Based On The Likelihood Ratio Test Control Charts

With the development of economic globalization, more and more emphasis is placed upon product quality. A product without excellent quality is difficult to have its foothold in the highly competitive market. Thus, techniques and methods to improve the quality of products have been extensively used and developed. Statistical Process Control (SPC for short) is now recognized as playing a very important role in modern industry. It can effectively detect the variability of the production process, resolve certain problems and control product quality.SPC was developed in 1920s and Walter A. Shewhart of Bell Telephone Laboratories developed the first sketch of a modern control chart in May 1924. SPC was used extensively in World War II both in the UK and in the USA, but lost its importance as industries converted to peacetime production. However, people in the West taught it to the Japanese, and W. E. Deming in particular made a big impact in Japan in the 1950s. Japanese industry applied SPC widely and proved that SPC saves money and attracts customers which can be verified by Japan's rapid industrial occupation of international market. At present, with the improvement of market economy system in China and the integration of the global economy, China's economy has been integrated into the world economy. If the quality of industrial products in China isn't paid much more attention, industrial products will not be exported abroad. Therefore, we must enhance product quality control and management.In general, there are three kinds of control charts. One is the well-known Shewhart chart which is the earliest used to monitor the process mean and variance. The advantage of this chart is that it is more efficient in detecting large shifts of process than others, but as to detect small shifts, it is not the case. The reason is that it uses only the information of the current sample but ignores the former samples. In order to overcome this disadvantage, and increases the sensi- tivity of Shewhart chart in detecting the small shift, Woodall and Champ(1987) gave a scheme of Shewhart chart with supplementary runs rules and the combined Shewhart-CUSUM charts and combined Shewhart-EWMA charts were proposed by Lucas(1982), Klein(1996), respectively. In addition, Page (1954) proposed the famous cumulative sum (CUSUM chart for short) chart which based on the Wald test. Another important chart, i.e., exponential weighted moving average (EWMA for short) chart, was first proposed by Roberts (1959). Both CUSUM and EWMA charts have the same character that they will use not only the information of the current sample but also will use the former samples. It is possible to infer whether the process is in control from these two charts based on all the samples that have been collected over time. These two kinds of charts are very effective in detecting small shifts of the process.After nearly seventy years development, the study range of SPC is therefore very broad and the new methods for the construction of the control chart are teeming out. The research results can be summarized briefly as follows:1. the study of Phase I and phase II control chart for independent data;2. the study of Phase I and phase II control chart for autocorrelated data;3. the study of multivariate control chart;4. the study of nonparametric control chart;5. the study of dynamic control chart;6. the economic design of control chart;7. the fine property of some control charts from the view of large sample theory.Recently, Woodall W. H. published a crucial paper in 2006 about the monitoring of public health-care. There are many applications of control charts in health-care monitoring and in public-health surveillance. He introduced some applications to industrial practitioners and discussed some of the ideas that arise that may be applicable in industrial monitoring. He considered the advantages and disadvantages of the charting methods proposed in the health-care and public-health areas. Some additional contributions in the industrial statistical process control literature relevant to this area are given in this paper. So there are many application and research opportunities available in the use of control charts for health-related monitoring.Monitoring linear profiles and its applications have been drawn more and more attention. This is a relatively new area of research, but it is growing rapidly. Profile monitoring is very useful in an increasing number of practical applications. Much of the work in the past few years has focused on the use of more effective charting methods, the study of more general shapes of profiles, and the study of the effects of variations of assumptions. There are many promising research topics yet to be pursued given the broad range of profile shapes and possible models.The structure of this dissertation is demonstrated as follows:In Chapter 1, we introduce the outline of the development of SPC, including the study methods and the hot issues in current SPC research. Also, the structure of this dissertation and its attributions are listed.In Chapter 2, we propose a new single chart which integrates the EWMA procedure with the likelihood ratio test statistics for jointly monitoring both the process mean and variance. Many authors have studied this problem. However, most of the work has their drawbacks. Some charts involve more parameters need to be determined and the performance of these control charts significantly depends on these parameters. So, its design is not easy. Others may not cope well with the monitoring of the decrease in the variance. This is a very important case in practice, because the variance decrease means that the quality of the product improved. In addition, most of the charts may not be appropriately used in the case that only an individual observation is available at one sampling point, which is quite common in many industrial processes. Our new chart has only tow parameters to be determined and it can be easily designed and constructed, and it has good average run length performance. It provides a quite satisfactory performance in various cases. In addition, it can solve the problems mentioned above effectively.In Chapter 3, based on the work of Chapter 2, we develop an adaptive chart. The property of the adaptive control chart is that the sampling interval and sample size depends on what is being observed from the prior data. The results can be summarized as: VSI (Variable Sampling Interval); VSS (Variable Sample Size); VSSI (Variable Sample Size and Sampling Interval); VP (Variable Parameter); VSSIFT (Variable Sample Size and Sampling Interval at Fixed Time). The results show that the AATS does decrease significantly and the efficiency of the chart improved.In Chapter 4, we discuss a self-starting control chart. In Chapter 2, we assume that the parameters of the process are known a prior. However, in most cases, these parameters are not known exactly a prior. Some authors have recommended using 20-30 samples with four to five observations each to estimate the process parameters for traditional control charts. They conclude that, when the number of reference samples is small, control charts with estimated parameters produce a large bias in the IC ARL and reduce the sensitivity of the chart in detecting process changes as measured by the out-of-control (OC) ARL. The obvious solution to this problem is to increase the Phase I sample size to reduce the variability in the sampling distribution of the estimates, especially the process variance. In fact that we need at least 200-300 samples to estimate the process parameters in order to get the similar performance as known parameters. In most cases, however, it may not be possible to wait for the accumulation of sufficiently large subgroups because the users usually want to monitor the process at the start-up stages and such huge samples are costly. Our new chart can solve the problems mentioned above, that is to say it is not necessary to assemble a large number of reference samples before the control scheme begins. The results show that the new chart has good performance.In Chapter 5, we extend our method to the multivariate case. Multivariate control chart is more complicate than the univariate one. Someone think that we can operate several control charts for the process parameters simultaneously and separately. Maybe this idea is accessible, but it is not feasible, because these variables are correlated to some extent. Multivariate theory have shown that even the quality characteristics are mutually independent, for the same significance level, the separated reliability for each variable is not equal to the combined reliability. In this Chapter, we propose a new single control chart which integrates the exponentially weighted moving average (EWMA) procedure with the likelihood ratio test for jointly monitoring both the multivariate process mean vector and the covariance matrix. Hawkins and Masoudou-Tchao (2008) also considered such a method to monitor the covariance matrix, but there are some differences between ours. For their chart, it turned out to be ARL-biased under some circumstance. However, for our chart, it is not ARL-biased any more. From the comparison with other charts, we can see that our new chart can be easily designed and performs better than others for the case in which the quality characteristics are bivariate normal random variables.In Chapter 6, we discuss the study and the applications of the monitoring of linear profiles. In many applications the quality of a process or product is best characterized and summarized by a functional relationship between a response variable and one or more explanatory variables. In some calibration applications, the profile can be represented adequately by a simple linear regression model, while in other applications more complicated model are needed. The monitoring of linear profile was firstly proposed and studied by Kand and Albin in 2000. By far, most of the models of linear profile contain only the intercept and the slope, and the monitoring methods are Shewhart X-bar control chart and the EWMA chart based on the estimation of the process parameters. In addition, the likelihood ratio test and the change-point methods were proposed for detecting changes in the parameters of a simple linear regression model. A self-starting method was proposed which avoids the distinction between Phase I and Phase II. In this Chapter, we propose a likelihood method which integrates the EWMA procedure. This chart performs better than others in detecting the process variance, including the increase and decrease which is very important in practical manufacturing production. Apart from this, it can also detect other parameters changes very well. At last, we apply the VSI procedure to our chart in order to reduce the time of detecting. It can be seen that the effect is remarkable.At last in Chapter 7, we summarize the main results and the technical approaches, and suggest some promising research topics yet to be pursued.In this dissertation, we proposed several creative control charts based on the likelihood ratio test statistics which integrates the EWMA procedure. The resulting ARL values can be obtained by Markov chain approximation for some charts, but others are obtained by Monte Carlo simulations.The original and creative ideas are represented as follows: 1. We propose a creative single chart which integrates the EWMA procedure with the likelihood ratio test statistics for jointly monitoring both the process mean and variance, which can be easily designed. The new chart can cope well with the monitoring of the decrease and increase in the variance. Also, it can cope with the individual case. Further, we provide the design of the chart under different IC ARL. At last, we provide details on the Markov-chain approximation of ARL of the new chart.2. Based on the efficiency of dynamic control chart, we address a VSI and VSS SLR chart based on the adjustment of control limits and the plotting statistics. We compare the difference between VSI and FSI control charts under the same sampling rate. Further, we provide details on the Markov-chain approximation of AATS of this chart.3. We propose a self-starting control chart when the process parameters are not exactly known a prior, , and so there is no need to assemble a large number of reference samples before the control scheme begins. We propose a diagnostic aids based on the change-point method which can locate the change-point in the process and isolate the type of parameters change.4. We propose a new single control chart for jointly monitoring both the multivariate process mean vector and the covariance matrix. The control limits are given and its performance is studied under different dimension, IC ARL and types of shifts. The design parameters are also given.5. We propose a creative control chart for monitoring the linear profile based on the likelihood ratio test statistics. It can isolate the type of parameters change in a profile using some statistics in the charting statistics, especially the direction of the shift in the variance. We also considered the corresponding design of the VSI control chart for linear profile.