FF-PROMETHEE - 2-Tuple Dilsel Fermatean Bulanık PROMETHEE (Akram-Bibi 2023)

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

1. Adım 1-2: Uzmanlar Tablo 2'deki 9 terimli dilsel ölçeği (EP→EG) ile yargı verir; 2TLFFN'ye dönüştürülür. Her KV için D^(r) matrisi.

   $$c_{pq}^{(r)}=(({\beta }_l^{(r)},L_{pq}^{(r)}),({\beta }_m^{(r)},M_{pq}^{(r)})),p=1..i,q=1..j,r=1..k$$
   LaTeX

   c^{(r)}_{pq} = ((β_l^{(r)}, L_{pq}^{(r)}), (β_m^{(r)}, M_{pq}^{(r)})), p=1..i, q=1..j, r=1..k

2. Adım 3: k tane D^(r) matrisini 2TLFFWA operatörü ile tek 2TLFFDM D = (c_{pq})_{ij}'ye birleştir.

   $$c_{pq}=2TLFFWA(c_{pq}^{(1)},\dots ,c_{pq}^{(k)})=\Delta (\tau ·∛(1-{\prod }_{r=1}^k(1-({\Delta }^{-1}({\beta }_l^{(r)},L^{(r)})/\tau {)}^3{)}^{{\varpi }_r}),\tau ·{\prod }_{r=1}^k({\Delta }^{-1}({\beta }_m^{(r)},M^{(r)})/\tau {)}^{{\varpi }_r})—Akram-Bibi2023Eq.(3)$$
   LaTeX

   c_{pq} = 2TLFFWA(c^{(1)}_{pq}, …, c^{(k)}_{pq}) = Δ( τ · ∛(1 − ∏_{r=1}^{k} (1 − (Δ^{-1}(β_l^{(r)},L^{(r)})/τ)^3)^{ϖ_r}) , τ · ∏_{r=1}^{k} (Δ^{-1}(β_m^{(r)},M^{(r)})/τ)^{ϖ_r} ) — Akram-Bibi 2023 Eq.(3)

3. Adım 4: KV-bazlı kriter ağırlığı 2TLFFN'leri ω^{(r)}_q, 2TLFFWA Eq.(4) ile birleştirilir → x = (x_1, …, x_n).

   $$x_q=2TLFFWA({\omega }_q^{(1)},\dots ,{\omega }_q^{(k)})perAkram-Bibi2023Eq.(4);defuzzifyviaS(x_q)beforeuseinpreferenceindexEq.(6)$$
   LaTeX

   x_q = 2TLFFWA(ω^{(1)}_q, …, ω^{(k)}_q) per Akram-Bibi 2023 Eq.(4); defuzzify via S(x_q) before use in preference index Eq.(6)

4. Adım 5: Skor fonksiyonu S(F), her birleştirilmiş 2TLFFN c_{pq}'yi 2-tuple dilsel skora dönüştürür (Eq.(1)).

   $$S_{pq}=S(c_{pq})=\Delta ((\tau /2)·(1+({\Delta }^{-1}({\beta }_l,L)/\tau {)}^3-({\Delta }^{-1}({\beta }_m,M)/\tau {)}^3))—Akram-Bibi2023Eq.(1)$$
   LaTeX

   S_{pq} = S(c_{pq}) = Δ( (τ/2) · (1 + (Δ^{-1}(β_l, L)/τ)^3 − (Δ^{-1}(β_m, M)/τ)^3 ) ) — Akram-Bibi 2023 Eq.(1)

5. Adım 6: Her kriter q ve sıralı (g,l) çifti için pairwise sapma D_q(T_g, T_l) = Δ^{-1}(S_q(T_g)) − Δ^{-1}(S_q(T_l)).

   $$D_q(T_g,T_l)={\Delta }^{-1}(S_q(T_g))-{\Delta }^{-1}(S_q(T_l))—Akram-Bibi2023Eq.(5)$$
   LaTeX

   D_q(T_g, T_l) = Δ^{-1}(S_q(T_g)) − Δ^{-1}(S_q(T_l)) — Akram-Bibi 2023 Eq.(5)

6. Adım 7: Genelleştirilmiş tercih fonksiyonu P_q(T_g, T_l) = F(D_q(T_g, T_l)). Fayda kriterleri için D_q > 0 iken P_q ≥ 0; maliyet kriterleri için işaret terslenir. Varsayılan Tip V Gauss σ=0.5.

   $$Gaussian(TypeV):P_q(T_g,T_l)=1-exp(-D_q²/(2\sigma ²))if{D}_q>0else0.Usual(TypeI):P_q=1if{D}_q>0else0.$$
   LaTeX

   Gaussian (Type V): P_q(T_g, T_l) = 1 − exp(−D_q² / (2σ²)) if D_q > 0 else 0. Usual (Type I): P_q = 1 if D_q > 0 else 0.

7. Adım 8: Çok kriterli tercih indeksi H(T_g, T_l) = ⊕_{q=1}^{j} x_q ⊗ P_q(T_g, T_l).

   $$H(T_g,T_l)={\oplus }_{q=1}^j{x}_q\otimes P_q(T_g,T_l)=(({\beta }_{l,H},L_H),({\beta }_{m,H},M_H))—Akram-Bibi2023Eq.(6)$$
   LaTeX

   H(T_g, T_l) = ⊕_{q=1}^{j} x_q ⊗ P_q(T_g, T_l) = ((β_{l,H}, L_H), (β_{m,H}, M_H)) — Akram-Bibi 2023 Eq.(6)

8. Adım 9-i: Pozitif Φ⁺(T_g) ve negatif Φ⁻(T_g) akışlar; PROMETHEE I kısmi sıralama Eq.(9-11).

   $$\Phi ⁺(T_g)={\oplus }_{l\ne g}H(T_g,T_l);\Phi ⁻(T_g)={\oplus }_{l\ne g}H(T_l,T_g)—Akram-Bibi2023Eqs.(7-8)$$
   LaTeX

   Φ⁺(T_g) = ⊕_{l≠g} H(T_g, T_l); Φ⁻(T_g) = ⊕_{l≠g} H(T_l, T_g) — Akram-Bibi 2023 Eqs.(7-8)

9. Adım 9-ii: PROMETHEE II net akış Φ(T_g) = Δ^{-1}(S(Φ⁺)) − Δ^{-1}(S(Φ⁻)); azalan Φ ile tam sıralama.

   $$\Phi (T_g)={\Delta }^{-1}(S(\Phi ⁺(T_g)))-{\Delta }^{-1}(S(\Phi ⁻(T_g)))—Akram-Bibi2023Eq.(12);rankbydescending\Phi —Eq.(13)$$
   LaTeX

   Φ(T_g) = Δ^{-1}(S(Φ⁺(T_g))) − Δ^{-1}(S(Φ⁻(T_g))) — Akram-Bibi 2023 Eq.(12); rank by descending Φ — Eq.(13)

Sezgi

Fermatean fuzzy outranking - 2-tuple linguistic Fermatean fuzzy sets (2TLFFS), μ³+ν³ ≤ 1. Output typically preference_flow (higher value = preferred).

Sonucu okuma: FF-PROMETHEE (specifically 2TLFF-PROMETHEE per Akram-Bibi 2023) extends crisp PROMETHEE to MAGDM under Fermatean fuzzy uncertainty encoded as 2-tuple linguistic terms. Use when (i) multiple experts express criterion-by-criterion judgments in linguistic terms, (ii) hesitancy is high enough that μ²+ν² ≤ 1 (intuitionistic) or μ²+ν² > 1 but μ³+ν³ ≤ 1 (Fermatean) is needed, and (iii) outranking-style ranking is preferred over compensatory aggregation. The Gaussian preference function (σ ≈ 0.5) is the paper's default. Run Methodology 1 (Gaussian) and Methodology 2 (Usual) in parallel for robustness - paper Section 5 reports identical top alternative under both.

Varsayımlar

• All decision-maker linguistic assessments decode to 2TLFFN satisfying μ³+ν³ ≤ 1
• Decision-maker weight vector ϖ is on the simplex
• Choice of preference function (Gaussian/Usual/…) is justified for each criterion's nature

Ne zaman kullanılmaz

• Single decision-maker with crisp judgments - use crisp PROMETHEE instead
• Hesitancy fits within intuitionistic constraint μ+ν ≤ 1 - IF-PROMETHEE may be more parsimonious
• Small dataset (m<3) - outranking machinery underutilised

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp PROMETHEE; cf. Brans-Mareschal 2005 Springer Ch.5)
• Assumes: All decision-maker linguistic assessments decode to 2TLFFN satisfying μ³+ν³ ≤ 1
• Assumes: Decision-maker weight vector ϖ is on the simplex
• Assumes: Choice of preference function (Gaussian/Usual/…) is justified for each criterion's nature

Sık yapılan hatalar

• Değer-uzayı ihlali: aggregation sonrası tüm 2TLFFN girişlerin μ³+ν³ ≤ 1 koşulunu sağladığından emin ol.
• Tercih fonksiyonu seçimi Φ⁺/Φ⁻'yi değiştirir; tepe alternatifin değişmesi gerekmez. Eşik tabanlı kriterler için Tip II/III, Gauss'a göre daha sadık olabilir.
• Maliyet kriteri işaret kuralı: P_q = F(−D_q); doğrudan F(D_q) uygulamak maliyet tercihini tersine çevirir.