{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T16:59:08Z","timestamp":1769705948960,"version":"3.49.0"},"reference-count":25,"publisher":"SAGE Publications","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2023,8,1]]},"abstract":"<jats:p>Multi-granulation decision-theoretic rough set effectively combines Bayesian decision approaches with multi-granulation rough set theory, and provides an important theoretical framework for studying rough set. In this paper, we explore several extensional models of multi-granulation decision-theoretic rough sets under the normal distribution of the decision loss function. Using the 3\u03c3 rule of normal distribution, we transform the decision loss of the multi-granulation decision-theoretic rough set into a set of interval values. We construct the upper and lower approximations from the optimistic, weakly optimistic, pessimistic, weakly pessimistic, optimistic-pessimistic, weakly optimistic-pessimistic, pessimistic-optimistic, and weakly pessimistic-optimistic viewpoints, and provide the decision rules of the proposed rough set models. The work in this paper brings the decision behavior based on a multi-granulation decision-theoretic rough set closer to the actual situation.<\/jats:p>","DOI":"10.3233\/jifs-224538","type":"journal-article","created":{"date-parts":[[2023,5,23]],"date-time":"2023-05-23T12:21:53Z","timestamp":1684844513000},"page":"2031-2046","source":"Crossref","is-referenced-by-count":0,"title":["Multi-granulation rough approximations under normal distribution"],"prefix":"10.1177","volume":"45","author":[{"given":"Jie","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, North Minzu University, Yinchuan, China"}]},{"given":"Renxia","family":"Wan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, North Minzu University, Yinchuan, China"}]},{"given":"Duoqian","family":"Miao","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Technology, Tongji University, Shanghai, 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