PF-TOPSIS: Pythagorean extension of TOPSIS

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

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •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 TOPSIS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'PF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Pythagorean Fuzzy numbers/tuples
• •Hatalı: 'PF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PF-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PF-TOPSIS'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: 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). Formül: R = (C_{j}(x_{i}))_{m \times n},\quad 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}} Anchor: Zhang & Xu 2014, §3.1 Eq. (3.1); Yager 2013 PFS Def.
2. 2.Adım 2 (F2): Step 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. Formül: \tilde{r}_{ij} = \begin{cases} (\mu_{ij}, \nu_{ij}) & j \in J_{B}\ \text{(benefit)} \\ (\nu_{ij}, \mu_{ij}) & j \in J_{C}\ \text{(cost)} \end{cases} Anchor: Standard PFS-TOPSIS extension; Zhang & Xu 2014 §4 example: all benefit (no complement exercised).
3. 3.Adım 3 (F3): Step 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. Formül: A^{+} = \{\langle C_{j},\ \arg\max_{i} s(\tilde{r}_{ij}) \rangle\}_{j=1}^{n},\quad A^{-} = \{\langle C_{j},\ \arg\min_{i} s(\tilde{r}_{ij}) \rangle\}_{j=1}^{n},\quad s(\mu,\nu) = \mu^{2} - \nu^{2} Anchor: Zhang & Xu 2014, Def 2.4 Eq.(2.5) score; §3.2 Eqs.(3.2),(3.6) PIS/NIS extraction.
4. 4.Adım 4 (F4): Step 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). Formül: D(x_{i}, x^{\pm}) = \tfrac{1}{2} \sum_{j=1}^{n} w_{j} \Bigl( |\mu_{ij}^{2} - \mu_{j}^{\pm 2}| + |\nu_{ij}^{2} - \nu_{j}^{\pm 2}| + |\pi_{ij}^{2} - \pi_{j}^{\pm 2}| \Bigr),\quad i = 1,\ldots,m Anchor: Zhang & Xu 2014, Def 3.1 Eq.(3.3) base distance; Eqs.(3.4),(3.7) weighted aggregation.
5. 5.Adım 5 (F5): Step 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. Formül: \zeta(x_{i}) = \dfrac{D(x_{i}, x^{-})}{D_{\max}(x_{i}, x^{-})} - \dfrac{D(x_{i}, x^{+})}{D_{\min}(x_{i}, x^{+})},\quad D_{\max}(x_{i}, x^{-}) = \max_{k} D(x_{k}, x^{-}),\ D_{\min}(x_{i}, x^{+}) = \min_{k} D(x_{k}, x^{+}) Anchor: Zhang & Xu 2014, Eq.(3.10); Hadi-Vencheh & Mirjaberi 2014, Inform Sci 277:263-272.
6. 6.Adım 6 (F6): Step 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. Formül: \text{rank}(x_{i}) = \text{argsort}_{\text{desc}}(\zeta(x_{i})),\quad x^{*} = \{x_{i}: \zeta(x_{i}) = \max_{k} \zeta(x_{k})\} Anchor: Zhang & Xu 2014, §3.3 Algorithm Step 5, Eq.(3.11).

Commonly paired with

• •n_a + PF-TOPSIS (common)

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