Page 635 - Design for Six Sigma a Roadmap for Product Development

588 Chapter Sixteen

2
variance of the high-level requirement y, Var(y) , follows the fol-
lowing relationships:

∂f
∂f
∂f
2
2
2
Var(y) 2 1 2 2 ... 2 i ...
2
∂x i ∂x 2 ∂x i

∂f
... 2 n 2
∂x n
For a nonlinear relationship y f(x 1 ,x 2 ,…,x i ,…,x n ) and
2
2
2
2
2
2
2
2
Var(y) a 1 1 a 2 2 ... a i i ... a n n 2
For a linear transfer function y a 1 x 1 a 2 x 2 ... a i x i ... a n x n .
2
2
Clearly, the reduction of Var(y) can be achieved by reducing i for
...
2
i 1 n. However, the reduction of i values will incur cost. This vari-
ance reduction cost might be different for different low-level character-
2
istics, that is, x i . However, the impact of reduction for each variance i ,
made on the reduction of Var(y), , depends on the magnitude of sen-
2
sitivities |∂f/∂x i |. The greater the sensitivity, the greater the impact of
2
2
the reduction of i on the reduction of . Therefore, it is more desir-
able to tighten the tolerances for those parameters that have high sen-
sitivity and low tolerance tightening costs. The objective of a cost-based
optimal tolerance design is to find an optimum strategy that results in
a minimum total cost (variability reduction cost quality loss).
The tolerance reduction cost is usually a nonlinear curve illustrated
by Fig. 16.7.
Much work has been done (Chase 1988) to the cost-based optimal
tolerance design. In this chapter, we discuss the cost-based optimal tol-
erance design approach proposed by Yang et al. (1994). In this approach,
the tolerance design problem is formulated as the following optimiza-
tion problem (Kapur 1993):
Cost

Figure 16.7 Cost versus tolerance.
Tolerance