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Economic Control Chart Models with Cycle Duration Constraints
Table 6 is determined to be unacceptable, it could be improved by supplementing the cycle duration constraint with an additional constraint.
Comparison to Gibra*s Approach
As mentioned in the Introduction, Gibra (1971) introduced the concept of constraining Tout, using an X -chart model and expressing the constraint in terms of the number of nonconforming items produced during a ąuality cycle. The constraint was then re-expressed in terms of a percentile of Tout. However, when considering the examples presented by Gibra, our numerical results differ substantially. This occurs because Gibra's analysis, in developing the distribution function of Tout, uses the exponential distribution to approximate the distribution of Si + S2 while we derive the exact expression. The approximation is ąuite inaccurate. For example, Figurę 2 shows the exact and approximate densities of S1 + S2 along with the corresponding means and standard deviations for Case 9. In using the exponential approximation in this example, Gibra's approach overstates the standard deviation of Sj + S2 by 47%. This leads to an economic control chart plan which is overly conservative with respect to the constraint and thus overly costly.
Specifically, in Case 9 the use of the exact expression for the distribution of T0ut leadst0 the plan n* = 12, L* = 1.99, and h* = 2.34 which effectively satisfies Pq 95 = 5.0 hours (the value specified in the constraint) and results in an hourly cost of $4.17. If Gibra's approximate distribution for T0ut is used to calculate the 95th percentile, we obtain the solution n* = 10, L* =
2.06, and h* = 1.67. This plan has a considerably shorter time between samples and, using the exact distribution, yields pQ 95 = 4.66 hours with an
hourly cost of $5.03. The constraint is exceeded rather than met, at a cost increase of 21% over the true optimal value.
Case 8 yields similar results: the 95th percentile obtained using
Gibra’s approach is 1.19 rather than 1.25 hours, with a cost penalty of 23%. Based on these results, we recommend using the exact distribution of Tout rather than the exponential approximation, in spite of the latter's computational advantages.
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