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is to find a way to define Ag and the
affinity
since there is no explicit Ag population
to be recognized. For SOP, since there is only one objective to be achieved the object-
tive itself can be viewed as Ag. Therefore, the
affinity
can be defined as the evaluation
of the objective function for a given Ab [5]. Such an implicit definition of Ag is reck-
oned to be more difficult to be used in a MOP context for the objectives are now mul-
tiple.
In [7], the authors argued that AIS has, in its elementary structure, the main fea-
tures required to solve MOP. There have been several attempts to address this in the
literature [8~12] but none of these presented a formal systematic framework due to
the aforementioned reasons. Some of them are coupled with other evolutionary
mechanisms [8, 12], and others sacrifice some biological metaphors in exchange for a
better performance [9, 10]. If one wishes to make AIS a new alternative computing
paradigm to solve MOP, clear definitions of each part of the immune metaphors and
their corresponding roles added to a general accepted framework are more pressing at
the moment than any specific algorithms. Furthermore, identifying the difference be-
tween AIS and the traditional evolutionary algorithms for solving MOP and what it is
the extra strength that AIS can offer is more meaningful than just providing relatively
better comparative results.
Based on such understanding, this paper presents a systematical AIS framework to
solve MOP with clear definitions and roles of the immune metaphors to be employed.
The new algorithm is mostly inspired by Clonal Selection [13] and Immune Network
[14, 15] theories, and is mainly based on the previous research in [3~5]. After com-
paring this algorithm to other state-of-the-art MOP algorithms using the ZDT1~ZDT4
benchmark functions, emphasis is placed on the following: 1) the difference between
AIS and traditional evolutionary algorithms, 2) the extra advantages that are exclu-
sively inherent in AIS and alike. Finally, it will be argued that if one considers each
objectives' combination as a unique antigen intruding on the immune system, MOP is
also an ideal test bed for the immune mechanism simulation.
2 Background
2.1 Multi-objective Optimization
Many real-world problems are inherently of a multi-objective nature with often con-
flicting issues. Generally, MOP consists of mini/maximizing the vector function:
()
()
()
()
.
f
x
=
[
f
x
,
f
x
,
K
,
f
x
]
T
(1)
1
2
m
subject to J inequality and K equality constraints as follows:
()
()
.
g
x
≥
0
j
=
1
,
K
J
;
h
x
=
0
k
=
1
,
K
K
(2)
j
k
where
is the
feasible region. There are two main methods that allow to deal with MOP, namely the
ideal multi-objective optimization procedure and the preference-based multi-objective
optimization procedure [16]. The fundamental difference between these two is that
is the vector of decision variables and
Ω
T
x
=
[
x
,
x
,
K
,
x
]
∈
Ω
1
2
n
