Page 644 - Design for Six Sigma a Roadmap for Product Development

Tolerance Design 597

linearized, or sensitivities can be found, then for each low-level char-
...
acteristic x i , for i 1 n, we have
∂f
y T (x i T i ) a i (x i T i ) (16.34)
∂x i
Then the Taguchi loss function is approximately

A 0
A 0
L(y) (y T) [a i (x i T i ) ] 2 (16.35)
2
2 2
0 0
Assume that when a low-level characteristic x i exceeds its tolerance
limit i , the cost for replacing it is A i ; then, equating A i with quality
loss in Eq. (16.35), we get
A 0
A i [a i i ] 2 (16.36)
2
0

A 1
0

i A 0 |a i | (16.37)
Example 16.14. Power Supply Circuit The specification of the output volt-
age of a power supply circuit is 9 1.5 V. If a power supply circuit is out of
specification, the replacement cost is $2.00. The resistance of a resistor
affects its output voltage; every 1 percent change in resistance will cause
output voltage to vary by 0.2 V. If the replacement cost for a resistor is $0.15,
what should be the tolerance limit for the resistor (in percentage)? By using
Eq. (16.37), we get

A 0.15 1.5
0
i 2.05
|a i | 0.22.0
A 0
So the tolerance limit for the resistor should be set at about 2 percent.
16.5.3 Tolerance allocation for
multiple parameters
Given the transfer function y f(x 1 ,x 2 ,…,x i ,…x n ), if we want to design
tolerance limits for all low-level characteristics x 1 ,x 2 ,…,x i ,…x n in
Taguchi’s approach, we can simply apply Eq. (16.37) to all parameters:

0
A n
A 2
A 1

; ,…,
0
0
n
1
2
|a 1 | |a 2 | A 0 |a n |
A 0
A 0
Therefore, the square of the range of the output y caused by the vari-
ation of x 1 ,x 2 ,…,x i ,…,x n is