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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}^ut^2t>

= ai|(l -a₂)a^N2^(t) 1 g(hN(t);t)

N(t)-1

+ (1 -a₂) E a^j2* 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-i^6*-* 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