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Hurwitz & Mathur

factors of complexity. On the other hand, too simple a set of rules—such as the famous Shewart set (±3-sigma from the target)-produce Average Run Lengths (ARLs) that are undesirably long.

Our purpose, then, is to introduce a new set of control rules that are both very simple, and have comparable ARL performance to the WECO rules. A summary of the rule sets is shown below in Table 1.

Average Run Lengths

We need to justify the use of any new control scheme by deriving its theoretical basis in terms of Average Run Length (ARL). ARL is the expected number of process points plotted on the chart before an 'out-of-control' signal is given, for different degrees of shift in the tiue process mean. We also need to examine the standard deviations of such ARLs.

Champ and Woodall (1987) give exact ARLs, for the WECO rules of run. These were used as the standard against which our simple rules could be compared. As it tumed out the simple rules gave ARL's remarkably close to the WECO ARL's. A Computer program by Champ and Woodall (1990) allows us to evaluate standard deviations of ARLs as well. As a check, ARLs for the 'new* rules were also derived algebraically. These results are shown below.

Assume that x is a random variable that is Normally distributed, with standard deviation s = 1 and mean p. Assume that a seąuence of independent drawings of x is madę, such as may be done for the familiar Shewart control chart with x the standardized sample subgroup mean. Using the notation of Champ and Woodall (1987), the runs rule which signals if k of the last m x-values fali in the interval (a,b), a < b, is written as T(k,m,a,b). Hence the traditional WECO rules are represented as: Ci u C₂ u C₃ u C₄ , where

Table 1. Process Control Rules for Out of Control "Signal"

SHEWART	WECO	NEW
1. A single point outside	1. A single point outside	1. A single point outside
±3 SD	±3 SD	±3 SD
	2. 2 out of 3 successive	2. 2 successive points at
	points at or beyond ±2 SD 3. 4 out of 5 successive points at or beyond ±1 SD 4. 8 successive points on one side of center linę.	or beyond ±1.5 SD

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