A Note on the Generalized Indices of Process Capability

Process capability indices are widely used to assess whether the outputs of an in-control process meet the specifications. The commonly used indices are Cp, Cpu, Cpl and Cpk. In most applications, the quality characteristics are assumed to follow normal distribution. But, in practice, many quality characteristics, e.g. count data, proportion defective etc. follow Poisson or binomial distributions, and these characteristics usually have one-sided specification limit. In these cases, computations of Cpu or Cpl using the standard formula is inappropriate. In order to alleviate the problem, some generalized indices (e.g. C index, Cf index, Cpc index and Cpy index) are proposed in literature. The variant of these indices for one-sided specification are Cu and Cl, Cfu and Cfl¸ Cpcu and Cpcl, Cpyu and Cpyl respectively. All these indices can be computed in any process regardless of whether the quality characteristics are discrete or continuous. However, the same value for different generalized indices and Cpu or Cpl signifies different capabilities for a process and this poses difficulties in interpreting the estimates of the generalized indices. In this study, the relative goodness of the generalized indices is quantifying capability of a process is assessed. It is found that only Cu or Cl gives proper assessment about the capability of a process. All other generalized indices give a false impression about the capability of a process and thus usages of those indices should be avoided. The results of analysis of multiple case study data taken from Poisson and binomial processes validate the above findings.