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

Overview

Z-PROMETHEE is PROMETHEE with two channels per cell ('how good is alternative a on criterion j' + 'how sure is the expert'). Reliability collapses to α via centroid, then stretches/shrinks the restriction TFN by √α (Kang transformation). After all cells are Z→TFN, classical PROMETHEE runs: for each pair (a, b) and each criterion j, compute the difference; pass through a step function (Type 4) that returns 0/0.5/1 based on q,p thresholds; weight by criterion importance; sum across criteria → π(a,b). Average π over all other alternatives → Φ+ (leaving, how much a outranks others) and Φ− (entering, how much a is outranked). Net Φ = Φ+ − Φ−; rank descending.

Strengths

• •Outranking respects intransitivity and non-compensation more than utility methods
• •Type-4 preference function captures both indifference and strict-preference thresholds explicitly
• •Z-numbers capture both vagueness and reliability of expert judgments
• •Net flow Φ produces clear total ordering: easier to communicate than partial ranking

Limitations

• •Sensitivity to q, p threshold choice: must be elicited carefully
• •Computationally O(m² × n): moderate scaling
• •Single dominant criterion can mask others if its weight is large (mitigated by careful weight elicitation)
• •PROMETHEE II loses incomparability information present in PROMETHEE I

Edge cases

• •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

Common pitfalls

• •Don't forget criterion direction (benefit/cost): flip differences for cost criteria
• •Don't confuse PROMETHEE I (partial) with PROMETHEE II (full ranking via net Φ)
• •Don't pick arbitrary q, p thresholds: elicit from decision-maker
• •Don't treat Z-numbers as regular fuzzy: Z→TFN via Eq.(6)-(8) first

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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