Preference ranking organization method for enrichment evaluation

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

The basic data related to such a problem can be written in a table containing ${\displaystyle n\times q}$ evaluations. Each line corresponds to an action and each column corresponds to a criterion.

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

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

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

${\displaystyle d_{k}(a_{i},a_{j})}$ is the difference between the evaluations of two actions for criterion ${\displaystyle f_{k}}$. Of course, these differences depend on the measurement scales used and are not always easy to compare for the decision maker.

As a consequence the notion of preference function is introduced to translate the difference into a unicriterion preference degree as follows:

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

where ${\displaystyle P_{k}:\mathbb {R} \rightarrow [0,1]}$ is a positive non-decreasing preference function such that ${\displaystyle P_{j}(0)=0}$. Six different types of preference function are proposed in the original Promethee definition. Among them, the linear unicriterion preference function is often used in practice for quantitative criteria:

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

where ${\displaystyle q_{j}}$ and ${\displaystyle p_{j}}$ are respectively the indifference and preference thresholds. The meaning of these parameters is the following: when the difference is smaller than the indifference threshold it is considered as negligible by the decision maker. Therefore, the corresponding unicriterion preference degree is equal to zero. If the difference exceeds the preference threshold it is considered to be significant. Therefore, the unicriterion preference degree is equal to one (the maximum value). When the difference is between the two thresholds, an intermediate value is computed for the preference degree using a linear interpolation.

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:

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

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

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

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

In order to position every action with respect to all the other actions, two scores are computed:

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

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

The positive preference flow ${\displaystyle \phi ^{+}(a_{i})}$ quantifies how a given action ${\displaystyle a_{i}}$ is globally preferred to all the other actions while the negative preference flow ${\displaystyle \phi ^{-}(a_{i})}$ quantifies how a given action ${\displaystyle a_{i}}$ is being globally preferred by all the other actions. An ideal action would have a positive preference flow equal to 1 and a negative preference flow equal to 0. The two preference flows induce two generally different complete rankings on the set of actions. The first one is obtained by ranking the actions according to the decreasing values of their positive flow scores. The second one is obtained by ranking the actions according to the increasing values of their negative flow scores. The Promethee I partial ranking is defined as the intersection of these two rankings. As a consequence, an action ${\displaystyle a_{i}}$ will be as good as another action ${\displaystyle a_{j}}$ if ${\displaystyle \phi ^{-}(a_{i})\geq \phi ^{-}(a_{j})}$ and ${\displaystyle \phi ^{-}(a_{i})\leq \phi ^{-}(a_{j})}$

The positive and negative preference flows are aggregated into the net preference flow:

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

Direct consequences of the previous formula are:

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

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

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:

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

Where:

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

The unicriterion net flow, denoted ${\displaystyle \phi _{k}(a_{i})\in [-1;1]}$, has the same interpretation as the multicriteria net flow ${\displaystyle \phi (a_{i})}$ but is limited to one single criterion. Any action ${\displaystyle a_{i}}$ can be characterized by a vector ${\displaystyle {\vec {\phi }}(a_{i})=[\phi _{1}(a_{i}),\ldots ,\phi _{k}(a_{i}),\phi _{q}(a_{i})]}$ in a ${\displaystyle q}$ dimensional space. The GAIA plane is the principal plane obtained by applying a principal components analysis to the set of actions in this space.

• Usual

${\displaystyle P_{j}(d_{j})={\begin{cases}0&{\text{if }}d_{j}\leq 0\\[4pt]1&{\text{if }}d_{j}>0\end{cases}}}$

• U-shape

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

• V-shape

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

• Level

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

• Linear

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

• Gaussian

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

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).