Supplier selection is the problem which deals with comparing two processes and selecting a better one that has a higher process capability. In this article, the generalization C pk T index is employed to deal with this problem. The generalization Cpk T index measures the overall yield for the processes with multiple independent quality characteristics. Based on the generalization Cpk T index, a hypothesis testing with two phases is developed to determine whether or not new supplier is better. It allows one to select the supplier with better quality. A test statistic by division method and its corresponding sampling distribution are derived. For various process minimum requirements, magnitudes of the difference between the two processes, unequal sample sizes of two processes, and α-risk, the critical values of the tests are tabulated. An application example is presented to illustrate the practicality of the proposed approach. Practitioners can use the proposed method to make reliable decisions in supplier selection problem. Copyright by ASTM Int'l (all rights reserved).