00470 05d5a5d1e980cfa3a7dd7d6256299b

476

Del Castillo

_ T 1 _T

F f(\-exp(-XTIf))

and 1 - P_a is the statistical power of the chart. For an X chart the power is given by

¹ - ^p. = C'*    ⁺La

where

= exp(-z²/2)/ Jlń and the Type I risk is

a-2j»*

Clearly, the optimal design of any other Shewhart Chart can be found by simply changing the defmitions of a and 1 - P_a in the model. Lorenzen and Vance (1986) generalized Duncan's model to other types of Shewhart charts in this way.

The production process is assumed to start in the in-control State with probability F₀ . Then, from Figurę 1, the costs and times during the cycle in which they are incurred can be derived. They are shown in Table 1.

Table 1. Costs Considered and Time Incurred

Cost type_Incurred during_

Sampling    F₀ min(R, T) +(1 - F₀)min(O, T) = cycle

False alarms    F₀ min(S, T) = A\

Producing out-of-control F^(_T-Sy-(T-RyY^-F₀)min{O,T)s A2