An ELECTRE approach for evaluating secondary education profiles: evidence from PISA survey in Serbia

Abstract

The use of multi-criteria decision making has been recognized as a useful tool for economic applications, especially in the last decade. The aim of our study is to promote the potential of the ELimination Et Choix Traduisant la REalité (ELECTRE) method in the field of economics of education. To evaluate the secondary education profiles in Serbia, we have used PISA data and the ELECTRE multi-level outranking (ELECTRE MLO) approach, as an alternative to commonly used econometric models. We introduce a new outranking relation, which is important for decision makers for two reasons. First, it allows the establishment of an acyclic relation, for which we provide a threshold value, that assures there will be no cycles in the relation graph. Second, the procedure does not require an excessive number of parameters as in other existing ranking methods, which enables an easier understanding, simplified use and less bias in the decision making process. After neutralizing the impact of socio-economic status and regional disparities, we obtain the performance levels for secondary education profiles in Serbia.

Appendices

Appendix 1: Proof of Theorem 1

Assume that alternative $a_i$ outranks alternative $a_j$. Then, let the parameter $M_i$ be defined as

$$\begin{aligned} M_i =\sum _{k=1}^n {\omega _k^*g_k^*(a_i )}. \end{aligned}$$

(14)

For alternatives $a_i$ and $a_j$,

$$\begin{aligned} M_i -M_j =\sum _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} \left( {g_k^*(a_i )-g_k^*(a_j )} \right) -\sum _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*} \left( {g_k^*(a_j )-g_k^*(a_i )} \right) . \end{aligned}$$

(15)

The following expression denotes $q$:

$$\begin{aligned} q=\frac{\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} \left( {g_k^*(a_i )-g_k^*(a_j )} \right) }{\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} }. \end{aligned}$$

(16)

From the definition of the Discordance index,

$$\begin{aligned} \sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*} \left( {g_k^*(a_j )-g_k^*(a_i )} \right) \le d_{ij} \sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*}. \end{aligned}$$

(17)

Note that $l_{ij} =\frac{d_{ij} }{q}$. Then,

$$\begin{aligned} M_i -M_j \ge q\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} -d_{ij} \sum _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*=} \,q\cdot \left( {\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} -l_{ij} \sum _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*} } \right) .\nonumber \\ \end{aligned}$$

(18)

Because $a_i$ outranks alternative $a_j$ by assumption,

$$\begin{aligned} \frac{\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} }{\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} +\sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*} }>\frac{l_{ij} }{l_{ij} +1}. \end{aligned}$$

(19)

Hence, we have

$$\begin{aligned} \sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^+ } \right. } \right\} } {\omega _k^*} >l_{ij} \sum \limits _{\left\{ {k\left| {g_k \in G_{ij}^- } \right. } \right\} } {\omega _k^*.} \end{aligned}$$

(20)

From Eqs. (18) and (20), we have $M_i -M_j >0.$ Overall, we have that if alternative $a_i$ outranks alternative $a_j$, then that implies that $M_i -M_j >0.$ Hence, there are no cycles because that would violate the transitivity property of the arithmetic relation “bigger than.”

Appendix 2: Proof of Proposition 1

We will show this by providing the example for which setting the cutting level $\lambda$ to be equal to the $AST$ will lead to the creation of cycles. Assume we have $m$ alternatives that we want to compare using $n$criteria of equal importance, and assume $m\ge n.$ The performance matrix is given in Table 8. In this example, the value of the AST is $\frac{n-1}{n}.$ By selecting $\lambda =\frac{n-1}{n},v=0.4$, the alternatives form a cycle, and we have $a_1 \succ a_2 \succ \ldots \succ a_n \succ a_1.$ For the case where $m<n$, a similar procedure leads us to a cycle of length $m$.

Table 8 Performance matrix for Proposition 1

Appendix 3: Robustness Check

See Table 9.

Table 9 Robustness check