PROMETHEE II - Preference Ranking Organisation Method for Enrichment of Evaluations II

1. 1. For each criterion j, compute pairwise difference matrix d_j(a,b): benefit criteria → x_aj − x_bj; cost criteria → x_bj − x_aj.

   $$d_j(a,b)=\{\begin{array}{ll}{x}_{aj}-x_{bj}& j\in J^{+}\\ x_{bj}-x_{aj}& j\in J^{-}\end{array}$$

2. 2. Apply preference function F_j to each pairwise difference to get unicriterion preference degree P_j(a,b) ∈ [0,1]. Common types: usual (0/1), vshape (linear ramp), ushape (step at q), level (combines p and q), vshape_2 (linear with q cutoff).

   $$P_j(a,b)=F_j(d_j(a,b))\in [0,1]$$

3. 3. Aggregate unicriterion preferences into global pairwise preference index π(a,b) = Σ_j w_j · P_j(a,b).

   $$\pi (a,b)=\sum _{j=1}^n{w}_j{P}_j(a,b)$$

4. 4. Compute positive outranking flow Φ⁺(a) and negative outranking flow Φ⁻(a).

   $${\Phi }^{+}(a)=\frac{1}{m-1}\sum _{b\ne a}\pi (a,b),{\Phi }^{-}(a)=\frac{1}{m-1}\sum _{b\ne a}\pi (b,a)$$

5. 5. Net flow Φ(a) = Φ⁺(a) − Φ⁻(a). Rank descending (higher net flow = better). Complete ranking (PROMETHEE II, as opposed to partial PROMETHEE I).

   $$\Phi (a)={\Phi }^{+}(a)-{\Phi }^{-}(a),\text{rank descending}$$