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Ranking

PF-COPRAS - Pythagorean extension of COPRAS

Pythagorean outranking/ranking - Pythagorean Fuzzy Number (PFN: μ, ν; μ²+ν² ≤ 1)

Yager, R. R.2014doi:10.1109/TFUZZ.2013.2278989 ↗

Overview

pf-copras extends COPRAS 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
Receive aggregated Pythagorean fuzzy decision matrix N = (μ_ij, ν_ij) with μ²+ν²≤1. Single-DM kernel skips group-aggregation via PFWA (paper Eq.9).
$I n p u t : N \in P F N^{m \times n}, w \in Δ^{n - 1}, cr i t er i a_{t} y p es \in b e n e f i t, cos t^{n} . C o n s t r ain t : μ_{i}^{2} j + ν_{i}^{2} j \leq 1\forall (i, j) .$
Thakur et al. 2022 §4.1, Tables 6-7 (aggregated decision matrix N_pq)
2
DUAL normalization (journal-as-published Thakur 2022 replikasyon). İki matris: (a) N'_swap Eq.10 literal cost-swap → h(U_i) benefit-index yolunu besler (Tablo 10 reproduce); (b) N_orig swap-yok → h(Y_i) cost-index yolunu besler (Tablo 11 h(Y_i) reproduce). Paper'ın gerçek hesap pratiği bu çiftli yoldur.
$p a t h w a y (a) : n_{i}^{'} j s w a p - E q .10; p a t h w a y (b) : n_{i} j = (μ, ν) or ij ina l$
Thakur 2022 Eq.10 + Tablo 10 + Tablo 11; as_stated_vs_as_used disclosure F2'de
3
Weighted decision matrix D = (d_ij). Weights w_j may be entropy-derived (Eq.11) or external. PFN scalar-mult: d_ij = w_j ⊙ n_ij.
$d_{i} j = w_{j} ⊙ n_{i} j = (\sqrt (1 - (1 - μ_{i}^{2} j)^{w_{j}}), ν_{i} j^{w_{j}}) . [T hak u r 2022 S t e p V I I f or m u l a]$
Thakur et al. 2022 Step VII (between Eqs.11 and 12)
4
Score function h(d_ij) = μ²(d_ij) − ν²(d_ij) (Eq.12). Defuzzifies each weighted PFN to a real-valued score.
$h (d_{i} j) = μ^{2} (d_{i} j) - ν^{2} (d_{i} j) . [E q .12]$
Thakur et al. 2022 Eq.12 (score function), Table 10
5
Max/Min indeksleri (Eqs.13-14, çiftli yol journal-as-published). h(U_i)=(1/|B|)Σ h(d^swap) over benefit (F2 yol-a); h(Y_i)=(1/|NB|)Σ h(d^noswap) over cost (F2 yol-b). NB ≠ ∅ zorunlu.
$h (U_{i}) = (1/∣ B ∣) Σ_{j \in B} h (d_{i} j^{s} w a p) [E q .13 y o l - a]; h (Y_{i}) = (1/∣ N B ∣) Σ_{j \in N B} h (d_{i} j^{n} os w a p) [E q .14 y o l - b]$
Thakur 2022 Eqs.13-14 + Table 11; pathway (a)+(b) ≤1e-5 reproduction kanıtı
6
Relative significance with exp-weighting (Eq.15): S_i = h(U_i) + (Σ_k e^{h(Y_k)}) / (e^{h(Y_i)}·Σ_k 1/e^{h(Y_k)}). Requires NB ≠ ∅.
$S_{i} = h (U_{i}) + (Σ_{k} e^{h (Y_{k})}) / (e^{h (Y_{i})} \cdot Σ_{k} (1/ e^{h (Y_{k})})) [E q .15]$
Thakur et al. 2022 Eq.15, Table 11 column S_i
7
Priority order T_i = S_i / max_k S_k × 100 (Eq.16). NOTE: when S_i is negative-valued (typical for score-based defuzzification on PFNs with ν>μ), max(S) is the least-negative; T_i ≥ 100 with higher T indicating better alternative. Descending rank by T.
$T_{i} = S_{i} / ma x_{k} S_{k} \times 100 [E q .16]; r ank = a r g sor t_{d} esc (T)$
Thakur et al. 2022 Eq.16, Table 11 columns T_i and Rank

Fits when / Look elsewhere when

Fits when

•Preserves pythagorean uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

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

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

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

•when S_i is negative-valued (typical for score-based defuzzification on PFNs with ν>μ), max(S) is the least-negative; T_i ≥ 100 with higher T indicating better alternative. Descending rank by T.

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

Commonly takes its weights from

n_a

How to cite

Yager, R. R. (2014). Pythagorean membership grades in multicriteria decision making. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2013.2278989

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

PF-COPRAS

Overview

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

Receive aggregated Pythagorean fuzzy decision matrix N = (μ_ij, ν_ij) with μ²+ν²≤1. Single-DM kernel skips group-aggregation via PFWA (paper Eq.9).

I n p u t : N \in P F N^{m \times n}, w \in Δ^{n - 1}, cr i t er i a_{t} y p es \in b e n e f i t, cos t^{n} . C o n s t r ain t : μ_{i}^{2} j + ν_{i}^{2} j \leq 1\forall (i, j) .

Thakur et al. 2022 §4.1, Tables 6-7 (aggregated decision matrix N_pq)

DUAL normalization (journal-as-published Thakur 2022 replikasyon). İki matris: (a) N'_swap Eq.10 literal cost-swap → h(U_i) benefit-index yolunu besler (Tablo 10 reproduce); (b) N_orig swap-yok → h(Y_i) cost-index yolunu besler (Tablo 11 h(Y_i) reproduce). Paper'ın gerçek hesap pratiği bu çiftli yoldur.

p a t h w a y (a) : n_{i}^{'} j s w a p - E q .10; p a t h w a y (b) : n_{i} j = (μ, ν) or ij ina l

Thakur 2022 Eq.10 + Tablo 10 + Tablo 11; as_stated_vs_as_used disclosure F2'de

Weighted decision matrix D = (d_ij). Weights w_j may be entropy-derived (Eq.11) or external. PFN scalar-mult: d_ij = w_j ⊙ n_ij.

d_{i} j = w_{j} ⊙ n_{i} j = (\sqrt (1 - (1 - μ_{i}^{2} j)^{w_{j}}), ν_{i} j^{w_{j}}) . [T hak u r 2022 S t e p V I I f or m u l a]

Thakur et al. 2022 Step VII (between Eqs.11 and 12)

Score function h(d_ij) = μ²(d_ij) − ν²(d_ij) (Eq.12). Defuzzifies each weighted PFN to a real-valued score.

h (d_{i} j) = μ^{2} (d_{i} j) - ν^{2} (d_{i} j) . [E q .12]

Thakur et al. 2022 Eq.12 (score function), Table 10

Max/Min indeksleri (Eqs.13-14, çiftli yol journal-as-published). h(U_i)=(1/|B|)Σ h(d^swap) over benefit (F2 yol-a); h(Y_i)=(1/|NB|)Σ h(d^noswap) over cost (F2 yol-b). NB ≠ ∅ zorunlu.

h (U_{i}) = (1/∣ B ∣) Σ_{j \in B} h (d_{i} j^{s} w a p) [E q .13 y o l - a]; h (Y_{i}) = (1/∣ N B ∣) Σ_{j \in N B} h (d_{i} j^{n} os w a p) [E q .14 y o l - b]

Thakur 2022 Eqs.13-14 + Table 11; pathway (a)+(b) ≤1e-5 reproduction kanıtı

Relative significance with exp-weighting (Eq.15): S_i = h(U_i) + (Σ_k e^{h(Y_k)}) / (e^{h(Y_i)}·Σ_k 1/e^{h(Y_k)}). Requires NB ≠ ∅.

S_{i} = h (U_{i}) + (Σ_{k} e^{h (Y_{k})}) / (e^{h (Y_{i})} \cdot Σ_{k} (1/ e^{h (Y_{k})})) [E q .15]

Thakur et al. 2022 Eq.15, Table 11 column S_i

Priority order T_i = S_i / max_k S_k × 100 (Eq.16). NOTE: when S_i is negative-valued (typical for score-based defuzzification on PFNs with ν>μ), max(S) is the least-negative; T_i ≥ 100 with higher T indicating better alternative. Descending rank by T.

T_{i} = S_{i} / ma x_{k} S_{k} \times 100 [E q .16]; r ank = a r g sor t_{d} esc (T)

Thakur et al. 2022 Eq.16, Table 11 columns T_i and Rank

Fits when / Look elsewhere when

Fits when

•Preserves pythagorean uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

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

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

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

•when S_i is negative-valued (typical for score-based defuzzification on PFNs with ν>μ), max(S) is the least-negative; T_i ≥ 100 with higher T indicating better alternative. Descending rank by T.

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.