PiF-ARAS - Picture Fuzzy extension of ARAS

Overview

pif-aras extends Zavadskas-Turskis 2010 ARAS to Picture fuzzy uncertainty via Cuong 2013 PiFS. Each criterion contributes via element-wise PiFN multiplication with the aggregated PiFN weight (Cuong 2013 ⊗ operator). The ideal alternative Ã_0 is synthetically composed from per-criterion ⟨max μ, min η, min ν⟩ of the weighted matrix. Alternatives are scored by the ratio of their defuzzified optimality D_i to the ideal D_0 (utility degree B_i = D_i/D_0). Higher B_i is better.

Output

utility, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-8 criteria works best

Used for

Picture Fuzzy MCDM, MAGDM with neutral/refusal stance modelling, 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 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.

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

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

2. 2

   Picture fuzzy criterion weights v̄_j = ⟨μ_j, η_j, ν_j⟩ are 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.

   $${\bar{v}}_j=⟨{\mu }_j,{\eta }_j,{\nu }_j⟩with{\mu }_j+{\eta }_j+{\nu }_j\le 1.Crispdefuzzification(Eq.(16)):w_j=[{\mu }_j+{\eta }_j/2+({\pi }_j/2)(1+{\mu }_j-{\nu }_j)]/{\Sigma }_k[{\mu }_k+{\eta }_k/2+({\pi }_k/2)(1+{\mu }_k-{\nu }_k)];informational.(Chowdhury2025\S 3.3Eq.(15)-(16))$$

   Chowdhury 2025 §3.3 Eq.(15)-(16)

3. 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.

   $${\tilde{q}}_{i}j={\bar{v}}_j\otimes {\tilde{x}}_{i}j=⟨{\mu }_{v_j}\cdot {\mu }_{x_{i}j},{\eta }_{v_j}+{\eta }_{x_{i}j}-{\eta }_{v_j}\cdot {\eta }_{x_{i}j},{\nu }_{v_j}+{\nu }_{x_{i}j}-{\nu }_{v_j}\cdot {\nu }_{x_{i}j}⟩(Cuong2013PiFNproduct;Chowdhury2025\S 3.2Eq.(4))$$

   Chowdhury 2025 §3.2 Eq.(4); §3.4 pre-text (CODAS); §3.5 pre-text (ARAS)

4. 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.

   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))

   Chowdhury 2025 §3.5 Step 1 Eq.(29)

5. 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.

   $${\tilde{S}}_i=⟨{\mu }_{S_i},{\eta }_{S_i},{\nu }_{S_i}⟩=\Sigma {\oplus }_{j=1}^n{\tilde{q}}_{i}j=⟨1-{\Pi }_{j=1}^n(1-{\mu }_{q_{i}j}),{\Pi }_{j=1}^n{\eta }_{q_{i}j},{\Pi }_{j=1}^n({\eta }_{q_{i}j}+{\nu }_{q_{i}j})-{\Pi }_{j=1}^n{\eta }_{q_{i}j}⟩(Chowdhury2025\S 3.5Step2Eq.(30))$$

   Chowdhury 2025 §3.5 Step 2 Eq.(30)

6. 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.

   $$D_i={\mu }_{S_i}+{\eta }_{S_i}/2+({\pi }_{S_i}/2)(1+{\mu }_{S_i}-{\nu }_{S_i});{\pi }_{S_i}=1-{\mu }_{S_i}-{\eta }_{S_i}-{\nu }_{S_i}(Chowdhury2025\S 3.5Step3Eq.(31);cf.Eq.(13)forgeneralPiFNdefuzzificationwith\eta /2term)$$

   Chowdhury 2025 §3.5 Step 3 Eq.(31); Eq.(13)

7. 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.

   $$B_i=D_i/D_0;ranking=argsor{t}_{d}esc(B);best=ranking[0].(Chowdhury2025\S 3.5Steps4-5Eq.(32))$$

   Chowdhury 2025 §3.5 Steps 4-5 Eq.(32)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• 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.

• Weighting-operator confusion: Chowdhury 2025 §3.4/§3.5 text refers to 'crisp weights' but the reported tables (Table 11) are reproducible only via PiFN ⊗ Eq.(4) with the aggregated PiFN weight v̄_j. The manifest follows the tables (PiFN ⊗) as canonical.

• Direction handling: Chowdhury 2025 encodes criterion direction at the linguistic-conversion stage (Table 6: lower TAP/SR → higher μ). Users supplying a raw PiFN matrix with criteria_types='min' should apply PiFN complement Ã^c = ⟨ν, η, μ⟩ to those columns before Step F3. The J fixture has all directions pre-encoded.

Works with

How to cite