Preference ranking organization method for enrichment evaluation

PROMETHEE & GAIA are multi-criteria decision analysis methods that belong to the family of outranking relations. They both offer a descriptive and a prescriptive approach.

On the one hand, the descriptive approach, called GAIA, allows the decision maker to visualize his problem. As a consequence, he is able to easily identify conflicts or synergies between criteria, to categorize alternatives and to highlight interesting performances. On the other hand, the prescriptive approach, called PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluation), provides the decision maker both a complete and a partial ranking of the alternatives.

The PROMETHEE method has been initiated by Professor Jean-Pierre Brans in the beginning of the eighties. Over the last 30 years, a number of researchers from the CoDE-SMG laboratory of the Université Libre de Bruxelles have also contributed to the development of this methodology. Among them, one can cite Professor Philippe Vincke or Professor Bertrand Mareschal. These research topics mainly cover the creation of visualization tools, the integration of uncertainty modeling, the development of robustness and sensitivity analysis tools or the group decision making aspects.

PROMETHEE has successfully been used in a number of complex decision making problems. Nowadays, hundreds of scientific papers describe its application to numerous area including finance, chemistry, environmental management, logistics ...

The PROMETHEE methods have been sequentially implemented in three main software: PROMCALC, DECISION LAB 2000 and more recently D-Sight.

The Model

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. At first, every pair of alternatives will be compared 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. 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.

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:

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

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

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.

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.