PiF-VIKOR: Picture extension of VIKOR

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: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •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 VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •if criterion j is cost; benefit criteria pass through unchanged.
• •Ties on r_j^+ broken by max d_H to (0,0,1). d_H(α,β) = ½(|μ_α−μ_β| + |η_α−η_β| + |ν_α−ν_β|).
• •default v = 0.5). Eq.(17).
• •If both hold, A^(1) is the unique compromise. If only T1 holds, compromise set = {A^(1), A^(2)}. If only T2 holds, compromise set = {A^(1), …, A^(L)} where Q(A^(L)) − Q(A^(1)) < 1/(m−1).

Common pitfalls

• •Hatalı: 'PIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy numbers/tuples
• •Hatalı: 'PIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PIF-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-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: Build the initial picture fuzzy decision matrix R_1 = [r_ij]_{m×n} where r_ij = ⟨μ_ij, η_ij, ν_ij⟩ with μ + η + ν ≤ 1. Direction (max/min) of each criterion is recorded. Formül: r_ij = ⟨μ_ij, η_ij, ν_ij⟩, μ_ij + η_ij + ν_ij ≤ 1; Fan 2023 §Phase 2 Step 1. Anchor: Fan, Cheng, Wu 2023, Phase 2 Step 1
2. 2.Adım 2 (F2): Step 2: Standardize R_1 → R_4 using picture fuzzy complement for cost criteria: r_ij ← ⟨ν_ij, η_ij, μ_ij⟩ if criterion j is cost; benefit criteria pass through unchanged. Formül: if direction_j = 'min': r_ij ← (ν_ij, η_ij, μ_ij); else r_ij unchanged. Fan 2023 §Phase 2 Step 2 (also Table 3 row→Table 4 row transform). Anchor: Fan, Cheng, Wu 2023, Phase 2 Step 2
3. 3.Adım 3 (F3): Step 3: Determine PIS r_j^+ and NIS r_j^- column-wise using Hamming distance to picture-fuzzy anchors (1,0,0) and (0,0,1). r_j^+ = arg min_i d_H(r_ij, (1,0,0)); r_j^- = arg min_i d_H(r_ij, (0,0,1)). Ties on r_j^+ broken by max d_H to (0,0,1). d_H(α,β) = ½(|μ_α−μ_β| + |η_α−η_β| + |ν_α−ν_β|). Formül: d_H(α,β) = ½(|μ_α−μ_β|+|η_α−η_β|+|ν_α−ν_β|); r_j^+ = arg min_i d_H(r_ij,(1,0,0)); r_j^- = arg min_i d_H(r_ij,(0,0,1)). Fan 2023 §Phase 2 Step 3 (Tables 5-6). Anchor: Fan, Cheng, Wu 2023, Phase 2 Step 3
4. 4.Adım 4 (F4): Step 4: Compute group utility S_i and individual regret R_i. S_i = Σ_j w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+); R_i = max_j { w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) }. Eqs.(15)-(16). Formül: S_i = Σ_j w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+): Eq.(15); R_i = max_j { w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) }: Eq.(16). Fan 2023. Anchor: Fan, Cheng, Wu 2023, Eqs.(15)-(16)
5. 5.Adım 5 (F5): Step 5: Compute Q_i = v · (S_i − S*)/(S^- − S*) + (1 − v) · (R_i − R*)/(R^- − R*), where S* = min_i S_i, S^- = max_i S_i, R* = min_i R_i, R^- = max_i R_i, and v = compromise coefficient (default v = 0.5). Eq.(17). Formül: Q_i = v · (S_i − S*)/(S^- − S*) + (1 − v) · (R_i − R*)/(R^- − R*); S* = min, S^- = max, R* = min, R^- = max, default v = 0.5. Fan 2023 Eq.(17). Anchor: Fan, Cheng, Wu 2023, Eq.(17)
6. 6.Adım 6 (F6): Step 6: Sort alternatives in ascending order of Q_i to obtain ranking. Apply Opricovic compromise tests: T1 (Acceptable advantage): Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1); T2 (Acceptable stability): A^(1) is also best in S or R ranking. If both hold, A^(1) is the unique compromise. If only T1 holds, compromise set = {A^(1), A^(2)}. If only T2 holds, compromise set = {A^(1), …, A^(L)} where Q(A^(L)) − Q(A^(1)) < 1/(m−1). Formül: ranking = argsort_asc(Q); T1: Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1); T2: A^(1) best in S or R; compromise_set per Opricovic rules. Fan 2023 Phase 2 Step 6 (also Opricovic 1998). Anchor: Fan, Cheng, Wu 2023, Phase 2 Step 6; Opricovic 1998

Commonly paired with

• •n_a + PIF-VIKOR (common)

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