A novel complex Fermatean fuzzy formalism with improved score function and aggregation operators

Why choosing the “best system” is so hard

Modern cities depend on invisible digital traffic: radio waves from phones, cars, sensors, and cameras. When authorities want to upgrade their signal processing and communication systems—for example, to keep traffic flowing safely and efficiently—they face a maze of options and conflicting criteria. Many of the numbers behind these choices are uncertain or fuzzy rather than clear-cut. This paper develops a new mathematical toolkit for reasoning carefully with such uncertainty, helping planners and engineers pick better systems in complex real-world settings.

From yes-or-no thinking to shades of maybe

Traditional decision methods assume that something either belongs in a category or it doesn’t: a system is “good enough” or “not good enough.” Decades of work in fuzzy logic relaxed this rigid view by allowing partial membership: a design might be judged “0.7 good” instead of simply yes or no. Later refinements, like intuitionistic, Pythagorean, and Fermatean fuzzy sets, added richer ways to describe not only how strongly something is supported, but also how strongly it is rejected and how much remains uncertain. Even these models, however, usually treat information as one-dimensional, like a single temperature reading, while many engineering signals are naturally two-dimensional, with a size (amplitude) and a timing or angle (phase) component.

Why complex-valued fuzziness matters

In fields such as wireless communication, radar, medical imaging, and power systems, engineers routinely work with complex-valued signals that combine magnitude and phase. For instance, baseband radio signals, radar echoes, and MRI data are all stored as complex samples. Real performance often depends on a pair of aspects: how strong the signal is and how stable or coherent it remains over time. The authors build on a relatively new structure called a complex Fermatean fuzzy set, which can encode these two aspects together while still capturing support, opposition, and hesitation. This richer description is essential when comparing advanced communication schemes that must simultaneously achieve high quality, stability, and robustness in noisy, changing environments.

Fixing an unreliable yardstick

To use fuzzy information in practice, decision systems need a way to assign an overall score to each candidate option. Earlier work had proposed a “score function” for complex Fermatean fuzzy numbers, but the authors show that it can fail: two different options can end up with exactly the same score and accuracy, making them indistinguishable even when one is clearly preferable. They analyze this weakness with a concrete numerical example and then introduce an improved scoring formula. The new function adds an extra interaction term between parts of the complex evaluation, allowing it to break ties that previously could not be resolved and to order alternatives more reliably.

New tools for combining fuzzy opinions

Real decisions almost never depend on a single criterion. A traffic communication system might be judged on signal quality, response speed, reliability, and energy use, each of which is itself uncertain. The heart of this paper is two new ways to combine such complex fuzzy assessments into a single overall evaluation: the complex Fermatean fuzzy ordered weighted averaging (CFFOWA) operator and the complex Fermatean fuzzy ordered weighted geometric (CFFOWG) operator. Both methods first sort the criteria for each option from most to least favorable, then blend them according to chosen weights that reflect their importance. The averaging version emphasizes typical performance, while the geometric version is more sensitive to very weak aspects. The authors prove that these operators behave sensibly: the result always remains a valid fuzzy description, stays within reasonable bounds, and changes monotonically as the inputs improve or worsen.

Putting the method to work in smart traffic control

To show how the framework operates, the authors model a realistic choice among four signal processing and communication approaches for traffic control: multiple-antenna systems, wavelet-based methods, fast Fourier transform modulation, and error-correcting codes. Each approach is evaluated on four attributes—signal-to-noise improvement, real-time responsiveness, reliability, and energy consumption—where every attribute is itself split into two complementary sub-features, such as average performance and stability. All of these judgments are encoded as complex Fermatean fuzzy numbers and then aggregated using the new operators. Both the averaging and geometric procedures independently rank the multiple-antenna system as the best overall choice. When compared against several established fuzzy decision techniques, the new methods agree on the top option while offering finer discrimination among the remaining contenders.

What this means for real-world choices

In simple terms, the paper equips decision-makers with a more nuanced calculator for weighing complicated, uncertain evidence. By refining the way scores are computed and by introducing two powerful combination rules tailored to complex-valued information, the authors make it easier to compare sophisticated technical systems fairly. Their demonstration in traffic signal processing suggests that the same approach could guide choices in fields like healthcare, energy planning, transportation design, and investment, wherever performance depends on multiple intertwined factors that cannot be captured with plain yes-or-no thinking.

Citation: Razaq, A., Komal, L., Alhamzi, G. et al. A novel complex Fermatean fuzzy formalism with improved score function and aggregation operators. Sci Rep 16, 13559 (2026). https://doi.org/10.1038/s41598-026-40838-3

Keywords: fuzzy decision making, signal processing, smart traffic systems, uncertainty modeling, multi-criteria analysis