This paper presents an algorithm for discovering pairs of an exception rule and a common sense rule under a prespecified schedule. An exception rule, which represents a regularity of exceptions to a common sense rule, often exhibits interestingness. Discovery of pairs of an exception rule and a common sense rule has been successful in various domains. In this method, however, both the number of discovered rules and time needed for discovery depend on the values of thresholds, and an appropriate choice of them requires expertise on the data set and on the discovery algorithm. In order to circumvent this problem, we propose two scheduling policies for updating values of these thresholds based on a novel data structure. The data structure consists of multiple balanced search-trees, and efficiently manages discovered patterns with multiple indices. One of the policies represents a full specification of updating by the user, and the other iteratively improves a threshold value by deleting the worst pattern with respect to its corresponding index. Preliminary results on four real-world data sets are highly promising. Our algorithm settled values of thresholds appropriately, and discovered interesting exception-rules from all these data sets.