# PROMETHEE

Pascal Francq

December 1, 2011 (January 20, 2004)

Table of Contents

This article contains a short introduction to the multi-criteria decision aid system PROMETHEE. For a full description, [1, 2] should be consulted.

## 1 The problem

Let be a set of solutions, each , represents the evaluation of a solution, , to a given criterion, . Table 1↓ represents a generic evaluation table.

## 2 Enrichment of the Preference Structure

The notion of generalized criteria is introduced in order to take into account the extent of the deviations between the evaluations.

For this purpose we define preference function as giving the degree of preference of solution over solution for a given criteria, . In most cases, we can assume that is a function of the deviation . We consider that the function is normalized, so that:

- ;
- , if , no preference or indifference;
- , if , weak preference;
- , if , strong preference;
- , if , strict preference.

It is clear that has to be a non-decreasing function of , with a shape similar to that in Figure 1↓. Parameters and are known respectively as the indifference and the preference threshold.

The generalized criterion associated with is then defined by the pair . The PROMETHEE method requires a generalized criterion to be associated with each criterion, .

## 3 Enrichment of the Dominance Relation

A valued outranking relation is built up that takes all the criteria into account. Let us now suppose that a generalized criterion is associated with each criterion .

The and values are the indifference and preference thresholds respectively. When the difference between the evaluations of and is lower than it is not significant, and the preference of over is thus equal to zero. When the difference between the evaluations of and is greater than it is considered to be very significant, and the corresponding preference is thus equal to one.

A multi-criteria preference index of over can then be defined, that takes all the criteria into account with the expression (↓).

where are weights associated with each criterion. These weights are positive real numbers that do not depend on the scales of the criteria.

It is interesting to note that if all the weights are equal, will simply be the arithmetical average of all the degrees.

expresses how, and to what extend, is preferred to , while expresses how is preferred to over all the criteria.

The values and are computed for each pair of alternatives . In this way, a complete and valued outranking relation is constructed on .

## 4 Exploitation for Decision Aid

Let us considered how each alternative, , faces the others and therefore defines the two following outranking flows:

- the positive outranking flow is given by:
- the negative outranking flow is given by:

The positive outranking flow expresses to what extent each alternative outranks all the others. The higher is, the better the alternative will be. represents the power of , i.e. its outranking character.

The negative outranking flow expresses to what extend each alternative is outranked by all the others. The smaller is, the better the alternative will be. represents the power of , i.e. its outranked character.

## 5 PROMETHEE I Ranking

Two rankings of the alternatives can be deduced naturally from the positive and the negative outranking flows. Let us denote them and respectively:

Partial PROMETHEE I ranking is the intersection of these two rankings:

where the index denotes that the ranking concerns the PROMETHEE I method.

Let us denote with , and the preference, the indifference and the incompatibility respectively between two alternatives. The results of the pairwise PROMETHEE I comparisons are the following:

is preferred to . In this case, the higher power of is associated with the lowest level of weakness of . The information given by both outranking flows is consistent and can be considered to be viable.

and are indifferent. The positive and the negative outranking flows of and are equal.

and cannot be compared. In this case, a higher power of one alternative is associated to a lower level of weakness of the other. This usually happens when is good on a set of criteria on which is weak, and vice-versa. As the information corresponding to the alternatives is not consistent, it seems natural that the method would not decide which one of the alternatives is better. In such a case, it is up to the decision-maker to assume the responsibility and to decided.

If a complete ranking of the alternatives is requested by the decision-maker, the net outranking flow can be considered: . This is the balance between the positive and the negative outranking flows. The higher the net flow is, the better the alternative will be.

## 6 PROMETHEE II Ranking

Complete PROMETHEE II ranking is defined by:

All the alternatives are now comparable.

## 7 An example of PROMETHEE

Let us illustrate the PROMETHEE method with the problem of choosing a car. Different criteria can be laid down, i.e. the price (as low as possible), the fuel consumption (as low as possible), the comfort, given as a number between and (as high as possible) and the power (as high as possible). Table 2↓ shows the values for different cars, with the best value for each criterion in bold: no one solution seems to be any better than any other. Moreover, each person may have his or her own preference, for example power may be very important compared to the other criteria.

The preference threshold, , is set for each criterion, i.e. if two solutions have a relative difference greater than for one criterion, one solution is always preferred to the other for this criterion. Furthermore, the indifference threshold, , is set at for each criterion, i.e. if two solutions have a relative difference below for one criterion, the two solutions will be considered to be equal for this criterion.

Let us first compute the PROMETHEE ranking of these solutions when all the criteria weights are set at (Table 3↓).

The results show that the best solution is , and that the worst is . Let us now compute the PROMETHEE ranking of these solutions when the weight of the power criterion is set at (Table 4↓).

The effect of this change is clearly to seen in the results, because, as expected, the turns out to be the best solution.

## 8 Implementation

The R Library proposes an implementation of the PROMETHEE method it the ROptimization library. The decision engine is implemented in the RPromKernel class : it takes a set of criteria (defined through the RPromCriterion class) and a set of solutions (defined with the RPromSol class. Finally, it is necessary to assign the values of each criterion for each solution.

## References

[1] Philippe Vincke, *Multicriteria Decision Aid*, John Wiley and sons, 1989.

[2] Jean-Pierre Brans & Bertrand Mareschal, ”The PROMCALC & GAIA Decision Support System for Multicriteria Decision Aid”, *Decision Support Systems*, 12(4‑5), pp. 297—310, 1994.