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Ranking
SF-WPM - Spherical extension of WPM
Spherical outranking/ranking - Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1)
Kutlu Gündoğdu, F., Yörükoğlu, M.2021doi:10.1007/978-3-030-45461-6_10 ↗
Overview
sf-wpm extends WPM 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
- 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
- •Multiplicative aggregation Eq.34 uses λ-power Eq.6 which preserves hesitancy through (1−(1−v²)^λ)^(1/2) term.
- •Steps 1-4 identical to SVSAW - easy to teach as "SAW vs WPM" comparison.
- •Robust to scale differences across criteria when weights are normalized.
Look elsewhere when
- •Crisp data sufficient - use base WPM 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
- •Strongly punishes weak criteria (geometric mean property) - may give counter-intuitive rankings if one criterion has structurally low μ.
- •Same score-form dependency as SF-SAW.
- •Less interpretable when w̃^s_j (defuzzified weights) are fractional/small.
Edge cases and pitfalls
- •Bir cell'de μ=0: λ-power Eq.6'da μ^λ=0 - o cell'in katkısı SF-çarpımda membership'i sıfırlar (multiplicative property). Hesitancy π=0: Eq.6'da (1−v²)^λ−(1−v²−0)^λ = 0 → hesitancy korunur (sıfır kalır). λ=0 sınırı: x̃^0 should be identity (1,0,0) ama Eq.6'da λ→0 limit alınmalı (Liu 2010 / Ashraf 2019 definition). Tüm ağırlıklar eşit (w̄^s_j hepsi 1/n): SF geometrik ortalama olur. Cost criteria için (SAW ile aynı): linguistik girdiyi tersine çevir, μ↔v swap yapma.
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
Kutlu Gündoğdu, F.; Yörükoğlu, M. (2021). Simple Additive Weighting and Weighted Product Methods Using Spherical Fuzzy Sets. 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_10
System ID, as it appears in reports and the API
SF-WPM