Fuzzy numbers & MCDM¶

Fuzzy numbers¶

fuzzytool.fuzzynum provides triangular (TFN) and trapezoidal (TrFN) fuzzy numbers with arithmetic, alpha-cuts, a crisp centroid, and a vertex distance.

from fuzzytool.fuzzynum import tfn, trfn, rank

a, b = tfn(1, 2, 3), tfn(2, 3, 4)
a + b              # TFN(3, 5, 7)
a * 2              # TFN(2, 4, 6)
a.centroid()       # 2.0
a.alpha_cut(0.5)   # (1.5, 2.5)
rank([tfn(0,1,2), tfn(5,6,7)])   # [1, 0] (largest first)

+/- are exact; *// use the standard positive-support approximation.

Fuzzy multi-criteria decision making¶

fuzzytool.mcdm offers two classic methods.

Fuzzy TOPSIS (Chen)¶

Rank alternatives by closeness to the fuzzy positive-ideal solution. The decision matrix and weights are triangular fuzzy numbers; benefit[j] says whether criterion j is maximized.

from fuzzytool.fuzzynum import tfn
from fuzzytool.mcdm import fuzzy_topsis

matrix = [
    [tfn(7, 8, 9), tfn(5, 6, 7)],   # alternative A: ratings per criterion
    [tfn(3, 4, 5), tfn(8, 9, 9)],   # alternative B
]
weights = [tfn(0.4, 0.5, 0.6), tfn(0.3, 0.4, 0.5)]
res = fuzzy_topsis(matrix, weights, benefit=[True, True])
res.closeness   # -> array([0.369, 0.316]), closeness coefficient per alternative
res.ranking     # -> [0, 1], alternative indices, best first

Fuzzy AHP (Chang's extent analysis)¶

Derive crisp criterion weights from a fuzzy pairwise-comparison matrix.

from fuzzytool.fuzzynum import tfn
from fuzzytool.mcdm import fuzzy_ahp

one = tfn(1, 1, 1)
matrix = [
    [one,                tfn(1, 2, 3)],
    [tfn(1/3, 1/2, 1),   one],
]
fuzzy_ahp(matrix)   # -> array([0.69, 0.31]), normalized crisp weights