Cairo University Faculty of Economics and Political Science Department of Statistics The Performance of the Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters Prepared by Nesma Ali Mahmoud Saleh Supervised by Dr. Mahmoud Al-Said Mahmoud Associate Professor of Statistics Department of Statistics Faculty of Economics & Political Science Cairo University A Thesis Submitted to Department of Statistics, Faculty of Economics and Political Science as Partial Fulfillment of the Requirements for the M.Sc. Degree in Statistics 2012
Abstract The Adaptive Exponentially Weighted Moving Average (AEWMA) control chart has the advantage of detecting in balance mixed range of mean shifts. Its performance has been studied under the assumption that the process parameters are known. Under this assumption, previous studies have shown that AEWMA provides superior statistical performance when compared to other different types of control charts. In practice, however, the process parameters are usually unknown and are required to be estimated. Using a Markov chain approach, it is shown that the performance of the AEWMA control chart is affected when parameters are estimated compared with the known parameters case. The effect of different standard deviation estimators on the chart performance is also investigated. Finally, a performance comparison is conducted between the EWMA chart and the AEWMA chart when the process parameters are unknown. We recommend the use of the AEWMA chart over the ordinary Exponentially Weighted Moving Average (EWMA) chart especially when a small number of Phase I samples is available to estimate the unknown parameters. Keywords: AEWMA, Estimation Effect, Markov Chain, Average Run Length, Statistical Process Control. Supervised by Dr. Mahmoud Al-Said Mahmoud Department of Statistics Faculty of Economics and Political Science Cairo University ii
Name: Nesma Ali Mahmoud Saleh. Nationality: Egyptian. Date and Place of Birth: 16 / 8 / 1988, Giza, Egypt. Degree: Master. Specialization: Statistics. Supervisor: Dr. Mahmoud Al-Said Mahmoud. Title of the Thesis: The Performance of the Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters. Summary This study is concerned with investigating the effect of estimating the mean and the standard deviation of a normally distributed process from m in-control Phase I samples of size n=5 on the performance of the Phase II AEWMA chart. Based on the fact that the standard deviation has higher effect on the chart performance, the effect of different types of process standard deviation estimators on the AEWMA chart performance is also investigated. Using the Markov chain approach, the in-control and out-of-control performance of the AEWMA chart is evaluated in terms of the ARL and SDRL measures. A performance comparison is then conducted between the EWMA and the AEWMA charts under the relaxation of the known parameters assumption. Generally, the performance of the AEWMA chart is seriously affected when process parameters are estimated, especially when a small number of Phase I samples is available. The difference between the estimators effect was found to be more significant on the SDRL values than the ARL values, and the unbiased estimators provide the chart with better performance than the biased ones. The chart requires a large number of Phase I samples to perform as if parameters are known, however, optimizing the chart design parameters help decreasing this required number. Finally, the comparison results emphasized on the preference of using the AEWMA chart over the EWMA chart to monitor the process mean when process parameters are estimated, especially when a small number of Phase I samples is available and the shift size is unpredictable. iii
The study consists of seven chapters and one appendix as follows: Chapter One defines the study problem and motive and highlights the main objectives of the study. Chapter Two gives a review of literature on the development of control charts up to the adaptive charting techniques. For each chart presented, a full description is provided along with its benefits and drawbacks. Another review of literature is given on control charts with estimated parameters. Chapter Three is about deriving a Markov chain model for the AEWMA control chart with estimated parameters. First, the different types of performance measures are presented, followed by a standardization technique for the AEWMA chart, and then the parameters estimators to be used are identified. Second, the Markov chain model is discussed in details. Chapter Four investigates the effect of estimating the parameters on the performance of the AEWMA chart, and the effect of different standard deviation estimators on the chart performance. Chapter Five presents a method for optimizing the AEWMA chart performance when parameters are estimated. New design values are also provided. Chapter Six conducts a performance comparison between the AEWMA and EWMA charts under the known and the unknown parameters cases. Chapter Seven gives the concluding remarks and recommendations. Appendix A is about deriving a Markov chain model for the EWMA control chart with estimated parameters. iv
Dedication To my beloved family, for always believing in me v
Acknowledgements First, I would like to express my deepest appreciation and gratitude to my supervisor, Dr. Mahmoud Al-Said, for his continuous support, patience, motivation, enthusiasm, and encouragement throughout the thesis work. Without his guidance, this work would not have been successful. Second, I would like to thank my family for all their love, support, and praying. I am truly blessed to have such people in my life. axáåt ftäx{ vi
Table of Contents List of Figures... List of Tables..... List of Abbreviations. Page # x xi xiii Chapter 1: Introduction 1 1.1 Overview... 1 1.2 Problem Definition and Motivation... 2 1.3 Objectives of the Study.. 4 1.4 Organization of the Thesis..... 5 Chapter 2: Control Charts: Literature Review and Background 6 2.1 Statistical Basis of Control Charts. 6 2.2 Traditional Control Charting Techniques.. 8 2.2.1 Shewhart Control Charts....... 8 2.2.2 EWMA Control Chart... 9 2.2.3 Notes on Shewhart and EWMA Control Charts... 12 2.3 Adaptive Control Charting Techniques..... 12 2.3.1 Adaptive Sampling Parameters Control Charts........ 12 2.3.2 Adaptive Design Parameters Control Charts.... 13 2.3.2.1 The AEWMA Control Chart 14 2.3.2.2 Notes on the AEWMA Control Chart.. 16 2.4 Previous Studies of Control Charts with Estimated Parameters....... 16 2.4.1 Shewhart Control Charts...... 17 2.4.2 EWMA Control Chart... 18 vii
2.4.3 Adaptive Control Charts... 18 2.4.4 Multivariate Control Charts.. 19 Chapter 3: Methodology 21 3.1 Control Charts Performance Measures...... 21 3.1.1 In-Control and Out-of-Control ARL....... 21 3.1.2 Types of ARL Measure. 22 3.2 Computing the Performance Measures... 22 3.3 Markov Chain Approach for Evaluating the RL Properties of the AEWMA Chart with Estimated Parameters.. 23 3.3.1 Standardizing the AEWMA Control Chart...... 23 3.3.2 Estimating the Process Parameters... 25 3.3.3 A Markov Chain Model for the AEWMA Chart with Estimated Parameters.... 26 3.3.3.1 Probability Density Function of Q (Case of the pooled sample standard deviations)......... 29 3.3.3.2 Probability Density Function of Q (Case of the averaged sample standard deviations and the averaged sample ranges)....... Chapter 4: Performance Evaluation of the AEWMA Control Chart with Estimated Parameters 31 33 4.1 In-control Performance.... 33 4.2 Out-of-control Performance..... 38 Chapter 5: Statistical Design of the AEWMA Chart with Estimated Parameters 45 5.1 Conceptual Framework of the Statistical Design... 45 viii
5.2 Statistical Design of the EWMA Control Chart.. 47 5.3 Statistical Design of the AEWMA Control Chart... 48 Chapter 6: Comparing the AEWMA Chart to the EWMA Chart under 52 the Relaxation of the Known Parameters Assumption 6.1 Performance Comparison..... 52 6.2 Known Parameters Case...... 53 6.3 Estimated Parameters Case...... 55 Chapter 7: Conclusions and Recommendations 61 References 63 Appendix A 68 الملخص العربي ix
List of Figures Figure Page # Figure 2.1 Graphical illustration of control charts 6 Figure 2.2 Shewhart control chart with undetected shift 9 Figure 2.3 EWMA control chart with undetected shift 11 Figure 3.1 Markov chain model 26 Figure 6.1 The logarithm of the out-of-control zero-state ARL values of the EWMA 1 (0.50), EWMA 2 (4), and AEWMA (0.50, 4) with known parameters when there is a sustained step shift in the process mean of size δ 54 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8 The logarithm of the out-of-control zero-state ARL values of the EWMA 1 (1.00), EWMA 2 (4), and AEWMA (1.00, 4) with known parameters when there is a sustained step shift in the process mean of size δ The logarithm of the out-of-control worst-case ARL values of the EWMA 1 (0.50), EWMA 2 (4), and AEWMA (0.50, 4) with known parameters when there is a sustained step shift in the process mean of size δ The logarithm of the out-of-control worst-case ARL values of the EWMA 1 (1.00), EWMA 2 (4), and AEWMA (1.00, 4) with known parameters when there is a sustained step shift in the process mean of size δ The logarithm of the out-of-control zero-state ARL values of the EWMA 1 (0.50), EWMA 2 (4), and AEWMA (0.50, 4) when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters: (a) m=20; (b) m=100; (c) m=500 The logarithm of the out-of-control zero-state ARL values of the EWMA 1 (1.00), EWMA 2 (4), and AEWMA (1.00, 4) when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters: (a) m=20; (b) m=100; (c) m=500 The logarithm of the out-of-control worst-case ARL values of the EWMA 1 (0.50), EWMA 2 (4), and AEWMA (0.50, 4) when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters: (a) m=20; (b) m=100; (c) m=500 The logarithm of the out-of-control worst-case ARL values of the EWMA 1 (1.00), EWMA 2 (4), and AEWMA (1.00, 4) when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters: (a) m=20; (b) m=100; (c) m=500 54 55 55 57 58 59 60 x
List of Tables Table Page # Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Optimal design parameters that produce an in-control ARL of 500 when the process parameters are known The in-control ARL comparison of the six standard deviation estimators when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters used are those producing an in-control ARL of 500 when the parameters are known The in-control SDRL comparison of the six standard deviation estimators when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters used are those producing an in-control ARL of 500 when the parameters are known The number of Phase I samples m of size n=5 required to produce an incontrol ARL of at least 490 when the design parameters used are those based on known parameters The corrected control limits h * required to produce an in-control ARL of 500 for each standard deviation estimator using m Phase I samples of size n=5. The values of λ and k are those of the known parameter case The out-of-control ARL and SDRL comparison of the six standard deviation estimators when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters λ and k used are those producing an in-control ARL of 500 when the parameters are known for the shift pair sizes (0.50, 4) with their corresponding h * The out-of-control ARL and SDRL comparison of the six standard deviation estimators when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters λ and k used are those producing an in-control ARL of 500 when the parameters are known for the shift pair sizes (1.00, 4) with their corresponding h * The out-of-control ARL and SDRL comparison of the six standard deviation estimators when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters λ and k used are those producing an in-control ARL of 500 when the parameters are known for the shift pair sizes (1.00, 5) with their corresponding h * 34 35 36 38 39 40 41 42 xi
Table 4.9 The out-of-control ARL and SDRL comparison of the six standard deviation estimators when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters λ and k used are those producing an in-control ARL of 500 when the parameters are known for the shift pair sizes (1.00, 6) with their corresponding h * 43 Table 5.1 The optimal values of λ and L that produce an in-control ARL of 500 when m subgroups each of size n=5 are used to estimate the parameters of the EWMA chart used to detect a sustained step shift in the process mean of size δ Table 5.2 The optimal values of λ, h and k that produce an in-control ARL of 500 when m subgroups each of size n=5 are used to estimate the parameters of the AEWMA chart used to detect in balance the shift pair sizes ( δ1, δ 2 ) in the process mean. (q=0.001 and α = 0. 05 ) 48 49 Table 5.3 The out-of-control ARL and SDRL when there is a sustained step shift in the process mean of size δ and when m subgroups each of size n=5 are used to estimate the unknown parameters. The design parameters λ, h, and k used are those the optimal parameters producing an in-control ARL of 500 when the parameters are estimated. 51 xii
List of Abbreviations AEWMA ARL CL CUSUM EWMA LCL MEWMA RL SDRL SPC SQC UCL VSI VSS VSSI Adaptive Exponentially Weighted Moving Average control chart Average Run Length Center Line Cumulative Sum control chart Exponentially Weighted Moving Average control chart Lower Control Limit Multivariate Exponentially Weighted Moving Average control chart Run Length Standard Deviation of Run Length Statistical Process Control Statistical Quality Control Upper Control Limit Variable Sampling Interval control chart Variable Sample Size control chart Variable Sample Size and Interval control chart xiii
Chapter 1 Introduction 1.1 Overview Statistical quality control (SQC) refers to the use of statistical techniques to measure and control the quality of processes. Statistical process control (SPC) is one of the SQC techniques that measures and analyzes variations in processes. SPC main intention is to assure that the product is manufactured as designed, i.e., to ensure that the process operates at the required targets. A valuable tool of SPC is the control chart. Control charts are graphical representation of certain descriptive statistics for specific quantitative measurements of the process. Their main role is to monitor production processes to detect any performance deterioration. Once the cause of deterioration is identified, it is eliminated if possible. Nowadays, control charting techniques are widely employed in various fields; such as manufacturing, service industries, software development, health care, public-health surveillance, and banking. Naturally, during the production, there exists variability in the process quality characteristics. This variability usually leads to undesirable changes in the process parameters, such as the process mean and/or standard deviation. The variability in process quality characteristics can happen due to common causes and/or special causes. Common causes of variation occur due to random effects. Such variation is small and thus does not affect the process distribution, i.e., the process parameters are not changed. When the process operates with common causes of variation, it is said to be in-control. On the other hand, special causes of variation occur due to human, raw material, machinery, or environmental effects. Such variation is substantial and thus alters the distribution of the quality characteristics, and consequently the process parameters are changed and said to be exposed to shifts in their values. When the process operates with 1
special causes of variation, it is said to be out-of-control. The main role of control charts is to identify assignable causes of variation that are presented in the form of shifts in the process parameters, and eliminate them if possible. 1.2 Problem Definition and Motivation Since the introduction of control charts in 1920s by Walter A. Shewhart, there were many contributions for developing control charts to overcome the major disadvantage of the Shewhart chart; that is the inefficiency of detecting small magnitudes of shifts. Roberts (1959) introduced the Exponentially Weighted Moving Average (EWMA) control chart and Page (1954) introduced the Cumulative Sum (CUSUM) chart. Both charts, EWMA and CUSUM, use all the information accumulated in a sequence of samples by combining previous and current observations. Their effectiveness in quickly detecting small and moderate magnitudes of shifts has been proven in several investigative studies. However, the design of each of the three charts, Shewhart, EWMA, and CUSUM, is based on being sensitive to a specific magnitude of parameter shift. Accordingly, if other magnitudes of shift occur in the process parameters, practitioners may face a delay in detecting them. In addition, Woodall and Mahmoud (2005) emphasized that the EWMA chart has a relative disadvantage compared with the Shewhart chart in that it severely suffers from the inertia problem; that is when the EWMA chart statistic is positioned near one of the control limits and the process shift occurs in the opposite direction of this limit (given in details in Chapter 2). Practically, the actual magnitude and direction of the shift are unpredictable and the chart statistic could be in a disadvantageous position directly before such shift occurs. Thus, in order to design better control schemes, researchers have suggested adding adaptive capabilities to control charts. Tsung and Wang (2010) summarized these adaptive capabilities into two categories: adaptable sampling parameters and adaptable design parameters. One of the charts proposed with adaptable design parameters is the Adaptive Exponentially Weighted Moving Average (AEWMA) control chart. The AEWMA chart is an EWMA chart with a variable smoothing parameter, i.e., adaptable weights. That is, the EWMA chart gives fixed weights for each of the previous and current observations based on the pre-specified magnitude of shift required to be 2
detected, while the AEWMA chart adapts the weights of the previous and current observations based on the magnitude of shifts the process is actually experiencing. The AEWMA chart has proven its superiority over several different types of control charts; see for example Capizzi and Masarotto (2003) and Costa et al. (2010). Woodall and Mahmoud (2005) also showed that the AEWMA chart has much better inertial properties than the ordinary EWMA chart. These conclusions reached about the AEWMA performance were all based on the assumption of known process parameters. However, in practice, process parameters are usually unknown and are required to be estimated. Many researchers have pointed out that control charts with estimated parameters provide inferior statistical performance than those with known parameters. See, for example, Quesenberry (1993), Chen (1997), Jones et al. (2001, 2004), and Castagliola et al. (2012). Jensen et al. (2006) emphasized that ignoring the unknown parameters while designing control charts, increases the variability in the monitoring scheme and thus affects the control chart performance. Yet, none of these researchers has investigated the effect of estimating the process parameters on the performance of the AEWMA control chart. Generally speaking, concern about estimating the process standard deviation is more than that of the process mean. This is because the process standard deviation has a higher effect on the control chart performance. Several standard deviation estimators are proposed in SPC literature. Yet, a problem that faces the practitioners is which of these estimators is better to be used. Most of the studies in literature evaluate the charts performances using one estimator for the process standard deviation, usually the average sample standard deviations or the average sample ranges. However, Chen (1997) evaluated the performance of the Shewhart 3-sigma X control chart when different estimators for the process standard deviation are used. Chen (1997) recommended the use of the pooled sample standard deviations estimator unless, for simplicity aspects, the range-based estimator is required. Mahmoud et al. (2010) recommended against the use of the range-based estimators when estimating the process standard deviation in quality control applications because of its low efficiency. In addition, they showed that pooling the sample standard deviations is uniformly better than averaging them in terms of 3
relative efficiency. In this dissertation, the researcher investigates the effect of these estimators on the statistical performance of the AEWMA control chart. 1.3 Objectives of the Study When the process parameters are unknown, the construction of control charts go through two phases; Phase I and Phase II. In Phase I (or retrospective phase), a historical data set (m samples each of size n) is accumulated and used to bring the process to the incontrol state by detecting and removing out-of-control samples. Then, the in-control process parameters are estimated once the out-of-control samples are discarded. In Phase II, the control chart intended to monitor the process is designed using the parameters estimates obtained from Phase I, with the aim of quickly detecting shifts in process parameters. Hence, the performance of Phase II control charts depends mainly on how well the parameters are estimated from Phase I. Consequently, the main objective of this study is to investigate the effect of estimating the process mean and standard deviation from an in-control Phase I data set on the performance of the Phase II AEWMA control chart used to monitor the process mean, under the normality assumption. The effect of different process standard deviation estimators on the chart performance is also investigated, with the aim of identifying the estimator that has the least effect on the chart performance. Using a Markov chain approach, the in-control and out-of-control performance of the AEWMA chart with estimated parameters are evaluated in this study. Two performance measures are considered; the average and standard deviation of the run length (RL) distribution. The RL is defined as the number of chart statistics plotted until the chart signals an out-of-control condition. In addition, the appropriate size of the Phase I data required for the estimation process to achieve a level of performance close to that when parameters are known is determined. Then, a statistical design for the AEWMA chart when process parameters are estimated is developed. A secondary objective of the study is to conduct a performance comparison between the EWMA chart and the AEWMA chart under the relaxation of the known parameters assumption. The importance of this comparison is to show whether the AEWMA chart 4
can maintain its superiority over the ordinary EWMA chart or not when the process parameters are estimated. 1.4 Organization of the Thesis Chapter 2 presents a literature review on the development of control charts in general, the AEWMA chart in particular, and some control charts with estimated parameters. The performance measures of the AEWMA chart with estimated parameters are derived in Chapter 3 using a Markov chain approach. Chapter 4 presents the results of the in-control and out-of-control performance of the AEWMA chart when different process standard deviation estimators are used. In Chapter 5, the way of optimizing the AEWMA chart performance under the unknown parameters assumption is presented. Chapter 6 provides a performance comparison between the EWMA and AEWMA charts. Finally, Chapter 7 gives the concluding remarks and recommendations. 5
Chapter 2 Control Charts: Literature Review and Background 2.1 Statistical Basis of Control Charts Control charts usually consist of three main elements; Center Line (CL), Lower Control Limit (LCL), and Upper Control Limit (UCL). The CL represents the target that the process should achieve, while the LCL and UCL represent the limits for the acceptable process variations. During the production process, successive samples are drawn at regular time intervals and some sample statistics are computed and plotted on the control chart. If the plotted statistic is located between the LCL and the UCL, then the process is considered to be in-control. If the plotted statistic exceeds either the LCL or the UCL, then the process is considered to be out-of-control. The case of a point plotted outside the control limits is referred to as a chart signal. Once a signal is declared, the production process is stopped in order to fix the problem if possible. Figure (2.1) illustrates a simple control charting technique. Chart Statistic Value Chart Statistic Signal UCL CL LCL Sample Number or Time Figure 2..1: Graphical illustration of control charts 6
Control charting technique is considered to be a hypothesis testing approach in a graphical form. Each time a sample is drawn, the calculated statistic is plotted to test whether the process is in- or out-of-control. Hence, a control chart represents a series of hypotheses testing, where every time we test H Process is in - control vs. H : Process is out - of - control 0 : 1. The null hypothesis refers to that the process parameter is not significantly different from the target value, while the alternative hypothesis refers to that the process parameter has deviated significantly from its target value. The test statistic in this case is the plotted chart statistic. The critical values for the chart statistic are the control limits. The power of the chart to detect out-of-control conditions, denoted by 1 β, is the probability to reject H0 every time the process is actually out-of-control, where β represents the probability of Type II error. On the other hand, the probability to reject H 0 when the process is in-control is denoted by α, and represents the probability of Type I error. In SPC literature, α is usually referred to as the false alarm probability. In this chapter, a review is given on the development of control charts. This review of literature is divided into three parts; the first part reviews two traditional control charts; Shewhart and EWMA charts. The second part reviews some of the adaptive control charts in general and the AEWMA chart in particular. The last part reviews some of the charts studied under the unknown parameters assumption. For all the presented charts, it is assumed that monitoring the mean of the quality characteristic X is of interest, and that the observed data (x i1, x i2,..., x in ), i = 1, 2, 3,..., are samples of size n taken at regular time intervals. For each sample, it is assumed that x i1, x i2,..., x in are independent and identically distributed (i.i.d) normal random variables with mean μ and standard deviation σ. The objective is then to detect any change in μ from an in-control or target value μ 0. 7