Page 306 - Design for Six Sigma a Roadmap for Product Development

276 Chapter Eight

Theorem 6: (Path Independence of Uncoupled Design)*
The information content of an uncoupled design is inde-
pendent of the sequence by which the DPs are changed to
satisfy the given set of FRs.
Theorem 7: (Path Dependency of Coupled and Decoupled Design)†
The information contents of coupled and decoupled
designs depend on the sequence by which the DPs are
changed to satisfy the given set of FRs.
Theorem 8: (Independence and Tolerance)
A design is an uncoupled design when the designer-
specified tolerance is greater than

p
∑ ⎛ ⎜ ∂FR i ⎞ ⎟ DP i 1,...., ;mj 1 ,..., p
Δ
Z
ij ⎜ ⎝ ∂DP ⎟ ⎠ j
i 1 j
in which case the non-diagonal elements of the design
matrix can be neglected from design consideration.
Theorem 9: (Design for Manufacturability)
For a product to be manufacturable, the design matrix for
the product, [A] (which relates the FR vector for the prod-
uct to the DP vector of the product) times the design
matrix for the manufacturing process, [B] (which relates
the DP vector to the PV vector of the manufacturing
process) must yield either a diagonal or triangular
matrix. Consequently, when either [A] or [B], represents
a coupled design, the independence of the FRs and robust
design cannot be achieved. When they are full triangular
matrices, either they must both be upper triangular or
both be lower triangular for the manufacturing process to
satisfy independence of functional requirements.
Theorem 10: (Modularity of Independence Measures)
Suppose that a design matrix [DM] can be partitioned
into square sub matrices that are nonzero only along
the main diagonal. Then the Reangularity, R, and
Semangularity‡, S, for [DM] are equal to the product of

*See Section 1.3, El-Haik (2005) for more details.
†See Section 1.3, El-Haik (2005) for more details.
‡Semangularity, S, on the other hand, is an angular measure of pair axes between DPs
and FRs. See Chapter , El-Haik (2005) for more details.