Progressively Interactive Evolutionary Multi-Objective Optimization ...

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