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
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.