PiF-COPRAS - COPRAS yönteminin Picture uzantısı

Formül adımları

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

1. Adım 1 — Her DM'in H^(k) matrislerini PFWA ile grup PF karar matrisine birleştir (Lu 2021 Eş.14): h_ij = (1 − ∏_k (1−μ^k)^ψ_k, ∏_k η^k^ψ_k, ∏_k (ν^k+η^k)^ψ_k − ∏_k η^k^ψ_k). Tek DM varsa H = H^(1).

   $$h_{i}j=(1-{\prod }_{k=1}^l(1-{\mu }_{ij}^k{)}^{{\psi }_k},{\prod }_{k=1}^l({\eta }_{ij}^k{)}^{{\psi }_k},{\prod }_{k=1}^l({\nu }_{ij}^k+{\eta }_{ij}^k{)}^{{\psi }_k}-{\prod }_{k=1}^l({\eta }_{ij}^k{)}^{{\psi }_k})$$
   LaTeX

   h_ij = (1 − ∏_{k=1}^l (1−μ_{ij}^k)^{ψ_k}, ∏_{k=1}^l (η_{ij}^k)^{ψ_k}, ∏_{k=1}^l (ν_{ij}^k + η_{ij}^k)^{ψ_k} − ∏_{k=1}^l (η_{ij}^k)^{ψ_k})

2. Adım 2 — CRITIC ile nesnel ağırlıklar q_j (Lu 2021 Eş.15-17): S(h_ij)=μ−ν, sütun-arası Pearson τ_jt, std σ_j, q_j = σ_j Σ_t(1−τ_jt) / Σ_j[σ_j Σ_t(1−τ_jt)].

   $${\tau }_{jt}={\Sigma }_i(S(h_{ij})-S{̄}_j)(S(h_{it})-S{̄}_t)/sqrt({\Sigma }_i(S(h_{ij})-S{̄}_j{)}^2·{\Sigma }_i(S(h_{it})-S{̄}_t{)}^2);{\sigma }_j=sqrt((1/m){\Sigma }_i(S(h_{ij})-S{̄}_j{)}^2);q_j={\sigma }_j{\Sigma }_t(1-{\tau }_{jt})/{\Sigma }_j[{\sigma }_j{\Sigma }_t(1-{\tau }_{jt})]$$
   LaTeX

   τ_{jt} = Σ_i (S(h_{ij})-S̄_j)(S(h_{it})-S̄_t) / sqrt(Σ_i (S(h_{ij})-S̄_j)^2 · Σ_i (S(h_{it})-S̄_t)^2); σ_j = sqrt((1/m) Σ_i (S(h_{ij})-S̄_j)^2); q_j = σ_j Σ_t (1-τ_{jt}) / Σ_j [σ_j Σ_t (1-τ_{jt})]

3. Adım 3 — Ağırlıklı normalleştirilmiş matris D (Lu 2021 Eş.18): d_ij = q_j × h_ij.

   $$d_{ij}=q_j\times h_{ij}=(1-(1-{\mu }_{ij}{)}^{q_j},{\eta }_{ij}^{q_j},({\nu }_{ij}+{\eta }_{ij}{)}^{q_j}-{\eta }_{ij}^{q_j})$$
   LaTeX

   d_{ij} = q_j × h_{ij} = (1 − (1−μ_{ij})^{q_j}, η_{ij}^{q_j}, (ν_{ij}+η_{ij})^{q_j} − η_{ij}^{q_j})

4. Adım 4 — Fayda (max) için R_i^+, maliyet (min) için R_i^- (Lu 2021 Eş.19-20).

   $$R_i^{+}={\oplus }_{j:max}{d}_{ij}=(1-{\prod }_{j:max}(1-{\mu }_{d_{ij}}),{\prod }_{j:max}{\eta }_{d_{ij}},{\prod }_{j:max}({\nu }_{d_{ij}}+{\eta }_{d_{ij}})-{\prod }_{j:max}{\eta }_{d_{ij}});R_i^{-}analogousovercostattributes$$
   LaTeX

   R_i^+ = ⊕_{j: max} d_{ij} = (1 − ∏_{j: max}(1−μ_{d_{ij}}), ∏_{j: max} η_{d_{ij}}, ∏_{j: max}(ν_{d_{ij}}+η_{d_{ij}}) − ∏_{j: max} η_{d_{ij}}); R_i^- analogous over cost attributes

5. Adım 5 — Önem skoru O_i (Lu 2021 Eş.21).

   $$O_i=S(R_i^{+})+{\Sigma }_{k=1}^{m}S(R_k^{-})/[S(R_i^{-})·{\Sigma }_{k=1}^m(1/S(R_k^{-}))]$$
   LaTeX

   O_i = S(R_i^+) + Σ_{k=1}^m S(R_k^-) / [S(R_i^-) · Σ_{k=1}^m (1/S(R_k^-))]

6. Adım 6 — Fayda derecesi G_i = O_i / max O_k (Lu 2021 Eş.22).

   $$G_i=O_i/ma{x}_{k=1,...,m}{O}_k(presentedas{G}_i\times 100$$
   LaTeX

   G_i = O_i / max_{k=1,...,m} O_k (presented as G_i × 100%)

Sezgi

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

Sonucu okuma: pif-copras extends COPRAS 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 COPRAS directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: Lu et al. 2021; cf. Belton-Gear 1983 for crisp COPRAS)
• 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.