PF-COPRAS: Pythagorean extension of COPRAS

Overview

Pythagorean outranking/ranking: Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Pythagorean outranking/ranking: Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1)
• •Preserves pythagorean 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: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Pythagorean 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 Pythagorean 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

• •when S_i is negative-valued (typical for score-based defuzzification on PFNs with ν>μ), max(S) is the least-negative; T_i ≥ 100 with higher T indicating better alternative. Descending rank by T.

Common pitfalls

• •Hatalı: 'PF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
• •Hatalı: 'PF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PF-COPRAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PF-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: Receive aggregated Pythagorean fuzzy decision matrix N = (μ_ij, ν_ij) with μ²+ν²≤1. Single-DM kernel skips group-aggregation via PFWA (paper Eq.9). Formül: Input: N ∈ PFN^{m×n}, w ∈ Δ^{n-1}, criteria_types ∈ {benefit,cost}^n. Constraint: μ²_ij + ν²_ij ≤ 1 ∀(i,j). Anchor: Thakur et al. 2022 §4.1, Tables 6-7 (aggregated decision matrix N_pq)
2. 2.Adım 2 (F2): Step 2: DUAL normalization (journal-as-published Thakur 2022 replikasyon). İki matris: (a) N'_swap Eq.10 literal cost-swap → h(U_i) benefit-index yolunu besler (Tablo 10 reproduce); (b) N_orig swap-yok → h(Y_i) cost-index yolunu besler (Tablo 11 h(Y_i) reproduce). Paper'ın gerçek hesap pratiği bu çiftli yoldur. Formül: pathway (a): n'_ij swap-Eq.10; pathway (b): n_ij = (μ,ν) orijinal Anchor: Thakur 2022 Eq.10 + Tablo 10 + Tablo 11; as_stated_vs_as_used disclosure F2'de
3. 3.Adım 3 (F3): Step 3: Weighted decision matrix D = (d_ij). Weights w_j may be entropy-derived (Eq.11) or external. PFN scalar-mult: d_ij = w_j ⊙ n_ij. Formül: d_ij = w_j ⊙ n_ij = (√(1 − (1 − μ²_ij)^{w_j}), ν_ij^{w_j}). [Thakur 2022 Step VII formula] Anchor: Thakur et al. 2022 Step VII (between Eqs.11 and 12)
4. 4.Adım 4 (F4): Step 4: Score function h(d_ij) = μ²(d_ij) − ν²(d_ij) (Eq.12). Defuzzifies each weighted PFN to a real-valued score. Formül: h(d_ij) = μ²(d_ij) − ν²(d_ij). [Eq. 12] Anchor: Thakur et al. 2022 Eq.12 (score function), Table 10
5. 5.Adım 5 (F5): Step 5: Max/Min indeksleri (Eqs.13-14, çiftli yol journal-as-published). h(U_i)=(1/|B|)Σ h(d^swap) over benefit (F2 yol-a); h(Y_i)=(1/|NB|)Σ h(d^noswap) over cost (F2 yol-b). NB ≠ ∅ zorunlu. Formül: h(U_i)=(1/|B|)Σ_{j∈B} h(d_ij^swap) [Eq.13 yol-a]; h(Y_i)=(1/|NB|)Σ_{j∈NB} h(d_ij^noswap) [Eq.14 yol-b] Anchor: Thakur 2022 Eqs.13-14 + Table 11; pathway (a)+(b) ≤1e-5 reproduction kanıtı
6. 6.Adım 6 (F6): Step 6: Relative significance with exp-weighting (Eq.15): S_i = h(U_i) + (Σ_k e^{h(Y_k)}) / (e^{h(Y_i)}·Σ_k 1/e^{h(Y_k)}). Requires NB ≠ ∅. Formül: S_i = h(U_i) + (Σ_k e^{h(Y_k)}) / (e^{h(Y_i)} · Σ_k (1/e^{h(Y_k)})) [Eq. 15] Anchor: Thakur et al. 2022 Eq.15, Table 11 column S_i
7. 7.Adım 7 (F7): Step 7: Priority order T_i = S_i / max_k S_k × 100 (Eq.16). NOTE: when S_i is negative-valued (typical for score-based defuzzification on PFNs with ν>μ), max(S) is the least-negative; T_i ≥ 100 with higher T indicating better alternative. Descending rank by T. Formül: T_i = S_i / max_k S_k × 100 [Eq. 16]; rank = argsort_desc(T) Anchor: Thakur et al. 2022 Eq.16, Table 11 columns T_i and Rank

Commonly paired with

• •n_a + PF-COPRAS (common)

How to cite