00115 Ë2a90e904fa483ecaa2c12e273f79e5

115

Economic Control Chart

h' = log h

FigurÄ™ 2. A Cost Function with no Finite Minimizing value of h. Parameter Values Come from Duncan (1956), Table 2, Example 23: X = 0.01, A = 0.5, E = 0.05, Tio = 71 = 0, F2 = 2, Ä„ = <% = 1, Co = $0, C\ = $2.25, Y = $500, W = $250, a = $0.50, b = $0.10. Hourly Cost Approaches C\ — $2.25 as h —> -Km.

$2.25 per hour! NotÄ™ that Duncan's approximations, which involve dropping terms expected to be negligible, Iead to his reported solution of n = 74, h = 25,and L = 3.1 for an hourly cost of $2.90. Since no finite solution for h exists for this problem, neither my approximation approach nor the search procedures gave a meaningful result.

The second problem encountered was a result of inaccuracy in calculating the probability of detecting a shift in the process parameter. In the X -chart case, this is given by:

p = 0(-Z, - AyjÅ„) + a>(-Z, +    )    [10]

where d>( ) represents the standard normal distribution function. Duncan recommends dropping the first term of this expression as it is expected to be negligible for practical values of L, A, and n. However, for Example 25 of Duncan's Table 2 (X = 0.01, A = 0.5, E = 0.05, 7o = T\ = 0, 72 = 2, Si = &i = 1, Co = $0, C\ = $2.25, Y = $50, W = $25, a = $0.50, b = $1.00), we have A = 0.5 standard deviations and the cost-minimizing values of L and n are relatively