ARL Calculation of EWMA Control Charts by using Markov Chain

 

 

Method 

There are many methods for quality control monitoring. Some of the, can be counted that we have focused on in during the course of this class are: EWMA, CUSUM and Shewhart charts.

Among those, EWMA is preferred with its non-normality robustness and being more sensitive to small shifts. EWMA studies begin with Roberts (1959) and Roberts (1966) find similarities of CUSUM and EWMA charts for detecting small shifts better than Shewhart control chart.

Then box, Jenkins (1974) , Hunter (1986), Crowder (1989), Runger (1999) and Testik (2003, 2008) contributed a lot regarding the designing EWMA chart. 

 

ARL can be calculated based on Markov chains after Brook and Evans published related paper. (1972) We will base on EWMA chart how ARL is calculated. Basically procedure is to divide the control interval between UCL and LCL into plenty of states. Each state is considered as a row in the Markov chain. 

 

Results 

 

It can be concluded from the results that as Lambda is smaller it is more sensitive to small shifts. For small values of lambda small shifts can be detected with very high performance comparing to Shewhart and high lambda-EWMA charts.

At high deviations all charts perform similar performance levels, where high lambda reacts a bit more quickly comparing smaller lambdas. Because, it is more close to Shewhart chart that takes individual values and can react quickly to high deviations from mean.