Summary: The issue of selecting suppliers or assessing the impact of process improvement has already been investigated based on several process capability indices. However, a statistical test for comparing the capability of two suppliers or processes based on the \(C_{pmk}\) index has not been developed due to the complexity of the sampling distribution. We apply the concept of generalized pivotal quantities to derive the generalized confidence intervals (GCI) for the capability ratio and the capability difference between two given suppliers or processes. A simulation study is conducted to investigate the performance of the proposed methods and compare them with the coverage rates and the average width of confidence intervals. The numerical results indicate that the GCI method on the ratio is quite satisfactory for comparing the capability of two suppliers or processes since the attained coverage rates of the confidence interval are very close to the nominal confidence level.