Optimization and Computational Fluid Dynamics - Department of ...

244 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto
8.7.2.1 Aggregating Functions
The most straightforward approach in handling multiple objectives is the use
of an arithmetical combination of all the objectives. Thus, the resulting single
function can be studied with any of the single-objective algorithms either
evolutionary or classical. Aggregating approaches are the oldest mathematical
programming methods found in literature [45].
Applied to EA, the aggregate function approach does not require any
change to the basic search mechanism. Therefore, it is efficient, simple, and
of easy implementation. It can be successfully used on simple multi-objective
optimizations that present continuous convex Pareto fronts. An example of
this approach is a linear sum of weights of the form:
�
m�
�
min wifi(X)
i=1
where the weight wi represent the relative importance of the m-th objective
function. The weighting coefficients are usually assumed to sum at 1:
m�
wi =1.
i=1
Aggregate function may be linear as in the previous example or nonlinear.
Both types of function have been used with evolutionary algorithms but,
generally speaking, aggregating methods have some limitations in generating
complex Pareto fronts, though nonlinear aggregating function do not necessarily
present such limitations [8]. Aggregating functions are widely used in
Multi-Criteria Decision Analysis (MCDA) or Multi-Criteria Decision Making
(MCDM) [5]. MCDA is a discipline aimed at supporting decision makers
who are faced with making numerous and conflicting evaluations. MCDM is
frequently used by EA users for a posteriori analysis of optimization results,
but the discipline can be applied in a much more sophisticated way for a
priori analysis. MCDM is briefly introduced in Sect. 8.7.3.
8.7.2.2 Population-based Approaches
In these techniques the population of an EA is used to diversify the search,
but the concept of Pareto dominance is not directly incorporated into the
selection process.
The first approach of this kind is the Vector Evaluation Genetic Algorithm
(VEGA) introduced by Schaffer [48]. At each generation this algorithm
performs the selection operation based on the objective switching rule,
i.e., selection is done for each objective separately, filling equal portions of
mating pool (the new generation) [12]. Afterwards, the mating pool is shuf-