PiF-ARAS - ARAS yönteminin Picture Bulanık uzantısı

Formül adımları

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

Adım 1 — Picture bulanık karar matrisini X̃ = [x̃_ij]_{m×n} oluştur; her hücre x̃_ij = ⟨μ_ij, η_ij, ν_ij⟩ PiFN koşulunu (μ_ij+η_ij+ν_ij ≤ 1) sağlar; reddetme π_ij = 1−μ_ij−η_ij−ν_ij. Kriter yönü PiFN kodlamasından önce dilbilimsel dönüşümde işlenir (Chowdhury 2025 §4.1 Tablo 6: maliyet kriterleri için düşük ham değer yüksek μ alır); böylece algoritma içinde matris tüm-benefit muamelesi görür.

$x ̃_{i} j = ⟨ μ_{i} j, η_{i} j, ν_{i} j ⟩; μ_{i} j, η_{i} j, ν_{i} j \in [0, 1]; μ_{i} j + η_{i} j + ν_{i} j \leq 1; π_{i} j = 1 - μ_{i} j - η_{i} j - ν_{i} j (C u o n g 2013 P i F S D e f 1; C h o w d h u r y 2025 § 3.2 E q . (1) - (2))$

LaTeX
x̃_ij = ⟨μ_ij, η_ij, ν_ij⟩; μ_ij, η_ij, ν_ij ∈ [0,1]; μ_ij+η_ij+ν_ij ≤ 1; π_ij = 1−μ_ij−η_ij−ν_ij (Cuong 2013 PiFS Def 1; Chowdhury 2025 §3.2 Eq.(1)-(2))
Adım 2 — Picture bulanık kriter ağırlıkları v̄_j = ⟨μ_j, η_j, ν_j⟩ dışsal verilir (DM toplanır, Chowdhury 2025 §4.1 Tablo 4-5) veya Picture bulanık ağırlıklandırma yönteminden türetilir. Karşılık gelen crisp ağırlık Eq.(16) ile: w_j = [μ_j + η_j/2 + (π_j/2)(1+μ_j-ν_j)] / Σ_k [...]. PiF-ARAS asıl olarak v̄_j (PiFN) kullanır.

$v ̄_{j} = ⟨ μ_{j}, η_{j}, ν_{j} ⟩ w i t h μ_{j} + η_{j} + ν_{j} \leq 1 . C r i s p d e f u z z i f i c a t i o n (E q . (16)) : w_{j} = [μ_{j} + η_{j} / 2 + (π_{j} / 2) (1 + μ_{j} - ν_{j})] / Σ_{k} [μ_{k} + η_{k} / 2 + (π_{k} / 2) (1 + μ_{k} - ν_{k})]; i n f o r m a t i o n a l . (C h o w d h u r y 2025 § 3.3 E q . (15) - (16))$

LaTeX
v̄_j = ⟨μ_j, η_j, ν_j⟩ with μ_j+η_j+ν_j ≤ 1. Crisp defuzzification (Eq.(16)): w_j = [μ_j + η_j/2 + (π_j/2)(1+μ_j-ν_j)] / Σ_k [μ_k + η_k/2 + (π_k/2)(1+μ_k-ν_k)]; informational. (Chowdhury 2025 §3.3 Eq.(15)-(16))
Adım 3 — Eleman-bazlı PiFN ağırlıklandırma: q̃_ij = v̄_j ⊗ x̃_ij (Cuong 2013 Eq.(4) PiFN çarpım). NOT: Chowdhury 2025 §3.4 metni 'crisp ağırlıkla çarpım' der; ancak Tablo 11 ancak agrege PiFN ağırlığı v̄_j ile Eq.(4) ⊗ kullanılarak yeniden üretilebilir — bkz. P.implementation_review.literature_disambiguation.weighting_operator. Ağırlıklı matris F4-F7'yi besler.

$q ̃_{i} j = v ̄_{j} \otimes x ̃_{i} j = ⟨ μ_{v_{j}} \cdot μ_{x_{i} j}, η_{v_{j}} + η_{x_{i} j} - η_{v_{j}} \cdot η_{x_{i} j}, ν_{v_{j}} + ν_{x_{i} j} - ν_{v_{j}} \cdot ν_{x_{i} j} ⟩ (C u o n g 2013 P i F N p r o d u c t; C h o w d h u r y 2025 § 3.2 E q . (4))$

LaTeX
q̃_ij = v̄_j ⊗ x̃_ij = ⟨μ_{v_j}·μ_{x_ij}, η_{v_j}+η_{x_ij}-η_{v_j}·η_{x_ij}, ν_{v_j}+ν_{x_ij}-ν_{v_j}·ν_{x_ij}⟩ (Cuong 2013 PiFN product; Chowdhury 2025 §3.2 Eq.(4))
Adım 4 — İdeal alternatif Ã_0'ı ağırlıklı matristen kriter başına sentetik en iyi PiFN çıkararak belirle: g̃_0j = ⟨max_i μ_q_ij, min_i η_q_ij, min_i ν_q_ij⟩. İdeal sentetiktir — bileşenleri aynı gerçek alternatife ait olmak zorunda değil.

LaTeX
Ã_0 = {g̃_01, g̃_02, ..., g̃_0n}; g̃_0j = ⟨max_i μ_q_ij, min_i η_q_ij, min_i ν_q_ij⟩ (Chowdhury 2025 §3.5 Step 1 Eq.(29))
Adım 5 — Her alternatif için (ve Ã_0 için) Picture bulanık optimallik fonksiyonu S̃_i'yi PiFN toplam operatörü ile hesapla: μ için 1-Π(1-μ) (komplement-çarpım), η için Πη, ν için Π(η+ν)-Πη (sonuç geçerli PiFN olarak kalır).

$S ̃_{i} = ⟨ μ_{S_{i}}, η_{S_{i}}, ν_{S_{i}} ⟩ = Σ \oplus_{j = 1}^{n} q ̃_{i} j = ⟨ 1 - Π_{j = 1}^{n} (1 - μ_{q_{i} j}), Π_{j = 1}^{n} η_{q_{i} j}, Π_{j = 1}^{n} (η_{q_{i} j} + ν_{q_{i} j}) - Π_{j = 1}^{n} η_{q_{i} j} ⟩ (C h o w d h u r y 2025 § 3.5 S t e p 2 E q . (30))$

LaTeX
S̃_i = ⟨μ_{S_i}, η_{S_i}, ν_{S_i}⟩ = Σ⊕_{j=1}^n q̃_ij = ⟨1 − Π_{j=1}^n (1−μ_{q_ij}), Π_{j=1}^n η_{q_ij}, Π_{j=1}^n (η_{q_ij}+ν_{q_ij}) − Π_{j=1}^n η_{q_ij}⟩ (Chowdhury 2025 §3.5 Step 2 Eq.(30))
Adım 6 — Picture bulanık optimallik fonksiyonu S̃_i'yi Picture defuzzifikasyon operatörü (Eq.(31)) ile crisp D_i'ye dönüştür: D_i = μ_{S_i} + η_{S_i}/2 + (π_{S_i}/2)(1+μ_{S_i}−ν_{S_i}). Aynı operatör Ã_0'a uygulanarak D_0 elde edilir.

$D_{i} = μ_{S_{i}} + η_{S_{i}} / 2 + (π_{S_{i}} / 2) (1 + μ_{S_{i}} - ν_{S_{i}}); π_{S_{i}} = 1 - μ_{S_{i}} - η_{S_{i}} - ν_{S_{i}} (C h o w d h u r y 2025 § 3.5 S t e p 3 E q . (31); c f . E q . (13) f o r g e n e r a l P i F N d e f u z z i f i c a t i o n w i t h η / 2 t e r m)$

LaTeX
D_i = μ_{S_i} + η_{S_i}/2 + (π_{S_i}/2)(1+μ_{S_i}−ν_{S_i}); π_{S_i} = 1−μ_{S_i}−η_{S_i}−ν_{S_i} (Chowdhury 2025 §3.5 Step 3 Eq.(31); cf. Eq.(13) for general PiFN defuzzification with η/2 term)
Adım 7 — Her alternatif için fayda derecesini B_i = D_i / D_0 hesapla (Eq.(32)). B_i'ye göre azalan sıralama yap — en büyük fayda derecesine sahip alternatif en arzulanır. Yapı gereği B_i ∈ [0, 1] olup B_0 = 1. İdeal alternatif genellikle sıralanan alternatifler arasında değildir; yalnızca payda görevi görür.

$B_{i} = D_{i} / D_{0}; r a n k i n g = a r g s o r t_{d} e s c (B); b e s t = r a n k i n g [0] . (C h o w d h u r y 2025 § 3.5 S t e p s 4 - 5 E q . (32))$

LaTeX
B_i = D_i / D_0; ranking = argsort_desc(B); best = ranking[0]. (Chowdhury 2025 §3.5 Steps 4-5 Eq.(32))

Sezgi

Picture utility-degree ranking - Picture Fuzzy Number (PiFN: μ, η, ν; μ+η+ν ≤ 1). Output typically utility (higher value = preferred).

Sonucu okuma: pif-aras extends Zavadskas-Turskis 2010 ARAS to Picture fuzzy uncertainty via Cuong 2013 PiFS. Each criterion contributes via element-wise PiFN multiplication with the aggregated PiFN weight (Cuong 2013 ⊗ operator). The ideal alternative Ã_0 is synthetically composed from per-criterion ⟨max μ, min η, min ν⟩ of the weighted matrix. Alternatives are scored by the ratio of their defuzzified optimality D_i to the ideal D_0 (utility degree B_i = D_i/D_0). Higher B_i is better.

Varsayımlar

Decision matrix entries are valid Picture Fuzzy Numbers (μ+η+ν ≤ 1)
Aggregated PiFN criterion weights provided or derivable
Criterion-direction inversion handled upstream (linguistic conversion) OR via PiFN complement on cost columns
All decision-maker(s) and experts use the same linguistic/PiFN scale

Ne zaman kullanılmaz

Neutral-stance modelling unnecessary - use base ARAS (crisp) or IF-ARAS instead
Data are already crisp - picture fuzzy layer adds noise without benefit
Linguistic scale does not distinguish neutral from refusal - single-component fuzzy or IFS suffices

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: inherited from crisp ARAS base; cf. Zavadskas-Turskis 2010)
Assumes: Decision matrix entries are valid Picture Fuzzy Numbers (μ+η+ν ≤ 1)
Assumes: Aggregated PiFN criterion weights provided or derivable
Assumes: Criterion-direction inversion handled upstream (linguistic conversion) OR via PiFN complement on cost columns
Assumes: All decision-maker(s) and experts use the same linguistic/PiFN scale

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin PiFN μ+η+ν ≤ 1 koşulunu sağladığından emin olun. PFN (Pythagorean) ve IFS girdileri geçerli PiFN DEĞİLDİR.
Ağırlık-operatörü karışıklığı: Chowdhury 2025 §3.4/§3.5 metni 'crisp ağırlık' der ama raporlanan tablolar (Tablo 11) ancak agrege PiFN ağırlığı v̄_j ile PiFN ⊗ Eq.(4) yoluyla yeniden üretilebilir. Manifest tabloları kanonik kabul eder.
Yön işleme: Chowdhury 2025 kriter yönünü dilbilimsel dönüşüm aşamasında kodlar (Tablo 6: düşük TAP/SR → yüksek μ). criteria_types='min' ile ham PiFN matris veren kullanıcılar bu sütunlara F3'ten önce PiFN komplemanı Ã^c = ⟨ν, η, μ⟩ uygulamalı.

Hesap adımları ve dayanakları

Construct the Picture fuzzy decision matrix X̃ = [x̃_ij]_{m×n} where each entry x̃_ij = ⟨μ_ij, η_ij, ν_ij⟩ is a PiFN satisfying μ_ij+η_ij+ν_ij ≤ 1, with refusal π_ij = 1−μ_ij−η_ij−ν_ij. Criterion-direction is handled at the linguistic-conversion step prior to PiFN encoding (per Chowdhury 2025 §4.1 Table 6: cost criteria are mapped so that lower raw values receive higher μ); the resulting PiFN matrix is therefore treated as all-benefit at the algorithm-internal level.

Dayanak: Chowdhury 2025 §3.2 Eq.(1)-(2); Cuong 2013
Picture fuzzy criterion weights v̄_j = ⟨μ_j, η_j, ν_j⟩ are either supplied externally (DM-elicited and aggregated across stakeholders per Chowdhury 2025 §4.1 Tables 4-5) or derived from a Picture fuzzy weighting method. The corresponding crisp weight is given by Eq.(16): w_j = [μ_j + η_j/2 + (π_j/2)(1+μ_j-ν_j)] / Σ_k [μ_k + η_k/2 + (π_k/2)(1+μ_k-ν_k)]. PiF-ARAS proper uses v̄_j (PiFN) - the crisp w_j is informational only.

Dayanak: Chowdhury 2025 §3.3 Eq.(15)-(16)
Element-wise PiFN weighting: compute the weighted Picture fuzzy decision matrix Q̃ = [q̃_ij] via q̃_ij = v̄_j ⊗ x̃_ij using the PiFN product operator (Cuong 2013 Eq.(4)). NOTE: Chowdhury 2025 §3.4 text refers to 'multiplication by the crisp criteria weights', but reproduction of Table 11 requires the PiFN ⊗ Eq.(4) operator with the aggregated PiFN weight v̄_j - see P.implementation_review.literature_disambiguation.weighting_operator. The weighted matrix feeds Steps F4-F7.

Dayanak: Chowdhury 2025 §3.2 Eq.(4); §3.4 pre-text (CODAS); §3.5 pre-text (ARAS)
Determine the ideal alternative Ã_0 by extracting per-criterion synthetic best PiFN from the weighted matrix: g̃_0j = ⟨max_i μ_q_ij, min_i η_q_ij, min_i ν_q_ij⟩. The ideal is synthetic - its components need not all belong to the same actual alternative.

Dayanak: Chowdhury 2025 §3.5 Step 1 Eq.(29)
Compute the Picture fuzzy optimality function S̃_i for each alternative (and for Ã_0) by aggregating per-criterion weighted PiFN cells via the PiFN sum operator. μ aggregation is via the complement product (1-Π(1-μ)); η aggregation is the simple product Π η; ν aggregation is Π(η+ν) − Π η, ensuring the result remains a valid PiFN.

Dayanak: Chowdhury 2025 §3.5 Step 2 Eq.(30)
Defuzzify the Picture fuzzy optimality function S̃_i to crisp D_i using the Picture defuzzification operator (Eq.(31)): D_i = μ_{S_i} + η_{S_i}/2 + (π_{S_i}/2)(1+μ_{S_i}−ν_{S_i}), where π_{S_i} = 1 − μ_{S_i} − η_{S_i} − ν_{S_i}. The same operator is applied to Ã_0 to obtain D_0.

Dayanak: Chowdhury 2025 §3.5 Step 3 Eq.(31); Eq.(13)
Compute the utility degree B_i = D_i / D_0 for each alternative (Eq.(32)). Rank in descending order of B_i - the alternative with the largest utility degree is the most desirable. By construction B_i ∈ [0, 1] with B_0 = 1 (the ideal). The ideal alternative itself is generally NOT among the ranked alternatives; it serves only as the denominator.

Dayanak: Chowdhury 2025 §3.5 Steps 4-5 Eq.(32)