ON EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART FOR TIME TRUNCATED LIFE TEST USING INVERSE WEIBULL DISTRIBUTION

Statistical control charts are widely used in industry for process and measurement control to monitor production processes in order to discover any problem or issues that may arise during the production process and to help in finding solutions for these issues. This research proposes the exponentially weighted moving average (EWMA) control chart, comparing it’s performance to NP and MA control chart for monitoring the number of nonconforming with the truncated lifetime of a product when the lifetimes of products follows inverse Weibull (IW) distribution. The number of failures during the life test is used as an indicator of the quality of the product. EWMA control chart was proposed for this specific situation, thus extending the applicability of control charts methodology to situations involving truncated life tests. The performance of these control charts was measured with the average run length (ARL). The simulation shows that the results obtained from the EWMA chart using IW distribution is more sensitive in detecting small shift and performs much better to monitoring shifts as when compared to other control charts considered in the study.


INTRODUCTION
To ensure the quality standards of products, every industry must develop some mechanism by adopting suitable statistical quality control techniques and procedures, Walter Shewhart introduced the idea of a control chart during 1920's to monitor the quality of the product, which is still popular and being used with some modification to eliminate abnormal variations through separation of variations caused by special causes of variation or common causes.Montgomery(2008).The process is deemed out-of control if the points fall outside the control limits that are calculated based on a mathematical formula or if a nonrandom pattern is detected, Baklizi and Ghannam (2022).The attribute control charts are particularly important in non-manufacturing quality improvement efforts in which the targeted quality characteristics are impossible to measure on a numerical scale, Montgomery (2009).The number of defects in an item produced by a manufacturing process is common to monitor for improving the quality of the product, Alghamdiet. al (2017).It assigns weight to the observations in geometrically decreasing order so that the most recent observations have higher contribution, while the contribution of the oldest ones is very little.Control charts are a fundamental tool of statistical process control (SPC), control charts are now widely used, not only in industry, but also in many other areas with real applications, such as health care manufacturing processes, environmental sciences etc. Shewhart (1931), developed the first control chart considered as the main tool of SPC using statistical principles in generating.It is sometimes called the Shewhart chart, which is a chart using the data of the previous production process to scatter a plot and consider the production process.Thus, the pattern of the scatter plot cannot be seen if the production process does not change significantly.For this reason, the Shewhart chart is good at detecting larger shifts in the process.Khoo (2004), studied the Moving average (MA) control chart for detecting the fraction of non-conforming observations and showed that the Moving average (MA) chart had better efficiency than the p chart.Roberts (1959), invented a control chart that could detect changes in the production process, even if the change was only slight, called the cumulative sum (CUSUM) chart and the exponentially weighted moving average (EWMA) chart, which makes them more sensitive than Shewhart chart for detecting the smaller and moderate shifts in the process.Many authors such as focus on the designing of Exponential weighted moving average(EWMA) and moving average(MA) charts for various situations.Abbas et. al(2012), proposed the Exponential weighted moving average cumulative sum(EWMA-CUSUM) charts for monitoring correlated data using the Average Run Length (ARL), extra quadratic loss, and relative Average Run Length (ARL) as criteria to measure the efficiency with Shewhart, CUSUM, EWMA, Shewhart-CUSUM, and Shewhart-EWMA charts.The newly proposed control charts have efficiency in detecting better than the compared charts, Zaman et.al (2014), proposed the CUSUM EWMA chart to detect the change of variation in the process using the ARL, extra quadratic loss, and relative Average Run Length as criteria to measure the efficiency with Shewhart, EWMA and CUSUM charts. Johnston (1993), Carson and Yeh (2008), shows how to modify the smoothing constant for use in an Exponentially Weighted Moving Average system(EWMA), when data has to be combined either across a number of periods, or is available from only a fraction of a normal forecast review interval.Lal and Kane (2018), To the best of our knowledge, there are no studies about the on exponentially weighted moving average control chart for time truncated life test using inverse weibull distribution.The specific structure of this paper is as follows.Section1 introduction, Section 2 introduces inverse weibull distribution and the control chart method of the study, 3 results and discussion, Section 4. Result and discussion, section 5 Conclusion.

MATERIAL AND METHODOLOGY Inverse Weibull Distribution and Control Chart
The Inverse Weibull distribution is a continuous distribution which deals with extreme events.The Inverse Weibull (IW) distribution can be readily applied to modeling processes in reliability, ecology, medicine, branching processes and biological studies.Youlong Wu et al. (2020), contributed that to monitor the failure time of systems or components, one can utilize LCL and UCL.If the time between failures plotted on the chart is below the LCL, it is an indication of increasing failure rate or system deterioration.If the plotted time is above the UCL, it shows improvement in the process.The properties and applications of IW distribution in several areas can be seen in the literature (Keller, et al (1985), Calabria and Pulcini (1989), (1990), (1994), Johnson (1984), Khan et al. (2008).A random variable X has an Inverse Weibull distribution with shape parameter α and scale parameter β having the Probability Density Function (PDF) given as; (1) and the Cumulative Distribution Function is given as; (2) Where α and β represents shape and scale parameter respectively.As mentioned in Aslam and Jun (2015), the shape parameter and scale parameter could be assumed to be known in practice.The shape and scale parameter can also be estimated from the data if it is unknown but in this case it is assumed to be known as  is 12.091 and  is 17.637.This assumption may follow from engineering experience we may use estimators from previous studies as  represents the life time of the products, Baklizi and Ghannam (2022).The average life time of the product is given as follows  = Γ (1 − 1  ) (3) where (. ) is the gamma function.Let µ 0 be the target mean life when the process is in control.We would like to design a control chart for monitoring the mean shift by observing the number of failed products by the specified time  0 (truncated time).It should be noted that the subscript 0 indicates the in-control process or the specified value.

Moving Average
The moving average statistic of size w at time i for the number of failures (Doi's) is computed by For the Out of Control

Exponential Weighted Moving Average
The EWMA -Exponentially Weighted Moving Average chart is used in statistical process control to monitor variables (or attributes that act like variables) that make use of the entire history of a given output.This is different from other control charts that tend to treat each data point individually.
25) Which can be recursively written as Therefore, the control limits of the EWMA control chart can be given as Where  2 the coefficient of control limit of EWMA control chart is,  0 is the mean of the process and variance is  2  ̅ .

Average Run Length
The ARL may be defined as the average number of samples to be plotted before the process indicates an out-of-control signal C. Chananet, (2014).

RESULT AND DISCUSSION
The descriptive statistics of the variables, the defectives of 90 samples, Exponential Moving Average data at  0.3, 0.2 and 0.1, Moving Average data at weight of 3 and 5 are presented.Goodness of fit tests, estimation of the parameters with Maximum Likelihood, estimation of the LCL and UCL of the variables with in-control and out of control test will be carried out in this section.

Discussion
From the ARL tables above, we observe that the out-ofcontrol ARLs are small when there is a small shift in the mean and the values increase as the mean shift increases.We noted that the simulated values of the ARL are very close to the target values (370) when the shift parameter c = 1.EWMA chart detects faster than the other charts after evaluating the performance using average run length.

CONCLUSION
A moving average (MA) and NP control charts was observed to be less effective than the proposed exponential weighted moving average (EWMA), when monitoring the number of failures under a time-truncated life test when the life of an item follows an Inverse Weibull distribution.It was observed that the EWMA control chart will be an efficient addition to the toolkit of the quality control practitioners when working on inverse Weibull distribution as it was noted to detect small process shifts more quickly than the other charts.
It is shown that the EWMA control chart outperforms the NP and MA control chart for all shift parameters after evaluating with the ARL.The result shows that the value of ARL decreases as n increases from 20 to 90 and there was also a rapid decrease as c decreases from 1.0 to 0.1.

Table : 1
shows the variable, sample size, mean, standard deviation, skewness and kurtosis of the data.