Consistencies of the capability indices based on the normal probability distribution

Consistências dos índices de capacidade baseados na distribuição normal de probabilidade

Abstract:

Capability analysis seeks to estimate the probability that a process will produce compliant products. The capability indices are dimensionless parameters that measure how well the process can meet specifications. In the literature, eight capability indices are listed, among others, considering a stable process under statistical control and based on the normal probability distribution, defined by: C_p, P_p, C_pk, P_pk, C_pm, P_pm, C_pmk, and P_pmk. Basically, the index formulas differ in the calculations of the variability within and total, and of the shifts of the mean in relation to the nominal value and the nearest specification limit. The objective of this article was to compare these capacity indexes, and for that, it was chosen the most consistent estimator, that is, the one that improved the accuracy and efficiency as the number of observations increased. Thus, a simulation of 30,000 values of a normal random variable with a mean equal to zero and a standard deviation equal to one was performed. This made it possible to sample this process 1,000 times using 5, 10, 15, 20, 25, and 30 rational subgroups with individual observations or sample elements. Subsequently, 20 mean shifts were provoked, with values ranging from 0.1 to 2 and varying by 0.1 unit. According to the results, it was concluded that the indexes C_pk and P_pk were the most consistent in presenting higher accuracy and efficiency for at least 15 rational subgroups or sample elements, regardless of the magnitude of the mean displacement in relation to the nominal value.

Keywords:
Capability index; Estimator; Quality control