Question

Determine the Average Run Length (ARL) of a x-bar chart with limits where the process has shifted one standard deviation in one direction

Answer 1

Once one begins to contemplate alternative schemes for issuing out-of-control signals based on process-monitoring data, the need quickly arises to quantify what a given scheme might be expected to do. For example, what are the pros and cons of adding the Western Electric set of alarm rules to a control charting scheme? The most effective means known for making this kind of prediction is the "Average Run Length" (ARL) notion.

It is useful to adopt the notation T = the period at which a process-monitoring scheme first signals T is called the (random) run length for the scheme. The probability distribution of T is called the run length distribution, and the mean or average value of this distribution is called the Average Run Length (ARL) for the process-monitoring scheme. That is, ARL = μT When one is setting up a process monitoring scheme, it is desirable that it produce a large ARL when the process is stable at standard values for process parameters and small ARLs under other conditions

Evaluating ARLs is not usually elementary. But there is one circumstance where an explicit formula for ARLs is possible and we can illustrate the meaning and usefulness of the ARL concept in elementary terms. That is the situation where • the process-monitoring scheme employs only the single alarm rule "signal the first time that a point Q plots outside control limits," and • it is sensible to think of the process as physically stable (though perhaps not at standard values for process parameters). Under the second condition, the values Q 1, Q 2, Q 3,... can be modeled as random draws from a fixed distribution, and the notation q = P[Q 1 plots outside control limits

will prove useful. In this simplest of cases, it follows that

ARL=1/q

1 (Some ARLs for Showchart x charts) Consider finding ARLs for a standards given Sh0wchart x chart based on samples of size Note that if standard values for the process mean and standard deviation are respectively μ and σ, the relevant control limits are

or

where

ARL=1/2

ARL=0.5

WHERE,

R is the runlength of the scheme

ARL The mean distribution of r is called average run length

therefore average run length ARL=0.5