PiF-CODAS: Picture extension of CODAS

Overview

Picture distance-based ranking: Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Picture distance-based 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

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

Edge cases

• •default 0.05 per §4.1) controls whether the Taxicab tie-breaker activates: ψ(x)=1 if |x|≥τ, else 0. Equivalent to: when two alternatives are Euclidean-close, defer to Taxicab; otherwise rely on Euclid

Common pitfalls

• •Hatalı: 'PIF-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy numbers/tuples
• •Hatalı: 'PIF-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PIF-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PIF-CODAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-CODAS'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 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 (Chowdhury 2025 §4.1 Table 6: cost criteria mapped so 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 ≤ 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: Build the weighted Picture fuzzy decision matrix q̃_ij = v̄_j ⊗ x̃_ij via Cuong 2013 Eq.(4) PiFN product: q̃_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⟩. This is the same weighted matrix used by Chowdhury 2025 PiF-ARAS Table 11 and PiF-WASPAS pre-step. 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 Eq.(4)) Anchor: Chowdhury 2025 §3.4 Eq.(21); Cuong 2013 Eq.(4)
3. 3.Adım 3 (F3): Step 3: Identify the Picture fuzzy negative-ideal solution per criterion. CANONICAL CHOICE (this manifest): ñs_j = ⟨min_i μ_{q̃_ij}, max_i η_{q̃_ij}, max_i ν_{q̃_ij}⟩ (worst across alternatives on each PiFN component for a benefit criterion: lowest positive, highest neutrality/ambivalence, highest negative). This convention reproduces Chowdhury 2025 §4.1 reported T1#1 and T10#16. ALTERNATIVE: Simic 2021 Eq.(27) explicit form: ñs_j = ⟨min μ, min η, max ν⟩; documented in L.alternative_formulations_known. Formül: ñs_j = ⟨min_i μ_{q̃_ij}, max_i η_{q̃_ij}, max_i ν_{q̃_ij}⟩ (Chowdhury 2025 §3.4 Eq.(22), canonical reading; cf. Simic 2021 Eq.(27) variant in L.alternative_formulations_known) Anchor: Chowdhury 2025 §3.4 Eq.(21)-(22); cf. Simic 2021 Eq.(27)
4. 4.Adım 4 (F4): Step 4: For each alternative i, compute (i) Euclidean PiFN distance to negative-ideal E_i and (ii) Taxicab PiFN distance T_i, summed component-wise across criteria and the three PiFN components (μ, η, ν). Both are non-negative; larger values indicate stronger dominance over the worst case (more desirable). Formül: E_i = √( Σ_{j=1}^n [(μ_{q̃_ij} − μ_{ñs_j})² + (η_{q̃_ij} − η_{ñs_j})² + (ν_{q̃_ij} − ν_{ñs_j})²] ) (Eq.(23)); T_i = Σ_{j=1}^n [|μ_{q̃_ij} − μ_{ñs_j}| + |η_{q̃_ij} − η_{ñs_j}| + |ν_{q̃_ij} − ν_{ñs_j}|] (Eq.(24)) Anchor: Chowdhury 2025 §3.4 Eq.(23)-(24); Keshavarz Ghorabaee 2016 (crisp CODAS distance pair)
5. 5.Adım 5 (F5): Step 5: Build the relative-assessment matrix h_ik with threshold indicator ψ on |E_i − E_k|; sum each row to H_i and rank alternatives by descending H_i. The threshold τ ∈ [0.01, 0.05] (paper default 0.05 per §4.1) controls whether the Taxicab tie-breaker activates: ψ(x)=1 if |x|≥τ, else 0. Equivalent to: when two alternatives are Euclidean-close, defer to Taxicab; otherwise rely on Euclidean alone. Formül: h_ik = (E_i − E_k) + ψ(E_i − E_k) · (T_i − T_k), ψ(x) = 1 if |x| ≥ τ else 0 (Eq.(26)-(27)); H_i = Σ_{k=1}^m h_ik (Eq.(28)); rank by H_i descending Anchor: Chowdhury 2025 §3.4 Eq.(25)-(28); Keshavarz Ghorabaee 2016 (crisp CODAS relative-assessment)

Commonly paired with

• •n_a + PIF-CODAS (common)

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