1.2. European school

1.2.1. PROMETHEE II

PROMEHTEE II is designed to evaluate decision alternatives according to the following steps:

Step 1. After defining the problem, calculate the preference function values. It is defined as (1.110) for profit criteria.

(1.110)\[\begin{equation} P(a,b) = F[d(a,b)], \; \forall a, b \in A \end{equation}\]

where \(d(a, b)\) is the difference between two actions (pairwise comparison):

(1.111)\[\begin{equation} d(a,b) = g(a) - g(b) \end{equation}\]

and the value of the preference function \(P\) is always between 0 and 1 and it is calculating for each criterion according to the Equation~(1.112):

(1.112)\[\begin{split}\begin{equation} P(d) = \left\{\begin{array}{cc} 0, & d \leq 0 \\ 1, & d > 0 \end{array}\right. \end{equation}\end{split}\]

Step 2. Calculate the aggregated preference indices (1.113).

(1.113)\[\begin{split}\begin{equation} \left\{\begin{array}{c} \pi(a,b) = \sum_{j=1}^{n} P_{j}(a,b)w_{j} \\ \pi(b,a) = \sum_{j=1}^{n} P_{j}(b,a)w_{j} \end{array}\right. \end{equation}\end{split}\]

where \(a\) and \(b\) are alternatives and \(\pi(a,b)\) shows how much alternative \(a\) is preferred to \(b\) over all of the criteria. There are some properties (1.114) which must be true for all alternatives set \(A\).

(1.114)\[\begin{split}\begin{equation} \left\{\begin{array}{c} \pi(a,a) = 0 \\ 0 \leq \pi(a,b) \leq 1 \\ 0 \leq \pi(b,a) \leq 1 \\ 0 \leq \pi(a,b) + \pi(b,a) \leq 1 \end{array}\right. \end{equation}\end{split}\]

Step 3. Calculate positive (1.115) and negative (1.116) outranking flows.

(1.115)\[\begin{equation} \phi^{+}(a) = \frac{1}{m-1}\sum_{x \in A} \pi(a,x) \end{equation}\]

(1.116)\[\begin{equation} \phi^{-}(a) = \frac{1}{m-1}\sum_{x \in A} \pi(x,a) \end{equation}\]

Step 4. Ranking is based on the net flow \(\Phi\) (1.117).

(1.117)\[\begin{equation} \Phi(a) = \Phi^{+}(a) - \Phi^{-}(a) \end{equation}\]

where larger value of \(\Phi(a)\) means better alternative.