Preference ranking organization method for enrichment evaluation: Difference between revisions

Revision as of 17:02, 7 June 2014

Revision as of 17:14, 7 June 2014

Line 43:

The basic data related to such a problem can be written in a table containing <math>n\times q</math> evaluations. Each line corresponds to an action and each column corresponds to a criterion.

: <math>

\begin{array}{|c|c|c|c|c|c|c|} \hline

& f_{1}(.\cdot) & f_{2}(.\cdot) & ...\cdots & f_{j}(.\cdot) & ...\cdots & f_{q}(.\cdot) \\ \hline

a_{1} & f_{1}(a_{1}) & f_{2}(a_{1}) & ...\cdots & f_{j}(a_{1}) & ...\cdots & f_{q}(a_{1}) \\

\hline

a_{2} & f_{1}(a_{2}) & f_{2}(a_{2}) & ...\cdots & f_{j}(a_{2}) & ...\cdots & f_{q}(a_{2}) \\ \hline

...\cdots & ...\cdots &...\cdots & ...\cdots & ...\cdots & ...\cdots & ...\cdots \\ \hline

a_{i} & f_{1}(a_{i}) & f_{2}(a_{i}) & ...\cdots & f_{j}(a_{i}) & ...\cdots & f_{q}(a_{i}) \\ \hline

...\cdots & ...\cdots & ...\cdots & ...\cdots& ...\cdots & ...\cdots & ...\cdots \\ \hline

a_{n} & f_{1}(a_{n}) & f_{2}(a_{n}) & ...\cdots & f_{j}(a_{i}) & ...\cdots&

f_{q}(a_{n})

\\ \hline

Line 79:

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:

:<math>\pi(a,b)=\displaystyle\sum_{k=1}^qP_{k}(a,b).\cdot w_{k}</math>

Where <math>w_k</math> represents the weight of criterion <math>f_k</math>. It is assumed that <math>w_k\ge 0</math> and <math>\sum_{k=1}^q w_{k}=1</math>. As a direct consequence, we have:

Line 119:

A}\{P_{k}(a_i,a_j)-P_{k}(a_j,a_i)\}</math>.

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),...\ldots,\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.

=== Promethee preference functions ===

*Usual

::<math>\begin{array}{cc} P_{j}P_j(d_{j}d_j)=\left\{

\begin{array}{lllcases}

0 & \text{if } & d_{j}d_j\leq 0 \\[4pt]

1 & \endtext{arrayif } d_j>0

\\

1 & \textend{ifcases} & d_{j}>0\\

</math>

\end{array}

\right.

\end{array}</math>

*U-Shapeshape

::<math>\begin{array}{cc} P_{j}(d_{j})=\left\{

\begin{array}{lll}

0 & \text{if} & |d_{j}| \leq q_{j} \\

Line 144 ⟶ 142:

\end{array}</math>

*V-Shapeshape

::<math>\begin{array}{cc} P_{j}(d_{j})=\left\{

\begin{array}{lll}

\frac{|d_{j}|}{p_{j}} & \text{if} & |d_{j}| \leq p_{j} \\

Line 157 ⟶ 155:

*Level

::<math>\begin{array}{cc} P_{j}(d_{j})=\left\{

\begin{array}{lll}

0 & \text{if} & |d_{j}| \leq q_{j} \\

Line 171 ⟶ 169:

*Linear

::<math>\begin{array}{cc} P_{j}(d_{j})=\left\{

\begin{array}{lll}

0 & \text{if} & |d_{j}| \leq q_{j} \\

Line 184 ⟶ 182:

*Gaussian

::<math>P_{j}(d_{j})=1-e^{-\frac{d_{j}^{2}}{2s_{j}^{2}}}</math>

== Promethee rankings ==