# ELECTRE methods

These decision analysis methods enable to identify a set of solutions to a given problem, to compare solutions or to classify them from the best to the worst. Available in Excel with XLSTAT.

## What are the ELECTRE methods?

The **ELECTRE** methods, whose acronym stands for **ELimination** and **Choice****Expressing****REality**, bring together a family of **decision aid methods** whose particularity is the partial aggregation based on the construction of relations of comparisons of the performances of each pair of solutions. Unlike classical optimization methods, which consist in formulating the problem in the form of a cost function and in searching its optimum, here we compare solutions 2 by 2, criterion by criterion putting forward a preference/indifference of a response to another and resulting in an over ranking matrix. These methods have the **advantage** of accepting situations of incomparability with qualitative and immeasurable criteria.

## What is the principle?

The different solutions of a decision-making problem are called potential** actions**, or alternatives. These actions are listed exhaustively or not and must be formulated by the user. The **consequences** of each of them are evaluated using **criteria**. A criterion can be **qualitative** or **quantitative** and must be defined by the user. When it is qualitative, the evaluation of the actions on this criterion must be reduced to a numerical scale defined by the user. For example, let’s consider the criterion "type of diploma" for the selection of a candidate in a recruitment process. This criterion will be reduced to an arbitrary numerical scale which can be: 0 for high-school diploma, 1 for associate degree, 2 for bachelor degree, and so on... To allow a different contribution of these criteria in the decisional problem, the user can provide a weight to each one increasing with its importance. In the example of the recruitment process let "age of the candidate" be a new criterion ranging from 25 to 50 years. Assuming that this criterion is less important than the criterion "type of diploma" then the weight value of the criterion "Age of the candidate" is set to 1 and the weight value of the criterion "type of diploma" is set to 2. Make sure to respect this rule also in the evaluation of your actions by criterion: if the criteria are evaluated in order of importance, so are the actions and vice versa.

In the end, in order to use an ELECTRE method, the user must provide at least: the list of actions, the list of criteria, the evaluation of each action by criterion and the weight of each criterion.

## ELECTRE options in XLSTAT

### Method choice: Electre 1

This method is used to identify a set of solutions to a decision-making problem. The criteria are true-criteria. Let a and b be two potential actions, Electre 1 gives an over ranking matrix that numerically translates the assertions "a over ranks b", noted aSb, meaning that the action a is privileged over the action b and the opposite assertion. To do this, we need to compute 2 matrices, one called concordance matrix and the second one called discordance matrix.

### Method choice: Electre 3

This method is used to classify a set of solutions from the best to the worst. Criteria are pseudo-criteria and in this case, thresholds are required to do the analysis. Compared to Electre 1, Electre 3 executes more computations in order to obtain the desired results. In a first step, the method computes matrix coefficients which summarize the information of concordance and discordance between actions of the problem. In a second step, the coefficients are used to build two pre-rankings, a first one which classifies solutions from the best to the worst and a second one which classifies from the worst to the best. The outranking matrix and the table rank are then deduced by crossing the two pre-rankings results.

## ELECTRE results in XLSTAT

**Concordance matrix**: This result displays the indexes of the concordance matrix computed with the equation given in the description section of the method.

**Discordance matrix**: This result displays the indexes of the discordance matrix computed with the equation given in the description section of the method.

**Over ranking matrix**: This result displays the matrix of 0 and 1 got with the over-ranking relation given in the description section of the method.

**Ranking table**: This result displays a table with the final rank of actions.

**Ranking tables of the concordance threshold sensitivity analysis**: This result displays 2 tables with the final rank of actions obtained with Electre 1 using a modified concordance threshold and a discordance threshold fixed to the user value (or set to the default value 0). The left table is the result with a 10% increase of the user value and the right table is the result with a 10% decrease.

**Ranking tables of the discordance threshold sensitivity analysis**: This result displays 2 tables with the final rank of actions obtained with Electre 1 using a modified concordance threshold and a discordance threshold fixed by the user (or set to the default value 0). The left table is the result with a 10% increase of the user value and the right table is the result with a 10% decrease.

Included in