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# Multi-Objective Optimization Problems: Part 1

Updated: Dec 8, 2023

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.