480

Del Castillo

needed for the optimization model is too difficult or if a starting solution for the optimization model is desired (other than a “classical” design, which may not be feasible).

For using the procedurę, the user needs to specify the two quantities, 0 < <j> < 1 and 0 < y < 1 such that

nf< $TP ATS < yT

In addition, we constrain the in-control average time to signal (average time to a false alarm), ATS*,. It is easy to show that

^ats*=mi-^a)^ł 1+M}~^aY^]

i=i

16]

and if a * 0 , then h

ATS. =

a

From (6), notę that ATS*, -> Jh = T if a -+ 0 and ATS*, -> h if a -> 1. Therefore, ATS*, ^ T, and we set the constraint

ATS.

or equivalently

>£,

This equation implies that the proportion of samples until a false alarm (or the proportion of time until a false alarm) has to be greater than £%. For example, if \ = 0.9 then the time until a false alarm must be on the average greater than

0. 9 T.

The procedurę for finding a feasible X chart design is as follows:

1.    Specify values for 4>, y, £, T, P, and 8.