Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 221
al. [13]. In this case, the inverse problem has been solved as a single-objective
optimization problem. Finally, a very recent application of evolution strategy
(ES), for the shape optimization of thermoelectric bodies exchanging heat
by convection and radiation, has been reported by Bia̷lecki el al. [3]. Like
GAs, ES belong to the general category of Evolutionary Algorithms [4], but
differ from the former mainly in the use of real-valued vectors of variables
instead of bit-strings. Moreover, in contrast to GA which relies heavily on
recombination to explore the design (search) space, ES use mutation as the
main search operator. An interesting result from this study was the presence
of very different shapes having practically the same value of the objective
function. As shown in the results section, we also found a similar trend, i.e.,
two families of shapes characterized by the same performance figures.
This short review of the available literature is followed by some remarks.
The first is that in the majority of cases, the authors have used in-house
and/or problem specific methods – solvers and in particular optimization
algorithms – that, while capable of guaranteeing high accuracy and computational
efficiency, however lack generality and robustness. Hence, they could
hardly be applied to the complex optimization tasks found in industrial applications.
In fact, this type of problems are frequently computationally demanding,
are affected by noise, and are characterized by several conflicting
objectives as well as numerous constraints.
The second remark is relative to the fact that in all cited works, the problem
was to optimize a single performance metric. In other words, it was
considered a single-objective optimization problem.
When there were several objectives such as in [23] and [29], these objectives
were incorporated into a single function using suitable weighting factors,
thereby reducing the problem into one of single-objective optimization again.
This approach, however, has several drawbacks, the first being that weights
must be provided apriori, which can influence the solution to a large degree.
Moreover, if the objectives are inherently very different such as in cost and
thermal efficiency, it can be difficult to define a single all-inclusive objective
function. Finally, the user should even take care of normalization, and this is
not always a simple task since the range of variation of each objective may
be unknown. True multi-objective optimization techniques overcome these
problems by keeping the objectives separate during the optimization process,
bearing in mind that there will frequently be no single optimum in cases
with opposing objectives, since any solution will be a compromise. This is
the case for the problem considered in this paper, where the two objectives,
maximization of heat transfer rate and minimization of pressure losses, are
clearly conflicting. We will show how to identify the solutions which lie on the
trade-off curve known as the Pareto Frontier (named after the Italian-French
economist, Vilfredo Pareto). These solutions, also known as non-inferior,
non-dominated or efficient solutions, all have the characteristic that none of
the objectives can be improved without prejudicing another. The advantage
of the use of the Pareto dominance concept is that, as it will be shown in