Z-PROMETHEE - Z-Number Preference Ranking Organization Method (II)

Overview

Z-PROMETHEE outputs alternative net flows Φ(a). Higher Φ = better alternative. Φ ranges roughly within [−(m−1), (m−1)] though practical values are smaller. Inspect Φ+ (strengths) and Φ− (weaknesses) for interpretation beyond final rank.

Data

Z-Number

Weights

Needs a weight source

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 q=p=0, Type-4 collapses to Type-1 Usual Criterion (strict 0/1 step)
• •When two alternatives are identical, Φ(a)=Φ(b)=0 → tie
• •When weights sum ≠ 1, engine auto-normalizes with warning
• •When K=1 expert, group aggregation step is identity
• •Sum of all Φ(a) is always exactly 0 (flow conservation) - useful sanity check

• Forgetting to apply criterion direction (benefit/cost) before computing d_{a,b}^j - leads to inverted preferences.

• Confusing PROMETHEE I (partial ranking via Φ+ and Φ− separately) with PROMETHEE II (full ranking via net Φ). Nuriyev uses PROMETHEE II.

• Type-4 thresholds (q, p) are problem-specific - must be elicited from decision-maker or set by domain knowledge; default q=0.05, p=0.15 are arbitrary placeholders.

• Treating Z-numbers as regular fuzzy directly - must apply Eq.(6)-(8) first.

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