Page 298 - Design for Six Sigma a Roadmap for Product Development

268 Chapter Eight

Reduction in the variance will reduce not only the probability of manufactur-
ing nonconforming gears but also the required information needed to man-
ufacture the part (Fig. 8.18). For the multivariate case, the joint density (DP)
is given by
exp [( ⁄2)(DP M)′ 1 (DP M)]
1

p
(2 | |
p
exp {( / ∑ [(DP μ )/σ ] }
−
1
)
2
2
i 1 i i i
or
() ∑
2π
p
where DP′ [DP 1 ,…,DP p ], M′ [ 1 ,…, ], and
[ 1 2 0 ]
2
0
2

0
(8.15)
0 p 2
Then, complexity is given by*

p
h(DP 1 ,…, DP p ) ln (2 e) | | nats (8.16)
For p 2, we have
h(DP 1 ,…, DP p ) ln 2 e 1 2 nats (8.17)
Using the concept of Boltzmann entropy, we were able to identify
variability as a source of complexity. However, variability is not the
only source of complexity, as we shall see later. In fact, sensitivity adds
to complexity as per the following theorem.

Theorem 8.1. The complexity of a design has two sources: variability
and sensitivity. The total design complexity in the case of linear design
is given by
h({FR}) h({DP}) ln |[A]| (8.18)

where |[A]| is the determinant of the nonsingular design matrix A.

Corollary 8.1 For process mapping (Fig. 8.2), in which h({DP})
h({PV}) ln |[B]|, then, by substitution in (8.18), the total design
complexity is given by

h({FR}) h({DP}) ln|[A]|
(8.19)
h({PV}) ln|[B]| ln|[A]|

*See El-Haik and Yang (1999) for proof.