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Building upon this foundation, the Pythagorean fuzzy\n                    <jats:italic>Z<\/jats:italic>\n                    -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\n                    <jats:italic>Z<\/jats:italic>\n                    -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\n                    <jats:italic>Z<\/jats:italic>\n                    -numbers, thereby promoting the advancement of improved arithmetic operations. A computational approach for arithmetic operations of Pythagorean fuzzy\n                    <jats:italic>Z<\/jats:italic>\n                    -numbers has been formulated to determine Pythagorean fuzzy\n                    <jats:italic>Z<\/jats:italic>\n                    -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\n                    <jats:italic>Z<\/jats:italic>\n                    -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\n                    <jats:italic>Z<\/jats:italic>\n                    -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.\n                  <\/jats:p>","DOI":"10.1007\/s12559-025-10541-y","type":"journal-article","created":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T08:37:14Z","timestamp":1771576634000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Extended Decision Making Method Based on Pythagorean Fuzzy Z-Numbers"],"prefix":"10.1007","volume":"18","author":[{"given":"Inayat","family":"Ullah","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Akram","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammet","family":"Deveci","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,20]]},"reference":[{"key":"10541_CR1","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh LA. 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