PF-VIKOR: Pythagorean extension of VIKOR

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 VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i.
• •If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Common pitfalls

• •Hatalı: 'PF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
• •Hatalı: 'PF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PF-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PF-VIKOR'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: Construct the Pythagorean fuzzy decision matrix X = [α_ij]_{m×n} where each entry α_ij = (μ_ij, ν_ij) is a PFN satisfying μ_ij²+ν_ij²≤1, with associated hesitancy π_ij² = 1−μ_ij²−ν_ij². Formül: α_ij = (μ_ij, ν_ij) ∈ Φ(U); μ_ij, ν_ij ∈ [0,1]; μ_ij²+ν_ij²≤1; π_ij² = 1−μ_ij²−ν_ij² (Abbas §4 Step 1; Yager 2013 PFS def) Anchor: Abbas 2024 §4 Step 1; Yager 2013 PFS definition
2. 2.Adım 2 (F2): Step 2: Weights w_j per criterion are either externally provided (e.g. from AHP, BWM, ENTROPY) or computed from PFN entropy. Abbas 2024 proposes E_SA-entropy then W_j = E_SA(C_j) / Σ_k E_SA(C_k). The PF-VIKOR algorithm proper is weight-source-agnostic; w_j with Σ w_j = 1 is the only requirement. Formül: w_j ≥ 0, Σ_j w_j = 1. Optional: W_j = E_SA(C_j)/Σ_k E_SA(C_k) where E_SA(C_j) = (1/(2m))Σ_i [|(1+μ_ij²+π_ij²)/2 − ν_ij²|]^(1/2) (Abbas 2024 Eq.(5)). Anchor: Abbas 2024 §4 Steps 2-3; Eq.(5)
3. 3.Adım 3 (F3): Step 3: Determine the synthetic best PFN f*_j and worst PFN f^-_j per criterion. For benefit criterion j: f*_j = (max_i μ_ij, min_i ν_ij); f^-_j = (min_i μ_ij, max_i ν_ij). For cost criterion j: swap (f*_j and f^-_j roles inverted). The reference points are synthetic constructions; they need not equal any actual alternative. Formül: If j benefit: f*_j = (max_i μ_ij, min_i ν_ij), f^-_j = (min_i μ_ij, max_i ν_ij). If j cost: f*_j = (min_i μ_ij, max_i ν_ij), f^-_j = (max_i μ_ij, min_i ν_ij). (Abbas 2024 §4 Step 4) Anchor: Abbas 2024 §4 Step 4
4. 4.Adım 4 (F4): Step 4: Compute per-cell normalised PF gap using PF distance. Abbas 2024 uses PF Hamming distance over squared components: d_H(α,β) = (|μ_α²−μ_β²| + |ν_α²−ν_β²| + |π_α²−π_β²|)/3. The normalised gap is Δ_ij = w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j). PF Euclidean distance d_E is a documented alternative. Formül: d_H(α,β) = (1/3)·[|μ_α²−μ_β²| + |ν_α²−ν_β²| + |π_α²−π_β²|]; Δ_ij = w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j). (Abbas 2024 §4 Step 5) Anchor: Abbas 2024 §4 Step 5
5. 5.Adım 5 (F5): Step 5: Compute group utility S_i (L_1-aggregated) and individual regret R_i (L_∞-aggregated) over the normalised gap matrix Δ. Formül: S_i = Σ_j Δ_ij = Σ_j [w_j · d_H(α_ij, f*_j) / d_H(f^-_j, f*_j)]; R_i = max_j Δ_ij. (Abbas 2024 §4 Step 5; Opricovic 1998 VIKOR L_1/L_∞ aggregation) Anchor: Abbas 2024 §4 Step 5; Opricovic 1998
6. 6.Adım 6 (F6): Step 6: Compute the VIKOR compromise index Q_i as a convex combination of normalised S and R, weighted by strategy parameter β (default β=0.5; β>0.5 favours group utility, β<0.5 favours individual regret). S* = min_i S_i, S^- = max_i S_i; R* = min_i R_i, R^- = max_i R_i. Formül: Q_i = β·(S_i − S*)/(S^- − S*) + (1−β)·(R_i − R*)/(R^- − R*). (Abbas 2024 §4 Step 7; Opricovic 1998) Anchor: Abbas 2024 §4 Step 7
7. 7.Adım 7 (F7): Step 7: Produce three rankings by sorting Q, S, R ascending. The alternative with the smallest Q is the candidate compromise solution. Formül: ranking_Q = argsort_asc(Q); ranking_S = argsort_asc(S); ranking_R = argsort_asc(R). Best alternative = ranking_Q[0]. (Abbas 2024 §4 Step 8) Anchor: Abbas 2024 §4 Step 8; Opricovic 1998 VIKOR
8. 8.Adım 8 (F8): Step 8: Apply the two-condition VIKOR compromise test. C1 (acceptable advantage): Q(A^(2)) − Q(A^(1)) ≥ DQ where DQ = 1/(m−1). C2 (acceptable stability in decision-making): A^(1) is also best ranked by S and/or R. If both hold, propose A^(1) as compromise solution. If C1 fails, return the longest prefix {A^(1)..A^(M)} satisfying Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}. Formül: C1: Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1). C2: A^(1) = ranking_S[0] OR A^(1) = ranking_R[0]. compromise_set = if C1 ∧ C2 then {A^(1)} else if ¬C1 then prefix(Q − Q^(1) < DQ) else if ¬C2 then {A^(1), A^(2)}. (Opricovic 1998 VIKOR compromise test) Anchor: Opricovic 1998 VIKOR compromise test (inherited)

Commonly paired with

• •n_a + PF-VIKOR (common)

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