Weighting method · Buckley, 1985

Fuzzy AHP

Fuzzy Analytic Hierarchy Process (Buckley's geometric-mean method)

AHP judgments as fuzzy bands: 'about 3 times more important' instead of exactly 3.

How it works

1. Fuzzify each Saaty judgment a into the triangular number (a−1, a, a+1), clipped to the 1–9 scale; reciprocals mirror.
2. Take the fuzzy geometric mean of each row (Buckley's method).
3. Normalize into fuzzy weights (l, m, u) and defuzzify by centroid (l + m + u) / 3.
4. Consistency is checked on the crisp modal matrix the CR < 0.10 gate and the inconsistency doctor still apply.

Use it when

• Decision-makers hedge their pairwise judgments and you want that hedging in the weights.
• You need fuzzy weights for a fuzzy ranking pipeline but still want a defensible crisp weight vector.

Watch out for

• Defuzzified weights usually land close to crisp AHP weights the value is the (l, u) band, not a different answer.
• Fuzzy consistency is an open research topic; the crisp-matrix CR is a pragmatic gate, not a fuzzy one.

Parameters

None beyond the pairwise judgments the TFN spread is the standard ±1 Saaty step.

Cite the method

Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17(3), 233-247.

@article{buckley1985fuzzy,
  author  = {Buckley, James J.},
  title   = {Fuzzy hierarchical analysis},
  journal = {Fuzzy Sets and Systems},
  volume  = {17},
  number  = {3},
  pages   = {233247},
  year    = {1985},
  doi     = {10.1016/0165-0114(85)90090-9}
}