PiF-EDAS: Picture extension of EDAS

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

Edge cases

• •See F.steps and D.parameters for PIF-EDAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'PIF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy numbers/tuples
• •Hatalı: 'PIF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PIF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PIF-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-EDAS'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: Form Picture Fuzzy decision matrix X̃ = [x̃_ij] with x̃_ij = (μ_ij, η_ij, ν_ij), μ_ij+η_ij+ν_ij ≤ 1. Formül: X̃ = [x̃_{ij}]_{m×n}, x̃_{ij} = (μ_{ij}, η_{ij}, ν_{ij}), μ_{ij}+η_{ij}+ν_{ij} ≤ 1 Anchor: Cuong 2013 Definition 1; Kamber 2026 Eq.(24)
2. 2.Adım 2 (F2): Step 2: Cost-complement normalize: for cost criteria, r_ij = (ν_ij, η_ij, μ_ij); for benefit criteria, r_ij = (μ_ij, η_ij, ν_ij) (unchanged). Formül: r_{ij} = (ν_{ij}, η_{ij}, μ_{ij}) \text{ if } j \in \text{Cost}; \quad r_{ij} = (μ_{ij}, η_{ij}, ν_{ij}) \text{ if } j \in \text{Benefit} Anchor: Khan vd. 2019; Kamber 2026 Eq.(25)
3. 3.Adım 3 (F3): Step 3: Per-criterion average solution AV_j via PFWA with equal weights 1/m: AV_j = (1 − ∏(1−μ_ij)^(1/m), ∏ η_ij^(1/m), ∏ ν_ij^(1/m)). Formül: AV_j = \left(1 - \prod_{i=1}^{m}(1-μ_{ij})^{1/m},\; \prod_{i=1}^{m} η_{ij}^{1/m},\; \prod_{i=1}^{m} ν_{ij}^{1/m}\right) Anchor: Wei 2017 Eq.(8) PFWA; Kamber 2026 Eq.(26)
4. 4.Adım 4 (F4): Step 4: Positive/Negative Distance from Average via Cuong score s(α) = μ − ν. PDA_ij = max(0, s(r_ij) − s(AV_j))/s(AV_j); NDA_ij = max(0, s(AV_j) − s(r_ij))/s(AV_j). NOTE: Kamber 2026 Eq.(27)/(28) print these with PDA and NDA labels swapped; canonical labeling is used here for compatibility with Step 6 normalization NSP=SP/maxSP, NSN=1−SN/maxSN. Formül: s(α) = μ − ν; \quad PDA_{ij} = \frac{\max(0, s(r_{ij}) - s(AV_j))}{s(AV_j)}; \quad NDA_{ij} = \frac{\max(0, s(AV_j) - s(r_{ij}))}{s(AV_j)} Anchor: Cuong 2013 Eq.(5); Kamber 2026 Eqs.(27)-(28) (with corrected labels)
5. 5.Adım 5 (F5): Step 5: Weighted sums SP_i = Σ_j w_j · PDA_ij and SN_i = Σ_j w_j · NDA_ij. Formül: SP_i = \sum_{j=1}^{n} w_j \cdot PDA_{ij}; \quad SN_i = \sum_{j=1}^{n} w_j \cdot NDA_{ij} Anchor: Kamber 2026 Eqs.(19)-(20); Keshavarz-Ghorabaee 2015 Eqs.(6)-(7)
6. 6.Adım 6 (F6): Step 6: Normalize: NSP_i = SP_i / max_i SP_i; NSN_i = 1 − SN_i / max_i SN_i. Formül: NSP_i = \frac{SP_i}{\max_i SP_i}; \quad NSN_i = 1 - \frac{SN_i}{\max_i SN_i} Anchor: Kamber 2026 Eqs.(21)-(22); Keshavarz-Ghorabaee 2015 Eqs.(8)-(9)
7. 7.Adım 7 (F7): Step 7: Appraisal Score AS_i = (NSP_i + NSN_i)/2; rank alternatives in descending order of AS_i (largest AS is best). Formül: AS_i = \tfrac{1}{2}(NSP_i + NSN_i); \quad \text{rank desc by } AS_i Anchor: Kamber 2026 Eq.(23); Keshavarz-Ghorabaee 2015 Eq.(10)

Commonly paired with

• •n_a + PIF-EDAS (common)

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