Complex Probabilistic Neutrosophic Sets for Multi-Criteria Decision-Making: New Aggregation Operators and Comparative Analysis
This paper introduces the novel complex probabilistic neutrosophic (CPN) set to better capture expert opinion acceptance levels in uncertain real-life scenarios, proposes new weighted averaging and geometric aggregation operators, and validates a corresponding multi-criteria decision-making framework through a car selection case study and comparative analysis.
Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
In the modern world, decisions are rarely simple. Whether choosing a new car, diagnosing a patient, or selecting a business strategy, we often face information that is incomplete, contradictory, or shifting over time. For decades, mathematicians have developed tools to handle this uncertainty. They began with simple ways to measure how much something belongs to a category, then added layers to account for doubt and neutrality. Later, they created systems that could track information that changes in cycles, like the rising and falling of tides or the fluctuation of stock prices. However, a significant gap remained. Existing tools could handle either the cyclical nature of data or the confidence level of an expert's opinion, but rarely both at the same time. This limitation meant that real-world scenarios, where opinions vary in strength and information repeats in patterns, were often oversimplified.
A team of researchers from the University of Anbar and King Faisal University has addressed this gap by creating a new mathematical framework called the complex probabilistic neutrosophic set. This system is designed to capture the full complexity of human judgment. It combines three distinct types of information: how much an option is accepted, how much it is rejected, and how much the evaluator is unsure. Crucially, it adds a fourth dimension: a measure of confidence or probability attached to each of those three judgments. Furthermore, it represents these judgments as complex values, allowing the system to track periodic changes or time-based patterns that traditional methods miss. The researchers did not just propose this idea; they built the mathematical machinery to make it work. They defined how to combine different pieces of information, how to compare them, and how to derive a single, clear ranking from a messy set of data.
To test their new system, the authors applied it to a practical scenario: a customer trying to choose the best car from five available options. The decision was based on three criteria: overall cost, quality, and exterior appearance. In a traditional approach, an expert might give a single score for each car on each criterion. In this new framework, the expert's input is much richer. For each car, the expert provides a value for how much they like the cost, how much they dislike it, and how unsure they are, all while indicating how confident they are in those specific feelings. The system also accounts for the fact that these feelings might have a cyclical nature, perhaps reflecting seasonal trends or recurring market conditions. The researchers then used two new mathematical tools, which they call aggregation operators, to process this complex data. These tools act like a sophisticated calculator that weighs the importance of each criterion—cost, quality, and appearance—and synthesizes the expert's multi-layered opinions into a single score for each car.
When the researchers ran the numbers, the system successfully identified a clear winner. Both of their new calculation methods pointed to the same car, labeled A5, as the best choice. The system ranked the cars in a specific order, placing A5 at the top and another car, A3, at the bottom. To ensure their results were not a fluke, the team tested how sensitive the outcome was to changes in the importance of the criteria. They adjusted the weights, making cost more important or making appearance more important, and observed the results. Even when the priorities shifted significantly, the system remained stable. The top choice, A5, consistently stayed at the top, and the bottom choice, A3, remained at the bottom. This stability suggests that the new method is robust and reliable, capable of handling the messy, uncertain, and cyclical nature of real-life decision-making without breaking down.
The researchers also compared their new method against older mathematical models to see where it stood. They found that previous systems, which could handle either probability or cyclical patterns but not both, were insufficient for this type of complex data. Their new framework acts as a comprehensive umbrella that includes those older methods as special cases. If you remove the probability aspect, the system simplifies to a known model for handling cycles. If you remove the cyclical aspect, it simplifies to a model for handling probability. This flexibility means the new tool can adapt to a wider range of problems than its predecessors. The authors acknowledge that their work is currently focused on individual decision-making and theoretical simulations. They note that future work will need to expand these tools to handle groups of experts, dynamic data that changes in real-time, and applications in fields like healthcare and finance. For now, however, the study demonstrates that by combining probability with complex, cyclical data, it is possible to create a more accurate and flexible way to navigate the uncertainties of the modern world.
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