PiF-CODAS - Picture extension of CODAS

Overview

pif-codas extends Keshavarz Ghorabaee et al. 2016 crisp CODAS to Picture fuzzy uncertainty via Cuong 2013 PiFS. For each alternative, computes Euclidean E_i and Taxicab T_i PiFN distances to a negative-ideal reference, then combines them in a relative-assessment matrix h_ik using a threshold indicator ψ that gates the Taxicab tie-breaker. Higher H_i = Σ_k h_ik is better. The dual-distance design with τ-threshold gives partial-compensation behaviour: when alternatives are Euclidean-close (within τ), the Taxicab metric resolves the tie; otherwise Euclidean dominates.

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 under epistemic uncertainty, expert-driven evaluation with linguistic terms

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.

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

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

2. 2

   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.

   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))

   Chowdhury 2025 §3.4 Eq.(21); Cuong 2013 Eq.(4)

3. 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.

   $$\tilde{n}{s}_j=⟨mi{n}_i{\mu }_{{\tilde{q}}_{i}j},ma{x}_i{\eta }_{{\tilde{q}}_{i}j},ma{x}_i{\nu }_{{\tilde{q}}_{i}j}⟩(Chowdhury2025\S 3.4Eq.(22),canonicalreading;cf.Simic2021Eq.(27)variantinL.alternativ{e}_{f}ormulation{s}_{k}nown)$$

   Chowdhury 2025 §3.4 Eq.(21)-(22); cf. Simic 2021 Eq.(27)

4. 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).

   $$E_i=\surd ({\Sigma }_{j=1}^n[({\mu }_{{\tilde{q}}_{i}j}-{\mu }_{\tilde{n}{s}_j}{)}^2+({\eta }_{{\tilde{q}}_{i}j}-{\eta }_{\tilde{n}{s}_j}{)}^2+({\nu }_{{\tilde{q}}_{i}j}-{\nu }_{\tilde{n}{s}_j}{)}^2])(Eq.(23));T_i={\Sigma }_{j=1}^n[|{\mu }_{{\tilde{q}}_{i}j}-{\mu }_{\tilde{n}{s}_j}|+|{\eta }_{{\tilde{q}}_{i}j}-{\eta }_{\tilde{n}{s}_j}|+|{\nu }_{{\tilde{q}}_{i}j}-{\nu }_{\tilde{n}{s}_j}|](Eq.(24))$$

   Chowdhury 2025 §3.4 Eq.(23)-(24); Keshavarz Ghorabaee 2016 (crisp CODAS distance pair)

5. 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.

   $$h_{i}k=(E_i-E_k)+\psi (E_i-E_k)\cdot (T_i-T_k),\psi (x)=1if|x|\ge \tau else0(Eq.(26)-(27));H_i={\Sigma }_{k=1}^m{h}_{i}k(Eq.(28));rankby{H}_{i}descending$$

   Chowdhury 2025 §3.4 Eq.(25)-(28); Keshavarz Ghorabaee 2016 (crisp CODAS relative-assessment)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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

• 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.

• Negative-ideal convention: PiFN component-wise selection rule is NOT uniformly defined across the PiF-CODAS literature. Simic 2021 Eq.(27) writes ⟨min μ, min η, max ν⟩; this manifest uses ⟨min μ, max η, max ν⟩ because it reproduces Chowdhury 2025 §4.1 reported ranking on the LA-JECM fixture. When applying to a new domain, document the convention and run sensitivity over both.

• Threshold τ choice: paper allows τ ∈ [0.01, 0.05]. Smaller τ → ψ activates more often → Taxicab affects more pairs. Closed-form on LA-JECM shows ranking is robust to τ within this range (verified at τ=0.01, 0.02, 0.05).

• Operational law variant: Simic 2021 uses Wang 2017 / Liang 2018 PiFN operations (different ⊕, ⊗, λ-power formulas) which yield numerically different weighted matrix q̃_ij than Cuong 2013 standard. This manifest follows Cuong 2013 (consistent with Chowdhury 2025 and the manifest family PIF-ARAS, PIF-WASPAS).

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