The concept of the Z-number offers a powerful framework for representing fuzzy information, enabling more accurate and reliable decision-making under uncertainty. Building upon this foundation, the Pythagorean fuzzy Z-number further enhances the capacity to model vagueness, accommodating a broader and more nuanced range of data, thus aligning more closely with human reasoning. To enhance this capability, this paper introduces continuous Pythagorean fuzzy Z-numbers, which allow for greater flexibility in expressing uncertainty and better reflect human judgment when dealing with vague or ambiguous data. Also, it proposes a comprehensive arithmetic framework to operate within this context. A probability measure for Pythagorean fuzzy events is established and utilized to assess the results of unary and binary operations involving Pythagorean fuzzy Z-numbers, thereby promoting the advancement of improved arithmetic operations. A computational approach for arithmetic operations of Pythagorean fuzzy Z-numbers has been formulated to determine Pythagorean fuzzy Z-numbers in a more efficient and practical manner, ensuring suitability for real-world applications. To facilitate practical application, a distance measure and a ranking function are developed, enabling more effective comparison and analysis. The study also extends the Weighted Aggregated Sum Product Assessment (WASPAS) method to the Pythagorean fuzzy Z-number environment and demonstrates its utility through a real-world case study. Comparative results and sensitivity analysis are presented to validate the robustness and effectiveness of the proposed approach. The proposed decision-making technique based on Pythagorean fuzzy Z-numbers provides enhanced reliability by jointly modeling uncertainty and expert hesitancy. The consistency of its results with established techniques underscores its superiority and real-world applicability.
Ullah et al. (Fri,) studied this question.