Page 382 - Six Sigma Demystified

362 Six SigMa DemystifieD

point may be either a maximum, a minimum, or a saddle point (also known as
a minimax). In n dimensions, a saddle point is a point where one or more
parameters are at a maximum and one or more parameters are at a minimum.
The stationary point may lie well outside the data range, in which case it is use-
ful for direction only.

Phase 1: Steepest Ascent Methodology
Data are collected (either single data points or replicated runs) along the path
of steepest ascent using the first-order regression equation developed in phase
0. Start at the design center: the point where, in coded units, (x , x ) = (0, 0).
2
1
Continuing the example provided under “Factorial Designs” above, we have a
first-order model developed in coded form:
y = –88,038 – 12,216x + 10,681x + 43,010x x
1 2 1 2
Given the levels of x (factor B) and x (factor E) in real terms (–50, 50) in
1
2
both cases, the design center is (0, 0). This will be our first test condition.
The path of steepest ascent is a simultaneous move of β (the coefficient of
1
the x term: –12,216) coded units in the x direction for every β (the coeffi-
1 1 2
cient of the x term: 10,681) coded units in the x direction. Sufficiently small
2
2
step changes provide reasonable resolution of the response. Given a specified
change of 10 (uncoded) units in x , for the coefficients specified in coded units,
1
the appropriate real (uncoded) change in x is calculated as
2
Uncoded change in x = (uncoded change in x )β /β 1
2
2
1
Change in x = 10 × 10,681/(–12,216) = –8.74
2

Thus each new design condition will be calculated as a step of 10 units in
the positive direction of factor B for every step of 8.8 units in the negative
direction of factor E.
We also can consider steepest ascent subject to constraints imposed by the
physical limitations of the system. For example, if x could not exceed 50, then
2
we proceed along the path of steepest ascent until the constraint is hit and then
proceed along its path (i.e., all subsequent data have the x factor set at 50).
2
We determine local maximum conditions by observing when the response
decreases. Data preceding an observed decrease in the response are likely to be
near a maximum condition.
We then conduct another experiment centered near the assumed maximum.
It is sufficient to run a suitable design to obtain a first-order model with one or