00113 ™008d52bed37335501528226c4f1d64

113

Economic Control Chart

=    +ARL2)

⁺ 12 ARL1 ⁺ a + bn

=

_ Cl

12/1

^b° ARLl

¹¹ (l-Ä„)Tb ⁰¹ X 2 ARLl

+ nE + T\ + Ti

_1_ ARL2 2A⁺ X

*3 - ^    m

A fuli derivation of these results is available from the author on request.

My justification of this approach is simple: it works! It was tested using input parameters from 102 published numerical examples: one from Chiu (1975), one from Chiu (1976), two from Chiu and Wetherill (1974), 25 from Duncan (1956), 48 extreme parameter combinations from Duncan (1978), two from Gibra (1978), 20 from Gibra (1981), two from Montgomery (1982), and one from Lorenzen and Vance (1986). For each example I calculated the cost-minimizing control chart parameters using the approximation and also using two slower search routines, one based on the Nelder-Mead (1965) simplex method and the other using IMSL® routines UVMIF (attributes sampling) or UMINF (variables sampling). I then compared the cost obtained using h calculated according to the approximation formula to the smaller of the costs obtained when using the search routines. Two examples were excluded for reasons discussed below. For the remaining 100 examples, the cost based on the approximate h was within 0.01% of that found using the search approach in 72 cases, and within 0.10% in 96 cases. In the worst case, the approximation-based cost exceeded the search-based cost by 0.71%. The optimal control chart parameters and costs were also generally consistent with those reported in the articles from which the examples were taken. It is difficult to precisely match other authors’ results siÅ„ce different approaches were used to approximate both the cost fimction and the optimal value of h. I calculate the exact cost fimction and onl^ approximate h , while in several earlier studies both the cost fimction and h were approximated.

Lorenzen and Vance (1986) also provide an approximation formula for h , recommending it as a starting point for a search procedurÄ™ rather than as a finaÅ‚ value. However, I was curious about how accurate this approximate h

b2 =

3a

12 ARLl