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308 A Unified Approach for Dual Response Surface Optimization
variance are:
k k k
2

ˆ y μ = a 0 + a i x i + a ii x + a ij x i x j ,
i
i=1 i=1 i< j
k k k
2

ˆ y σ = b 0 + b i x i + b ii x + b ij x i x j
i
i=1 i=1 i< j
where ˆy μ is the estimated value of the mean and ˆy σ is the estimated value of the
standard deviation.
Various optimization schemes have been proposed in past 15 years. Here, we pro-
pose a unified formulation, which includes some existing techniques as special cases.
The approach will thus facilitate comparison and interpretation of results obtained
via different techniques. Through the use of two examples from published work, we
shall demonstrate the versatility of the proposed scheme by replicating results given
by other researchers. The results are obtained by adjusting the weight parameters in
the constraints so that the trade-off between meeting the targets for the dual responses
can be explicitly incorporated. All computations are carried out in Excel Solver which
can easily be implemented and be used by most practitioners.
This chapter is organized as follows. A review of existing techniques is presented in
Section 20.2. This is followed by a description of the proposed optimization scheme.
In Section 20.4, the method is tested and illustrated using two examples, one of
which has been extensively investigated by various researchers. A detailed com-
parison and a sensitivity analysis are also presented. Finally, a brief conclusion is
given.

20.2 REVIEW OF EXISTING TECHNIQUES FOR DUAL RESPONSE
SURFACE OPTIMIZATION

Various optimization schemes have been proposed in the past 15 years. The scheme
2
proposed by Vining and Myers (referred to as VM) is to optimize a primary response
subject to an appropriate secondary response constraint using the Lagrangian multi-
plier approach:

min (or max) y primary
X
subject to y secondary = ε,

where ε is a specific target value.
3
Del Castillo and Montgomery (referred to as DM) solved the problem using the
generalized reduced gradient algorithm, which is available in some software packages
such as Excel Solver. It has been demonstrated that the approach can handle the cases
of target-is-best, larger-is-better, and smaller-is-better.