Human Development Index Based on ELECTRE TRI-C Multicriteria Method: An Application in the City of Recife

Abstract

The Human Development Index (HDI) is an indicator that measures individuals’ welfare through three dimensions: health, education and income. Since its conception, the HDI has been a focus of attention for various segments of society such as politicians, professionals in the media, policy makers, academics and ordinary citizens. The index, however, has received several criticisms over the years, the compensatory effect between the dimensions being the main one. In this context, this paper puts forward an alternative approach for calculating the Municipal Human Development Index, thereby mitigating some criticisms of the index and supporting public decision making. For this purpose, the ELECTRE TRI-C multicriteria method was used, in order to attenuate the compensatory effect, to reduce calculation problems and to allow comparison year by year. An application was conducted in the city of Recife, Pernambuco, in order to demonstrate the benefits of the proposed approach. As a result, a more adequate classification of the regions in four levels of human development was obtained.

Appendices

Appendix 1: Recife Human Development Units

Table 6 gives information about the Recife Human Development Units, showing their codes and the neighborhoods which are part of them.

Table 6 Recife Human Development Units

Appendix 2: ELECTRE TRI-C Method

This appendix offers additional information about ELECTRE TRI-C multicriteria method, such as its formulas.

The ELECTRE TRI-C is a sorting method that involves ordered classes and each one of these classes are represented for only one reference action. Basically, outranking relations between alternatives and reference actions are explored, with the intend to find suitable classes to each alternative. The method results in two classifications and these classifications must be analyzed together in the recommendation of the decision process.

The set of alternatives is denoted by A and must be known a priori. So, a set of alternatives ${\text{A}} = \{ {\text{a}}_{{1,}} {\text{a}}_{2} , \ldots ,{\text{a}}_{{\text{i}}} \}$ is given and should be distributed in completely ordered classes $\{ {\text{C}}_{{1,}} \ldots ,{\text{C}}_{{\text{h}}} , \ldots ,{\text{C}}_{{\text{q}}} \}$, where ${\text{C}}_{1}$ is the worst class and ${\text{C}}_{\text{q}}$ is the best. The objective of the method is assign the alternatives to the set of classes, according to a set of criteria ${\text{F}} = \{ {\text{g}}_{1} , \ldots ,{\text{g}}_{{\text{j}}} , \ldots ,{\text{g}}_{{\text{n}}} \}$. The set of criteria has a vector of weights that can be interpreted as a voting power, denoted as${\text{w}}_{\text{j}}$, such that ${\text{w}}_{\text{j}} > 0,j = 1, \ldots , n$ and assuming$\mathop \sum \nolimits_{{{\text{j}} = 1}}^{\text{n}} {\text{w}}_{\text{j}} = 1$.

The set of reference actions ${\text{B}} = \{ {\text{a}}_{0}^{\prime } ,{\text{a}}_{1}^{\prime } , \ldots ,{\text{a}}_{{\text{h}}}^{\prime } , \ldots ,{\text{a}}_{{\text{q}}}^{\prime } ,{\text{a}}_{{{\text{q}} + 1}}^{\prime } \}$ should be determined a priori too, and they represent the classes in which the alternatives will be distributed. Note that the number of reference actions is equal to ${\text{q}} + 2$ and actions ${\text{a}}_{0}^{\prime }$ and ${\text{a}}_{{{\text{q}} + 1}}^{\prime }$ represent the worst and the best performances in each criterion, respectively.

Firstly, the comprehensive concordance index ${\text{C}}\left( {a,a^{\prime } } \right)$ should be calculated, as shown in Eqs. 3 and 4, respectively. This index considers all criteria in which the relation a outranks a′ is valid. Equation 3 involves the preference threshold (${\text{p}}_{\text{j}}$) and the indifference threshold (${\text{q}}_{\text{j}}$).

$${\mathbf{C}}_{{\mathbf{j}}} \left( {{\mathbf{a}},{\mathbf{a}}^{'} } \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{if}}\quad {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) \ge {\mathbf{p}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right),} \hfill \\ 1 \hfill & {{\text{if}}\quad {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) \le {\mathbf{q}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right),} \hfill \\ {\frac{{{\mathbf{p}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) + {\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right)}}{{{\mathbf{p}}_{{{\mathbf{j}}}} \left( {{\mathbf{a}}^{'} } \right) - {\mathbf{q}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right)}}} \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right.$$

(3)

$${\mathbf{C}}\left( {{\mathbf{a}},{\mathbf{a}}^{'} } \right) = \frac{{\mathop \sum \nolimits_{{{\mathbf{j}} \in {\mathbf{F}}}} {\mathbf{w}}_{{\mathbf{j}}} {\mathbf{C}}_{{\mathbf{j}}} \left( {{\mathbf{a}},{\mathbf{a}}^{'} } \right)}}{{\mathop \sum \nolimits_{{{\mathbf{j}} \in {\mathbf{F}}}} {\mathbf{w}}_{{\mathbf{j}}} }}$$

(4)

Subsequently, the partial discordance index ${\text{d}}_{\text{j}} (a,a^{\prime } )$ must be calculated, as shown in Eq. 5. This index considers all criteria against the affirmation a outranks a′, and take in account the veto threshold (${\text{v}}_{\text{j}}$).

$${\mathbf{d}}_{{\mathbf{j}}} \left( {{\mathbf{a}},{\mathbf{a}}^{'} } \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\mathbf{if}}\quad {\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) < - {\mathbf{v}}_{{\mathbf{j}}} ,} \hfill \\ {\frac{{{\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) + {\mathbf{p}}_{{\mathbf{j}}} }}{{{\mathbf{p}}_{{{\mathbf{j}}}} - {\mathbf{v}}_{{\mathbf{j}}} }}} \hfill & {{\mathbf{if}}\quad - {\mathbf{v}}_{{\mathbf{j}}} \le {\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) < - {\mathbf{p}}_{{\mathbf{j}}} ,} \hfill \\ 0 \hfill & {{\mathbf{if}}\quad {\mathbf{g}}_{{\mathbf{j}}} \left( {\mathbf{a}} \right) - {\mathbf{g}}_{{\mathbf{j}}} \left( {{\mathbf{a}}^{'} } \right) \ge - {\mathbf{p}}_{{\mathbf{j}}} .} \hfill \\ \end{array} } \right.$$

(5)

After, the credibility index σ(a,a′) can be measured as per Eq. 6. This index is intended to measure the strength of the assertion that alternative a outranks reference action a′,

$${\mathbf{\sigma }}\left( {{\mathbf{a}},{\mathbf{a}}^{'} } \right) = {\mathbf{c}}\left( {{\mathbf{a}},{\mathbf{a}}^{'} } \right)\mathop \prod \limits_{{{\mathbf{j}} = 1}}^{{\mathbf{n}}} {\mathbf{T}}_{{\mathbf{j}}} ({\mathbf{a}},{\mathbf{a}}^{'} ),\quad {\text{where}}\,{\text{T}}_{{\text{j}}} \left( {{\text{a}},{\text{a}}^{\prime } } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{1 - {\text{d}}_{{\text{j}}} ({\text{a}},{\text{a}}^{\prime } )}}{{1 - {\text{c}}_{{\text{j}}} {\text{}}({\text{a}},{\text{a}}^{\prime } )}}} & {{\text{if}}\quad {\text{d}}_{{\text{j}}} (a,a^{\prime } ) > {\text{c}}_{{\text{j}}} (a,a^{\prime } )} \\ 1 & {{\text{otherwise}}} \\ \end{array} } \right.$$

(6)

Given the credibility index, a credibility level, denoted by λ, should be defined, representing the minimum value of σ(a,a′) in order to validate or not the outranking relation.

The recommendation of the ELECTRE TRI-C method is a classification, which is formed from two joint rules: descending and ascending. These classifications are made in accordance with the two procedures described below:

Descending rule: evaluating the worst to the best class, the first class ${\text{a}}_{\text{t}}^{\prime }$ that satisfies $\upsigma\left( {{\text{a}},{\text{a}}_{\text{t}}^{\prime } } \right) \ge {\lambda }$ must be found. Given t:

• For ${\text{t}} = {\text{q}}$, select ${\text{C}}_{\text{q}}$ as a possible class to which to assign alternative a.

• For $0 < {\text{t}} < q$, if $\uprho\left( {{\text{a}},{\text{a}}_{\text{t}}^{\prime } } \right) > \rho \left( {{\text{a}},{\text{a}}_{{{\text{t}} + 1}}^{\prime } } \right)$, select ${\text{C}}_{\text{t}}$ as a possible class to which to assign alternative a. Otherwise, select ${\text{C}}_{{{\text{t}} + 1}}$. Consider $\uprho\left( {{\text{a}},{\text{a}}_{\text{k}}^{\prime } } \right) = {\text{min}}\left\{ {\upsigma\left( {{\text{a}},{\text{a}}_{\text{k}}^{\prime } } \right), \upsigma\left( {{\text{a}}_{\text{k}}^{\prime } ,{\text{a}}} \right)} \right\}$.

• For ${\text{t}} = 0$, select ${\text{C}}_{1}$ as a possible class to which to assign alternative a.

Ascending rule: evaluating the best to the worst class, the first class ${\text{b}}_{\text{k}}$ that satisfies $\upsigma\left( {{\text{a}}_{\text{k}}^{\prime } ,{\text{a}}} \right) \ge\uplambda$ must be found. Given k:

• For ${\text{k}} = 1$, select ${\text{C}}_{1}$ as a possible class to which to assign alternative a.

• For 1 < k < (q + 1), if $\uprho\left( {{\text{a}},{\text{a}}_{\text{k}}^{\prime } } \right) > \rho \left( {{\text{a}},{\text{a}}_{{{\text{k}} - 1}}^{\prime } } \right)$, select ${\text{C}}_{\text{k}}$ as a possible class to which to assign alternative a. Otherwise, select ${\text{C}}_{{{\text{k}} - 1}}$. Consider $\uprho({\text{a}},{\text{a}}_{\text{k}}^{\prime } ) = \hbox{min} \left\{ {\upsigma({\text{a}},{\text{a}}_{\text{k}}^{\prime } ),\upsigma({\text{a}}_{\text{k}}^{\prime } ,{\text{a}})} \right\}$.

• For ${\text{k}} = ({\text{q}} + 1)$, select ${\text{C}}_{\text{q}}$ as a possible class to which to assign alternative a.