PiF-MOORA - Picture extension of MOORA

Overview

pif-moora extends MOORA to handle Picture uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1) algebra. The final scores are defuzzified via score function S = μ − ν before ranking.

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 picture fuzzy decision matrix D = (α_ij)_{m×n} where α_ij = ⟨μ_ij, η_ij, ν_ij⟩ with μ_ij + η_ij + ν_ij ≤ 1.

   $$D=({\alpha }_{i}j{)}_{m\times n};{\alpha }_{i}j=⟨{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j⟩;{\mu }_{i}j+{\eta }_{i}j+{\nu }_{i}j\le 1$$

   Cuong 2013 Def.1; Tian et al. 2022 Def.1

2. 2

   Per alternative i, aggregate PFNs across BENEFIT criteria (j ∈ J⁺) using PFWA with original weights w_j (no re-normalization). Yields B_i = ⟨μ_B, η_B, ν_B⟩.

   $$B_i=⟨1-\prod _{j\in J^{+}}(1-{\mu }_{i}j{)}^{w_j},\prod _{j\in J^{+}}({\eta }_{i}j{)}^{w_j},\prod _{j\in J^{+}}({\nu }_{i}j{)}^{w_j}⟩$$

   Wei 2017 PFWA; Brauers-Zavadskas 2006 Eq.4 benefit term

3. 3

   Per alternative i, aggregate PFNs across COST criteria (j ∈ J⁻) using PFWA with original weights w_j. Yields C_i = ⟨μ_C, η_C, ν_C⟩.

   $$C_i=⟨1-\prod _{j\in J^{-}}(1-{\mu }_{i}j{)}^{w_j},\prod _{j\in J^{-}}({\eta }_{i}j{)}^{w_j},\prod _{j\in J^{-}}({\nu }_{i}j{)}^{w_j}⟩$$

   Wei 2017 PFWA; Brauers-Zavadskas 2006 Eq.4 cost term

4. 4

   Defuzzify B_i and C_i via Tian-Peng 2020 score function S(p) = (1/3)(δ + 1 - ξ + 1 - γ).

   $$s_B(i)=S(B_i),s_C(i)=S(C_i);S(p)=(1/3)(\delta +1-\xi +1-\gamma )$$

   Tian-Peng 2020 (cited in Tian et al. 2022 Def.3)

5. 5

   MOORA ratio system: y_i = s_B(i) - s_C(i). Larger y_i ⇒ better alternative.

   $$y_i=s_B(i)-s_C(i)$$

   Brauers-Zavadskas 2006 Eq.4 (RS); Tian et al. 2022 Eq.2 (PF analog)

6. 6

   Rank alternatives in descending order of y_i.

   $$ranking=argsor{t}_{d}esc(y_i)$$

   Brauers-Zavadskas 2006

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

• Value-space violation: ensure all entries satisfy PiFS: μ (membership), η (neutral), ν (non-membership); μ+η+ν ≤ 1 before computation.

• Defuzzification method affects ranking: score function S = μ − ν is the canonical choice but alternatives exist.

• Bu manifestin B.extensions bloğunda kaynak gösterilen Wei (2017), 'Picture fuzzy aggregation operators...' (DOI 10.3233/JIFS-161798) GERİ ÇEKİLMİŞTİR (RETRACTED); bu kaynağı PFWA/agregasyon gerekçesi olarak KULLANMAYIN.

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