PF-ELECTRE-II - Pythagorean fuzzy ELECTRE-II for group MCDM

Overview

PF-ELECTRE-II takes a per-DM stack of Pythagorean fuzzy decision matrices plus per-DM PFN importance and per-criterion per-DM PFN weights, aggregates them in Phase I via PFWA, and then in Phase II constructs three-tier concordance/discordance sets, the concordance/discordance indices ψ and δ, strong and weak outranking relations, and finally a forward/reverse/average distillation of the strong+weak outranking graph to produce a ranking. Smaller φ(a) is a better rank.

Output

rank position, lower is better

Data

Pythagorean Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-8 criteria works best

Used for

Pythagorean Fuzzy MAGDM, Supplier / vendor selection with multi-expert panels, Project portfolio prioritization under epistemic uncertainty, Sustainable energy planning with stakeholder groups

How it works

1. 1

   Her DM_f için PFN önemi (μ_f, ν_f, π_f) verildiğinde, normalized score-style formülle ϕ_f hesapla; Σ_f ϕ_f = 1.

2. 2

   Her hücre için PFWA_ϕ((μ^(1),ν^(1)),…,(μ^(K),ν^(K))) = (√(1-Π(1-μ^(f)²)^ϕ_f), Π ν^(f)^ϕ_f) ile aggregated PFDM Y kur.

3. 3

   Maliyet sütunları için (μ,ν)→(ν,μ); fayda sütunları olduğu gibi. Ỹ üret.

4. 4

   Her kriter için DM-bazlı PFN ağırlıkları PFWA ile aggregate et → ω_j^PFN; sonra Eq.19 normalize → skalar η_j, Σ_j η_j = 1.

5. 5

   Her hücreye Def 2 op 2 (PFN scalar-power) ile η_j uygula: y*_ij = (√(1-(1-μ̃_ij²)^η_j), ν̃_ij^η_j). Y* tamamlandı.

6. 6

   Her (a,b) için BΨ (strong: μ_a²>μ_b² ∧ ν_a²<ν_b²), BΨ' (medial), BΨ'' (weak), B= (indifferent), BΔ/BΔ'/BΔ'' (discordance ayna) kümelerini kur.

7. 7

   ψ_ab = ωBΨ·Σ_{j∈BΨ}η_j + ωBΨ'·Σ_{j∈BΨ'}η_j + ωBΨ''·Σ_{j∈BΨ''}η_j + ωB=·Σ_{j∈B=}η_j. Paper §5 ψ_15 = 1·(0.2056+0.1960+0.1960) + 0.25·0.1960 = 0.6466 - bu hesap el ile birebir doğrulanır.

8. 8

   δ_ab = max(ωBΔ·max_{j∈BΔ}d(y*_aj,y*_bj), …) / max_{j∈1..n} d(y*_aj,y*_bj); d normalized Euclidean Eq.10.

9. 9

   Strong: (ψ≥ψ* ∧ δ≤δ*) ∨ (ψ≥ψ⁰ ∧ δ≤δ⁰). Weak: (ψ≥ψ⁻ ∧ δ≤δ⁰). Paper §5: (0.6, 0.7, 0.8) ve (0.8, 0.7) eşikleri.

10. 10

    G_s ve G_w üzerinde ileri distillation φ', G_s ve G_w'nin ayna görüntüsünde geri distillation φ''; final φ = (φ' + φ'')/2; argmin sıralama.

Fits when / Look elsewhere when

Assumptions to verify

• All per-DM cells are valid PFNs (μ²+ν²≤1)
• DM importance is available as PFN linguistic ratings (or crisp ϕ override is provided)
• Criteria are independent (additive outranking assumption)
• Threshold ladders are strictly monotone (E-3, E-4)

Edge cases and pitfalls

• •Single DM (K=1): Phase I trivial, ϕ_1=1, PFWA = identity. J fixture uses this case.
• •All-benefit criteria (R_N=∅): Eq.17 noop. Akram §5 uses this. Ỹ = Y.
• •Flat threshold ladder (ψ⁻ = ψ⁰ = ψ*): two clauses of Eq.11 collapse → manifest's E-3 rejects.
• •Component-wise dominance: if μ_a² > μ_b² AND ν_a² < ν_b² on all j, BΨ(a,b) = {all criteria}, ψ_ab=1, δ_ab=0 → guaranteed strong outranking (matches synthetic J fixture).
• •Equal PFN cells: B=(a,b) absorbs them; with default ωB= = 0.25 (paper §5) they contribute ¼ to concordance - neither favorable nor adversarial.

• Value-space violation: ensure all per-DM and aggregated cells satisfy PFN: μ ∈ [0,1], ν ∈ [0,1], μ²+ν² ≤ 1 before computation; reject otherwise (E-2).

• Threshold ladder violation: enforce 0 < ψ⁻ < ψ⁰ < ψ* < 1 and 1 > δ⁰ > δ* > 0 strictly (E-3, E-4); a flat ladder collapses Eq.11/12 to a single condition and breaks the three-tier semantics.

• Set-weight degeneracy: if (ωBΨ', ωBΨ'', ωB=) and (ωBΔ', ωBΔ'') are all 1.0, Eq.8/Eq.9 collapse to the IFS-ELECTRE-II Devadoss-Rekha 2017 form; paper §5 uses (1, 2/3, 1/3, 1/4) and (1, 3/4, 2/4) to obtain a graded contribution. Document the choice in any application.

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