Statistical Performance of Shewhart S - Chart with Variable Sample Size and Estimated Parameters

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Cairo University Faculty of Economics and Political Science Department of Statistics - English Section Graduation Project Statistical Performance of Shewhart S - Chart with Variable Sample Size and Estimated
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Cairo University Faculty of Economics and Political Science Department of Statistics - English Section Graduation Project Statistical Performance of Shewhart S - Chart with Variable Sample Size and Estimated Parameters Presented by: Madonna Magdy Besada Maria Ashraf Sobhy Marina Maher Joseph Maureen Hany Sadek Under the Supervision of Prof. Mahmoud Al-Said Mahmoud Submission date: 24/6/2012 0 Abstract In this report we discuss the behaviour of the variable sample size Shewhart S- chart when the parameters of the underlying process are unknown and thus have to be estimated. We focus on the effect of estimating the process standard deviation. The in-control average run length (ARL0) of the control chart with estimated parameters is compared with the ARL0 of the chart of known parameters case, fixing the in-control average sample size (ASS0) for both. Additionally, we give some recommendations on the choice of Phase I sample size and number of Phase I samples in the context of Shewhart S-charts with variable sample size and estimated parameters. 1 Abbreviations SPC QC CL UCL UWL LCL LWL RL ARL ARL 0 ARL 1 ASS ASS 0 ASS 1 EWMA CUSUM VSS VSI n h k n s n L m n 0 SAS Statistical Process Control Quality characteristic Centre line Upper control limit Upper warning limit Lower control limit Lower warning limit Run length Average run length In-control average run length Out-of-control average run length Average sample size In-control Average Sample Size Out-of-control average sample size overall probability of false alarm Exponentially weighted moving average Cumulative sum control chart Variable sampling size Variable sampling interval Sample size Sampling interval Control limit coefficient Small sample size Large sample size number of Phase I samples Phase I sample size Statistical package for statistical analysis system 2 Table of contents Section 1: Introduction Definition of control charts Types of control charts Adaptive control chart The objective The methodology literature review Section 2: VSS S-Chart The design of S-chart Measures of performance..1 Section : The Methodology (Simulation) Simulation steps..1.2 Simulation results Section 4: An illustrative example...24 Section : Conclusion and Recommendations Conclusion 2.2 Recommendation...2 References List of Bibliography 1 Appendices Appendix (1): Simulation results tables.. 2. Appendix (2): Example data. 4. Appendix (): Simulation results figures Appendix (4): SAS programs...1 Index of Figures 1.1 Shewhart control chart for monitoring the process parameter Adaptive VSS Shewhart control chart for monitoring the process mean parameter ARL 0 vs. m at ASS 0 = and n S =4 and n L = ARL 0 vs. n 0 at ASS 0 = and n S =4 and n L = 20. ARL 0 vs. m at ASS 0 = and n S = and n L = ARL 0 vs. n 0 at ASS 0 = and n S = and n L = Monitoring company (A) performance (at m=0) Monitoring company (B) performance (at m=100)...2 4 Index of Tables.1 Best n 0 and m for each combination at ASS 0 = Best n 0 and m for each combination at ASS 0 = Best n 0 and m for each combination at ASS 0 =.. 2 Section 1: Introduction Recently, controlling and improving quality have been considered one of the most important business strategies to achieve customer satisfaction globally. Statistical Process Control (SPC) is a collection of statistical and analytical tools that can be used to achieve process stability and variability reduction about the process target value. The most important tool in SPC is the control charts which was first introduced by Shewhart in the 1920's. 1.1 Definition of control charts: Control chart is an online graphical representation used to plot sample points of certain quality characteristic (QC) in a production process. QC is an important physical or temporal characteristic of a product or service such as; weight of a product or the time consumer has to wait to get a service. In any production process we need to make sure that one or more of the process parameters has/have to satisfy a target value(s) (which represents the optimal value of the interested QC). Practically, we expect variability around the target value. There are two reasons for this variability; common and special causes. First, common causes are natural or random variation as their effect exists in all the process output. This variation is small and cannot be removed by statistical tools for quality control. For example, the same worker in the same production condition does not fill the same amount of the company product in the designated package. As long as the process operates with only common causes of variation, the process is said to be in-control. 6 Second, special causes of variation are occurred due to something wrong has happened in the process. The reasons for special causes of variations are usually classified into four reasons; which are machine, workers, raw material, and environment. For example, the machine might become less effective due to the destruction of one of its components. The variability caused by special causes is substantial. It is usually assumed that when special causes occur, the distribution of the QC is changed (shifted). A process is said to be out-of-control if it operates in the presence of special causes. This type of variability can be detected by control charts. In fact, the main purpose of a control chart is to separate common causes from special causes. When a special cause of variation is detected by a control chart, the process is stopped, and investigations are carried on to determine the reason and eliminate it (if possible). The process is resumed when the special cause of variation is removed. Since special causes lead to changes in the process parameters, control charts are used to detect any changes in one or more of these parameters. The control chart is built up by taking samples every fixed time interval, say h hours. Then plot the appropriate sample statistic for each sample. For example, if we are interested in monitoring changes in the process mean, we plot the sample average. Figure (1.1) shows the design of a Shewhart control chart for monitoring a process parameter. The horizontal axis represents the sample number or the time when the sample was drawn. The vertical axis represents the value of the sample statistic. Shewhart chart includes three main lines which are the centre line (CL), the lower control limit (LCL) and the upper control limit (UCL). The process is considered to be in-control if the sample point lies between the LCL and the UCL. However, if the sample point lies outside these limits, the process is deemed out-of-control. As shown in Figure (1.1), all sample points are plotted between the LCL and UCL except the th sample point which indicates that the illustrative process has gone out-ofcontrol at this time. Figure (1.1): Shewhart control chart for monitoring a process parameter Chart statistic UCL CL Sample number LCL 1.2 Types of control charts: There are many classifications for the control charts. First, based on number of quality characteristics under investigation. We have two control charts, the univariate control chart which is used to monitor one quality characteristic and the multivariate control chart which is used to monitor more than one quality characteristic. Second, based on the type of quality characteristic, we have variable control chart on which the QC can be measured on numerical scale. If a QC cannot be represented numerically, we use the attribute control chart where the items are classified as conforming and nonconforming to the specifications on the QC, as the QC is said to be attribute. Third, based on whether or not the process parameters (location and dispersion parameters) are known. When the parameters are known we use Phase II control charts to monitor any change in the process parameter. To measure the performance of these charts we use 8 the average run length ARL which is the average number of points plotted until the chart gives a signal. There are two types of ARL; in-control average run length ARL 0 and out-of-control average run length ARL 1. First, ARL 0 is the average number of points plotted before the chart gives a false alarm i.e. despite being in-control, the chart signals. Second, ARL 1 is the average number of points plotted until the chart detect a shift. When comparing between Phase II charts, we fix the ARL 0 for all of them and compare the performance based on the ARL 1 values. Then the chart with the least ARL 1 is the best. Practically, the process parameters are usually unknown so we estimate them using historical data set of m in-control samples and the control charts used in this case are called Phase I control charts. The overall probability of a signal is used to measure the performance of Phase I control charts. Similarly, there are two types for the overall probability of a signal; overall probability of false alarm ( ) and overall power which are defined as: ( ),, where and are the marginal probabilty of Type I and Type II error, respectivily. To compare the performance of Phase I charts, we fix the overall probability of false alarm and compare according to the overall power. The chart with the largest power is the best. Fourth, based on the type of plotted statistics we have Shewhart and Non-Shewhart charts. Shewhart control charts take decisions based on the current chart statistic. The Shewhart control charts for variable include -chart (the sample means are plotted in order to control the mean of the QC), R-chart (the sample ranges are plotted in order to control the variability of the QC), S-chart (the sample standard deviations are plotted in order to control the variability of the QC), S 2 -chart (the sample variances are plotted in order to control the variability of the QC). The Shewhart control charts for attribute include C-chart (plotting the number of defectives per item), U-chart (plotting the rate of defectives per item), np-chart (plotting the number of defective items in a sample), P-chart (plotting the percentage of defective items in a sample). 9 On the other hand, Non-Shewhart control charts are based on the current and the previous chart statistics. In the 190s, the exponentially weighted moving average (EWMA) control chart and the cumulative sum control chart (CUSUM) were introduced. It was proved in several studies that they are effective in detecting small and moderate shifts quickly. The EWMA statistic is defined as; ( ), 0 1 where is the current sample statistic and is the previous chart statistic. is called the smoothing parameter, small values for are chosen to detect small shifts faster, while large values for to detect large shifts faster. It should be noted that Shewhart chart is a special case from the EWMA chart when = 1. Shewhart chart is better for Phase I analysis because the out-of-control samples can be removed as the decision is based on the current statistic. On the other hand, EWMA and CUSUM are not recommended for Phase I analysis as they are based on the current and the previous statistics. As mentioned before, EWMA and CUSUM are better for Phase II in detecting small shifts, while Shewhart charts are better for detecting large shifts. Sometimes in Phase II analysis, a combination of Shewahrt and Non-Shewhart charts is used to assure detecting small and large shifts taking into account the increase of the probability of false alarm. 1. Adaptive control charts: Control chart usually has three design parameters: the sample size (n), the sampling interval (h) and the control limit coefficient (k). Standard Shewhart charts are used with fixed design parameters. There were many contributions in control charts; such as Non-Shewhart charts (EWMA and CUSUM) or adaptive control charts, which are used to overcome the major disadvantage of the Shewhart chart; that is the inefficiency of detecting small shifts. In this report we focused on the later type of charts. 10 Adaptive control chart has at least one variable design parameter (variable sampling size VSS, variable sampling interval VSI, or variable control limit coefficient), where switching among different parameters depends on the location of the current plotted sample statistic in the chart. It consists of three regions which are action region, warning region and central region. Action region is the region where we get a signal (the process is out-of-control), which occurs when the sample point falls outside the interval (, ), where is the largest allowable value for the process parameter, is the smallest allowable value for the process parameter. Warning region is the region where sample point falls between ( ) or ( ), which means that the process is in-control, but there is an evidence that a signal might occur. Central region is the region where the sample point falls between (, ). Here, the process is in-control and this is the best region where there is no evidence that any signal might occur as it includes the central line which is the optimal value for the process parameter. In adaptive control chart, if the current plotted statistic lies in the central region then there is no indication that the process parameters have changed so the next sample will be taken with a smaller sample size and/or a larger sampling interval, and/or a larger control limit coefficient. On the contrary, there is an indication that the process parameters have changed if the current sample statistic lies in the warning region, therefore the next sample will be taken from the process with larger sample size, smaller sampling interval, and/or smaller control limit coefficient will be used. It is shown in Figure (1.2), point A lies in the central region; therefore the next sample should be taken with smaller sample size (n s ), while point B lies in the warning region so the next sample size should be larger (n L ), at the 8 th sample point, the process has gone out-ofcontrol. 11 Figure (1.2): Adaptive VSS Shewhart control chart for monitoring the process mean parameter X j X chart c c W 1 c CL A c c W 2 B c K 2 c c 9 10 Sample number K The objective of this project: The process performance depends on the location and dispersion parameters but the priority is for controlling dispersion parameter first. The majority of studies in this area focused on control charts for location parameters, such as the process mean. However, only few studies focused on monitoring the dispersion parameters, such as the process standard deviation.if the process standard deviation is not stable, then the accuracy of the control chart for the location parameter is questionable. Consequently, this report focuses on Shewhart S-Chart with Variable Sample Size to monitor the standard deviation of an intended QC. The performance of the VSS S-chart has been studied in the literature assuming known parameter case. Practically, in most cases the parameters are unknown and have to be estimated from an in-control Phase I data set. The performance of the VSS S-chart with estimated parameters has not been studied, yet. Our report is similar to that made by Castagliola P. et al (2011) who studied VSS -chart with estimated parameters. The VSS -chart is used to monitor the process mean. Our aim is to study the statistical performance of the VSS S-chart with estimated parameters. 12 1. Methodology: In this report we used a Monte Carlo simulation technique to evaluate the performance of the VSS S-chart with estimated parameters from an in-control Phase I data set. We will show the effect of some important Phase I factors, such as the number of samples and the sample size, on Phase II performance. We used the SAS software to perform the necessary calculations in our simulation. Exactly 0,000 data sets are used to evaluate the performance in terms of the incontrol Average Sample Size ( ) and in-control Average Run Length ( ). Data sets are generated from Normal distribution. 1.6 Literature Review: The properties of the variable sampling interval (VSI) -Chart were first studied by Renyolds et al. (1988). See also Reynolds (1990), Runger and Montgomery (199), Amin and Miller (199). Moreover, the properties of -chart with variable sample size and variable sampling interval were studied by Prabhu et al. (1994) who studied it under the assumption that the process starts in a state of out-of-control, in 199 Costa studied the properties of VSSI - Chart when the process mean is out-of-control (assuming exponential distribution), while in 1996, he extended the VSSI control charts to the joint X-bar and R-chart. Zhang and Hua (2002) also studied the np chart with variable sample size or variable sampling interval. The -Charts with estimated parameters were studied by Del Castillo (1996). Jones and Champ (2002) studied the design of EWMA charts with estimated parameters. While the dispersion-type control charts (S 2, S, and R control charts) were studied by Chen (1998) and Shahriari et al. (2009). The rest of this report is organized as follows; we will discuss the VSS S-chart in Section 2. The simulation results are presented in Section. In Section 4, we will illustrate the use of the VSS S-chart with estimated parameters using an illustrative example. Finally, conclusion and recommendation are given in Section. 1 Section 2: VSS S-chart 2.1 The design of the VSS S-chart: Let, i= 1, 2,... and j= 1,..., denote certain quality characteristic of a process. We assume that s are independent and normally distributed Phase II samples each of size with mean (a+ ) and standard deviation (b ). If a=0 and b=1 the process is in-control, otherwise the process is out-of-control. (S i ) on S-chart Process standard deviation can be monitored by plotting the sample standard deviation S i = ( ) ( ) i= /. In this report we will consider only the upper control limit to detect the increase in the standard deviation. We will assume that the lower control limit is zero; we rely on the fact that in S-chart the standard deviation reduction is corresponding to a desirable improvement in the quality, so we care only about its increase. Some control charts users care about the lower limit to detect the decrease in the standard deviation and investigate the reason of this reduction to use it for improving the production process. Similarly, in this report we do not have a lower warning limit. 14 2.2 Measures of performance: Control charts are used to monitor the process parameters to achieve two objectives. First, when the process is in-control, we want the chart to signal infrequently. Statistically, if the process is in-control, we want the probability that the computed statistic is plotted as out-ofcontrol to be as small as possible, i.e. probability of false alarm is as small as possible. Second, when the process is out-of-control, we want the chart to signal as soon as possible. Statistically, if the process is out-of-control, we want the probability that the computed statistic is plotted as in-control to be as small as possible, i.e. the probability of true signal is as large as possible. Concerning the previous two objectives, the measures of performance of VSSchart that we will use are in-control average run length ARL 0 and in-control average sample size ASS 0. RL is the number of samples until the chart signals. Consequently, RL follows geometric distribution with probability of success p. Therefore, ARL = 1/p where p is the probability of signal. The in-control average run length ARL 0 is 1/p where p=, i.e. = Pr (signal/ in-control process). On the other hand, the out-of-control average run length ARL 1 = 1/p where p=1-, i.e. 1- = Pr (signal/ out-of-control process). Decreasing will increase ARL 0 which is a desired result, but also ARL 1 will increase resulting in undesired consequences; this is due to the inverse relation between and. Most statisticians consider ARL 0 = 0 is the desired value for ARL 0 as it achieves a balance between and. ASS is the average sample size until th
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