SF-PROMETHEE: Spherical extension of PROMETHEE

Overview

SF-PROMETHEE (Spherical Fuzzy Preference Ranking Organization METHod for Enrichment Evaluations), Brans-Vincke 1985'in "preference fonksiyonlu pairwise outranking" mantığını SF'ye taşır. Kritik teknik problem: SF'de SUBTRACTION TANIMSIZ (Eq.3 ⊕ var, Eq.4 ⊗ var ama ⊖ yok): klasik PROMETHEE'de d=c_j(x_i)−c_j(x_k) gerekirken, SF-PROMETHEE TOPSIS'ten ödünç alır: Eq.15 d_j(x_i,x_k)=RC[c̃_j(x_i)]−RC[c̃_j(x_k)] (RC=relative closeness, TOPSIS Eq.16 D⁻/(D⁻+D⁺)). Bu RC artık scalar, fark alınabilir. Akış: PIS/NIS belirle Eqs.20-21 (Eq.21'de √(1−...) feasibility-correction); RC hesapla; pairwise deviation d_j; Eq.19 linguistic preference function f_j (EP/WP/MP/SP/VSP/STP) ile preference SFS'ye çevir (p=0.5 strict threshold, q=0.1 indifference); Eq.24 global preference index r̃_ik = SWAM ile ağırlıklarla agg; Eq.25 outranking flows φ⁺(x_i)=Σ_k r̃_ik/(n−1), φ⁻(x_i)=Σ_k r̃_ki/(n−1); Score(φ⁺) ve Score(φ⁻) ile sırala (klasik PROMETHEE I-II kuralları); incomparability varsa Eq.25 net flow φ(x_i)=Score(φ⁺)−Score(φ⁻).

Strengths

• •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).

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.

Assumptions to verify

• •Thresholds (q, p, v) can be expressed in Spherical Fuzzy scale
• •Veto + concordance semantics adapted to fuzzy arithmetic

When not to use

• •Small dataset (m<3): outranking machinery underutilised

Edge cases

• •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.

Common pitfalls

• •(1) SF subtraction ⊖ kullanma: TANIMSIZ; Eq.15 RC-deviation kullan. (2) RC TOPSIS'ten gelir: PIS/NIS belirleme adımı YAPILMASI ZORUNLU (Eqs.20-21). (3) Eq.21'de feasibility correction √(1−(μ²+v²)) yalnız μ²_N+v²_N>1 durumunda: her relative NIS'e değil. (4) Eq.19 preference function 6 seviyeli: EP/WP/MP/SP/VSP/STP: boundary değerleri kontrol et (d=0.1 hangi sınıf? 0.1≤d<0.2 ⇒ WP). (5) Preference matrix R̃_S^j cell'leri SF triple, NOT linguistic terms: Eq.22 R̃_L^j linguistic, Eq.23 R̃_S^j SF çevrim. (6) Outranking flows Eq.25: φ⁺ "outranks others" (satır toplamı), φ⁻ "outranked by others" (sütun toplamı): karıştırma. (7) PROMETHEE I outranks Score(φ⁺_i)≥Score(φ⁺_k) AND Score(φ⁻_i)≤Score(φ⁻_k); PROMETHEE II yalnız net flow φ. (8) Crisp criteria için (Example 2): RC için Δ⁺/Δ⁻ direkt hesapla (Sharaf §4 Example 2), SF'ye çevirmeye gerek yok.

Commonly paired with

• •n_a + SF-PROMETHEE (common)

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