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is obviously problem-contingent. This leads to following legitimate question: how can
one know that the population size is sufficient for a more complicated problem? And
on the other hand, how can one be sure that the population size is not redundant for a
simpler problem? If one failed in the first scenario the true Pareto front can never be
approached; or if they failed in the second case one could end up with many
redundant evaluation times, which is more severe than it looks since in real life it is
expensive and time consuming to evaluate objective functions [20]. However, due to
its mating scheme and selection mechanism population-based GA has to have its
population size fixed.
Can AIS, as a new computing paradigm, offer a solution? This paper gives the an-
swer by addressing the following two questions:
1.
Does one still need to fix the size of the population?
2.
Can the population size adapt to the problem so that the initial population size is no
longer crucial to the success of the algorithm?
If the answers to both questions are 'yes', then another problem to be addressed is
how one can control the population size during the search. The problem is tackled by
emulating the third immune metaphor discussed in Section 2.2. The accomplishment
of aim 2 makes aim 3 automatically achieved since only the necessary Abs are pre-
served during each step.
3.1 The PAIA Algorithm
The basic definitions are first given so that one can describe the algorithm with
clarity:
−
Antigen (Ag):
Ag is the problem to be optimized.
−
Antibody (Ab):
Ab is the candidate solutions of the problem to be optimized.
−
Ag
-
Ab affinity:
for SOP, it is defined as the objective (fitness) value; for MOP, it
is determined by using the non-dominance concept, i.e. solutions in the first non-
dominated front have the highest affinity, then the second front, and so on.
−
Ab
-
Ab affinity (Abs' affinity):
it is defined as the distance (refer to Eqs. (3)) in
the decision variable space between one randomly chosen Ab in the first non-
dominated front and the one in the remaining population.
−
Ab
-
Ab suppression (Abs' suppression/Network suppression):
when two Abs
are very close to each other, they can recognize each other. The result is that one of
them is suppressed and deleted. Unlike Abs' affinity, this term is defined as the
Euclidian
distance in the objective space.
The PAIA algorithm can be described via the following steps:
1.
Initialization:
a random Ab population is first created.
2.
Identify_Ab:
one random Ab in the first non-dominated front is identified.
3.
Activation:
the identified Ab is used to activate the remaining dominated Abs.
Dominated Abs' affinity value (NB: affinity is the inverse of the affinity value) is
calculated according to Eqs. (3), where
n
is the dimension of the decision variables.
