PiF-VIKOR - VIKOR yönteminin Picture uzantısı

Formül adımları

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

Adım 1 — Başlangıç resim bulanık karar matrisi R_1 = [r_ij]_{m×n} oluştur; r_ij = ⟨μ_ij, η_ij, ν_ij⟩ ve μ + η + ν ≤ 1. Her kriterin yönü (max/min) kaydedilir.

$r_{i} j = ⟨ μ_{i} j, η_{i} j, ν_{i} j ⟩, μ_{i} j + η_{i} j + ν_{i} j \leq 1; F a n 2023 § P h a s e 2 S t e p 1 .$

LaTeX
r_ij = ⟨μ_ij, η_ij, ν_ij⟩, μ_ij + η_ij + ν_ij ≤ 1; Fan 2023 §Phase 2 Step 1.
Adım 2 — Maliyet kriterleri için resim bulanık tümleme uygulayarak R_1 → R_4 standardize et: maliyet kriteri j için r_ij ← ⟨ν_ij, η_ij, μ_ij⟩; fayda kriterleri değişmez.

$i f d i r e c t i o n_{j} =^{'} m i n^{'} : r_{i} j \leftarrow (ν_{i} j, η_{i} j, μ_{i} j); e l s e r_{i} j u n c h a n g e d . F a n 2023 § P h a s e 2 S t e p 2 (a l s o T a b l e 3 r o w \to T a b l e 4 r o w t r a n s f o r m) .$

LaTeX
if direction_j = 'min': r_ij ← (ν_ij, η_ij, μ_ij); else r_ij unchanged. Fan 2023 §Phase 2 Step 2 (also Table 3 row→Table 4 row transform).
Adım 3 — Sütun bazında PIS r_j^+ ve NIS r_j^-'yi (1,0,0) ve (0,0,1) çapalarına Hamming uzaklığıyla belirle. r_j^+ = arg min_i d_H(r_ij, (1,0,0)); r_j^- = arg min_i d_H(r_ij, (0,0,1)). r_j^+ eşitliği max d_H(·,(0,0,1)) ile çözülür. d_H(α,β) = ½(|μ_α−μ_β| + |η_α−η_β| + |ν_α−ν_β|).

$d_{H} (α, β) = ½ (| μ_{α} - μ_{β} | + | η_{α} - η_{β} | + | ν_{α} - ν_{β} |); r_{j}^{+} = a r g m i n_{i} d_{H} (r_{i} j, (1, 0, 0)); r_{j}^{-} = a r g m i n_{i} d_{H} (r_{i} j, (0, 0, 1)) . F a n 2023 § P h a s e 2 S t e p 3 (T a b l e s 5 - 6) .$

LaTeX
d_H(α,β) = ½(|μ_α−μ_β|+|η_α−η_β|+|ν_α−ν_β|); r_j^+ = arg min_i d_H(r_ij,(1,0,0)); r_j^- = arg min_i d_H(r_ij,(0,0,1)). Fan 2023 §Phase 2 Step 3 (Tables 5–6).
Adım 4 — Grup faydası S_i ve bireysel pişmanlık R_i hesapla. S_i = Σ_j w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+); R_i = max_j { w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) }. Eq.(15)-(16).

$S_{i} = Σ_{j} w_{j} \cdot d_{H} (r_{i} j, r_{j}^{+}) / d_{H} (r_{j}^{-}, r_{j}^{+}) — E q . (15); R_{i} = m a x_{j} w_{j} \cdot d_{H} (r_{i} j, r_{j}^{+}) / d_{H} (r_{j}^{-}, r_{j}^{+}) — E q . (16) . F a n 2023 .$

LaTeX
S_i = Σ_j w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) — Eq.(15); R_i = max_j { w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) } — Eq.(16). Fan 2023.
Adım 5 — Q_i = v · (S_i − S*)/(S^- − S*) + (1 − v) · (R_i − R*)/(R^- − R*) hesapla; S* = min_i S_i, S^- = max_i S_i, R* = min_i R_i, R^- = max_i R_i, v = uzlaşma katsayısı (varsayılan v = 0.5). Eq.(17).

$Q_{i} = v \cdot (S_{i} - S *) / (S^{-} - S *) + (1 - v) \cdot (R_{i} - R *) / (R^{-} - R *); S * = m i n, S^{-} = m a x, R * = m i n, R^{-} = m a x, d e f a u l t v = 0.5 . F a n 2023 E q . (17) .$

LaTeX
Q_i = v · (S_i − S*)/(S^- − S*) + (1 − v) · (R_i − R*)/(R^- − R*); S* = min, S^- = max, R* = min, R^- = max, default v = 0.5. Fan 2023 Eq.(17).
Adım 6 — Alternatifleri Q_i'nin artan sırasına göre sırala. Opricovic uzlaşma testlerini uygula: T1 (Kabul edilebilir avantaj) — Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1); T2 (Kabul edilebilir kararlılık) — A^(1) aynı zamanda S veya R sıralamasında da en iyi. Her ikisi sağlanırsa A^(1) tek uzlaşık çözüm. Yalnız T1 sağlanırsa uzlaşma kümesi = {A^(1), A^(2)}. Yalnız T2 sağlanırsa uzlaşma kümesi = {A^(1), …, A^(L)} (Q(A^(L)) − Q(A^(1)) < 1/(m−1)).

$r a n k i n g = a r g s o r t_{a} s c (Q); T 1 : Q (A^{(} 2)) - Q (A^{(} 1)) \geq 1 / (m - 1); T 2 : A^{(} 1) b e s t i n S o r R; c o m p r o m i s e_{s} e t p e r O p r i c o v i c r u l e s . F a n 2023 P h a s e 2 S t e p 6 (a l s o O p r i c o v i c 1998) .$

LaTeX
ranking = argsort_asc(Q); T1: Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1); T2: A^(1) best in S or R; compromise_set per Opricovic rules. Fan 2023 Phase 2 Step 6 (also Opricovic 1998).

Sezgi

Picture outranking/ranking - Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1). Output typically utility (higher value = preferred).

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

Varsayımlar

Decision matrix entries are valid Picture 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 Picture 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 PiFS: μ (membership), η (neutral), ν (non-membership); μ+η+ν ≤ 1 koşulunu sağladığından emin olun.
Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = μ − ν kanonik seçimdir.

Hesap adımları ve dayanakları

Build the initial picture fuzzy decision matrix R_1 = [r_ij]_{m×n} where r_ij = ⟨μ_ij, η_ij, ν_ij⟩ with μ + η + ν ≤ 1. Direction (max/min) of each criterion is recorded.

Dayanak: Fan, Cheng, Wu 2023, Phase 2 Step 1
Standardize R_1 → R_4 using picture fuzzy complement for cost criteria: r_ij ← ⟨ν_ij, η_ij, μ_ij⟩ if criterion j is cost; benefit criteria pass through unchanged.

Dayanak: Fan, Cheng, Wu 2023, Phase 2 Step 2
Determine PIS r_j^+ and NIS r_j^- column-wise using Hamming distance to picture-fuzzy anchors (1,0,0) and (0,0,1). r_j^+ = arg min_i d_H(r_ij, (1,0,0)); r_j^- = arg min_i d_H(r_ij, (0,0,1)). Ties on r_j^+ broken by max d_H to (0,0,1). d_H(α,β) = ½(|μ_α−μ_β| + |η_α−η_β| + |ν_α−ν_β|).

Dayanak: Fan, Cheng, Wu 2023, Phase 2 Step 3
Compute group utility S_i and individual regret R_i. S_i = Σ_j w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+); R_i = max_j { w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) }. Eqs.(15)-(16).

Dayanak: Fan, Cheng, Wu 2023, Eqs.(15)-(16)
Compute Q_i = v · (S_i − S*)/(S^- − S*) + (1 − v) · (R_i − R*)/(R^- − R*), where S* = min_i S_i, S^- = max_i S_i, R* = min_i R_i, R^- = max_i R_i, and v = compromise coefficient (default v = 0.5). Eq.(17).

Dayanak: Fan, Cheng, Wu 2023, Eq.(17)
Sort alternatives in ascending order of Q_i to obtain ranking. Apply Opricovic compromise tests: T1 (Acceptable advantage) - Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1); T2 (Acceptable stability) - A^(1) is also best in S or R ranking. If both hold, A^(1) is the unique compromise. If only T1 holds, compromise set = {A^(1), A^(2)}. If only T2 holds, compromise set = {A^(1), …, A^(L)} where Q(A^(L)) − Q(A^(1)) < 1/(m−1).

Dayanak: Fan, Cheng, Wu 2023, Phase 2 Step 6; Opricovic 1998