Q24:
Why are some design spaces multidimensional rectangular and others not?
Figure 12: Overlay plot displaying functional and rectangular design spaces (Factors A, B).

A:
The shape of the design space depends on the relationships between the factors and the responses as well as any physical or
computational restrictions. Historically, the proven acceptable region has been best understood as a rectangle. For example,
in Figure 12, a rectangular design space is presented two ways, in blue and black. This concept can be extended to more than
two dimensions, creating a multidimensional rectangle.
In most cases, some feasible areas of operation will be excluded if the design space is specified by several ranges for individual
factors. This approach will result in a multidimensional rectangle that will not be equal to the nonrectangular design space.
Defining the design space as functional equations with restrictions as shown in Table III will enable one to take advantage
of the largest possible design space region. The yellow region in Figure 12 is the space defined by equations with restrictions
or specifications on the quality characteristics. The blue and black rectangles can be used as design space representations
but neither one provides the largest design space possible. For easeofuse in manufacturing, it may be practical to use a
multidimensional rectangle within the design space as an operating region.
Evaluating a design space
This section addresses variability within the design space and the implications on the probability of passing specifications,
in particular, when the process operates toward the edge of the design space. Alternative methods to specify the design space
to account for this variability are discussed using the example data provided in previous sections of this article (1, 2).
Suggestions on the placement of the normal operating region (NOR) and confirmation of design space are provided as well.
Q25:
How do I take into account uncertainty in defining the design space for one factor and one response?
Figure 13: Empirical model of degradate (Y) on Factor B. Solid line represents Degradate = 0.72 +0.57 * Factor B. Dotted line
represents the upper statistical limit.

A:
The regression line (shown in Figure 13) provides an estimate of the average degradate level. When Factor B is equal to 0.49,
the regression line indicates that on the average, degradate will be 1.00%. This means that when operating at 0.49, there
is roughly a 50% chance of observing degradate levels failing the specification by exceeding the upper limit of 1.00%. If
one were to define the design space as the entire green region below 0.49 in Figure 13, it would come with a significant
risk of failing specification at the edge, and thus not provide assurance of quality.
Figure 13 also illustrates an approach that makes use of a statistical interval to protect against uncertainty. In the region
described by the striped rectangle in Figure 13, the probability of passing the specification increases from 50%, at B = 0.49,
to a higher probability, at B = –0.08. Thus, a range for Factor B that protects against the uncertainty and provides higher
assurance of the degradate being less than or equal to 1.00% is between –1 to –0.08. This range corresponds to the solid green
region in Figure 13. The width of the interval and the increased probability will change based on the interval that is selected.
There are multiple ways to establish intervals that can be calculated to protect against uncertainty. Question 26b provides
more details on this increased assurance of quality.
