Volume 38, Issue 1

Reliability is a popular concept that has been used in the manufacturing industry. In this paper, we consider a parallel system containing n non-identical and independent components in which each component is repairable except when all components are failed. As a special case, estimating the reliability of the system with identical components is considered. In real life, the data obtained for repair rate and failure rate could be subject to uncertainty. Here, to address this situation, failure and repair rates are considered as fuzzy numbers to estimate the reliability of the system. Fuzzy system reliability is estimated using fuzzy failure and repair rates, which are obtained by using confidence intervals and point estimators of failure rate and repair rate.

Independence between successive counts is not a sensible premise while dealing, for instance, with very high sampling rates. After assessing the impact of falsely assuming independent binomial counts in the performance of np -charts, such as the one with 3-σ control limits, we propose a modified np -chart for monitoring first-order autoregressive counts with binomial marginals. This simple chart has an in-control average run length (ARL) larger than any out-of-control ARL, i.e., it is ARL-unbiased. Moreover, the ARL-unbiased modified np -chart triggers a signal at sample t with probability one if the observed value of the control statistic is beyond the lower and upper control limits L and U . In addition to this, the chart emits a signal with probability γ L {\gamma_{L}} (resp. γ U {\gamma_{U}} ) if that observed value coincides with L (resp. U ). This randomization allows us to set the control limits in such a way that the in-control ARL takes the desired value ARL 0 {\operatorname{ARL}_{0}} , in contrast to traditional charts with discrete control statistics. Several illustrations of the ARL-unbiased modified np -chart are provided, using the statistical software R and resorting to real and simulated data.

One of the most important issues in the operation of any one- or multi-variable control chart is to determine the design parameters. Because in practice, production processes are affected by several assignable causes, several papers have been published to determine the design parameters. In all the papers presented so far, it has been assumed that after the occurrence of one of the assignable causes until the issuance of the true alarm, no other assignable cause occurs. Contrary to popular opinion, this paper argues that the formulas presented under this assumption for the average cost and quality cycle time in previous papers are incorrect and shows how the formula can be corrected. Therefore, this paper theoretically and numerically examines the conditions of occurrence of this assumption and its relationship with the design parameters in the design of multivariate control charts. A new economic model for determining design parameters is also presented. Numerical results show that the old formulas have a significant under-estimation of the average cost per unit time of the quality cycle. Also, a numerical study for economic and economic-statistical design of T 2 {T^{2}} control chart is presented under the proposed model.