PiF-WASPAS - Picture Fuzzy extension of WASPAS

Overview

pif-waspas extends Zavadskas et al. 2012 crisp WASPAS to Picture fuzzy uncertainty via Cuong 2013 PiFS. Computes BOTH a WSM component (Q^(1) via PiFN-weighted sum of cells) AND a WPM component (Q^(2) via PiFN-power product of cells using crisp criteria weights as exponents), then combines them via PiFN convex combination Q̃ = 0.5·Q^(1) ⊕ 0.5·Q^(2). Final score F_i = (μ + (1-η) + (1-ν))/3 ∈ [0,1]. Higher F_i is better. The dual WSM+WPM design provides compromise between full-compensation (sum) and limited-compensation (product) aggregation styles.

Output

utility, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Picture Fuzzy MCDM, MAGDM with dual-aggregation (WSM+WPM) compromise behaviour, manufacturing process optimization under expert linguistic ratings

How it works

1. 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.

   $${\tilde{x}}_{i}j=⟨{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j⟩;{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j\in [0,1];{\mu }_{i}j+{\eta }_{i}j+{\nu }_{i}j\le 1;{\pi }_{i}j=1-{\mu }_{i}j-{\eta }_{i}j-{\nu }_{i}j(Cuong2013PiFSDef1;Chowdhury2025\S 3.2Eq.(1)-(2))$$

   Chowdhury 2025 §3.2 Eq.(1)-(2); Cuong 2013

2. 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.

   $${\bar{v}}_j=⟨{\mu }_j,{\eta }_j,{\nu }_j⟩,{\mu }_j+{\eta }_j+{\nu }_j\le 1.Crisp:w_j=[{\mu }_j+{\eta }_j/2+({\pi }_j/2)(1+{\mu }_j-{\nu }_j)]/{\Sigma }_k[{\mu }_k+{\eta }_k/2+({\pi }_k/2)(1+{\mu }_k-{\nu }_k)](Chowdhury2025Eq.(15)-(16))$$

   Chowdhury 2025 §3.3 Eq.(15)-(16)

3. 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.

   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))

   Chowdhury 2025 §3.3 Eq.(17)

4. 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.

   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))

   Chowdhury 2025 §3.3 Eq.(18)

5. 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).

   $${\tilde{Q}}_i=0.5\cdot Q^{(}1{)}_i\oplus 0.5\cdot Q^{(}2{)}_i;0.5\cdot A=⟨1-(1-{\mu }_A{)}^{0.5},{\eta }_A^{0.5},{\nu }_A^{0.5}⟩(Eq.(5));\oplus perEq.(3)(Chowdhury2025\S 3.3Eq.(19))$$

   Chowdhury 2025 §3.3 Eq.(19)

6. 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.

   $$F_i=({\mu }_{{\tilde{Q}}_i}+(1-{\eta }_{{\tilde{Q}}_i})+(1-{\nu }_{{\tilde{Q}}_i}))/3;ranking=argsor{t}_{d}esc(F);best=ranking[0](Chowdhury2025\S 3.3Eq.(20))$$

   Chowdhury 2025 §3.3 Eq.(20)

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •default in Zavadskas 2012 crisp WASPAS).

• Value-space violation: ensure all entries satisfy PiFN μ+η+ν ≤ 1 BEFORE computation. PFN (Pythagorean, μ²+ν²≤1) and IFS (μ+ν≤1, no η) inputs are NOT valid PiFN.

• Dual weighting: PiF-WASPAS uses BOTH PiFN weights v̄_j (in WSM Eq.(17) as ⊗ operand) AND crisp weights w_j (in WPM Eq.(18) as PiFN-power exponent). Supplying only one will fail; derive the other via Eq.(16) defuzzification.

• Direction handling: criterion direction is encoded at linguistic-conversion stage (Chowdhury 2025 Table 6). For raw PiFN with criteria_types='min', apply PiFN complement Ã^c = ⟨ν, η, μ⟩ to cost columns BEFORE F3.

• Paper Table 9 WPM μ column unreliable: Chowdhury 2025 Table 9 WPM μ values were apparently miscomputed (e.g., T1 reports ⟨0.690, 0.174, 0.170⟩ with sum 1.034 > 1, violating PiFN). The η and ν columns of paper Table 9 WPM are correct. Manifest implements canonical Eq.(18) per Cuong 2013 Eq.(6)+Eq.(4) and uses canonical-computed F_i and ranking - first/last alternatives match paper, middle positions diverge.

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