PiF-ARAS: Picture Fuzzy extension of ARAS

Overview

Picture utility-degree ranking: Picture Fuzzy Number (PiFN: μ, η, ν; μ+η+ν ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Picture utility-degree ranking: Picture Fuzzy Number (PiFN: 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 ARAS base; cf. Zavadskas-Turskis 2010)
• •Assumes: Decision matrix entries are valid Picture Fuzzy Numbers (μ+η+ν ≤ 1)
• •Assumes: Aggregated PiFN criterion weights provided or derivable
• •Assumes: Criterion-direction inversion handled upstream (linguistic conversion) OR via PiFN complement on cost columns
• •Assumes: All decision-maker(s) and experts use the same linguistic/PiFN scale

Assumptions to verify

• •Decision matrix entries are valid Picture Fuzzy Numbers (μ+η+ν ≤ 1)
• •Aggregated PiFN criterion weights provided or derivable
• •Criterion-direction inversion handled upstream (linguistic conversion) OR via PiFN complement on cost columns
• •All decision-maker(s) and experts use the same linguistic/PiFN scale

When not to use

• •Neutral-stance modelling unnecessary: use base ARAS (crisp) or IF-ARAS instead
• •Data are already crisp: picture fuzzy layer adds noise without benefit
• •Linguistic scale does not distinguish neutral from refusal: single-component fuzzy or IFS suffices

Edge cases

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

Common pitfalls

• •Hatalı: 'PIF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy Numbers (μ+η+ν ≤ 1)
• •Hatalı: 'PIF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Aggregated PiFN criterion weights provided or derivable
• •Hatalı: 'PIF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion-direction inversion handled upstream (linguistic conversion) OR via PiFN complement on cost columns
• •Hatalı: 'PIF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/PiFN scale
• •Hatalı: PIF-ARAS'yi 'Neutral-stance modelling unnecessary' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-ARAS'yi 'Data are already crisp' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-ARAS'yi 'Linguistic scale does not distinguish neutral from refusal' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct the Picture fuzzy decision matrix X̃ = [x̃_ij]_{m×n} where each entry x̃_ij = ⟨μ_ij, η_ij, ν_ij⟩ is a PiFN satisfying μ_ij+η_ij+ν_ij ≤ 1, with refusal π_ij = 1−μ_ij−η_ij−ν_ij. Criterion-direction is handled at the linguistic-conversion step prior to PiFN encoding (per Chowdhury 2025 §4.1 Table 6: cost criteria are mapped so that lower raw values receive higher μ); the resulting PiFN matrix is therefore treated as all-benefit at the algorithm-internal level. Formül: x̃_ij = ⟨μ_ij, η_ij, ν_ij⟩; μ_ij, η_ij, ν_ij ∈ [0,1]; μ_ij+η_ij+ν_ij ≤ 1; π_ij = 1−μ_ij−η_ij−ν_ij (Cuong 2013 PiFS Def 1; Chowdhury 2025 §3.2 Eq.(1)-(2)) Anchor: Chowdhury 2025 §3.2 Eq.(1)-(2); Cuong 2013
2. 2.Adım 2 (F2): Step 2: Picture fuzzy criterion weights v̄_j = ⟨μ_j, η_j, ν_j⟩ are either supplied externally (DM-elicited and aggregated across stakeholders per Chowdhury 2025 §4.1 Tables 4-5) or derived from a Picture fuzzy weighting method. The corresponding crisp weight is given by Eq.(16): w_j = [μ_j + η_j/2 + (π_j/2)(1+μ_j-ν_j)] / Σ_k [μ_k + η_k/2 + (π_k/2)(1+μ_k-ν_k)]. PiF-ARAS proper uses v̄_j (PiFN): the crisp w_j is informational only. Formül: v̄_j = ⟨μ_j, η_j, ν_j⟩ with μ_j+η_j+ν_j ≤ 1. Crisp defuzzification (Eq.(16)): w_j = [μ_j + η_j/2 + (π_j/2)(1+μ_j-ν_j)] / Σ_k [μ_k + η_k/2 + (π_k/2)(1+μ_k-ν_k)]; informational. (Chowdhury 2025 §3.3 Eq.(15)-(16)) Anchor: Chowdhury 2025 §3.3 Eq.(15)-(16)
3. 3.Adım 3 (F3): Step 3: Element-wise PiFN weighting: compute the weighted Picture fuzzy decision matrix Q̃ = [q̃_ij] via q̃_ij = v̄_j ⊗ x̃_ij using the PiFN product operator (Cuong 2013 Eq.(4)). NOTE: Chowdhury 2025 §3.4 text refers to 'multiplication by the crisp criteria weights', but reproduction of Table 11 requires the PiFN ⊗ Eq.(4) operator with the aggregated PiFN weight v̄_j: see P.implementation_review.literature_disambiguation.weighting_operator. The weighted matrix feeds Steps F4-F7. Formül: q̃_ij = v̄_j ⊗ x̃_ij = ⟨μ_{v_j}·μ_{x_ij}, η_{v_j}+η_{x_ij}-η_{v_j}·η_{x_ij}, ν_{v_j}+ν_{x_ij}-ν_{v_j}·ν_{x_ij}⟩ (Cuong 2013 PiFN product; Chowdhury 2025 §3.2 Eq.(4)) Anchor: Chowdhury 2025 §3.2 Eq.(4); §3.4 pre-text (CODAS); §3.5 pre-text (ARAS)
4. 4.Adım 4 (F4): Step 4: Determine the ideal alternative Ã_0 by extracting per-criterion synthetic best PiFN from the weighted matrix: g̃_0j = ⟨max_i μ_q_ij, min_i η_q_ij, min_i ν_q_ij⟩. The ideal is synthetic: its components need not all belong to the same actual alternative. Formül: Ã_0 = {g̃_01, g̃_02, ..., g̃_0n}; g̃_0j = ⟨max_i μ_q_ij, min_i η_q_ij, min_i ν_q_ij⟩ (Chowdhury 2025 §3.5 Step 1 Eq.(29)) Anchor: Chowdhury 2025 §3.5 Step 1 Eq.(29)
5. 5.Adım 5 (F5): Step 5: Compute the Picture fuzzy optimality function S̃_i for each alternative (and for Ã_0) by aggregating per-criterion weighted PiFN cells via the PiFN sum operator. μ aggregation is via the complement product (1-Π(1-μ)); η aggregation is the simple product Π η; ν aggregation is Π(η+ν) − Π η, ensuring the result remains a valid PiFN. Formül: S̃_i = ⟨μ_{S_i}, η_{S_i}, ν_{S_i}⟩ = Σ⊕_{j=1}^n q̃_ij = ⟨1 − Π_{j=1}^n (1−μ_{q_ij}), Π_{j=1}^n η_{q_ij}, Π_{j=1}^n (η_{q_ij}+ν_{q_ij}) − Π_{j=1}^n η_{q_ij}⟩ (Chowdhury 2025 §3.5 Step 2 Eq.(30)) Anchor: Chowdhury 2025 §3.5 Step 2 Eq.(30)
6. 6.Adım 6 (F6): Step 6: Defuzzify the Picture fuzzy optimality function S̃_i to crisp D_i using the Picture defuzzification operator (Eq.(31)): D_i = μ_{S_i} + η_{S_i}/2 + (π_{S_i}/2)(1+μ_{S_i}−ν_{S_i}), where π_{S_i} = 1 − μ_{S_i} − η_{S_i} − ν_{S_i}. The same operator is applied to Ã_0 to obtain D_0. Formül: D_i = μ_{S_i} + η_{S_i}/2 + (π_{S_i}/2)(1+μ_{S_i}−ν_{S_i}); π_{S_i} = 1−μ_{S_i}−η_{S_i}−ν_{S_i} (Chowdhury 2025 §3.5 Step 3 Eq.(31); cf. Eq.(13) for general PiFN defuzzification with η/2 term) Anchor: Chowdhury 2025 §3.5 Step 3 Eq.(31); Eq.(13)
7. 7.Adım 7 (F7): Step 7: Compute the utility degree B_i = D_i / D_0 for each alternative (Eq.(32)). Rank in descending order of B_i: the alternative with the largest utility degree is the most desirable. By construction B_i ∈ [0, 1] with B_0 = 1 (the ideal). The ideal alternative itself is generally NOT among the ranked alternatives; it serves only as the denominator. Formül: B_i = D_i / D_0; ranking = argsort_desc(B); best = ranking[0]. (Chowdhury 2025 §3.5 Steps 4-5 Eq.(32)) Anchor: Chowdhury 2025 §3.5 Steps 4-5 Eq.(32)

Commonly paired with

• •PIF-CIMAS + PIF-ARAS (common)
• •PIF-SWARA + PIF-ARAS (occasional)
• •CRITIC + PIF-ARAS (occasional (via crisp-w defuzzification Eq.(16)))

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