Abstract
AbstractMeasurement of control chart efficiency by comparison of average run length (ARL) is widely implemented in quality control. The aim of this study is to evaluate the ARL, which is a solution of the integral equation obtained from the Exponential Weighted Moving Average (EWMA) statistic with a long-memory Autoregressive Fractionally Integrated Moving Average (ARFIMA) process. The derivation of the analytical ARL of the EWMA control chart and proof of the existence and uniqueness of the analytical ARL by Fixed Point theory are shown. Moreover, the numerical ARL carried out by the Composite Midpoint Rule technique of the EWMA control chart is demonstrated. A comparison between the analytical and numerical ARL is also illustrated. The findings indicated that analytical ARL of the EWMA control chart is more quickly computational than the numerical ARL. Therefore, the analytical ARL is an alternative method for measuring the efficiency of the EWMA control chart with the long-memory ARFIMA process.
Highlights
Entrepreneurs need to modernize their business processes in order to achieve competitive advantages that are the lowest price and highest quality
Using Equations (8) and (15) to evaluate the average run length (ARL) of the Exponential Weighted Moving Average (EWMA) control chart of the long-memory Autoregressive Fractionally Integrated Moving Average (ARFIMA) process with exponential white noise and the shift parameter β = β0(1 + δ), where δ is shift size, the numerical value of the analytical ARL, numerical ARL, and Absolute Percentage Relative Error (APRE) are shown as Tables 1 and 2
The research indicated that the Numerical Integral Equation (NIE) method shows good agreement with the exact solution of the analytical ARL of the integral equation because the numerical ARL converges to the analytical ARL when the shift size increases
Summary
The statistical quality control plays important roles as an effective technique for controlling production process. Several products from the process are controlled by the statistical quality control chart before delivering to the customers. The gathered data are often formed by the time series, so quality control charts such as the EWMA control chart are developed for detecting the defective points on the time series data in order to be the great quality and reliable products. This is the advantages in application of the statistical quality control to solve the problems in real situations
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