PF-VIKOR - VIKOR yönteminin Pythagorean uzantısı

Formül adımları

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

Adım 1 — Pisagoryan bulanık karar matrisini X = [α_ij]_{m×n} oluştur; her hücre α_ij = (μ_ij, ν_ij) PFS koşulunu (μ_ij²+ν_ij²≤1) sağlar; tereddüt π_ij² = 1−μ_ij²−ν_ij².

$α_{i} j = (μ_{i} j, ν_{i} j) \in Φ (U); μ_{i} j, ν_{i} j \in [0, 1]; μ_{i} j ² + ν_{i} j ² \leq 1; π_{i} j ² = 1 - μ_{i} j ² - ν_{i} j ² (A b b a s § 4 S t e p 1; Y a g e r 2013 P F S d e f)$

LaTeX
α_ij = (μ_ij, ν_ij) ∈ Φ(U); μ_ij, ν_ij ∈ [0,1]; μ_ij²+ν_ij²≤1; π_ij² = 1−μ_ij²−ν_ij² (Abbas §4 Step 1; Yager 2013 PFS def)
Adım 2 — Kriter ağırlıkları w_j ya dışsal (AHP, BWM, ENTROPY) ya da PFN entropisinden hesaplanır. Abbas 2024 E_SA entropi ile W_j = E_SA(C_j) / Σ_k E_SA(C_k) önerir. PF-VIKOR ağırlık-kaynağından bağımsızdır; tek koşul Σ w_j = 1.

$w_{j} \geq 0, Σ_{j} w_{j} = 1 . O p t i o n a l : W_{j} = E_{S} A (C_{j}) / Σ_{k} E_{S} A (C_{k}) w h e r e E_{S} A (C_{j}) = (1 / (2 m)) Σ_{i} [| (1 + μ_{i} j ² + π_{i} j ²) / 2 - ν_{i} j ² |]^{(} 1 / 2) (A b b a s 2024 E q . (5)) .$

LaTeX
w_j ≥ 0, Σ_j w_j = 1. Optional: W_j = E_SA(C_j)/Σ_k E_SA(C_k) where E_SA(C_j) = (1/(2m))Σ_i [|(1+μ_ij²+π_ij²)/2 − ν_ij²|]^(1/2) (Abbas 2024 Eq.(5)).
Adım 3 — Kriter başına sentetik en iyi PFN f*_j ve en kötü PFN f^-_j belirle. Fayda kriteri j için: f*_j = (max_i μ_ij, min_i ν_ij); f^-_j = (min_i μ_ij, max_i ν_ij). Maliyet için roller ters çevrilir. Referans noktalar sentetiktir.

$I f j b e n e f i t : f *_{j} = (m a x_{i} μ_{i} j, m i n_{i} ν_{i} j), f_{j}^{-} = (m i n_{i} μ_{i} j, m a x_{i} ν_{i} j) . I f j c o s t : f *_{j} = (m i n_{i} μ_{i} j, m a x_{i} ν_{i} j), f_{j}^{-} = (m a x_{i} μ_{i} j, m i n_{i} ν_{i} j) . (A b b a s 2024 § 4 S t e p 4)$

LaTeX
If j benefit: f*_j = (max_i μ_ij, min_i ν_ij), f^-_j = (min_i μ_ij, max_i ν_ij). If j cost: f*_j = (min_i μ_ij, max_i ν_ij), f^-_j = (max_i μ_ij, min_i ν_ij). (Abbas 2024 §4 Step 4)
Adım 4 — Hücre başına normalleştirilmiş PF boşluk PF uzaklığı ile hesaplanır. Abbas 2024 kare bileşenler üzerinde PF Hamming uzaklığı kullanır: d_H(α,β) = (|μ_α²−μ_β²| + |ν_α²−ν_β²| + |π_α²−π_β²|)/3. Δ_ij = w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j). PF Öklid uzaklığı belgelenmiş alternatiftir.

$d_{H} (α, β) = (1 / 3) \cdot [| μ_{α} ² - μ_{β} ² | + | ν_{α} ² - ν_{β} ² | + | π_{α} ² - π_{β} ² |]; Δ_{i} j = w_{j} \cdot d_{H} (α_{i} j, f *_{j}) / d_{H} (f_{j}^{-}, f *_{j}) . (A b b a s 2024 § 4 S t e p 5)$

LaTeX
d_H(α,β) = (1/3)·[|μ_α²−μ_β²| + |ν_α²−ν_β²| + |π_α²−π_β²|]; Δ_ij = w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j). (Abbas 2024 §4 Step 5)
Adım 5 — Normalleştirilmiş Δ üzerinde grup faydası S_i (L_1) ve bireysel pişmanlık R_i (L_∞) hesapla.

$S_{i} = Σ_{j} Δ_{i} j = Σ_{j} [w_{j} \cdot d_{H} (α_{i} j, f *_{j}) / d_{H} (f_{j}^{-}, f *_{j})]; R_{i} = m a x_{j} Δ_{i} j . (A b b a s 2024 § 4 S t e p 5; O p r i c o v i c 1998 V I K O R L_{1} / L_{\infty} a g g r e g a t i o n)$

LaTeX
S_i = Σ_j Δ_ij = Σ_j [w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j)]; R_i = max_j Δ_ij. (Abbas 2024 §4 Step 5; Opricovic 1998 VIKOR L_1/L_∞ aggregation)
Adım 6 — VIKOR uzlaşı endeksi Q_i'yi normalize S ve R'nin konveks kombinasyonu olarak strateji parametresi β (varsayılan β=0.5) ile hesapla. S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.

$Q_{i} = β \cdot (S_{i} - S *) / (S^{-} - S *) + (1 - β) \cdot (R_{i} - R *) / (R^{-} - R *) . (A b b a s 2024 § 4 S t e p 7; O p r i c o v i c 1998)$

LaTeX
Q_i = β·(S_i − S*)/(S^- − S*) + (1−β)·(R_i − R*)/(R^- − R*). (Abbas 2024 §4 Step 7; Opricovic 1998)
Adım 7 — Q, S, R'yi artan sırada sırala (üç sıralama). En küçük Q'lu alternatif aday uzlaşı çözümüdür.

$r a n k i n g_{Q} = a r g s o r t_{a} s c (Q); r a n k i n g_{S} = a r g s o r t_{a} s c (S); r a n k i n g_{R} = a r g s o r t_{a} s c (R) . B e s t a l t e r n a t i v e = r a n k i n g_{Q} [0] . (A b b a s 2024 § 4 S t e p 8)$

LaTeX
ranking_Q = argsort_asc(Q); ranking_S = argsort_asc(S); ranking_R = argsort_asc(R). Best alternative = ranking_Q[0]. (Abbas 2024 §4 Step 8)
Adım 8 — İki koşullu VIKOR uzlaşı testi. C1 (kabul edilebilir avantaj): Q(A^(2)) − Q(A^(1)) ≥ DQ = 1/(m−1). C2 (kararlılık): A^(1) S ve/veya R sıralamasında da birinci. Her ikisi sağlanırsa A^(1) önerilir. C1 sağlanmazsa Q farkı < DQ kalan prefix, yalnız C2 sağlanmazsa {A^(1), A^(2)}.

$C 1 : Q (A^{(} 2)) - Q (A^{(} 1)) \geq 1 / (m - 1) . C 2 : A^{(} 1) = r a n k i n g_{S} [0] O R A^{(} 1) = r a n k i n g_{R} [0] . c o m p r o m i s e_{s} e t = i f C 1 \land C 2 t h e n A^{(} 1) e l s e i f \neg C 1 t h e n p r e f i x (Q - Q^{(} 1) < D Q) e l s e i f \neg C 2 t h e n A^{(} 1), A^{(} 2) . (O p r i c o v i c 1998 V I K O R c o m p r o m i s e t e s t)$

LaTeX
C1: Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1). C2: A^(1) = ranking_S[0] OR A^(1) = ranking_R[0]. compromise_set = if C1 ∧ C2 then {A^(1)} else if ¬C1 then prefix(Q − Q^(1) < DQ) else if ¬C2 then {A^(1), A^(2)}. (Opricovic 1998 VIKOR compromise test)

Sezgi

Pythagorean outranking/ranking - Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1). Output typically utility (higher value = preferred).

Sonucu okuma: pf-vikor extends VIKOR to handle Pythagorean uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Varsayımlar

Decision matrix entries are valid Pythagorean 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

Ne zaman kullanılmaz

Classical data sufficient - use base VIKOR directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
Assumes: Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
Assumes: Underlying crisp method's compensation assumption holds in uncertain space
Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin PFN: μ ∈ [0,1], ν ∈ [0,1], μ²+ν² ≤ 1 koşulunu sağladığından emin olun.
Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = μ² − ν² kanonik seçimdir.

Hesap adımları ve dayanakları

Construct the Pythagorean fuzzy decision matrix X = [α_ij]_{m×n} where each entry α_ij = (μ_ij, ν_ij) is a PFN satisfying μ_ij²+ν_ij²≤1, with associated hesitancy π_ij² = 1−μ_ij²−ν_ij².

Dayanak: Abbas 2024 §4 Step 1; Yager 2013 PFS definition
Weights w_j per criterion are either externally provided (e.g. from AHP, BWM, ENTROPY) or computed from PFN entropy. Abbas 2024 proposes E_SA-entropy then W_j = E_SA(C_j) / Σ_k E_SA(C_k). The PF-VIKOR algorithm proper is weight-source-agnostic; w_j with Σ w_j = 1 is the only requirement.

Dayanak: Abbas 2024 §4 Steps 2-3; Eq.(5)
Determine the synthetic best PFN f*_j and worst PFN f^-_j per criterion. For benefit criterion j: f*_j = (max_i μ_ij, min_i ν_ij); f^-_j = (min_i μ_ij, max_i ν_ij). For cost criterion j: swap (f*_j and f^-_j roles inverted). The reference points are synthetic constructions; they need not equal any actual alternative.

Dayanak: Abbas 2024 §4 Step 4
Compute per-cell normalised PF gap using PF distance. Abbas 2024 uses PF Hamming distance over squared components: d_H(α,β) = (|μ_α²−μ_β²| + |ν_α²−ν_β²| + |π_α²−π_β²|)/3. The normalised gap is Δ_ij = w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j). PF Euclidean distance d_E is a documented alternative.

Dayanak: Abbas 2024 §4 Step 5
Compute group utility S_i (L_1-aggregated) and individual regret R_i (L_∞-aggregated) over the normalised gap matrix Δ.

Dayanak: Abbas 2024 §4 Step 5; Opricovic 1998
Compute the VIKOR compromise index Q_i as a convex combination of normalised S and R, weighted by strategy parameter β (default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.

Dayanak: Abbas 2024 §4 Step 7
Produce three rankings by sorting Q, S, R ascending. The alternative with the smallest Q is the candidate compromise solution.

Dayanak: Abbas 2024 §4 Step 8; Opricovic 1998 VIKOR
Apply the two-condition VIKOR compromise test. C1 (acceptable advantage): Q(A^(2)) − Q(A^(1)) ≥ DQ where DQ = 1/(m−1). C2 (acceptable stability in decision-making): A^(1) is also best ranked by S and/or R. If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Dayanak: Opricovic 1998 VIKOR compromise test (inherited)