Since the problem in this study is of multi-label classification, we intend to provide flexible information by predicting some candidate metabolic pathway classes for the query compounds, rather than just the most likely metabolic pathway class. Therefore, instead of Eq. 5, let us consider the following equation containing 11 scores in a one-column vector: where D ; is a descending operator that sorts the 11 scores of Eq. 4 for S(C q [j) according to the descending order (S 1 §S 2 § Á Á Á §S j § Á Á Á §S 11 ). If there is a tie among these scores, a random order will be made among those with a tie. Consequently, the predicted metabolic pathway classes for the query compound can be derived according to the descending order of Eq. 6; i.e., if S 1~S (P k [6), S 2~S (P k [1), S 3~S (P k [10) , then it follows that the query compound C q is involved in the 6 th metabolic pathway class (''Metabolism of Other Amino Acids'') will be ranked as the highest in the likelihood, that C q in the 1 st metabolic pathway class (''Carbohydrate Metabolism'') as the 2 nd , and that C q in the 10 th metabolic pathway class (''Biosynthesis of Other Secondary Metabolites'') as the 3 rd . The corresponding results thus obtained are, respectively, called the 1 st -order, 2 nd -order, and 3 rd -order predicted metabolic pathway classes. And so forth.