PF-TOPSIS - Pythagorean extension of TOPSIS

Overview

pf-topsis extends TOPSIS 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

revised closeness, 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

   Construct the Pythagorean fuzzy decision matrix R = (C_j(x_i))_{m×n}; each entry is a PFN (μ, ν) with μ²+ν²≤1, π = √(1−μ²−ν²).

   $$R=(C_j(x_i){)}_{m\times n},C_j(x_i)=({\mu }_{ij},{\nu }_{ij}),{\mu }_{ij}^2+{\nu }_{ij}^2\le 1,{\pi }_{ij}=\sqrt{1-{\mu }_{ij}^2-{\nu }_{ij}^2}$$

   Zhang & Xu 2014, §3.1 Eq. (3.1); Yager 2013 PFS Def.

2. 2

   Cost-criterion complement normalisation: keep benefit criteria as-is; for cost criteria swap membership and non-membership: (μ,ν) → (ν,μ). Note: Zhang & Xu 2014 §4 example uses all-benefit criteria; complement is a widely adopted PFS extension.

   $${\tilde{r}}_{ij}=\{\begin{array}{ll}({\mu }_{ij},{\nu }_{ij})& j\in J_B\text{(benefit)}\\ ({\nu }_{ij},{\mu }_{ij})& j\in J_C\text{(cost)}\end{array}$$

   Standard PFS-TOPSIS extension; Zhang & Xu 2014 §4 example: all benefit (no complement exercised).

3. 3

   Identify PF-PIS A⁺ and PF-NIS A⁻ directly from the normalised matrix R via the score function s(p) = μ² − ν²: per column, A⁺ takes the PFN with maximum score, A⁻ the PFN with minimum score. Zhang & Xu's canonical path applies weights inside the distance (F4), not via prior scalar weighting - so PIS/NIS are extracted from R, not from a weighted matrix.

   $$A^{+}=\{⟨C_j,\arg \underset{i}{\max }s({\tilde{r}}_{ij})⟩{\}}_{j=1}^n,A^{-}=\{⟨C_j,\arg \underset{i}{\min }s({\tilde{r}}_{ij})⟩{\}}_{j=1}^n,s(\mu ,\nu )={\mu }^2-{\nu }^2$$

   Zhang & Xu 2014, Def 2.4 Eq.(2.5) score; §3.2 Eqs.(3.2),(3.6) PIS/NIS extraction.

4. 4

   Compute weighted Hamming distance from each alternative to PF-PIS and PF-NIS over (μ², ν², π²) components, with criterion weights w_j inside the sum (Zhang-Xu canonical, no separate scalar weighting step).

   $$D(x_i,x^{\pm })=\frac{1}{2}\sum _{j=1}^n{w}_j(|{\mu }_{ij}^2-{\mu }_j^{\pm 2}|+|{\nu }_{ij}^2-{\nu }_j^{\pm 2}|+|{\pi }_{ij}^2-{\pi }_j^{\pm 2}|),i=1,\dots ,m$$

   Zhang & Xu 2014, Def 3.1 Eq.(3.3) base distance; Eqs.(3.4),(3.7) weighted aggregation.

5. 5

   Compute the revised closeness index ζ (Zhang-Xu 2014 Eq.3.10, originating from Hadi-Vencheh & Mirjaberi 2014). ζ ≤ 0 always; ζ = 0 attained iff an alternative simultaneously achieves the maximum D⁻ and the minimum D⁺. Used in preference to the classical relative closeness D⁻/(D⁺+D⁻) because the latter cannot guarantee simultaneous PIS-closest / NIS-farthest selection.

   $$\zeta (x_i)=\frac{D(x_i,x^{-})}{D_{\max }(x_i,x^{-})}-\frac{D(x_i,x^{+})}{D_{\min }(x_i,x^{+})},D_{\max }(x_i,x^{-})=\underset{k}{\max }D(x_k,x^{-}),D_{\min }(x_i,x^{+})=\underset{k}{\min }D(x_k,x^{+})$$

   Zhang & Xu 2014, Eq.(3.10); Hadi-Vencheh & Mirjaberi 2014, Inform Sci 277:263-272.

6. 6

   Rank alternatives in descending order of ζ. The optimal alternative is the one with maximum ζ; the global optimum ζ=0 is attained when (and only when) one alternative dominates both ideal-set distances.

   $$\text{rank}(x_i)={\text{argsort}}_{\text{desc}}(\zeta (x_i)),x^{\ast }=\{x_i:\zeta (x_i)=\underset{k}{\max }\zeta (x_k)\}$$

   Zhang & Xu 2014, §3.3 Algorithm Step 5, Eq.(3.11).

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

• •when (and only when) one alternative dominates both ideal-set distances.

• 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.

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