Page 218 - Design for Six Sigma a Roadmap for Product Development

190 Chapter Six

reason for studying the ideal function is to have a physics-based math-
ematical model of the system under consideration before testing.
This allows the DFSS team to evaluate various DP levels in spite
of the presence of noise factors. Ideal function is written as FR
βM, where β is sensitivity with respect to the signal. This sensitivity
could be constant or a function of design and noise factors [i.e.,
β(DP 1 ,…,DP p ;N 1 ,N 2 ,…,N L )].
In the following sections, we will assume that transfer functions can be
approximated by a polynomial additive model with some modeling and
experimental error, that is, FR i
P j 1 A ij DP j
K k 1 β ik M k error
(noise factors), where the A ij and β ik are the sensitivities of the FR (or
any response) with respect to design parameters and signal factors,
respectively, P is the number of DPs, and K is the number of signals.
This approximation is valid in any infinitesimal local area or volume
of the design space. The noise terms will be further modeled as an
error term to represent the difference between the actual transfer
function and the predicted one. The additivity is extremely desired in
the DPs and all design mappings. As the magnitude of the error term
reduces, the transfer function additivity increases as it implies less
coupling and interaction. In additive transfer functions, the signifi-
cance of a DP is relatively independent from the effect of other DPs,
indicating uncoupled design per DFSS axiom 1. Physical solution enti-
ties that are designed following axiom 1 will have an additive transfer
function that can be optimized easily, thus reducing the DFSS project
cycle time. From an analysis standpoint, this additivity is needed in
order to employ statistical analysis techniques like parameter design,
design of experiment (DOE), and regression. Nonlinearity is usually
captured in the respective sensitivities.

6.3.2 Block diagram and synthesis
Let {FR 1 ,FR 2 ,FR 3 } and {f 1 , f 2 , f 3 } be the set Y′, the set of FRs; and F′,
the set of hypothesized or proven transfer functions, respectively.
Each f i can be written in the form f i (M i , DPs i ), i 1,2,3, where M is
the signal (revisit Sec. 5.9.1). The mapping f(M,DP) will be assumed
to be additive. In addition, assume that the three mappings are com-
plete and constitute a design project. The objective of the synthesis or
block diagramming activity is to piece together the solution entity
identified for each function in order to have a graphical representa-
tion of the design. This requires the identification of the operational
relationships as depicted by the transfer functions (this step is a
design analysis step) and the precedence or logical relationships in
the design hierarchy that govern the P-diagram and the transfer