Multi-Objective Problem
Definition: A general multi-objective optimization problem (MOOP) can be defined as, minimize a set of functions f(x), subject to p inequality and q equality constraints.
(1) where
and
(2) where
m is number of objectives;
D is feasible search space;
is the set of
n-dimensional decision variables (continuous, discrete or integer);
R is the set of real numbers;
Rn is
n-dimensional hyper-plane or space;
li and
ui are lower and upper limits of
i-th decision variable.
In MOOP, the desired goals are often conflicting against each other and it is not possible to satisfy all the goals at a time. Hence it gives a set of non-inferior solutions also known as Pareto optimal solutions. The Pareto optimal solution refers to a solution, around which there is no way of improving any objective without degrading at least one other objective (Deb et al., 2002).
Pareto Front: Pareto front is a set of nondominated solutions, being chosen as optimal, if no objective can be improved without sacrificing at least one other objective. On the other hand a solution x* is referred to as dominated by another solution x, if and only if, x is equally good or better than x* with respect to all objectives. The definition of Pareto optimality is very much useful in MOEAs to classify the population of solutions into dominated and non-dominated members, thereby helping in the selection of member solutions from one generation to next generation.