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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 T_out, 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 T_out. 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 T_out, 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 leads^t0 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 T_out rather than the exponential approximation, in spite of the latter's computational advantages.