00025 9a441c8ac791b0dbc63b51d79ba885

24

Molnau

[31

K_b = [ln(u_d)-ln(u_a)]/[u_d-u_al

where u_a represents the time-between-events acceptable level and u_d the detectable level. Both u_a and u_d are assumed to be means of a Poisson process, therefore the time between events is exponentially distributed.

After the K_b value has been computed, the decision interval, h, can be determined from using Average Run Length (ARL) tables, such as those presented in Lucas (1985). The ARL values represent the average number of observations taken before an out-of-control signal is given. A long ARL is desired when a process is in control, converseIy a short ARL is preferred when the process is out-of-control. The value of h is determined using a normalized reference value K^:

|4]

Kt-Kb*".

Based on K^, the u_a/u_d ratio, and the acceptance freąuency of false out-of-control signals (Type II error), the ARL tables are used to choose the appropriate value of h_t.

Since h_t is normalized, the decision interval value of h_b needs to be further calculated:

[5]

^hb ^{= h}t^/ua

Finally, once the values of h_b and K_b have been defined, the CUSUM procedurę can be defined by equation 1. The statistic is evaluated after every new defect observation, Yj. Any observation leading to Sj > h_b signals an out-of-control situation.

The CUSUM scheme, because of its use of old data and sensitivity to upcoming trends, is a useful test for monitoring manufacturing processes.

Seguential Probability Ratio Test

The Seąuential Probability Ratio (SPR) test, developed by Wald (1947), has been traditionally used in acceptance sampling. However, the test is also useful in detecting changes in defect levels. The SPR is the ratio of the probability of reaching a particular decision assuming the altemate hypothesis is correct (i.e., the true fraction defective is actually greater than the specified value) to the probability of reaching the same decision if the original hypothesis was correct (i.e., the true fraction defective is equal to the specified value). The ratio is expressed in terms of number of items examined. Items are examined until either an accept or a reject decision is madę. At every examination there