PiF-WASPAS: Picture Fuzzy extension of WASPAS

Overview

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

Strengths

• •Method-specific: Picture WSM-WPM hybrid 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 WASPAS base; cf. Belton-Gear 1983, Wang-Luo 2009; Zavadskas et al. 2012)
• •Assumes: Decision matrix entries are valid PiFNs (μ+η+ν ≤ 1)
• •Assumes: Both PiFN weights v̄_j and crisp w_j provided/derivable
• •Assumes: Criterion-direction inversion handled upstream OR via PiFN complement
• •Assumes: Same linguistic/PiFN scale across all decision-maker(s)

Assumptions to verify

• •Decision matrix entries are valid PiFNs (μ+η+ν ≤ 1)
• •Both PiFN weights v̄_j and crisp w_j provided/derivable
• •Criterion-direction inversion handled upstream OR via PiFN complement
• •Same linguistic/PiFN scale across all decision-maker(s)

When not to use

• •Single-aggregation behaviour wanted (use PiF-SAW for sum-only or PiF-ARAS for ratio)
• •Data are crisp: base WASPAS suffices
• •Neutral-stance modelling unnecessary: IF-WASPAS or fuzzy-WASPAS lighter alternative

Edge cases

• •default in Zavadskas 2012 crisp WASPAS).

Common pitfalls

• •Hatalı: 'PIF-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid PiFNs (μ+η+ν ≤ 1)
• •Hatalı: 'PIF-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Both PiFN weights v̄_j and crisp w_j provided/derivable
• •Hatalı: 'PIF-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion-direction inversion handled upstream OR via PiFN complement
• •Hatalı: 'PIF-WASPAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Same linguistic/PiFN scale across all decision-maker(s)
• •Hatalı: PIF-WASPAS'yi 'Single-aggregation behaviour wanted (use PiF-SAW for sum-only or PiF-ARAS for ratio)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-WASPAS'yi 'Data are crisp' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-WASPAS'yi 'Neutral-stance modelling unnecessary' 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 ∈ [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 supplied externally (DM-elicited, aggregated per Chowdhury 2025 §4.1 Tables 4-5) or derived from a Picture fuzzy weighting method. The corresponding crisp weight via Eq.(16): w_j = [μ_j + η_j/2 + (π_j/2)(1+μ_j-ν_j)] / Σ_k [...]. PiF-WASPAS uses BOTH: v̄_j (PiFN) in Q^(1) WSM (Eq.17), and w_j (crisp) in Q^(2) WPM (Eq.18) as PiFN-power exponents. Formül: v̄_j = ⟨μ_j, η_j, ν_j⟩, μ_j+η_j+ν_j ≤ 1. Crisp: 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)] (Chowdhury 2025 Eq.(15)-(16)) Anchor: Chowdhury 2025 §3.3 Eq.(15)-(16)
3. 3.Adım 3 (F3): Step 3: Compute Q^(1) WSM component per Eq.(17): Q^(1)_i = Σ⊕_{j=1}^n (v̄_j ⊗ x̃_ij). Each weighted cell v̄_j ⊗ x̃_ij is a PiFN computed via Cuong 2013 Eq.(4) PiFN product; cells are aggregated across criteria via Cuong 2013 Eq.(3) PiFN sum. NOTE: The Σ⊕ aggregation here is the same one as in PiF-ARAS Eq.(30): μ_Q1 = 1−Π(1−μ), η_Q1 = Π η, ν_Q1 = Π(η+ν)−Π η. Reproduction-verified: Chowdhury 2025 Table 9 WSM column reproduces EXACTLY (16/16) under this implementation. Formül: Q^(1)_i = Σ⊕_{j=1}^n (v̄_j ⊗ x̃_ij) = ⟨1 − Π_j (1 − μ_v_j·μ_x_ij), Π_j (η_v_j+η_x_ij−η_v_j·η_x_ij), Π_j ((η_v_j+η_x_ij−η_v_j·η_x_ij)+(ν_v_j+ν_x_ij−ν_v_j·ν_x_ij)) − Π_j (η_v_j+η_x_ij−η_v_j·η_x_ij)⟩ (Chowdhury 2025 §3.3 Eq.(17); Cuong 2013 Eq.(3)-(4)) Anchor: Chowdhury 2025 §3.3 Eq.(17)
4. 4.Adım 4 (F4): Step 4: Compute Q^(2) WPM component per Eq.(18): Q^(2)_i = Π⊗_{j=1}^n (x̃_ij)^{w_j}. Each per-cell PiFN-power (x̃_ij)^{w_j} via Cuong 2013 Eq.(6): A^λ = ⟨μ^λ, 1−(1−η)^λ, 1−(1−ν)^λ⟩; powered cells aggregated across criteria via Cuong 2013 Eq.(4) PiFN product. CRITICAL NOTE: Chowdhury 2025 Table 9 WPM column has computational errors: reported T1 = ⟨0.690, 0.174, 0.170⟩ has sum=1.034>1 (violates PiFN constraint). Canonical Eq.(18) gives T1 = ⟨0.348, 0.174, 0.170⟩ (sum=0.692, valid PiFN). Verified: η AND ν columns of canonical match Table 9 EXACTLY for all 16 rows; μ column differs because paper's μ values were apparently miscomputed (then propagated into Table 10 F_i and ranking). Manifest implements canonical Eq.(18) per the published operational laws. Formül: Q^(2)_i = Π⊗_{j=1}^n (x̃_ij)^{w_j} where (x̃_ij)^{w_j} = ⟨μ_x_ij^{w_j}, 1−(1−η_x_ij)^{w_j}, 1−(1−ν_x_ij)^{w_j}⟩ (Eq.(6)); aggregation = Π via PiFN ⊗ Eq.(4): μ_Q2 = Π_j μ_x_ij^{w_j}, η_Q2 = combined via η ⊗ recursion, ν_Q2 likewise (Chowdhury 2025 §3.3 Eq.(18); Cuong 2013 Eq.(4),(6)) Anchor: Chowdhury 2025 §3.3 Eq.(18)
5. 5.Adım 5 (F5): Step 5: Combined WASPAS PiFN score Q̃_i per Eq.(19): Q̃_i = 0.5·Q^(1)_i ⊕ 0.5·Q^(2)_i. Implementation uses Cuong 2013 Eq.(5) scalar multiplication (λA = ⟨1−(1−μ)^λ, η^λ, ν^λ⟩) with λ=0.5 to halve each component, then PiFN sum Eq.(3) ⊕ to combine. The convex combination weight λ=0.5 is the Chowdhury 2025 choice (matches default in Zavadskas 2012 crisp WASPAS). Formül: Q̃_i = 0.5·Q^(1)_i ⊕ 0.5·Q^(2)_i; 0.5·A = ⟨1−(1−μ_A)^{0.5}, η_A^{0.5}, ν_A^{0.5}⟩ (Eq.(5)); ⊕ per Eq.(3) (Chowdhury 2025 §3.3 Eq.(19)) Anchor: Chowdhury 2025 §3.3 Eq.(19)
6. 6.Adım 6 (F6): Step 6: Defuzzify each combined Q̃_i to crisp F_i via Eq.(20): F_i = (μ_{Q̃_i} + (1 − η_{Q̃_i}) + (1 − ν_{Q̃_i})) / 3. F_i ∈ [0, 1]; higher is better. Rank alternatives in descending order of F_i. Formül: F_i = (μ_{Q̃_i} + (1 − η_{Q̃_i}) + (1 − ν_{Q̃_i})) / 3; ranking = argsort_desc(F); best = ranking[0] (Chowdhury 2025 §3.3 Eq.(20)) Anchor: Chowdhury 2025 §3.3 Eq.(20)

Commonly paired with

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

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