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== History==
The basic elements of the PROMETHEEPromethee method have been first introduced by Professor Jean-Pierre Brans (CSOO, VUB Vrije Universiteit Brussel) in 1982.<ref name="Brans">{{Cite news|author=J.P. Brans|title=L’ingénierie de la décision: élaboration d’instruments d’aide à la décision. La méthode PROMETHEE.|year=1982|publisher=Presses de l’Université Laval}}</ref> 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 GAIAGaia,<ref name="Gaia">{{Cite news|title=Geometrical representations for MCDA. the GAIA module|author=B. Mareschal, J.P. Brans|year=1988|publisher=European Journal of Operational Research}}</ref> 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 PROMETHEEPromethee,<ref name="Promethee">{{Cite news|title=A preference ranking organisation method: The PROMETHEE method for MCDM|author=J.P. Brans and P. Vincke|publisher=Management Science|year=1985}}</ref> provides the decision maker with both complete and partial rankings of the actions.
PROMETHEEPromethee 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 PROMETHEEPromethee methods<ref name="applications">{{Cite news|author=M. Behzadian, R.B. Kazemzadeh, A. Albadvi and M. Aghdasi|title=PROMETHEE: A comprehensive literature review on methodologies and applications|year=2010|publisher=European Journal of Operational Research}}</ref> haswas recentlypublished beenin published2010.
== Uses and applications ==
While it can be used by individuals working on straightforward decisions, the PROMETHEEPromethee & GAIAGaia is most useful where groups of people are working on complex problems, especially those with several multi-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 PROMETHEEPromethee &and GAIAGaia can be applied include:
* [[Choice]] – The selection of one alternative from a given set of alternatives, usually where there are multiple decision criteria involved.
* [[Prioritization]] – Determining the relative merit of members of a set of alternatives, as opposed to selecting a single one or merely ranking them.
* [[Conflict resolution]] – Settling disputes between parties with apparently incompatible objectives
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The applications of PROMETHEEPromethee &and GAIAGaia 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.
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Some uses of PROMETHEEPromethee &and GAIAGaia have becomedbecome case-studies. Recently these have included:
* Deciding which resources are the best with the available budget to meet SPS quality standards (STDF – [[WTO]]) [See more in External Links]
* Selecting new route for train performance ([[Italferr]])[See more in External Links]
:<math>\pi_k(a_i,a_j)=P_k[d_k(a_i,a_j)]</math>
where <math>P_k:\R\rightarrow[0,1]</math> is a positive non-decreasing preference function such that <math>P_j(0)=0</math>. Six different types of preference function are proposed in the original PROMETHEEPromethee definition. Among them, the linear unicriterion preference function is often used in practice for quantitative criteria:
:<math>P_k(x) \begin{cases} 0, & \text{if } x\le q_k \\ \frac{x-q_k}{p_k-q_k}, & \text{if } q_k<x\le p_k \\ 1, & \text{if } x>p_k \end{cases}</math>
:<math>\phi(a_i)=\displaystyle\sum_{k=1}^q\phi_{k}(a_i).w_{k}</math>
Where:
where:
:<math>\phi_{k}(a_i)=\frac{1}{n-1}\displaystyle\sum_{a_j
The unicriterion net flow, denoted <math>\phi_{k}(a_i)\in[-1;1]</math>, has the same interpretation as the multicriteria net flow <math>\phi(a_i)</math> but is limited to one single criterion. Any action <math>a_i</math> can be characterized by a vector <math>\vec \phi(a_i) =[\phi_1(a_i),...,\phi_k(a_i),\phi_q(a_i)]</math> in a <math>q</math> dimensional space. The GAIA plane is the principal plane obtained by applying a principal components analysis to the set of actions in this space.
=== PROMETHEEPromethee preference functions ===
*Usual
:<math>P_{j}(d_{j})=1-e^{-\frac{d_{j}^{2}}{2s_{j}^{2}}}</math>
== PROMETHEEPromethee rankings ==
===Promethee I===
PROMETHEEPromethee 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===
PROMETHEEPromethee II is a complete ranking of the actions. It is based on the multicriteria net flow. It includes preferences and indifferences (preorder).
==See also==
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