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Outranking
SF-PROMETHEE - Spherical extension of PROMETHEE
Spherical outranking/ranking - Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1)
Sharaf, I. M.2021doi:10.1007/978-3-030-45461-6_16 ↗
Overview
sf-promethee extends PROMETHEE to handle Spherical uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.
- Output
- preference flow, higher is better
- Data
- Spherical Fuzzy, uncertainty tuples complete
- Weights
- Needs a weight source
- Size
- 2+ alternatives, 3-10 criteria works best
- Used for
- Spherical Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms
Fits when / Look elsewhere when
Fits when
- •6-level linguistic preference function Eq.19 (EP/WP/MP/SP/VSP/STP) maps deviation→SFS naturally.
- •Outranking flows φ⁺/φ⁻ give both ranking and incomparability information.
- •Compatible with crisp and SF weights (book Example 2 demonstrates crisp criterion weights → SFS conversion).
Look elsewhere when
- •Small dataset (m<3) - outranking machinery underutilised
Assumptions to verify
- Thresholds (q, p, v) can be expressed in Spherical Fuzzy scale
- Veto + concordance semantics adapted to fuzzy arithmetic
Limitations
- •Uses TOPSIS RC as deviation surrogate - inherits some TOPSIS dependence on PIS/NIS choice.
- •p=0.5, q=0.1 thresholds are book defaults - sensitivity should be checked for borderline cases.
- •Score-based comparison of φ⁺/φ⁻ can produce ties or incomparability requiring net-flow Eq.25 fallback.
Edge cases and pitfalls
- •p=0.5 strict threshold, q=0.1 indifference - sensitivity'siz bu değerler kullanılır. d≤0 (negatif fark): preference 0'a yaklaşır (Eq.19 başlangıç koşulu d<0.1 EP=(0.5,0.5,0.5) yarı-tarafsız). d≥p=0.5: strict preference (1,0,0). Conjugate (negatif d için): complement SFS kullanılır (Sharaf §3.2 not). Eq.21 relative NIS'te μ²_N+v²_N>1 durumunda √(1−(μ²+v²)) hesitancy düzeltmesi devreye girer - μ ve v'nin küre yüzeyinde olduğundan emin olmak için. Incomparability: Score(φ⁺(x_i))>Score(φ⁺(x_k)) AMA Score(φ⁻(x_i))>Score(φ⁻(x_k)) - partial ranking PROMETHEE I yetersiz; Eq.25 net flow φ=Score(φ⁺)−Score(φ⁻) ile complete ranking yapılır (PROMETHEE II). SF-cell SCORE comparison: Eq.8 score fonksiyonu kullanılır.
Value-space violation: ensure all entries satisfy SFS: μ,ν,π ∈ [0,1]; μ²+ν²+π² ≤ 1 before computation.
Defuzzification method affects ranking: score function S = μ² − ν² is the canonical choice but alternatives exist.
Works with
Commonly takes its weights from
How to cite
Sharaf, I. M. (2021). Evaluating Geothermal Energy Systems Using Spherical Fuzzy PROMETHEE. In: Kahraman C., Kutlu Gündoğdu F. (eds.) Decision Making with Spherical Fuzzy Sets - Theory and Applications. Studies in Fuzziness and Soft Computing vol. 392. Springer. https://doi.org/10.1007/978-3-030-45461-6_16
System ID, as it appears in reports and the API
SF-PROMETHEE