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ŽŶĐůƵĚŝŶŐƌĞŵĂƌŬƐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϯϴ ŚĂƉƚĞƌϯ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϯϵ DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϯϵ ϯ͘ϭ DŝdžĞĚtDͲh^hDĐŚĂƌƚĨŽƌůŽĐĂƚŝŽŶ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϰϬ ĞƐŝŐŶƐƚƌƵĐƚƵƌĞŽĨƚŚĞƉƌŽƉŽƐĞĚĐŚĂƌƚ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϰϭ  ŽŵƉĂƌŝƐŽŶƐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϰϰ  /ůůƵƐƚƌĂƚŝǀĞĞdžĂŵƉůĞ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϱϬ  ϯ͘ϭ͘ϭ ϯ͘ϭ͘Ϯ ϯ͘ϭ͘ϯ ϯ͘Ϯ DŝdžĞĚtDͲh^hDĐŚĂƌƚĨŽƌĚŝƐƉĞƌƐŝŽŶ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϱϯ
 ͲtDĐŽŶƚƌŽůĐŚĂƌƚ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϱϰ  h^hDͲ
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 ĐŽŶƚƌŽůĐŚĂƌƚ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϭϬ &ůŽĂƚŝŶŐ 
 ĐŽŶƚƌŽůĐŚĂƌƚ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϭϮ ŽŵƉĂƌŝƐŽŶƐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϭϱ  /ůůƵƐƚƌĂƚŝǀĞĞdžĂŵƉůĞ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϭϴ  ϱ͘Ϯ͘ϭ ϱ͘Ϯ͘Ϯ ϱ͘Ϯ͘ϯ ϱ͘Ϯ͘ϰ ϱ͘ϯ ŽŶĐůƵĚŝŶŐƌĞŵĂƌŬƐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϮϭ ƉƉĞŶĚŝdžϱ͘ϭ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϮϮ ƉƉĞŶĚŝdžϱ͘Ϯ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϮϮ ZĞĨĞƌĞŶĐĞƐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϮϯ ^ĂŵĞŶǀĂƚƚŝŶŐ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϮϳ ƵƌƌŝĐƵůƵŵsŝƚĂĞ͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘͘ϭϮϵ  DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ŚĂƉƚĞƌϭ  Šƒ’–‡”ͳ –”‘†— –‹‘ This chapter provides an introduction to statistical process control (
) and its main technique: the control chart. A brief description of the so called memory control charts, including cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, is also given. Finally, a synopsis of the thesis containing the inspiration towards the proposals and an outline are presented. ͳǤͳ –ƒ–‹•–‹ ƒŽ’”‘ ‡•• ‘–”‘Ž Production processes are subject to variations, e.g. in the process of filling bottles with cooking oil the amount of oil filled will not be exactly same; in the process of making tube light rods the diameter or length of any two rods will not be the same. These variations are mainly classified into two types, namely common cause variation and special cause variation. Common cause variation always exists even if the process is designed very well and maintained very carefully. This variation should be relatively small in magnitude and is, uncontrollable and due to many small unavoidable causes. A process is said to be in statistical control if only common cause variation is present. The variations outside this common cause pattern are called special cause variations. These variations are subject to some problem in the system, like poor tuning of equipment, controller fell asleep or got absent, computer stopped working, poor lot of raw material, machine break down. A process working under both types of variation is said to be out of control. The increase of variation (or the inclusion of special cause variation) in the process generally changes the process parameters like DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭ Ϯ  location or/and dispersion parameters. Change in the process location can lead to a greater number of nonconforming items. Similarly, a change in the process dispersion is also important to be detected as an increase in the process dispersion shows a straightforward drop in the quality of process, as a larger spread in the data leads to lower uniformity in the process. On the other hand, the detection of the decrease in the process dispersion may improve the quality of the process, if the underlying special cause can be detected as early as possible.
 possesses some of the most extensively used techniques to detect the presence of special cause variation in processes. The control chart is one of those techniques and it started with Shewhart control charts containing the mean () chart for process location and the range ( ), the standard deviation (
) and the variance (
 ) charts for process dispersion. The structure of these control charts is based on a statistic plotted against three additional lines: the center line ( ), the upper control limit ( ) and the lower control limit (  ). The two control limits (i.e.  and  ) are basically the parameters of a control chart which are selected in such a way that there is a very small probability (generally referred as False Alarm Rate ( ) in the quality control literature and denoted by ) of the in control data points falling outside these limits. Similarly, the probability of the out of control data points falling outside the control limits is called the power (used as a performance measure) of a control chart. Another performance measure for the control charts is the average run length  we define a random variable  . If equal to the number of samples until the first out of control signal occurs then the probability distribution of this random variable  is known as the run length distribution. The average of this distribution is called average run length and is denoted by control . The in control is denoted by of a control chart is denoted by . DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů , while out of /ŶƚƌŽĚƵĐƚŝŽŶ  A shortcoming of a Shewhart control chart is that its conclusion is merely based on the present sample which means that it pays no attention to the past data resulting into a relatively bad performance for small disturbances in the process. In contrast, the CUSUM control charts and the EWMA control charts (also referred as memory control charts) are based (in different) ways on past information along with current. Due to this feature these charts are more efficient to detect small and moderate shifts. The present study is based on providing new memory control charting techniques (by modifying the existing structures and also by designing some new structures) that perform relatively better than the existing ones, especially for small and moderate shifts in the process parameters. In the pursuing sections, we provide the detailed structures of CUSUM and EWMA charts for monitoring the process parameters. ͳǤʹ ——Žƒ–‹˜‡•— ‘–”‘Ž Šƒ”–• The CUSUM chart was originally introduced by Page (1954) and is suited to detect small and sustained shifts in a process. The chart measures a cumulative deviation from the mean or a target value. There exist two versions of the CUSUM chart, used to monitor the process location: the V-mask CUSUM and the tabular CUSUM. The V-mask procedure, which is not very common in use, normalizes the deviations from the mean (or target) and plots these deviations. As long as these deviations are plotted around the target value the process is said to be in control, otherwise out of control. The tabular method of evaluating a CUSUM chart works by accumulating the deviations up and down from a target value for which we use the notations   and   , respectively. The quantities   and   are known as upper and lower CUSUM statistics, respectively, and these are defined as:          ", !           " !  (1.1) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϯ ϰ where # is the sample number,  is the study variable,  is the target mean of study variable  , is the reference value of CUSUM scheme, often taken equal to the half of the amount of shift which we are interested to detect (cf. Ewan and Kemp (1960)). The starting value for both plotting statistics is taken equal to zero, i.e.     . Now we plot these two statistics against the control limit $ and it is concluded that the process mean has moved upward if  % $ for any value of # whereas the process mean is said to be shifted downwards if  % $ for any value of #. The CUSUM chart is defined by two parameters i.e. and $. These two parameters are used in the standardized manner (cf. Montgomery (2009)) given as:  &' , $  (' (1.2) where ' is the in control standard deviation of the study variable  and & and ( are the two constants which have to be chosen very carefully because the performance of the CUSUM chart is very sensitive to these constants. Several CUSUM structures are also recommended for monitoring the process dispersion. Page (1963) introduced the CUSUM chart for monitoring an increase in process dispersion using sample ranges. Following him, Hawkins (1981), Tuprah and Ncube (1987), Chang and Gan (1995), Acrosta-Mejia et al. (1999) and Castagliola et al. (2009) proposed several improved versions of CUSUM charts for process dispersion. These charts are based on transforming the sample variance such that the new transformed form may be closely approximated by a normally distributed variable and hence applying the usual CUSUM structures (recommended by Page (1954)) on it. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů /ŶƚƌŽĚƵĐƚŝŽŶ  ͳǤ͵ š’‘‡–‹ƒŽŽ›™‡‹‰Š–‡†‘˜‹‰ƒ˜‡”ƒ‰‡ ‘–”‘Ž Šƒ”–• The EWMA control chart was introduced by Roberts (1959) to particularly address the shifts of small and moderate magnitude. Like the CUSUM scheme, EWMA also utilizes the past information along with the current, but the weights attached to the data are exponentially decreasing as the observations become less recent. An EWMA control chart for monitoring the location of a process is based on the statistic: )  * ! +  *) (1.3) where # is the sample number and * is a constant such that  , * - +. The quantity ) is the starting value and it is taken equal to the target mean  or the average of initial data in case when the information on the target mean is not available. The control limits for the EWMA statistic given in (1.3) are given as: .     where    ' / 0 0 +  +  *  4 2    3 0  +  +  * 2   ! ' / 0 1 (1.4) is the control limit coefficient. Like CUSUM charts, EWMA control charts also have two parameters (* and ). * determines the decline of weights, while of the control limits, so jointly these two parameters determine the determines the width performance of the EWMA charts. The above mentioned limits given in (1.4) are called time varying limits of the EWMA charts. For large values of # these limits converge to constant limits which are given as:     ' / 0 0 ,    ,    ! ' / 0 0 (1.5) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϱ ϲ Hence, the factor 5+  +  * 6 in (1.4) tends to 1 if the sample number tends to infinity.  For monitoring the process dispersion, EWMA charts are also based on normalizing the sample variance. Wortham and Ringer (1971) suggested an EWMA control chart for monitoring the process dispersion. Ng and Case (1989), Crowder and Hamilton (1992), Castagliola (2005) and Huwang et al. (2010) followed them and proposed improved versions of EWMA chart for monitoring process variance. ͳǤͶ —–Ž‹‡‘ˆ–Š‡–Š‡•‹• After the development of Shewhart, CUSUM and EWMA charts by Shewhart (1931), Page (1954) and Roberts (1959), respectively, several modifications of these charts have been presented in order to further enhance the performance of these charts. Klein (2000), Khoo (2004), Koutras et al. (2007) and Antzoulakos and Rakitzis (2008) proposed the application of different runs rules with the Shewhart structure. Riaz (2008a) and Riaz (2008b) proposed the auxiliary based control charts for monitoring the process variability and location respectively, where both of these charts are based on regression-type estimators. Lucas (1982) presented the combined Shewhart-CUSUM quality control scheme in which Shewhart limits and CUSUM limits are used simultaneously. Lucas and Crosier (1982) recommended the use of the fast initial response (FIR) CUSUM which gives a head start to the CUSUM statistic by setting the initial values of the CUSUM statistic equal to some positive value (non-zero). This feature gives better  performance but at the cost of a decrease in . Yashchin (1989) presented the weighted CUSUM scheme which gives different weights to the previous information used in CUSUM statistic. Similarly, on the EWMA side, Lucas and Saccucci (1990) presented the combined Shewhart-EWMA quality control scheme which gives better  performance for both small and large shifts. Steiner (1999) provided the DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů /ŶƚƌŽĚƵĐƚŝŽŶ  FIR EWMA which gives a head start to the initial value of the EWMA statistic (like FIR CUSUM) and hence improves the  performance of the EWMA charts. It is hard to find an application of the runs rules schemes with the CUSUM and EWMA charts in the literature. Chapter 2 proposes the application of some runs rules schemes with the control structure of CUSUM and EWMA charts for process location. The performance of these runs rules based CUSUM and EWMA charts is evaluated in terms of . Comparisons of the proposed schemes are made with some existing representative CUSUM- and EWMA-type counterparts used for small and moderate shifts. The findings reveal that the proposed schemes are able to perform better than the other schemes under investigation. The work of Chapter 2 has been published in Quality and Reliability Engineering International as Riaz, Abbas and Does (2011) and Abbas, Riaz and Does (2011). Chapter 3 introduces a new control structure named as mixed EWMA-CUSUM control chart for monitoring the process location. The core of this idea is to mix the effects of EWMA and CUSUM charts into a single structure such that the resulting mixed chart perform better than the classical ones (i.e. CUSUM by Page (1954) and EWMA by Roberts (1959)). This is done by applying the CUSUM structure over the EWMA statistic. An obvious counterpart of this mixed chart is also developed for monitoring the process dispersion which is named as CS-EWMA chart as its plotting statistic is based on cumulatively summing the exponentially weighted moving averages. Some additional material in the form of comparisons and illustrative examples are also provided. From this chapter, an article on the mixed EWMA-CUSUM chart for location is published in Quality and Reliability Engineering International as Abbas, Riaz and Does (2012a), while another article on CS-EWMA chart for process dispersion has been accepted for publication in Quality and Reliability Engineering International as Abbas, Riaz and Does (2012b). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϳ ϴ  Following the approach of Riaz (2008a) and Riaz (2008b), chapter 4 proposes several new control structures using the information from auxiliary variable(s). These charts include the CUSUM and EWMA charts (for monitoring the process location) based on the information of one or more auxiliary variables. The regression estimation technique for the mean is used in defining the control structure of the proposed charts. Comparisons with univariate as well as bi-variate EWMA and CUSUM charts are provided. An article on auxiliary based EWMA chart for location has been accepted for publication in Communications in Statistics - Theory and Methods as Abbas, Riaz and Does (2012c). Chapter 5 proposes an alternative to the CUSUM and EWMA charts, named as the progressive mean (7) control chart. This newly developed control chart is not only outperforming the existing memory charts, but also, its control structure is very simple as compared to the CUSUM and EWMA charts. An article on 7 control chart has been published in Quality and Reliability Engineering International as Abbas, Zafar, Riaz and Hussain (2012). Using the idea of a progressive statistic, two new control charts are also developed (named as floating control charts) for monitoring the process dispersion. These floating charts also surpass the existing CUSUM and EWMA charts for monitoring the process standard deviation. An article on the floating charts has been submitted for publication in International Journal of Production Research as Abbas, Riaz and Does (2012d).   DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ŚĂƉƚĞƌϮ  Šƒ’–‡”ʹ —•”—Ž‡•„ƒ•‡†ƒ† The control chart is an important statistical technique that is used to monitor the quality of a process. One of the charting procedures is the Shewhart-type control charts which are used mainly to detect large shifts. Two alternatives to the Shewhart-type control charts are the cumulative (CUSUM) control charts and the exponentially weighted moving average (EWMA) control charts which are especially designed to detect small and moderately sustained changes in quality. Enhancing the ability of design structures of control charts is always desirable and one may do it in different ways. Runs rules schemes are generally used to enhance the performance of Shewhart control charts. In this chapter we propose the use of runs rules schemes for the CUSUM and EWMA charts and evaluated their performance in terms of the . Comparisons of the proposed schemes are made with some existing representative CUSUM and EWMA-type counterparts used for small and moderate shifts. The comparisons revealed that the proposed schemes perform better for small and moderate shifts while they reasonably maintain their efficiency for large shifts as well. This chapter is based on two articles for monitoring the process location i.e. Riaz, Abbas and Does (2011) and Abbas, Riaz and Does (2011). ʹǤͳ –”‘†— –‹‘ The CUSUM and EWMA chart structures discussed in Chapter 1 are known as the classical CUSUM and EWMA charts. The detailed study on the performance of DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϵ ϭϬ  CUSUM chart was done by Hawkins and Olwell (1998), whereas Steiner (1999) evaluated the 8 of the classical EWMA. These respectively. @ 8 are provided in Table 2.1 and Table 2.2, TABLE 2.1: 9:; Values for the classical CUSUM chart with <  => ? (A (B 0 0.25 0.5 0.75 1 1.5 2 2.5 3 168 74.2 26.6 13.3 8.38 4.75 3.34 2.62 2.19 465 139 38.0 17.0 10.4 5.75 4.01 3.11 2.57  TABLE 2.2: 9:; values for the classical EWMA chart at 9:;=  ?== C 0 D  => E ;  F> GFH 499.89 D  => F? ;I 500.81 D  => ? ;  I> =JF 499.36 D  => J? ;  I> =GG 0.25 102.99 169.49 255.96 321.3 0.5 28.86 47.5 88.75 139.87 0.75 13.56 19.22 35.55 62.46 1 8.22 10.4 17.09 30.57 1.5 4.17 4.77 6.27 9.8 2 2.66 2.94 3.4 4.46 499.36 The results of Tables 2.1 and 2.2 are based on the test that one point falling outside the limits indicates an out of control situation (the classical scheme of signaling). This test may be further extended to a set of rules named as sensitizing rules and runs rules schemes which help to increase the sensitivity of the charts to detect out of control situations. The common set of sensitizing rules are (cf. Nelson (1984)): one or more points outside the control limits; two out of three consecutive points outside the 2 sigma warning limits but still inside the control limits; four out of five consecutive points beyond the 1 sigma limits but still inside the control limits; a run of eight consecutive points on one side of the center line but still inside the control limits; six points in a row steadily increasing or decreasing but still inside the control limits; fourteen points in a row alternating up and down but still inside the DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  control limits. The basic principle underlying these runs rules is twofold. Firstly, specific patterns of out of control conditions might be detected earlier, such as a small but persistent trend. Secondly, the decision rules are designed to have roughly the same (marginal) false alarm probability. To enhance the performance of control charts, many researchers have used the idea of using different sensitizing rules and runs rules schemes with the Shewhart-type control charts, e.g. see Klein (2000), Khoo (2004), Koutras et al. (2007), and Antzoulakos and Rakitzis (2008). The application of sensitizing rules causes an increase in false alarm rates, whereas the runs rules schemes take care of this issue. Klein (2000), Khoo (2004), and Antzoulakos and Rakitzis (2008) suggested different runs rules schemes, namely K out of L and modified K out of L, to be used with the Shewhart-type control charts. They studied their performance and found that these runs rules schemes perform better as compared to the usual Shewhart-type control charts. There is a variety of literature available on CUSUM and EWMA charts. e.g. see Lucas and Crosier (1982), Yashchin (1989), and Hawkins and Olwell (1998) for CUSUM and Lucas and Saccucci (1990), Steiner (1999), and Capizzi and Masarotto (2003) for EWMA. All the existing approaches use only the usual scheme of signaling an out of control situation. It is hard to find an application of the runs rules schemes with the CUSUM and EWMA charts in the literature. However, Westgard et al. (1977) studied some control rules using combined Shewhart-CUSUM structures. They proved superiority of this combined approach on the separate Shewhart’s approach but ignored any comparison with the separate CUSUM application. Also their control rules considered only one point at a time for testing an out of control situation. The false alarm rates of their control rules kept fluctuating and no attempt was made to keep them fixed at a pre-specified level which is very important for valid comparisons among different control rules/schemes. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϭ ϭϮ In this chapter we analyze some of the K out of L runs rules schemes (like MNM, MOP and  modified MOP schemes) with CUSUM and EWMA charts, following Klein (2000), Khoo (2004), and Antzoulakos and Rakitzis (2008), and compared their performance (in terms of the ) with some other schemes meant particularly for small shifts. Section 2.2 contains the detailed discussion about the proposed runs rules schemes applied on the CUSUM chart, while the discussion on the schemes applied on the EWMA chart are included in Section 2.3. ʹǤʹ Š‡’”‘’‘•‡†• Š‡‡•ˆ‘”–Š‡ Šƒ”–• A process is called to be out of control when a point falls outside the control limits. Specific runs rules or extra sensitizing rules can be used in addition to enhance the power of detecting out of control situations. The CUSUM charts can also take benefit out of these runs rules schemes if properly applied with the CUSUM structures. Following Klein (2000), Khoo (2004), and Antzoulakos and Rakitzis (2008), we propose here two runs rules schemes to be used with the CUSUM charts to monitor the location parameter. The proposed schemes are based on the following terms and definitions. Action Limit (9;): This is a threshold level for the value of CUSUM chart statistic. If some value of CUSUM statistic exceeds the of the , the process is called to be out of control. The value would be greater than the classical CUSUM critical limit $ for a fixed . Warning Limit (Q;): This is a level for the value of the CUSUM chart statistic beyond which (but not crossing the ) some pattern of consecutive points indicate an out of control situation. The value of the R would be smaller than the classical CUSUM critical level $ for a fixed . DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  Using the above definitions we propose the two runs rules schemes for the CUSUM chart as: Scheme I: A process is said to be out of control if one of the following four conditions is satisfied: 1. 2. 3. 4. One point of   falls above the One point of   falls above the . . Two consecutive points of   fall between the R and the . Two consecutive points of   fall between the R and the . Scheme II: A process is said to be out of control if one of the following four conditions is satisfied: 1. 2. 3. 4. One point of   falls above the One point of   falls above the Two out of three consecutive points of   fall between the R and the given shift; i.e. the are proportional to the value of the is higher if the values of the R and the There are infinite pairs of R and The objective is to find those pairs of level and at the same time minimize the The . Two out of three consecutive points of   fall between the R and the Note that the values of the R and the versa. .  . for a are higher and vice which fix the in control and R that maintain the .   at a desired level. value at the desired value. computations may be carried out using different approaches, like integral equations, Markov chains, approximations and Monte Carlo simulations. Details regarding the first two may be seen in Brook and Evans (1972) and Lucas and Crosier (1982) and the references therein. An lower-sided CUSUM ( approximation for the upper-sided CUSUM (   ) and the ) is given as (cf. Siegmund (1985)): DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϯ ϭϰ    and  The  S TUV>WW ! M@  &( ! +>+XX  + M@  & S TUV>WW ! M@  &( ! +>+XX  + M@  & 8 for a two-sided CUSUM can be obtained by the following relation:  +Y  + ! +Y  Monte Carlo simulation is also a standard option to obtain approximations for the and we have adopted this approach in our study. For that purpose we have developed a simulation algorithm using an add-in feature of Excel software which helps calculating the 8. ʹǤʹǤͳ ‡”ˆ‘”ƒ ‡‡˜ƒŽ—ƒ–‹‘‘ˆ–Š‡’”‘’‘•‡†• Š‡‡• The performance of the two proposed schemes for the CUSUM chart has been evaluated in terms of under different in control and out of control situations. To meet the desired objective, we have used our simulation algorithm to find the for the pairs of R and size Z from [ \ ! @ ]^ _`  and  values . We have generated for different values of @ 100,000 samples of  ' a and we have calculated the statistics  and  for all samples. Here  and ' refer to the mean and the standard deviation of the process under study and @ is the amount of shift in  . Here ' is assumed to be in control i.e. '  ' . The value of @ indicates the state of control for our process mean, i.e. @   implies that the process mean  is in control (i.e.    ) and @ b  that the process mean  is out of control (i.e.    ). Without loss of generality we have taken    and '  + in our simulations. After obtaining 100,000 samples we have applied all four conditions of the two proposed runs rules schemes (i.e. Schemes I & II). In this way the run lengths are found for the two DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  proposed schemes. This procedure is repeated 5,000 times and each time the run lengths are computed for both schemes. By taking the average of these run lengths we obtain the for the two schemes. 8 To evaluate the performance of the two proposed schemes I and II, we will report the  results for the values of the equal to 168, 200 and 500. Other values of the  can be easily obtained. The choices made will show the performance of the two schemes and enable us to make comparisons with the results of other schemes and approaches from the literature.  By fixing the pairs of R obtained the the 8 and  at a desired level for the proposed schemes I and II, we are able to obtain using our algorithm. Then for these pairs of R and , we have at different values of @ for both the schemes. The results of the R and along with their corresponding  values for the above mentioned pre-specified are provided in Tables 2.3 – 2.8 for both schemes. In Tables 2.3 – 2.8 the first two columns contain the R and pairs which fix the and the remaining columns give the corresponding   value at a specified desired level values. Some researchers (e.g. Antzoulakos and Rakitzis (2008)) suggest also to report the standard deviations of the run lengths along with the values to describe more about the run length behavior. Moreover, Palm (1990) and Shmueli and Cohen (2003) highlighted the importance of percentile points of the run length distribution and suggested to report them for the interest of practitioners. Therefore the standard deviations (denoted by
c ) and the percentile points (denoted by  ) of the run length distribution are also computed for proposed schemes I and II. The results of
c and  (for #  + MB B dB e) are provided in Riaz, Abbas and Does (2011) as Tables VIII – XI for the two proposed schemes at   +Xf. For the other values of  similar tables can be easily obtained. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϱ ϭϲ  TABLE 2.3: Q;, 9; and 9:;E values for the proposed scheme I at 9:;=  EgG Q;h Limits 0.25 0.5 0.75 3.42 9;h 4.8 71.872 25.564 3.44 4.6 72.258 3.48 4.4 3.53 4.2 C 1 1.5 2 13.539 8.66 5.078 3.679 25.653 13.5 8.568 5.013 3.607 71.936 25.593 13.496 8.516 4.936 3.525 71.399 25.3 13.332 8.404 4.828 3.423  TABLE 2.4: Q;, 9; and 9:;E values for the proposed scheme II at 9:;=  EgG Q;h Limits 0.25 0.5 0.75 3.5 9;h 4.44 71.489 25.379 3.6 4.19 72.938 25.368 3.7 4.08 73.11 25.369 3.8 4.03 73.589 25.403 C 1 1.5 2 13.398 8.462 4.941 3.541 13.352 8.383 4.83 3.424 13.306 8.344 4.777 3.376 13.277 8.316 4.75 3.347  TABLE 2.5: Q;, 9; and 9:;E values for the proposed scheme I at 9:;=  F== Q;h Limits 0.25 0.5 0.75 3.9 9;h 4.24 82.952 28.697 3.8 4.29 84.537 3.7 4.4 3.6 4.77 i 3.57 C 1 1.5 2 13.801 8.904 4.909 3.492 28.736 14.065 8.661 5.002 3.523 82.115 28.496 13.822 8.812 4.993 3.561 84.15 28.716 13.961 8.981 5.228 3.725 79.474 28.94 14.262 9.213 5.51 4.076  TABLE 2.6: Q;, 9; and 9:;E values for the proposed scheme II at 9:;=  F== Q;h Limits C 9;h 0.25 0.5 0.75 1 1.5 2 3.9 4.23 82.975 28.206 13.77 8.869 4.978 3.435 3.8 4.28 81.403 28.356 13.903 8.699 4.966 3.503 3.7 4.6 82.732 28.824 14.078 8.874 5.14 3.687 81.518 29.138 14.266 9.114 5.457 4.12 3.64 i  DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  TABLE 2.7: Q;, 9; and 9:;E values for the proposed scheme I at 9:;=  ?== Q;h Limits 0.25 0.5 0.75 4.8 9;h 5.12 141.111 38.599 4.7 5.2 150.372 4.6 5.39 i 4.49 C 1 1.5 2 17.392 10.518 5.905 4.057 38.594 17.529 10.599 5.898 4.14 145.189 38.195 17.468 10.558 6.007 4.237 146.564 38.492 17.725 10.857 6.333 4.689  TABLE 2.8: Q;, 9; and 9:;E values for the proposed scheme II at 9:;=  ?== Q;h Limits 0.25 0.5 0.75 4.8 9;h 5.11 139.705 38.856 4.7 5.19 142.159 4.6 5.5 4.54 i C 1 1.5 2 17.459 10.506 5.822 4.078 37.975 17.267 10.583 5.872 4.104 145.787 38.334 17.394 10.734 6.053 4.273 149.035 39.904 17.568 10.966 6.451 4.873 The standard errors of the results reported in Tables 2.3 – 2.8 are expected to remain around 1% (in relative terms) as we have checked by repeating our simulation results. We have also replicated the results Table 2.1 for (  A of the classical CUSUM scheme using our simulation routine and obtained almost the same results which ensures the validity of the algorithm developed in Excel and the simulation results obtained from it. We have observed for the two proposed schemes I and II that: i) many pairs of R and may be found which fix the but the optimum choice helps minimizing the ii)   at a desired level value (cf. Table 2.3 – 2.8); the two proposed schemes perform very good at detecting small and moderate shifts while maintaining their ability to address the large shifts as well (cf. Tables 2.3 – 2.8); DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϳ ϭϴ  iii) the two proposed schemes I and II are almost equally efficient for the shifts of small, moderate and large magnitude and hence may be used as a replacement of each other at least for normally distributed processes;  iv) with an increase in the value of @ the ; schemes, at a fixed value of  the   decreases rapidly for both the v) with a decrease in the value of vi) the proposed schemes I and II may be extended to more generalized schemes schemes for a given value of @ (cf. Tables 2.3 – 2.8); decreases quickly for both the (as given in Klein (2000), Khoo (2004), and Antzoulakos and Rakitzis (2008)); vii) the
c decreases if the value of @ increases for both the schemes I and II (cf. Riaz, Abbas and Does (2011)); viii) the run length distributions of both the schemes are positively skewed (cf. Riaz, Abbas and Does (2011)). ʹǤʹǤʹ ‘’ƒ”‹•‘• In this section we compare the performance of the proposed schemes I and II with some existing schemes for detecting small, moderate and large shifts. The is used as a performance measure for all the schemes under discussion. The existing schemes we have considered for comparison purpose include the classical CUSUM scheme of Page (1954), the weighted CUSUM scheme of Yashchin (1989), the EWMA scheme given in Steiner (1999) and the fast initial response (FIR) CUSUM scheme of Lucas and Crosier (1982). The  results for the above mentioned schemes are provided in the Table 2.1 – 2.2 and Tables 2.9 – 2.10 at some selective values of  which will be used for the comparisons. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  Now we present a comparative analysis of the proposed schemes with the existing schemes one by one. Proposed versus the classical CUSUM: The classical CUSUM scheme of Page (1954) accumulates the up and down deviations from the target and is quite efficient at detecting small shifts. Table 2.1 provides the Tables 2.3 – 2.4 provide the performance of the classical CUSUM scheme. performances of the two proposed schemes. The results of these tables advocate that the proposed schemes are better compared with the classical CUSUM scheme for small shifts while for moderate and large shifts their performances almost coincide. Proposed versus the weighted CUSUM: Yashchin (1989) presented a class of weighted control schemes that generalize the basic CUSUM technique by assigning different weights to the past information used in the classical CUSUM statistic. The performance of the terms as defined earlier in this article. Tables 2.7 and 2.8 provide the performance of the weighted CUSUM scheme is given in Table 2.9 where j represents the weight and the other proposed schemes at   B so these tables can be used to compare the proposed schemes with the weighted CUSUM scheme. TABLE 2.9: 9:; values for the symmetric two-sided weighted CUSUM scheme at 9:;=  ?== k <  => ? 0.5 1 0.7 l 3.16 86.30 0.8 3.46 0.9 1.0 C 1.5 2 15.90 6.08 3.52 70.20 13.30 5.66 3.50 3.97 54.40 11.40 5.50 3.60 5.09 39.00 10.50 5.81 4.02 By comparing the results of Tables 2.7, 2.8 and 2.9 we can see that the proposed schemes perform better than the weighted CUSUM for small and moderate shifts. Particularly, when @ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϵ ϮϬ  is small the performance of our proposed schemes is significantly better than that of the weighted CUSUM scheme. However for j  + the weighted CUSUM scheme is the same as the classical CUSUM so the comments of the proposed versus the classical CUSUM scheme hold here as well. Proposed versus the classical EWMA: Steiner (1999) gave a simple method for studying the run length distribution of the classical EWMA chart. Table 2.2 presents some selective 8 of the EWMA chart where * is the weighting constant and  coefficient. As the proposed schemes have is the control limits  B in Tables 2.7 and 2.8 we use these tables for a comparison with the EWMA chart. From the Tables 2.2, 2.7 and 2.8 we see that for @  >MB the classical EWMA chart has an schemes are minimizing the same   value of +d+>e whereas the proposed value around +A. This shows that the proposed schemes perform better than the classical EWMA scheme for @  >MB. The same superiority also holds for all >MB - @ - +. However, for @ % + the proposed schemes and the classical EWMA scheme have almost the same behavior as can be easily seen from the corresponding tables. Proposed versus the FIR CUSUM: Lucas and Crosier (1982) presented the Fast Initial Response (FIR) CUSUM which gives a head start value, say  , to the classical CUSUM statistic. A standard CUSUM has      while an FIR CUSUM sets  and  to 8 for the FIR CUSUM at (  A and   + some nonzero value. Table 2.10 presents the for discussion and comparison purposes. The FIR CUSUM scheme decreases the values as compared to those of the classical CUSUM scheme at the cost of reduction in   value from +Xf to +XP (see Table 2.1 vs. Table 2.10) which is generally undesirable in sensitive processes (e.g. those directly related to intensive care units which are highly time sensitive, cf. Bonetti et al. (2000)). The  results given in Tables 2.3 and 2.4 of the DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  proposed schemes are also obtained for purposes here. Looking at the  same amount of reduction in    +Xf and hence can be used for comparison results of Tables 2.3 and 2.4 we can see that almost the may be achieved, as obtained by FIR CUSUM, using the proposed schemes without paying any cost in terms of a decrease in  value and the need of a head start value. C TABLE 2.10: 9:;m for FIR CUSUM scheme with n=  lOH and <  => ? l  H, n=  E 0 0.25 0.5 0.75 1 1.5 2 163 71.1 24.4 11.6 7.04 3.85 2.7 In brief the proposed schemes have shown better performance for the smaller values of @ (i.e. small shifts), which is the main concern of CUSUM charts, while for larger values of @ the proposed schemes can perform equally well as the other schemes. The better performance can be further enhanced with the help of other runs rules schemes of Khoo (2004) and Antzoulakos and Rakitzis (2008). ʹǤʹǤ͵ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ To illustrate the application of the proposed CUSUM schemes we use the same method as in Khoo (2004). Two datasets are simulated consisting of some in control and some out of control sample points. For dataset 2.1 we have generated 50 observations in total, of which the first 20 observations are from [+ (showing the in control situation) and the remaining 30 observations are generated from [>MB+ (showing a small shift in the mean level) while for dataset 2.2 we have generated 30 observations in total, of which the first 20 observations are same as for dataset 2.1 and the remaining 10 observations are generated from [++ (showing a moderate shift in the mean level). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů Ϯϭ ϮϮ  The two proposed CUSUM schemes of this study (i.e. schemes I and II) are applied to the above mentioned two datasets. Additionally, the classical CUSUM scheme is also applied to these two datasets for illustration and comparison purposes. The CUSUM statistics are computed for the two datasets and are plotted against the respective control limits used with the three CUSUM schemes by fixing the  (  B>d is used as the control limit to have R  A>f and at B. For the classical CUSUM scheme,   B. For the proposed scheme I,  B>+M are used, while for the proposed scheme II R  A>f and  B>++ are used to have the  value equal to B for both schemes. The graphical displays of the three CUSUM schemes for the two datasets are given in the following two figures. Figure 2.1 exhibits the behavior of dataset 2.1, where a small mean shift was introduced. Figure 2.2 illustrates the behavior of dataset 2.2, where a moderate mean shift was introduced. Figure 2.1: CUSUM chart of the classical scheme and the proposed scheme I and II for dataset 2.1 н ϲ ,;ĐůĂƐƐŝĐĂůͿ t>ĂŶĚ>;ƐĐŚĞŵĞ/Ϳ t>ĂŶĚ>;ƐĐŚĞŵĞ//Ϳ ϱ ϰ ϯ Ϯ ϭ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ϰϭ ϰϯ ϰϱ ϰϳ ϰϵ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  Figure 2.2: CUSUM chart of the classical scheme and the proposed scheme I and II for dataset 2.2 ϲ н ,;ĐůĂƐƐŝĐĂůͿ t>ĂŶĚ>;ƐĐŚĞŵĞ/Ϳ t>ĂŶĚ>;ƐĐŚĞŵĞ//Ϳ ϱ ϰ ϯ Ϯ ϭ Ϭ ϭ Ϯ ϯ ϰ ϱ ϲ ϳ ϴ ϵ ϭϬ ϭϭ ϭϮ ϭϯ ϭϰ ϭϱ ϭϲ ϭϳ ϭϴ ϭϵ ϮϬ Ϯϭ ϮϮ Ϯϯ Ϯϰ Ϯϱ Ϯϲ Ϯϳ Ϯϴ Ϯϵ ϯϬ ^ĂŵƉůĞEƵŵďĞƌ In Figure 2.1 we see that an out of control signal is received at sample points # 49 and 50 by the proposed scheme I (i.e. two out of control signals) and at sample points # 49 and 50 by the proposed scheme II (i.e. two out of control signals). Similarly, in Figure 2.2 the out of control signals are received at sample point # 26 by proposed scheme I (i.e. one out of control signal) and at sample points # 25, 26 and 27 by the proposed scheme II (i.e. two out of control signal). For both datasets, the classical CUSUM scheme failed to detect any shift in the process mean. It is evident from the above figures that the proposed schemes have detected out of control signals, which are not spotted by the classical CUSUM scheme for the dataset 2.1, where a small mean shift was present, while the situation is almost identical for dataset 2.2, where a moderate shift we introduced. It is to be noted that these signaling performances of the proposed schemes versus the classical CUSUM scheme are in accordance with the DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů Ϯϯ Ϯϰ  findings of subsection 2.2.2, where we found that the proposed schemes are more efficient than the classical CUSUM scheme for small shifts, while almost equally good for other shifts. ʹǤ͵ Š‡’”‘’‘•‡†• Š‡‡•ˆ‘”–Š‡ Šƒ”–• Shewhart control charts are good in detecting large disturbances in the process, but it takes too long for Shewhart-type charts to detect a small or moderate shift. To overcome this problem some sensitizing rules are designed but their implementation inflates the prespecified false alarm rate. This issue may be resolved by the introduction of the runs rules schemes as we have mentioned in Section 2.2. Klein (2000), Khoo (2004) and Antzoulakos and Rakitzis (2008) presented runs rules schemes applied on the Shewhart control charts to enhance their performance for small and moderate shifts, keeping the false alarm rate at the pre-specified level. The application of these runs rules schemes is not commonly used with the CUSUM and EWMA control charts. Taking inspiration from the application of runs rules on CUSUM charts, we propose two runs rules schemes in this section for the design structure of the EWMA control chart named as “simple MOM EWMA scheme” and “modified MOP EWMA scheme”. The procedural and conceptual framework of these two proposed schemes is defined as: Simple FOF EWMA scheme: According to the simple MOM EWMA scheme a process is said to be out of control if two consecutive points are plotted either below a lower signaling limit (
) or above an upper signaling limit (
). Modified FOI EWMA scheme: According to the modified MOP EWMA scheme a process is said to be out of control if one of the following two conditions is satisfied. i. At least two out of three consecutive points fall below an
and the point above the
(if any) falls between the  and the
. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD At least two out of three consecutive points fall above a 
and the point below the  ii. 
(if any) falls between  and the 
. The signaling limits
and 
mentioned above in the definitions of our two proposals are especially set limits chosen for the two schemes separately, depending upon the desired , while  is the same as defined in (1.4). The control structure for the proposed schemes is given as: .

  where    0 o ' /0 +     +  *  4 2 3 0 +  +  * 2   ! o ' / 0 1 o (2.1) is the signaling limit coefficient of the proposed schemes and the other terms are the same as defined in Section 1.3. It has to be mentioned that the above mentioned signaling limits coefficient o . is set according to the pre-specified value of Moreover, a signaling limit on either side may be split into two lines (as is done in Section 2.2) to reach at some optimum pair. We opted the choice where the outer split of the line is taken at infinity. However, one may take some different appropriately chosen outer splits other than infinity. The parameters of these two proposed schemes are the central line and two signaling limits as given in (2.1) (i.e.  ,
and 
). The upper and lower signaling limits are symmetric around the  positive relation between  . Using the and width of the signaling limits (depending upon we fix and will vary according to the pre-specified at the desired level and find the corresponding pair of symmetric signaling limits. Based on these especially set signaling limits we carry out our  o ),  study at the desired values. The calculation of may be carried out using different approaches such as integral equations, Markov chains, approximations and Monte Carlo simulations. We have chosen to DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů Ϯϱ Ϯϲ  use Monte Carlo simulations to obtain calculation of values. The simulation algorithms for the values of the proposed schemes are developed in Excel using an Add-In feature 7
#L. ʹǤ͵Ǥͳ ‡”ˆ‘”ƒ ‡‡˜ƒŽ—ƒ–‹‘‘ˆ–Š‡’”‘’‘•‡†• Š‡‡• To investigate the performance of our proposed EWMA schemes we have considered different in control and out of control situations. A suitable number of samples (say 100,000) of a fixed size Z are generated from [ \ ! @ ]^ _`  ' a. The EWMA statistics for these samples are then calculated and the conditions of the two proposed EWMA schemes (as listed in Section 2.3) are applied on them using the signaling limits given in (2.1) through our simulation algorithm. By executing this process repeatedly we obtain different run length hvalues and other properties as well. It is to be values which ultimately help computing noted that the value of @  , the  @ b , the  o is worked out such that the desired  value is achieved. For values are evaluated with the help of their corresponding o and then, for values are computed by introducing different shifts in the process. To evaluate the performance of the two proposals we fix the pre-specified  values, in this section, at 168, 200 and 500. These choices will suffice to exhibit the behavior of our proposed schemes and will enable us to make valid comparisons with their already existing counterparts. On similar lines other choices of the   can also be obtained. By fixing values at the above mentioned levels (using their corresponding obtained the  at different values of @. These 2.16 for the aforementioned desired values) at different choices of *͘   o) we have values are provided in Tables 2.11 – preferences (along with their corresponding DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů o ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  TABLE 2.11: 9:; values for the simple FOF EWMA scheme at 9:;=  EgG 0 *  >+  M>+AB 169.8676 *  >MB o  M>+fA 169.4769 *  >B  M>PA 169.6763 169.7112 0.25 54.5771 73.4836 94.8246 110.7766 0.5 19.8026 26.6284 37.9883 49.063 0.75 10.5927 12.9547 17.408 23.2892 1 6.9435 7.9456 9.8833 13.0739 1.5 4.1117 4.3476 4.7293 5.5777 2 2.9796 3.0954 3.1231 3.3368 @ o *  >dB o  +>fP o TABLE 2.12: 9:; values for the modified FOI EWMA scheme at 9:;=  EgG 0 *  >+  +>fd 167.3173 *  >MB o  +>ePX 169.9927 *  >B o  +>fB 168.5416 170.9464 0.25 34.367 43.3236 55.4569 62.6484 0.5 14.0389 17.8611 23.0836 27.6195 0.75 8.1338 9.4968 11.9229 14.2207 1 5.7064 6.4798 7.6037 8.5804 1.5 3.7755 3.9823 4.2103 4.5232 2 3.2047 3.2708 3.3136 3.4072 @ o *  >dB o  +>Xd TABLE 2.13: 9:; values for the simple FOF EWMA scheme at 9:;=  F== 0 *  >+  M>M++ 200.5694 *  >MB o  M>MA 199.8855 *  >B o  M>e 200.8923 201.1229 0.25 60.9801 80.7515 107.709 126.1691 0.5 20.9561 28.6011 42.0774 55.7335 0.75 11.2452 13.7306 19.0848 25.862 1 7.1859 8.1962 10.6258 13.9457 1.5 4.198 4.4825 4.9002 5.7061 2 3.0577 3.1331 3.1899 3.4272 @ o The standard deviation of the run lengths (denoted by
c by  (#  +, MB, B, dB and e) are also provided at  ) and the # pV percentiles denoted  B in Abbas, Riaz and Does (2011). Similar results can be easily obtained for other values of with *  >dB o  +>fdB . These measures along may help studying the behavior of the run length distribution. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů Ϯϳ Ϯϴ  TABLE 2.14: 9:; values for the modified FOI EWMA scheme at 9:;=  F== 0 *  >+  +>feB 201.9206 *  >MB o  M>f 200.5236 *  >B  +>eM 200.5969 *  >dB o  +>d+B 0.25 39.1578 50.0565 63.1482 70.9068 0.5 15.4204 19.5705 25.1144 30.4725 0.75 8.6061 10.1922 12.8642 15.3578 1 5.981 6.7621 7.7437 9.1272 1.5 3.8655 4.1115 4.3324 4.6841 2 3.2523 3.3186 3.3386 3.4238 @ o o 200.886 TABLE 2.15: 9:; values for the simple FOF EWMA scheme at 9:;=  ?== @ *  >+  M>BBX o *  >MB o  M>BBA *  >B o  M>PX *  >dB o  M>++B 0 501.7558 505.5284 501.2598 502.0725 0.25 103.3109 169.1349 235.1138 280.6187 0.5 29.5748 47.0105 78.0771 108.8792 0.75 14.3216 19.2776 30.8742 45.3405 1 8.9561 10.5964 15.1992 22.1033 1.5 4.9197 5.2578 6.1014 7.7862 2 3.4498 3.5527 3.6815 4.0883 TABLE 2.16: 9:; values for the modified FOI EWMA scheme at 9:;=  ?== 0 *  >+ o  M>P 502.883 *  >MB o  M>PAB 499.6153 505.3564 501.9698 0.25 66.6864 97.0108 133.7117 155.7078 0.5 21.4251 31.2023 46.3541 57.7739 0.75 11.7427 14.4295 20.6223 26.0312 1 7.5539 8.6761 11.0991 13.8363 1.5 4.4676 4.7066 5.1336 5.7812 2 3.4534 3.549 3.6276 3.7787 @ *  >B  M>MM o *  >dB o  +>efM The relative standard errors of the results reported in Tables 2.11 – 2.16 are also calculated and are found to be around 1%. We have also replicated the results of the classical EWMA DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  chart and found almost the same results as Steiner (1999) which ensures the validity of our simulation algorithm. Mainly, the findings for the two proposed schemes are: i. the two proposed schemes are performing very well at detecting small and moderate shifts while their performance for large shifts is not bad either (cf. Tables 2.11– 2.16); ii. with an increase in the value of @ the  a given iii. iv. v. vi. vii. viii.  decreases rapidly for both schemes, at (cf. Tables 2.11– 2.16); with a decrease in the value of  the  for a given value of @ (cf. Tables 2.11– 2.16); decreases quickly for both schemes the modified MOP scheme is performing significantly better than the simple MOM scheme for all choices of * (cf. Tables 2.11– 2.16); performance of the two proposed schemes is generally better for smaller choices of * (cf. Tables 2.11– 2.16); the modified MOP scheme has the ability to perform well even for moderately large values of *; the application of both the schemes is quite simple and easily executable; the performance of the EWMA type charts can further be enhanced by extending the proposed schemes with the help of other runs rules schemes; ix. the
c decreases for both schemes as the value of @ increases (cf. Abbas, Riaz and Does (2011)); x. the run length distribution of both schemes is positively skewed (cf. Abbas, Riaz and Does (2011)). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů Ϯϵ ϯϬ  ʹǤ͵Ǥʹ ‘’ƒ”‹•‘• In this section we provide a detailed comparison of the proposed schemes with their already existing counterparts meant for detecting small shifts, i.e. EWMA- and CUSUM-type charts. The performance of all the control charting schemes is compared in terms of . The control schemes used for the comparison purposes include the classical EWMA, the classical CUSUM, the FIR CUSUM, the FIR EWMA, the weighted CUSUM, the double CUSUM, the distribution-free CUSUM and the runs rules schemes based CUSUM. Proposed versus the classical EWMA: The classical EWMA is defined by Roberts (1959). values for the classical EWMA are given in Table 2.2. The classical EWMA refers to one out of one (+O+) scheme. The comparison of the three schemes (i.e. the classical EWMA and the two proposed schemes) shows that both proposed EWMA schemes of Section 2.3 are performing better than the classical scheme in terms of (cf. Tables 2.15 & 2.16 vs. Table 2.2). Moreover the modified MOP scheme is outperforming the simple MOM scheme with a great margin for the small shifts (i.e. >MB - @ - +>B). The performance of the two proposed schemes almost coincide for larger values of @. Proposed versus the classical CUSUM: The classical CUSUM is defined by Page (1954). The values of the classical CUSUM are given in Table 2.1 at  168 and 465. The comparison of the classical CUSUM with the proposed schemes reveals that both schemes are outperforming the classical CUSUM scheme at all the values of @ (cf. Tables 2.11 & 2.12 vs. Table 2.1). Particularly, comparing the three schemes at @  >MB, we observe that the modified MOP scheme is performing the best with scheme with  fixed at +Xf.  PA>A followed by the simple MOM  BA>X, whereas the classical CUSUM has the modified MOP scheme is giving almost half      dA>M which mean that than the classical CUSUM scheme with DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  Proposed versus the FIR CUSUM: The FIR CUSUM presented by Lucan and Crosier (1982) gives a head start to the CUSUM statistic rather than setting it equal to zero. The 8 of FIR CUSUM with two different values for the head start ( ) are given in Table 2.10. Comparing the performance of FIR CUSUM with the proposed schemes we can see that the modified MOP scheme is performing better than the FIR CUSUM even though that there is a problem with the FIR CUSUM because its  value is less than 168 (the desired level).  Moreover, we see that if the value of Co increases than the value of decreases, which is not recommended in case of sensitive processes (cf. Bonetti et al. (2000)). The proposed schemes are not only fixing the  at the pre-specified level (so that valid comparison can be made) but also performing better in terms of  minimizing the (with a fixed )  8, i.e. the proposed schemes are without a decrease in  and without the need of any head start value (cf. Tables 2.11 & 2.12 vs. Table 2.10). Proposed versus the FIR EWMA: Lucas and Saccucci (1990) proposed the application of the FIR feature with the EWMA control chart (especially with small values of *). The values of the EWMA control chart with FIR feature are provided in Table 2.17. @ TABLE 2.17: 9:; values for the FIR EWMA scheme % Head Start *  >+  M>f+A *  >MB  M>eef *  >B  P>d+ *  >dB  P>fd 0 25 50 487 468 491 483 497 487 498 496 0.5 25 50 28.3 24.2 46.5 43.6 87.8 86.1 140 139 1 25 50 8.75 6.87 10.1 8.79 16.9 15.9 30.2 29.7 2 25 50 3.57 2.72 3.11 2.5 3.29 2.87 4.33 4.09 Comparing the FIR EWMA with the proposed schemes we observe that the proposed schemes are not only having smaller 8 but they also fix the  value at desired level DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϯϭ ϯϮ  which is not the case with the FIR EWMA (cf. Tables 2.15 & 2.16 vs. Table 2.17). The other comments made in favor of the proposed schemes versus the FIR CUSUM are also valid here with the same spirit and strength. Proposed versus the weighted CUSUM: Yashchin (1989) proposed a class of weighted CUSUM charts which generalize the classical CUSUM charts by giving weights to the past information and can be viewed as the EWMA version of the CUSUM charts. The 8 for the weighted CUSUM are given in Table 2.9 where the weights given to the past information are represented by j. Comparing the weighted CUSUM with the proposed schemes we notice that the proposed schemes are performing better than the weighted CUSUM for all the values of @, which shows the uniform superiority of the proposed schemes over the weighted CUSUM (cf. Tables 2.15 & 2.16 vs. Table 2.9). Proposed versus the double CUSUM: Waldmann (1995) has shown that the simultaneous use of two classical CUSUMs improves the performance of the CUSUM chart. This simultaneous use of the two CUSUM charts is being given the name of double CUSUM. The performance of the double CUSUM is given in Table 2.18 in which parameters of the 1st CUSUM are and $ and parameters of the 2nd CUSUM are q and $q . TABLE 2.18: 9:;m for the double CUSUM with r  I> I, sq  g> G and rq  I> Iat 9:;=  ?== C $  M>X 0 0.5 1 1.5 2 507 27.1 9.85 5.55 3.57 Comparison of the double CUSUM with the proposed schemes shows that the double CUSUM performs better than the simple MOM scheme for @  >B but the modified MOP scheme performs better than both the simple MOM scheme and the double CUSUM. For all other values of @, the modified MOP scheme is performing the best followed by the simple MOM scheme (cf. Tables 2.15 & 2.16 vs. Table 2.18). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  Proposed versus the distribution-free CUSUM: Chatterjee and Qiu (2009) proposed a class of distribution-free CUSUM charts. The three non-parametric control charts named as B1, B2 and B3 depend upon the variable  which is defined as: hhhuvh     t { whhhuvh b   b  x x  y b z w  +M x x  Z where  is the number of samples since the last time the statistic  was zero. The performance of these non-parametric charts is given in Table 2.19. TABLE 2.19: 9:; values for different distribution-free CUSUM schemes with nominal 9:;=  F== 0 @ 0.5 1 B1 B2 B3 178.43 173.78 201.86 25.31 18.37 27.38 12.54 7.94 9.14 30 B1 B2 B3 202.92 194.44 197.79 18.89 18.68 19.20 6.60 6.43 6.45 40 B1 B2 B3 195.04 198.98 201.87 22.40 20.52 21.36 5.66 5.70 5.77 50 B1 B2 B3 190.88 199.35 202.79 16.96 18.73 17.51 6.59 6.84 6.50 w|}~ Chart 5 For @  >B the best ARL performance is at w|}~  B by chart B1. In this case the +d> whereas the    for the simple MOM and modified MOP schemes is M+> and +B>A, respectively, which shows superiority of the modified MOP scheme. For @  + the distribution-free charts perform slightly better for w|}~  A but for all other values of w|}~ , the modified MOP scheme is again performing better. This proves the dominance of the modified MOP scheme as compared to the distribution-free CUSUM charts in general (cf. Tables 2.13 & 2.14 vs. Table 2.19). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϯϯ ϯϰ  Proposed versus the runs rules based CUSUM: In Section 2.2 we proposed two runs rules schemes, namely CUSUM scheme I and CUSUM scheme II, on the CUSUM charts and computed the  values for the two schemes which are given in Tables 2.5 and 2.6 for  M. Comparing the proposed EWMA schemes with these runs rules based CUSUM schemes I and II we see that the two proposed EWMA schemes are performing better than the CUSUM schemes of Section 2.1 (cf. Tables 2.13 & 2.14 vs. Tables 2.5 & 2.6). Moreover, for an overall comparison of the proposed schemes with their existing counterparts mentioned and compared above we have made some graphs showing curves of different schemes. It is evident from the Figures 2.3 – 2.5 that the curves of the two proposed EWMA schemes exhibit dominance in general as compared to all the other schemes covered in this chapter. Figure 2.3: 9:; curves for the simple FOF and modified FOI EWMA schemes, the classical CUSUM and the FIR CUSUM at 9:;=  EgG tDƐŝŵƉůĞϮͬϮ;ʄсϬ͘ϭͿ ĐůĂƐƐŝĐĂůh^hD;ŬсϬ͘ϱͿ ϴϬ tDŵŽĚŝĨŝĞĚϮͬϯ;ʄсϬ͘ϭͿ &/Zh^hD;ŬсϬ͘ϱ͕ŽсϭͿ ϳϬ ϲϬ Z>Ɛ ϱϬ ϰϬ ϯϬ ϮϬ ϭϬ Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ į DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭ ϭ͘ϱ ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD Figure 2.4: 9:; curves for the simple FOF and modified FOI EWMA schemes, the runs rules scheme I for CUSUM and the the runs rules scheme II for CUSUM at 9:;=  F== tDƐŝŵƉůĞϮͬϮ;ʄсϬ͘ϭͿ h^hD/;ŬсϬ͘ϱͿ tDŵŽĚŝĨŝĞĚϮͬϯ;ʄсϬ͘ϭͿ h^hD//;ŬсϬ͘ϱͿ ϵϬ ϴϬ ϳϬ Z>Ɛ ϲϬ ϱϬ ϰϬ ϯϬ ϮϬ ϭϬ Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ ϭ ϭ͘ϱ į Figure 2.5: 9:; curves for the simple FOF and modified FOI EWMA schemes, the classical EWMA, the FIR EWMA, the weighted CUSUM and the double CUSUM at 9:;=  ?== tDƐŝŵƉůĞϮͬϮ;ʄсϬ͘ϭͿ tD;ʄсϬ͘ϭͿ tDŵŽĚŝĨŝĞĚϮͬϯ;ʄсϬ͘ϭͿ &/ZtD;ʄсϬ͘ϭͿ ϲϬ ϱϬ ϰϬ Z>Ɛ  ϯϬ ϮϬ ϭϬ Ϭ Ϭ͘ϱ ϭ Ϯ į DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϯϱ ϯϲ  Particularly, the curve of the modified MOP EWMA scheme is on the lower side compared to all other schemes. This shows the best performance of the modified MOP EWMA scheme compared to all others. For the small shifts, the gap between the curves of the proposed schemes with those of the other schemes is large, whereas this gap reduces as the size of the shift increases. This implies that the proposals of the study (particularly modified MOP EWMA scheme) are generally more beneficial for small shifts. ʹǤ͵Ǥ͵ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ This section presents an illustrative example to show how the proposed schemes can be applied in real situations. For this purpose we have used dataset 2.1 and dataset 2.2 from subsection 2.2.3. The EWMA statistics are calculated with *  >+ and the three schemes (i.e. the two proposed schemes and the classical scheme with   B) are applied to the datasets. The graphical display of the control chart with all the three schemes applied to the datasets 2.1 and 2.2 are given in Figures 2.6 and 2.7 respectively. From Figure 2.6 we can see that the first 20 points are plotted around the central line whereas an upward shift in the points can be seen afterwards. The classical scheme is not signaling any shift whereas the simple MOM scheme is signaling at points # 49 and 50. The modified MOP scheme is giving 3 out of control signals and these are at points # 45, 49 and 50. This clearly indicated that the modified MOP scheme is not only signaling earlier than the classical scheme but also is giving more number of signals. The situation is not much different in Figure 2.7 where the classical and the simple MOM schemes failed to detect any shift while the modified MOP scheme gives out of control signals at points # 25, 26 and 27. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZƵŶƐƌƵůĞƐďĂƐĞĚh^hDĂŶĚtD  Figure 2.6: EWMA chart of the classical scheme and the the simple FOF and modified FOI schemes for the dataset 2.1 ŝ >>;ĐůĂƐƐŝĐĂůͿ >>;ƐŝŵƉůĞϮͬϮͿ >>;ŵŽĚŝĨŝĞĚϮͬϯͿ Ϭ͘ϴ Ϭ͘ϲ Ϭ͘ϰ Ϭ͘Ϯ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ϰϭ ϰϯ ϰϱ ϰϳ ϰϵ ͲϬ͘Ϯ ͲϬ͘ϰ ͲϬ͘ϲ ͲϬ͘ϴ Figure 2.7: EWMA chart of the classical scheme and the the simple FOF and modified FOI schemes for the dataset 2.2 ŝ >>;ĐůĂƐƐŝĐĂůͿ >>;ƐŝŵƉůĞϮͬϮͿ >>;ŵŽĚŝĨŝĞĚϮͬϯͿ Ϭ͘ϴ Ϭ͘ϲ Ϭ͘ϰ Ϭ͘Ϯ Ϭ ϭ Ϯ ϯ ϰ ϱ ϲ ϳ ϴ ϵ ϭϬ ϭϭ ϭϮ ϭϯ ϭϰ ϭϱ ϭϲ ϭϳ ϭϴ ϭϵ ϮϬ Ϯϭ ϮϮ Ϯϯ Ϯϰ Ϯϱ Ϯϲ Ϯϳ Ϯϴ Ϯϵ ϯϬ ͲϬ͘Ϯ ͲϬ͘ϰ ͲϬ͘ϲ ͲϬ͘ϴ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϯϳ ϯϴ  The above example indicates that the modified MOP EWMA scheme is giving the advantage in terms of run length as well the number of signals for both small and moderate shifts. The outcomes of these two illustrative examples are completely in accordance with the findings of subsection 3.3.1. ʹǤͶ ‘ Ž—†‹‰”‡ƒ”• For small shifts CUSUM charts and EWMA charts are considered most effective. The efficiency of these charts can also be increased by using different sensitizing rules and runs rules schemes with their usual design structure. We have proposed two runs rules schemes each for CUSUM and EWMA charts for the location parameter. By investigating the performance of the these proposed schemes and by comparing them with some existing schemes we found that the proposed schemes have the ability to perform better for small and moderate shifts while reasonably maintaining their efficiency for large shifts as well. To make the CUSUM and EWMA charts even more efficient, some other sensitizing rules/runs rules schemes can be used with their respective structures on the similar lines as followed in this chapter. The proposals and the recommendations of this chapter can also be extended for the attribute control charts based on CUSUM and EWMA patterns. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ŚĂƉƚĞƌϯ  Šƒ’–‡”͵ ‹š‡†Ǧ Šƒ”–• Shewhart-type control charts are sensitive for large disturbances in the process, while CUSUM- and EWMA-type control charts are intended to spot small and moderate disturbances. In this chapter we propose a mixed EWMA-CUSUM control chart for detecting a shift in the process mean and evaluating its 8. Comparisons of the proposed control chart are made with some representative control charts including the classical CUSUM, classical EWMA, Fast Initial Response (FIR) CUSUM, FIR EWMA, adaptive CUSUM with EWMA based shift estimator, weighted CUSUM and runs rules based CUSUM and EWMA. The comparisons reveal that the mixing of the two charts makes the proposed scheme even more sensitive to small shifts in the process mean than the other schemes designed for detecting small shifts. Following the mixed EWMA-CUSUM chart for location, we also propose a new control chart for monitoring the process dispersion. This chart is named the CS-EWMA chart as its plotting statistic is based on a cumulative sum of the exponentially weighted moving averages. Comparisons with other memory charts used to monitor the process dispersion are done by means of the . An illustration of the proposed technique is done by applying the CS-EWMA chart on a simulated dataset. This chapter is based on two papers; one for monitoring the location parameter (cf. Abbas, Riaz and Does (2012a)) and the other for monitoring the dispersion parameter (cf. Abbas, Riaz and Does (2012b)). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϯϵ ϰϬ  ͵Ǥͳ ‹š‡†Ǧ Šƒ”–ˆ‘”Ž‘ ƒ–‹‘ After the development of CUSUM and EWMA charts, several modifications of these charts have been presented in order to further enhance the performance of these charts. Lucas (1982) presented the combined Shewhart-CUSUM quality control scheme in which Shewhart limits and CUSUM limits are used simultaneously. Lucas and Crosier (1982) recommended the use of the FIR CUSUM which gives a head start to the CUSUM statistic by setting the initial values of the CUSUM statistic equal to some positive value (non-zero). This feature gives better  performance but at the cost of a decrease in . Yashchin (1989) presented the weighted CUSUM scheme which gives different weights to the previous information used in CUSUM statistic. Section 2.2 introduced the runs rules schemes to the CUSUM charts and shown that the runs rules based CUSUM performs better than the classical CUSUM for small shifts. Similarly, on the EWMA side, Lucas and Saccucci (1990) presented the combined Shewhart-EWMA quality control scheme which gives better  performance for both small and large shifts. Steiner (1999) provided the FIR EWMA which gives a head start to the initial value of the EWMA statistic (like FIR CUSUM) and hence improves the  performance of the EWMA charts. Section 2.3 discussed the runs rules schemes to the EWMA charts and showed that the runs rules based EWMA performs better than the classical EWMA for small shifts. In the next subsection we present a mixed EWMACUSUM quality control scheme for monitoring the mean of a normally distributed process. The inspiration is to get an improved performance by combining the features of EWMA and CUSUM charts in a single control structure. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  ͵ǤͳǤͳ ‡•‹‰•–”— –—”‡‘ˆ–Š‡’”‘’‘•‡† Šƒ”– In this subsection we propose an assortment of the classical EWMA and CUSUM schemes by combining the features of their design structures. The said proposal mainly depends on two statistics named as 7 and 7 which are defined as: 7  €      . 7  €      where q ‚ q ‚  ! 7 ƒ q ‚ „  ! 7 ƒ (3.1) is a time varying reference value for the proposed charting structure, the quantities 7 and 7 are known as the upper and lower CUSUM statistics which are initially set to zero (i.e. 7  7  ) and are based on the EWMA statistic  which is defined as:   *‚  ! 5+  *‚ 6 (3.2) In (3.2), *‚ is the constant like * in (1.3) such that  , *‚ - + and the initial value of the  statistic is set equal to the target mean i.e.    . Now the mean and variance of statistic  is given as: †‡     ˆ‰   'Š ‹ 0Œ 0Œ \+  5+  *‚ 6 a  (3.3) and this will be used later in the calculation of the parameters of the proposed chart. In (3.1) and (3.2) we are considering the case of individual observations Z  + which may be extended easily for the subgroups. Now the statistics 7 and 7 are plotted against the control limit, say $‚q . As long as the values of 7 and 7 are plotted inside the control limit, the process is said to be in control, otherwise out of control. It is to be noted here that if the statistic 7 is plotted above $‚q the process mean is said to be shifted above the target value and if the statistic 7 is plotted above $‚q the process is said to be shifted below the target value. The control limit $‚q is selected according to a prefixed . A large DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϰϭ ϰϮ  value of the prefixed q ‚ . q ‚ and $‚q are defined as: , will give a larger value of $‚q and vice versa. The two quantities  &‚ Ž ‘K   &‚ Ž 'Š / 0Œ 0Œ \+  5+  *‚ 6 a 4  \+  5+  *‚ 6 a3 0Œ 1 $‚q  (‚ Ž ‘K   (‚ Ž 'Š / 0Œ  (3.4) where &‚ and (‚ are the constants like & and (, respectively, in the classical set up for the CUSUM (cf. Section 1.2). The time varying values ‚ and $‚ are due to the variance of the EWMA statistic in expression (3.3). For a fixed value of &‚ , we can select the value of (‚ from the tables (that are given later in this subsection) that fix the In general q ‚  at our desired level. is chosen equal to half of the shift (in units of the standard deviation of  ). Hence, we choose &‚  >B as it makes the CUSUM structure more sensitive to the small and moderate shifts (cf. Montgomery (2009)), to which memory charts actually target. To evaluate the performance of a control scheme, we have used the Monte Carlo simulation approach in this chapter. An algorithm in R language (provided in Appendix 3.1) is developed to calculate the run lengths. The algorithm is run 50,000 times to calculate the average of those 50,000 run lengths. A detailed study on the performance of the proposed EWMA-CUSUM control chart to monitor the mean of a normally distributed process is provided in Tables 3.1 – 3.3 for some selective choices of @, *‚ and (‚ . For this purpose  ’8 other values of are fixed at 168, 400 and 500 which are the commonly used choices. For  ’8 one may easily obtain the results on similar lines. The relative standard errors for the results provided in Tables 3.1 – 3.3 are also calculated and found to be less than 1.2%. Moreover, we have also replicated the results of the classical CUSUM and the classical EWMA using our simulation algorithm and found almost DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  similar results as by Hawkins and Olwell (1998) and Lucas and Saccucci (1990), respectively, ensuring the validity of the simulation algorithm used. TABLE 3.1: 9:; values for the proposed EWMA-CUSUM scheme with <“  => ? at 9:;=  EgG C 0 D“  => E l“  FE> I 168.0441 D“  => F? l“  EI> F” 168.0652 D“  => ? l“  G> EF 169.8763 D“  => J? l“  ?> HG 171.0422 0.25 52.6449 54.1752 59.7829 68.15245 0.5 24.85945 22.40665 22.54895 24.12865 0.75 17.0208 14.0235 12.85555 12.60565 1 13.3323 10.4832 8.9565 8.2741 1.5 9.743 7.3272 5.78565 4.99665 2 7.90705 5.8231 4.4341 3.7365 C 0 D“  => E l“  II> ?H 402.0894 D“  => F? l“  EG> J 397.404 D“  => ? l“  E=> ?F 398.6486 D“  => J? l“  g> ”H 0.25 73.31955 78.02035 90.45915 108.0086 0.5 33.06085 29.0845 28.94885 31.44015 0.75 22.39445 17.79425 15.75695 15.66785 1 17.63975 13.2232 10.94695 10.17115 1.5 12.88105 9.113 6.9587 6.03875 2 10.45315 7.2235 5.2808 4.4203 C 0 D“  => E l“  IJ> HF 498.3882 D“  => F? l“  F=> EG 502.018 D“  => ? l“  EE> F 507.9555 D“  => J? l“  J> IF 0.25 80.13585 83.7529 100.2635 121.9883 TABLE 3.2: 9:; values for the proposed EWMA-CUSUM scheme with <“  => ? at 9:;=  H== 400.8962 TABLE 3.3: 9:; values for the proposed EWMA-CUSUM scheme with <“  => ? at 9:;=  ?== 507.5152 0.5 35.524 30.88825 30.7466 33.5054 0.75 24.0522 18.8755 16.6399 16.5139 1 18.8637 13.8816 11.45835 10.6107 1.5 13.79075 9.6036 7.29565 6.3101 2 11.19775 7.59055 5.52345 4.589 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϰϯ ϰϰ  The main findings about our proposed EWMA-CUSUM quality control scheme for monitoring the mean of a normally distributed process are given as: i. mixing of the EWMA and CUSUM schemes really boosts the performance of the resulting combination of the two charts especially for small and moderate shifts in the process (cf. Tables 3.1 – 3.3); ii. for detecting small shifts in the process, the performance of the proposed scheme is iii. the proposed scheme is iv. better with smaller values of *‚ and vice versa (cf. Tables 3.1 – 3.3); , the decreases with a decrease in the value of @ and vice versa (cf. Tables 3.1 – 3.3); for a fixed value of @, the  v. unbiased, i.e. for a fixed value of   of the proposed scheme decreases with a decrease in (cf. Tables 3.1 – 3.3); for a fixed value of , in *‚ (cf. Tables 3.1 – 3.3). the control limit coefficient (‚ decreases with the increase ͵ǤͳǤʹ ‘’ƒ”‹•‘• In this subsection we present a comprehensive comparison of the proposed mixed EWMA-CUSUM scheme with some existing representative EWMA and CUSUM control charts available in the literature. The performance of the control chart is compared in terms of . The set of the schemes considered for the comparison consist of the classical CUSUM, the classical EWMA, the FIR CUSUM, the FIR EWMA, the adaptive CUSUM with EWMA based shift estimator, the weighted CUSUM and the runs rules based CUSUM and EWMA. Proposed versus the classical CUSUM: The values for the classical CUSUM control scheme proposed by Page (1954) are given in Table 2.1. Comparison of the classical DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  CUSUM with the proposed schemes reveal that the proposed scheme is performing really good for all the values of *‚ , particularly for small values of *‚ . We can see that, for all values of *‚ , the proposed scheme has better performance as compared to the classical Proposed versus the classical EWMA: The values for the classical EWMA with time CUSUM (cf. Table 2.1 vs. Table 3.1). varying limits, given by Steiner (1999), are provided in Table 2.2. Comparing the classical EWMA *  >MBh with the proposed scheme we observe that the proposed scheme has better  ’8 performance with its respective values of *‚ (cf. Table 2.2 vs. Table 3.3). Proposed versus the FIR CUSUM: The FIR CUSUM presented by Lucas and Crosier (1982) provides a head start to the CUSUM statistic. The 8 of the CUSUM with FIR feature are given in Table 2.10 in which head start is represented by  . The FIR feature decreases the  of the CUSUM chart and more importantly this decreased very small for the larger values of  (for   +,   becomes  +XP) which is not recommended in case of sensitive processes like in health care (cf. Bonetti et al. (2000)). Comparing the proposed scheme with the FIR CUSUM we see that for smaller values of *‚ the proposed scheme has a better not have the fixed 3.1). performance than the FIR CUSUM, even if the FIR CUSUM does  at +Xf but has smaller  value, i.e. +XP (cf. Table 2.10 vs. Table Proposed versus the FIR EWMA: FIR EWMA presented by Steiner (1999) is similar to the FIR CUSUM as it also gives a head start to the EWMA statistic. The control limits for the FIR based EWMA chart are given as:  • '– 5+  +  —}p 6˜ * +  +  *  M* DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϰϱ ϰϲ where ‘  MN™š›+  —  +O+e. The  8 for the FIR EWMA with *  >+ and the proposed chart with *‚  >+ are given in Table 3.4. Comparing the proposed scheme with the FIR EWMA we see that the proposed scheme is performing better than the FIR EWMA for smaller shifts i.e. @ , >B. For moderate and larger shifts, FIR EWMA seems superior as compared to the proposed chart. TABLE 3.4: 9:; values for the FIR EWMA scheme and the proposed chart C 0 *  >+,  P —  >A 515.6 *  >+,  P —  >B 613.8 *‚  >+, ‘œ  >B  œ  Pd>eA 516.48 *‚  >+, ‘œ  >B  œ  A>f 0.25 83.1 99.2 81.03 85.79 FIR EWMA EWMA-CUSUM 613.62 0.5 18.5 22.1 35.76 37.55 0.75 7.3 8.8 24.2 25.41 1 3.8 4.6 19.01 19.94 1.5 1.7 2.1 13.9 14.55 2 1.3 1.4 11.29 11.8 3 1 1 8.48 8.88 4 1 1 6.96 7.29 Proposed versus the adaptive CUSUM with EWMA based shift estimator: Jiang et al. (2008) proposed the use of adaptive CUSUM with EWMA-based shift estimator. They used the concept of adaptively updating the reference value of the CUSUM chart using the EWMA estimator and then using a suitable weighting function. The values for the adaptive  CUSUM are given in Table 3.5 in which @|` , *, j and ( are the parameters of the chart. Comparing the performance of the proposed scheme we notice that the proposed scheme is outperforming the adaptive CUSUM for small values of @. For moderate and large values of @, both the proposed scheme and adaptive CUSUM have almost the same (cf. Table 3.5 vs. Table 3.2). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů performance DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  TABLE 3.5: 9:; values for adaptive CUSUM with C žŸ  => ? and D  => I at 9:;=  H== C 0 j  +>B (  B>B 0.25 399.7 ji jM jP jA j  M>B (  A>dP (  A>BB (  A>PeA (  A>PPd (  A>PPA 400.85 400.19 399.29 399.39 399.97 92.82 91.65 88.96 87.02 85.81 85.8 0.5 30.52 30.1 29.32 28.79 28.46 28.45 0.75 14.7 14.5 14.2 14 13.89 13.88 1 9.07 8.96 8.81 8.72 8.67 8.66 1.5 4.89 4.87 4.84 4.83 4.83 4.82 2 3.23 3.25 3.28 3.31 3.35 3.34 There is also an adaptive EWMA chart (cf. Capizzi and Masarotto (2003)), but its performance is inferior to the adaptive CUSUM, so the results of our proposal are superior to the adaptive EWMA as well. Proposed versus the weighted CUSUM: Weighted CUSUM presented by Yashchin (1989) gives weights to the past information in the CUSUM statistic. The 8 for the weighted CUSUM are given in Table 2.9 in which the weights given to the past information are represented by j. The comparison of the proposed scheme with the weighted CUSUM shows that the proposed scheme is performing better than the weighted CUSUM for the small and moderate shifts (like @ , +>B). For larger values of @, the weighted CUSUM almost coincide with the proposed scheme (cf. Table 2.9 vs. Table 3.3). Proposed versus the runs rules based CUSUM: Section 2.2 introduced the use of the runs rules schemes with the design structure of the CUSUM charts. The 8 for the two runs rules based CUSUMs are given in Tables 2.3 and 2.4 in which WL and AL are representing the warning limits and action limits, respectively. The comparison of the proposed scheme with both the runs rules based CUSUM schemes shows that the proposed scheme has the ability to perform better than the runs rules based CUSUM for all the choices of *‚ (cf. Tables 2.3 and 2.4 vs. Table 3.1). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϰϳ ϰϴ  Proposed versus the runs rules based EWMA: Section 2.3 introduced the use of the runs rules schemes EWMA structure. The 8 for the two runs rules based EWMAs are given in Tables 2.15 and 2.16. The comparison of the proposed schemes with both the runs rules based EWMA schemes shows that the proposed scheme is performing better as long as * % >+ for the runs rules based EWMA schemes. For *  >+, modified MOP EWMA scheme becomes a bit superior to the proposed chart (cf. Tables 2.15 and 2.16 vs. Table 3.3). Overall View: In order to provide an overall comparative view of the proposed scheme with the other existing counterparts we have made some graphical displays, in the form of curves. Three selective graphs of different charts/schemes are given in Figures 3.1 – 3.3. In these figures RR CUSUM (EWMA) stands for the runs rules based CUSUM (EWMA) schemes and the other terms/symbols used are self-explanatory. Figure 3.1: 9:; curves for the proposed scheme, the Classical CUSUM, the FIR CUSUM and Runs Rules based CUSUMs at 9:;=  EgG WƌŽƉŽƐĞĚ;ʄсϬ͘ϱͿ &/Zh^hD;ŽсϭͿ ZZh^hD//;t>сϯ͘ϱ͕>сϰ͘ϰϰͿ ϴϬ ůĂƐƐŝĐĂůh^hD;ŬсϬ͘ϱ͕ŚсϰͿ ZZh^hD/;t>сϯ͘ϱϯ͕>сϰ͘ϮͿ ϳϬ ϲϬ Z>Ɛ ϱϬ ϰϬ ϯϬ ϮϬ ϭϬ Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ į ϭ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭ͘ϱ Ϯ DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  Figure 3.2: 9:; curves for the proposed scheme and adaptive CUSUM at 9:;=  H== WƌŽƉŽƐĞĚ;ʄсϬ͘ϭͿ WƌŽƉŽƐĞĚ;ʄсϬ͘ϱͿ ĚĂƉƚŝǀĞh^hD;ɶсϮ͘ϱͿ ϭϬϬ ϵϬ ϴϬ Z>Ɛ ϳϬ ϲϬ ϱϬ ϰϬ ϯϬ ϮϬ ϭϬ Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ į ϭ ϭ͘ϱ Ϯ Figure 3.3: 9:; curves for the proposed scheme, the classical EWMA, the FIR EWMA, the runs rules based EWMA and the weighted CUSUM at 9:;=  ?== WƌŽƉŽƐĞĚ;ʄсϬ͘ϱͿ ZZtD/;ʄсϬ͘ϱͿ ůĂƐƐŝĐĂůtD;ʄсϬ͘ϱͿ ZZtD//;ʄсϬ͘ϱͿ tĞŝŐŚƚĞĚh^hD;ɶсϬ͘ϵͿ ϭϬϬ ϵϬ ϴϬ Z>Ɛ ϳϬ ϲϬ ϱϬ ϰϬ ϯϬ ϮϬ ϭϬ Ϭ Ϭ͘ϱ ϭ į ϭ͘ϱ Ϯ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϰϵ ϱϬ  By examining the graphs of curves of different schemes under study we see that the curve of the proposed schemes are on the lower side which shows evidence for the dominance of the proposed scheme over the other schemes. For small values of @, the difference between the of the proposed scheme and the other schemes is larger whereas for the moderate values of @ this difference almost disappears. For large values of @, the curve of the proposed chart seems above the curves of some other charts showing the poor performance of the proposed chart for large shifts. To sum up, we may infer that in general the proposed chart is superior for small and moderate shifts while for larger shifts its performance is inferior to some of the other schemes under investigation. ͵ǤͳǤ͵ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ Besides exploring the statistical properties of a method it is always good to provide its application on some data for illustration purposes. Here we present an illustrative example to show how the proposed scheme can be applied in the real situation. For this purpose a dataset is generated containing 40 observations. The first 20 observations are generated from the in control situation (i.e. [+ so that the target mean is 0) and the remaining 20 observations are generated from an out of control situation with a small shift introduced in the process (i.e. [>B+). The classical CUSUM, the classical EWMA and the proposed scheme are applied to this dataset and the parameters are selected to be &  >B and (  B>d for the classical CUSUM scheme, *  >MB and  P for the classical EWMA scheme, *‚  >MB, &‚  >B and (‚  M>+f for the proposed scheme to guarantee that  DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů  B. DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ Figure 3.4: The classical CUSUM chart for the simulated dataset using <  => ? and l  ?> =” at 9:;=  ?== н Ͳ Ś ϲ ϱ ŝ ϰ ϯ Ϯ ϭ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ^ĂŵƉůĞEƵŵďĞƌ Figure 3.5: The classical EWMA chart for the simulated dataset using D  => F? and ;  I at 9:;=  ?== ŝ  ŽŶƚƌŽů>ŝŵŝƚƐ Ϭ͘ϴ Ϭ͘ϯ ŝ  ϱϭ ͲϬ͘Ϯ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ͲϬ͘ϳ Ͳϭ͘Ϯ ^ĂŵƉůĞEƵŵďĞƌ  DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϱϮ  Figure 3.6: The proposed scheme for the simulated dataset using D“  => F?, <“  => ? and l“  F=> EG at 9:;=  ?== Dн DͲ ďŝ ϭϮ ϭϬ ŝ ϴ ϲ ϰ Ϯ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ^ĂŵƉůĞEƵŵďĞƌ  The calculations for the proposed scheme are given in Table 3.6 and the graphical display of all three control structures are provided in Figures 3.4 – 3.6 with the statistics  and  plotted against the control limit $ for the classical CUSUM scheme; ) plotted against the control limits given in (1.4) for the classical EWMA scheme; and 7 and 7 plotted against the control limit $‚ for the proposed scheme. From Table 3.6 and Figure 3.6 it is obvious that out of control signals are received at samples # 32, 33, 34, 35, 36, 37, 38, 39 and 40 by the proposed scheme (giving 8 out of control signals). Figures 3.4 – 3.5 show that the separate applications of the classical CUSUM and EWMA schemes fail to detect any out of control situation for the given dataset. This clearly indicates superiority of the proposed scheme over the classical CUSUM and EWMA schemes and it is exactly in accordance with the findings of subsection 3.1.1. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ TABLE 3.6: Application example of the proposed scheme using D“  => F?, ¡œ  => ? and ¢œ  F=> EG at 9:;=  ?==  Sample    No.  q ‚ 7 0 7 0 $‚q 5.045 Sample    No.  q ‚ 7 7 $‚q 1 -0.113 -0.028 0.125 21 0.781 0.452 0.189 3.175 0 7.627 2 -1.906 -0.498 0.156 0 0.341 6.306 22 -0.016 0.335 0.189 3.321 0 7.627 3 -1.891 -0.846 0.171 0 1.016 6.915 23 -0.061 0.236 0.189 3.368 0 7.627 4 0.508 -0.508 0.179 0 1.344 7.235 24 0.332 0.26 0.189 3.439 0 7.627 5 1.374 -0.037 0.184 0 1.198 7.409 25 1.391 0.543 0.189 3.793 0 7.627 6 0.05 -0.015 0.186 0 1.027 7.506 26 1.89 0.879 0.189 4.483 0 7.627 7 0.401 0.089 0.187 0 0.751 7.559 27 0.709 0.837 0.189 5.131 0 7.627 8 0.692 0.239 0.188 0.051 0.323 7.589 28 -0.82 0.423 0.189 5.364 0 7.627 9 0.851 0.392 0.188 0.255 0 7.606 29 1.481 0.687 0.189 5.863 0 7.627 10 0.927 0.526 0.189 0.593 0 7.615 30 0.314 0.594 0.189 6.268 0 7.627 11 2.187 0.941 0.189 1.346 0 7.621 31 2.231 1.003 0.189 7.082 0 7.627 12 0.02 0.711 0.189 1.868 0 7.623 32 0.802 0.953 0.189 7.846* 0 7.627 13 0.12 0.563 0.189 2.242 0 7.625 33 -1.25 0.402 0.189 8.059* 0 7.627 14 2.138 0.957 0.189 3.01 0 7.626 34 0.351 0.389 0.189 8.260* 0 7.627 15 0.183 0.764 0.189 3.585 0 7.627 35 1.362 0.632 0.189 8.703* 0 7.627 16 -2.389 -0.024 0.189 3.371 0 7.627 36 -0.529 0.342 0.189 8.856* 0 7.627 17 -0.269 -0.086 0.189 3.097 0 7.627 37 2.59 0.904 0.189 9.571* 0 7.627 18 0.317 0.015 0.189 2.923 0 7.627 38 0.287 0.75 0.189 10.132* 0 7.627 19 0.055 0.025 0.189 2.759 0 7.627 39 1.676 0.981 0.189 10.924* 0 7.627 20 1.293 0.342 0.189 2.912 0 7.627 40 -0.303 0.66 0 7.627 0.189 11.395* * indicates proposed scheme giving out of control signal ͵Ǥʹ ‹š‡†Ǧ Šƒ”–ˆ‘”†‹•’‡”•‹‘ Page (1963) introduced the CUSUM chart for monitoring the increase in process dispersion using sample ranges. Following him, Tuprah and Ncube (1987), Chang and Gan (1995) and Acosta-Mejia et al. (1999) proposed several improved versions of the CUSUM chart for process dispersion. On the other hand, Wortham and Ringer (1971) suggested an EWMA control chart for monitoring the process dispersion. Ng and Case (1989) and Crowder and Hamilton (1992) proposed improved versions of the EWMA chart for DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϱϯ ϱϰ  monitoring process variance. Castagliola (2005) and Castagliola et al. (2009) proposed EWMA respectively CUSUM control charts for monitoring the process variance based on a logarithmic transformation of the sample variance. This article proposes a new memory-type control chart based on the same transformation, named as CS-EWMA chart, for monitoring the process dispersion by mixing the effects of EWMA and CUSUM charts. After presenting the basic structures of the EWMA and CUSUM charts for monitoring the process dispersion in the next subsection, we present the details of our proposed CS-EWMA chart for process standard deviation in the subsequent subsection. ͵ǤʹǤͳ F Ǧ ‘–”‘Ž Šƒ”– Castagliola (2005) proposed an
 -EWMA control chart for monitoring the process dispersion. This control structure is based on a three parameter logarithmic transformation which is given as:   ‘ £ ! £ ¤‡
 ! ¥£  (3.5) where
 is the sample variance for # pV sample defined as
  © ¦ª̈«¬5–§¨ –§ 6 ` , y represents the wpV observation from the # pV sample of size Z from a normal distribution with mean , standard deviation ' and  is the average of the # pV sample. The constants ‘ £ , £ and ¥£ are defined as £  ­£ Z, ¥£  £ Z' and ‘ £  £ Z  M­£ Z ¤‡'  as in Castagliola (2005). He derived the distribution of  and showed that if the constants ‘ £ , £ and ¥£ are judiciously selected, then the distribution of variable  becomes very close to the normal distribution with mean  £ Z and variance '£ Z, i.e.  ® [5 £ Z '£ Z6 (cf. Appendix A in Castagliola (2005)). Table 3.7 reproduces the values of £ Z, '£ Z for Z  PAB x > +B from Table I in Castagliola (2005). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ­£ Z, £ Z,  £ Z and DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ Now using the approximately normally distributed variable  from Castagliola (2005), the  plotting statistic for the
 -EWMA chart is defined as: )  * ! +  *) (3.6) where * is the sensitivity parameter chosen as  , * - + and the initial value of ) is taken as )  given as: £ Z ! ­£ Z ¤‡5+ ! £ Z6. The control limits for the statistic given in (3.6) are    £ Z  / where The 0 0 '£ Z,    £ Z,    £ Z ! / 0 0 '£ Z (3.7) is the control limit coefficient that determines the distance between  and  . Table 3.7: Values of ¯°  , ±°  , 9°  , ²°   and n°   3 9°   -0.6627 ²°   1.8136 n°   0.6777 ¯°   ±°   0.02472 0.9165 4 -0.7882 2.1089 0.6261 0.01266 0.9502 5 -0.8969 2.3647 0.5979 0.00748 0.9670 6 -0.9940 2.5941 0.5801 0.00485 0.9765 7 -1.0827 2.8042 0.5678 0.00335 0.9825 8 -1.1647 2.9992 0.5588 0.00243 0.9864 9 -1.2413 3.1820 0.5519 0.00182 0.9892 10 -1.3135 3.3548 0.5465 0.00141 0.9912 11 -1.3820 3.5189 0.5421 0.00112 0.9927 12 -1.4473 3.6757 0.5384 0.00090 0.9938 13 -1.5097 3.8260 0.5354 0.00074 0.9947 14 -1.5697 3.9705 0.5327 0.00062 0.9955 15 -1.6275 4.1100 0.5305 0.00052 0.9960 values of
 -EWMA are given in Table 3.8 for different values of *, where ³ represents the amount of shift in the standard deviation (i.e. ³  the shifted standard deviation) and the in control work on this subject. The ' Y' with ' representing is fixed at M, as is done in earlier 8 in this section are evaluated through simulation procedures by running +´ replications. The program is developed in R language. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϱϱ ϱϲ  Table 3.8: 9:; values for the µF -EWMA chart with 0.5 D  => =? ;  F> Fg”h 9.257 D  => E ;  F> H?Fh 0.6 11.679 8.931 0.7 16.101 13.03 0.8 26.108 23.679 29.961 ¶ 6.866 D  => F ;  F> ?”Fh  ? and 9:;=  F== D  => I ;  F> gIHh 5.448 D  => H ;  F> gHIh 5.928 D  => ? ;  F> gI”h 7.856 8.481 10.735 16.909 13.064 16.699 25.58 48.763 46.058 80.157 166.467 5.616 7.606 0.9 63.459 70.501 107.839 169.543 274.071 474.331 0.95 133.554 153.817 204.856 264.011 327.699 392.103 1 199.781 200.702 200.756 200.262 200.59 199.224 1.05 78.77 92.938 98.675 98.004 97.342 96.541 1.1 32.542 41.746 47.688 49.736 50.94 51.373 1.2 11.986 15.382 17.449 18.537 19.274 20.022 1.3 7.064 8.766 9.571 9.909 10.226 10.527 1.4 5.054 6.09 6.419 6.517 6.6 6.753 1.5 3.983 4.722 4.835 4.801 4.787 4.82 2 2.133 2.418 2.343 2.225 2.148 2.098 3 1.338 1.456 1.395 1.336 1.294 1.267 Note that the results from Table 3.8 coincide with the results of Table III of Castagliola (2005). ͵ǤʹǤʹ Ǧ F  ‘–”‘Ž Šƒ”– Taking inspiration from the
 -EWMA chart, Castagliola et al. (2009) proposed a CUSUM-
 chart for monitoring the process dispersion which is based on the statistic  given in (3.5). The CUSUM-
 chart uses two plotting statistics, named as   and   , and given as:   € 5   £ Z6  .   € 5   £ Z6  where  !  ƒ „  !  ƒ (3.8) is the reference value and the sensitivity parameter of the CUSUM-
 chart. The initial values for the plotting statistics given in (3.8) are taken equal to zero, i.e.     DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ . These plotting statistics are plotted against a control limit $ and an out of control signal is  received if either of the two statistics (i.e.   and   ) is plotted above $. and $ are jointly the two parameters of CUSUM-
 chart and their standard forms are given as:  &'£ Z, $  ('£ Z (3.9) where k and h are constants which determine the properties of the chart. The values for the CUSUM-
 chart for different choices of its parameters are given in Table 3.9 with fixed at M. Table 3.9: 9:; values for the CUSUM-µF chart with 0.5  >+ $  +>BPh 0.6 11.382 8.452 0.7 15.489 12.169 0.8 24.394 21.619 29.699 60.662 170.474 0.9 54.649 63.997 116.766 226.253 475.443 0.95 114.332 143.17 213.662 294.554 379.161 1 199.846 199.241 199.841 200.769 199.625 1.05 102.864 104.01 104.806 102.974 100.707 1.1 52.641 50.675 53.502 54.227 54.99 1.2 25.057 20.867 20.373 20.777 21.353 1.3 16.451 12.763 11.255 11.092 11.18 1.4 12.434 9.256 7.654 7.2 7.115 1.5 10.068 7.341 5.832 5.26 5.126 2 5.509 3.885 2.873 2.41 2.197 3 3.347 2.379 1.686 1.404 1.3 6.516  >B $  P>fBBh  ? and 9:;=  F== ³ 9.059  >MB $  X>AdXh  >dB $  M>XMh 5.071 + $  +>eXh 7.303 8.264 13.223 12.295 18.314 43.032 5.199  6.121 ͵ǤʹǤ͵ ‡•‹‰•–”— –—”‡‘ˆ–Š‡’”‘’‘•‡† Šƒ”– In this subsection we propose a memory-type control chart which is based on mixing the effects of EWMA and CUSUM charts into a single control chart structure. For the location parameter this idea was explored in Section 3.1. Again let y (i.e. the wpV DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϱϳ ϱϴ observation of # pV sample with w  +M x > >  Z and #  +M x x>) be distributed normally with  mean  and variance ' under an in control situation. Then the two plotting statistics (named as 7 and 7 ) for the proposed CS-EWMA chart are given as: 7  € 5   £ Z6  . 7  € 5   £ Z6  where q ‚ q ‚  ! 7 ƒ q ‚ „  ! 7 ƒ (3.10) is the reference value for the proposed chart, like K in (3.8). The initial value for both plotting statistics is taken equal to zero, i.e. 7  7  .  is the EWMA statistic which is defined as:   *‚  ! 5+  *‚ 6 (3.11) where *‚ is the smoothing constant like * in (3.6) and is chosen as  , *‚ - +.  is the statistic defined in (3.5). The initial value for the statistic  is taken as   £ Z ! ­£ Z ¤‡5+ ! £ Z6. The statistics 7 and 7 are now plotted against the control limit $‚q and an out of control signal is received if either of the two plotting statistics given in (3.10) is plotted above $‚q . If 7 is plotted above $‚q that would indicate a positive shift in the process standard deviation and if the value of 7 gets larger than $‚q then it would be declared that the process standard deviation has shifted downwards. The standard forms of depending upon the variance of  (i.e. ·‰u‡¸†   '£ Z ¹ . q ‚  &‚ ¹'£ Z/ $‚q  (‚ ¹'£ Z/ where 0Œ 0Œ ‚ 0Œ 0Œ º 0Œ ‚ /0 Œ 4 3 0Œ 1 º  $‚ / 0Œ 0Œ 0Œ q ‚ and $‚q º, cf. (3.7)) are given as: (3.12)  &‚ '£ Z and $‚  (‚ '£ Z. Here in (3.12), we have used the asymptotic standard deviation of the statistic  but the practitioner may use the exact standard deviation as discussed by Steiner (1999). Note that the CUSUM-
 is a special case of the proposed DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ CS-EWMA chart with *‚  +. Finally, a detailed study on the  performance of the proposed CS-EWMA chart is given in Tables 3.10 – 3.15, where the and n = 5. The algorithm developed in R language for computing the Appendix 3.2.  is fixed at M 8 is given in Values of $‚ for sample sizes other than B can be found by the relation $‚`»´  $‚`¼´ \ i. ii. ]½ `»´ a ]½ `¼´  for a fixed for fixed values of Z,  M. From Tables 3.10 – 3.15, we can conclude that: and , the value of $‚ decreases with an increase in the for fixed values of Z, *‚ and , large values of ‚ value of *‚ and vice versa; ‚ are giving small  values for detecting a positive shift in the process dispersion; Table 3.10: 9:; values for the CS-EWMA chart with D“  => =?, ³ 0.5 ‚  >+ $‚  XM>Xh 23.156 ‚  >MB $‚  Ad>Xh 0.6 26.652 23.825 0.7 32.317 28.983 0.8 43.072 38.98 34.238 20.626 ‚  >B $‚  Me>Bh  ? and 9:;=  F== ‚  >dB $‚  +f>+Bh 15.133 ‚ + $‚  +>XMh 20.32 17.749 15.758 24.993 22.144 19.969 31.092 28.876 17.482 13.289 0.9 73.716 68.713 63.844 61.61 61.053 0.95 127.175 123.052 120.583 121.6 123.108 1 199.718 200.262 200.951 200.69 199.752 1.05 94.196 88.525 84.496 82.271 81.696 1.1 52.536 47.005 41.056 37.801 35.939 1.2 31.726 27.331 22.219 18.725 16.32 1.3 24.789 21.241 16.942 13.848 11.534 1.4 21.141 18.078 14.341 11.569 9.429 1.5 18.774 16.091 12.706 10.183 8.233 2 13.481 11.562 9.112 7.224 5.712 3 10.16 8.729 6.877 5.442 4.271 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϱϵ ϲϬ  Table 3.11: 9:; values for the CS-EWMA chart with D“  => E,  ? and 9:;=  F== ³ 0.5 ‚  >+ $‚  Af>+h 17.682 ‚  >MB $‚  PB>Bh 15.366 ‚  >B $‚  MM>Mdh 12.769 ‚  >dB $‚  +A>+h 10.995 ‚ + $‚  f>f+h 0.6 20.542 17.895 14.96 13.009 11.626 0.7 25.299 22.109 18.707 16.487 15.072 0.8 34.747 30.731 26.633 24.372 23.188 0.9 63.816 58.764 55.477 55.787 57.655 0.95 115.831 113.49 115.278 120.601 126.425 9.717 1 199.304 199.885 200.85 199.751 200.113 1.05 102.296 96.919 95.415 95.943 96.935 1.1 55.637 50.376 45.925 44.641 44.409 1.2 30.526 26.334 22.075 19.62 18.345 1.3 22.596 19.244 15.62 13.331 11.874 1.4 18.601 15.786 12.672 10.624 9.178 1.5 16.218 13.729 10.956 9.064 7.73 2 11.075 9.385 7.424 6.039 4.997 3 8.098 6.881 5.438 4.395 3.559 Table 3.12: 9:; values for the CS-EWMA chart with D“  => F,  ? and 9:;=  F== ³ 0.5 ‚  >+ $‚  PB>Mh 14.017 ‚  >MB $‚  MA>eXh 11.652 ‚  >B $‚  +B>Adh 9.421 ‚  >dB $‚  +>Ph 8.106 ‚ + $‚  X>BPh 0.6 16.572 13.807 11.243 9.796 8.858 0.7 20.988 17.597 14.579 12.978 12.141 0.8 30.235 25.828 22.383 21.258 21.411 0.9 59.291 54.54 54.423 59.188 65.493 0.95 112.046 111.302 120.099 132.689 145.693 1 200.201 199.572 200.733 200.695 200.392 1.05 104.94 100.801 100.762 103.131 104.378 1.1 56.775 51.104 48.576 48.77 49.734 1.2 29.533 24.981 21.284 19.8 19.186 1.3 20.761 17.182 13.998 12.367 11.485 1.4 16.545 13.588 10.88 9.367 8.401 1.5 14.048 11.505 9.131 7.731 6.791 2 9.041 7.414 5.805 4.797 4.078 3 6.369 5.252 4.117 3.372 2.797 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů 7.222 DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  Table 3.13: 9:; values for the CS-EWMA chart with D“  => I,  ? and 9:;=  F== ³ 0.5 ‚  >+ $‚  Mf>+Ph 12.39 ‚  >MB $‚  +e>Pdh 9.956 ‚  >B $‚  ++>fdh 7.911 ‚  >dB $‚  d>dfh 6.822 ‚ + $‚  B>+Xh 0.6 14.876 11.99 9.621 8.444 7.772 0.7 19.235 15.661 12.875 11.729 11.362 0.8 28.467 23.896 21.086 21.167 22.79 0.9 57.66 53.647 56.972 66.231 78.395 0.95 111.3 112.994 128.065 147.259 164.091 6.134 1 200.262 199.154 199.005 200.568 199.814 1.05 105.433 101.759 102.912 105.905 106.502 1.1 56.456 50.968 49.573 51.114 52.289 1.2 28.726 23.988 20.749 19.841 19.724 1.3 19.793 16.096 13.122 11.908 11.206 1.4 15.467 12.395 9.873 8.664 7.917 1.5 12.948 10.325 8.125 6.969 6.265 2 7.991 6.376 4.933 4.108 3.535 3 5.473 4.409 3.412 2.789 2.366 Table 3.14: 9:; values for the CS-EWMA chart with D“  => H,  ? and 9:;=  F== ³ 0.5 ‚  >+ $‚  MP>APh 11.425 ‚  >MB $‚  +B>fPh 8.956 ‚  >B $‚  e>XMh 7.014 ‚  >dB $‚  X>Pfh 6.056 ‚ + $‚  A>MAh 0.6 13.878 10.937 8.707 7.703 7.26 0.7 18.23 14.579 12.041 11.259 11.47 0.8 27.332 22.911 20.802 22.072 25.63 0.9 56.545 53.599 61.015 76.068 94.607 0.95 110.015 115.617 138.526 161.045 184.439 1 199.007 200.336 200.485 199.411 199.888 1.05 104.542 102.206 105.385 106.762 107.729 1.1 55.908 50.87 50.536 52.258 53.902 1.2 28.069 23.363 20.537 19.92 20.051 1.3 19.113 15.296 12.611 11.57 11.075 1.4 14.786 11.625 9.3 8.193 7.625 1.5 12.235 9.581 7.494 6.484 5.892 2 7.32 5.716 4.373 3.647 3.158 3 4.884 3.852 2.965 2.414 2.134 5.526 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϲϭ ϲϮ Table 3.15: 9:; values for the CS-EWMA chart with D“  => ?,  iii. ³ 0.5 ‚  >+ $‚  M>+h 10.819 ‚  >MB $‚  +P>MBh 8.244 ‚  >B $‚  d>eeh 6.394 ‚  >dB $‚  B>Mdh 5.563 ‚ + $‚  P>Bdh 0.6 13.25 10.215 8.096 7.292 7.082 0.7 17.575 13.835 11.575 11.222 12.113 0.8 26.704 22.207 21.05 23.96 30.154 0.9 56.118 53.989 65.611 86.907 113.788 0.95 111.12 118.942 146.808 177.365 205.588 1 200.775 199.405 199.3062 199.499 200.476 1.05 104.617 102.52 105.77 107.044 106.577 1.1 55.436 50.59 51.073 53.56 54.65 1.2 27.554 22.734 20.376 20.218 20.334 1.3 18.626 14.656 12.186 11.379 11.054 1.4 14.272 11.049 8.842 7.888 7.449 1.5 11.76 9.015 7.041 6.138 5.654 2 6.863 5.226 3.98 3.31 2.91 3 4.487 3.464 2.601 2.22 1.87 for fixed values of Z, *‚ and , ‚ like >MB - shifts in the negative direction, large values of for fixed values of Z, ‚ and , 5.16 negative shifts of small magnitude are detected efficiently using moderate values of iv.  ? and 9:;=  F== ‚ ‚ - >B, whereas for large are recommended; large values of *‚ are recommended for detecting large positive shifts and vice versa while for negative shifts, varied behavior is seen; v. for fixed values of Z, *‚ and value of ‚ and vice versa; , the value of $‚ decreases with an increase in the In this section we have used the statistic  to design the control structure of our proposed chart. Many other transformations of
 , that result into a statistic which is distributed approximately normal, can be used. Acosta-Mejia et al. (1999) proposed two such transformations named as ] and ¾. Castagliola et al. (2010) proposed a four parameter Johnston transformation and named the resulting variable as  . Huwang et al. (2010) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ 
 proposed a logarithmic transformation and showed that   ¤‡ ¿  À  Á follows ' approximately a normal distribution which can be used to design a memory control chart to monitor the process dispersion. In the next subsection we compare the control charts based on these transformations with our proposal. ͵ǤʹǤͶ ‘’ƒ”‹•‘• This subsection contains the comparison of the proposed CS-EWMA chart with the
 -EWMA, CUSUM-
 and some other recently proposed CUSUM and EWMA charts for monitoring the process dispersion. CS-EWMA versus µF -EWMA: The values for the
 -EWMA chart (discussed in Section 2.1) are given in Table 3.8. Comparison reveals that the performance of the proposed chart with ‚  + is almost the same as compared to the
 -EWMA for the positive shifts, but for negative shifts, the performance of the CS-EWMA chart is far more superior than the
 -EWMA. Moreover, the performance of
 -EWMA becomes very poor for moderate and large values of *, even the  values become larger than the prefixed  for negative shifts. This is not the case with the proposed CS-EWMA chart, as for large values of *‚ , the CUSUM factor in the proposed chart still makes it remain better in terms of 3.8 vs. Tables 3.10 – 3.15). CS-EWMA versus CUSUM-µF : Table 3.9 contains the 8 (cf. Table values for the CUSUM-
 chart proposed by Castagliola et al. (2009). This CUSUM-
 chart is a special form of our proposed CS-EWMA chart with *‚  +. The performance of the proposed chart is better than CUSUM-
 chart, especially for small values of smoothing constant *‚ . The additional DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϲϯ ϲϰ parameter (i.e. *‚ ) in CS-EWMA chart makes sure that the performance of the proposed chart  is not deflated for large values of of ‚ like it occurs with the CUSUM-
 chart for large values (cf. Table 3.9 vs. Tables 3.10 – 3.15). CS-EWMA versus some other EWMA and CUSUM charts: Acosta-Mejia et al. (1999) proposed some CUSUM-type charts (named as ¾ CUSUM, ] CUSUM and  CUSUM) for monitoring process dispersion and compared the performance of their proposed charts with the CUSUM chart by Page (1963), CUSUM
chart by Tuprah and Ncube (1987) and CUSUM ¤‡
 chart by Chang and Gan (1995). They showed through comparison that their proposed  CUSUM is performing better (in terms of values) than the other charts discussed. Similarly, Huwang et al. (2010) proposed two new EWMA type charts (named as $$R+ CUSUM and $$RM CUSUM) for monitoring the standard deviation of a process. They also compared the performance of their proposed chart with some other competitors, like the $ EWMA chart by Crowder and Hamilton (1992) and the
 EWMA chart by Shu and Jiang (2008), and showed that their proposed charts are more sensitive (in detecting shifts) than the other competitors. The values of the charts discussed by Acosta-Mejia et al. (1999) are given in Table 3.16, while Table 3.17 contains the by Huwang et al. (2010). 8 of the charts discussed Table 3.16: 9:; values for some one-sided CUSUM-type charts for detecting variance increases with  ? and 9:;=  F== 199.93  M>BX $  A>ffh 201.80 ¾ CUSUM  >Pf $  A>Mfh 200.70 ] CUSUM  >Pf $  A>Mfh 201.10 CUSUM
 +>+PA $  +>eh 200.60 h CUSUM  +>+eP $  +f>ABh 42.94 40.40 41.04 41.04 38.80 34.60 1.2 18.07 17.60 17.17 17.15 16.85 14.14 1.3 10.75 10.82 10.23 10.21 10.36 8.42 1.4 7.63 7.81 7.26 7.24 7.50 5.93 1.5 5.98 6.13 5.66 5.65 5.85 4.58 2 3.18 3.13 2.90 2.98 3.01 2.20 1 CUSUM ¤‡
  >Xf $  M>XXh 1.1 ³ CUSUM DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů 200.76 DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  ϲϱ Table 3.17: 9:; values for some one-sided EWMA-type charts for detecting variance increases with  ? and 9:;=  F== ³ *  >B *  >+ CH-EWMA (  +>BB) SJ-EWMA (  +>BXf) 200.75 HHW1EWMA (  +>fMf) 200.92 HHW2EWMA (  +>fdM) 1.1 43.24 32.26 28.89 27.28 44.26 35.15 34.32 32.05 1.2 18.09 14.43 11.69 10.78 18.23 14.96 14.1 12.69 1.3 10.77 9.17 6.85 6.2 10.56 9.09 8.2 7.21 1.4 7.63 6.73 4.75 4.24 7.35 6.53 5.65 4.89 1.5 5.98 5.38 3.62 3.22 5.68 5.13 4.28 3.68 2 3.18 2.93 1.8 1.62 2.95 2.72 2.03 1.76 CH-EWMA (  +>B+P) SJ-EWMA (  M>Md) 199.48 HHW1EWMA (  M>MBP) 199.43 HHW2EWMA (  M>PBB) CH-EWMA (  +>Bef) SJ-EWMA (  M>APP) 199.67 HHW1EWMA (  M>PM) 200.22 HHW2EWMA (  M>Add) 1.1 46.63 39.73 41.18 37.87 48.48 43.45 46.14 41.79 1.2 18.79 16.05 16.66 14.7 19.52 17.25 18.65 16.2 1.3 10.54 9.21 9.45 8.16 10.67 9.56 10.35 8.82 1.4 7.16 6.4 6.45 5.49 7.09 6.43 6.9 5.81 1.5 5.41 4.89 4.83 4.07 5.24 4.8 5.11 4.26 2 2.67 2.45 2.24 1.88 2.47 2.3 2.32 1.93 1 ³ 1 200.33 200.64 *  >M 199.57 200.65 CH-EWMA (  +>PP) SJ-EWMA (  +>eAP) 200.36 HHW1EWMA (  M>de) 199.51 HHW2EWMA (  M>+Pe) 200.02 199.4 Table 3.18: 9:; values for one-sided CS-EWMA chart with r“  E, *  >P 200.35 199.45  ? and 9:;=  F== ³ 1 *‚  >B $‚  B>Peh 200.4035 *‚  >+ $‚  B>+Ph 200.9845 *‚  >M $‚  A>BAh 200.7247 *‚  >P $‚  P>eMfh 200.1826 *‚  >A $‚  P>AMh 200.621 *‚  >B $‚  P>+h 1.1 23.963 31.478 36.48 39.44 41.226 43.263 1.2 11.261 13.798 15.39 16.031 16.599 17.147 1.3 8.006 9.1 9.485 9.553 9.635 9.684 199.6103 1.4 6.53 7.041 7.012 6.837 6.721 6.641 1.5 5.685 5.941 5.689 5.432 5.224 5.112 2 3.975 3.834 3.423 3.08 2.858 2.673 3 3.04 2.732 2.347 2.143 1.894 1.674 All charts presented in Tables 3.16 – 3.17 are one-sided, i.e. designed to detect just positive shift in the process dispersion. For a valid comparison of the proposed chart with these charts, we have evaluated the values of the proposed chart with the one-sided structure for DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϲϲ 8 are given in Table 3.18  monitoring an increase in process standard deviation. These ‚ with  + and   M. It can be noticed through the comparison that the proposed chart is outperforming all its competitors in the form of EWMA and CUSUM type charts (cf. Tables 3.16 – 3.17 vs. Table 3.18). Finally, before concluding this subsection, we provide the discussed above. Figure 3.7 contains the EWMA chart with *‚  >M, ‚ curves of the different charts curves of the two-sided charts containing: CS-  >MB and $‚  MA>eX;
 -EWMA chart with *  >M and  M>BeM; CUSUM-
 chart with  >MB and $  X>AdX. Figure 3.8 shows the curves of the one-sided charts for positive shifts containing: CS-EWMA chart with *‚  >B, ‚  + and $‚  B>Pe;  CUSUM chart with chart with  +>+eP and $  +f>AB; CUSUM
 +>+PA and $  +>e; $ EWMA chart with *  >B and EWMA chart with *  >B and  +>fdM.  +>BB;
  +>BXf; and $$RM EWMA chart with *  >B and Figure 3.7: 9:; curves for two-sided structures of CS-EWMA, µF -EWMA and CUSUM-µF charts with 9:;=  F== ^ͲtD h^hD tD ϮϱϬ ϮϬϬ Z>Ɛ ϭϱϬ ϭϬϬ ϱϬ Ϭ Ϭ͘ϱ Ϭ͘ϲ Ϭ͘ϳ Ϭ͘ϴ Ϭ͘ϵ Ϭ͘ϵϱ ϭ à ϭ͘Ϭϱ ϭ͘ϭ ϭ͘Ϯ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭ͘ϯ ϭ͘ϰ ϭ͘ϱ Ϯ ϯ DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  Figure 3.8: 9:; curves for one-sided structures of CS-EWMA, nÄ CUSUM, CUSUM µ, µÅ EWMA, ns EWMA and ssQF EWMA charts with 9:;=  F== ^ͲtD Wh^hD h^hD^ ϱϬ ϰϬ Z>Ɛ ϯϬ ϮϬ ϭϬ Ϭ ϭ͘ϭ ϭ͘Ϯ ϭ͘ϯ à ϭ͘ϰ ϭ͘ϱ Ϯ Figure 3.7 shows that the performance of all three charts is almost the same for positive shifts but for negative shifts, the proposed chart is giving a better Figure 3.8, the performance. Similarly, in curve of the proposed chart seems lower than all other curves for small values of ³ and thus showing a better performance for small positive shifts. ͵ǤʹǤͷ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ An application of CS-EWMA chart on a simulated dataset is provided in this subsection to show the implementation of the proposal. For this purpose, two datasets are generated having 40 samples of size Z  B for both the samples. First 20 samples are generated from [+A referring to an in control situation with   + and '  A. For dataset 3.1, the remaining 20 observations are generated from [+B showing a positive DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϲϳ ϲϴ shift in the process standard deviation with ³  _BÀ  +>++f. Similarly, for dataset 3.2, _A  the remaining 20 observations are generated from [+P showing a negative shift in dispersion with ³  >fXX. Now the
 -EWMA chart, CUSUM-
 chart and their mixture named as CS-EWMA chart are applied to the given datasets. The chart output of the
 - EWMA chart with *  >M and  M>BeM is shown in Figure 3.9. Figure 3.10 shows the graphical display of the CUSUM-
 with the proposed chart with *‚  >M, ‚  >B and $  P>fBB. The calculation details for  >B Æ q ‚  >+Xd and $‚  +B>Ad Æ $‚q  B>+Bd are given in Table 3.19, whereas the chart output is given in Figure 3.11. Figure 3.9: Graphical display of the µF -EWMA chart with D  => F and ;  F> ?”F ũ;ĚĂƚĂƐĞƚϯ͘ϭͿ ũ;ĚĂƚĂƐĞƚϯ͘ϮͿ ĐŽŶƚƌŽůůŝŵŝƚƐ ϭ Ϭ͘ϴ Ϭ͘ϲ Ϭ͘ϰ Ϭ͘Ϯ Ϭ ͲϬ͘Ϯ ͲϬ͘ϰ ͲϬ͘ϲ ͲϬ͘ϴ Ͳϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ Figures 3.9 – 3.11 clearly indicate that all three charts are giving out of control signals at samples 39 and 40 for dataset 3.1 (i.e. positive shift in the process dispersion). Moreover, the proposed CS-EWMA chart detected a negative shift for dataset 3.2 at samples 37, 38, 39 and DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  40 but the other two charts did not signal the downward shift in the process standard deviation. Figure 3.10: Graphical display of the CUSUM µF chart with r  => ? and s  I> G?? н;ĚĂƚĂƐĞƚϯ͘ϭͿ Ͳ ;ĚĂƚĂƐĞƚϯ͘ϮͿ , ϱ ϰ͘ϱ ϰ ϯ͘ϱ ϯ Ϯ͘ϱ Ϯ ϭ͘ϱ ϭ Ϭ͘ϱ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ^ĂŵƉůĞEƵŵďĞƌ Figure 3.11: Graphical display of the CS-EWMA chart with D“  => F, r“  => ? and s“  E?> HJ Dн;ĚĂƚĂƐĞƚϯ͘ϭͿ DͲ ;ĚĂƚĂƐĞƚϯ͘ϮͿ ,ƋΖ ϴ ϳ ϲ ϱ ϰ ϯ Ϯ ϭ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϲϵ ϳϬ  Table 3.19: Calculation details of the proposed CS-EWMA chart for dataset 3.1 Sample No. 1
 5.61  0.74  0.32 7 0.14 7 0 Sample No. 21
 0.81  -1.42  -0.44 7 7 0 0.32 2 4.48 0.38 0.33 0.3 0 22 11.47 2.04 0.06 0 0.1 3 2.58 -0.38 0.19 0.31 0 23 4.69 0.45 0.14 0 0 4 1.7 -0.84 -0.02 0.12 0 24 3.21 -0.1 0.09 0 0 5 7.04 1.13 0.21 0.16 0 25 6.42 0.97 0.26 0.09 0 6 5.96 0.84 0.34 0.32 0 26 3.84 0.15 0.24 0.16 0 7 3.84 0.15 0.3 0.45 0 27 4.94 0.54 0.3 0.29 0 8 3.29 -0.07 0.23 0.5 0 28 9.35 1.65 0.57 0.68 0 9 8.62 1.5 0.48 0.81 0 29 4.21 0.29 0.51 1.02 0 10 10.33 1.84 0.75 1.39 0 30 10.5 1.87 0.79 1.64 0 11 3.33 -0.05 0.59 1.81 0 31 1.45 -0.99 0.43 1.89 0 12 0.65 -1.54 0.17 1.8 0 32 10.1 1.8 0.7 2.42 0 13 3.99 0.21 0.17 1.8 0 33 3.16 -0.12 0.54 2.79 0 14 2.21 -0.57 0.03 1.65 0 34 5.16 0.61 0.55 3.16 0 15 0.88 -1.38 -0.25 1.22 0.1 35 7.84 1.32 0.71 3.7 0 16 9.86 1.75 0.15 1.19 0 36 3.14 -0.13 0.54 4.06 0 17 5.48 0.71 0.26 1.28 0 37 3.44 -0.01 0.43 4.32 0 18 1.1 -1.22 -0.04 1.07 0 38 6.96 1.11 0.57 4.71 0 19 3.48 0.01 -0.03 0.86 0 39 11.04 1.97 0.85 5.38* 0 20 1.67 -0.86 -0.19 0.5 0.03 40 6.47 0.98 0.87 6.08* 0 * indicates an out of control signal by CS-EWMA chart  ͵Ǥ͵ ‘ Ž—†‹‰”‡ƒ”• CUSUM control charts and EWMA control charts are the two most commonly used memory control charts in the literature. These control schemes do not only use the current observation but also accumulate the information from the past to give a quick signal if the process is slightly off-target. In this chapter we have combined the CUSUM and EWMA control schemes into a single control structure and proposed a mixed EWMA-CUSUM control scheme for monitoring the process location. Performance of the proposed scheme is compared with other CUSUM and EWMA type control charts which are meant to detect small and moderate shifts in the process. The comparisons revealed that the proposed scheme DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  is really good at detecting the smaller shifts in the process as compared to the other schemes under study. Following our proposal for location, this chapter also proposes a new memory-type control chart, named as the CS-EWMA chart, for monitoring the process standard deviation. The design of the proposed chart is based on mixing the effects of EWMA and CUSUM control charts into a single control structure. The performance of the proposed CS-EWMA chart is evaluated using as indicator. In terms of values, the proposed chart is compared with existing CUSUM and EWMA control charts and it is noticed that the proposed chart has better performance for both positive as well as negative shifts in the process dispersion. Finally, an illustrative example is provided to show the application of the proposed chart. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϳϭ ϳϮ  ’’‡†‹š͵Ǥͳ x=c(); Q=c(); sdp=c(); K=c(); H=c(); Mp=c(); Mn=c(); rl=c(); k=0.5; h=37.42; ld=0.1 mu=0; sig=1 for(j in 1:50000) { for(i in 1:1000000) { x[i]=rnorm(1,mu,sig) if(i==1) {Q[i]=ld*x[i]+(1-ld)*mu;} else{Q[i]=ld*x[i]+(1-ld)*Q[i-1];} sdp[i]=sqrt((ld/(2-ld))*(1-(1-ld)^(2*i))) K[i]=k*sdp[i] H[i]=h*sdp[i] if(i==1) {Mp[i]=max(0,Q[i]-K[i]);} else{Mp[i]=max(0,Q[i]-K[i]+Mp[i-1]);} if(i==1) {Mn[i]=max(0,-Q[i]-K[i]);} else{Mn[i]=max(0,-Q[i]-K[i]+Mn[i-1]);} if(Mp[i]>H[i] | Mn[i]>H[i]) {rl[j]=i; break;} else{rl[j]=0;} } } mean(rl)  ’’‡†‹š͵Ǥʹ Q=c(); Mp=c(); Mn=c(); rl=c(); Tk=c() n=5; a=-0.8969; b=2.3647; c=0.5979; ETi=0.00748; STi=0.9670 Z0=a+b*log(1+c,exp(1)) ld=0.05; K=0.1*sqrt(ld/(2-ld)); H=62.6*sqrt(ld/(2-ld)) for(j in 1:50000) { for(i in 1:1000000) { x=rnorm(n,0,1) T[i]=a+b*log(var(x)+c,exp(1)) if(i==1) {Q[i]=ld*T[i]+(1-ld)*Z0;} else{Q[i]=ld*T[i]+(1-ld)*Q[i-1];} if(i==1) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů DŝdžĞĚtDͲh^hDĐŚĂƌƚƐ  {Mp[i]=max(0,Q[i]-ETi-K);} else{Mp[i]=max(0,Q[i]-ETi-K+Mp[i-1]);} if(i==1) {Mn[i]=max(0,-Q[i]+ETi-K);} else{Mn[i]=max(0,-Q[i]+ETi-K+Mn[i-1]);} if(Mp[i] > H | Mn[i] > H) {rl[j]=i;break;} else{rl[j]=0;} } } mean(rl) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϳϯ ϳϰ     DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ŚĂƉƚĞƌϰ  Šƒ’–‡”Ͷ —š‹Ž‹ƒ”›‹ˆ‘”ƒ–‹‘„ƒ•‡†ƒ†  ‘–”‘Ž Šƒ”–• Two popular categories of control charts are CUSUM and EWMA charts, which are good at quickly detecting the presence of small and moderate disturbances. Targeting on small and moderate shifts in the process mean, this chapter proposes EWMA- and CUSUM-type control charts which utilize the information of auxiliary variable(s). The regression estimation technique for the mean is used in defining the control structure of the proposed charts. It is shown that the proposed charts are performing better than their competitors which are also designed for detecting small shifts. This chapter is based on an article by Abbas, Riaz and Does (2012c), consisting of the EWMA-type control charts based on auxiliary information. ͶǤͳ ‘–”‘Ž Šƒ”–•—•‹‰ƒ—š‹Ž‹ƒ”›‹ˆ‘”ƒ–‹‘ Information accessible at the stage of estimation other than that in the sample is called auxiliary information. The concept of using auxiliary information is frequently used in the field of survey sampling and estimation techniques. Auxiliary information can be used at either or both the design and estimation stage. Sampling techniques like probability sampling (cf. Fuller (2009)) and rank set sampling (cf. McIntyre (1952)) are examples of the utilization of auxiliary information at the sample selection stage. Ratio, product and regression-type estimators are examples of utilization of auxiliary information at the estimation stage (cf. Cochran (1977) and Fuller (2002)). These estimators are designed in such a way that they not DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϳϱ ϳϲ  only utilize the sample information but also make use of the information available other than that. This makes these estimators more efficient than the conventional ones. This auxiliary information is also used in control charting techniques in order to enhance their performance. Examples are the regression control chart proposed by Mandel (1969) and cause-selecting control charts proposed by Zhang (1985). The control structure of these control charts is based on regressing the study variable on the auxiliary variable. The residuals obtained from that regression are used for monitoring the process (see also Wade and Woodall (1993)). Riaz (2008a) introduced the concept of using auxiliary information at the time of estimating the plotting statistic of a control chart. He proposed a control chart which uses a regression-type estimator as the plotting statistic to monitor the variability of the process and showed the dominance (in terms of power) of his proposed control chart over the well-known Shewhart-type control charts for the same purpose (i.e. ,
and
 charts). Riaz (2008b) proposed a regression-type estimator to monitor the location of the process. He not only showed the superiority of his proposal over the Shewhart’s  chart but also over the regression charts (cf. Mandel (1969)) and the cause-selecting charts (cf. Zhang (1985)). Later Riaz and Does (2009) proposed another variability chart based on a ratio-type estimator and proved the dominance of their proposed chart over the one based on regression-type estimator. Following these authors, this chapter proposes the use of auxiliary information with the control structure of EWMA and CUSUM charts. The regression estimation technique is used to exploit the information from the auxiliary variable and without loss of generality the case of individual observations have been considered. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  ͶǤʹ  ‘–”‘Ž Šƒ”–•—•‹‰ƒ—š‹Ž‹ƒ”›‹ˆ‘”ƒ–‹‘ Let an auxiliary variable  be correlated with the variable of interest  and let us denote the correlation between these two variables by NJ– . The observations of  and  are obtained in the paired form for each sample and the population mean and variance of  (i.e. – and '– respectively) are assumed to be known. Also we assume bivariate normality of  and , i.e. Š 'Š  \ a È[ É\ a  Ê  – NJ– 'Š '– NJ– 'Š '– ËÌ '– (4.1) where [ represents the bivariate normal distribution. The regression estimate of the population mean Š (cf. Cochran (1977)) is given as: 7–§   ! ͊– –    (4.2) where ͊– is the change in  due to one unit change in  and is ͊–  NJ– \ Î a. The mean and variance of the statistic 7– in (4.2) are given as:   Ð7–   Š , 7–   'Ñ  'Š +  NJ– ] ]Ï (4.3) Equation (4.3) implies that 7– is also an unbiased estimator of Š and 'Ñ , 'Š as long as  NJ– % . Based on the regression estimator in (4.2), the plotting statistic for the proposed EWMA chart based on a single auxiliary variable (named as MXEWMA chart) is defined as: q )q  *q 7–§ ! +  *q ) (4.4) where *q is the smoothing constant for the proposed statistic and 7–§ is the value of statistic q 7– for the # pV sample. ) represents the past information (like ) ) and its initial value DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϳϳ ϳϴ (i.e. )q ) is also taken equal to the target mean  , i.e. the in control mean of . Now based on  (4.3) the time varying control limits for the proposed chart are: .     q 'Ñ / 0Ò 0Ò +  +  *q   4 2    3 0Ò 2 q q     ! 'Ñ / Ò +  +  *   0 1 where q (4.5) determines the width of the control limits for the proposed MXEWMA chart. The values for the proposed MXEWMA chart with time varying limits are given in Tables 4.1 – 4.5 for some selective choices of NJ– in which @ represents the amount of shift in the study variable, i.e. @  ÓÔ¬ Ô^ Ó ]Î where  is the out of control mean of . The program (developed in R language) for evaluating the 8 is given in Appendix 4.1 which is replicated 50,000 for each simulated value. Note that – remains constant. Table 4.1: 9:; values for the proposed MXEWMA chart with time varying limits, ÕÖ×  => =? and 9:;=  ?== @ 0 *q  >P q  M>AfP 500.3011 *q  >B q  M>XPe 500.8313 *q  >+  M>fMA 499.7023 *q  >MB q P 0.25 66.3103 77.5273 103.1312 168.8081 254.324 321.6227 0.5 21.2101 23.6211 28.7434 47.2173 88.1445 139.8713 0.75 10.7125 11.8433 13.592 19.2345 35.401 62.2502 1 6.632 7.2905 8.2072 10.3617 17.1148 30.3995 1.5 3.4443 3.7573 4.1644 4.7647 6.2628 9.7633 2 2.244 2.4205 2.6563 2.9313 3.3701 4.4521 2.5 1.6555 1.7719 1.9147 2.0831 2.2565 2.6079 3 1.3401 1.4104 1.5112 1.6109 1.6904 1.8151 4 1.0642 1.0882 1.1224 1.1609 1.184 1.1946 5 1.0049 1.0085 1.0131 1.0221 1.0265 1.0281 q 499.6045 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů *q  >B  P>dM q 500.9026 *q  >dB q  P>ff 499.9704 ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  Table 4.2: 9:; values for the proposed MXEWMA chart with time varying limits, ÕÖ×  => F? and 9:;=  ?== @ 500.0592 *q  >B q  M>XPe 500.5358 *q  >+  M>fMA 499.1371 *q  >MB q P q 0 *q  >P q  M>AfP 499.8049 *q  >B  P>dM 499.951 *q  >dB q  P>ff 0.25 63.1554 73.8038 97.5858 161.9697 245.9632 313.2722 0.5 20.1268 22.4316 27.1393 44.1797 83.0098 132.8202 0.75 10.1982 11.2488 12.8277 18.0008 32.9028 57.9059 1 6.3049 6.9254 7.7548 9.7748 15.8281 28.0123 1.5 3.2871 3.5816 3.9532 4.5005 5.832 8.9444 2 2.1384 2.3119 2.5261 2.7962 3.1754 4.0979 2.5 1.5945 1.6997 1.8406 1.9877 2.1497 2.4354 3 1.2952 1.363 1.4519 1.5523 1.6178 1.7158 4 1.0495 1.0687 1.0955 1.132 1.1476 1.1565 5 1.0044 1.0067 1.0108 1.0149 1.0178 1.0183 q 500.5678 Table 4.3: 9:; values for the proposed MXEWMA chart with time varying limits, ÕÖ×  => ? and 9:;=  ?== @ 0 *q  >P q  M>AfP 500.7792 *q  >B q  M>XPe 499.5635 *q  >+  M>fMA 499.8114 *q  >MB q P 499.692 *q  >B  P>dM 500.7859 *q  >dB q  P>ff 0.25 52.6523 61.1011 80.6591 135.7683 216.0974 285.4555 0.5 16.6817 18.5391 22.044 34.5785 65.3989 108.1412 0.75 8.4242 9.3261 10.5432 14.1099 24.7989 44.3004 1 5.2477 5.7592 6.4306 7.8001 11.9099 20.6193 1.5 2.7663 3.0129 3.3183 3.7182 4.5573 6.5619 2 1.8392 1.9673 2.1455 2.3486 2.5852 3.136 2.5 1.3979 1.48 1.5834 1.7044 1.7964 1.9476 3 1.17 1.2168 1.2786 1.3501 1.3947 1.4325 4 1.0155 1.0253 1.038 1.0513 1.0606 1.064 5 1.0007 1.0012 1.0019 1.002 1.0028 1.0056 q q 500.9686 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϳϵ ϴϬ  Table 4.4: 9:; values for the proposed MXEWMA chart with time varying limits, ÕÖ×  => J? and 9:;=  ?== @ 0 *q  >P q  M>AfP 500.0659 *q  >B q  M>XPe 499.5868 *q  >+  M>fMA 500.7509 *q  >MB q P 500.4051 *q  >B  P>dM 499.2859 *q  >dB q  P>ff 0.25 33.8532 38.4587 48.8401 84.1265 146.8905 211.6496 0.5 10.6126 11.7078 13.4213 18.9673 34.9019 61.2737 0.75 5.398 5.9343 6.6291 8.0816 12.4585 21.7022 1 3.4068 3.7177 4.1045 4.7175 6.1522 9.5862 1.5 1.8835 2.0268 2.2095 2.4126 2.6787 3.2731 2 1.3299 1.4043 1.4969 1.5949 1.6784 1.7904 2.5 1.1006 1.1309 1.1772 1.2269 1.2577 1.2721 3 1.0199 1.0296 1.0429 1.0608 1.0733 1.0742 4 1.0001 1.0004 1.0006 1.0011 1.0014 1.0017 5 1 1 1 1 1 1 q q 499.7069 Table 4.5: 9:; values for the proposed MXEWMA chart with time varying limits, ÕÖ×  => ”? and 9:;=  ?== @ 0 *q  >P q  M>AfP 500.1483 *q  >B q  M>XPe 499.3424 *q  >+  M>fMA 500.5895 *q  >MB q P 500.2327 *q  >B  P>dM 500.0067 *q  >dB q  P>ff 0.25 9.6039 10.6139 12.1289 16.7024 30.2287 53.736 0.5 3.1274 3.4017 3.7577 4.2576 5.389 8.1474 0.75 1.75 1.8739 2.0315 2.2142 2.4181 2.8603 q q 500.7743 1 1.2547 1.3136 1.3951 1.4846 1.5468 1.622 1.5 1.0102 1.0162 1.0216 1.034 1.0418 1.0432 2 1 1.0003 1.003 1.0005 1.0006 1.0013 2.5 1 1 1 1 1 1 3 1 1 1 1 1 1 4 1 1 1 1 1 1 5 1 1 1 1 1 1 The  in Tables 4.1 – 4.5 is fixed at B which will enable us to make comparison of the proposed control chart with some other charts/schemes. Tables 4.1 – 4.5 refer to a situation where the information about the population correlation coefficient is DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  assumed to be known, because the information about the population correlation coefficient NJ– is known in many practical situations (cf. Garcia and Cebrian (1996)). However, this may not be the case in every situation. Then we have to estimate the value of NJ– from preliminary samples (L). The estimators used to estimate NJ– and ͊– are: Ç؊–  ¦Ù §«¬Š§ Š–§ – © © Ù /¦Ù §«¬Š§ Š ¦§«¬–§ – ¦ Š§ Š–§ – Íڊ–  §«¬ ¦Ù – © Ù , §«¬ § – (4.6) From Tables 4.1 – 4.5, the main findings about our proposed MXEWMA control chart for monitoring the location of a process are given as: i. the use of auxiliary information in the form of a regression estimator boosts the ii. for fixed values of NJ– and @, the performance of the proposed chart with time iii. iv. performance of EWMA control chart, especially for large values of NJ– ; varying limits is better for small values of *q ; for fixed values of *q , large values of NJ– ; for all choices of *q , exceeds v.  q q and @, the performance of the proposed chart is better for the and NJ– the proposed chart is for any value of @; for small values of NJ– , the unbiased, i.e.  never for the proposed scheme decreases gradually with an increase in the value of @ but for the large values of NJ– , the scheme decreases rapidly with an increase in the value of @. for the proposed Note that if we apply the same set up with samples sizes Z % + instead of Z  +, the results will be the same with the obvious adjustments in the control limits in (4.2) and (4.3). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϴϭ ϴϮ  ͶǤʹǤͳ ‘’ƒ”‹•‘• In this subsection we provide a broad comparison of our proposed MXEWMA chart with the classical CUSUM, the classical EWMA and some of their recent modifications. Below, we present the one by one comparison of the proposed scheme with its counterparts: MXEWMA versus classical EWMA: 8 for the classical EWMA chart with time varying are given in Table 2.2. Comparing the MXEWMA chart (with *q  >MB) with the classical EWMA chart we can see that for all the values of NJ– the performance of the MXEWMA chart is better than the classical EWMA for a fixed value of @ (cf. Table 2.2 vs. Tables 4.1 – 4.5). Moreover, an important point here is that the classical EWMA is a special case of the MXEWMA chart, i.e. applying MXEWMA chart to a process where NJ–   is equivalent to applying the classical EWMA. From Table 4.1 we see that for NJ–  >B the results almost coincides with the results of Table 2.2 as was to be expected. It is to be noted here that the results of proposed MXEWMA chart with time varying limits are on the same pattern as compared to the classical EWMA with time varying limits while the proposed chart with asymptotic limits (computational results are not provided here) is mainly following the pattern of classical EWMA with asymptotic limits. MXEWMA versus classical CUSUM: The classical CUSUM proposed by Page (1954) is discussed briefly in Section 1.2. A comprehensive study on the CUSUM charts is given by Hawkins and Olwell (1998). The  fixed at B. 8 for the CUSUM chart are given in Table 4.6 with Comparing the MXEWMA chart with the classical CUSUM chart we observe that the newly proposed chart is outperforming the classical CUSUM even for small values of NJ– (cf. Table 4.6 vs. Tables 4.1 – 4.5). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ TABLE 4.6: 9:; values for the classical CUSUM scheme  & ( 0.25 @ 0 0.25 0.5 0.75 1 1.5 2 2.5 3 4 5 8.585 500 94.8 31.08 17.54 12.17 7.58 5.55 4.41 3.69 2.84 2.26 0.5 5.071 500 145.5 38.87 17.32 10.52 5.82 4.06 3.15 2.6 2.03 1.72 0.75 3.539 500 200.7 57.07 22.13 11.6 5.45 3.55 2.67 2.18 1.63 1.24 1 2.665 500 249.5 81.44 30.9 14.67 5.75 3.41 2.45 1.94 1.38 1.09 MXEWMA versus runs rules based CUSUM and EWMA: The 8 for the runs rules based CUSUM are provided in Tables 2.7 – 2.8 and for runs rules based EWMA these are given in Tables 2.15 – 2.16. Comparing the MXEWMA chart with runs rules based CUSUM and EWMA we can see that the proposed chart is uniformly surpassing both the CUSUM schemes I & II and the EWMA simple MOM scheme. The EWMA modified MOP scheme is performing better than the proposed chart as long as NJ– , >B, but once we have NJ– Û >B, the proposed chart outperforms the EWMA modified MOP scheme as well (cf. Tables 2.7 – 2.8 & 2.15 – 2.16 vs. Tables 4.1 – 4.5). MXEWMA versus MEWMA: Lowry et al. (1992) introduced a multivariate extension of the EWMA chart named as MEWMA chart. For the bivariate case, the MEWMA statistic is )+ )+  ¹  º  K ¹  º ! +  K ¹  º  )M )M and the chart gives an out of control signal if    )+ (Ü . Here   Ý © 5Þ© 6 Ý5Ý©§ 6]Ω ]Ï ÎÏ )M  Ê '– 'Š– 'Š– )+ ˹ º % )M 'Š . Lowry et al. (1992) assumed, that the mean of both the variables are equal (i.e. Š  –  ). According to this assumption the MEWMA only depend upon the shift parameter @  /  ÞÎÏ values of . We have evaluated the values of this bivariate EWMA chart through Monte Carlo simulation by running +´ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϴϯ ϴϰ  replications. To validate our simulation code, we have replicated Table 1 of Lowry et al. (1992) article and found the same results. Now using our simulation code, we find the values of the bivariate EWMA with the mean of  fixed and defining the shift in the mean of  as we have defined in Tables 4.1 8 are given in Table 4.7 where K  >+ and (Ü  +>fPP with – 4.5. These   B. TABLE 4.7: 9:; values for MEWMA charts with ß  => E and lH  E=> GII at 9:;=  ?== @ 0 NJ–  >B 500.57 NJ–  >MB 501.6147 NJ–  >B 501.0223 NJ–  >dB 498.4386 NJ–  >eB 0.5 36.81699 34.50276 27.63295 16.425 4.41316 1 9.8781 9.31899 7.66392 4.85723 1.55707 1.5 4.91344 4.65957 3.88276 2.54956 1.05029 2 3.08706 2.94745 2.47956 1.6849 1.00066 2.5 2.19454 2.10018 1.79616 1.27854 1.00001 3 1.6936 1.6269 1.41069 1.08457 1 500.4125 Comparing the performance of the bivariate EWMA with the proposed chart, it can be noticed that the proposed chart has smaller  values as compared to the bivariate EWMA chart for all the corresponding values of NJ– (cf. Table 4.7 vs. Tables 4.1 – 4.5). ͶǤʹǤʹ Š‡ ƒ•‡‘ˆ–™‘ƒ—š‹Ž‹ƒ”›˜ƒ”‹ƒ„Ž‡•  Let us consider three variables, named as ,  and R, following a trivariate normal distribution ([à ) out of which  is the study variable, while  and R are two auxiliary variables, i.e. 'Š Š  É  Ì È[à áÉ – Ì  ¿ '–Š â R '⊠'Š– '– '–â 'Šâ '–â Áã  'â DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů (4.7) ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  Based on the above trivariate normal distribution, Kadilar and Cingi (2005) provided a regression estimator for estimating the population mean of the study variable  which are given as: 7£§   ! ͊– –    ! ͊â â  R  where ͊–  ]ÎÏ © ]Ï and͊â  ]Îä © ]ä (4.8) . The mean and variance of 7£ is given as:   Ð7£   Š , 7£   '£  'Š +  NJ–  NJâ ! MNJ– NJâ ǖâ  (4.9) where NJ– is the Pearson’s correlation coefficient between  and , and similarly NJâ and ǖâ are defined. Now the plotting statistic and the control limits of the EWMA chart based on two auxiliary variables is given as: å )å  *å 7£§ ! 5+  *å 6) .     å 'Ñ / 0å 0å (4.10) +  +  *å   4 2    3 2 0å å å  / +  +  *      ! 'Ñ 1 0å (4.11) å where *å is the smoothing constant for the proposed statistic. ) represents the past information and its initial value (i.e. )å ) is also taken equal to the target mean  i.e. the in control mean of . å determines the width of the control limits for the proposed EWMA chart based on two auxiliary variables. Finally, after a simulation study like we have done with one auxiliary variable, the key conclusions about the EWMA chart based on two auxiliary variables are: i.  values for the proposed chart decrease with an increase in values of either or both NJ– and NJâ but they increase as the value of ǖâ increases; DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϴϱ ϴϲ  ii. iii. iv. v. the combinations of NJ– , NJâ and ǖâ (e.g. NJ–  NJâ  ǖâ   >B>dB and NJ–  NJâ  ǖâ   >dB>B) for which the values of plotting statistic 7£ and its variance '£ are the same, yield the same for fixed values of *å and , 8; the value of combination of NJ– , NJâ and ǖâ ; å remains the same for any possible the performance of the proposed chart is usually better for the smaller values of *å ; auxiliary information really improves the performance (which is not a surprise!). ͶǤʹǤ͵ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ In this subsection, we give an illustrative example for which we generate a dataset (named as dataset 4.1) containing 30 observations in total. The first 20 observations are generated from a trivariate normal distribution given as:  + + >B >B É  Ì È[à ¿É B Ì  É>B +  ÌÁ R B >B  + Hence the in control mean of the study variable  equals +. The remaining 10 observations are generated from an out of control situation with a shift of one sigma in the mean of the study variable  (i.e. @  +) given as:  ++ + >B >B É  Ì È[à ¿É B Ì  É>B +  ÌÁ R B >B  + Now the variable  is used to setup the classical EWMA control chart (with plotting statistic represented by ) and parameters *  >MB and  P), variables  and  are used to setup the MXEWMA control chart (with plotting statistic represented by )q and parameters *q  >MB and q  P) and variables ,  and R are used to setup the MTEWMA control chart (with plotting statistic represented by )å and parameters *å  >MB and DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů å  P). The ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  calculation steps for applying the MXEWMA chart are given in Table 4.7, whereas the graphical display of the three charts is given in Figure 4.1. Table 4.8: Simulated dataset 4.1 and calculation steps of MXEWMA chart Sample No.  1 10.07 2 11.21 3 8.32 4 9.63 5 7.4 6  R 5.22 7– )q 4.34 9.97 9.99 7.15 4.15 10.13 10.03 4.91 4.46 8.37 9.61 2.95 4.32 10.65 9.87 4.08 4.26 7.86 9.37 10.56 4.16 5.86 10.98 7 9.52 4.55 4.69 8 8.85 5.23 4.02 9 9.58 5.45 10 10.98 11 10.11 12   Sample No. 10.65 16 11.06 10.81 17 10.15 10.89 18 9.12 10.93 19 10.5 10.95 20 8.67 9.77 10.97 21 9.74 9.76 10.97 8.74 9.51 10.98 4.62 9.36 9.47 5.04 4.48 10.96 5.52 5.68 9.85 9.6 4.06 4.76 13 10.7 5.25 14 8.35 4.33 15 11.09 5.88  5.86 R 7– )q  6.67 10.63 10.12 10.98 5.48 3.77 9.92 10.07 10.98 5.18 4.58 9.03 9.81 10.98 4.97 6.67 10.51 9.99 10.98 5.5 3.55 8.42 9.59 10.98 10.41 5.29 4.05 10.26 9.76 10.98 22 10.11 3.33 3.91 10.95 10.06 10.98 23 12.59 5.2 7.1 12.49 10.67 10.98 10.98 24 11.59 4.98 4.65 11.6 10.9 10.98 9.84 10.98 25 11.24 6.42 4.27 10.53 10.81 10.98 9.84 10.98 26 10.65 4.81 4.66 10.74 10.79 10.98 10.07 9.9 10.98 27 11.09 5.29 3.56 10.94 10.83 10.98 6.63 10.57 10.07 10.98 28 10.48 5.25 4.36 10.36 10.71 10.98 4.24 8.68 9.72 10.98 29 11.92 5.04 4.14 11.9 11.01 10.98 5.02 10.65 9.95 10.98 30 11.17 4.32 5.53 11.51 11.13 10.98 Figure 4.1: Graphical display of MTEWMA, MXEWMA and classical EWMA charts for the simulated dataset 4.1 DdtD DdžtD >ŝŵŝƚƐ;DdtDͿ >ŝŵŝƚƐ;DdžtDͿ ϭϭ͘Ϯ ϭϬ͘ϴ ϭϬ͘ϰ ϭϬ ϵ͘ϲ ϵ͘Ϯ ϴ͘ϴ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϴϳ ϴϴ  Figure 4.1 indicates that the classical EWMA control chart failed to detect any shift in the mean of the study variable . MXEWMA detected the shift at samples # 29 and 30 (cf. Table 4.8), whereas MTEWMA detected the shift at samples # 24, 25, 26, 27, 28, 29 and 30. ͶǤ͵  ‘–”‘Ž Šƒ”–•—•‹‰ƒ—š‹Ž‹ƒ”›‹ˆ‘”ƒ–‹‘ In this section we utilize the efficiency of regression estimator to design a CUSUMtype structure and try to study the effect of this efficient estimator on the performance of CUSUM chart. Now the plotting statistics for the proposed MXCUSUM chart (which is based on the estimator given in (4.2)) is given as: [  € 57–§   6  q  ! [ ƒ, [  € 57–§   6  q  ! [ ƒ (4.12) Initial values for the statistics given in (4.12) are taken equal to zero i.e. [  [  . The decision rule for the proposed chart is given as: the statistics [ and [ are plotted against the control limit $q . For any value of #, if the value of [ exceeds the value of $q than the process mean is declared to be shifted upwards and if the value of [ exceeds the value of $q than the process mean is said to be moved downwards. q  & q 'Ñ , $q  (q 'Ñ q and $q are defined as: (4.13) The values of & q and (q need to be selected carefully because the properties of MXCUSUM chart mainly depend on these two constants (along with the value of NJ– ). The specifications of MXCUSUM chart makes it the generalized form of the classical CUSUM chart given in Section 1.2, i.e. for NJ–  , the proposed MXCUSUM chart becomes equivalent to the classical CUSUM in terms of plotting statistic as well For the proposed chart Tables 4.9 – 4.11 contain the 8 with DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů  performance. fixed at B. ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  Table 4.9: 9:; values for the proposed MXCUSUM chart with <q  => F? and lq  G> ?G? at 9:;=  ?== 0.05 0.25 NJ– 0.5 0.75 0.95 0 499.3466 499.1864 499.2012 500.9817 500.6632 0.25 94.6109 89.987 74.984 48.1801 16.1035 0.5 31.0236 29.6155 25.1541 17.3604 7.0516 0.75 17.5129 16.8127 14.5609 10.4637 4.5919 1 12.1546 11.7041 10.2408 7.5157 3.4645 1.5 7.5715 7.3144 6.4712 4.8689 2.3575 2 5.5412 5.3636 4.7791 3.6584 1.9967 2.5 4.4047 4.2701 3.8258 2.9861 1.7963 3 3.6817 3.5743 3.2238 2.5107 1.2199 @ Table 4.10: 9:; values for the proposed MXCUSUM chart with <q  => ? and lq  ?> =JE at 9:;=  ?== 0.05 0.25 NJ– 0.5 0.75 0.95 0 499.015 500.0021 500.8843 499.0574 500.1008 0.25 145.2716 138.2263 114.5163 68.9901 15.3663 0.5 38.777 36.3617 28.9044 17.0683 5.3405 0.75 17.2793 16.3181 13.3748 8.6464 3.2904 1 10.4965 9.9897 8.4132 5.7578 2.4388 1.5 5.8086 5.5756 4.8327 3.5082 1.7847 2 4.0503 3.9052 3.4373 2.5806 1.2021 2.5 3.1451 3.0414 2.7049 2.1126 1.0074 3 2.5978 2.5187 2.2684 1.8667 1 @ The algorithm used for calculating the where 50,000 simulations are used. 8 in Tables 4.9 – 4.11 is given in Appendix 4.2 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϴϵ ϵϬ  Table 4.11: 9:; values for the proposed MXCUSUM chart with <q  E and lq  F> gg? at 9:;=  ?== 0.05 0.25 NJ– 0.5 0.75 0.95 0 499.0466 499.1701 500.5972 499.4241 500.9488 0.25 249.1542 240.639 209.9628 139.8514 26.0941 0.5 81.2212 75.9419 58.9892 30.2808 5.0512 0.75 30.8019 28.4313 21.2702 10.7289 2.5918 1 14.6235 13.5187 10.2682 5.6554 1.795 1.5 5.7314 5.3872 4.3635 2.814 1.1276 2 3.4063 3.2422 2.7406 1.9231 1.0031 2.5 2.4483 2.3484 2.0355 1.4745 1 3 1.9387 1.8674 1.6356 1.1929 1 @ Before concluding this section, we present the main findings about our proposed MXCUSUM control chart. i. the use of an auxiliary variable with the control structure of CUSUM chart is really  advantageous in terms of the Tables 4.9 – 4.11); ii. iii. iv. values if the value of NJ– is reasonably large (cf. to attain a fixed value of , the value of (q has to remain fixed for all the value of to attain a fixed value of , the value of (q decrease with an increase in the value NJ– (cf. Tables 4.9 – 4.11); of & q and vice versa (cf. Tables 4.9 – 4.11); for a fixed value of , the  values decrease rapidly with a decrease in the values of either or both NJ– and @ (cf. Tables 4.9 – 4.11). Like MXCUSUM, a CUSUM chart based on the information of two auxiliary variables (named as MTCUSUM) can also be easily designed by following the procedure given in subsection 4.2.2. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  ͶǤ͵Ǥͳ ‘’ƒ”‹•‘• Now we provide a comparison of MXCUSUM chart with some of the revisions and extensions of CUSUM and EWMA-type control charts. MXCUSUM versus the classical CUSUM: values of & with  8 for two-sided CUSUM chart for different fixed at B are given in Table 4.6. Collectively Tables 4.9 – 4.11 and Table 4.6 validate that the classical CUSUM is a special case of the proposed MXCUSUM chart. The performance of the proposed chart with NJ–  >B (i.e. close to zero) is almost same as compared to the value of NJ– for the proposed chart, its performance of classical CUSUM. As we increase the  values decrease. MXCUSUM versus the classical EWMA: Table 2.2 contains the where * and 8 of classical EWMA are the parameters of EWMA control chart. The comparison of MXCUSUM chart with the classical EWMA show that the proposed chart is performing better in general for NJ– Û >MB (cf. Table 2.2 vs. Tables 4.9 – 4.11). For large values of NJ– (like NJ– Û >dB) the proposed chart (with any value of & q ) outperfoms the classical EWMA (with any value of *). MXCUSUM versus MXEWMA: Comparing the performance of MXCUSUM chart with MXEWMA chart for a specific value of NJ– (e.g. NJ–  >B) it can be observed that MXEWMA (with *q  >+) has a slightly better performance as compared to the proposed chart but once *q is greater than >+ for the MXEWMA chart, the proposed chart becomes better than MXEWMA (cf. Table 4.1 – 4.5 vs. Tables 4.9 – 4.11). The MTCUSUM control chart, which is based on two auxiliary variables, has even better properties as was to be expected (results not given here). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϵϭ ϵϮ  ͶǤ͵Ǥʹ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ An example to illustrate the application of the proposed chart with a real dataset is provided in this subsection. The three CUSUM-type control charts are applied to the simulated dataset 4.1 with &  >B and (  B>d+ for the classical CUSUM; & q  >B and (q  B>d+ for the MXCUSUM; and & å  >B and (å  B>d+ for MTCUSUM chart with  4.2. fixed at B for all three charts. The chart output for all three charts is given in Figure Figure 4.2: Graphical display of MTCUSUM, MXCUSUM and classical CUSUM charts for simulated dataset 4.1 Ddh^hD ,;Ddžh^hDͿ ϭϬ ,;Ddh^hDͿ h^hD Ddžh^hD ,;h^hDͿ ϴ ϲ ϰ Ϯ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ^ĂŵƉůĞEƵŵďĞƌ Figure 4.2 shows that MTCUSUM chart is detecting the positive shift at the 24th sample and onwards, whereas the classical CUSUM and MXCUSUM are detecting the shift at the 27th and the 29th sample, respectively. This verifies the observations in subsection 4.3.1. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  ͶǤͶ ‘ Ž—†‹‰”‡ƒ”• This chapter proposes the use of auxiliary information with the control structures of the EWMA and CUSUM charts. The regression estimation technique is used to exploit the auxiliary information. Note that the proposed MXEWMA and MXCUSUM charts are the extended forms of the classical EWMA and CUSUM charts respectively, i.e. the proposed charts are equal to the classical ones when the correlation between the study variable and auxiliary variable(s) is equal to . The performance of the proposed charts is evaluated in terms of for different values of the correlation between the study variable and the auxiliary variable(s). A comparison of the proposed charts with the classical CUSUM, the classical EWMA and some of their recent modifications is also made. The comparisons showed that the proposed charts are good at detecting small to moderate shifts in the process location, while their ability to detect large shifts is not bad either. Finally, results are also supported by the illustrative examples. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϵϯ ϵϰ  ’’‡†‹šͶǤͳ library(MASS) Mx=c(); Zp=c(); ucl=c(); lcl=c(); rl=c() muy=0; mux=0; sigyy=1; sigxx=1; sigyx=0.5 betayx=sigyx/sigxx; rho=sigyx/sqrt(sigyy*sigxx) meanv=c(muy,mux) sigmam=matrix(c(sigyy,sigyx,sigyx,sigxx),2,2) muM=muy; sigM=sqrt(sigyy*(1-rho^2)) ld=0.25; L=3 for(j in 1:10000) { for(i in 1:1000000) { w=mvrnorm(1,meanv,sigmam) y=w[1]; x=w[2] Mx[i]=y-betayx*x if(i==1) {Zp[i]=ld*Mx[i]+ (1-ld)*muM;} else{Zp[i]=ld*Mx[i]+(1-ld)*Zp[i-1];} ucl[i]=muM+L*sigM*sqrt((ld/(2-ld))*(1-(1-ld)^(2*i))) lcl[i]=muM-L*sigM*sqrt((ld/(2-ld))*(1-(1-ld)^(2*i))) if(Zp[i]>ucl[i]|Zp[i]<lcl[i]) {rl[j]=i;break;} else{rl[j]=0;} } } mean(rl)   ’’‡†‹šͶǤʹ library(MASS) Mx=c(); Np=c(); Nn=c(); rl=c() muy=0; mux=0; sigyy=1; sigxx=1; sigyx=0.5; betayx=sigyx/sigxx; rho=sigyx/sqrt(sigyy*sigxx) meanv=c(muy,mux) sigmam=matrix(c(sigyy,sigyx,sigyx,sigxx),2,2) muM=muy; sigM=sqrt(sigyy*(1-rho^2)) K=0.5*sigM; H=5.071*sigM for(j in 1:100000) { for(i in 1:1000000) { w=mvrnorm(1,meanv,sigmam) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵdžŝůŝĂƌLJŝŶĨŽƌŵĂƚŝŽŶďĂƐĞĚh^hDĂŶĚtDĐŽŶƚƌŽůĐŚĂƌƚƐ  y=w[1]; x=w[2] Mx[i]=y-betayx*x if(i==1) {Np[i]=max(0,(Mx[i]-muM)-K);} else{Np[i]=max(0,(Mx[i]-muM)-K+Np[i-1]);} if(i==1) {Nn[i]=max(0,-(Mx[i]-muM)-K);} else{Nn[i]=max(0,-(Mx[i]-muM)-K+Nn[i-1]);} if(Np[i]>H|Nn[i]>H) {rl[j]=i;break;} else{rl[j]=0;} } } mean(rl) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϵϱ ϵϲ    DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ŚĂƉƚĞƌϱ  Šƒ’–‡”ͷ ”‘‰”‡••‹˜‡ ‘–”‘Ž Šƒ”–‹‰ This chapter proposes a control chart for monitoring the process location, named as Progressive Mean (7) control chart, in which a progressive mean is used as a plotting statistic. Taking the inspiration from the 7 chart, two new memory control charts for monitoring the process dispersion, named as floating  
 and floating  
 control charts, are also proposed. The proposed charts are designed such that they utilize not only the current information but also the past information. Therefore, the proposed charts are natural competitors for the classical CUSUM, the classical EWMA and some recent modifications of these two charts. This chapter is based on two papers; one for monitoring the location parameter (cf. Abbas, Zafar, Riaz and Does (2012)) and the other for monitoring the dispersion parameter (cf. Abbas, Riaz and Does (2012d)). ͷǤͳ Š‡’”‘’‘•‡†’”‘‰”‡••‹˜‡‡ƒ ‘–”‘Ž Šƒ”– Let  be a quality characteristic of which the mean will be monitored. It is assumed that we use individual observations from a normal distribution. If  , #  + M P x x> is the sequence of independent and identically distributed observations from the process under investigation, then progressive mean 7 is defined as the cumulative average over time. Mathematically, we may define the 7 statistic as: DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϵϳ ϵϴ  7  ¦§¨«¬ Š¨  (5.1) 7 is an unbiased estimator of population mean  and its variance is given by ]^©  . According to the typical three sigma control limits, the control charting structure based on the 7 statistic and its variance may be defined as:      P ^ h ] _    h     ! P ]^ _ (5.2) where all the parameters used in (5.2) are as explained earlier. It is obvious that the control limits given in (5.2) are time varying and exploit the past and present information using equal weights. Note that the design structure of the proposed chart is relative simple and easy to execute compared with the CUSUM and EWMA control charts. A problem with the control structure in (5.2) is that the control limits remain too wide for the large values of # (wide relative to the plotting statistic). It turns out that it is almost impossible for the plotting statistic in (5.1) to cross the control limits in (5.2), in case of shifted mean. We have solved this issue by imposing a penalty on the control limits such that the control limits are a bit narrower for large values of #. We choose this penalty function equal to —#  # æ . Hence, the penalized limits for the proposed 7 chart are given by:       q ]^  ^>çèé hh    h     !  q where  q is a constant that is used to control the run lengths. ]^  ^>çèé (5.3) In (5.3) we have used different possibilities of ê and we have searched for a suitable constant  q for each possibility to fix the in control process properties in terms of optimum out of control properties in terms of suitable choice in terms of optimizing the .  and We identified ê  >M as the most properties. For this optimum choice of —# (i.e. # > ) we have worked out the values of constants  q by fixing DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů . These constants for some WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ  commonly used  values are provided in Table 5.1. For these constants and  ϵϵ values given in Table 5.1 we have carried out a RL study and the resulting properties in terms of   and are provided in Table 5.2. Appendix 5.1 contains the details regarding the 8. The standard errors algorithm, that is made in R language, used for the computation of for the results of Tables 5.1 – 5.2 remain less than 1.3%. TABLE 5.1: Values of the control limit constant nq for different choices of 9:;= for the proposed Äë control chart  q  168 200 3 3.129 400 500 3.568 3.639 3.846  TABLE 5.2: 9:; values for the proposed chart with different shifts Pre-fixed  370 0 0.25 0.5 0.75 1 500 498.14 47.2353 19.032 11.1564 400 400.938 44.474 17.872 370 369.00 17.452 200 200.82 168 170.33 @ 1.5 2 3 4 5 7.5504 4.5204 3.1455 1.9803 1.4248 1.1186 10.313 7.200 4.261 2.985 1.877 1.360 1.0847 10.166 10.166 7.0941 4.184 2.931 1.721 1,236 1.043 34.7366 14.6914 8.6191 5.9814 3.5996 2.5415 1.6124 1.1981 1.0338 33.0695 12.1479 8.1779 5.659 3.4086 2.4437 1.5555 1.1598 1.0263 We conclude from Tables 5.1 – 5.2: i. The proposed 7 control chart is really good in detecting small and moderate shifts and is still good in detecting large shifts (cf. Table 5.2); ii. iii.  decreases fastly with an increase in @ (cf. Tables 5.1 and 5.2); The control structure of the proposed chart is very easy compared to the CUSUM and EWMA charts. Note that if we apply the same set up with samples sizes Z % + instead of Z  +, the results will be the same with the obvious adjustments in the control limits in (5.3). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϬϬ  ͷǤͳǤͳ ‘’ƒ”‹•‘• To detect small and moderate shifts, EWMA and CUSUM charts and some of their modifications are available. We have introduced in this chapter a rather simple alternative to these charts, namely the 7 control chart, and in this section we compare the performance of our proposed chart with some of its counterparts in terms of . We compare the performance of the proposed chart with the classical EWMA, the classical CUSUM, the fast initial response (FIR) CUSUM, the FIR EWMA, the runs rules based CUSUM, the runs rules based EWMA and the adaptive EWMA. We use for the  the values +Xf, M, Pd, A and B, so that valid comparisons with each chart can be made. Below, we present one by one the comparison of the proposed chart with its competitor. Proposed versus the classical CUSUM: values for the classical CUSUM are given by in Table 2.1. Comparison of the classical CUSUM with the proposed 7 chart clearly shows that the proposed chart almost outperforms the classical CUSUM for all the values of @ (cf. Table 2.1 vs. Table 5.2). Proposed versus the classical EWMA: Table 2.2 contains the values for the classical EWMA chart. Comparison of Table 2.2 with Table 5.2 shows the uniform superiority of the proposed 7 chart over the classical EWMA chart. Proposed versus FIR CUSUM and FIR EWMA: values for the FIR CUSUM presented by Lucas and Crosier (1982) and FIR EWMA by Lucas and Saccucci (1990) are given in Table 2.10 and Table 2.17, respectively. The comparison of FIR CUSUM and the proposed 7 chart indicates that the performance of proposed control chart is better for small and moderate shifts even when the  for the FIR CUSUM is not fixed at +Xf (cf. Table 2.10 vs. Table 5.2). We may conclude that the proposed chart performs really well for small DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ  shifts. The same is the case for the FIR EWMA, i.e. for a 25% head start the performance of the proposed chart is superior to the FIR EWMA for every choice of *, but as we increase the head start to 50%, the performance of the FIR EWMA becomes better than the proposed chart for the larger shifts with *  >+ (cf. Table 2.17 vs. Table 5.2). Proposed versus runs rules based CUSUM and EWMA: In chapter 2 we introduced the 8 for the runs rules based CUSUM and the runs rules based EWMA, respectively. The runs rules based CUSUM are provided in Tables 2.7 – 2.8 and for runs rules based EWMA these are given in Tables 2.15 – 2.16. The comparison of the runs rules based CUSUM with the proposed chart shows that the proposed chart performs better than the runs rules based CUSUM uniformly for both schemes (cf. Tables 2.7 – 2.8 vs. Table 5.2). Similarly, the comparison of the runs rules based EWMA with the proposed chart also reveals that the proposed chart is superior to the runs rules based EWMA for most of the values of @ (cf. Tables 2.15 – 2.16 vs. Table 5.2). Proposed versus adaptive EWMA: The adaptive EWMA of Capizzi and Masarotto (2003) is designed so that it performs better for small and large shifts at the same time by giving weights to past information using a suitable function of the current error. Three functions of error represented by ìVí > , ìîo >  and ìïíî >  are used in their article. values for the adaptive EWMA with these 3 functions of errors are given in Table 5.3, where the value of @ is targeted between >MB  A. TABLE 5.3: 9:; values for the adaptive EWMA at 9:;=  ?== Error function ìVí >  ìîo >  ìïíî >  0.25 0.5 0.75 1 98.51 40.94 25.04 135.01 42.72 97.03 41.54 @ 1.5 2 3 4 17.59 10.11 6.08 2.29 1.26 21.99 13.91 7.12 4.25 2.01 1.28 25.68 18.16 10.52 6.36 2.35 1.27 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϬϭ ϭϬϮ  The comparison of the proposed chart with the adaptive EWMA shows that the proposed chart also outperforms the adaptive EWMA for all the values of @ (cf. Table 5.3 vs. Table 5.2). Proposed versus adaptive CUSUM: Jiang et al. (2008) proposed the use of an adaptive CUSUM with EWMA based shift estimator.  8 for the adaptive CUSUM with @|`  >B and *  >P are given in Table 3.5. Comparison of Table 5.2 and Table 3.5 shows that the proposed 7 chart is uniformly superior compared to the adaptive CUSUM with EWMA based shift estimator. curves of the proposed 7 chart against To summarize the results, we have made some its existing counterparts. These are given in Figures 5.1 – 5.3. curves for the proposed chart, the classical EWMA (with *  >+), the adaptive EWMA (with error function ìVí > ), the runs rules based CUSUM scheme II (with R  A>f and  B>++) and the runs rules based EWMA modified MOP scheme (with *  >+) are presented in Figure 5.1. FIGURE 5.1: 9:; Curves for proposed chart, classical EWMA, adaptive EWMA, runs rules based CUSUM and runs rules based EWMA at 9:;=  ?== WƌŽƉŽƐĞĚ tD Ϭ͘Ϯϱ Ϭ͘ϱ ĚĂƉƚŝǀĞtD ZZh^hD// ZZtD// ϭϱϬ ϭϮϬ Z> ϵϬ ϲϬ ϯϬ Ϭ Ϭ͘ϳϱ ϭ į DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭ͘ϱ Ϯ  WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ ϭϬϯ FIGURE 5.2: 9:; curves for proposed chart and adaptive CUSUM at 9:;=  H==  WƌŽƉŽƐĞĚ ĚĂƉƚŝǀĞh^hD;ɶсϰ͕Śсϰ͘ϯϰϴͿ ĚĂƉƚŝǀĞh^hD;ɶсϮ͕Śсϱ͘ϭϬϱͿ ϴϬ Z> ϲϬ ϰϬ ϮϬ Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ ϭ į ϭ͘ϱ Ϯ ϯ ϰ  FIGURE 5.3: 9:; curves for proposed chart, classical CUSUM and FIR CUSUM at 9:;=  EgG WƌŽƉŽƐĞĚ h^hD &/Zh^hD;ŽсϭͿ ϴϬ Z> ϲϬ ϰϬ ϮϬ Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ į ϭ ϭ͘ϱ Ϯ   DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϬϰ  In Figure 5.2 comparison of the proposed chart and the adaptive CUSUM are provided, whereas Figure 5.3 contains the curves for the proposed chart, the classical CUSUM and the FIR CUSUM (with head start   +). Figures 5.1 – 5.3 clearly show the best performance of the proposed chart against the other methods. ͷǤͳǤʹ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ In this section we demonstrate the application of the proposed 7 chart. Also the classical EWMA and the classical CUSUM charts are included in the example to validate the superiority of the proposal. For this purpose we have generated two datasets of 40 and 30 observations, respectively, such that in dataset 5.1 the first 20 observations are generated from [+ (i.e. the in control situation) and the second set of the 20 observations from [>B+ (i.e. the out of control situation having a shift of >B' (small shift)). Similarly, in dataset 5.2 the first 20 observations are generated from [+ and the second set of 10 observations from [+>B+ (i.e. an out of control situation having a shift of +>B' (moderate shift)). The 7 statistic for the proposed chart, the EWMA statistic with *  >MB and the CUSUM statistic with &  >B are calculated. To fix the +>MXd for the 7 chart,  at B we have used  q   P for the classical EWMA and (  B>d for the classical CUSUM. The graphical displays of the proposed 7, EWMA and CUSUM charts are presented in the Figures 5.4, 5.5 and 5.6 for dataset 5.1 and in Figures 5.7, 5.8 and 5.9 for dataset 5.2, respectively. From Figure 5.4 we can see that the proposed chart gives out of control signals at samples # 35, 36, 37, 38, 39 and 40, thus giving a total of 6 out of control signals. Figure 5.5 shows that the classical EWMA control chart gives one out of control signal at sample # 38 and Figure 5.6 depicts that the classical CUSUM control chart gives out of control signals at DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ  samples # 38, 39 and 40, thus giving 3 signals. An upward shift occurred after sample # 20, which is detected by the proposed chart more quickly than the EWMA and the CUSUM. It illustrates the ability of the proposed chart to quickly detect small shifts in the process. FIGURE 5.4: Graphical display of the proposed Äë chart for dataset 5.1 WDƐƚĂƚŝƐƚŝĐƐ ϰ h> >> ϯ Ϯ ϭ Ϭ Ͳϭ ͲϮ Ͳϯ Ͳϰ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ^ĂŵƉůĞEƵŵďĞƌ FIGURE 5.5: Graphical display of the classical EWMA chart for dataset 5.1 tDƐƚĂƚŝƐƚŝĐ ϭ͘ϱ h> >> ϭ Ϭ͘ϱ Ϭ ͲϬ͘ϱ Ͳϭ Ͳϭ͘ϱ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ϯϵ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϬϱ ϭϬϲ  FIGURE 5.6: Graphical display of the classical CUSUM for dataset 5.1 h^hDƐƚĂƚŝƐƚŝĐ Ś ϲ ϱ ϰ ϯ Ϯ ϭ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϭ ϯϯ ϯϱ ϯϳ ^ĂŵƉůĞEƵŵďĞƌ FIGURE 5.7: Graphical display of the proposed Äë chart for dataset 5.2 WDƐƚĂƚŝƐƚŝĐ ϰ h> >> ϯ Ϯ ϭ Ϭ Ͳϭ ͲϮ Ͳϯ Ͳϰ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů Ϯϯ Ϯϱ Ϯϳ Ϯϵ ϯϵ WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ The situation is not much different in dataset 5.2, where the 7 chart detects the shift at  samples # 23, 24, 25, 26, 27, 28, 29 and 30 (cf. Figure 5.7). The classical EWMA and the CUSUM detect the shift at samples # 27, 28, 29 and 30 (cf. Figures 5.8 and 5.9). FIGURE 5.8: Graphical display of the classical EWMA chart for dataset 5.2 Ϯ tDƐƚĂƚŝƐƚŝĐ h> >> ϭ͘ϱ ϭ Ϭ͘ϱ Ϭ ͲϬ͘ϱ Ͳϭ Ͳϭ͘ϱ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ^ĂŵƉůĞEƵŵďĞƌ FIGURE 5.9: Graphical display of the classical CUSUM for dataset 5.2 h^hDƐƚĂƚŝƐƚŝĐ Ś ϭϮ ϭϬ ϴ ϲ ϰ Ϯ Ϭ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ Ϯϳ Ϯϵ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϬϳ ϭϬϴ  In both small and moderate shifts we see that the proposed chart detect the shift more quickly than the others. The number of signals given by the proposed chart is also greater than the classic ones. These outcomes are exactly in accordance with the findings of subsection 5.1.2. ͷǤʹ Ž‘ƒ–‹‰ ‘–”‘Ž Šƒ”–•ˆ‘”’”‘ ‡••†‹•’‡”•‹‘ There is a lot of literature available on CUSUM and EWMA-type control charts for monitoring the process dispersion, e.g. see Page (1963), Hawkins (1981), Acosta-Mejia et al. (1999) and Chang and Gan (1995) for CUSUM-type charts and Ng and Case (1989), Crowder and Hamilton (1992) and Huwang et al. (2010) for EWMA-type charts. Additionally, Wu and Tian (2005) and Zhang and Chang (2008) also provided the CUSUM and EWMA charts, respectively, for monitoring the process mean and variance simultaneously. Most of these charts are based on transforming the sample variance such that the new transformed form may be closely approximated by a normally distributed variable and hence applying the usual CUSUM and EWMA structures (recommended by Page (1954) and Roberts (1959), respectively) on it. In a similar direction, Castagliola (2005) proposed a new
 EWMA chart for monitoring the process dispersion. He used a logarithmic three parameter transformation to obtain a normal approximation for sample variance. A similar transformation is used by Castagliola et al. (2009) to setup a CUSUM
 chart for monitoring process dispersion. The details regarding the EWMA and CUSUM charts based on the logarithmic transformation are given in Section 3.2. Castagliola et al. (2010) proposed another similar type of transformation based on a four parameter Johnson
ð transformation. They claimed that this four parameter transformation gives a better approximation to the normal distribution as compared to the DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ  three parameter logarithmic transformation. With the notation of Castagliola et al. (2010), it follows that   ‘ñ ! ñ ¤‡ ¹ where ‘ñ  ò§© ïó ôó ïó ò§© ñ Z, º (5.4) ñ  ­ñ Z, ¥ñ  ñ Z' and õñ  cñ Z' . Variable  in (5.4) follows approximately a normal distribution with mean ñ Z and variance 'ñ Z where the values of ñ Z, 'ñ Z, in Table 5.4. ñ Z, ­ñ Z, ñ Z and cñ Z for all Z  P A B x  +B are given Table 5.4: Values of ¯ö  , ±ö  , 9ö  , ²ö   and nö   3 ¯ö   0.0184 ±ö   0.9475 9ö   3.1936 ²ö   1.1952 nö   -0.2588 ÷ö   4 0.0078 0.9739 3.3657 1.3983 -0.2438 12.591 5 0.0039 0.9852 3.5402 1.5727 -0.2352 11.312 6 0.0022 0.9908 3.7111 1.7281 -0.2295 10.530 7 0.0014 0.994 3.8768 1.8698 -0.2254 10.000 8 0.0009 0.9958 4.0369 2.0010 -0.2224 9.618 15.077 9 0.0006 0.9970 4.1918 2.1238 -0.2200 9.328 10 0.0004 0.9978 4.3417 2.2396 -0.2181 9.100 11 0.0003 0.9983 4.4869 2.3495 -0.2166 8.917 12 0.0002 0.9987 4.6279 2.4544 -0.2152 8.766 13 0.0002 0.9989 4.7648 2.5549 -0.2141 8.640 14 0.0001 0.9991 4.8981 2.6515 -0.2132 8.532 15 0.0001 0.9993 5.0279 2.7446 -0.2123 8.440 Note that in case of õñ ! ¥ñ 
 - , the transformation given in (5.4) is not possible, but Castagliola et al. (2010) showed that the probability of occurrence of this event is so close to zero that it can be neglected. From the values of ¥ñ and õñ it can be noticed that õñ ! ¥ñ 
 -  implies a very large value of
 as compared to the value of ' so it can be taken as an out of control situation with a large positive shift. Details about the distributional properties of  can be found in Castagliola et al. (2010). DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϬϵ ϭϭϬ  Furthermore, it should be noted that any change in the process standard deviation will change the mean of the normalized variables given in (3.5) and (5.4). So based on the these two (approximately) normalized statistics, we are now able to define our new control structures, named as floating  
 and floating  
 charts, respectively. These charts monitor basically the mean of the transformed statistics in (3.5) and (5.4) and hence control the process dispersion. F ͷǤʹǤͳ Ž‘ƒ–‹‰  ‘–”‘Ž Šƒ”– The first proposed chart, named as floating  
 chart, is based on the three parameter logarithmic transformation given in (3.5). The plotting statistic is given as:   ¦§ø«¬ £ø  (5.5) The statistic in (5.5) is a cumulative average of the three parameter logarithmic transformation given in (3.5). According to the probability distribution theory we have that, if  follows (approximately) a normal distribution with mean  £ Z and variance '£ Z that   ¦U¼ U N# will have mean  £ Z and variance ]½© `  . This implies that the control limits (including the upper control limit ( ), center line ( ) and lower control limit (  )) for the floating statistic given in (5.5) can be defined as:    £ Z  £ ]½ ` _    £ Z, ,    £ Z ! where the width of the control limits is determined by adjusting this constant ( £) as the £. £ The ]½ ` _ (5.6)  can be controlled by 8 for a control chart with wider limits are larger and vice versa. The same problem (like the control limits in (5.2)) also persists with the control structure in (5.6), i.e. the control limits remain too wide for the larger values of # (wide relative to the plotting statistic). Therefore, following (5.3), the penalized control limits for DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ the proposed floating  
 chart are given as:   £ where   £ Z  q £ q ]½ ` £ y éè^>ç ,  £   £ Z,  £   £ Z ! q ]½ ` £ y éè^>ç (5.7) is the adjusted control limit coefficient. Table 5.5: 9:; values for the floating °  µF chart with rq°  I> EF” and 9:;=  F== Ch 0.5  Ih 6.041  ?h 3.779  Jh 2.890  ”h 2.417 0.6 8.069 5.003 3.810 3.164 0.7 11.874 7.316 5.546 4.585 0.8 20.468 12.625 9.533 7.835 0.9 49.392 31.652 23.918 19.851 0.95 98.625 70.337 56.347 47.163 1.05 92.869 67.545 54.129 46.352 1.1 47.906 31.469 24.452 20.236 1.2 21.249 13.379 10.268 8.496 1.3 12.892 8.130 6.259 5.213 1.4 9.113 5.778 4.461 3.740 1.5 7.040 4.482 3.485 2.941 2 3.316 2.205 1.774 1.527 3 1.834 1.323 1.148 1.067 Table 5.6: 9:; values for the floating °  µF chart with rq°  I> ?gG and 9:;=  IJ= Ch  Ih  ?h  Jh  ”h 0.5 7.157 4.425 3.391 0.6 9.629 5.892 4.478 3.696 0.7 14.123 8.674 6.542 5.371 0.8 24.638 15.057 11.310 9.280 0.9 61.342 38.450 28.858 23.761 0.95 135.241 91.241 71.136 59.045 1.05 128.999 89.714 69.639 58.704 1.1 60.472 38.941 29.562 24.367 1.2 25.692 16.054 12.208 10.039 1.3 15.356 9.646 7.356 6.110 1.4 10.757 6.792 5.220 4.333 1.5 8.234 5.229 4.049 3.377 2 3.813 2.506 2.001 1.703 3 2.030 1.434 1.206 1.103 2.807 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϭϭ ϭϭϮ  Table 5.7: 9:; values for the floating °  µF chart with rq°  I> GHg and 9:;=  ?== 0.5 Ch 7.896  Ih 4.870  ?h 3.712  Jh 3.085  ”h 0.6 10.599 6.491 4.912 4.054 0.7 15.639 9.538 7.207 5.913 0.8 27.327 16.588 12.501 10.242 0.9 69.082 42.830 32.216 26.335 0.95 158.678 104.768 80.317 66.427 1.05 153.847 102.791 79.859 66.697 1.1 69.135 43.487 33.006 27.250 1.2 28.649 17.766 13.468 11.097 1.3 17.037 10.578 8.085 6.666 1.4 11.847 7.443 5.696 4.728 1.5 9.008 5.711 4.405 3.684 2 4.133 2.719 2.148 1.838 3 2.179 1.519 1.259 1.133 Note thaththe control limits given in (5.6) are a special case of the limits in (5.7) with ê  . We have tested several values of ê and ê  >M was found to be optimal, i.e. ê  >M gives  smaller values for a fixed proposed floating  
 chart. The . Tables 5.5 – 5.7 contain the values for the 8 for the proposed chart are evaluated by running +´ simulations. The simulation program is developed in R language. ͷǤʹǤʹ Ž‘ƒ–‹‰ F  ‘–”‘Ž Šƒ”– The plotting statistic for the second proposed chart (based on a four parameter Johnson
ð transformation) to monitor the process dispersion is given as:   ¦§ø«¬ ñø  Like  in (5.5), here  also have mean ñ Z and variance (5.8) © ` ]ó  . Therefore the control limits for this second proposed chart, named as floating  
 chart, are given as: DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ   ñ where  ñ Z  q ñ q ]ó ` ñ y éè^>ç ,  ñ  ñ Z,  ñ  ñ Z ! q ]ó ` ñ y éè^>ç is the control limit coefficient for this second proposed chart. The the floating  
 chart are given in Tables 5.8 – 5.10. (5.9) values for Table 5.8: 9:; values for the floating ö  µF chart with rqö  I> EF” and 9:;=  F== 0.5 Ch 5.737  Ih 3.557  ?h 2.737  Jh 2.305  ”h 0.6 7.877 4.830 3.676 3.056 0.7 11.766 7.201 5.447 4.472 0.8 20.684 12.632 9.476 7.789 0.9 49.990 31.946 24.242 19.811 0.95 101.245 71.182 56.534 47.38 1.05 94.654 67.969 54.991 46.672 1.1 48.593 32.012 24.519 20.390 1.2 21.784 13.621 10.427 8.551 1.3 13.266 8.241 6.315 5.227 1.4 9.410 5.883 4.505 3.737 1.5 7.176 4.549 3.496 2.924 2 3.363 2.206 1.754 1.520 3 1.846 1.316 1.141 1.063 From Tables 5.5 – 5.10 we may conclude that: i. both floating charts are performing good, not only for positive shifts but also for negative shifts in the process standard deviation; ii. iii. for a fixed , the proposed floating  
 chart is performing better for small shifts, like @ Û >e and @ - +>P, whereas the performance of floating  
 chart is better for large shifts, like @ - >f and @ Û +>A; for the fixed values of ê and , the values of the control limit coefficients are the same for both proposed charts; iv. for large values of Z, the values for both charts are more symmetric with respect to @ as the distribution of both  and  becomes very close to normal as Z increases. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϭϯ ϭϭϰ  Table 5.9: 9:; values for the floating ö  µF chart with rqö  I> ?gG and 9:;=  IJ= 0.5 Ch 6.809  Ih 4.176  ?h 3.183  Jh 2.649  ”h 0.6 9.339 5.692 4.301 3.562 0.7 14.052 8.522 6.413 5.266 0.8 24.861 14.998 11.250 9.176 0.9 62.300 38.592 29.075 23.830 0.95 138.501 92.615 71.719 59.133 1.05 131.474 89.931 70.533 59.158 1.1 62.083 39.362 29.939 24.634 1.2 26.376 16.284 12.254 10.120 1.3 15.833 9.753 7.411 6.090 1.4 11.044 6.888 5.247 4.337 1.5 8.401 5.280 4.027 3.364 2 3.821 2.473 1.946 1.667 3 2.015 1.402 1.187 1.093 Table 5.10: 9:; values for the floating ö  µF chart with rqö  I> GHg and 9:;=  ?== Ch 0.5  Ih  ?h  Jh  ”h 7.511 4.583 3.480 2.884 0.6 10.318 6.257 4.716 3.889 0.7 15.587 9.379 7.044 5.769 0.8 27.370 16.606 12.389 10.122 0.9 70.351 43.080 32.349 26.301 0.95 161.226 105.404 80.668 66.694 1.05 155.272 103.961 80.665 66.948 1.1 70.928 44.268 33.470 27.389 1.2 29.405 18.016 13.538 11.106 1.3 17.526 10.787 8.137 6.688 1.4 12.167 7.525 5.735 4.723 1.5 9.239 5.746 4.397 3.652 2 4.125 2.654 2.085 1.771 3 2.119 1.460 1.226 1.110 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ  ͷǤʹǤ͵ ‘’ƒ”‹•‘• In this section, we compare the performance of the proposed floating charts with some recently proposed CUSUM and EWMA-type control charts for monitoring the process dispersion. The control charts selected for the comparison purpose include the
 -EWMA by Castagliola (2005), EWMA Â 
 by Castagliola et al. (2010) and CUSUM 
 by Castagliola et al. (2009) directly, while we have also compared the performance of our proposed charts with the Shewhart  chart, a CUSUM chart for process dispersion proposed by Page (1963) and an EWMA chart proposed by Crowder and Hamilton (1992), indirectly. Proposed versus µF -EWMA and EWMA Å  µF : Castagliola (2005) proposed an EWMA chart for monitoring the process dispersion based on the same logarithmic transformation as in (3.5), named as
 -EWMA. Following him, Castagliola et al. (2010) proposed another EWMA chart based on the same four parameter Johnson
ð transformation as in (5.4), named as EWMA Â 
 for controlling the process standard deviation. The two parameters of these charts are the smoothing parameter * and the control limit coefficient these two charts for the optimal choices of * and . The values of are given in Table 5.11. Comparing the performance of proposed charts with these EWMA-type charts, we can notice that both proposed charts have smaller  values for a fixed   Pd. Moreover, the proposed charts are showing more dominance for small shifts as compared to large values of @ (cf. Table 5.11vs. Tables 5.6 & 5.9). Castagliola (2005) showed in his article that the
 -EWMA control chart performs better than the Shewhart  chart for small shifts like @ - M. He also proved the dominance of his proposed chart over the CUSUM chart proposed by Page (1963) and the EWMA charts DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϭϱ ϭϭϲ  proposed by Crowder and Hamilton (1992). Therefore, we can state that the performance of our proposed charts is better than these charts too. Table 5.11: 9:; values for the µF -EWMA and EWMA Å  µF charts with 9:;=  IJ= C µF -EWMA  Ih EWMA Å  µF 0.5 10 5.6  ?h  Jh 4 3.1  ”h 9.4  Ih 5.2  ?h 3.7  Jh 2.9  ”h 0.6 13.9 8 5.7 4.5 13.6 7.7 5.5 4.3 0.7 21.3 12.6 9.1 7.1 21.2 12.4 8.9 7 0.8 40.8 23.1 17 13.5 40.8 23 16.8 13.4 0.9 130.3 68.9 48.5 38.2 126.5 68.3 48.4 38.1 0.95 289.7 184.8 137.6 110.3 274.3 179.9 135.8 109.4 1.05 173.2 142.3 115.1 96.8 195.9 148 118.1 98.8 1.1 91.9 59.8 45.2 36.9 97 61.6 46.1 37.5 1.2 36.5 22.8 17 13.8 38.8 23.5 17.4 14 1.3 20.3 12.4 9.3 7.5 21.3 13 9.6 7.7 1.4 13.4 8.1 6.1 4.9 13.8 8.4 6.3 5.1 1.5 9.8 5.8 4.4 3.6 9.8 6 4.5 3.7 2 3.9 2.3 1.8 1.5 3.8 2.3 1.8 1.5 Proposed vs. CUSUM µF : Castagliola et al. (2009) proposed a CUSUM 
 chart based on the three parameter logarithmic transformation as in (3.5). The chart with optimal parameter choices are given in Table 5.12. 8 of the CUSUM 
 Table 5.12: 9:; values for the CUSUM µF chart with 9:;=  IJ= C 0.5 0.6 0.7 0.8 0.9 0.95 1.05 1.1 1.2 1.3 1.4 1.5 2 3 10.8 15.4 24.1 44 108.9 216.9 183.3 98.6 39.5 21.7 14.1 10.2 3.8 5 5.6 8.3 13.4 25.4 68.4 154.8 145.7 64.6 24.3 13.1 8.3 5.9 2.3 7 3.8 5.7 9.4 18.2 51.1 122.9 117.5 49.2 18 9.6 6.1 4.3 1.8 9 2.9 4.4 7.3 14.3 41.3 103.1 99.5 40.3 14.5 7.7 4.9 3.5 1.5 DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ The performance of this CUSUM 
 control chart is more or less similar to that of 
 -EWMA and EWMA Â 
 charts. Comparing the performance of our proposed charts with the CUSUM 
 , we may conclude that the proposed charts are performing better than the CUSUM 
 chart for almost all values of @ (cf. Table 5.12 vs. Tables 5.6 & 5.9). Apart from the tabular comparison, Figures 5.10 and 5.11 provide the curves of the charts discussed in this subsection for an increase and a decrease, respectively, in the process dispersion. It is clear from Figures 5.10 – 5.11 that the curves of both proposed charts  are on the lower side of other curves. This shows that the proposed charts have smaller values for a fixed   Pd. In addition, both proposed charts are showing almost the same performance as their curves coincide in both figures. Figure 5.10: 9:; curves for floating °  µF , floating ö  µF , µF -EWMA, EWMA Å  µF and CUSUM µF charts for increase in the process dispersion &ůŽĂƚŝŶŐdͲ^Ϯ &ůŽĂƚŝŶŐhͲ^Ϯ ^ϮͲtD tD:Ͳ^Ϯ h^hD^Ϯ ϭϲϬ ϭϰϬ ϭϮϬ ϭϬϬ ϴϬ ϲϬ ϰϬ ϮϬ Ϭ ϭ͘Ϭϱ ϭ͘ϭ ϭ͘Ϯ ϭ͘ϯ ϭ͘ϰ ϭ͘ϱ Ϯ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϭϳ ϭϭϴ  Figure 5.11: 9:; curves for floating °  µF , floating ö  µF , µF -EWMA, EWMA Å  µF and CUSUM µF charts for decrease in the process dispersion &ůŽĂƚŝŶŐdͲ^Ϯ &ůŽĂƚŝŶŐhͲ^Ϯ ^ϮͲtD tD:Ͳ^Ϯ h^hD^Ϯ ϮϬϬ ϭϴϬ ϭϲϬ ϭϰϬ ϭϮϬ ϭϬϬ ϴϬ ϲϬ ϰϬ ϮϬ Ϭ Ϭ͘ϵϱ Ϭ͘ϵ Ϭ͘ϴ Ϭ͘ϳ Ϭ͘ϲ Ϭ͘ϱ ͷǤʹǤͶ ŽŽ—•–”ƒ–‹˜‡‡šƒ’Ž‡ For illustrating the application of the proposed charts, we generate two datasets (namely dataset 5.3 and dataset 5.4) of 25 subgroups each of size Z  B, i.e. one for an increase and the other for a decrease in the process standard deviation. For dataset 5.3, the first 15 subgroups are generated from [+ showing an in control standard deviation while the remaining 10 subgroups are generated from [+>M referring to an out of control standard deviation with @  +>M. Similarly, for dataset 5.4 the first 15 subgroups are the same as for dataset 5.3, whereas the remaining 10 subgroups are taken from [>f showing an negative shift in the process dispersion with @  >f. Both proposed charts are applied to the datasets with parameters;  £ Z  >dAf, '£ Z  >eXd, M>PXAd, £ Z  >BeXe, ê  >M and >Pe, 'ñ Z  >efBM, ñ Z q £ £ Z  >feXe, ­£ Z   P>BXf for the floating  
 chart; ñ Z   P>BAM, ­ñ Z  +>BdMd, ñ Z  >MPBM, cñ Z  DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ ++>P+M, ê  >M and  q ñ  P>BXf for the floating  
 chart. The calculations for both proposed charts with dataset 5.3 are given in Table 5.13. Figure 5.12 shows the chart output of the floating  
 chart for both datasets while the chart output of the floating  
 chart is given in Figure 5.13. Table 5.13: Calculation details of the proposed charts for dataset 5.3 Subgroup Number µFŸ °Ÿ ù°Ÿ ;n;° ön;° öŸ 3.458 -0.045 2.131 -0.998 ùöŸ -0.045 ;n;ö -3.511 ön;ö -0.521 -2.16 2.168 1.633 1 0.815 -0.079 -0.079 -3.443 2 0.363 -0.992 -0.536 -2.116 3.519 3 1.567 0.929 -0.047 -1.592 1.607 0.924 -0.04 -1.625 4 0.594 -0.482 -0.156 -1.3 1.315 -0.45 -0.142 -1.328 1.336 5 1.565 0.927 0.061 -1.111 1.126 0.922 0.071 -1.135 1.143 6 2.384 1.687 0.332 -0.977 0.992 1.654 0.335 -0.999 1.007 7 0.659 -0.356 0.234 -0.876 0.891 -0.321 0.241 -0.896 0.904 8 0.069 -1.857 -0.028 -0.797 0.812 -2.106 -0.052 -0.816 0.824 9 0.709 -0.264 -0.054 -0.734 0.749 -0.228 -0.072 -0.751 0.759 10 0.564 -0.543 -0.103 -0.681 0.696 -0.513 -0.116 -0.697 0.705 11 0.529 -0.614 -0.149 -0.637 0.651 -0.588 -0.159 -0.652 0.66 12 0.696 -0.287 -0.161 -0.598 0.613 -0.251 -0.167 -0.613 0.621 13 0.875 0.018 -0.147 -0.565 0.58 0.051 -0.15 -0.58 0.588 14 2.485 1.765 -0.01 -0.536 0.551 1.731 -0.015 -0.55 0.558 15 0.431 -0.829 -0.065 -0.511 0.526 -0.818 -0.069 -0.524 0.532 16 2.123 1.47 0.031 -0.488 0.503 1.442 0.025 -0.501 0.509 17 3.219 2.27 0.163 -0.467 0.482 2.247 0.156 -0.48 0.488 18 0.332 -1.068 0.094 -0.449 0.464 -1.085 0.087 -0.461 0.469 19 2.552 1.816 0.185 -0.432 0.447 1.782 0.176 -0.444 0.451 20 0.148 -1.592 0.096 -0.416 0.431 -1.731 0.081 -0.428 0.436 21 3.935 2.677 0.219 -0.402 0.417 2.694 0.205 -0.413 0.421 22 1.592 0.957 0.253 -0.389 0.404 0.95 0.239 -0.4 0.408 23 3.744 2.575 0.354 -0.377 0.392 2.579 0.341 -0.388 0.395 24 1.834 1.205 0.389* -0.366 0.38 1.187 0.376 -0.376 0.384 25 1.215 0.51 0.394* -0.355 0.37 0.525 0.382** -0.365 0.373 * indicates an out of control signal by floating  
 chart ** indicates an out of control signal by floating  
 chart DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϭϵ ϭϮϬ  It can be seen from Figures 5.12 – 5.13 that the proposed charts are effectively detecting both positive and negative shifts. This can also be confirmed from Table 5.13, where the floating  
 chart is signaling at subgroups # 24 and 25, while the floating  
 chart is detecting the positive shift at subgroup # 25. Figure 5.12: Chart output of floating °  µF chart for datasets 5.3 & 5.4 ĚĂƚĂƐĞƚϱ͘ϯ ĚĂƚĂƐĞƚϱ͘ϰ ŽŶƚƌŽů>ŝŵŝƚƐ ϰ ϯ Ϯ ϭ Ϭ Ͳϭ ͲϮ Ͳϯ Ͳϰ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ Ϯϭ Ϯϯ Ϯϱ ^ĂŵƉůĞEƵŵďĞƌ Figure 5.13: Chart output of floating ö  µF chart for datasets 5.3 & 5.4 ĚĂƚĂƐĞƚϱ͘ϯ ĚĂƚĂƐĞƚϱ͘ϰ ŽŶƚƌŽů>ŝŵŝƚƐ ϰ ϯ Ϯ ϭ Ϭ Ͳϭ ͲϮ Ͳϯ Ͳϰ ϭ ϯ ϱ ϳ ϵ ϭϭ ϭϯ ϭϱ ϭϳ ^ĂŵƉůĞEƵŵďĞƌ DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϵ Ϯϭ Ϯϯ Ϯϱ WƌŽŐƌĞƐƐŝǀĞĐŽŶƚƌŽůĐŚĂƌƚŝŶŐ  ͷǤ͵ ‘ Ž—†‹‰”‡ƒ”• To monitor the location of a process two main types of charts, named as Shewharttype control charts and memory control charts (like EWMA and CUSUM) are available. Former are recommended for large shifts P - @ - B while latter are good at detecting small and moderate shifts (like >MB - @ - +>B). In this chapter, we have proposed another type of memory control chart, the so called 7 control chart. The performance of the new chart is evaluated in terms of 8 and we have compared the performance of proposed control chart with different existing memory control charts. Comparisons revealed that the newly proposed chart performs very good and outperforms its competitors for small and moderate shifts but also shows good performance for large shifts. Following the 7 chart for location, two memory-type control charts for process dispersion, named as the floating  
 control chart (based on a three parameter logarithmic transformation) and the floating  
 control chart (based on a four parameter Johnson
ð transformation) are proposed. The performance evaluation of the proposed charts is done by calculating the values using simulation procedures. These 8 are compared with some EWMA- and CUSUM-type control charts for monitoring the process standard deviation. The comparisons show that the proposed charts are dominating the other charts in terms of values. Moreover, an inter-proposed charts comparison shows that the floating  
 chart is better for small shifts, whereas the floating  
 chart is superior for large shifts in the process dispersion. At the end, an illustrative example is provided which shows the application of the proposed charts on simulated datasets. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϮϭ ϭϮϮ  ’’‡†‹šͷǤͳ y=c(); PM=c(); ucl=c(); lcl=c(); rl=c() mu=0; sig=1 p=0.2; K=3.129 for(j in 1:100000) { for(i in 1:1000000) { y[i]=rnorm(1,mu,sig) PM[i]=mean(y[1:i]) ucl[i]=mu+K*(sig/i^(0.5+p)) lcl[i]=mu-K*(sig/i^(0.5+p)) if(PM[i]>ucl[i]|PM[i]<lcl[i]) {rl[j]=i;break;} else{rl[j]=0;} } } mean(rl)   ’’‡†‹šͷǤʹ rl=c(); Tj=c(); FT=c(); ucl=c(); lcl=c() n=5; a=-0.8969; b=2.3647; c=0.5979; ETj=0.00748; STj=0.9670 p=0.2; K=3.129 mu=0; sig=1 for(j in 1:100000) { for(i in 1:1000000) { x=rnorm(n,mu,sig) Tj[i]=a+b*log(var(x)+c,exp(1)) FT[i]=mean(Tj[1:i]) ucl[i]=ETj+K*(STj/i^(0.5+p)) lcl[i]=ETj-K*(STj/i^(0.5+p)) if(FT[i]>ucl[i]|FT[i]<lcl[i]) {rl[j]=i; break;} else{rl[j]=0;} } } mean(rl) DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZĞĨĞƌĞŶĐĞƐ  ‡ˆ‡”‡ ‡• Abbas, N., Riaz, M. and Does, R.J.M.M. (2011). Enhancing the Performance of EWMA Charts. Quality and Reliability Engineering International, 27(6), 821 – 833. Abbas, N., Riaz, M. and Does, R.J.M.M. (2012a). Mixed Exponentially Weighted Moving Average – Cumulative Sum charts for Process Monitoring. Quality and Reliability Engineering International, DOI: 10.1002/qre.1385. Abbas, N., Riaz, M. and Does, R.J.M.M. (2012b). CS-EWMA Chart for Monitoring Process Dispersion. Quality and Reliability Engineering International, DOI: 10.1002/qre.1414. Abbas, N., Riaz, M. and Does, R.J.M.M. (2012c). An EWMA-type Control Chart for Monitoring the Process Mean Using Auxiliary Information. Accepted for publication in Communications in Statistics – Theory and Methods. Abbas, N., Riaz, M. and Does, R.J.M.M. (2012d). New Memory-type Control Charts for Monitoring the Process Dispersion. Submitted to International Journal of Production Research. Abbas, N., Zafar, R.F., Riaz, M. and Hussain, Z. (2012). Progressive Mean Control Chart for Monitoring Location Parameter. Quality and Reliability Engineering International, DOI: 10.1002/qre.1386. Acosta-Mejia, C., Pigniatiello, J. and Rao, B. (1999). A comparison of Control Charting Procedures for Monitoring Process Dispersion. IIE Transactions, 31(6), 569-579. Antzoulakos, D.L. and Rakitzis, A.C. (2008). The Modified r out of m Control Chart. Communications in Statistics – Theory and Methods, 37(2), 396-408. Bonetti, P.O., Waeckerlin, A., Schuepfer, G. and Frutiger, A. (2000). Improving TimeSensitive Processes in the Intensive Care Unit: the Example of ‘Door-to-Needle Time’ in Acute Myocardial Infarction. International Journal for Quality in Health Care, 12(4), 311317. Brook, D. and Evans, D.A. (1972). An Approach to the Probability Distribution of CUSUM Run Length. Biometrika, 59(3), 539 - 549. Capizzi, G.and Masarotto, G. (2003). An Adaptive Exponentially Weighted Moving Average Control Chart. Technometrics, 45(3), 199-207. Castagliola, P. (2005). A New S2-EWMA Control Chart for Monitoring Process Variance. Quality and Reliability Engineering International, 21(8), 781-794. Castagliola, P., Celano, G. and Fichera, S. (2009). A New CUSUM-S2 Control Chart for Monitoring the Process Variance. Journal of Quality in Maintenance Engineering, 15(4), 344-357. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϮϯ ϭϮϰ  Castagliola, P., Celano, G. and Fichera, S. (2010). A Johnson's Type Transformation EWMAS2 Control Chart. International Journal of Quality Engineering and Technology, 1(3), 253275. Chang, T.C. and Gan, F.F. (1995). A Cumulative Sum Control Chart for Monitoring Process Variance. Journal of Quality Technology, 27(2), 109-119. Chatterjee, S. and Qiu, P. (2009). Distribution-Free Cumulative Sum Control Charts Using Bootstrap-Based Control Limits. Annals of Applied Statistics, 3(1), 349-369. Cochran, W.G. (1977). Sampling Techniques. 3rd ed. New York: John Wiley and Sons. Crowder, S.V. and Hamilton, M.D. (1992). An EWMA for Monitoring a Process Standard Deviation. Journal of Quality Technology, 24(1), 12-21. Ewan, W.D. and Kemp, K.W. (1960). Sampling inspection of continuous processes with no autocorrelation between successive results. Biometrika, 47(3-4), 363-380. Fuller, W.A. (2002). Regression Estimation for Survey Samples. Survey Methodology, 28(1), 5-23. Fuller, W.A. (2009). Sampling Statistics. John Wiley and Sons Inc., Hoboken, New Jersey. Garcia, M.R. and Cebrian, A.A. (1996). Repeated Substitution Method: The Ratio Estimator for the Population Variance. Metrika, 43(1), 101-105. Hawkins, D.M. (1981). A CUSUM for a Scale Parameter. Journal of Quality Technology, 13(4), 228-231. Hawkins, D.M. and Olwell, D.H. (1998). Cumulative Sum Charts and Charting Improvement. Springer, New York. Huwang, L., Huang, C. J. and Wang, Y.H.T. (2010). New EWMA Control Charts for Monitoring Process Dispersion. Computational Statistics and Data Analysis, 54(10), 23282342. Jiang, W., Shu, L. and Aplet, D.W. (2008). Adaptive CUSUM Procedures with EWMABased Shift Estimators. IIE Transactions, 40(10), 992-1003. Kadilar, C. and Cingi, H. (2005). A New Estimator Using Two Auxiliary Variables. Applied Mathematics and Computation, 162(2), 901–908. Khoo, M.B.C. (2004). Design of Runs Rules Schemes. Quality Engineering, 16(1), 27–43. Klein, M. (2000). Two Alternatives to the Shewhart X Control Chart. Journal of Quality Technology, 32(4), 427–431. Koutras, M.V., Bersimis, S. and Maravelakis, P.E. (2007). Statistical Process Control using Shewhart Control Charts with Supplementary Runs Rules. Methodology and Computing in Applied Probability, 9(2), 207–224. Lowry, C.A., Woodall, W.H., Champ, C.W. and Rigdon, S.E. (1992). A Multivariate Exponentially Weighted Moving Average Control Chart. Technometrics, 34(1), 46 – 53. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ZĞĨĞƌĞŶĐĞƐ  Lucas, J.M. (1982). Combined Shewhart-CUSUM Quality Control Schemes. Journal of Quality Technology, 14(2), 51-59. Lucas, J.M. and Crosier, R.B. (1982). Fast Initial Response for CUSUM Quality-Control Scheme. Technometrics, 24(3), 199-205. Lucas J.M. and Saccucci M.S. (1990). Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics, 32(1), 1–12. Mandel, B.J. (1969). The Regression Control Chart. Journal of Quality Technology, 1(1), 1-9. McIntyre, G.A. (1952). A Method for Unbiased Selective Sampling, Using Ranked Sets. Australian Journal of Agricultural Research, 3(4), 385-390. Montgomery, D.C. (2009). Introduction to Statistical Quality Control. 6th ed. New York: John Wiley & Sons. Nelson, L.S., (1984), The Shewhart Control Chart – Tests for Special Causes, Journal of Quality Technology, 16(4), 237-239 Ng, C.H. and Case, K.E. (1989). Development and Evaluation of Control Charts Using Exponentially Weighted Moving Averages. Journal of Quality Technology, 21(4), 242-250. Page, E.S. (1954). Continuous Inspection Schemes. Biometrika, 41(1-2), 100-115. Page, E.S. (1963). Controlling the Standard Deviation and Warning Lines by CUSUM. Technometrics, 5(3), 307-309. Palm, A.C. (1990). Tables of Run Length Percentiles for Determining the Sensitivity of Shewhart Control Charts for Averages with Supplementary Runs Rules. Journal of Quality Technology, 22(4), 289–298. Riaz, M. (2008a). Monitoring Process Computational Statistics, 23(2), 253-276. Variability using Auxiliary Information. Riaz, M. (2008b). Monitoring Process Mean Level using Auxiliary Information. Statistica Neerlandica, 62(4), 458-481. Riaz, M., Abbas, N. and Does, R.J.M.M. (2011). Improving the Performance of CUSUM Charts. Quality and Reliability Engineering International, 27(4), 415 – 424. Riaz, M. and R.J.M.M. Does (2009). A process variability control chart. Computational Statistics, 24(2), 345–368. Roberts, S.W. (1959), Control Chart Tests Based on Geometric Moving Averages, Technometrics, 1(3), 239-250. Shewhart, W.A. (1931). Economic Control of Quality of Manufactured Product, New York: 1931. Reprinted by ASQC, Milwaukee, 1980. Shmueli, G. and Cohen, A. (2003). Run-Length Distribution for Control Charts with Runs and Scans Rules. Communications in Statistics – Theory and Methods, 32(2), 475–495. Shu, L.J. and Jiang, W. (2008). A New EWMA Chart for Monitoring Process Dispersion. Journal of Quality Technology, 40(3), 319-331. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϮϱ ϭϮϲ  Siegmund, D.O. (1985). Sequential Analysis: Test and Confidence Intervals. New York: Springer-Verlag. Steiner, S.H. (1999). EWMA control charts with time varying control limits and fast initial response. Journal of Quality Technology, 31(1), 75 – 86. Tuprah, K. and Ncube, M. (1987). A Comparison of Dispersion Quality Control charts. Sequential Analysis, 6(2), 155-163. Wade, M.R. and Woodall, W.H. (1993). A Review and Analysis of Cause-Selecting Control Charts. Journal of Quality Technology, 25(3), 161-169. Waldmann, K.H. (1995). Design of double CUSUM quality control schemes. European Journal of Operational Research, 95(3), 641-648. Westgard, J.O., Groth, T., Aronsson, T. and De Verdier. C.H. (1977). Combined ShewhartCUSUM Control Chart for Improved Quality Control in Clinical Chemistry. Clinical Chemistry, 23(10), 1881-1887. Wortham, A.W. and Ringer, L.J. (1971). Control via Exponentially Smoothing. The Logistics and Transportation Review, 7(1), 33-39. Wu, Z. and Tian, Y. (2005). Weighted-loss-Function CUSUM chart for Monitoring Mean and Variance of a Production Process. International Journal of Production Research, 43(14), 3027–3044. Yashchin, E. (1989). Weighted Cumulative Sum Technique. Technometrics, 31(1), 321-338. Zhang, G.X. (1985). Cause-Selecting Control Charts-A New Type of Quality Control Charts. The QR Journal, 12, 221-225. Zhang, G. and Chang, S.I. (2008). Multivariate EWMA Control Charts Using Individual Observations for Process Mean and Variance Monitoring and Diagnosis. International Journal of Production Research, 46(24), 6855–6881. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ^ĂŵĞŶǀĂƚƚŝŶŐ  ƒ‡˜ƒ––‹‰ Het doel van statistische procesbeheersing is om een procesvoering te krijgen die geschoeid is op een kwantitatieve leest. De regelkaart is het belangrijkste gereedschap van statistische procesbeheersing en werd in 1924 door Shewhart geïntroduceerd in het bedrijfsleven. Het is een grafiek van metingen van een kwaliteitskarakteristiek van het proces op de verticale as uitgezet tegen de tijd op de horizontale as. De grafiek wordt aangevuld met regelgrenzen die de procesinherente variatie markeren. Zodra een meting buiten de regelgrenzen valt dan noemen we het proces niet beheerst. In de loop der jaren is hier veel onderzoek naar gedaan. In begin vijftiger jaren van de vorige eeuw ontwikkelde Page een nieuw type regelkaart: de gecumuleerde som regelkaart, beter bekend geworden als de CUSUM regelkaart. Kort daarna, in 1959, presenteerde Roberts de exponentieel gewogen voortschrijdend gemiddelde regelkaart, die bekend werd onder de naam van EWMA regelkaart. Beide regelkaarten gebruiken naast de laatst gemeten waarneming ook de voorafgaande waarnemingen om te oordelen of het proces beheerst is. In die zin hebben deze regelkaarten een geheugen. In hoofdstuk 1 worden beide regelkaarten geïntroduceerd. De standaard signaleringsregel is dat als een waarneming buiten de regelgrenzen valt, het proces niet beheerst is. Door vervanging van de regelgrenzen door waarschuwings- en/of actiegrenzen kunnen ook patronen van opeenvolgende waarnemingen als signaleringsregels gebruikt worden. In hoofdstuk 2 worden deze aanpassingen gebruikt om de CUSUM - en EWMA regelkaarten tot snellere signalering te laten komen bij met name kleine verschuivingen in het gemiddelde. Hiervoor wordt als criterium genomen de gemiddelde run DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ϭϮϳ ϭϮϴ  lengte, waarbij als definitie van de run lengte gebruikt wordt: het aantal opeenvolgende waarnemingen dat nodig is om tot een signaal te komen. Het blijkt dat de aanpassingen in hoofdstuk 2 tot substantiële verbetering leiden voor het sneller signaleren van kleine verschuivingen in het gemiddelde. In hoofdstuk 3 worden de CUSUM – en EWMA regelkaarten gecombineerd tot een gemeenschappelijke regelkaart. Deze gecombineerde regelkaart wordt ontwikkeld voor zowel het monitoren van de locatie als van de spreiding. Vergelijkingen op basis van de gemiddelde run lengten worden gemaakt met bestaande regelkaarten. Het blijkt dat de gecombineerde regelkaart tot betere resultaten leidt. Een betere beheersing van de procesparameters kan ook verkregen worden door gebruik te maken van aanvullende informatie uit de data. Door het benutten van de correlatie tussen de kwaliteitskarakteristiek en de aanvullende kenmerken, zijn nieuwe CUSUM – en EWMA regelkaarten ontwikkeld. In hoofdstuk 4 worden deze nieuwe regelkaarten vergeleken met bestaande regelkaarten die gebruikt worden voor hetzelfde doel. Het blijkt dat aanvullende informatie de prestaties van de regelkaarten substantieel verbetert. Tot slot wordt in het laatste hoofdstuk een nieuw type regelkaart voorgesteld die wordt gebaseerd op het progressief voortschrijdend gemiddelde. Deze regelkaart wordt zowel voor de locatie parameter als voor de spreidingsparameter ontwikkeld. De prestaties, wederom in termen van de gemiddelde run lengte, blijken in de praktijk uitstekend te zijn.     DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů ƵƌƌŝĐƵůƵŵsŝƚĂĞ  —””‹ —Ž—‹–ƒ‡ Nasir Abbas was born in Rawalpindi (Pakistan) on 21st June, 1987. He completed his schooling in 2003 from F.G. Secondary School, Mehfooz Road, Rawalpindi, Pakistan. He earned his F.Sc. and B.Sc in 2005 and 2007, respectively, from the F.G. Sir Syed College, The Mall, Rawalpindi, Pakistan with the major subjects as Mathematics, Statistics and Physics at F.Sc. level, and Mathematics, Statistics and Economics as B.Sc. level. He got his M.Sc. in Statistics from the Department of Statistics Quaid-i-Azam University Islamabad Pakistan in 2009 where he secured the top position in his department and M.Phil in Statistics from the Department of Statistics Quaid-i-Azam University Islamabad Pakistan in 2011. Additionally, he is also serving as Assistant Census Commissioner in Pakistan Bureau of Statistics from July 2011-present. His current research interests include Statistical Quality Control particularly control charting based on the classical, Bayesian as well as nonparametric methodologies. DĞŵŽƌLJͲƚLJƉĞŽŶƚƌŽůŚĂƌƚƐŝŶ^ƚĂƚŝƐƚŝĐĂůWƌŽĐĞƐƐŽŶƚƌŽů View publication stats ϭϮϵ