Preference ranking organization method for enrichment evaluation

The Preference Ranking Organization Method for Enrichment of Evaluations and its descriptive complement Geometrical Analysis for Interactive Aid are better known as the PROMETHEE & GAIA methods. They are multi-criteria decision aid methods that belong to the family of the outranking methods initiated by Professor Bernard Roy with the ELECTRE methods. An original aspect of the PROMETHEE and GAIA methods is that they offer complementary descriptive and prescriptive approaches to the analysis of discrete multicriteria problems including a number of actions (decisions) evaluated on several criteria.

The descriptive approach, named GAIA, allows the decision maker to visualize the main features of a decision problem: he/she is able to easily identify conflicts or synergies between criteria, to identify clusters of actions and to highlight remarkable performances.

The prescriptive approach, named PROMETHEE, provides the decision maker with both complete and partial rankings of the actions.

The basic elements of the PROMETHEE method have been first introduced by Professor Jean-Pierre Brans (CSOO, VUB Vrije Universiteit Brussel) in 1982. It was later developed and implemented by Professor Jean-Pierre Brans and Professor Bertrand Mareschal (Solvay Brussels School of Economics and Management, ULB Université Libre de Bruxelles), including extensions such as GAIA.

PROMETHEE has successfully been used in many decision making contexts worlwide. For a non-exhaustive list of scientific publications about extensions, applications and discussions related to the PROMETHEE methods see http://biblio.promethee-gaia.net

The PROMETHEE methods have been implemented in several interactive computer software: PROMCALC (MS-DOS program developed by B. Mareschal, 1990), DECISION LAB 2000 (MS-WIN program developed by the Canadian company Visual Decision in collaboration with Prof. B. Mareschal, 2000), D-Sight (MS-WIN program developed by the Decision Sights company, 2010) and PROMETHEE (MS-WIN program developed by Prof. B. Mareschal, 2011).

The Model

Assumptions

Let $A=\{a_{1},..,a_{n}\}$ be a set of n alternatives and let $F=\{f_{1},..,f_{q}\}$ be a consistent family of q criteria. Without loss of generality, we will assume that these criteria have to be maximized.

The data related to such a problem can be written in a table containing $n\times q$ evaluations. Each lines represents an alternative and each column represents a criterion.

${\begin{array}{|c|c|c|c|c|c|c|}\hline &f_{1}(.)&f_{2}(.)&...&f_{j}(.)&...&f_{q}(.)\\\hline a_{1}&f_{1}(_{a}{1})&f_{2}(a_{1})&...&f_{j}(a_{1})&...&f_{q}(a_{1})\\\hline a_{2}&f_{1}(a_{2})&f_{2}(a_{2})&...&f_{j}(a_{2})&...&f_{q}(a_{2})\\\hline ...&...&...&...&...&...&...\\\hline a_{i}&f_{1}(a_{i})&f_{2}(a_{i})&...&f_{j}(a_{i})&...&f_{q}(a_{i})\\\hline ...&...&...&...&...&...&...\\\hline a_{n}&f_{1}(a_{n})&f_{2}(a_{n})&...&f_{j}(a_{i})&...&f_{q}(a_{n})\\\hline \end{array}}$

Pair Wise Comparisons

At first, pairwise comparisons will be made between the alternatives for each criterion:

$d_{k}(a_{i},a_{j})=f_{k}(a_{i})-f_{k}(a_{j})$

$d_{k}(a_{i},a_{j})$ represents the difference between two alternatives for criterion $f_{k}$. Of course, this quantity depends of the units used and not every difference is meaningful for the decision maker.

Preference Degree

As a consequence the next step consists to transform this difference into a unicriterion preference degree as follows:

$\pi _{k}(a_{i},a_{j})=P_{k}[d_{k}(a_{i},a_{j})]$

where $P_{k}:\mathbb {R} \rightarrow [0,1]$ is a positive non-decreasing function such that $P_{j}(0)=0$. Usually six different types of unicriterion preference functions are considered. Among them, the linear unicriterion preference function is often used in practice:

$P_{k}(x){\begin{cases}0,&{\text{if }}x\leq q_{k}\\{\frac {x-q_{k}}{p_{k}-q_{k}}},&{\text{if }}q_{k}<x\leq p_{k}\\1,&{\text{if }}x>p_{k}\end{cases}}$

where $q_{j}$ and $p_{j}$ are respectively called the indifference and the preference thresholds. The meaning of these parameters is the following: when the difference is lower than the indifference threshold it is considered to be negligible. Therefore the unicriterion preference value is equal to zero. If the difference exceeds the preference threshold it is considered to be significant. Therefore the unicriterion preference value is equal to one (which is the maximal value). When the difference lies between these two values, a linear interpolation is computed.

Global Preference Degree

Once every comparison between pairs of alternatives are performed for every criterion, a natural process is to compute an aggregated score to globally compare every couple of alternatives:

$\pi (a,b)=\displaystyle \sum _{k=1}^{q}P_{k}(a,b).w_{k}$

Where $w_{k}$ represents the weight of criterion $f_{k}$. It is assumed that $w_{k}\geq 0$ and $\sum _{k=1}^{q}w_{k}=1$. As a direct consequence, we have:

$\pi (a_{i},a_{j})\geq 0$

$\pi (a_{i},a_{j})+\pi (a_{j},a_{i})\leq 1$

Global Scores

In order to characterize every alternative a_i with respect to all the other alternatives, one computes two scores:

$\phi ^{+}(a)={\frac {1}{n-1}}\displaystyle \sum _{x\in A}\pi (a,x)$

$\phi ^{-}(a)={\frac {1}{n-1}}\displaystyle \sum _{x\in A}\pi (x,a)$

The positive flow $\phi ^{+}(a_{i})$ quantifies how a given alternative $a_{i}$ is globally preferred to all the other alternatives while the negative flow $\phi ^{-}(a_{i})$ quantifies how a given alternative $a_{i}$ is being globally preferred by all the other alternatives. An ideal alternative would have a positive flow equal to 1 and a negative flow equal to 0. These two scores induce two complete rankings. The first one allows ranking the alternatives according to the decreasing values of the positive flow score. The second one allows ranking the alternatives according to the increasing values of the negative flow score. The Promethee I partial ranking consists in the intersection of these two rankings. As a consequence, an alternative $a_{i}$ will be as good as another alternative $a_{j}$ if $\phi ^{-}(a_{i})\geq \phi ^{-}(a_{j})$ and $\phi ^{-}(a_{i})\leq \phi ^{-}(a_{j})$

The positive and negative scores may be aggregated into a net flow score:

$\phi (a)=\phi ^{+}(a)-\phi ^{-}(a)$

Direct consequences of the previous formula are:

$\phi (a_{i})\in [-1;1]$

$\sum _{a_{i}\in A}\phi (a_{i})=0$

The Promethee II complete ranking is obtained by ordering the alternatives according to the decreasing values of the net flow score.

Other expression of the net flow

Finally, the net flow score can be decomposed as follows:

$\phi (a_{i})=\displaystyle \sum _{k=1}^{q}\phi _{k}(a_{i}).w_{k}$

where :

$\phi _{k}(a_{i})={\frac {1}{n-1}}\displaystyle \sum _{a_{j}\in A}\{P_{k}(a_{i},a_{j})-P_{k}(a_{j},a_{i})\}$.

The unicriterion net flow score, denoted $\phi _{k}(a_{i})\in [-1;1]$, has exactly the same interpretation as $\phi (a_{i})$ but is limited to a given criterion. Thus any alternative $a_{i}$ can be characterize by a vector ${\vec {\phi }}(a_{l})=[\phi _{1}(a_{i}),...,\phi _{k}(a_{i}),\phi _{q}(a_{i})]$ in a $q$ dimensional space. GAIA is the projection plane obtained by applying a principal component analysis to this space.

PROMETHEE Preference Functions

• Usual

${\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&d_{j}\leq 0\\\\1&{\text{if}}&d_{j}>0\\\end{array}}\right.\end{array}}$

• U-Shape

${\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&|d_{j}|\leq q_{j}\\\\1&{\text{if}}&|d_{j}|>q_{j}\\\end{array}}\right.\end{array}}$

• V-Shape

${\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}{\frac {|d_{j}|}{p_{j}}}&{\text{if}}&|d_{j}|\leq p_{j}\\\\1&{\text{if}}&|d_{j}|>p_{j}\\\end{array}}\right.\end{array}}$

• Levels

${\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&|d_{j}|\leq q_{j}\\\\{\frac {1}{2}}&{\text{if}}&q_{j}<|d_{j}|\leq p_{j}\\\\1&{\text{if}}&|d_{j}|>p_{j}\\\end{array}}\right.\end{array}}$

• Linear

${\begin{array}{cc}P_{j}(d_{j})=\left\{{\begin{array}{lll}0&{\text{if}}&|d_{j}|\leq q_{j}\\\\{\frac {|d_{j}|-q_{j}}{p_{j}-q_{j}}}&{\text{if}}&q_{j}<|d_{j}|\leq p_{j}\\\\1&{\text{if}}&|d_{j}|>p_{j}\\\end{array}}\right.\end{array}}$

• Gaussian

$P_{j}(d_{j})=1-e^{-{\frac {d_{j}^{2}}{2s_{j}^{2}}}}$

Ranking

PROMETHEE I

PROMETHEE I gives a partial ranking of the alternatives and is based on the positive and negative flows. It includes the notions of Preference, Indifference and Incomparability.

PROMETHEE II

PROMETHEE II gives a complete ranking of the alternatives and is base on the net flows. It includes the notions of Preference and Indifference.

See also

• Decision making
• Decision making software
• D-Sight
• Multi-Criteria Decision Analysis
• Pairwise comparison
• Preference

External links

• PROMETHEE & GAIA web site
• D-Sight: PROMETHEE based software