Economic Control Chart Models with Cycle Duration Constraints

183

ARL2 =

P =

ECL =

Sl =

S2 =

S3 =

M-i» ^CTi ^-N(t) = Tout ⁼ Po.95 ⁼ CTout ⁼

1/a when the measured statistics are independent,

wherea = Pr{exceeding control limits | process in control}

average run length while out of control

1/(1—P) when the measured statistics are independent,

where p = Pr{not exceeding control limits | process out of

control}

1-P

expected length of a Ä…uality cycle

time between occurrence of assignable cause and the next sampling point

time between first sampling point following occurrence assignable cause and sampling point at which the out of control State is detected

time required to locate and correct the assignable cause (E{S₃} = Ti+T₂)

mean and standard deviation of Sj, i = 1, 2, 3

number of samples taken as of t hours = [t/h]

total time spent in an out of control State (T_out = Si + S₂ + S3)

95th percentile of the distribution of T_out

standard deviation of T_out

Appendix B: Derivations

Proof ofTheorem 1

Recall that T_out = Si + S₂ + S3. Beginning with Sj + S₂, notÄ™ that Si + S₂ = t if and only if the assignable cause is undetected for N(t) = (t/h]

samples, where [•] represents the greatest integer function. Thus, for Si + S₂ to equal t it must be true that S₂ = h[t/h] and Si = t — h[t/h]. Then

fSi+S₂(0

= fi(t-hN(t))P{S₂ = hN(t)}

= fl(t-hN(t))p(l -p)^N(t)

- Mt-hN(t))

— PU-P)^N® e^- 1

t^ 0.

Now S3 has an Erlangian distribution with density: