In recent years, the problem of learning a real-valued function that induces a ranking over an instance space has gained importance in machine learning literature. Here, we propose a supervised algorithm that learns a ranking function, called ranking instances by maximizing the area under the ROC curve (RIMARC). Since the area under the ROC curve (AUC) is a widely accepted performance measure for evaluating the quality of ranking, the algorithm aims to maximize the AUC value directly. For a single categorical feature, we show the necessary and sufficient condition that any ranking function must satisfy to achieve the maximum AUC. We also sketch a method to discretize a continuous feature in a way to reach the maximum AUC as well. RIMARC uses a heuristic to extend this maximization to all features of a data set. The ranking function learned by the RIMARC algorithm is in a human-readable form; therefore, it provides valuable information to domain experts for decision making. Performance of RIMARC is evaluated on many real-life data sets by using different state-of-the-art algorithms. Evaluations of the AUC metric show that RIMARC achieves significantly better performance compared to other similar methods.