Function Based Adaptive Exponentially Weighted Moving Average Control Charts

Abstract

The statistical process control provides tools to enhance production stability and meet quality standards. In this regard, statistical quality control charts are the most proximal tools. As control charts are responsible to give the earliest detection of any change/deviation from an in-control parametric state to an out-of-control state so, monitoring control chart designs must efficiently fulfill this need. Deploying an efficient control chart means a tool that gives quick detection of shifts/deviations to provide a quality production unit. The manufacturers welcome modifications in the monitoring charts with minimum risk, fewer defects, and the use of a reasonable number of resources. In the beginning, simple pair of control limits-based Shewhart control charts were introduced which were capable to capture the large process shifts. Over time, demands enrich to modified efficient chart designs in terms of small shift detection ability. The exponentially weighted moving average (EWMA) control charts were developed to capture small to moderate process shifts efficiently. The EWMA charts prove efficient by using the weightage (smoothing constant) to recent and previously computed values as part of the current plotting statistic computations, characterized as memory control charts. By introducing this idea of developing memory control charts, the performance of the charts drastically improved in terms of providing small to moderate shift detection rapidly. As time passes, the researchers pointed out that the shift in any production process may vary with any magnitude at any time. The respective designs of Shewhart and EWMA control charts are designed for the known process shifts scenario which contradicts the reality that the process already knows the magnitude of the shift before its occurrence. To handle this issue, the researchers developed adaptive control charts which adapt the smoothing constant (SC) value as per the estimated shift magnitude. This idea, to deal with real-time shift sizes revolutionized the statistical process control chart designs. These designs proved more efficient than the existing charts and were named the adaptive EWMA (AEWMA) charts. In AEWMA charts shift estimation-based selection of SC value to compute EWMA statistic is the main factor that made them more sensitive. So, if there is a small process shift then a small SC value should be assigned to get a quick detection or otherwise vice versa. In the presented research, a function is developed to determine the SC value as per the estimated shift size on a real-time basis. It gives improved results in comparison with its counterparts for providing quick detection over a wide range of shifts. The proposed design is also better in terms of adapting efficiently for any magnitude of shift with any sample size as this feature was also lacking in the existing AEWMA control charts. The proposed design is used to construct a function-based mean monitoring control chart, variance monitoring control chart, coefficient of variation monitoring control chart, and the joint monitoring of mean and variance using a single statistic strategy. These designs are named as the ‘Function based adaptive exponentially weighted moving average control charts’. The proposed function is continuous and designed for the normal population characteristics, whereas the counterpart designs operate with the allocation/selection of the SC value. The efficacy of the presented chart designs is evaluated in terms of the smaller run length characteristics over the wide range of process shifts. Various parametric value settings are used to conduct a simulation study by using the monte carlo simulation method for each control chart design in R language software with 50000 iteration runs for both the proposed and the existing counterpart plotting statistics. The respective computed results are exhibited in extensive tables. The tabular analysis strengthens the argument that the proposed chart is exclusively efficient in providing quick detection of shifts than the existing counterpart: mean, variance, CV, and joint monitoring plotting statistics. Moreover, the methodology of each chart is accompanied by an illustrative study, a real-life example, and algorithmic steps to provide a deep insight into the concept implementation.