Example 19: Interval type-2 AHP and MARCOS

Interval type-2 trapezoidal judgements weigh the criteria with the DTraT score, and two experts' ratings are ranked with MARCOS. The 7-term scale is illustrative, not a published table.

Source

examples/19_it2_ahp_marcos.py

"""Interval type-2 AHP and MARCOS with an illustrative 7-term linguistic scale.

The scale below is **illustrative**, in the style of the interval type-2 literature:
its term values are not a transcription of any published table (the table of Chen
and Lee is not reproduced here). Seven IT2TrFN terms (upper trapezoid with height 1,
lower trapezoid with height 0.9) rate the criteria pairwise and the alternatives;
`ahp_fuzzy_numbers` weighs the criteria with the DTraT score and `rank_fuzzy`
ranks with MARCOS, which needs strictly positive entries (the DTraT score is).
"""

from bulanik import IT2TrFN
from bulanik.mcdm import DecisionMatrix, ahp_fuzzy_numbers, rank_fuzzy

# (upper trapezoid, lower trapezoid); heights 1 above, 0.9 below
_TERMS = {
    "VL": ((0, 0, 0, 0.1), (0, 0, 0, 0.05)),
    "L": ((0, 0.1, 0.1, 0.3), (0.05, 0.1, 0.1, 0.2)),
    "ML": ((0.1, 0.3, 0.3, 0.5), (0.2, 0.3, 0.3, 0.4)),
    "M": ((0.3, 0.5, 0.5, 0.7), (0.4, 0.5, 0.5, 0.6)),
    "MH": ((0.5, 0.7, 0.7, 0.9), (0.6, 0.7, 0.7, 0.8)),
    "H": ((0.7, 0.9, 0.9, 1.0), (0.8, 0.9, 0.9, 0.95)),
    "VH": ((0.9, 1.0, 1.0, 1.0), (0.95, 1.0, 1.0, 1.0)),
}
SCALE = {k: IT2TrFN((*u, 1, 1), (*l, 0.9, 0.9)) for k, (u, l) in _TERMS.items()}

criteria = ["cost", "quality", "service"]
alternatives = ["Vendor 1", "Vendor 2", "Vendor 3"]
types = ["cost", "benefit", "benefit"]

# Pairwise comparison of the criteria. IT2TrFN has no reciprocal method, so the lower
# triangle uses the trapezoid reciprocal (1/d, 1/c, 1/b, 1/a) written out here; the
# scale terms are shifted by 1 so that they read as ratios of at least 1.
def shifted(term):
    up, lo = term.upper, term.lower
    return IT2TrFN((*(1 + x for x in up[:4]), *up[4:]), (*(1 + x for x in lo[:4]), *lo[4:]))


def recip(x):
    u, l = x.upper, x.lower
    return IT2TrFN((1 / u[3], 1 / u[2], 1 / u[1], 1 / u[0], u[4], u[5]),
                   (1 / l[3], 1 / l[2], 1 / l[1], 1 / l[0], l[4], l[5]))


cq, cs, qs = shifted(SCALE["MH"]), shifted(SCALE["H"]), shifted(SCALE["M"])
one = IT2TrFN((1, 1, 1, 1, 1, 1), (1, 1, 1, 1, 0.9, 0.9))
comparisons = [
    [one, cq, cs],
    [recip(cq), one, qs],
    [recip(cs), recip(qs), one],
]
weights = ahp_fuzzy_numbers(comparisons, criteria)
print("weights:", {k: round(v, 4) for k, v in weights.weights.items()})
print("score:", weights.operators["score"])

# two experts rate the alternatives with words of the scale
words = [
    [["M", "H", "MH"], ["H", "M", "H"], ["ML", "VH", "M"]],
    [["MH", "H", "M"], ["H", "MH", "H"], ["M", "H", "ML"]],
]
experts = [[[SCALE[w] for w in row] for row in table] for table in words]
matrix = DecisionMatrix(alternatives, criteria, experts)
result = rank_fuzzy(matrix, weights, types, method="marcos")
print("IT2 ranking:", " > ".join(result.ranking))
for name in result.ranking:
    print(f"{name}: f(K) {result.scores[name]:.4f}")
print("score used:", result.operators["score"])

assert abs(sum(weights.weights.values()) - 1.0) < 1e-9
assert weights.ranking[0] == "cost"
expected_w = {"cost": 0.4687, "quality": 0.3056, "service": 0.2257}
assert all(abs(weights.weights[k] - v) < 5e-5 for k, v in expected_w.items())
assert result.ranking == ("Vendor 3", "Vendor 1", "Vendor 2")
expected_f = {"Vendor 3": 0.7471, "Vendor 1": 0.6437, "Vendor 2": 0.5416}
assert all(abs(result.scores[k] - v) < 5e-5 for k, v in expected_f.items())
assert result.operators["score"] == "dtrat"

Output

Run when this page is built:

weights: {'cost': 0.4687, 'quality': 0.3056, 'service': 0.2257}
score: dtrat
IT2 ranking: Vendor 3 > Vendor 1 > Vendor 2
Vendor 3: f(K) 0.7471
Vendor 1: f(K) 0.6437
Vendor 2: f(K) 0.5416
score used: dtrat

What to notice

• The linguistic scale is illustrative: its values are not a transcription of any published table.
• The reciprocal of an IT2TrFN is written out by hand (trapezoid points reversed and inverted, heights kept); the library has no reciprocal method for it.
• The default IT2 score, DTraT, is positive, so MARCOS accepts it without a score= argument.

Go further

• Guide: Interval type-2 numbers
• Guide: Weighting
• Guide: Ranking
• API: bulanik.mcdm

Run it yourself: python examples/19_it2_ahp_marcos.py.