In real life, the best solution that fulfills a single objective function might not be the best solution for others. It's rare to obtain one solution that beats all objectives. This gives us the motivation to solve multi-objective optimization problems.
Considering a cantilever beam subjected to loading at the far end, the design objective might be
O1: Minimum weight
O2: Minimum deflection
The beam design problem is usually subjected to the following constraints
C1: Limit stress at any section to the allowable value
C2: Limit deflection value at the far end
Applying multi-objective optimization solver, we can get solutions with the multiple values of objective functions, says O1 & O2 plotting below.
The solutions are classified as dominant (not useful) and non-dominant (useful or Pareto's) solutions. We can choose one or any non-dominant solution sets as one of our best solutions. The choice of the best solution might be based on the decision maker or other pre-defined criteria.
By definition for any minimizing problem, x is non-dominant solution iif
In other words, the solution is non-dominant or Pareto's solution if none of the objective functions can’t be improved in value without degrading some of the other objective values.