promethee_ii
mcda.promethee_ii(normalized, weights)
Compute PROMETHEE II net outranking flows per tool.
PROMETHEE II (Brans and Vincke 1985) compares every ordered pair of tools on each metric and aggregates the comparisons into one net flow per tool. For an ordered pair (a, b) and metric j the difference is d_j(a, b) = x_aj - x_bj on a higher-is-better column. A preference function P maps that difference to a degree of preference for a over b in [0, 1]. The default is the Type I (usual) preference function, which expresses strict dominance per criterion:
P(d) = 1 if d > 0, else 0so a strictly outscores b on metric j as soon as it scores higher, regardless of the size of the gap, and an equal score yields no preference. The aggregated preference of a over b is the weighted sum over metrics:
pi(a, b) = sum_j w_j * P(d_j(a, b))The positive flow phi_plus(a) averages pi(a, b) over the other tools, measuring how much a outranks the rest. The negative flow phi_minus(a) averages pi(b, a), measuring how much the rest outrank a. Both divide by n_tools - 1. The net flow is
phi(a) = phi_plus(a) - phi_minus(a)and is the PROMETHEE II preference score. It is already higher is better (the most-preferred tool has the largest net flow) and lies in [-1, 1], so this function returns phi unchanged and beam.mcda.rank (1 = highest score) gives the ranking.
The net flow is computed by pymcdm.methods.PROMETHEE_II with the "usual" preference function and all criteria typed as profit (+1), because the matrix is oriented higher is better. The native pairwise loop has been replaced by that call.
The diagonal pair (a, a) has d = 0, so P(0) = 0 contributes nothing and a tool never outranks itself. With a single tool there is no other tool to compare against; the function returns a single zero in that case.
Extension point. Brans and Vincke define six preference functions. The other five (U-shape, V-shape, level, linear with indifference and preference thresholds, Gaussian) introduce an indifference threshold q, below which a difference is treated as no preference, and a preference threshold p, above which preference is full, with a shape in between. These soften the all-or-nothing behavior of the usual function and need per-metric q and p in the metric’s own units. To add them, pass the chosen preference function name and its thresholds to PROMETHEE_II through a new argument; the rest of this function does not change. The usual function is the default because it needs no thresholds and so makes no scale assumption beyond the higher-is-better orientation already imposed on the input.
Parameters
| Name | Type | Description | Default |
|---|---|---|---|
normalized |
np.ndarray | Shape (n_tools, n_metrics), values in [0, 1], higher is better. |
required |
weights |
np.ndarray | Shape (n_metrics,), non-negative; typically sums to 1. |
required |
Returns
| Type | Description |
|---|---|
| np.ndarray | Shape (n_tools,), the net outranking flow phi, higher is better. |
References
Brans, J.-P. and Vincke, P. A preference ranking organisation method: the PROMETHEE method for multiple criteria decision-making. Management Science 31(6), 647-656 (1985).