Something Else about Decision Making

8 min readOct 6, 2023

ELECTRE

Notably, decision-making is an action that occurs at all times in our lives. In terms of administration, the position of decision-maker is directly linked to the role that the individual responsible for answering questions such as:

What to do?
What’s the best decision?
What problem are you facing?
And the solution alternatives?

The answer to these questions is crucial for the performance of those involved in decision-making processes.

So we can say that the decision-making process in the sense of administration is a step that precedes taking action to solve a problem.

Several thinkers in science have addressed this organizational issue through decision theory and mathematical modeling. A typical problem is the transportation of goods, in which the decision maker needs to choose the transport companies considering attributes such as price, delivery time, level of delay, damage to the cargo.

This type of more complex question, which involves several criteria or objectives, appears to have been approached initially motivated by questions in military applications through the mathematical method called Operational Research and later passed to be used in various economic activities. The evolution of this methodology led to the conception of several specific approaches and applications. Then arise methodologies called Multi Criteria Decision Making (MCDM) tools.

In this context, the Elimination Et Choix Traduisant la Réalité (ELECTRE) is presented, which appears in the literature on MCDM as a methodology belonging to the French school. ELECTRE encompasses a family of methodologies.

With this method, originally presented by Benayoun in 1966 and
later by Roy in 1968, it is possible to mathematically model a question involving a decision based on a multicriteria choice. The ELECTRE method is an analysis method applied in the social sciences of Administration and Production Engineering, which is supported by Applied Mathematics resources. It is a deterministic (non-probabilistic) method applicable to the decision-making process of problems classifiable as complex or ill-defined

When looking for a solution to a question, with the support of a multicriteria method (MADM), the first step to consider is to frame the question, or the dilemma, within a problematic. Based on Roy and Benayoun’s theory, we present the four types of problems worked on by MADM and its methodologies (MAD).

Problematic P.α

The intended result is a choice or a selection procedure.

💭 Methodology: clarify the decision by choosing a subset as restricted as possible, in view of the final choice of a single action. This set will contain the best actions or the satisfactory actions.

Problematic P.β

The intended outcome is screening, or a grading procedure.

💭 Methodology: clarify the decision by a screening resulting from the allocation of each action to a category. The different categories are defined a priori based on the rules applicable to the set of actions.

Problematic P.γ

The intended result is an array, or sorting procedure.

💭 Methodology: clarify the decision for an arrangement obtained by regrouping all shares of shares into equivalence classes. These classes are sorted in full or in part, depending on preferences.

Problematic P.δ

The intended result is a description or a cognitive procedure.

💭 Methodology: clarify the decision by a description, in appropriate language, of the actions and their consequences.

MAD

Real systems are involved in complex situations, a range of variables that are integral and intervening elements in a decision-making process, so the role of MAD is to serve as a means of rationalizing decision-making. The MAD represent the result of applying the Theory of Mathematics in the decision-making process.

The approach of a need for action, as proposed by mathematical methods of multicriteria evaluation, has precisely the purpose of unifying the understandings of all intervening decision-makers in the decision-making process. The definition of a need for action and the consequent unification of understandings are parts related to the structuring of the problem.

After obtaining the preferences of the “decider”, the problem is reduced to a classification process of mathematical aggregation, which is what defines the type of multicriteria decision support method to be applied. Based on the preference aggregation and classification procedure, MADs are distinguished into three types:

Aggregation methods to a single synthesis criterion;
Subordination methods;
Interactive, alternative and hybrid methods.

👉Aggregation methods to a single synthesis criterion

The synthesis single criterion methods assume that the preferences of the “deciders” can be represented by a Utility or a Value Function. These must be evaluated by the analyst through the use of linear models: additive, multiplicative, among others.

These methods adopt the principle of transitivity and do not admit the incompatibility of potential actions. They consider, in general, only the situations of preference and indifference, which results in total rankings of the alternatives. Some examples of these methods cited in the relevant literature are: UTA, PREFCALC, UTASTAR, MINORA, AHP,
MACBETH, MAVT, SMART, EVAMIX and TOPSIS.

👉Subordination methods (outranking)

The outranking methods are also known as Multicriteria Decision Aiding Methods (MCDA). The mathematical outranking relation is defined by Roy as being binary. It compares the arguments for and against the hypothesis that action or alternative a is at least as good as action b. This is equivalent to saying that a is “no worse than” b, through the notation:

A subordination or outranking relationship (S) allows the treatment of incompatibility between shares. Situations of incompatibility can occur in practice, due to the uncertainty and imprecision of the data used and the characteristics of the “decision maker”.

The outranking models are applied when the basic axioms of ordinality and transitivity may not be fully respected.

👉Interactive, alternative and hybrid methods

Hybrid methods are called those methods that use both the concepts of the American School and those of the French School.

The ELECTRE I algorithm

The first method of the ELECTRE I family aims to choose alternatives that are preferred by most of the criteria and do not cause any unacceptable level of dissatisfaction for any of the criteria.
analyzed.

ELECTRE I is an aggregation method based on the concept of overclassification or categorization. Its aggregation formulas that define the system are among the simplest and help to elucidate the most complex Multicriteria Decision Support Problems (MDSP) that come in sequence in the other methods of the ELECTRE family. In applying the concept of overclassification, the ELECTRE I method presents the following axioms:

Concordance: It consists in the fact that a significant subset of the criteria considers that alternative a is (weakly) preferable to alternative b.

Disagreement: It consists in the fact that there are no criteria in which the intensity of preference for b, in relation to alternative a, exceeds an acceptable limit.

The algorithm of the method consists of representing the quantities attributed to the concepts of agreement (or dominance) and disagreement in the form of matrices: the agreement matrix and the disagreement matrix.

Representations in mathematical language begin by considering the alternatives as elements of a set A:

where aᵢ represents a given alternative of order i, and each of them evaluated according to a set of criteria, G:

where n is the number of criteria and gⱼ is the performance or evaluation in the criterion of order j.

The parameters w₁, w₂, w₃, …, wᵢ, …, wₙ are defined as the weights assigned by the decision maker, for all n validated criteria for all k alternatives, and it is assumed that the weights of the criteria do not change during the application of the method (they are invariant).

Here we introduce the concept of concordance index, C(a,b), between any two alternatives called a and b as:

a measure of the number of criteria for which action a is preferred to action b, (aPb); or,
action a is equal to action b, (a=b).

The concordance index can be represented by the following relationship:

where:

where q is an arbitrary value for the indifference threshold; and value factor used in the calculation of may be equal to 1 or 2.

The concordance index can be structured as a matrix where each Cᵢⱼ element corresponds to the weighted percentage of the criteria for which the alternative index i is preferred over the alternative index j. By definition, 0≤Cᵢⱼ≤1.

The concept of disagreement is complementary to that of concordance, as it represents the “discomfort” experienced in choosing alternative a over alternative b.

To determine the matrix of discordance indices, it is necessary to define, first, a scale of intervals common to all criteria, since each criterion can have different scales. This scale is used in the comparison between an alternative i instead of a j regarding the discomfort caused by choosing one instead of the other, through all the evaluation criteria.

The criterion that has the highest upper value on its scale will be the one in which the “decider” experiences the greatest discomfort when going from a better level to a worse one.

With these considerations, the disagreement i-th is determined by:

and

Once obtained, this value will be used to determine the disagreement indices which will structure the disagreement matrix, whose each element is expressed by: