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OTE/SPH
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August 31, 2006
Review of Existing Techniques 311
returns a near-global one for degenerate problems within a radial region of exper-
imentation. A comprehensive discussion and comparison of various algorithms are
also given in the papers cited.
20.2.2 Comparison of proposed scheme with existing techniques
The proposed method provides a general framework that unifies some of the existing
techniques. This can be seen as follows.
4
Consider LT’s formulation in which the MSE is minimized:
2 2
min MSE = ( ˆy μ − T) + ˆy . (20.2)
σ
X
*
From equation (20.1), if we set ω μ = ω σ = 1 and T σ = 0, we have
2
min δ + δ 2
μ σ
*
subject to ˆy μ (x) − δ μ = T μ , ˆ y σ (x) − δ σ = 0. (20.3)
*
It follows that δ μ = ˆy μ (x) − T μ and δ σ = y σ (x). Thus equation (20.3) is equivalent to
(20.2), since
2

2 2 * 2
δ + δ = ˆy μ (x) − T μ + y σ (x) .
μ
σ
Next, consider VM and DM’s target-is-best formulation, 2,3 which can be formulated
as
min ˆ y σ
*
subject to ˆ y μ = T μ . (20.4)
*
From (20.1), if we set ω μ = 0, ω σ = 1, and T σ = 0, then δ μ has no bearing on (20.1)
and δ σ = ˆy σ (x). It follows that (20.1) is equivalent to
2
min ˆ y + constant
σ
*
subject to ˆ y μ (x) = T μ ,
which is the same as (20.4).
*
It is clear that by using suitable settings for ω μ ,ω σ , and T , other existing formu-
i
lations can be replicated. These are summarized in Table 20.1, where T max and T min
μ μ
are respectively the maximum and minimum possible values for the mean response
max 2 min 2
so that we have min ˆy μ − T ≡ max ( ˆy μ ) and min ˆy μ − T ≡ min ( ˆy μ ). In all
μ μ
T
2
the above schemes, the solution space is restricted to xx ≤ r or x l ≤ x ≤ x u .
5
Although an equivalent formulation for the CN approach cannot be readily ob-
tained from the proposed scheme, we shall show in the numerical example that iden-
tical result can be obtained by tuning the weights. It is worth noting that our scheme
not only provides the slackness for the mean target as in LT and CN, but also includes
the slackness from the variance response target, which others do not consider. We
discuss later how the trade-off between meeting the mean and variance targets can
be expressed explicitly by tuning the weights. Such flexibility, which is not available
in previous approaches, enhances the practical appeal of the current formulation.