Page 220 - Design for Six Sigma a Roadmap for Product Development

192 Chapter Six

{ }{ } ε (noise factors) (6.2)
β 2β 3 A 11 ′ DP 1
{FR 3} β 1β 2β 3M 1 β 3 A 22 DP 2
ideal function
A 33 DP 3
design mapping
In this mathematical form, FR 3 is called a dependent variable, while
M, DP 1,DP 2, and DP 3 are independent variables. Note that we made no
assumption about the sensitivity coefficients, A ij values and it can be a
linear or a nonlinear function of the DPs. The global parent function is
function 1. Function 2 is the parent of function 3, the lower level func-
tion in the chain. The equation represents a redundant design where
P, the number of DPs, is greater than m, the number of FRs. This
redundant design may be changed to ideal design by fixing two DPs
(the least sensitive DPs) and using the most sensitive DP as a robust-
ness parameter.
2. The pure parallel synthesis. In the pure parallel arrangement
(Fig. 6.4), we have the same input signal across the same hierarchical
functional entities. For the case of three entities, the transfer function
equation is given by

DP 1
DP 2
A 11 0 0 β 1 00 { }
FR 1 DP 3
0 A 22 0 0 β 2 0 ] (6.3)
FR 2 M 1
{ } [ 0 0
FR 3 A 33 00 β 3 M 2
M 3
β 1 00 M 1 A 11 0 0 DP 1
0 β 2 0 M 2 0 A 22 0 ]{ }
DP 2 error
0
[ ] { } [ 0 0 A 33 DP 3 (noise factors)
M 3
0 β 3
ideal functions design mapping
with the constraints: M
3 i 1 M i and FR ≤
3 i 1 FR i .
The pure parallel structure is an uncoupled design. We should expect
a structure that is a combination of pure series and pure parallel
arrangement to be a redundant design that may be reduced to achieve
uncoupled design if certain DPs can be fixed.
3. In-between synthesis. At a certain level of hierarchy, we may have
the following arrangement of Fig. 6.5.We refer to this arrangement as the
“in-between” hierarchy since it is neither pure series nor pure parallel.
This is due to the lack of an input relationship between the higher-
level functions (functions 1 and 2). Both of these functions are global
parents, thus violating the requirement of a pure series system.