Seminar: Fuzzy Optimization
Abstract
Optimization means deriving the maximum benefit from a given
system
subject to system constraints. In real life design problems, the
objective
functions, the constraints, and the outcomes of possible events are
imprecisely known. This imprecision is more pronounced in situations of
increasing complexity. To derive maximum benefit from the system, this
imprecision calls for an adequate representation in our optimization
models. Fuzzy sets, i.e. sets with a continuum of grades of membership,
can adequately model these imprecise constraints and goals. Thus the
Fuzzy
Set
Theory proposes to be an appropriate modelling tool
for handling the system imprecision. In case of multi-objective
optimization, the trade-off between two or more conflicting goals and
constraints may not be known beforehand. Fuzzy Optimization may be used
in
this situation to get an overall compromise optimum solution by
considering the grade of membership as the degree of desirability of the
constraint or objective function.
Conclusion
As the system complexity increases, our ability to make precise
and yet significant statements about its behaviour diminishes. As a
result most design problems are linguistically posed for greater
freedom,
thereby causing imprecision in the objective functions. In most
real-life
situations, the constraints are vaguely known. We look forward to the
Theory of Fuzzy Sets,i.e. sets with a continuum of grade of membership,
to
model this imprecision.
We have looked into two methods for the solution
of the ``Fuzzy Optimization'' problem. The $\alpha$-cut method used when
only the constraints are fuzzy, gives solution in the parametric form.
The
$\lambda$-formulation approach gives an overall compromise solution when
both the goals and constraints are fuzzy. The results show an
improvement over the previous crisp optimization solutions in the cases
when, the goals are imprecisely known, imprecise or soft constraints are
involved, and in case of multi-objective optimization, when the
trade-off
between the conflicting goals and constraints is not known
beforehand. However the computational effort involved may be greater
than
the effort in the crisp optimization problems. We have looked into the
importance of
appropriate assessment of membership functions and the use of
appropriate
aggregation model depending on the application.