Optimizing biometric system selection via complex spherical fuzzy einstein aggregation operators

Why smarter security choices matter

Banks increasingly rely on biometric systems—fingerprints, face scans, iris readers—to protect accounts and curb fraud. But picking the “best” device is far from simple. Real-world conditions like changing lighting, sensor glitches, busy branches, or shifting customer habits make performance highly uncertain and time-dependent. This article presents a new mathematical decision framework designed to help banks choose biometric systems that stay reliable under such messy, fluctuating conditions, aiming to reduce financial mishaps while reflecting genuine human and technical uncertainty.

Seeing uncertainty in three shades

Traditional decision tools often assume we can assign a single clear-cut score to each option: good, bad, or somewhere in between. Real life is murkier. Earlier generations of fuzzy logic tried to capture this by allowing partial membership in a category and, later, by adding non-membership and neutrality. Still, they typically relied on simple, one-dimensional numbers. The authors build on a more expressive idea called complex spherical fuzzy sets, which track three aspects at once—support for an option, opposition to it, and neutral hesitation—and encode each not just as a magnitude but also as a phase, a kind of timing or cyclic behavior. This allows the model to represent patterns like “this device works very well most of the day but struggles during peak evening traffic” in a precise yet flexible way.

A new way to merge many noisy opinions

In banking practice, biometric choices depend on many criteria: ease of use, data backup, battery life, location tracking and more, all judged by experts who may disagree or be unsure. Combining these scattered, sometimes extreme opinions into one balanced verdict is tricky. Linear averaging can let one unusually high or low score dominate and can blur important interactions between criteria. To address this, the authors introduce two new “Einstein” aggregation operators tailored to the complex spherical fuzzy setting. These operators use nonlinear mathematical rules that encourage criteria to support each other but prevent one strong score from fully compensating for weaknesses elsewhere. As a result, the combined assessment is smoother, stays within realistic bounds, and is less sensitive to outliers or unreliable inputs.

Ranking choices without ties

Even with a sophisticated language for uncertainty and smart aggregation, decision makers still need a clear ranking of alternatives. Earlier scoring formulas for complex spherical fuzzy data sometimes treated distinct options as equal, making it impossible to say which biometric system was truly better. The paper proposes an improved score function that more carefully weighs support, neutrality, and opposition, along with their phase behavior. This refined measure stretches the numerical range and reduces ties, so two systems that behave differently across time or conditions are much less likely to end up with the same overall score. That, in turn, gives managers an unambiguous ordering of candidates.

Putting the method to work in a bank

To show how the method performs in practice, the authors study a bank that suspects a high risk of fraudulent activity and wants to upgrade its biometric access. Four commercial systems are compared across four key criteria: user friendliness, data backup provisions, battery backup, and employee tracking via location services. Expert judgments on each device–criterion pair are encoded as complex spherical fuzzy numbers, capturing both how strongly each requirement is met and how that performance varies in different situations. The new Einstein aggregation operators then combine these evaluations into overall scores for each device. Both versions of the operator agree on the top choice: a system referred to as “CP plus,” which consistently outperforms the others once uncertainty and interdependence among criteria are fully accounted for.

How this helps banks and beyond

From a lay perspective, the study’s main message is that picking security technology should not rely on simple averages or lab-only test scores. By modeling what experts know—and do not know—about a system’s behavior over time, and by merging those opinions in a controlled, nonlinear way, the proposed framework delivers rankings that are more stable, more transparent, and better aligned with real-world risk. Although the case study focuses on biometric systems in banking, the same approach could guide complex choices in areas like environmental policy, healthcare technology, or cyberdefense, wherever decisions must be made under deep uncertainty and changing conditions.

Citation: Kanwal, S., Shuaib, U., Alhamzi, G. et al. Optimizing biometric system selection via complex spherical fuzzy einstein aggregation operators. Sci Rep 16, 13183 (2026). https://doi.org/10.1038/s41598-026-39908-3

Keywords: biometric security, fuzzy logic, multi-criteria decision making, bank fraud prevention, uncertainty modeling