Progressively Interactive Evolutionary Multi-Objective Optimization ...
Progressively Interactive Evolutionary Multi-Objective Optimization ...
Progressively Interactive Evolutionary Multi-Objective Optimization ...
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for a maximization problem. Mathematically, this curve is f(x ∗ ) : x ∗ ∈ X ∗<br />
which are all the optimal points for the two objective optimization problem.<br />
A number of points are also plotted in the figure, which constitute<br />
a finite set. Among this set of points, the points connected by broken<br />
lines are the points which are not dominated by any point in the finite<br />
set. Therefore, these points constitute a non-dominated set within the finite<br />
set. The other points which do not belong to the non-dominated set<br />
are dominated by at least one of the points in the non-dominated set.<br />
In the field of <strong>Multi</strong>-Criteria Decision Making, the terminology slightly<br />
differs. For a given set of points in the objective space, the points which<br />
are not dominated by any other point belonging to the set are referred<br />
as non-dominated points, and their corresponding images in the decision<br />
space are referred as efficient. Based on the definition of weak and strong<br />
domination for a pair of points, the concept of weak efficiency and strong<br />
efficiency can be developed for a point within a set. A point x ∗ ∈ X, is<br />
weakly efficient if and only if there does not exist another x ∈ X such<br />
that fi(x) > fi(x ∗ ) for i ∈ {1, 2, . . . , M}. Weak efficiency should be distinguished<br />
from strong efficiency which states that a point x ∗ ∈ X, is<br />
strongly efficient if and only if there does not exist another x ∈ X such<br />
that fi(x) ≥ fi(x ∗ ) for all i and fi(x) > fi(x ∗ ) for at least one i.<br />
The terminologies, efficiency and non-domination, are used differently<br />
in different fields. The researchers in the field of Data Envelopment Analysis<br />
tend to call the points in the objective space as efficient or inefficient.<br />
Some researchers prefer to call only the pareto-optimal points as efficient<br />
or non-dominated points. To avoid any confusion, we shall not be differentiating<br />
between efficiency and non-domination and the terminologies<br />
will be used only in reference to points belonging to a set. The two terminologies<br />
will be used synonymously for points in the objective space as<br />
well as the decision space, based on domination comparisons performed<br />
in the objective space. If the set in which domination comparisons are<br />
made, encompasses the entire feasible region in the objective space, then<br />
the efficient or non-dominated points for that set will be referred as paretooptimal<br />
points.<br />
In Figure 1.3, for a set of points {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, the points<br />
{1, 2, 3, 4, 5, 6, 7, 8, 9} are weakly efficient and the points {1, 2, 3, 4, 5} are<br />
strongly efficient. Note that the set of all strongly efficient points is a subset<br />
of the set of all weakly efficient points. The point {10} is inefficient as<br />
it is dominated by at least one other point in the set. It should be noted<br />
that the notion of efficiency arises while comparing points within a set.<br />
Here the set in consideration consists of 10 number of points with few as<br />
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