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Baxley

parameters for this algorithm. K is normally set at 3, consistent with normal control charting practice, but L can be smałler in order to compensate for drifts in a morę timely manner. As discussed by Box and Jenkins (1963) and again by Box, Jenkins, and MacGregor (1974), this is appropriate when the cost of being off target is large relative to the cost of an adjustment. In fact, for L = 0 this algorithm reduces to a minimum-mean-squared-error time senes controller, as described by Box and Jenkins (1976) , where smali adjustments are madę for every sample. In this case, Eąuation 3 can be changed to Var(^ -Target^ = 0) =    . In other words, the control error variance equals the

forecast error variance if no special causes are present. As shown by Box and Jenkins (1963) and in a simulation study by Baxley (1990), when L increases, the average number of sample intervals between adjustments increases but so does the root mean sąuared control error, oy This is demonstrated in Figures 3a, 3b, and 3c, which are simulations of a process under EWMA control with different values for L, giving the following results:

Average Interval

L    ^a/cs between Adjustments

3.0	1.31	25
1.5	1.09	5.3
0	0.99	1

Another question concems robustness of controller performance to mis-estimation of X or g. This was addressed by a simulation study of the type documented by Baxley (1990) but with the X and gain used for the controller not in agreement with the same parameters for the process model of Eąuation 1. A series of 20 simulations was run according to a response surface experiment design with g = 1, X = .4, and a_a = 1. The controlled factors for this simulation study were L, the value of X used for the controller ( X_c), and a multiplier for the adjustment called the controller gain (g_c) such that x, = gd-Z/g). The settings for L, X_c> and g_c ranged from 0 to 3, .2 to .6, and .5 to 1.5 respectively. The root mean sąuared control error ( a_y) was calculated for each simulation, and then a fuli ąuadratic response surface model was fit to these data. Figurę 4 is a contour plot of the model predictions for a _r versus X_ę and g_c with L = 1.5. The controller performance is seen to be unaffected by changing tuning parameters except for the lower-left and upper-right regions. The former represents a case where the data are over-filtered and the adjustments are too smali. This results in the sluggishness in retuming the