Z-COPRAS - Z-Number extension of COPRAS

Z-Number outranking/ranking - Z-Number (Z = (A, B): restriction A, reliability B; both fuzzy)

Zadeh, L. A.2011doi:10.1016/j.ins.2011.02.022 ↗

Overview

z-copras extends COPRAS to handle Z-Number uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Z-Number (Z = (A, B): restriction A, reliability B; both fuzzy) algebra. The final scores are defuzzified via convert to regular fuzzy: Ã = B·A, then centroid before ranking.

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

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 Z-Number 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 Z = (A, B) where A restricts values, B indicates reliability of A; both TFNs typically before computation.
Defuzzification method affects ranking: convert to regular fuzzy: Ã = B·A, then centroid is the canonical choice but alternatives exist.
If B (reliability) is identical in every cell, the sqrt(alpha) constant cancels exactly in the column-sum normalisation ratio (R_ij = x'_ij / Σx'_kj); ranking AND scores stay unchanged. This is a mathematical property, not a defect. B changes the result only when reliability differs between alternatives.

Works with

Commonly takes its weights from

n_a

How to cite

Zadeh, L. A. (2011). A note on Z-numbers. Information Sciences. https://doi.org/10.1016/j.ins.2011.02.022

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

Z-COPRAS

Overview

Output

utility, higher is better

Data

Z-Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Z-Number MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

Edge cases and pitfalls

Value-space violation: ensure all entries satisfy Z = (A, B) where A restricts values, B indicates reliability of A; both TFNs typically before computation.

Defuzzification method affects ranking: convert to regular fuzzy: Ã = B·A, then centroid is the canonical choice but alternatives exist.

If B (reliability) is identical in every cell, the sqrt(alpha) constant cancels exactly in the column-sum normalisation ratio (R_ij = x'_ij / Σx'_kj); ranking AND scores stay unchanged. This is a mathematical property, not a defect. B changes the result only when reliability differs between alternatives.