Multivariate control charts for nonconformities

When the nonconformities are independent, a multivariate control chart for nonconformities called a demenit control chart using a distribution approximation technique called an Edgeworth Expansion, is proposed. For a demerit control chart, an exact control limit can be obtained in special cases, but not in general. A proposed demerit control chart uses an Edgeworth Expansion to approximate the distribution of the demerit statistic and to compute the demerit control limits. A simulation study shows that the proposed method yields reasonably accurate results in determining the distribution of the demerit statistic and hence the control limits, even for small sample sizes. The simulation also shows that the performances of the demerit control chart constructed using the proposed method is very close to the advertised for all sample sizes.; Since the demerit control chart statistic is a weighted sum of the nonconformities, naturally the performance of the demerit control chart will depend on the weights assigned to the nonconformities. The method of how to select weights that give the best performance for the demerit control chart has not yet been addressed in the literature. A methodology is proposed to select the weights for a one-sided demerit control chart with and upper control limit using an asymptotic technique. The asymptotic technique does not restrict the nature of the types and classification scheme for the nonconformities and provides an optimal and explicit solution for the weights.; In the case presented so far, we assumed that the nonconformities are independent. When the nonconformities are correlated, a multivariate Poisson lognormal probability distribution is used to model the nonconformities. This distribution is able to model both positive and negative correlations among the nonconformities. A different type of multivariate control chart for correlated nonconformities is proposed. The proposed control chart can be applied to nonconformities that have any multivariate distributions whether they be discrete or continuous or something that has characteristics of both, e.g., non-Poisson correlated random variables. The proposed method evaluates the deviation of the observed sample means from pre-defined targets in terms of the density function value of the sample means. The distribution of the control chart test statistic is derived using an approximation technique called a multivariate Edgeworth expansion. For small sample sizes, results show that the proposed control chart is robust to inaccuracies in assumptions about the distribution of the correlated nonconformities.