Some new methods for comparing the capability of two processes using process capability index Cpm

1. Introduction

Process capability indices constitute one of the most broadly used tools of statistical process control. They have been introduced in order to measure whether a production process is capable of producing according to some specifications, which are associated with a measurable characteristic of the items produced from it. A plethora of books and articles on this issue appeared in the literature after the appearance of the pioneering paper by Kane (1986). These cover several aspects on process capability indices, such as introduction of new and more effective indices, distributional properties of estimators, construction of confidence intervals and testing of hypotheses. Kotz and Johnson (1993, 2002), Kotz and Lovelace (1998), Pearn and Kotz (2006), Wu et al. (2009) and Spiring (2010) provide reviews of all the important contributions on this issue, while Spiring et al. (2003) and Yum and Kim (2011) give extensive lists of the bibliography on process capability indices. Finally, Ahmad et al. (2019) give a bibliometric analysis of published literature on process capability indices, which indicates a continuous increase in the number of publications continuously since 1978.

Among the process capability indices for univariate normally distributed processes that one may find in the literature, the most broadly used remain, undoubtedly, the indices C_p, C_pk, C_pm and C_pmk when the specifications of the process are bilateral and CPL and CPU when the specifications of the process are unilateral. All these indices arise as functions of process parameters, i.e. process mean (μ) and standard deviation (σ), and process specifications, i.e. lower specification limit (L), upper specification limit (U) and target value (T) and their use is meaningful only in the case where the examined process is in statistical control (i.e. is operating with only chance causes of variation present) and the data produced from it are independent (see, e.g. Kotz and Lovelace, 1998). They are defined as follows:$$C_p=\frac{U-L}{6\sigma },\text{CPL}=\frac{\mu -L}{3\sigma },\text{CPU}=\frac{U-\mu }{3\sigma },C_{pk}=\min \left\{\frac{\mu -L}{3\sigma },\frac{U-\mu }{3\sigma }\right\},C_{pm}=\frac{U}{}$$

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