Z-AHP - Z-Number Analytic Hierarchy Process

Overview

Z-AHP outputs criterion importance weights (sum=1). Higher weight = more important. Pair with any alternative-ranking method (Z-TOPSIS, Z-PROMETHEE per Nuriyev's hybrid) that consumes a weight vector. For multi-level hierarchies, compute weights level-by-level and combine via Saaty's hierarchical aggregation.

Data

Z-Number

Weights

Derived internally, no weight source needed

How it works

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

Fits when

Edge cases and pitfalls

• •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)

• Forgetting reciprocity Eq.(5) when entering only upper triangle - implementation must auto-fill lower triangle.

• Confusing Z-AHP COA defuzzification (Eq.30, simple average of l,m,u) with GMIR (weighted average using m·4). Nuriyev uses COA, NOT GMIR.

• Applying Buckley geometric mean to AGGREGATED Z-vectors before Z→TFN - must do Z→TFN FIRST then geometric mean.

• Treating reliability TFN as restriction TFN - they are TWO separate fuzzy components per Z-number.

• Collapsing each expert's judgement to a crisp number (or averaging the raw Z-numbers) before Eq.(25): the paper converts every expert's Z-number to a regular TFN first (Eq.6-8), averages the TFNs cell by cell, and only defuzzifies the final fuzzy weights (Eq.30).

Works with

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