Z-AHP: Z-Number Analytic Hierarchy Process

Overview

Z-AHP is AHP with two channels per judgment: 'how much more important' (restriction) AND 'how sure I am' (reliability). The reliability TFN is collapsed to a single number α via centroid; then it stretches/shrinks the restriction TFN by √α. After all cells are Z→TFN, Buckley's geometric mean (instead of Saaty's eigenvector) computes row priorities; ratio of each row's geometric mean to the sum gives fuzzy weights; COA defuzzifies; normalize. Hierarchical structure is handled by Saaty's level-by-level composition.

Strengths

• •Hierarchical decomposition enables complex problems (Nuriyev demonstrates goal → 4 main criteria → 9 sub-criteria)
• •Captures both vagueness (TFN) and reliability (second TFN): richer than Buckley fuzzy AHP
• •Buckley's geometric mean is well-established and avoids iterative eigenvector computation
• •Group decision-making built-in via per-cell arithmetic averaging (Eq.25)

Limitations

• •Full n×n pairwise scales poorly: n(n-1)/2 unique comparisons + K experts × levels
• •COA defuzzification ignores TFN shape information (vs GMIR which weights m by 4)
• •Reciprocity assumption may not hold when experts disagree systematically
• •Saaty's classical consistency check (CR) requires post-defuzzification crisp matrix; not natively Z-number aware

Edge cases

• •When all cells are (1,1,1)+(VH,VH), weights are uniform 1/n
• •When K=1, group aggregation is identity: pipeline unchanged
• •Two-level hierarchy: compute weights per level, then composite = main × sub (Nuriyev Table 12)
• •Reciprocity slightly violated: auto-correct or warn
• •n very large (>12): pairwise burden: switch to Z-BWM (2n-3 comparisons)

Common pitfalls

• •Don't apply Buckley geometric mean to raw Z-numbers: do Z→TFN first
• •Don't confuse COA (Eq.30, simple average) with GMIR ((l+4m+u)/6 used in Z-BWM)
• •Don't forget reciprocity Eq.(5) on the lower triangle
• •Restriction TFN and reliability TFN are SEPARATE: never average them

Worked example

1. 1.Step 1
2. 2.Step 2
3. 3.Step 3
4. 4.Step 4
5. 5.Step 5
6. 6.Step 6
7. 7.Step 7
8. 8.Step 8
9. 9.Step 9

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