00164 ‹c986b3f04708d4e974fc7da0f1f328

00164 ‹c986b3f04708d4e974fc7da0f1f328



165


Economic Contro! Chart Models with Cycle Duration Constraints

detected at any sampling point, we find P{S2 = jh} = p(l - p)J for j = 0, 1, 2, . . . so S2 has mean p2 = h(l - p)/p and variance a] =    (1 - p)/p^. By

assumption, S3 has mean p.3 = r/0 and variance <73 = r/0^.

Theorem 1: Let N(t) represent the number of samples taken as of time t, namely [t/h]. Then the distribution function of T0ut >s given by:

FT0ut(t) = P^ut2t>

= ai|(l -a2)aN2(t) 1 g(hN(t);t)

N(t)-1

+ (1 -a2) E aj2* g(hj;t) + a^(t) g(t;t) - g(0;t)    (5)

where

A,p

ai = TiTT ;

e - 1

a2 = (1 -P)e‘

and

1 X\ J g(x;t) = i e -j

r r-l

I-i6*-* z i L j=o

-Xh


(6)


i']}


(7)

The proof of this Theorem is given in Appendix B. While the expression for Fy t(0 is complex, a program subroutine can easily be written

for function evaluation.

There are of course altemative approaches to constrained optimization. We define the out of control time in a broad sense, meaning the time period within a Ä…uality cycle during which the process parameter retains a shifted value, regardless of whether the process continues to operate. An altemative, in cases where the process shuts down during searches and/or repairs, is to


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