Data-based explanatory fault diagnosis methods are of great practical significance to modern industrial systems due to their clear elaborations of the cause and effect relationship. Based on Boolean logic, logical analysis of data (LAD) can discover discriminative if-then rules and use them to diagnose faults. However, traditional LAD algorithm has a defect of time-consuming computation and extracts only the least number of rules, which is not applicable for high-dimensional large data set and for fault that has more than one independent causes. In this paper, a novel fast LAD with multiple rules discovery ability is proposed. The fast data binarization step reduces the dimensionality of the input Boolean vector and the multiple independent rules are searched using modified mixed integer linear programming (MILP). A Case Study on Tennessee Eastman Process (TEP) reveals the superior performance of the proposed method in reducing computation time, extracting more rules and improving classification accuracy.