00187 Åf391b5c3ed26430f581079dee5f20f

188

McWilliams

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, . . t_n 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,..., t_n | 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