FF-MARCOS - Fermatean extension of FF-MARCOS

Fermatean outranking/ranking - Fermatean Fuzzy Set (FFS: μ, ν; μ³+ν³ ≤ 1)

Senapati, T., Yager, R. R.2020doi:10.1007/s12652-019-01377-0 ↗

Overview

ff-marcos extends FF-MARCOS to handle Fermatean uncertainty. All arithmetic operations (normalisation) are performed using Fermatean Fuzzy Set (FFS: mu, nu; mu^3+nu^3 <= 1) algebra. Each normalised cell (mu_hat, nu_hat) holds TWO independent closeness-to-ideal ratios (cubing recovers the crisp MARCOS ratios mu/mu_AI and nu_AI/nu, both 'higher is better'); F3 averages their cubes -- S=(mu_hat^3+nu_hat^3)/2 -- weights and sums row-wise before the crisp MARCOS utility-ratio ladder (K+/K-, f(K+)/f(K-), f(K)) runs unchanged.

Output: utility, higher is better
Data: Fermatean Fuzzy, uncertainty tuples complete
Weights: Needs a weight source
Size: 2+ alternatives, 3-10 criteria works best
Used for: Fermatean Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

Look elsewhere when

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

Assumptions to verify

Decision matrix entries are valid Fermatean 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

Value-space violation: ensure all entries satisfy FFS: mu^3+nu^3 <= 1 (q=3 special case of q-ROF) before computation.
Do NOT aggregate a row's normalised cells into one Fermatean number (e.g. via FFWA) and score it once at the end, and do NOT score a normalised cell with the raw/shifted DIFFERENCE score mu^3-nu^3: F2's mu_hat=(mu/mu_AI)^(1/3) and nu_hat=(nu_AI/nu)^(1/3) are BOTH already 'higher is better' closeness-to-ideal ratios (unlike a raw Fermatean pair, where nu is a badness dial), so they must be AVERAGED - S=(mu_hat^3+nu_hat^3)/2 - not differenced or aggregated via a product-form operator. Both mistakes were made and fixed 2026-09-13 (see manifest J._note + karar.md): FFWA-then-score silently zeroed an alternative's utility whenever any single criterion tied the ideal; the shifted difference score kept values positive but still inverted monotonicity on cost criteria.

Works with

Commonly takes its weights from

n_a

How to cite

Senapati, T.; Yager, R. R. (2020). Fermatean fuzzy sets. Journal of Ambient Intelligence and Humanized Computing. https://doi.org/10.1007/s12652-019-01377-0

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

FF-MARCOS

Overview

Output

utility, higher is better

Data

Fermatean Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

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

Edge cases and pitfalls

Value-space violation: ensure all entries satisfy FFS: mu^3+nu^3 <= 1 (q=3 special case of q-ROF) before computation.

Do NOT aggregate a row's normalised cells into one Fermatean number (e.g. via FFWA) and score it once at the end, and do NOT score a normalised cell with the raw/shifted DIFFERENCE score mu^3-nu^3: F2's mu_hat=(mu/mu_AI)^(1/3) and nu_hat=(nu_AI/nu)^(1/3) are BOTH already 'higher is better' closeness-to-ideal ratios (unlike a raw Fermatean pair, where nu is a badness dial), so they must be AVERAGED - S=(mu_hat^3+nu_hat^3)/2 - not differenced or aggregated via a product-form operator. Both mistakes were made and fixed 2026-09-13 (see manifest J._note + karar.md): FFWA-then-score silently zeroed an alternative's utility whenever any single criterion tied the ideal; the shifted difference score kept values positive but still inverted monotonicity on cost criteria.