Inequalities between retirees and workers: an empirical model to capture the effect of taxation. Bérangère Legendre

Introduction

French pensioners benefit from tax deductions. In the context of pension system reform and given the impact of the current financial and economic crisis on government deficits, the tax preference given to senior citizens raises questions. [START_REF] Lavigne | Retraite, épargne et fiscalité[END_REF] asks, "Should we give tax benefits to the elderly?" The author explains that the tax benefits given to retirees imply considerable public tax expenditures that are unjustifiable in terms of equity. Current retirees enjoy an average standard of living, nearly equivalent to that of persons in the labour force (See El Mekkaoui De Freitas et al., 2008, andConseil d'Orientation des retraites, 2009). According to the French statistical institute (INSEE), the ratio of retired individuals' income per consumption unit to that of workers was approximately 0.89 in 2003 (0.96 including capital income). Moreover, pensioners' incomes are, on average, 102% of the average income of the population. Inter-cohort inequalities do not seem to justify these tax exemptions. Pensions are more equally distributed than income received from employment (See Brown and Prus, 2006), and intra-cohort inequality does not seem to be a more convincing explanation.

The existence of an increased risk of poverty among pensioners is also a questionable justification for this preferential tax treatment. According to the INSEE, the poverty rate is greater among labour force participants than among retirees. With a poverty threshold set at 60% of the median income, the poverty rate for pensioners amounted to 9.6% in 2006 compared to 9.8% for those in the labour force. With a threshold set at 50% of the median income, the INSEE estimates the poverty rate for men aged 60-74 to be 3.6% compared to 6.4% for 50-to 59-year-old men. However, according to Eurostat, the poverty rate for people over 60 years of age was 6.9% in 2007 but only 6.5% among those under the age of 60. Eurostat's results show an increasing poverty trend among retirees since 2009: 7.4% among people aged 60 and over compared to 4.8% for those under 60.

Several phenomena with opposing effects influence the living standards of pensioners. Young people and future retirees have or will have experienced more setbacks than middle aged people during their careers (See [START_REF] Cloarec | Les passages de l'emploi à la retraite[END_REF], Colin, Iehlé and Mahieu, 2000, and Briard et al., 2009). It is therefore more difficult for individuals of these generations to meet all of the requirements for a full pension beginning at retirement age. Moreover, several reforms have contributed to increasing the contributory demands of the French pension system: -Increased career duration, -A higher legal retirement age, -The establishment of a discounting mechanism.

These phenomena have resulted in an increased risk of poverty for future generations of retirees (See Franco et al., 2009). However, this risk is offset by the increased participation of women from younger generations in the labour market and by an increase in the average skill level.

In this preliminary research, we examine the adequacy of tax arrangements for French retirees. What is the impact of differential taxation on the inequality between retirees and workers?

Taxation in France

French pensioners benefit from specific tax rules concerning income taxation, social contributions, housing and property taxes. These are the most important examples:

-Elderly people benefit from a 10% reduction in taxes if they are retired. The age of a taxpayer may also prompt special discounts. For example, for retirees over the age of 65, an allowance of 1,138 euros is provided if the overall net income is between 14,010 and 22,590 euros. This allowance may be up to 2,276 euros if the total net income is less than or equal to 14,010 euros. This policy has no age limit for disabled individuals.

-The taxation of a life annuity depends on the age when its holder first received the entitlement. If the holder was under 50 at enrolment, 70% of the amount of the annuity is taxable. If the beneficiary waits until age 60, the taxable portion drops to 40%. -An exemption from the property tax on buildings is planned for retirees older than 75 years. This exemption is subject to the individual's income. Retirees aged 65 to 75 years are also eligible for a rebate of 100 euros. -Residents living in a nursing home do not pay the housing tax. People over the age of 60 can be exempted from housing tax, subject to the income of the individual. If retirees over 60 live with one or more adult children seeking employment, they receive automatic relief from paying the housing tax. -Concerning social contributions, retirees may receive exemptions depending on their income level. When they are subject to taxation, they often benefit from lower tax rates.

According to [START_REF] Ferrand | Réformer les retraites: quelles solutions progressistes?[END_REF], some of these tax dispositions are no longer relevant in France. Recent studies show that retired and active households have equivalent standards of living (See COR, 2009, and[START_REF] Legendre | Inégalités de niveaux de vie des actifs et des retraités : une multidécomposition de l'indice de Gini appliquée au cas Français[END_REF]. Other tax rules remain useful when they respond to a precise social need. For instance, expenses related to a person's disability are eligible for specific tax cuts.

Survey and empirical model

We We create two representative and exhaustive samples of the population:

-The retired sample includes individuals declaring that they are retired, -The sample of workers includes the active population of employed and unemployed people seeking a job.

Retaining only those individuals from either exclusively active households or exclusively retired households, we propose an indicator for the inequality between the active population and retirees:

where act t c y , is the average equivalent income of the active population ranged by percentile, c, and ret t c y , is the equivalent income of retirees ranged by percentiles (see figure 1).

The figure 1 depicts the equivalent income over the distribution among retirees and workers. We note that individuals belonging to the lowest percentiles of workers receive a higher equivalent income than individuals belonging to the highest percentiles of retirees. But from the 76th to the 84th percentile, retirees receive a higher equivalent income than workers.

In the following procedure, we explain the difference between act t c y , and ret t c y , , depicted in the figure 1. This difference is taken as a proxy for inequality between workers and retirees, and is represented by the indicator t c y , . This indicator t c y , allows us to compare the retired population and the active population despite the different sizes of the samples.

Percentiles Annual equivalent income in €

Reti rees Workers

Within the households, we assign each member an equivalent disposable income. The OECD (Organisation for Economic Cooperation and Development) equivalence scale is used in which one unit of consumption is attributed to the first household adult member, 0.5 to other members over the age of 14, and 0.3 to children under 14. We first propose an OLS specification using data for the years 2004 to 2007. c y indicates the inequality between retirees and workers and is the dependant variable in the following specification: where c X includes the explanatory variables. As explained above, French retirees benefit from specific tax rules. Consequently, we introduce the following tax variables:

-Social contributions paid by retirees, as a percentage of the equivalent disposable income, -Social contributions paid by workers, -Income tax and housing tax paid by retirees, -Income tax and housing tax paid by workers, -Property tax paid by retirees, -Property tax paid by workers, -Solidarity tax on wealth (STW) paid by retirees, -STW paid by workers.

Next, we introduce variables to control for the impact of socio-economic phenomena:

-The mean age of the samples (retired and active), -The mean age at the end of the studies, -The mean number of household members by percentile, -The proportion of women in the percentiles, -The proportion of homeowners, -The proportion of foreign-born persons, -The proportion of executives, -The proportion of farmers, -The proportion of white collar workers, -The proportion of blue collar workers, -The proportion of employees.

Given the presence of outliers in our data, the classical least squares estimator may be distorted. To correct for this bias, we propose different robust-to-outliers methods from the literature. First, we calculate the Cook distances to indicate data points that are particularly worth checking for validity. Data points with large outliers and/or high leverage may distort the outcome and accuracy of the OLS regression. Consequently, observations associated with a Cook distance larger than 1 receive a zero weight. We then employ a Tukey biweight loss function.

Cook distances are able to identify isolated outliers, but they are inappropriate in cases with clusters of outliers. Rousseew and Van Zommeren (1990) show that one outlier can mask the presence of another. Full robustness can be achieved using the Salibian- [START_REF] Salibian-Barrera | A fast algorithm for s-regression estimates[END_REF] estimator. It consists of randomly selecting N subsets with p observations (p-subsets), where p is the number of estimated parameters. Each p-subset is defined such that it does not contain outliers. The number of N subsets is generated to guarantee that at least one p-subset where α is the expected proportion of outliers (equal to 0.2), p is the number of estimated parameters and Pclean is the desired probability of having at least one p-subset without outliers among N subsets. This probability is fixed at 0.99.

We also propose a model with panel data from 2004 to 2007. Given the short temporal dimension, we prefer a random effects model. The Hausman test confirms that this specification is a better fit for our data (see table 5 in the appendix).

Assuming that the entity's error term is not correlated with the regressors, the empirical model is formulated as follows:

where the dependent variable, t c y , , represents the difference between the equivalent incomes of active people and retirees across percentiles.

Results

The different specifications proposed indicate the impacts of the income tax, social contributions, property taxes and the housing tax on the inequality between workers and retirees (see tables 1 to 4 in the appendix). Using linear regressions, robustness analysis and panel regressions, we highlight the significant and positive impact of the social security contributions paid by the active population on the size of the inequality. Housing taxes paid by workers also have a significant and positive impact on the inequality between workers and retirees. In other words, the correlation between the impact of these taxes and our dependent variable is positive, indicating an increasing trend in the income gap between the working population and retirees. In contrast, we demonstrate the significant and negative impact of the social contributions paid by pensioners, their housing and income taxes. When these taxes increase, the gap between the disposable income of working people and retirees tends to decrease.

These results show that a reduction in tax deductions specifically for retirees could reduce the gap in living standards between the active population and the retired population. While some

(3) ( ) ( ) [ ]       - - - = P clean P N α 1 1 log 1 log (4) (5) t c t c t c X y , , , ε α + = t c c t c , , ρ τ ε + =
tax relief involves targeted services (personal services) with social objectives, other types do not have an economic or social justification. For example, the 10% income tax deduction for retirees and the 10% deduction allocated to workers for business expenses have no economic justification, especially when the retired household has a high standard of living. When the income tax paid by retirees increases by one percentage point relative to disposable income, the gap in living standards between workers and pensioners seems to fall by approximately 0.39 euro.

Decreasing the housing tax paid by workers or reducing the income tax paid by working households would also reduce inequality. The results of the robust estimation using the Salibian-Barrera method (see table 3 in the appendix) show that a one percentage point increase in the property tax paid by active persons would imply an increase in the gap between the living standards of workers and retirees of 0.2 euro. We observe a significant and positive impact of the proportion of homeowners among the active population. The higher the proportion of homeowners among workers, the greater the income gap. Retirees are more frequently homeowners, and this allows them to relax their budget constraints. On the contrary, the mortgage or rent payments constitute a heavy burden for active households. If the average proportion of homeowners among workers increases by one percentage point, the income gap between workers and retirees increases by 9 euros, on average (see table 3 in the appendix).

Concluding remarks

Different tax rules are applied to members of the labour force and retired people. However, the degree of inequality between the two groups and the living standards of pensioners do not seem to justify the existence of unequal treatment (See [START_REF] Legendre | Inégalités de niveaux de vie des actifs et des retraités : une multidécomposition de l'indice de Gini appliquée au cas Français[END_REF]. Our empirical analysis suggests that an increase in social contributions for pensioners and a reduction in some of their income tax advantages could reduce the income gap between the active and retired populations.

  use data from the European survey EU-SILC (Community Statistics on Income and Living Conditions) for 2004 to 2007. Specifically, we use the French data from the SRCV (Statistics on Resources and Living Conditions) included in the European survey and conducted by the National Institute of Statistics and Economic Studies (INSEE). The French SILC (Community Statistics on Income and Living Conditions) survey addresses issues relating to poverty and the living conditions of individuals and households. It provides information on the taxes and social security costs incurred by households as well as social benefits.

Figure

  Figure 1 Income distribution: equivalent incomes by percentile, 2007

  selected with high probability (See Salibian-[START_REF] Salibian-Barrera | A fast algorithm for s-regression estimates[END_REF][START_REF] Salibian-Barrera | A fast algorithm for s-regression estimates[END_REF] Verardi and[START_REF] Croux | Robust regression in Stata, Stata Journal[END_REF]:

Appendix


Jun Zheng
email: jun.zheng@ecp.fr

Olivier Cailloux
email: olivier.cailloux@ecp.fr

Vincent Mousseau
email: vincent.mousseau@ecp.fr

Constrained Multicriteria Sorting Method Applied to Portfolio Selection

Keywords: Multicriteria decision aiding, Portfolio selection, Preference elicitation

The paper focuses on portfolio selection problems which aim at selecting a subset of alternatives considering not only the performance of the alternatives evaluated on multiple criteria, but also the performance of portfolio as a whole, on which balance over alternatives on specific attributes is required by the Decision Makers (DMs). We propose a two-level method to handle such decision situation. First, at the individual level, the alternatives are evaluated by the sorting model Electre Tri which assigns alternatives to predefined ordered categories by comparing alternatives to profiles separating the categories. The DMs' preferences on alternatives are expressed by some assignment examples they can provide, which reduces the DMs' cognitive efforts. Second, at the portfolio level, the DMs' preferences express requirements on the composition of portfolio and are modeled as constraints on category size. The method proceeds through the resolution of a Mixed Integer Program (MIP) and selects a satisfactory portfolio as close as possible to the DMs' preference. The usefulness of the proposed method is illustrated by an example which integrates a sorting model with assignment examples and constraints on the portfolio definition. The method can be used widely in portfolio selection situation where the decision should be made taking into account the performances of individual alternatives and portfolio simultaneously.

Introduction

Let us consider the student enrollment in universities every year. Universities want to select students with good performances on several criteria (such as GPAs, motivation, maturity, . . . ). At the same time, the selected students should satisfy some specific requirements at a collective level. For instance, the number of students in each department should be more or less balanced. Each department tries to achieve a gender (nationality, etc.) diversity. Moreover, the positions available are limited. Therefore, the universities face a decision which consists of selecting a certain number of students, designing a waiting list and rejecting the other students (see similar example in universities [START_REF] Cardinal | Multiple criteria sorting: An application to student selection[END_REF], other examples are available in the book [START_REF] Salo | Portfolio Decision Analysis[END_REF]). Another example of such portfolio selection problems concerns allocating grants to research proposals. The committee evaluates the merit of the proposal, including originality, novelty, rigor and the ability of the researchers to carry out the research individually. On a whole level, they try to balance the funding among disciplines, institutions and even regions. Therefore, a decision is to be made to select certain research proposals within limited budget.

The two problems above share some characteristics. Firstly, they involve evaluating individual alternatives according to their performances on multiple criteria. Secondly, a portfolio is to be selected based not only on individual alternative's performance, but also on the performance of the whole portfolio. Such situation typically corresponds to a portfolio selection problem.

There is a large number of methods in literature for evaluating and selecting portfolios [START_REF] Golabi | Selecting a portfolio of solar energy projects using multiattribute preference theory[END_REF][START_REF] Rao | A balance model for evaluating firms for acquisition[END_REF][START_REF] Hall | A model for making project funding decisions at the national cancer institute[END_REF][START_REF] Archer | An integrated framework for project portfolio selection[END_REF][START_REF] Chien | A portfolio-evaluation framework for selecting R&D projects[END_REF]. Cost-benefit analysis [START_REF] Phillips | Transparent prioritisation, budgeting and resource allocation with multi-criteria decision analysis and decision conferencing[END_REF], multiattribute utility theory [START_REF] Duarte | Developing a projects evaluation system based on multiple attribute value theory[END_REF], weighted scoring [START_REF] Coldrick | An R&D options selection model for investment decisions[END_REF] are widely used. Some researchers combine preference programming with portfolio selection considering incomplete preference information [START_REF] Liesiö | Preference programming for robust portfolio modeling and project selection[END_REF][START_REF] Liesiö | Robust portfolio modeling with incomplete cost information and project interdependencies[END_REF]. However, to our knowledge, Multiple Criteria Decision Aiding (MCDA) outranking methods have not been applied to portfolio selection problem. Furthermore, the ability of the methods to express sophisticated preference on portfolios has little been explored. A balance model [START_REF] Farquhar | A balance model for evaluating subsets of multiattributed items[END_REF] is developed which measures the distribution of specific attributes by dispersion and uses such measurement to select subsets of multiattribute items. [START_REF] Golabi | Selecting a portfolio of solar energy projects using multiattribute preference theory[END_REF] uses constraints to eliminate the ones which do not fit in the requirement on whole portfolio.

We propose a two-level method for such portfolio selection problems. At individual level, the paper uses Electre Tri method [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF][START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF] to evaluate the alternatives on multiple criteria, which assigns alternatives to predefined ordered categories by comparing an alternative with several profiles. The DMs' preference on individual evaluation can be taken into account by some assignment examples. At portfolio level, a wide class of preferences on portfolios (resource limitation, balance of the selected items over an attribute. . . ) are represented using general category size constraints. An optimization procedure is performed by solving a MIP to infer the values of preference parameters and to identify a satisfactory portfolio.

The paper is organized as follows. Section 2 formulates portfolio selection problem as a constrained multicriteria sorting problems. Section 3 presents a mathematical program which computes the portfolio that best matches the DMs' preferences. Section 4 illustrates the proposed method with an example. The last section groups conclusions.

Problem Formulation

Evaluating Alternatives with Electre Tri Method

Alternatives to be included on a portfolio are evaluated by an outranking method Electre Tri [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF][START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF]. This method assigns alternatives to predefined ordered categories by comparing the alternatives to profiles which define the frontiers of two successive categories. For example, for enrollment problem described in Section 1, the DMs want to sort the students into three categories: accepted, waiting list or rejected according to students' performances on multiple criteria. Thus the two profiles could be two frontiers which separate these three categories.

Formally, Electre Tri assigns each alternative of a set A = {a 1 , a 2 , ..., a n } to k pre-defined ordered categories Cat 1 ≪ Cat 2 ≪ ... ≪ Cat k . K denotes the set of indices of the k categories (K = {1, 2, ..., k}). The alternatives are evaluated on m criteria. Let J denote the set of the indices of the criteria g 1 , g 2 , ..., g m (J = {1, 2, ..., m}). For all j ∈ J, a ∈ A, g j (a) represents the evaluation of a with respect to the jth criteria. In what follows we assume, without loss of generality, that preference increases with the value on each criterion. b h is the upper limit of category h and the lower limit of category h + 1, h = 1, 2, ..., k -1. In other words, the frontier separating two categories is represented by the evaluations on the set of criteria of the profile b h : g j (b h ), ∀j ∈ J. The assignment of an alternative a results from the comparison of a to the profiles b 1 , b 2 , ..., b k-1 .

Electre Tri uses an outranking relation which represents assertions a b h whose meaning is "a is at least as good as b h ". In order to validate the assertion a b h , a "sufficient" majority of criteria should be in favor of this assertion. A set of weights coefficients (w 1 , w 2 , ..., w m ) which sum to 1 representing relative importance of the criteria is additively used in the concordance test when computing the strength of the coalitions of criteria being in favor of the assertion a b h . We consider a simplified Electre Tri method which ignores discrimination thresholds (preference and indifference threshold) and veto threshold involved in the standard non-discordance condition [START_REF] Mousseau | A user-oriented implementation of the ELECTRE TRI method integrating preference elicitation support[END_REF]. Such simplification is in line with the axiomatic study of Bouyssou and Marchant [START_REF] Bouyssou | An axiomatic approach to noncompensatory sorting methods in MCDM, I: The case of two categories[END_REF][START_REF] Bouyssou | An axiomatic approach to noncompensatory sorting methods in MCDM, II: more than two categories[END_REF].

Electre Tri builds a concordance index C(a, b h ) ∈ [0, 1], ∀a ∈ A, ∀h ∈ K, defined as C(a, b h ) = j∈J|gj
Given the outranking relation , alternatives are assigned to categories on the basis of the way they compare to profiles b h . Electre Tri proposes two assignment procedures (so-called pessimistic and optimistic rule). In this paper we consider only the pessimistic rule which assigns alternative a to the highest category Cat h for which a b h-1 and not a b h . So as to implement Electre Tri, an elicitation process is necessary to determine the values of preference parameters (profiles b h , weights w j and majority level λ). In a portfolio selection perspective, we consider DMs' preference at two levels. At alternative level, the DMs express preferences on alternatives individually. At a portfolio level, they express preferences on portfolios as a whole (resource limitation, balance of the selected items over an attribute, . . . ). These two preference levels are distinguished, as they are elicited in different ways, and could be provided by different DMs who have expertise and understanding of the portfolio selection at different levels.

DMs' Preference on Alternatives

The DMs have little understanding of the precise semantics of the preference parameters involved in Electre Tri. On the contrary, they can easily express their expertise on which category an alternative should be assigned to. Therefore, we propose to elicit the DMs' preference in an indirect way, in accordance with the disaggregation-aggregation paradigm. Instead of providing precise values for the parameters, the DMs provide assignment examples, i.e. alternatives which they are able to assign confidently to a category. For instance, in a student selection problem, the DMs may state that one particular student should be assigned to the best category (the set of accepted students). Inference procedure can thus be used to compute values for the preference parameters that best match the assignment examples. Several authors have proposed disaggregation methodologies from assignment examples expressed by the DMs. Mousseau and S lowiński use non-linear programming to infer all the parameters simultaneously [START_REF] Mousseau | Inferring an ELECTRE TRI model from assignment examples[END_REF], and some suggest to infer weights only assuming the profiles are fixed [START_REF] Mousseau | Using assignment examples to infer weights for ELECTRE TRI method: Some experimental results[END_REF]. Researchers also proposed to compute robust assignment categories to which an alternative is possible to be assigned, considering all combinations of values compatible with the DMs' preference statements [START_REF] Dias | An aggregation/disaggregation approach to obtain robust conclusions with ELECTRE TRI[END_REF] and developed corresponding software [START_REF] Dias | IRIS: a DSS for multiple criteria sorting problems[END_REF]. Recently, an evolutionary approach has been presented to infer all parameters of Electre Tri model [START_REF] Doumpos | An evolutionary approach to construction of outranking models for multicriteria classification: The case of the ELECTRE TRI method[END_REF]. In this paper, we assume all the preference parameters are variables and infer them by solving a MIP.

DMs' Preference Information on Portfolios

The DMs' preferences can also be expressed at the portfolio level (resource limitation, balance on the composition of categories w.r.t. an attribute, . . . ). We formalize such preferences as general constraints on category size. For example, in the student enrollment case, let us denote the category of rejected students Cat 1 , the category of waiting list Cat 2 and the category of admitted students Cat 3 . Suppose the university only have 100 positions available, and such constraint can be modeled as the number of students in Cat 1 cannot exceed 100. Moreover, balancing gender in the selected students (100 students in total) can also be modeled as a constraint that the number of female students in Cat 1 should not be lower than 30. Adding such constraints to the selection process may result in rejecting some male students whose performances are better than those of the accepted female students. However, such portfolio is more satisfactory for the DMs in terms of gender balance. Modeling the DMs' preference as constraints eliminates some portfolios which don't satisfy their requirements on the whole portfolio.

Mathematical Program Formulation

Stating the Problem and Decision Variables

Given a set of alternatives A, a set of criteria indices J, evaluations of the alternatives g j (a), ∀a ∈ A, j ∈ J, a set of category indices K = {1, 2, ..., k}, a set of profiles b h , 1 ≤ h ≤ k -1, the goal of the program is to determine the performances of profiles g j (b h ), ∀j ∈ J, 1 ≤ h ≤ k -1, weights w j and majority threshold λ, satisfying all the constraints given by the DMs in the form of assignment examples and portfolio constraints. The MIP also defines additional variables involved in the way Electre Tri assigns alternatives to categories. The binary variables C j (a, b h ), ∀a ∈ A, j ∈ J, 1 ≤ h ≤ k -1 represent the partial concordance indices such that C j (a, b h ) = 1 if and only if the performance of the alternative a on the criterion j is at least as good as the performance of the profile b h . The continuous variables σ j (a, b h ) represent the weighted partial concordance indices, they are such that σ j (a, b h ) = w j if and only if C j (a, b h ) = 1. Finally, binary variables n(a, h), ∀a ∈ A, h ∈ K are defined so that n(a, h) = 1 if and only if alternative a is assigned to category h. A slack variable s is used in the objective function which appreciates the ability of the Electre Tri model to reproduce the assignment examples in a robust way.

The constraint j∈J w j = 1 is posed, and the following constraints are used to ensure a correct ordering of the profiles defining the categories:

∀j ∈ J, 2 ≤ h ≤ k -1 : g j (b h-1 ) ≤ g j (b h ).

Constraints Stemming from Preferences at Individual Level

The set of assignment examples E is the set of pairs (a, h) ∈ A × K specifying that alternative a is assigned to Cat h . Recall that satisfying an assignment example (a, h) amounts to satisfy both j∈J:gj (a)≥gj (b h-1 ) w j ≥ λ and j∈J:gj (a)≥gj (b h ) w j < λ. The sum of support in favor of the outranking of an alternative a over a profile b h , j∈J:gj (a)≥gj (b h ) w j , can also be written j∈J C j (a, b h )w j with C j (a, b h ) equal to one iff g j (a) ≥ g j (b h ). Constraints (1) define the binary variables C j (a, b h ), ∀j ∈ J, a ∈ A, 1 ≤ h ≤ k -1, where ε is an arbitrary small positive value, and M is an arbitrary large value. See also Fig. 1.

1 M ((g j (a) -g j (b h )) + ε) ≤ C j (a, b h ) ≤ 1 M (g j (a) -g j (b h )) + 1 . ( 1
) 0 1 -M 0 M gj (a) -gj(b h ) Cj(a, b h ) Cj(a, b h ) = 1/M * (gj(a) -gj(b h )) + 1 Cj (a, b h ) = 1/M * (gj(a) -gj(b h )) ⑦ ⑥ ✲ ✻ Fig. 1. constraining Cj(a, b h ) to the appropriate value
The following constraints define the variables σ j (a, b h ) representing the sum of the support in favor of the assertion "a is at least as good as b h " while avoiding the non-linear expression σ j (a, b h ) = C j (a, b h )w j [START_REF] Meyer | Disaggregation of bipolar-valued outranking relations[END_REF]. See also Fig. 2.

∀j ∈ J, a ∈ A, 1 ≤ h ≤ k -1 :          σ j (a, b h ) ≤ w j σ j (a, b h ) ≥ 0 σ j (a, b h ) ≤ C j (a, b h ) σ j (a, b h ) ≥ C j (a, b h ) + w j -1 . (2)
We also define, for simplicity of use in the next constraints, ∀j ∈ J, a ∈ A: σ j (a, b 0 ) = w j and σ j (a, b k ) = 0.

0 wj 1 0 1 Cj(a, b h ) σj(a, b h ) σj(a, b h ) = Cj(a, b h ) σj (a, b h ) = Cj(a, b h ) + wj -1 • ( ✻ ❄ ✲ ✻ Fig. 2.

constraining σj (a, b h ) to the appropriate value

Finally, we need to ensure that each assignment example is assigned to the class specified by the DMs. The variable s is a slack variable used in the objective function.

∀(a, h) ∈ E :

σ j (a, b h ) + s + ε ≤ λ ≤ j∈J σ j (a, b h-1 ) -s .
(3)

Constraints Stemming from Preferences at Portfolio Level

Suppose the DMs want to impose, in a student selection problem, that at least 30 students in the best category (i.e. Cat k ) are females. To model this, we define a function Gender on the set of alternatives that equals one if the student a is a female student and zero otherwise, and set as a constraint that the sum of Gender(a) on each alternative a assigned to Cat k should be at least 30 ( a→Cat k Gender(a) > = 30). In a project selection problem, suppose the DMs want to make sure that the sum of the costs of the selected projects (say, the projects in the best category) do not exceed the available budget x. A function Cost would be defined on the set of alternatives representing their cost attribute, and a constraint is added to ensure that the sum of Cost(a) on alternatives a assigned to the best category should be no greater than the budget ( a→Cat k Cost(a) ≤ x).

More generally, portfolio preferences are represented as a set N of tuples h, n h , n h , P , 1 ≤ h ≤ k, n h , n h ∈ IR, P a function from A to IR, representing the constraint that the preferential model inferred by the program should be such that the number of alternatives from A assigned to Cat h weighted by their attribute P should be at least n h and at most n h : n h ≤ a→Cat h P (a) ≤ n h .

The following constraints define the binary variables n(a, h), ∀a ∈ A, 1 ≤ h ≤ k, so that n(a, h) equals one iff a is assigned to category Cat h , that is, j∈J σ j (a, b h-1 ) ≥ λ and j∈J σ j (a, b h ) < λ. The first constraints force that n(a, h) = 1 requires that a goes to category h, and the last ones force that exactly one n(a, h) among all h equals one.

∀a ∈ A, 1 ≤ h ≤ k :          n(a, h) ≤ 1 + j∈J σ j (a, b h-1 ) -λ , n(a, h) ≤ 1 + λ - j∈J σ j (a, b h ) -ε . (4) ∀a ∈ A : 1≤h≤k n(a, h) = 1.
(

) 5
These variables permit to guarantee the desired group sizes.

∀ h, n h , n h , P ∈ N : n h ≤ a∈A n(a, h)P (a) ≤ n h . (6)

Objective Function and Resolution Issues

In order to maximize the separation between the sum of support and the majority threshold, the objective of the MIP is set to maximize the slack variable s as defined in Constraints (3). The slack variable evaluates the ability of the Electre Tri model to "reproduce" the assignment examples in a robust way. However the preference information of the DMs does not lead univocally to a single compatible portfolio. The optimization procedure finds out one of the compatible portfolios. In an interactive perspective, the DMs can provide further preference information considering the results of the MIP, and the information can be added to the optimization procedure to get a more satisfactory portfolio. The decision aiding process can proceed with several interactions until the DMs are content with the selected portfolio.

Illustrative Example

Let us illustrate the method with the following hypothetical decision situation. A government board has the responsibility to choose which research projects to finance among a list of 100 research proposals. The selection process involves sorting these proposals into three categories: projects that are considered very good and should be funded (category Good ); projects that are good and should be funded if supplementary budget can be found (category Average); projects that are of insufficient quality and should not be funded (category Bad ). To sort these projects in these three categories, the board agrees to use the following six criteria.

sq The project's scientific quality, evaluated on a 5 points ordinal scale. rq The proposal's writing quality, evaluated on a 5 points ordinal scale. a The proposal's adequacy with respect to the government priorities, evaluated on a 3 points ordinal scale. te The experience of the researcher teams submitting the project, evaluated on a 5 points ordinal scale. ic Whether the proposal includes international collaboration, a binary assessment. ps The researchers' publication score evaluated by an aggregate measure of the total quality of publications of the researchers involved in the proposal (evaluated on a [0,100] scale).

The scales on all criteria are defined such that a greater value corresponds to a better evaluation. Supplementary to these six criteria, the 100 projects to be evaluated are described by three attributes: the research domain to which the project belongs (Operational Research (OR), Artificial Intelligence (AI) or Statistics); the budget the project asks funding for; the originating country. Table 1 shows the data about the first 7 projects in the list (complete data lists for the whole example are available at http://www.lgi.ecp.fr/ ~mousseau/ADT2011/). In order to determine an appropriate preference model, the board gives as a first stage 30 examples of past research proposals whose performances on the six criteria and final quality evaluation are known. A part of this data is shown in Table 2. The inference program is run with these assignment examples, and without supplementary portfolio constraints. Table 3 lists the resulting profiles and weights. Note that the profiles performances values in all our tables have been rounded up. Because each alternative used in this example has integer performance values on all criteria, doing so does not impact the way each alternative compares to these profiles. The resulting preference model is used to evaluate the 100 research projects, which leads to 22 projects being evaluated as good projects. The board is not satisfied with this set of projects because accepting these projects induces a total funding cost of 718 which exceeds the available budget (400). The program is thus run again with a supplementary constraint on the sum of the budget of the projects being assigned to the Good category to ensure that it stays below the available budget. This second stage inference yields other profiles and weights, given in Table 4, and a new list of assignments of which a part is displayed in Table 5. At this stage 11 projects are assigned to category Good and therefore are to be financed, leading to a total cost below 400. However the board is not fully satisfied yet because one domain is largely favored by this result, as the AI domain has 7 projects selected whereas only 1 project in the OR domain is to be financed. In a third stage, the inference program is thus run again with a new constraint requiring that the domain OR has at least 2 projects in the category Good. The final assignment results, shown partly in Table 6, are considered satisfactory. The process could have continued had the board wished a better balance among the originating countries, or had they wished to consider more closely also the Average category. In case an infeasible problem had been reached at some point during the process, some constraints would have had to be relaxed or deleted. The reader will find in Mousseau et al. [START_REF] Mousseau | Dealing with inconsistent judgments in multiple criteria sorting models[END_REF] algorithms on how to proceed for constraints relaxation.

The proposed approach has been implemented as a free, open source Java library [START_REF] Cailloux | ELECTRE and PROMETHEE MCDA methods as reusable software components[END_REF][START_REF] Cailloux | J-MCDA: free/libre java libraries for MCDA[END_REF]. The mathematical programs implementation relies on the JLP software package, a free and open source Java Linear Programming wrapper on top of CPLEX (the solver we used) and other commercial and free solvers. Solving the problems used in this illustrative example takes less than one minute. A study of the solving time of a related problem is available for the interested reader [START_REF] Cailloux | Eliciting ELECTRE TRI category limits for a group of decision makers[END_REF]. That study does not take into account the portfolio constraints and thus examines a simpler problem. It shows that small to medium size problems (consisting of less than eight criteria, of three categories and of less than one hundred alternatives) are solvable within ninety minutes, which is a reasonable time provided that this kind of approach is primarily used in an off-line mode. Analysis of the solving time of the problem studied here, thus with the added portfolio constraints, is left as a future work.

Conclusion

The method applies constrained Electre Tri model to portfolio selection problems in order to select a satisfactory portfolio considering DMs' preferences both at individual and portfolio level. Using a sorting model, the alternatives are evaluated by their intrinsic performances on criteria. Unsatisfactory portfolios which do not meet the DMs' requirements on portfolios as a whole are screened out by adding category size constraints to Electre Tri model. Because of such category size constraints, the assignment of an alternative is dependent on its evaluation but also on other alternatives.

Our formalization permits to tackle the challenges the DMs may face during the decision of portfolio selection. (1) At individual level, an alternative is evaluated on multiple criteria which can be qualitative or quantitative criteria. Moreover, the DMs easily express their preferences on alternatives by assignment examples. (2) At portfolio level, the best alternatives do not necessarily compose the best portfolio. Our method takes into account the overall portfolio performance by modeling the DMs' preference on portfolio as constraints. [START_REF] Bouyssou | An axiomatic approach to noncompensatory sorting methods in MCDM, II: more than two categories[END_REF] The preference information at the two levels (individual classification of alternatives and preference at the portfolio level) can be elicited from different stakeholders. (4) The proposed method involves the DMs deeply by asking them preference in an intuitive way.

The proposed method can be widely used in portfolio selection situations where the decision should be made taking into account the individual alternative and portfolio performance simultaneously. The proposed syntax of category size constraints has a broad descriptive ability for portfolio decision modeling. The method can be extended by providing robust recommendation to the DMs as a result of incomplete preference information. Moreover, the preference on portfolio level can be modeled as objectives rather than constraints of the optimization procedure, which would lead to a multiobjective problem.

  (a)≥gj (b h ) w j . It represents the degree of credibility of the assertion a b h . The assertion a b h is considered to be valid if C(a, b h ) ≥ λ, λ being the "majority level" such that λ ∈ [0.5, 1] .

Table 1 .

 1 Some of the research projects to be evaluated. The budget is in tens of Ke.

		evaluations criteria		descriptive attributes
	Project rq ps a sq te ic	budget domain country
	Pr001	2 47 2 3 1 0	27	Stat.	Germany
	Pr002	2 3 2 4 4 0	29	Stat.	France
	Pr003	5 63 1 5 1 0	20	Stat.	Italy
	Pr004	1 92 3 5 5 1	34	AI	Germany
	Pr005	4 13 2 4 2 0	32	Stat.	Germany
	Pr006	5 5 3 5 1 0	22	Stat.	Netherlands
	Pr007	1 27 3 2 5 1	34	OR	Germany
	. . .

Table 2 .

 2 Some research project examples and their respective assignments.

	Project rq ps a sq te ic	Cat
	Ex01	4 50 2 3 3 0	Average
	Ex02	4 85 3 1 5 1	Good
	Ex03	3 95 1 2 5 1	Average
	Ex04	5 91 2 2 5 1	Good
	Ex05	5 89 1 5 3 0	Good
	Ex06	3 5 3 2 2 1	Average
	. . .

Table 3 .

 3 Profiles, weights and majority threshold inferred during the first stage.

		rq ps a sq te ic λ
	b1	2 73 4	1	2	1
	b2	4 96 4	5	3	1
	w 0.2 0.2 0 0.2 0.2 0.2 0.5

Table 4 .

 4 Profiles, weights and majority threshold inferred with supplementary budget constraint.

		rq	ps	a	sq	te	ic	λ
	b1	2	2	2	1	2	1
	b2	3	84	2	4	3	2
	w 0.143 0.143 0.143 0.143 0.286 0.143 0.643

Table 5 .

 5 A part of the assignment of the research projects with the preference model inferred during the second stage.

	Project rq ps a sq te ic	budget domain country	Cat
	Pr001	2 47 2 3 1 0	27	Stat.	Germany	Bad
	Pr002	2 3 2 4 4 0	29	Stat.	France	Average
	Pr003	5 63 1 5 1 0	20	Stat.	Italy	Bad
	Pr004	1 92 3 5 5 1	34	AI	Germany	Good
	Pr005	4 13 2 4 2 0	32	Stat.	Germany	Average
	Pr006	5 5 3 5 1 0	22	Stat.	Netherlands	Bad
	Pr007	1 27 3 2 5 1	34	OR	Germany	Average
	. . .

Table 6 .

 6 A part of the assignment of the research projects with the preference model inferred during the third stage.

	Project rq ps a sq te ic	budget domain country	Cat
	Pr001	2 47 2 3 1 0	27	Stat.	Germany	Average
	Pr002	2 3 2 4 4 0	29	Stat.	France	Average
	Pr003	5 63 1 5 1 0	20	Stat.	Italy	Bad
	Pr004	1 92 3 5 5 1	34	AI	Germany	Good
	Pr005	4 13 2 4 2 0	32	Stat.	Germany	Average
	Pr006	5 5 3 5 1 0	22	Stat.	Netherlands	Average
	Pr007	1 27 3 2 5 1	34	OR	Germany	Average
	. . .