Page 324 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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JWBK119-20
Review of Existing Techniques 309
4
Lin and Tu (referred to as LT) noted that the optimization scheme based on
Lagrangian multipliers may be unrealistic, since it forces the estimated mean to a
specific value. They thus propose minimizing the mean squared error (MSE) instead:
2
2
min MSE = ( ˆy μ − T) + ˆy .
σ
X
5
Copeland and Nelson (referred to as CN) pointed out that minimizing the MSE places
no restriction on how far the resulting value of ˆy μ might be from the target value T and
suggested the use of direct minimization of an objective function. For target-is-best
situation, the proposed scheme is to minimize the function ˆy σ + ε, where

2 2 2
( ˆy μ − T) , if (y μ − T) > ,
ε = 2
2
0, if (y μ − T) ≤ .
6
A quality lost function model proposed by Ames et al. for multiple response sur-
faces can also be applied to the dual response case. The scheme is:
m
2
min QLP = ω i ( ˆy i − T i ) ,
i=1
where ˆy i is the estimated response for quality criterion i, with respective target value,
T i , and W i is the corresponding weight for each response.
7
Recently, Kim and Lin (referred to as KL) proposed a fuzzy approach for dual
8
response surface optimization based on the desirability function formulation, a pop-
9
ular multiresponse optimization technique. Vining extended the work of Khuri and
6
11
10
Conlon, Pignatiello, and Ames et al. and proposed an optimization scheme using
a squared error loss function that explicitly incorporates both the variance--covariance
structure of both of the responses.
20.2.1 Optimization scheme
For a dual response system in which targets can be specified, it is natural to try to
achieve the ideal targets as closely as possible. While there are several ways to achieve
such objective, we propose a formulation using the idea of goal programming, 12,13 in
which both the objective function and constraints are of quadratic form as follows:
2 2
min δ + δ
μ σ
*
*
subject to y μ (x) − ω μ δ μ = T μ , y σ (x) − ω σ δ σ = T σ ,
(20.1)
T
and either xx ≤ r 2 or x l ≤ x ≤ x u .
2
2
The function to be minimized δ + δ , is the objective function where δ μ ,δ σ are un-
μ
σ
restricted scalar variables (i.e. they can be positive or negative). The subsequent
two equations are the constraints where ω μ ,ω σ (≥0) are user-defined weights;
*
*
T * ={T μ , T σ } are the ideal response mean and standard deviation associated with a
ˆ
set of response functions Y(x) ={ ˆy μ (x), ˆy σ (x)}. which are typically of quadratic form;
T
and x = [x 1 , x 2 , . .., x a ] is set of control factors. The typical solution space of control
T
2
factors that is of interest can be either xx ≤ r for spherical region, where r is the