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Ranking
SF-TOPSIS - Spherical extension of TOPSIS
Spherical outranking/ranking - Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1)
Kutlu Gündoğdu, F., Kahraman, C.2019doi:10.3233/JIFS-181401 ↗
Overview
SF-TOPSIS (Kutlu Gündoğdu & Kahraman 2019) extends TOPSIS to Spherical Fuzzy Sets (μ²+ν²+π² ≤ 1, hesitancy π independent). Pipeline: SWAM/SWGM aggregation of DM judgments → weighted SF matrix via SF multiplication ⊗ → defuzzify by score S(α) = (μ−π)² − (ν−π)² → SF-PIS / SF-NIS extraction (max/min score per criterion) → normalized Euclidean SF distance D(X, X*) and D(X, X⁻) (factor 1/(2n)) → revised closeness ratio ξ(X) = D(X,X*)/D_min(X,X*) − D(X,X⁻)/D_max(X,X⁻) → ASCENDING rank (smaller ξ = better).
- Output
- utility, 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
- •Wider preference domain than IFS/PFS (π independent allows μ+v+π>1 as long as squared sum ≤1).
- •Familiar TOPSIS pipeline (PIS/NIS + closeness ratio) reduces learning burden.
- •Two distance options provided (Euclidean Szmidt-Kacprzyk and Zhang-Xu) for robustness checks.
Look elsewhere when
- •Crisp data sufficient - use base TOPSIS directly (avoid unnecessary uncertainty layer)
- •Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous
Assumptions to verify
- Decision matrix entries are valid Spherical Fuzzy numbers/tuples
- Underlying crisp method's compensation assumption holds in uncertain space
- All decision-maker(s) and experts use the same linguistic/uncertainty scale
Limitations
- •Rank reversal possible when alternatives are added/removed (inherited from crisp TOPSIS).
- •Sensitive to choice of score function (book Eq.20 "2μ−π/2" form differs from JIFS-181401 Eq.25 "μ−π" form - different rankings possible).
- •No formal way to handle interactive criteria (independence assumed).
Edge cases and pitfalls
- •Tüm cell'lerin score'u eşitse (SF-PIS=SF-NIS dejenere durumu): closeness ratio 0/0 → tanımsız; rastgele tie-breaker veya accuracy fonksiyonu Eq.24 ile karar verilir. Bir cell'in μ²+v²+π²>1 olması yasak (SF tanım ihlali); validation'da reddedilmeli. Çok düşük μ + çok yüksek v cell'lerinde Eq.20 score negatif olur - ranking için normaldir (negatif=NIS'e yakın). Ağırlık SF-cell'ler için Σw=1 KISITI YOKTUR (SF weights triple olarak agrege edilir, defuzzification sırasında normalize edilir). Tek DM durumunda SWAM trivial (w=1) - algoritma aynı. Cost-criteria için: book konvansiyonu "DMs assign lower linguistic term" (Step 4 sonrası not) - cost cell'ler için conjugate (μ↔v) almak yerine linguistik girdiyi tersine çevir.
Value-space violation: ensure μ² + ν² + π² ≤ 1 (NOT μ + ν + π ≤ 1 - that is IFS).
Score function: SF uses S = (μ−π)² − (ν−π)², NOT the PFS-style S = μ² − ν².
Ranking direction: ASCENDING by ξ (Eq. 37 revised), NOT descending - opposite of classical CC.
Works with
Commonly takes its weights from
How to cite
Kutlu Gündoğdu, F.; Kahraman, C. (2019). Spherical fuzzy sets and spherical fuzzy TOPSIS method. Journal of Intelligent & Fuzzy Systems. https://doi.org/10.3233/JIFS-181401
System ID, as it appears in reports and the API
SF-TOPSIS