A novel approach to deriving the lower confidence limit of indices C_pu, C_pl, and C_pk in assessing process capability

Abstract

Process capability indices (PCIs) are widely used as a measure of process potential and process performance. Unfortunately, the use of sample data to estimate PCIs means that any error in the sampling can introduce considerable uncertainty into the assessment of process capability. This necessitates the use of the lower confidence limit (LCL) in the estimation of minimum process capability. Furthermore, the complexity of sampling distributions of the PCIs greatly hinders interval estimation, such that only an approximate or asymptotic LCL can be achieved. This paper proposes a novel approach to deriving the LCL of indices C_pu, C_pl and C_pk using Boole’s inequality and DeMorgan’s theorem. This approach is based on subsample data collected from a stable process. Hypothesis testing is also used to determine whether the process is capable of satisfying the quality requirements of customers. We calculated the critical values of the PCIs for various significance levels, capability requirements and sample sizes. Finally, we present analysis of two cases to demonstrate the applicability of the proposed approach.

Keywords:

• process capability indices
• lower confidence limit
• process performance
• Boole’s inequality
• DeMorgan’s theorem

Acknowledgement

The authors would like to thank the Editor and three anonymous referees for their constructive comments and careful reading, which significantly improved the presentation of this paper.