ASIAN JOURNAL OF MANAGEMENT RESEARCH Online Open Access publishing platform for Management Research © Copyright 2010 All rights reserved Integrated Publishing association
Research Article
ISSN 2229 – 3795
Exponentially weighted moving average control chart
Kalgonda A.A 1 , Koshti.V.V 2 , Ashokan. K. V 3 1 Department of Statistics, New College, Kolhapur, Maharashtra 2 Department of Statistics, P. V. P. College, Kavathe Mahankal, Sangli, Maharashtra 3Department of Biological science, P. V. P. College, Kavathe Mahankal, Sangli, Maharashtra
[email protected]
ABSTRACT The exponentially weighted moving average (EWMA) control chart was introduced by Roberts in 1959, which is a good alternative to the Shewhart control chart when one is interested in small shifts. Several studies were made for the properties of ARL of EWMA control chart. Roberts (1959), using simulation developed monographs of ARL s for normally distributed observations. Robinson and Ho (1978) used a numeric procedure to determine the ARL, presenting several combinations of L and λ for change in the process mean with the help of an Edgeworth series expansion. Crowder (1987, 1989), presented tables for ARL of the EWMA chart, by solving a system of integral equations. Crowder (1987) has given a computer program that calculates the ARL of the EWMA chart for controlling the mean of a normal process. Lucas and Saccucci (1990) presented table and graph of ARL values for different values of L and λ. They have evaluated the run length properties of EWMA control schemes by representing the EWMA statistic as a continuous Markov chain. In the present paper, simulation is carried out to calculate the ARL values using Cprograms. Observing these values it is seen that approximately the same values of ARL are obtained by simulation method using C programming. That is, the Markov chain approach by Lucas, Saccucci and the present simulation technique yields the same ARL results. Keywords: Average run length, CUSUM control chart, EWMA control chart, Statistical process control, simulation. 1. Introduction The control charts are classified according whether to use or not to use the past values of control statistic. Shewhart control charts are based on the information about the process contained in the current observation only and it ignores any information given by the entire sequence of points. Hence Shewhart control chart is classified as control chart without memory. Consequently, Shewhart control chart are found to be less sensitive in detecting smaller shifts, particularly smaller than 1.5 times standard deviation. (Montgomery 2001).When the small shifts are of interest, the effective alternatives to Shewhart control chart are Cumulative Sum (CUSUM) control chart and Exponentially Weighted Moving Average (EWMA) control chart. Both these charts are based on memory and perform better than Shewhart chart while detecting smaller shifts. In these charts, information from the past samples are cumulated up to the current sample ASIAN JOURNAL OF MANAGEMENT RESEARCH Volume 2 Issue 1, 2011
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Exponentially weighted moving average control chart Kalgonda A.A, Koshti.V.V, Ashokan. K. V
and then the decision about the process is taken. Cumulative sum (CUSUM) control chart was first proposed by Page (1954) for detecting small shifts in the process mean. Some authors namely Duncan (1974), Lucas (1976), Hawkins (1990) stated that the CUSUM control chart is much more efficient than the usual X control chart for detecting smaller variations in the average. Robert (1959) introduced the EWMA control chart along with its ARL properties using simulation. Further it has been shown that the EWMA chart is useful for detecting small shifts in the process mean. Other authors namely, Crowder (1987, 1989), Lucas and Saccucci (1990) presented EWMA chart as a better alternative to detect smaller changes in the process average. 2. Control Chart with Memory In the Shewhart control chart, the control statistic is always a function g(x) of the present sample say X1 , X2 , , Xn. There are good arguments in favor of using the result of previous samples. If the control statistic of the control chart considers all the previous sample information, then the control chart is said to have an unlimited length. In moving average control chart; the test statistic only regards the last k samples (k>1). Such control chart has memory of limited length. If k=1, that is only most recent sample unit is considered, the control chart is said to be without memory. The control charts with memory are further subdivided into two categories according to uniform and non uniform memory. The chart with uniform memory, assigns the equal weights. In charts with non uniform memory, weights go on decreasing as the observation proceeds. In this situation; the general form of the control statistic is given below. Let j be the time point, which takes integer values 1,2, ….,t. The first sampling point is denoted by j =1 and the most recent sampling point by j = t. The observation vector Xj be defined as, Xj = ( Xj1, Xj2, ……….. , Xjn) , j = 1,2,……..,t. Where for every time point j, Xj1, Xj2, ……….. , Xjn is the sample subgroup. Let g(Xj) be a suitable statistic which can be used in control chart without memory.( Mittag H.J. and Rinne H., (1993)). Define general linear scalar test statistic as, t
Yt = a t + å b j g( Xj ) ( 1.1 ) i=1
Where a t and b 1 , b 2,………….., bt are constants to be chosen . The performance of memory control chart depends on the choice of the coefficients at and b1,b2,………….., bt. The various special cases of the control statistic in ( 1.1) for selected values of at and b1, b2,………….., bt are considered as below. Case ( i ) : By setting a t = 0 and b j = 1 , j = t. 0 , j = 1,2,………,t1. ( 1.1 ) becomes, Yt = g( Xt ) ( 1.2 ). This gives the test statistic for Shewhart control chart. If the function g(Xt ) is sample mean of the sample drawn at time t, then one gets Shewhart X chart. Case (ii): Let Y0 is given target value for process mean or dispersion. Substituting a t = t Y0 and b j = 1 for j = 1,2,….,t in equation ( 1.1 ) gives the control statistic of CUSUM chart as , ASIAN JOURNAL OF MANAGEMENT RESEARCH Volume 2 Issue 1, 2011
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Exponentially weighted moving average control chart Kalgonda A.A, Koshti.V.V, Ashokan. K. V t
Yt = t Y0 +
t
å i=1
b j g( Xj ) = å [ b j g( Xj ) Y0 ] ( 1.3 ). i=1
That is the cumulative sum of the deviations from target value Y0, where accumulation procedure to all preceding samples. Since the coefficients bj’ s are constants, CUSUM chart has uniform memory. If g( Xj ) = X , then such CUSUM chart can be used to monitor process mean. Case (iii ) : The EWMA control chart operates with recursively defined control statistic, Yt = (1 l ) Yt1 + l g( Xt) , 0< l £ 1 …….. ( 1.4 ). Where Y0 is given target value for process mean or dispersion. This control statistic is an exponentially weighted moving average of all samples drawn until time t. By reverse substitution, Y1 = (1 l ) Y0 + l g( X1) Y2 = (1 l ) Y1 + l g( X2) = (1 l ) [(1 l ) Y0 + l g( X1) ] + l g( X2) = (1 l ) 2 Y0 + l [ (1 l ) g( X1) + g( X2) ] = (1 l ) 2 Y0 + l
2
å
(1 l ) 2j g( Xj )
j =1
In general, Yt = (1 l ) t Y0+ l
t
å
(1 l ) tj g( Xj) ,
( 1.5 ).
j =1
This test statistic Yt , of the EWMA control chart results from ( 1.1 ) by putting, a t = (1 l ) t Y0 and bj = l (1 l ) tj for j = 1,2,….,t The EWMA chart has memory of unlimited length. Since bj’s are non zero and non uniform decreasing geometric sequence , EWMA is a chart with non uniform memory. If l =1, the EWMA control statistic ( 1.4 ) is identical to the Shewhart chart with test statistic Yt = g( Xt) . If g( Xj) is sample mean X , then one get EWMA X chart where 0< l £ 1. In the next section, EWMA chart is introduced. 3. Exponentially Weighted Moving Average (EWMA) control chart A quality control chart based on Exponentially Weighted Moving Average ( EWMA ) was first developed by Roberts in 1959. The EWMA chart uses the most recent and the past observations. The performance of the EWMA control chart is similar to that of the CUSUM chart and it is easier to set up and operate, in some way than the CUSUM. The EWMA is an exponentially weighed moving average of current and past observations. The EWMA control procedure can be made sensitive to a small shift by choosing the weighing factor. The EWMA statistic is defined as, Zi = λ Xi + ( 1λ ) Zi1 (1.6) where 0
