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McCarville & Montgomery

When applying the optimum guard bands to the defective level model and the Gage Loss model, Equations (9) and (14), the defective level becomes 105.663 ppm and the gage loss becomes 74371.7 ppm, or 7.437%.

Numerical Optimization

Naturally, with morÄ™ than two gages, the graphical optimization method becomes morÄ™ difficult. Numerical methods can be utilized to optimize these linear models. Nonlinear programming techniÄ…ues are an obvious choice where the constraint model is the ppm(Def) and the model to be minimized is the Gage Loss. G. C. Derringer and R. Suich (1980) developed a nonlinear programming method that is Ä…uite effective for solving problems such as these. They suggest that by developing a single measure of desirability and by finding the maximum desirability level, an optimal solution to the problem will be found.

For example, if the objective for a response variable is to achieve a target of Bj while staying within the specification limits of Ai and Ci such that Ai²Bi²Q, then the desirability would increase from Ai to Bj and decrease from Bi to Q. The desirability functions for optimizing around a target as well as minimizing and maximizing are shown in FigurÄ™ 19 along with the associated effects of their weights.

Once the desirability, dj, is determined for each of m responses, an overall desirability is calculated as follows:

D = (d₁d₂d₃...d_m)¹/^m

The overall desirability can now be optimized by finding its maximum value. Design-Expert utilizes a simplex steepest ascent method for finding the optimum. A starting point is randomly selected and the steepest ascent begins. Several optimizations should be performed starting from randomly selected points in the design region. Design-Expert performs this automatically. If the Solutions are fairly close together, then one optimum exists within the region; however, if the Solutions are widely different, then multiple optimals exist.

With all of the weights set at one, ten cycles of non-linear programming was performed which yielded optimal values of 642.5 for GB1 and 613.4 for GB2. The predicted values for the ppm(Def) and Gage Loss are 100 and 74,600 respectively. When applying these values for GB1 and GB2 to the original models developed earlier, EÄ…uations (9) and (14), the calculated ppm(Def) becomes 99.9988 and the Gage Loss is 75,162.6, or 7.516%.