A monitoring system (MS) has been used to monitor products’ job cycles. It is indicated that by incorporating the job cycle into the product’s life cycle, warrantors can devise novel warranty models and consumers can define and model random maintenances sustaining the reliability of the product through warranty. In this study, by incorporating limited job cycles and a refund into the traditional free repair warranty, a two-dimensional free repair warranty with a refund (2DFRW-R) is devised for guaranteeing the product reliability to consumers. Under the condition that 2DFRW-R is planned to guarantee product reliability, a bivariate random periodic replacement (BRPR) (i.e., a random periodic replacement where the accomplishment of the Nth job cycle and the replacement time T are designed as replacement limits) is modeled to sustain the post-warranty reliability from the point of view of the consumer. From the point of view of the warrantor, the warranty cost related to 2DFRW-R is derived, and the characteristics of 2DFRW-R are explored. From the point of view of consumers, the expected cost rate related to BRPR is constructed, and the existence and uniqueness of the optimal BRPR are summarized as well. By discussing parameters, several special cases are derived. The characteristics of the proposed models are analyzed in numerical examples.

Control chart pattern recognition (CCPR) can quickly recognize anomalies in charts, making it an important tool for narrowing the search scope of abnormal causes. Most studies assume that the observations are normal, independent and identically distributed (NIID), while the assumption of independence cannot always be satisfied under continuous manufacturing processes. Recent research has considered the existence of autocorrelation, but the recognition rate is overestimated. In this paper, a novel scheme is proposed to recognize control chart patterns (CCPs) in which the inherent noise is autocorrelated. By assuming that the inherent noise follows a first-order autoregressive (AR (1)) process, the one-dimensional convolutional neural network (1DCNN) is applied for extracting features in the proposed scheme, while the grey-wolf-optimizer-based support vector machine (GWOSVM) is employed as a classifier. The simulation results reveal that the proposed scheme can effectively identify seven types of CCPs. The overall accuracy is 89.02% for all the autoregressive coefficients, and the highest accuracy is 99.43% when the autoregressive coefficient is on the interval (−0.3, 0]. Comparative experiments indicate that the proposed scheme has great potential to identify CCPs in autocorrelated processes.

This paper considers the bivariate copula-based first-order Markov models to describe nonlinear dependence in autocorrelated data. Then the cumulative sum (CUSUM) charts based on log-likelihood ratio are developed to monitor the mean shifts. The performance of the proposed CUSUM charts is studied in both linear and nonlinear time series under weak, moderate and strong dependence by numerical simulations. Comparisons with the AR(1) model-based CUSUM and EWMA charts are present under the same range of scenarios. Simulation results show that the Clayton copula-based CUSUM charts are more effective to almost all the mean shifts and further have an obvious advantage in strong dependence. An example from chemical production is given to illustrate the implementation of the proposed control scheme.

Control chart pattern recognition was initially focused on single patterns with the assumption of normal, independent, and identical distribution. In practice, though, these assumptions are rarely valid in manufacturing processes, due to numerous influencing factors and short intervals in data collecting. It is necessary to consider that the inherent disturbance is autocorrelated and that two single patterns appear at the same time. This study presents a novel framework integrating Empirical Mode Decomposition (EMD) and one-dimensional Convolutional Neural Networks (1DCNN) with feature component selection for recognizing concurrent control chart patterns in autocorrelated manufacturing processes. We assume the inherent disturbance follows a first-order autoregressive (AR (1)) process and simulate eleven concurrent patterns. Then, the EMD method decomposes the concurrent pattern into a series of feature components, wherein the correlation coefficient is employed as the index by which to select the two feature components. Finally, the selected feature components and raw data are combined to create a feature vector that acts as the input for the 1DCNN model. The simulation results demonstrate that the proposed model achieves a recognition accuracy of 92.39%, outperforming both the singular spectrum analysis–support vector machine (SSA-SVM) and the singular spectrum analysis–random forest (SSA-RF) methods in terms of accuracy and robustness.