PIF-ARTASI - Picture Fuzzy ARTASI

Formül adımları

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

Adım 12: Uzman-bazlı PFS karar matrisi.

$W_{t} i l d e_{i j}^{(h)} = (m u_{i j}^{(h)}, e t a_{i j}^{(h)}, n u_{i j}^{(h)}); c o n s t r a i n t m u + e t a + n u < = 1$

LaTeX
W_tilde^{(h)}_{ij} = (mu^{(h)}_{ij}, eta^{(h)}_{ij}, nu^{(h)}_{ij}) ; constraint mu+eta+nu <= 1
Adım 13 (Eq 15): PFWA toplama (Wang 2017).

$m u_{a} g g = 1 - p r o d_{h} (1 - m u_{h})^{w_{h}}; e t a_{a} g g = p r o d_{h} e t a_{h}^{w_{h}}; n u_{a} g g = p r o d_{h} (n u_{h} + e t a_{h})^{w_{h}} - p r o d_{h} e t a_{h}^{w_{h}}$

LaTeX
mu_agg = 1 - prod_h (1 - mu_h)^{w_h} ; eta_agg = prod_h eta_h^{w_h} ; nu_agg = prod_h (nu_h + eta_h)^{w_h} - prod_h eta_h^{w_h}
Adım 14 (Eq 16): PFS → crisp via Sc.

$U_{i j} = (m u_{i j} + (1 - e t a_{i j}) + (1 - n u_{i j})) / 3$

LaTeX
U_{ij} = (mu_{ij} + (1 - eta_{ij}) + (1 - nu_{ij})) / 3
Adım 15: nitel U + nicel T → L.

$L_{i j} = U_{i j} (q u a l i t a t i v e); L_{i j} = T_{i j} (q u a n t i t a t i v e)$

LaTeX
L_{ij} = U_{ij} (qualitative) ; L_{ij} = T_{ij} (quantitative)
Adım 16: uyarlanabilir sınırlar.

$S_{j}^{m} a x = m a x_{i} L_{i j} + (m a x_{i} L_{i j})^{1 / m}; S_{j}^{m} i n = m i n_{i} L_{i j} - (m i n_{i} L_{i j})^{1 / m}$

LaTeX
S_j^max = max_i L_{ij} + (max_i L_{ij})^{1/m} ; S_j^min = min_i L_{ij} - (min_i L_{ij})^{1/m}
Adım 17a-b: iki seviyeli standartlaştırma.

$R_{i j} = (L_{i j} - S_{j}^{m} i n) / (S_{j}^{m} a x - S_{j}^{m} i n) [b e n e f i t]; R_{i j} = (S_{j}^{m} a x - L_{i j}) / (S_{j}^{m} a x - S_{j}^{m} i n) [c o s t]; C_{i j} = R_{i j} * (b e t a_{u} - b e t a_{l}) + b e t a_{l}$

LaTeX
R_{ij} = (L_{ij} - S_j^min)/(S_j^max - S_j^min) [benefit] ; R_{ij} = (S_j^max - L_{ij})/(S_j^max - S_j^min) [cost] ; C_{ij} = R_{ij}*(beta_u - beta_l) + beta_l
Adım 18a-b: ideal/anti-ideal V^+, V^-.

$V_{i j}^{+} = (C_{i j} / m a x_{i} C_{i j}) * w_{j} * b e t a_{u} [b e n e f i t]; V_{i j}^{+} = (m i n_{i} C_{i j} / C_{i j}) * w_{j} * b e t a_{u} [c o s t]; U_{i j} = (m i n_{i} C_{i j} / C_{i j}) * w_{j} * b e t a_{u} [b e n e f i t] o r (C_{i j} / m a x_{i} C_{i j}) * w_{j} * b e t a_{u} [c o s t]; V_{i j}^{-} = - U_{i j} + m a x_{i} U_{i j} + m i n_{i} U_{i j}$

LaTeX
V_{ij}^+ = (C_{ij}/max_i C_{ij})*w_j*beta_u [benefit] ; V_{ij}^+ = (min_i C_{ij}/C_{ij})*w_j*beta_u [cost] ; U_{ij} = (min_i C_{ij}/C_{ij})*w_j*beta_u [benefit] or (C_{ij}/max_i C_{ij})*w_j*beta_u [cost] ; V_{ij}^- = -U_{ij} + max_i U_{ij} + min_i U_{ij}
Adım 19: N+ = Σ V+, N− = Σ V−.

$N_{i}^{+} = s u m_{j} V_{i j}^{+}; N_{i}^{-} = s u m_{j} V_{i j}^{-}$

LaTeX
N_i^+ = sum_j V_{ij}^+ ; N_i^- = sum_j V_{ij}^-
Adım 20 (Eq 26): K_i son fayda; multiplikatif.

$K_{i} = (N_{i}^{+} + N_{i}^{-}) * (p s i * f (N_{i}^{+})^{t} a u + (1 - p s i) * f (N_{i}^{-})^{t} a u)^{(} 1 / t a u); f = i d e n t i t y; d e f a u l t s p s i = 0.5, t a u = 1$

LaTeX
K_i = (N_i^+ + N_i^-) * ( psi * f(N_i^+)^tau + (1 - psi) * f(N_i^-)^tau )^(1/tau) ; f = identity ; defaults psi=0.5, tau=1

Sezgi

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

Sonucu okuma: PIF-ARTASI extends ARTASI (Kara et al. 2024) to Picture Fuzzy Sets. PFS-based qualitative criteria are aggregated via PFWA, converted to crisp via score function Sc, and combined with quantitative criteria before the ARTASI standardisation pipeline. Higher K is better.

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 ARTASI directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

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

Hatalı: 'PIF-ARTASI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy numbers/tuples
Hatalı: 'PIF-ARTASI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
Hatalı: 'PIF-ARTASI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
Hatalı: PIF-ARTASI'yi 'Classical data sufficient' durumunda kullanmak - recommendation_metadata.not_recommended_when alternatif öneriyor.
Hatalı: PIF-ARTASI'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak - recommendation_metadata.not_recommended_when alternatif öneriyor.

Hesap adımları ve dayanakları

Step 12: Construct per-expert PFS decision matrix W̃^(h)_{ij} for each expert h=1..d. Each cell is a PFS triple (μ, η, ν) with μ+η+ν ≤ 1.

Dayanak: Kara et al. 2024, Step 12
Step 13 (Eq 15): Aggregate expert matrices via PFWA with expert weights w_h: PFWA(p_1..p_d) = ⟨1−∏(1−μ_h)^{w_h}, ∏ η_h^{w_h}, ∏(ν_h+η_h)^{w_h} − ∏ η_h^{w_h}⟩ (Wang 2017 form, preserves μ+η+ν ≤ 1).

Dayanak: Kara et al. 2024, Eq (15); Wang et al. 2017 PFWA
Step 14 (Eq 16): Collapse PFS to crisp via score function U_{ij} = Sc(W̃_{ij}) = (μ + (1−η) + (1−ν))/3.

Dayanak: Kara et al. 2024, Eq (16)
Step 15: Combine crisp qualitative U_{ij} (from PFS) with quantitative T_{ij} (crisp from start) into L_{ij}. If only qualitative criteria, L = U.

Dayanak: Kara et al. 2024, Step 15
Step 16 (Eqs 17-18): Adaptive bounds per criterion j: S_j^max = max_i L_{ij} + (max_i L_{ij})^{1/m}; S_j^min = min_i L_{ij} − (min_i L_{ij})^{1/m}. m = number of alternatives.

Dayanak: Kara et al. 2024, Eqs (17)-(18)
Step 17a-b (Eqs 19-20): Two-level standardization. First-level rate R_{ij} ∈ [0,1] direction-aware: for benefit R_{ij} = (L_{ij} − S_j^min)/(S_j^max − S_j^min); for cost R_{ij} = (S_j^max − L_{ij})/(S_j^max − S_j^min). Second-level scale C_{ij} = R_{ij}·(β^u − β^l) + β^l ∈ [β^l, β^u]; default (β^l, β^u) = (1, 100).

Dayanak: Kara et al. 2024, Eqs (19)-(20)
Step 18a-b (Eqs 21-22): Ideal/anti-ideal weighted utility via Tatar 2025 reversal-on-cost-only convention. Benefit: V_{ij}^+ = (C_{ij}/max_i C_{ij})·w_j·β^u; intermediate U_{ij} = (min_i C_{ij}/C_{ij})·w_j·β^u. Cost: swap roles of max and min. Then V_{ij}^- = −U_{ij} + max_i U_{ij} + min_i U_{ij}. (Tatar 2025 correction explicitly forbids the simple v_{ij}/v_j^- form.)

Dayanak: Kara et al. 2024, Eqs (21)-(22); Tatar 2025 §2 corrections
Step 19 (Eqs 24-25): Row sums N_i^+ = Σ_j V_{ij}^+; N_i^- = Σ_j V_{ij}^-.

Dayanak: Kara et al. 2024, Eqs (24)-(25)
Step 20 (Eq 26): Final utility K_i = (N_i^+ + N_i^-) · [ψ·f(N_i^+)^τ + (1−ψ)·f(N_i^-)^τ]^{1/τ}. Defaults ψ=0.5, τ=1 (Tatar 2025 application example), f = identity. Multiplicative juxtaposition between (N+ + N−) and bracket per math convention; higher K_i is better.

Dayanak: Kara et al. 2024, Eq (26); Tatar 2025 application example