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114

McWilliams

would be if used as the finał value, and tested it on the numerical examples discussed above. This approximation did not perform nearly as well: the approximation-based optimal cost was within 0.01% of the search-based cost in only one case, exceeded it by 5% in 7 cases, and had a maximum error of 12.59%.

In testing my approximation, I did encounter two numerical examples for which Solutions were not readily obtainable. These examples illustrate situations where the cost-minimizing strategy is to choose to not implement statistical process control, and also serve to point out problems with the approximations suggested by Duncan (1956). Typically, the cost, as a function of h, takes the shape shown in Figurę 1. However, for the combination of parameters chosen by Duncan (1956) for his Example 23 of Table 2 (X = 0.01, A = 0.5, E = 0.05, To = T\ = 0, 72 = 2, 5\ = &i = 1, Co = $0, Ci = $2.25, Y = $500, W = $250, a = $0.50, b = $0.10), the function has the shape shown in Figurę 2 and is not minimized at any finite value of h. With this combination of a Iow out of control penalty cost and high false alarm and repair costs, it is less expensive to let the process continue in an out of control State than to control it, so the cost-minimizing strategy is to use h = +oo at a cost of Ci =

h' = log h

Figurę 1. Typical Relation between Cost and Sampling Interval h. Parameter Values Come from Duncan (1956), Table 2, Example 1: X = 0.01, A = 2.0, E = 0.05, To = T\ = 0, Tl = 2, Ą = Ą = 1, Co = $0, Ci = $100, Y = $50, W = $25, a = $0.50, b - $0.10. Calculated Optimal Control Chart

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Parameters Are n = 5, L = 3.08, and h = 1.411 for a Minimum Hourly Cost of $4.01.