PF-TODIM - Pythagorean extension of TODIM

Overview

pf-todim extends TODIM to handle Pythagorean uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Output

utility, higher is better

Data

Pythagorean Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Pythagorean Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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).

   $$R=(r_{i}j{)}_{m\times n},r_{i}j=P({\mu }_{i}j,{\nu }_{i}j),{\mu }_i{j}^2+{\nu }_i{j}^2\le 1;w_j\in [0,1],{\Sigma }_{j=1}^n{w}_j=1;\theta >0$$

   Ren 2016 §3.3 Step 1, Eq.(3.3)

2. 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. l_ij = (r_ij)^c = P(ν_ij, μ_ij) if C_j is cost

   $$l_{i}j=r_{i}jif{C}_{j}isbenefit$$

   Ren 2016 §3.3 Step 2, Eq.(3.4)

3. 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).

   $$w_{j}r=w_j/w_r,w_r=ma{x}_{j\in \mathrm{1..}n}{w}_j,w_{j}r\in (0,1]$$

   Ren 2016 §3.3 Step 3, Eq.(3.5)

4. 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. 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.

   $$d({\beta }_1,{\beta }_2)=\surd (½\cdot [({\mu }_1^2-{\mu }_2^2{)}^2+({\nu }_1^2-{\nu }_2^2{)}^2+({\pi }_1^2-{\pi }_2^2{)}^2])[Eq.3.2]$$

   Ren 2016 §3.3 Step 4, Eq.(3.2), Eq.(3.6), Def 3.2

5. 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.

   $$\delta (A_i,A_t)={\Sigma }_{j=1}^n{\phi }_j(A_i,A_t),fori,t\in 1,...,m$$

   Ren 2016 §3.3 Step 5, Eq.(3.8)

6. 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.

   $${\xi }_i=({\Sigma }_{t=1}^m\delta (A_i,A_t)-mi{n}_i{\Sigma }_{t=1}^m\delta (A_i,A_t))/(ma{x}_i{\Sigma }_{t=1}^m\delta (A_i,A_t)-mi{n}_i{\Sigma }_{t=1}^m\delta (A_i,A_t)),i=1,...,m$$

   Ren 2016 §3.3 Step 6, Eq.(3.10)

7. 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).

   $$Ranking:sortalternativesbydescending{\xi }_i;A_{(1)}>A_{(2)}>...>A_{(m)}$$

   Ren 2016 §3.3 Step 7

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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

• Value-space violation: ensure all entries satisfy PFN: μ ∈ [0,1], ν ∈ [0,1], μ²+ν² ≤ 1 before computation.

• Defuzzification method affects ranking: score function S = μ² − ν² is the canonical choice but alternatives exist.

Works with

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