fuzzy subsets A and B is a fuzzy set defined by the membership function:

and the intersection of
Þ . The complement of A is a fuzzy set defined by the membership function: l A y i ð Þ ¼ 1 À l A y i ð Þ.

Fuzzy lorenz curve, fuzzy gini coefficient
The Gini coefficient, that is usually defined mathematically based on the Lorenz curve, measures the inequality among values of a frequency distribution (for example, levels of income). A Gini coefficient of zero expresses perfect equality, where all values are the same. A Gini coefficient of one (or 100%) expresses maximal inequality among values. Statistics methods used to calculate Gini are based on certain axiomatic assumptions that who generally ignore the differential convexity of the Lorenz curve. The differential convexity is to extend the convexity concept of function by using a geometrical approach on its epigraph (the set of points on or above the graph of the function). More specifically, since the convexity of a function is equivalent to the convexity of its epigraph, the extension is obtained by substituting a line joining two points in the epigraph by a curve, namely a solution of a fixed secondorder ordinary differential equation.

Traditional gini's index
The Gini coefficient (Gini 1912) is based on the Lorenz curve, which plots the proportion of the total income of the population (y-axis) that is cumulatively earned by the bottom x% of the population (Fig. 1). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (A in Fig. 1) over the total area under the line of equality (A and B in Fig. 1); i.e., IG ¼ A= A þ B ð Þ. The Gini coefficient can theoretically range from 0 (complete equality) to 1 (complete inequality). It is based on the income levels of individuals and is represented by P m k¼1 n j n k w j À w k , j ¼ 1; ::::; m distinct income groups; n ¼ P m j¼1 n j is the total number of people and w is the average/mean of any income w distribution.
The Gini coefficient is also related to the Lorenz curve in a diagrammatic way. The more 'bowed' the Lorenz curve and the higher the degree of income inequality; hence, the higher the Gini. In the next section, an alternative approach to calculate Gini coefficient is proposed considering the differential convexity of the Lorenz curve. It's about a fuzzy version of the Gini coefficient calculated as agreement index for fuzzy numbers (see Belhadj et al. (2021) for another proposal). Fuzzy version of the Gini coefficient is a measure of inequality. It can be used to indicate how the distribution of income has changed within a country over a period of time, thus it is possible to see if inequality is increasing or decreasing.

Agreement gini's index for fuzzy numbers
The Lorenz curve is a probability plot comparing the distribution of a variable against a hypothetical uniform distribution of that variable. The Lorenz curve for a probability distribution is a continuous function. However, Lorenz curves representing discontinuous functions can be constructed as the limit of Lorenz curves of probability distributions, the line of perfect inequality being an example.
For the case of discontinuous functions, the imprecision is inherent to the analyst. It is brought by the limits set by the income intervals that 'discretize the Lorenz curve' and make it not smooth (the smooth function is locally well approximated as a linear function at each interior point) and does contain break, angle, or cusp.
For a long time, the probability theory has been the predominant theory to model uncertainties of reality. It is based on certain axiomatic assumptions, which are hardly ever tested, when these theories are applied to reality. Fuzzy Theory is one of other theories, which was initially intended to be an extension of classical logic. It uses degrees of truth as a mathematical model of vagueness: If X is a collection of objects denoted generically by x, then a fuzzy set T in X is a set of ordered pairs: It is a generalization of the indicator function for classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. Its range is the subset of nonnegative real numbers whose supremum is finite. For sup l T x ð Þ ¼ 1: normalized fuzzy set.
Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field. This includes Euclidean spaces, which are affine spaces. Let subsets H ¼ y j À Á 1=a j ; n 0 y j 1; j ¼ 1; :::; mg and L ¼ y j ; È 0 y j 1; j ¼ 1; :::; mg of S. Subsets H and L are convex. We represent the function H by a fuzzy subset H & R. Let us now consider a fuzzy number D & R, D is an isosceles right triangle, which we call the agreement index of D with regard to H (Kaufmann and Gupta 1991), the ratio being defined in the following way: as shown in Fig. 2. Consider the triangle fuzzy number D & R shown in Fig. 2, 8y j 2 R; j ¼ 1; :::; m groups or attributes : with the following function H: l D and l H denote the degrees of truth i.e., membership of group or attribute j ¼ 1; ::::; m, respectively, to subsets D and H (Fig. 2), y j is incomes ratio and y ij the income levels of individuals i ¼ 1; :::; n by attribute j ¼ 1; ::::; m; F À1 : ð Þ is the quantile function and a j is the degree of convexity of the Lorenz curve (Fig. 2).
Equations (2 and 3) are curves do constitute, respectively, membership function of equality line and membership function of attributes inequality.
Equation (3) is a convex function to real-valued defined on an interval with the property that his epigraph is convex set; this is a fuzzy version of Lorenz curve. And, Eq. (2) is the fuzzy line of equality.
We must now describe the surface A \ H and compute its area as follows, gives y j ¼ 1 8y j 2 R; j ¼ 1; :::; m groups or attributes (see appendix for the demonstration of the following equations): and and Area D ¼ Thus, we find from (6) and (8) that and from (7) and (9), (1) becomes: 8j ¼ 1; :::; m groups or attributes: t j D; H ð Þ is also the ratio of the area (A in Fig. 1) under the line of equality (A ? B in Fig. 1). It summarizes the information in a fuzzy Lorenz curve: if a j ! 0 the fuzzy Lorenz curve moves outward, producing greater inequality and then t D; H ð Þ j ! 1 and vice versa if a j ! 1 that show t D; H ð Þ j ! 0. We find from (11) that t j D; H ð Þ is very relative to the curvature of the Lorenz curve. This result is important because it proves that the agreement index of D with regard to H cannot be absolute. a j is then the proportional variation of FGI j . Comparing income distributions among countries may be difficult because benefits systems may be different in different countries. The fuzzy Lorenz curve l H y j À Á can account for all of these differences. Indeed, for economies with very different income distributions, the fuzzy Lorenz curve can have different shapes and still yields, therefore, different FGI coefficients.
It is claimed that the FGI is sensitive to the income of the middle classes and to that of the extremes.
In Sect. 4 we combine in a single measure, membership values of poor individuals to attributes subsets, and income attributes inequality ratios of individuals. We obtain an integrated fuzzy multidimensional poverty. Besides, it will be substantiated that in the multidimensional approach, a rigorous and comprehensive analysis and measurement of poverty can be achieved applying the fuzzy set theory. And that the application of this theory provides basic information for the design of socioeconomic policies addressed to the gradual elimination in time of the causes that produce and reproduce intergenerational states of poverty.
Our methodology, in this section, is: (1) we present and discuss Traditional monetary and non-monetary approaches, and then we propose (2) Fuzzy monetary and nonmonetary approaches, and (3) the Fuzzy integrated nonmonetary poverty.

Measuring multidimensional poverty
Poverty covers both monetary (unidimensional) and nonmonetary (multidimensional) aspects. Sen (1992) believes that poverty is not the mere lack of income to meet basic needs, but deprivations in basic human capabilities. An income poverty line,z, well captures the monetary aspect of poverty but cannot accurately reflect the non-monetary aspects.

Traditional monetary and non-monetary approaches
The traditional monetary approach to poverty uses income or consumption expenditure y i as an indicator of well-being. This approach starts by identifying the poor as those with insufficient income to attain minimum basic needs z.
In the traditional monetary approach to poverty, the individual poverty function is Then, this approach aggregates the poor shortfall from the poverty line into a poverty index. Poverty headcount, poverty gap, and severity of poverty are the most common indices used in the literature. All these indices belong to the family of Foster-Greer-Thorbecke (FGT) poverty measures (Foster et al. 1984) given by: where 1 : ð Þ is the indicator function, i ¼ 1; :::; n individual and s is a parameter indicating the sensitivity of the index to the distribution among the poor. The higher s, the more sensitive the index is to the poorest persons in the economy. For s ¼ 0, FGT is the headcount, for s ¼ 1 it is the poverty gap, and for s ¼ 2 it represents the severity of poverty.
The traditional monetary approach to poverty demands a multidisciplinary analysis. Three main socioeconomic conceptual developments were introduced in the last three decades. The first one is the concept of social exclusion including in particular long term and recurrently unemployed and employed in precarious and unskilled jobs (Lenoir 1974). The second one was introduced by Sen (1999) is the concepts of functioning, capabilities and entitlement. The UNDP (1997,1998) developed a third concept to the analysis and measurement of poverty. Unlike the social exclusion and Sen's approaches, the UNDP approach was made operational because it ends with a proposed measure of poverty. Its annual Human Development Report publishes two Human Poverty Indexes, one for developing countries and another for industrialized countries including deprivation in longevity, deprivation in knowledge and deprivation in living standard.
The choice of the set of socioeconomic attributes related to the state of poverty is based on the information available. We select the socioeconomic attributes whose lack of, or partial possession of any of those attributes, contributes to the state of a household poverty. They are represented by the m-order vector X ¼ X 1 ; :::; X j ; :::; X m À Á of attributes. As attributes considered in X we find: income of the household; years of schooling of the household head; age, job status and gender of the household head; years of schooling of the parent of the household head; occupation of the household head, ownership and typology of the household residence; drinkable water; sanitary (bathroom, shower, sewage) services; and number of senior and handicapped persons in the household.
A multivariate measure of poverty purports to arrive at a poverty index as a function of the m attributes included in X as follows: where, S ij ¼ z j Ày ij z j 1 y ij z j À Á . z j is the insufficient income to attain minimum basic needs by attribute j, y ij is the individual income or consumption expenditure by attribute j and w j is the share of attribute j.
The index in (14) measures the relative deprivation and poverty corresponding to each attribute included. It identifies the most important attributes or dimensions that need to be addressed to achieve a structural reduction in poverty.
We note that if the individual is the household, we must modified Eq. (14) as well as the rest of the equations as follows: where n h ik and t k denote, respectively, the equivalent size and the share of socio-economic group (for example area).
h is the elasticity size that corresponds to the elasticity of equivalence scale compared to the size of household (for the calculation method see Belhadj and Limam (2012)). h allows to pass from the income (usually, disposable income) of a household of any size to its equivalent level of income.

Fuzzy monetary and non-monetary approaches
This section presents an alternative approach to the derivation of monetary and non-monetary poverty indices using instruments from FST. The FST is a rigorous method to operationalize a multivariate analysis of poverty. Cerioli and Zani (1990) applied fuzzy set theory to estimate the poverty in Italy. Dagum et al. (1992), Cheli et al. (1994), Cheli and Lemmi (1995), Belhadj and Limam (2012) and Betti et al. (2017), among others made further contributions and applications.

Fuzzy monetary poverty
Let P be the set of poor households defined as follows P ¼ i : y i zg f . However, if i 2 P and there is an uncertainty in the definition of the poverty line, and thereby of the poor, we may specify a membership function which indicates the degree to which a household is considered as poor.
The membership function being continuous differentiable and non-increasing (Fig. 1), to be used in the empirical illustration, is proposed as follows: where d is the concavity of the underlying individual poverty function and is related to the extreme poverty aversion parameter involved in Foster et al. (1984). As noted in Atkinson (2003), it seems reasonable in practice to use the traditional values 1 and 2 for d. A value of 3 for d is sometimes used in the unidimensional context to get a measure which is more sensitive to transfers involving the poorest members of the population. If d ¼ 0, we get the standard definition of a crisp set with the membership functions taking a value of 1 for all those with an income below z (dotted chart, Fig. 3). The membership function in (15) is convex i.e., defined by its lower limit z, its upper limit y 1 , and a point of inflection whose a typical value is y 1 þ z ð Þ=2. The decline is slower when the distance y 1 -z increases. Convex function is the most well-understood functional in the calculus of variations. And so, the convex membership function adequately reflects the variation in poverty following a variation in income.
The aggregator function across individuals is with _ is the maximal value of l P y i ð Þ and l P y i ð Þ because the set of poor still remains P [ P, P ¼ i : z\y i y Ã g f : This measure complies with extended strong focus, monotonicity, restricted strong monotonicity, restricted continuity, non-decreasingness in poverty domain, Subgroup consistency, anonymity and population invariance. It can be shown that it satisfies the transfer and transfer sensitivity axioms, while being continuous and even decomposable (Zheng 1997).

Fuzzy non-monetary poverty
Let A u;v::: be the deprivation space related to the attributes u; f v:::g and corresponds to the set of points y i 2 R m þ such that y ij \z j ; 8j 2 u; v:::g f . A k-deprivation space A k is the P P subset of R m þ corresponding to the set of vectors y i such that y ij \z j for at least k attributes (Fig. 4) where Y is an n Â m strictly positive real-valued matrix whose element y ij represents the ith individual income in the jth attribute, p u;v::: [ 0, are attributes' weights.
Membership functions of attributes (Fig. 4), each means the degree of deprivation of an attribute, are defined as follows: l Au;v::: y iu;v::: À Á ¼ 1 i fy iu;v::: \y 1u;v::: z u;v::: À y iu;v::: z u;v::: À y 1u;v::: d if y 1u;v::: y iu;v::: \z u;v::: 0 i fy iu;v::: ! z u;v::: And the aggregator function (across individuals) is given by: X m u;v:::¼1 p u;v::: _l Au;v;:: y iv ð Þ h i b 1 S iu;v:: The parameter b in (17) and (19) defines the degree of substitutability between different attributes. When b tends to infinity, relative deprivations are non-substitutes and poverty is defined unidimensionally by the attribute deprivation with the highest level of deprivation. But when b ¼ 1, attributes are perfect substitutes and poverty is also defined unidimensionally as a simple weighted sum of attributes. Convexity of attributes, that is concavity in the space of deprivations, requires b ! 1.
For the non-monetary poverty it is necessary to deal with situations in which transfers do not alter the marginal distribution of each attribute (Bresson 2009).
The construction of a measure of non-monetary poverty must then necessarily verify a certain number of properties called axioms (Belhadj and Limam 2012). We show that our proposition satisfies several axioms in particular attribute decomposability, restricted strong simple transfer, independent transfer and non-ambiguous transfer sensitivity.

Fuzzy integrated non-monetary poverty
Poverty is due to inadequate income or education, or poor health, or insecurity, or low self-confidence, or a sense of powerlessness, or the absence of rights such as freedom of speech (Haughton and Khandker 2009). However, poverty measures need to be accompanied by inequality measures by attribute to distinguish for example inadequate availability of schools or a corrupt health service. Inequality can be understood at different levels. It occurs between different groups or attributes, either taking into account of their weighting or not (Milanovic 2012). This different understanding of inequality has different consequences for establishing changes in inequality levels. Inequality refers to discrepancies in income, wealth, education, health, nutrition, space, politics and social identity.
Poverty is related to, yet distinct from, inequality (Haughton and Khandker 2009). This finding brings us back to integrating a measure of inequality in a poverty measurement as follows: X m u;v:::¼1 p u;v::: _l A u;v;:: y iv ð Þ Â FGI u;v::: FGI is the inequality measure of (10); it relates the full distribution of wellbeing. Poverty,l A y ð Þ, is focused on the lower end of the distribution only-those who fall below a poverty line. The product of the two measures the compound effect (Fig. 5).
The measure in (20) is the so-called Fuzzy integrated non-monetary poverty. It is a summary measure of deprivation that includes 'poor' and the inequalities that come across economic, education and health/nutrition-related dimensions (attributes). This measure allows measuring and considering dimensions or attributes composition and also attributes inequality. It is joint distribution of deprivation to reach the poor and poorest. It is Robust to wide range of attributes.

Inequality in tunisia
In this section, we illustrate the use of the proposed attributes (dimensions) inequality measures based on Tunisian household data of 2015 and we make remarks concerning its implementation and practical issues. This exercise highlights the inevitability of making value judgments when analyzing any different income distributions.

Inequality by dimension
Dimensions (socio-economic factors) used here are area and environment. Hence, Tunisia is divided into three parts: The Greater Tunis and two homogenous sets, namely the Littoral and the Interior. The Greater Tunis area with 25% of the total population is characterized by very special administrative, social and economic properties. The Tunisian Littoral (Bizerte, Cap-Bon, Sahel, Sfax and Gabes) had known since independence in 1956 an economic and social prosperity. If we compare the per capita expenditure during 2015 we notice that this subdivision is justified. In addition to this regional decomposition, it is necessary to take into account the rural-urban distinction. We also aggregated the rural part of the Greater Tunis and the littoral. Two reasons support this aggregation. First, the size of the rural Greater Tunis is very small, only 167 households and second, the rural of Greater Tunis and those of the rest of the littoral are very similar and can be lumped together to form a homogenous spatial set. This gives five homogenous regions, namely the Greater Tunis, the Urban Littoral, the Urban Interior, the Rural Littoral and Rural Interior.
We base our analysis on the household survey data conducted by the Tunisian National Institute of Statistics (TNIS) in 2015, involving 25,144 households. The sampling scheme and the results of the survey are detailed in TNIS (2015). The survey provides information about total annual expenditures in dinars per household as well as the size of each household and other information. A brief summary of the total annual expenditure is given in Table 1.
The Interquartile Range (IQR) tells us that the middle 50% of values in the dataset have a spread between 261,248 dinars (' 128,063$) and 533,498 dinars (' 261,519$) per household and per year (line 5, Table 1). Larger values of Rural Littoral and Rural Interior indicate that the central portion of data of these areas spread out further. Conversely, the middle values cluster more tightly for the rest of areas.
According to Table 2, using FGI or GI, Urban Interior, Rural Interior and Rural Littoral regions have the highest income inequality in 2015. The remaining regional averages were: Greater Tunis and Urban Littoral. According to environment, the rural environment has the highest income inequality score in 2015 (Table 3). The value 0.5 for a guarantees the convexity of the fuzzy Lorenz curve and justifies the presence of an average inequality in Tunisia.
We note, using FGI or GI, that income inequality in Tunisian regions varies less in geographic terms but disparities and dispersions according to FGI differ with respect to the Gini coefficient GI. Contrary to the Gini coefficient, FGI means that inequality in Tunisia is than 50% greater. However, the difficulty for Tunisia will be to reduce interpersonal inequalities and to raise the level of development and living standards. In addition, the reduction in disparities between the different regions of the country, through enhanced development of the most backward regions, and dispersions intra-regions, should be the priorities of responsible policies.
We highlight that the distribution of income is important for making progress towards the income poverty and the headcount, and when considering the depth of poverty. Indeed, poverty and inequality although theoretically distinct concepts (Atkinson 1987) are very closely linked as they summarize different aspects of the same. Inequality considers the entire spread of a distribution, whereas poverty mainly focuses on the lower part of the distribution and is mainly concerned with identifying the poor and summarizing this into an indicator that shows levels of poverty in a society (Foster et al. 2013).

Inequality by attribute, poverty
We calculate, in this section, fuzzy attributes inequality and fuzzy multidimensional poverty integrated. We use the attributes available in the survey which are: expenditures on food and beverages at home and outside home (Alimentation), expenditures of housing including interest on rental or mortgages payments and other expenditures related to periodic maintenance of property (Shelter), expenditures on medicines, health insurance coverage and medical service (Health care), expenditures related to the purchase, maintenance and repair of vehicles as well as public transportation and other related expenses (Transportation), expenditures corresponding to the purchase and phone communication as well as to Internet connection and other related expenses (Telecommunication), expenses relative to the payment of fees and the purchase of books and equipments related to public and private schools or universities (Education). A brief summary of the total annual expenditure by attributes is given in Table 4. Large values of IQR are for education, shelter and food attributes.
Using a poverty line per day and per individual equal to $1.90 (World Bank 2015), we calculate a threshold per attribute proportional to the share of each attribute (Table 5, column 4). Inequality and poverty by attribute, respectively, appear in columns (2) and (3) of Table 5. b; d ¼ 2 mean, respectively, attributes are substitutes and poverty is defined multidimensionally, and that the variation of poverty is convex as in Fig. 3. There are different specifications of membership functions or variations of poverty (see e.g., Belhadj and Kaabi (2020) and Belhadj (2011)). All specifications verify the same property: A negative variation in poverty follows a positive variation in income; it is only the rate of variation that changes according to the specification. For example, for d ¼ 1, the convex curve in Fig. 3 becomes a line, and for a straight lines the variation is constant.
IMP ¼ 0:152 (Table 5, end column 3) is the percentage of deprivations poor people experience, as a share of the possible deprivations that would be experienced if all people were deprived in all attributes.
The poverty headcount ratio tells us which people are poor. IMP also shows how people are poor. But IMP can be unfolded and folded in different ways. So you see how it's  made and how it can be changed. For example, to see the contribution of each attribute, we multiply the censored poverty measure by the weight of that attribute. The multidimensional inequality (Table 5, end column 2) is equal to 0.565. Marginal distributions of attributes (Table 5, column 2) show a situation where two persons switch achievements either one for a single attribute, in which both are deprived, or for two substitutable attributes. The resulting transfer specifies conditions under which an alternative form of progressive transfer among the poor that preserves the marginal distributions of attributes and lowers inequality should lower poverty, or at least not raise it.
The dimensional transfer is verified by the class of measures IMP that respect dimensional breakdown and to supplement these with associated inequality measures.
From the results of Table 5, we find that the measure IMP in (20) can be used for additional purposes. Beyond measuring poverty and wellbeing, this measure can be adapted to target services and conditional cash transfers or to monitor the performance of programmes. However, the more policy-relevant information there is available on poverty; the better-equipped policymakers will be to reduce it. For example, individuals or households that are deprived in education require a different poverty reduction strategy from that are deprived in housing conditions. We can deduce that an efficient targeting must depend on whether it is unidimensional or multidimensional poverty that policy is intended to reduce because the value of targeting prescriptions depends on the structure of the transfers.
Indeed, the appropriate indicator to use to design efficient targeting schemes is not the poverty measure themselves. It is the degree of deprivation for each attribute that should be used to identify which group it is most efficient to target. This makes it necessary inter alia to consider important trade-offs between the effect of targeting on the poorest of the poor and the effect of targeting on the speed of income increase among the not-so-poor.

Concluding remarks
In this paper, we propose an integrated non-monetary poverty combining an attributes inequality measure. In this non-monetary poverty measure we introduce population poverty-inequality ratios by attributing using membership functions and determining their contributions to the total poverty ratio. These ratios provide basic information about the causes of poverty. They are of paramount importance to the design and implementation of a structural socioeconomic policy to abate the causes of poverty. Hence, they purport to break the mechanism responsible for its intergenerational transmission. This policy, being structural, not cyclical, should aim at a process of structural change with the scope of generating stable, efficient and equitable socio-economic processes.
An application using individual well-being data from Tunisian households in 2015 is presented to illustrate proposed concepts. Results show that the choice of poverty measures that policy is intended to help in efficient targeting. The fuzziness involved in choosing one specific poverty index and one specific poverty frontier and the possible sensitivity of targeting prescriptions to that choice make it desirable to use targeting membership function tools. These tools also help assess the normative strength of targeting prescriptions.