Advanced monitoring techniques of statistical process control for normal and non-normal distributed processes

Abstract

This study investigates efficient computational methods for designing and evaluation phases of the Shewhart type control charts under runs rules. The efficient computational methods considered are exact equations or formulas for computing the probability of single point and run length properties of control charts. Properties of run length include average run length, variance of run length, standard deviation of run length, coefficient of variation of run length, and moments of run length. Issues in control charts can be handled by a generalized skewness correction structure that depends on the amount of skewness instead of the assumption of normality. However, one of the limitations of existing control charts is that actual false alarm rate deviates severely from intended level when probability distribution is unknowingly skewed and/or limited number of samples are used for estimation purposes. To handle the situation when inspection units are selected under ranked set schemes, multivariate control charts are proposed under bivariate ranked set schemes. In addition, a numerical technique is employed for computing power and average run length of mean control chart under ranked set schemes instead of involving Monte Carlo simulation procedures. Besides this, a comparative analysis between false alarm rate based control charts and average run length based control charts with runs rules are conducted. The proposed method is demonstrated on the following applications: generalized skewness correction structure to monitor the chemical properties, Hotelling’s T2 and cumulative sum control chart to monitor the quality of irrigation water, as well as mean control charts to monitor the quality of petrochemical process and groundwater. There are several interesting findings from this study. These include the following outcomes: the current study are interesting because exact equations for computing the probability of single point and run length properties are considered as an alternative to Markov Chain approach for similar purposes; proposed skewness correction structure of mean outperformed the existing mean control charts when process parameters and probability distribution are unknown; numerical method for computing the power and average run length of mean control chart under ranked set schemes is found more time efficient than existing methods based on Monte Carlo simulation; multivariate control charts under bivariate ranked set schemes are found more proficient than existing multivariate control charts under simple random sampling; performance order of runs rules with false alarm rate based control charts are persistent, whereas performance order of runs rules with average run length based control charts are dependent on the circumstances, that is, sample size, size of variation, type of control chart, and side of control limit (upper-sided and lower-sided). For the real data applications, cumulative sum control chart performs outstandingly in detecting small variations in calciummagnesium and residual sodium contents of irrigation water. Likewise, skewness correction structure has been proven to be excellent in monitoring product purity.