00187 Åf391b5c3ed26430f581079dee5f20f
188
This concludes the proof of Theorem 1.
Appendii C: Estimating Erlang Parameters r and q
In addition to requiring input values for the economic control chart parameters specified by Lorenzen and Vance, the cycle duration constraint approach requires speciflcation of parameters r (shape) and 0 (scalÄ™) of the Erlang distribution used as a model for the time required to discover and repair the assignable cause. The Erlang distribution is simply a gamma distribution with the shape parameter restricted to be an integer value, and has density function
f(t)
Qr r-1 -0t 8 t e
(r-1)!
FigurÄ™ 3 illustrates this distribution for shape parameters r = 1, 2, 5, and 10 with the distribution mean held constant. NotÄ™ that as r increases the distribution becomes morÄ™ symmetric and variability decreases.
If observations tj, t2, . . tn of the discovery and repair time are available, maximum likelihood estimates of r and 0 are easy to calculate. The log likelihood function is given by
f(ti, t2,..., tn | r, 0) = nr ln 0 + (r - 1) ln (fi ti) - 0 £ ti - n ln [(r - 1)!] (1)
With no restriction on r (the gamma distribution case), finding maximum likelihood estimates for r and 0 involves simultaneously solving two nonlinear equations with no closed form solution available (Choi and Wette, 1969). If r were known, the maximum likelihood estimate of 0 is easily shown
to be 0 = (nr)/l tj. Since in our application r is restricted to integer values,
A
we recommend evaluating 0 and the log likelihood function (1) at plausible
A
values of r, and then choosing the r, 0 combination which maximizes log likelihood. This could be easily done on a spreadsheet, for example, as is illustrated in FigurÄ™ 4. We assume that n = 10 observations ti, t2, . . ., tio are
A
available and calculate 0 and log likelihood at r = 1, 2,.. ., 6. Log likelihood
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