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An Algorithm and a Graphical Approach for Short Run Processes

2.    Pick a value of n and a value off such that nf<łTP

3.    Using Figurę 2 choose a value of k such that /<*

4.Depending on 8, use Figures 3 or 4 to select the value of k and from the feasible region defined by the two previous constraints and

„ 1 Power ^ —

1I

5. If the feasible region is empty, return to step 1 and redefine y, <{> or £ to be smaller.

We illustrate the procedurę with the following example. Suppose T = 10, P = 10, <j> = 0.25, y = 0.25, £ = 0.99, 5 = 2.O.. Since TP = 100 we select n = 5 and/= 5.. From Figurę 2, we see that for/= 5, k has to be greater than 2.8 (approximate!y) in order that ATS_M > 0.997. Since 8 = 2.0, we use Figurę 4 where we have that \/Jy = 0.80. Then the third constraint limiting the feasible region is given by Power £ 0.80. The shaded region in Figurę 4 shows the resulting feasible design zonę. From it we can choose, for example, (n,kf,T) = (4,3,5,10). Notę that if a shift of size 8=1 was to be detected instead, the feasible region is empty and we need to redefine <j>, y,or

Conclusions

We have presented a constrained optimization algorithm for the economic design of control charts used in finite-length manufacturing processes. Statistical and production-related constraints that link the chart design variables with the production process were included in the model formulation. The constraints can be solved graphically to provide a simple method for finding feasible chart designs. This approach is useful in the cases when estimating the parameters of the model is difficult in practice.