PF-TODIM: Pythagorean extension of TODIM

Overview

Pythagorean outranking/ranking: Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Pythagorean outranking/ranking: Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1)
• •Preserves pythagorean uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Assumes: Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

Assumptions to verify

• •Decision matrix entries are valid Pythagorean 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

When not to use

• •Crisp data sufficient: use base TODIM directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •tie-breaking by h(·). The magnitude uses PFN Euclidean distance d(l_ij,l_tj) per Eq.(3.2). Gain branch: φ_j = √(w_jr · d(l_ij,l_tj) / Σ_j w_jr). Loss branch (with attenuation θ): φ_j = −(1/θ) · √((Σ_j

Common pitfalls

• •Hatalı: 'PF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
• •Hatalı: 'PF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PF-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PF-TODIM'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Identify the Pythagorean fuzzy decision matrix R = (r_ij)_{m×n}, r_ij = P(μ_ij, ν_ij), with μ_ij²+ν_ij²≤1 (Ren 2016 Eq.3.3); supply criterion weight vector w = (w_1,...,w_n)ᵀ with Σw_j=1, and prospect-theory attenuation factor θ>0 (typical θ∈[1,5]; θ=2.25 Tversky-Kahneman default). Formül: R = (r_ij)_{m×n}, r_ij = P(μ_ij, ν_ij), μ_ij²+ν_ij²≤1; w_j∈[0,1], Σ_{j=1}^n w_j = 1; θ > 0 Anchor: Ren 2016 §3.3 Step 1, Eq.(3.3)
2. 2.Adım 2 (F2): Step 2: Transform R into normalized PFN matrix L=(l_ij)_{m×n} by applying PFN complement to cost criteria (Ren 2016 Eq.3.4): l_ij = r_ij for benefit criterion C_j; l_ij = (r_ij)^c = P(ν_ij, μ_ij) for cost criterion C_j. Formül: l_ij = r_ij if C_j is benefit l_ij = (r_ij)^c = P(ν_ij, μ_ij) if C_j is cost Anchor: Ren 2016 §3.3 Step 2, Eq.(3.4)
3. 3.Adım 3 (F3): Step 3: Compute relative weights w_jr = w_j / w_r where w_r = max_j{w_j} is the largest criterion weight; the reference criterion has w_jr=1 (Ren 2016 Eq.3.5). Formül: w_jr = w_j / w_r, w_r = max_{j∈{1..n}} w_j, w_jr ∈ (0,1] Anchor: Ren 2016 §3.3 Step 3, Eq.(3.5)
4. 4.Adım 4 (F4): Step 4: Compute pairwise prospect-theoretic dominance φ_j(A_i, A_t) of alternative A_i over A_t with respect to criterion C_j (Ren 2016 Eq.3.6). PFN comparison l_ij vs l_tj uses score-then-accuracy rule (Def 3.2): l_ij>l_tj ⇔ s(l_ij)>s(l_tj), or tie-breaking by h(·). The magnitude uses PFN Euclidean distance d(l_ij,l_tj) per Eq.(3.2). Gain branch: φ_j = √(w_jr · d(l_ij,l_tj) / Σ_j w_jr). Loss branch (with attenuation θ): φ_j = −(1/θ) · √((Σ_j w_jr) · d(l_ij,l_tj) / w_jr). Equal branch: φ_j = 0. Formül: d(β_1, β_2) = √( ½ · [ (μ_1² − μ_2²)² + (ν_1² − ν_2²)² + (π_1² − π_2²)² ] ) [Eq.3.2] φ_j(A_i, A_t) = { + √( w_jr · d(l_ij, l_tj) / Σ_{j=1}^n w_jr ), if l_ij > l_tj (gain) 0, if l_ij = l_tj − (1/θ) · √( (Σ_{j=1}^n w_jr) · d(l_ij, l_tj) / w_jr ), if l_ij < l_tj (loss) } [Eq.3.6] PFN comparison rule (Def 3.2): l_ij > l_tj ⇔ s(l_ij) > s(l_tj), with score s(β)=μ²−ν² [Eq.2.6] and accuracy h(β)=μ²+ν² [Eq.3.1] as tie-breaker. Anchor: Ren 2016 §3.3 Step 4, Eq.(3.2), Eq.(3.6), Def 3.2
5. 5.Adım 5 (F5): Step 5: Compute overall dominance δ(A_i, A_t) of A_i over A_t by summing per-criterion dominances (Ren 2016 Eq.3.8): δ(A_i, A_t) = Σ_{j=1}^n φ_j(A_i, A_t). Yields m×m dominance matrix δ. Formül: δ(A_i, A_t) = Σ_{j=1}^n φ_j(A_i, A_t), for i,t ∈ {1,...,m} Anchor: Ren 2016 §3.3 Step 5, Eq.(3.8)
6. 6.Adım 6 (F6): Step 6: Compute overall prospect value ξ_i of each alternative A_i by max-min normalization of row-sum overall dominance (Ren 2016 Eq.3.10): ξ_i = (Σ_t δ(A_i,A_t) − min_i{Σ_t δ(A_i,A_t)}) / (max_i{Σ_t δ(A_i,A_t)} − min_i{Σ_t δ(A_i,A_t)}). Each ξ_i ∈ [0,1]; best alternative attains ξ=1, worst ξ=0. Formül: ξ_i = ( Σ_{t=1}^m δ(A_i, A_t) − min_{i} { Σ_{t=1}^m δ(A_i, A_t) } ) / ( max_{i} { Σ_{t=1}^m δ(A_i, A_t) } − min_{i} { Σ_{t=1}^m δ(A_i, A_t) } ), i = 1,...,m Anchor: Ren 2016 §3.3 Step 6, Eq.(3.10)
7. 7.Adım 7 (F7): Step 7: Rank alternatives in descending order of ξ_i. The greater ξ_i, the better A_i (Ren 2016 §3.3, sentence after Eq.3.10). Formül: Ranking: sort alternatives by descending ξ_i; A_{(1)} > A_{(2)} > ... > A_{(m)} Anchor: Ren 2016 §3.3 Step 7

Commonly paired with

• •n_a + PF-TODIM (common)

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