In this work we study inverse and consensus analysis for PROMETHEE II ranking when the performance table is fixed. Existing sensitivity and consensus studies usually vary only the criterion weights $w_k$ while keeping the preference functions $P_k(\cdot)$ unchanged. In fact both sets of parameters determine the net flow scores $\phi_i$. We introduce a linear re‑parameterisation based on weighted preference increments over the observed performance differences:
$$ \phii = \frac{1}{n-1}\sum{j\neq i}\sum_{k=1}^m w_k\bigl(P_k(a_i,a_j)-P_k(a_j,ai)\bigr) = \sum{k=1}^m wk\,\Delta{ik} + \sum{k=1}^m \Delta p{ik}, $$ where $\Delta_{ik}$ are fixed increments derived from the data and $\Delta p_{ik}$ are adjustable parameters of the preference functions. This expression linearises the net flows and makes the interaction between weights and preference functions explicit.
Based on this representation we formulate three linear programming models:
minimise the deviation of weights $\|w-w^0\|_1$ while allowing the preference parameters $\Delta p$ to vary so that the target net flows are reproduced; minimise the combined deviation of weights and preference parameters $\|w-w^0\|_1+\|\Delta p-\Delta p^0\|_1$; perform multi‑decision‑maker consensus analysis, with objectives either the total deviation $\sum_d \|w^{(d)}-w\|_1$ or the maximal individual deviation $\max_d \|w^{(d)}-w\|_1$, thus balancing efficiency and fairness.
An illustrative example shows that allowing preference functions to change can dramatically alter the conclusions of inverse analysis compared with weight‑only models, and that the proposed group formulations can distinguish efficient consensuses from fairer ones.
Review