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310 A Unified Approach for Dual Response Surface Optimization
radius of the zone of interest, or x l ≤ x ≤ x u for rectangular region. The products ω i δ i
(i = μ, σ) introduce an element of slackness into the problem, which would otherwise
require that the targets T i (i = μ, σ) be rigidly met; which will happen if ω μ and ω σ
are both set to zero. The setting of ω μ ,ω σ will thus enable the user to control the rel-
ative degree of under- or over-achievement of the targets for the mean and standard
deviation, respectively. The relative magnitudes of the ω i also represents the user’s
perception of the relative ease with which the optimal solution of y i (x) will achieve
*
the targets T . By using different parameter values, alternative Pareto solutions with
i
varying degrees of under-attainment can be found. A Pareto solution 14 (also called a
noninferior or efficient solution) is one in which an improvement in the meeting of
one target requires a degradation in the meeting of another.
The above formulation can be run in Excel Solver using a simple template shown in
Figure 20.1. In general, it is recommended that the optimization program be run with
different starting points. Although the scheme may not guarantee a global optimum,
it is a simple and practical technique that gives good results that are close to the global
optimal solution in many instances. The issues involved in using Solver are discussed
3
by Del Castillo and Montgomery. An algorithm (DRSALG) and its ANSI FORTRAN
implementation that guarantees global optimality, in a spherical experimentation re-
gion, are presented by Del Castillo et al. 15,16 Recently, Fan 17 presented an algorithm,
DR2, that can guarantee a global optimal solution for nondegenerate problems and

Figure 20.1 Screen shot for implementing the proposed formulation in Excel Solver.