Desirability Functions continue to attract attention of scientists and researchers working in
the area of multi-response optimization. There are many versions of such functions, differing
mainly in formulations of individual and overall desirability functions. Derringer and
Suich&rsquo / s desirability functions being used throughout this thesis are still the most preferred
ones in practice and many other versions are derived from these. On the other hand, they have
a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current
approaches to their optimization, which are based on derivative-free search techniques and
modification of the functions by higher-degree polynomials, need to be diversified considering
opportunities offered by modern nonlinear (global) optimization techniques and related
softwares. A first motivation of this work is to develop a new efficient solution strategy for the
maximization of overall desirability functions which comes out to be a nonsmooth composite
constrained optimization problem by nonsmooth optimization methods.
We observe that individual desirability functions used in practical computations are of mintype,
a subclass of continuous selection functions. To reveal the mechanism that gives rise to
a variation in the piecewise structure of desirability functions used in practice, we concentrate
on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of
desirability functions via piecewise max-type functions.
In this thesis, we introduce adjusted desirability functions based on a reformulation of the
individual desirability functions by a binary integer variable in order to deal with their piecewise
definition. We define a constraint on the binary variable to obtain a continuous optimization
problem of a nonlinear objective function including nondifferentiable points with
the constraints of bounds for factors and responses. After describing the adjusted desirability
functions on two well-known problems from the literature, we implement modified subgradient
algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for
solving the corresponding optimization problems. Moreover, BARON solver of GAMS is
used to solve these optimization problems including adjusted desirability functions. Numerical
applications with BARON show that this is a more efficient alternative solution strategy
than the current desirability maximization approaches.
We apply negative logarithm to the desirability functions and consider the properties of the
resulting functions when they include more than one nondifferentiable point. With this approach
we reveal the structure of the functions and employ the piecewise max-type functions
as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure
of the individual functions over their compact and connected interval that yield our
so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have
efficient structural and topological properties. We present the structural stability, optimality
and constraint qualification properties of GDFs using that of max-type functions.
As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach
for multi-objective optimization problems, based on a separation of the parameters:
in y-space (optimization) and in x-space (representation). This approach is about calculating
the factor variables corresponding to the ideal solutions of each individual functions in y, and
then finding a set of compromised solutions in x by considering the convex hull of the ideal
factors. This is an early attempt of a new multi-objective optimization method. Our first results
show that global optimum of the overall problem may not be an element of the set of
compromised solution.
The overall problem in both x and y is extended to a new refined (disjunctive) generalized
semi-infinite problem, herewith analyzing the stability and robustness properties of the objective
function. In this course, we introduce the so-called robust optimization of desirability
functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability
functions.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12612649/index.pdf |
| Date | 01 July 2010 |
| Creators | Akteke-ozturk, Basak |
| Contributors | Weber, Gerhard-wilhelm |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for METU campus |
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