The Preference Ranking Organization METHod for Enrichment of Evaluations and its descriptive complement geometrical analysis for interactive aid are better known as the Promethee and Gaia[1] methods.
Based on mathematics and sociology, the Promethee and Gaia method was developed at the beginning of the 1980s and has been extensively studied and refined since then.
It has particular application in decision making, and is used around the world in a wide variety of decision scenarios, in fields such as business, governmental institutions, transportation, healthcare and education.
Rather than pointing out a "right" decision, the Promethee and Gaia method helps decision makers find the alternative that best suits their goal and their understanding of the problem. It provides a comprehensive and rational framework for structuring a decision problem, identifying and quantifying its conflicts and synergies, clusters of actions, and highlight the main alternatives and the structured reasoning behind.
The basic elements of the Promethee method have been first introduced by Professor Jean-Pierre Brans (CSOO, VUB Vrije Universiteit Brussel) in 1982.[2] 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.
The descriptive approach, named Gaia,[3] 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,[4] provides the decision maker with both complete and partial rankings of the actions.
Promethee has successfully been used in many decision making contexts worldwide. A non-exhaustive list of scientific publications about extensions, applications and discussions related to the Promethee methods[5] was published in 2010.
While it can be used by individuals working on straightforward decisions, the Promethee & Gaia is most useful where groups of people are working on complex problems, especially those with several criteria, involving a lot of human perceptions and judgments, whose decisions have long-term impact. It has unique advantages when important elements of the decision are difficult to quantify or compare, or where collaboration among departments or team members are constrained by their different specializations or perspectives.
Decision situations to which the Promethee and Gaia can be applied include:
The applications of Promethee and Gaia to complex multi-criteria decision scenarios have numbered in the thousands, and have produced extensive results in problems involving planning, resource allocation, priority setting, and selection among alternatives. Other areas have included forecasting, talent selection, and tender analysis.
Some uses of Promethee and Gaia have become case-studies. Recently these have included:
Let
A=\{a1,..,an\}
F=\{f1,..,fq\}
The basic data related to such a problem can be written in a table containing
n x q
\begin{array}{|c|c|c|c|c|c|c|}\hline &f1( ⋅ )&f2( ⋅ )& … &fj( ⋅ )& … &fq( ⋅ )\ \hline a1&f1(a1)&f2(a1)& … &fj(a1)& … &fq(a1)\\ \hline a2&f1(a2)&f2(a2)& … &fj(a2)& … &fq(a2)\ \hline … & … & … & … & … & … &. … \ \hline ai&f1(ai)&f2(ai)& … &fj(ai)& … &fq(ai)\ \hline … & … & … & … & … & … & … \ \hline an&f1(an)&f2(an)& … &fj(an)& … & fq(an) \ \hline \end{array}
At first, pairwise comparisons will be made between all the actions for each criterion:
dk(ai,aj)=fk(ai)-fk(aj)
dk(ai,aj)
fk
As a consequence the notion of preference function is introduced to translate the difference into a unicriterion preference degree as follows:
\pik(ai,aj)=Pk[dk(ai,aj)]
where
Pk:\R → [0,1]
Pk(0)=0
Pk(x)\begin{cases}0,&ifx\leqk\
x-qk | |
pk-qk |
,&ifqk<x\lepk\ 1,&ifx>pk\end{cases}
where
qj
pj
When a preference function has been associated to each criterion by the decision maker, all comparisons between all pairs of actions can be done for all the criteria. A multicriteria preference degree is then computed to globally compare every couple of actions:
qP | |
\pi(a,b)=\displaystyle\sum | |
k |
(a,b) ⋅ wk
Where
wk
fk
wk\ge0
q | |
\sum | |
k=1 |
wk=1
\pi(ai,aj)\ge0
\pi(ai,aj)+\pi(aj,ai)\le1
In order to position every action with respect to all the other actions, two scores are computed:
\phi+(a)=
1 | |
n-1 |
\displaystyle\sumx\pi(a,x)
\phi-(a)=
1 | |
n-1 |
\displaystyle\sumx\pi(x,a)
The positive preference flow
\phi+(ai)
ai
\phi-(ai)
ai
ai
aj
\phi+(ai)\ge\phi+(aj)
\phi-(ai)\le\phi-(aj)
The positive and negative preference flows are aggregated into the net preference flow:
\phi(a)=\phi+(a)-\phi-(a)
Direct consequences of the previous formula are:
\phi(ai)\in[-1;1]
\sum | |
ai\inA |
\phi(ai)=0
The Promethee II complete ranking is obtained by ordering the actions according to the decreasing values of the net flow scores.
According to the definition of the multicriteria preference degree, the multicriteria net flow can be disaggregated as follows:
\phi(ai)=\displaystyle\sum
q\phi | |
k |
(ai).wk
Where:
\phik
(a | ||||
|
\displaystyle\sum | |
aj \in A |
\{Pk(ai,aj)-Pk(aj,ai)\}
The unicriterion net flow, denoted
\phik(ai)\in[-1;1]
\phi(ai)
ai
\vec\phi(ai)=[\phi1(ai),\ldots,\phik(ai),\phiq(ai)]
q
Pj(dj)= \begin{cases} 0&ifdj\leq0\\[4pt] 1&ifdj>0 \end{cases}
\begin{array}{cc}Pj(dj)=\left\{ \begin{array}{lll} 0&if&|dj|\leqqj\\ \\ 1&if&|dj|>qj\\ \end{array} \right. \end{array}
\begin{array}{cc}Pj(dj)=\left\{ \begin{array}{lll}
|dj| | |
pj |
&if&|dj|\leqpj\\ \\ 1&if&|dj|>pj\\ \end{array} \right. \end{array}
\begin{array}{cc}Pj(dj)=\left\{ \begin{array}{lll} 0&if&|dj|\leqqj\\ \\
1 | |
2 |
&if&qj<|dj|\leqpj\\ \\ 1&if&|dj|>pj\\ \end{array} \right. \end{array}
\begin{array}{cc}Pj(dj)=\left\{ \begin{array}{lll} 0&if&|dj|\leqqj\\ \\
|dj|-qj | |
pj-qj |
&if&qj<|dj|\leqpj\\ \\ 1&if&|dj|>pj\\ \end{array} \right. \end{array}
Pj(dj
| ||||||||||||||||
)=1-e |
Promethee I is a partial ranking of the actions. It is based on the positive and negative flows. It includes preferences, indifferences and incomparabilities (partial preorder).
Promethee II is a complete ranking of the actions. It is based on the multicriteria net flow. It includes preferences and indifferences (preorder).