Univariate parametric and nonparametric double generally weighted moving average control charts

Abstract

Statistical process control (SPC) is a collection of scientific tools developed and engineered to diagnose unnecessary variation in the output of a production process and eliminate it or perhaps accommodate it by adjusting process settings. The task of quality control (QC) is of fundamental importance in manufacturing processes when a change in the process causes misleading results, this alteration should be detected and corrected as soon as possible. Statistical QC charts originated in the late 1920s by Dr. W. A. Shewhart provide a powerful tool for monitoring production lines in manufacturing industries. They are also have been implemented in various disciplines, such as sequential monitoring of internet traffic flows, health care systems, and more. Shewhart-type charts are effective in detecting large shifts in the process but ineffective in detecting small to moderate shifts. This blind spot allows small shifts (smaller than one standard deviation) to continue undetected in the process, thereby incurring larger total costs for manufacturers.

This thesis addresses this issue by augmenting current time-weighted charts (charts that use all the information from the start of a process until the most recent sample/observation) with a Double Generally Weighted Moving Average (DGWMA) chart, leading to more effective process monitoring. The objective of this thesis is to provide the fundamentals and introduce the researcher/practitioner to the essentials of the univariate DGWMA chart from both parametric and nonparametric perspectives. Numerous concepts and characteristics of proposed DGWMA charts are discussed comprehensively. Theoretical expressions and detailed calculations have been provided to aid the interested reader to familiarize and study the topic more thoroughly. This thesis paints a bigger picture of the DGWMA chart in a sense that other time-weighted charts such as the Generally Weighted Moving Average (GWMA), Exponentially Weighted Moving Average (EWMA), Double Exponentially Weighted Moving Average (DEWMA) and Cumulative Sum (CUSUM) fall under this umbrella. Both real-life data and simulated examples have been embedded throughout the thesis. We make use of R and Mathematica software packages to calculate numerical results related to the run length distribution and its associated characteristics in this thesis.

We only consider control charts for monitoring the process location parameter. However, our conclusions and recommendations are extendable for the process dispersion parameter. In this thesis, we consider the DGWMA chart as the main chart and the EWMA, DEWMA, GWMA, and CUSUM charts as special cases. The thesis consists of the following chapters with a short description for each chapter as follows:

Chapter 1 provides a brief introduction to SPC concepts and gives a literature review in terms of background information for the research conducted in this thesis. The scope and objectives of the present research are highlighted in detail.

Chapter 2 provides an overview and a theoretical background on the design and implementation of the DGWMA chart derived from the SPC literature review. The properties of the DGWMA chart, including the plotting statistic, the structure for the weights, the control limits (exact/steady-state), etc. are considered in detail. The weighting structure of the DGWMA chart and its special case are discussed and pictured to emphasize the impact of weights in increasing the detection capability of time-weighted charts. Three approaches are described and investigated for calculating the run length distribution and its associated characteristics for the DGWMA chart and its special case the DEWMA chart; this includes: (i) exact approach; (ii) Markov chain approach; (iii) Monte Carlo simulation.

In Chapter 3 we develop a one-sided generalized parametric chart (denoted by DGWMA-TBE) for monitoring the time between events (TBE) of nonconformities items originating from the high-yield processes when the underlying process distribution is gamma and the parameter of interest is known (Case K) and unknown (Case U). A Markov chain approach is implemented to derive the run length distribution and its associated characteristics for the DGWMA and DEWMA charts. An exact approach is also used to derive closed-form expressions for the run length distribution of the proposed chart. Performance analysis has been undertaken to execute a comparative study with several existing time-weighted charts. The proposed chart encompasses one-sided GWMA-TBE, EWMA-TBE, DEWMA-TBE, and Shewhart-type charts as limiting or special cases. The CUSUM-TBE chart is also included in the performance comparison. The necessary design parameters are provided to aid the implementation of the proposed chart and finding the optimal design and near optima design that is useful for practitioners. Alternative discrete distributions are considered for the weights of the GWMA-TBE chart and a discussion is provided to address the connection between new weights originating from the suggested distributions and the chart’s capability in detecting shifts. As a result, one can design an optimal GWMA-TBE chart by replacing weights from the discrete Weibull distribution without the implementation of the double exponential smoothing technique.

Chapter 4 focuses on developing a two-sided nonparametric (distribution-free) DGWMA control chart based on the exceedance (EX) statistic, denoted as DGWMA-EX when the parameter of interest is unknown (Case U) and the underlying process distribution is continuous and symmetric. An exact approach and a Markov chain approach are considered to calculate the run length distribution and its associated characteristics for the proposed chart. A performance comparison has been undertaken to execute analysis with other nonparametric time-weighted charts available in the SPC literature. The proposed chart en-compasses two-sided GWMA-EX, EWMA-EX, DEWMA-EX, and Shewhart-type charts as limiting or special cases. The CUSUM-EX chart is also included in the performance comparison Also, the performance of the proposed DGWMA-EX chart has been evaluated under different symmetric and skewed distributions in comparison with its main counterparts, and the necessary results and recommendations are provided for practitioners to design an optimal chart.

Chapter 5 encloses this thesis with a summary of the research conducted and provides concluding remarks concerning future research opportunities.