PiF-COPRAS: Picture extension of COPRAS

Overview

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

Strengths

• •Method-specific: Picture outranking/ranking: Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1)
• •Preserves picture uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •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

Assumptions to verify

• •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

When not to use

• •Crisp data sufficient: use base COPRAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •If only a single DM is provided, H = H^(1).

Common pitfalls

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

Worked example

1. 1.Adım 1 (F1): Step 1: Aggregate per-DM matrices H^(k) (k=1,...,l) into group PF decision matrix H = [h_ij] via PFWA (Lu 2021 Eq.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), where ψ_k is the expert weight (Σψ_k = 1). If only a single DM is provided, H = H^(1). Formül: 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}) Anchor: Lu 2021 Eqs.12-14; Wang 2017 PFWA
2. 2.Adım 2 (F2): Step 2: Determine objective criteria weights q_j via CRITIC (Lu 2021 Eqs.15-17) using score S(h_ij) = μ − ν. Compute Pearson correlation τ_jt between columns; standard deviation σ_j; then q_j = σ_j Σ_t (1−τ_jt) / Σ_j [σ_j Σ_t (1−τ_jt)]. Formül: 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})] Anchor: Lu 2021 Eqs.15-17; Diakoulaki et al. 1995 (CRITIC origin)
3. 3.Adım 3 (F3): Step 3: Weighted normalized matrix D = [d_ij] via scalar multiplication (Lu 2021 Eq.18; Wang 2017 scalar-mult Eq.4): d_ij = q_j × h_ij = (1−(1−μ)^{q_j}, η^{q_j}, (ν+η)^{q_j} − η^{q_j}). Formül: 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}) Anchor: Lu 2021 Eq.18; Wang 2017 Eq.4 (scalar mult)
4. 4.Adım 4 (F4): Step 4: Sum weighted PFNs over benefit (max-direction) attributes → R_i^+, and over cost (min-direction) attributes → R_i^- via Wang 2017 ⊕ (Lu 2021 Eqs.19-20): R_i = (1−∏(1−μ_d), ∏η_d, ∏(ν_d+η_d)−∏η_d). Formül: 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 Anchor: Lu 2021 Eqs.19-20; Wang 2017 Eq.2 (⊕)
5. 5.Adım 5 (F5): Step 5: Significance O_i (Lu 2021 Eq.21): O_i = S(R_i^+) + Σ_k S(R_k^-) / [S(R_i^-) · Σ_k 1/S(R_k^-)], where S(α)=μ−ν (Cuong 2014 Def.3, Eq.6). Formül: O_i = S(R_i^+) + Σ_{k=1}^m S(R_k^-) / [S(R_i^-) · Σ_{k=1}^m (1/S(R_k^-))] Anchor: Lu 2021 Eq.21
6. 6.Adım 6 (F6): Step 6: Utility degree G_i = O_i / max_k O_k (Lu 2021 Eq.22; reported as percent). Higher G_i ⇒ better. Rank alternatives by descending G_i. Formül: G_i = O_i / max_{k=1,...,m} O_k (presented as G_i × 100%) Anchor: Lu 2021 Eq.22

Commonly paired with

• •n_a + PIF-COPRAS (common)

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